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1996PhLB..388...51D
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Absorption versus decay of black holes in string theory and T-symmetry
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1996-01-01
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Classically a black hole can absorb but not emit energy. We discuss how this T-asymmetric property of black holes arises in the recently proposed (T-symmetric) microscopic models of black holes based on bound states of D-branes. In these string theory based models, the nonvanishing classical absorption is made possible essentially by the exponentially increasing degeneracy of quantum states with mass of the black hole. The classical limit of the absorption cross section computed in the microscopic model agrees with the result obtained from a classical analysis of a wave propagating in the background metric of the corresponding black hole (upto a numerical factor).
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https://arxiv.org/pdf/hep-th/9605234.pdf
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{'No Header': 'TIFR-TH-96/26 hep-th/9605234', 'ABSORPTION VS DECAY OF BLACK HOLES IN STRING THEORY AND T-SYMMETRY': 'Avinash Dhar, Gautam Mandal and Spenta R. Wadia Tata Institute of Fundamental Research Homi Bhabha Road, Bombay 400 005, INDIA \ne-mail: adhar, mandal, [email protected]', 'ABSTRACT': 'Classically a black hole can absorb but not emit energy. We discuss how this T-asymmetric property of black holes arises in the recently proposed (T-symmetric) microscopic models of black holes based on bound states of D-branes. In these string theory based models, the nonvanishing classical absorption is made possible essentially by the exponentially increasing degeneracy of quantum states with mass of the black hole. The classical limit of the absorption crosssection computed in the microscopic model agrees with the result obtained from a classical analysis of a wave propagating in the background metric of the corresponding black hole (upto a numerical factor).', '0 Introduction': "Recent rapid developments in string theory have opened up the exciting possibility of a microscopic derivation of the physics of black holes. Based on progress in understanding bound states of D-branes [1, 2, 3, 4, 5, 6], microscopic models of black holes have been constructed in 4+1 [7, 8, 9, 10, 11] and 3+1 [12, 13] dimensions. Perhaps the most promising feature of these models is that a counting of microscopic states correctly reproduces black hole degeneracy as required by the Bekenstein-Hawking formula S = 1 4 A [15]. There are, of course, many other aspects of black hole physics which one would like to derive from these simple microscopic models. The very existence of an event horizon is one such aspect. This implies that classically a black hole can absorb but not emit energy. Can one possibly understand this apparent lack of time-reversal symmetry 1 in terms of the proposed microscopic models which are based on a manifestly time-reversal symmetric microscopic theory? \nA priori it would seem that this question must remain unanswered at present. This is because while the microscopic models of black holes as bound states of D-branes have been constructed in the weakly coupled string theory regime, the semiclassical picture of the black hole is expected to be valid in the strong coupling regime [7], and exploring this regime is at present beyond our technical abilities. It is, therefore, surprising that the simple microscopic model of the Hawking emission proposed in [8, 18] yields an expression for the decay rate that agrees with the standard formula in all its essential details! Even though we don't quite understand why this works, we are encouraged enough by this agreement to address the question about the classical absorption by a black hole as a first step towards exploring the horizon physics in the microscopic models. In this paper we will consider the microscopic model of [8, 18] and show that the time-reversed process of the Hawking emission leads to absorption by the black hole which is indeed nonzero in the classical limit, unlike the Hawking emission which vanishes in this limit. This leads to classical absorption but not decay by this microscopic model of the black hole. \nThis paper is organized as follows. In Section 1 we review the model \nof [8, 18] for Hawking emission. In Section 2 we consider the time-reversed process and calculate the absorption coefficient. At the microscopic level the magnitude of the matrix element leading to decay is identical to that leading to absorption. Nevertheless, the classical limit of absorption by the macroscopic black hole is nonzero while the decay vanishes. In Section 4 we discuss the classical propagation of a massless scalar particle in the geometry of the black hole under discussion. We calculate the absorption crosssection and compare with the result of Section 3. In Section 5 we discuss possible strong coupling effects. We argue that the essential details of the result obtained here are expected to survive even in the strong coupling limit.", '1 Hawking Decay': "In this section we will review the model of Hawking decay of the 4+1dimensional charged black hole considered in [8, 18]. We will use a notation that will be useful later for discussing absorption in the next section. \nIn the microscopic model the decay of a black hole is interpreted as the annihilation of two massless open string excitations on a D-brane, each with energy ω/ 2, into a massless closed string quantum of energy ω . As discussed in [8], these open string degrees of freedom live in the 2-dimensional spacetime whose space part is the S 1 of radius R along x 5 that is common to D-onebranes (which wrap around the compact coordinate x 5 in the 10-dim. space-time) and D-fivebranes (which wrap around the compact 5-dim. space S 1 × T 4 labelled by x 5 , x 6 , x 7 , x 8 and x 9 ). There are N B species of bosonic (and as many fermionic) open string levels for each value of momentum in the x 5 direction. N B = Q 1 Q 5 for the extremal holes where Q 1 is the number of D-onebranes and Q 5 is the number of D-fivebranes. As discussed in [18], however, there are problems in applying this naive picture to realistic black holes, the so-called 'fat' black holes. These problems are resolved in a modified model proposed in [18], which uses an observation made in [19]. In this modified model the D-onebranes and D-fivebranes arrange themselves inside the bound state in such a way that effectively the number of bosonic species N B = 1, while the effective radius of the S 1 along the x 5 direction in which these bosons live is L = RQ 1 Q 5 . Since the charged black holes considered in [8] and under discussion here, especially in sections 3 and 4 where we discuss the classical limit, are of the 'fat' type, we will henceforth \nwork within the effective model of [18]. \nWe may write a low-energy effective action for the interaction of a closed massless (in the 4+1 noncompact space-time, /vectorx = ( x 1 , x 2 , x 3 , x 4 ) and t ) string with two massless (in the 2-dim. space-time x 5 and t ) open strings: \nS int ∝ g st ∫ dt ∫ 2 πL 0 dx 5 ∂ a φ ( t, x 5 ) ∂ a φ ( t, x 5 ) h ( t, /vectorx = 0) , a = ( t, x 5 ) (1) \nSuch an interaction can be inferred from the analyses of [20, 21, 22, 23]. In the above, dimensional reduction has been applied to the directions x 6 , x 7 , x 8 and x 9 and hence the fields are independent of these coordinates. The field φ ( t, x 5 ) describes a massless open string excitation 2 moving along x 5 . The field h ( t, /vectorx ) is a massless (scalar) closed string - its masslessness in the (4+1)dimensional noncompact space-time ( /vectorx, t ) implies that it is independent of x 5 . Finally, g st is the string coupling in 10-dimensions and L = RQ 1 Q 5 is the effective radius of the S 1 along x 5 . \nThe normal mode expansion of the field φ ( t, x 5 ) is \nφ ( t, x 5 ) = + ∞ ∑ h = -∞ 1 √ 2 ωL ( a n exp[ inx 5 L -iωt ] + c.c. ) , ω = | n | L (2)", 'Matrix element for decay': "Let us now consider an initial state | i 〉 with total left-moving (along x 5 ) momentum ˜ N L L and total right moving momentum ˜ N R L . There are, of course, many states | i 〉 corresponding to a given choice of ˜ N L and ˜ N R . These are characterized microscopically by the set of number { ˜ N L ( n ) , n = 1 , 2 , · · · , ∞} and { ˜ N R ( n ) , n = 1 , 2 , · · · , ∞} where ˜ N L ( n ) = a + n a n , ˜ N R ( n ) = a + -n a -n , n = 1 , 2 , · · · , ∞ . In other words, \n| i 〉 = ∞ ∏ n =1 ( ˜ N L ( n )! ˜ N R ( n )! ) -1 2 ( a † n ) ˜ N L ( n ) ( a † -n ) ˜ N R ( n ) | 0 〉 (3) \nClearly, ˜ N L,R = ∞ n =1 n ˜ N L,R ( n ). \nIt is a matter of simple one-dimensional thermodynamics to compute the number of microstates | i 〉 corresponding to a given choice of ˜ N L and ˜ N R . Including contribution from fermionic open string excitations also, we get \n∑ \nΩ = e S , S = 2 π (√ ˜ N L + √ ˜ N R ) . (4) \nThis is related to the expression for degeneracy of nonextremal states given in [9, 8] by the relation [18] \n˜ N L,R = Q 1 Q 5 N L,R . (5) \nThe final state | f 〉 that we are interested in is obtained from the initial state | i 〉 by the annihilation of a left-moving open string of momentum m L L = ω 2 with a right-moving open string of momentum m R L = ω 2 into a massless closed string quantum of energy ω . The remaining gas of open string excitations in the final state | f 〉 is, therefore, characterized by ˜ N ' L = ˜ N L -m L , ˜ N ' R = ˜ N R -m R : \n| f 〉 = h † ω ⊗ ∞ ∏ n =1 ( ˜ N ' L ( n )! ˜ N ' R ( n )! ) -1 2 ( a † n ) ˜ N ' L ( n ) ( a † -n ) ˜ N ' R ( n ) | 0 〉 (6) \nSince we will be interested in closed strong quanta with zero momentum parallel to the branes, for us m L = m R = m . \nAs in (4), (5) we can easily compute the number of microstates | f 〉 corresponding to a given choice of ˜ N ' L and ˜ N ' R : \nΩ ' = e S ' , S ' = 2 π (√ ˜ N ' L + √ ˜ N ' R ) (7) \nThe S -matrix element for decay from the initial state | i 〉 to the final state | f 〉 , to 1st order in string coupling g st , can be computed using (1) following standard perturbation theory rules and is given by \n〈 f | S | i 〉 ∝ g st √ m L √ m R √ ωV 4 m L L · m R L · Lδ m L ,m R δ ( ω -m L L -m R L ) ( ˜ N L ( m L ) ˜ N R ( m R )) 1 / 2 (8) \nwhere m L /L = ω/ 2 = m R /L and V 4 is the volume of the 4-dimensional noncompact space (box normalization). \nNow, recall that the nonextremal black holes under discussion are characterized by six parameters [9] which label the corresponding D-brane bound states. These are denoted N 1 , N ¯ 1 , N 5 , N ¯ 5 , N L and N R and stand for respectively the number of D-onebranes, anti-D-onebranes, D-fivebranes, anti-Dfivebranes, total left moving momentum and total right moving momentum. (Note that Q 1 ≡ N 1 -N ¯ 1 and Q 5 ≡ N 5 -N ¯ 5 .) All microscopic states | i 〉 which have a common value of these parameters refer to the 'same' macroscopic black hole ('no-hair' 3 ). Therefore, the microscopic model of the black hole is a density matrix \nρ = 1 Ω ∑ { i } | i 〉〈 i | (9) \nwhere the sum { i } is over all possible distributions { ˜ N L ( n ) } and { ˜ N R ( n ) } keeping N L and N R fixed. It is this formula that leads to the entropy S = Trρ ln ρ = -{ i 1 Ω ln 1 Ω = ln Ω. \n- \n-{ Density matrices like the one in (9) are not unfamiliar in particle physics. They arise, e.g. in calculating the decay rate of an unpolarized particle into unpolarized products. As there, in the present case also, the total 'unpolarized'transition probability is given by \n∑ \n} \nP decay ( i → f ) = 1 Ω ∑ { i } , { f } |〈 f | S | i 〉| 2 (10) \nThe division by Ω represents averaging over initial states, while the final states are simply summed over. The passage to the decay rate d Γ is usual and one gets \nd Γ ∝ d 4 /vector kG N m 〈 ˜ N L ( m ) 〉〈 ˜ N R ( m ) 〉 (11) \nwhere G N is Newton's constant in 4-dimensional noncompact space, ω = 2 m L is the energy of the emitted massless closed string, /vector k is its momentum in 4-dimensional space ( | /vector k | = ω ) and 〈 ˜ N L,R ( m ) 〉 is the average distribution in the initial state (with fixed total momenta ˜ N L and ˜ N R ). For large values of ˜ N L and ˜ N R one can compute these average distributions by approximating \nthe microcanonical ensemble by a canonical ensemble (in the 1-dimensional thermodynamics that gave rise to (4) and (7)). One gets the standard BoseEinstein distributions \n〈 ˜ N L,R ( m ) 〉 = ( e β L,R ω/ 2 -1 ) -1 (12) \nβ L,R = πL/ √ ˜ N L,R = πL √ Q 1 Q 5 N L,R (13) \nwhere \nand we have used that m = ωL/ 2.", '2 Absorption': "Consider now the absorption of a massless closed string quantum by the black hole. The elementary process here is just the reverse of the decay process of the previous section. In fact, let us consider the absorption of a massless quantum of energy ω = 2 m/L by the initial state | i ' 〉 labelled by the total momentum ˜ N ' L and ˜ N ' R (as in the final state | f 〉 of the previous section). The final state | f ' 〉 of the black hole in this case, then, contains an additional left (right) moving open string mode of momentum m L /L = ω/ 2 ( m R /L = ω/ 2 ) (just like in the initial state | i 〉 of the previous section). Thus, in this absorption process the initial and final states of the previous section just get interchanged. Furthermore, it is trivial to see from (1) (or by using perturbative unitarity of string theory) that to first order is g st perturbation theory, 〈 f ' | S | i ' 〉 = 〈 i | S | f 〉 = -〈 f | S | i 〉 ∗ . It follows, therefore, that for the macroscopic black hole the absorption probability is \nP abs ( i ' → f ' ) = 1 Ω ' ∑ { i ' } , { f ' } |〈 f ' | S | i ' 〉| 2 = 1 Ω ' ∑ { i } , { f } |〈 f | S | i 〉| 2 (14) \nThe division by Ω ' signifies averaging over the microscopic initial states | i ' 〉 whose degeneracy is the same as that of the state | f 〉 and is given by Ω ' in (7). From (10) and (14) we see that at the macroscopic level the absorption probability is related to the decay probability by the equation \nP abs ( i ' → f ' ) = Ω Ω ' P decay ( i → f ) (15) \nThus the absorption probability is larger than the decay probability by the factor Ω / Ω ' (recall that Ω increases exponentially with the mass of the black hole and that Ω ' refers to the black hole with mass smaller than that to which Ω refers). As we shall see it is this enhancement factor that is responsible for a nonzero classical absorption by the black hole. \nThe passage from the absorption probability to the absorption crosssection σ A is usual. We get \nσ A ∝ Ω Ω ' G N ωL 2 〈 ˜ N L ( m ) 〉〈 ˜ N R ( m ) 〉 (16) \nwhere m = ωL/ 2 and 〈 ˜ N L,R ( m ) 〉 are given in (12), (13).", '3 Classical Limit': 'We would now like to discuss the results of the previous two sections in the classical limit. This limit is taken by letting the mass of the black hole become very large, i.e., M /greatermuch 1 (in Planck units). Actually, the classical limit is more subtle in the case of charged black holes [26, 27, 28]. One also needs to ensure that the black hole is not too close to extremality. More quantitatively, for the 5-dimensional charged black holes under discussion, the appropriate conditions are \nM /greatermuch ∆ M /greatermuch 1 M 2 , M /greatermuch 1 (17) \nThe second part of the first condition ensures consistency of thermal description (∆ M is mass deviation from the extremal limit) [26], while the first part ensures that deviations from extremality are small in the macroscopic sense. \nNow, let us assume that the nonextremal charged black holes under discussion are obtained by perturbing N R away from its extremal value of zero. Let us take the classical limit by scaling Q 1 , Q 5 and N ( ≡ N L -N R ) as follows \nQ 1 → λQ 1 , Q 5 → λQ 5 , N → λN, λ /greatermuch 1 (18) \nkeeping the ratio N R /N L fixed and small. Such a scaling is natural for Reissner-Nordstrom black holes which satisfy the condition Q 1 R/g st = Q 5 RV/g st = N/R . These scalings have the following effect on the mass M of the black hole, ∆ M and the effective radius L of the S 1 in the x 5 direction: \nM → λM, ∆ M → λ ∆ M, L → λ 2 L. (19) \nThe conditions in (17) are then automatically satisfied for λ /greatermuch 1 in the case of absorption, while in the case of decay they are satisfied at least in the early stages of decay. Thus, a consistent way of taking the classical limit is to do the scalings (18), and let λ become large. \nNow, under the scalings (18) and (19), we see from (13) that β L,R scale as √ \nβ L → λβ L , β R → √ λβ R (20) \n√ \nNote that it follows from (13) that β R is always much larger than β L (because of the condition N R /lessmuch N L ), and remains so under the scalings (20).', 'Vanishing classical decay rate :': 'We are now ready to discuss the classical limits of (11) and (16). Let us consider the decay first. Because of (20) the decay rate peaks at ω ∼ 1 /β R in the classical limit. In (11) we may, therefore, expand 〈 ˜ N L ( m ) 〉 and retain only the first term : \n〈 ˜ N L ( m ) 〉 ∼ 1 β L ω (21) \nThus, the decay rate becomes [8] \nd Γ ∝ d 4 /vector kA h ( e β R ω/ 2 -1) -1 (22) \nwhere A h ∼ G N √ Q 1 Q 5 N L is the area of the horizon of the black hole. Now, using (18) and the second of (20), we find that the decay rate vanishes exponentially in the classical limit: \nd Γ ∼ λ 3 / 2 e -√ λ (23) \nwhere in the last equation we have displayed only the λ -dependence.', 'Nonvanishing classical absorption :': "To see what happens to the absorption crosssection, (16), in this limit, we also need to compute the enhancement factor Ω Ω ' . Using that ˜ N ' L,R = ˜ N L,R -m and ω = 2 m L , we get \nΩ Ω ' = e [ ω 2 β L + o ( ω 2 )] e [ ω 2 β R + o ( ω 2 )] (24) \nThe coefficient of the ω 2 term in the first exponent is the derivative of β L ∼ √ L/M with respect to M , and under the scalings in (19) vanishes as λ -1 / 2 for large λ . Similarly the corresponding coefficient in the second exponent involves the derivative of β R ∼ √ L/ ∆ M with respect to ∆ M , which also vanishes as λ -1 / 2 for large λ . The coefficients of higher powers of ω in both the exponents vanish even faster as λ becomes large. 4 \nUsing (24) and (12) in (16), we get \nσ A ∝ G N ωL 2 (1 -e -β L ω/ 2 ) -1 (1 -e -β R ω/ 2 ) -1 (25) \nwhich is clearly nonvanishing in the classical limit. \nWe will now restrict the above formula to frequencies ω satisfying ω /lessmuch β -1 L . This is done for the following reason. In the classical calculation of the absorption crosssection in the next section, we have restricted ourselves to small values of ω . The corrections are order ωr 0 , where r 0 ∼ ( G N M ) -1 / 2 ∼ β L is the radius of the horizon. The corrections are, therefore, higher order in g st . To include this consistently one must, therefore, also include higher order g st corrections in the microscopic model, which we have not done here. \nNow under the condition ω /lessmuch β -1 L , we may expand the first factor in brackets in (25) in powers of ωβ L . Retaining only the first term, we get \nσ A ∝ A h (1 -e -β R ω/ 2 ) -1 (26) \nNow we let λ become large after doing the appropriate scalings given in (20). For any given fixed ω /lessmuch β -1 L , β R ω will eventually become very large as λ becomes large 5 . Therefore in this limit (26) gives \nσ A ∝ A h (27) \nThus, the enhancement factor Ω / Ω ' has ensured that the absorption coefficient remains nonzero in the classical limit! As we shall see in the next section, the result we have obtained above in (27) from a microscopic calculation matches in all its essential details with that obtained from a classical calculation of wave propagation in the appropriate black hole geometry. \n→", '4 Classical Wave Analysis and Absorption': "In this section we consider classical propagation of a massless field in the geometry of the 4+1 dimensional black hole. We take the massless field to be one of the scalar moduli which has a simple propagation equation [29] \nD µ ∂ µ φ = 0 (28) \nHere the metric defining the Laplacian is [9] \nds 2 = -f -2 / 3 ( r ) g ( r ) dt 2 + f 1 / 3 ( r )[ g ( r )] -1 dr 2 + f 1 / 3 r 2 [ dχ 2 + sin 2 χdθ 2 +sin 2 χ sin 2 θdφ 2 ] g ( r ) = (1 -r 2 0 /r 2 ) f ( r ) = (1 + r 2 0 r 2 sinh 2 α )(1 + r 2 0 r 2 sinh 2 γ )(1 + r 2 0 r 2 sinh 2 σ ) (29) \nThe parameters r 0 , α, γ, σ appearing in the metric can be related to various parameters of the microscopic model by the relations \nQ 1 = V r 2 0 2 g st sinh2 α, Q 5 = r 2 0 2 g st sinh2 γ, N = R 2 V r 2 0 2 g 2 st sinh2 σ, (30) \nM = RV r 2 0 2 g 2 st (cosh 2 α +cosh2 γ +cosh2 σ ) . (31) \nIn order to calculate the absorption coefficient from (28), we will follow a procedure similar to the one used in [30] for the 3+1 dimensional Schwarzschild black hole. \nEquation (28) admits the following separation of variables: \nφ ( r, t, χ, θ, φ ) = e -iωt R ωl ( r ) Z l ( χ ) (32) \nWe will be interested in the low frequency behaviour. It is then enough for us to concentrate on the s wave. The corresponding radial function R ω ≡ R ωl | l =0 satisfies the differential equation \n[ g r 3 d dr gr 3 d dr + ω 2 f ] R ω = 0 (33) \nThis equation can be alternatively written, in terms of ψ ω ≡ r 3 / 2 R ω , as \n[ -d 2 dr 2 ∗ + V ω ( r ∗ )] ψ ω = 0 V ω ( r ∗ ) ≡ -ω 2 f + 3 4 1 r 2 (1 -r 2 0 r 2 )(1 + 3 r 2 0 r 2 ) (34) \nwhere r ∗ ≡ ∫ dr/ (1 -r 2 0 /r 2 ) = r +(1 / 2) r 0 ln | ( r -r 0 ) / ( r + r 0 ) | is the 'tortoise coordinate'.", 'Solution in the Far Region ( r /greatermuch r 0 , ( r 0 /ω 2 ) 1 / 3 )': 'In this region we can keep terms only upto 1 r 2 in (34). The solution, given in terms of Coulomb wave functions, has the following asymptotic expansions: (i) ωr /greatermuch 1 \nR ω ∼ r -3 / 2 ( -2 iω ) -a ( e -iωr Γ(2 a ) Γ( a ) A + e iωr [ e -iπa Γ(2 a ) Γ( a ) A + B ] ) [1 + o ( ωr ) -1 ] (35) \n(ii) ωr /lessmuch 1 \nR ω ∼ r a -3 / 2 e iωr [( A + B Γ(1 -2 a ) Γ(1 -a ) ) -( -2 iωr ) 1 -2 a B Γ(1 -2 a ) Γ(1 -a ) ](1 + o ( ωr )) (36) \nHere \na = 1 2 + √ 1 -( ωr 0 ) 2 (2 + s 1 ) , s 1 = sinh 2 α +sinh 2 γ +sinh 2 σ (37)', 'Solution in the Near Region ( r → r 0 )': 'Here (33) reduces to \n( g d dr g d dr + ω 2 f 0 ) R ω = 0 ⇒ [( d dr ∗ ) 2 + ω 2 f 0 ] R ω = 0 (38) \nwhere \nf 0 = f | r = r 0 = cosh 2 α cosh 2 γ cosh 2 σ (39) \nThe solution to (38), using the boundary condition that there is no outgoing exponential at the event horizon, is \nR ω ∼ A 0 exp[ -i ( ω √ f 0 r ∗ + δ )] (40) \nBesides these solutions, it is also easy to derive the following exact ω = 0 solution (at any r ) \nR ω =0 = A 1 + B 1 2 ln | 1 -r 2 0 r 2 | (41) \nMatching (36) and (40) with the r 0 /r → 0 and r 0 /r → 1 limits of (41) ( cf. [30]) we get the following relations between various coefficients at low frequency ( ωr 0 << 1): \nB = βA 0 A = [1 -β Γ(1 -2 a ) Γ(1 -a ) ] A 0 β ≡ 2 i ( ωr 0 ) 3 √ f 0 Γ( a )Γ(2 -2 a ) Γ(2 a )Γ(1 -2 a ) (42) \nChoosing A to be such that the coefficient of e iωr r 3 / 2 in (35) is 1, we get \nR ω /similarequal e -iωr r 3 / 2 + R e iωr r 3 / 2 (43) \nThe absorption coefficient is, therefore, to leading order in ωr 0 \n|A| 2 ≡ 1 -|R| 2 = π 2 ( ωr 0 ) 3 √ f 0 = 1 4 π ω 3 A h (44) \nwhere we have used (39) and the expression for the area of the event horizon \nA h = 2 π 2 r 3 0 cosh α cosh γ cosh σ (45) \nIt is easy to show that the s wave absorption crosssection σ A is related to the absorption coefficient by \nσ A = 4 π ω 3 |A| 2 (46) \nwhich in this case, therefore, is \nσ A = A h (47) \nas claimed in the previous section. Like (44), (47) is also calculated to the leading order in ωr 0 .', '5 Concluding Remarks': "In summary, in this letter we have computed the absorption crosssection of massless quanta by a near extremal 4+1 dimensional charged black hole within the context of the string theory based microscopic model proposed in [7, 8, 18]. The authors of [8] have correctly reproduced the Hawking radiation formula for a near extremal black hole (modulo a numerical coefficient). Our microscopic computation of the absorption crosssection agrees (modulo a numerical coefficient) with the classical calculation from the analysis of a massless wave proapagating in the background metric of the appropriate black hole. \nThe basic reason why we get a nonzero absorption crosssection is the presence of the enhancement factor Ω / Ω ' in this calculation relative to the Hawking decay, which vanishes in the classical limit. The factor Ω / Ω ' depends only on the counting of the microscopic quantum states of a near extremal black hole and does not depend on the details of the matrix element calculation. This is just like the factors 〈 ˜ N L ( m ) 〉 and 〈 ˜ N R ( m ) 〉 , which also depend only on the counting of states and in the decay calculation and give rise to the universal black body nature of the Hawking decay formula. The precise cancellation of these factors with the enhancement factor Ω / Ω ' , in the classical limit, is then what gives a nonzero result for the classical absorption crosssection, as opposed to the Hawking decay, which vanishes in the classical limit. \nWe believe that it is reasonable to expect that the above feature of our calculation will not be modified by taking strong coupling effects into account, at least for near extremal black holes. On the other hand, one may, a priori, not have expected to get a detailed agreement of the clasical limit of the microscopic calculation with classical absorption crosssection calculation. That the former agrees with the latter in all its essential details is, therefore, a surprise. The magic here is the same as the one that gives the Hawking decay coefficient proportional to the area of the horizon in the calculation of [8]. This is because the magnitude of the microscopic matrix element that is responsible for absorption is the same as the one that gives the decay, at this order of string coupling. It is possible that with a better understanding of the microscopic models of black holes we might understand why certain physical situations are insensitive to the strong coupling effects of the 'dense horizon soup' [18]. In this context, it would be very interesting to compute \nthe numerical coefficient in front of the decay rate in (22) and the absorption coefficient in (27) and to see whether they agree with their expected values. \nOne of the essential features of the existence of a horizon is that classically it acts as a one way valve for particles and energy. It seems to us that the microscopic models which incorporate this feature must 'know' about the existence of a horizon in the strong coupling regime. There are, of course, many other aspects of the physics of horizon that need to be explored. Hopefully, further study will provide a better and more detailed understanding of this and other aspects of black hole physics within the context of the microscopic models. \nAcknowledgement: One of us (SW) would like to acknowledge the Japan Society for the Promotion of Science (JSPS) for a fellowship and Professors Yoneya, Kawai and Ninomiya for their excellent hospitality at the University of Tokyo (Komaba), KEK and the Yukawa Institute for Theoretical Physics, Kyoto University, respectively. We would like to thank S.R. Das for a discussion and for pointing out a numerical error in equation (46) in an earlier version of the paper. We would also like to thank T.P. Singh for pointing out reference [17].", 'References': '- [1] J. Dai, R.G. Leigh and J. Polchinski, Mod. Phys. Lett. A4 (1989) 2073; J. Polchinski, Phys. Rev. Lett. 75 (1995) 4724, hep-th 9510017\n- [2] E. Witten, Nucl. Phys. B460 (1996) 335, hep-th 9510135\n- [3] A. Sen, Phys. Rev. D53 (1996) 2874, hep-th 9511026\n- [4] C. Vafa, Nucl. Phys. B463 (1996) 415, hep-th 9511088 ; Nucl. Phys. B463 (1996) 435, hep-th 9512078\n- [5] M. Bershadsky, V. Sadov and C. Vafa, Nucl. Phys. B463 (1996) 420, hep-th 9511222 ; Nucl. Phys. B463 (1996) 398, hep-th 9510225\n- [6] M. Douglas, hep-th 9512077\n- [7] A. Strominger and C. Vafa, hep-th 9601029 \n- [8] C. Callan and J. Maldacena, hep-th 9602243\n- [9] G. Horowitz, J. Maldacena and A. Strominger, hep-th 9603109\n- [10] Breckenridge, D. Lowe, R. Myers, A. Peet, A. Strominger and C. Vafa, hep-th 9603078\n- [11] G. Horowitz and A. Strominger, hep-th 9602051\n- [12] J. Maldacena and A. Strominger, hep-th 9603060\n- [13] C. Johnson, R. Khuri and R. Myers, hep-th 9603061\n- [14] G. Horowitz, D. Lowe and J. Maldacena, hep-th 9603195\n- [15] J. Bekenstein, Lett. Nuovo Cimento 4 (1972) 737; Phys. Rev. D7 (1973) 2333; Phys. Rev. D9 (1974) 3292; S. Hawking, Phys. Rev. Lett. 26 (1971) 1344; Comm. Math. Phys. 43 (1975) 199\n- [16] S. Hawking, Phys. Rev. D13 (1976) 191; Phys. Rev. D14 (1976) 2460\n- [17] R. Penrose, p. 581 in General Relativity , Ed. by S. Hawking and W. Israel (Cambridge, 1979)\n- [18] J. Maldacena and L. Susskind, hep-th 9604042\n- [19] S. Das and S. Mathur, hep-th 9601152\n- [20] Garousi and R. Myers, hep-th 9603194\n- [21] A. Hashimoto and I. Klebanov, hep-th 9604065\n- [22] I. Klebanov and L. Thorlacius, Phys. Lett. B 371 (1996) 51, hep-th 9510200\n- [23] S.S. Gubser, A. Hashimoto, I. Klebanov and J. Maldacena, hep-th 9601057\n- [24] G. Mandal and S.R. Wadia, Phys. Lett. B 372 (1996) 34, hep-th 9511218\n- [25] F. Larsen and F. Wilczek, hep-th 9604134 \n- [26] J. Preskill, P. Schwarz, A. Shapere, S. Trivedi and F. Wilczek, Mod. Phys. Lett. A6 (1991) 2353\n- [27] C. Holzhey and F. Wilczek, Nucl. Phys. B380 (1992) 447\n- [28] P. Krauss and F. Wilczek, Nucl. Phys. B433 (1995) 403\n- [29] See, e.g., A. Sen, Nucl. Phys. B450 (1995) 103, hep-th 9504027\n- [30] W. Unruh, Phys. Rev. D 14 (1976) 3251'}
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1994PhRvL..72..183C
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Supersymmetry of the (2+1)-dimensional black holes
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1994-01-01
| 4 | 0.45 | 164 |
['-', '-', '-', '-', '-', '-', '-', '-', '-', '-', '-']
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[]
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The supersymmetry properties of the asymptotically anti-de Sitter (adS) black holes of Einstein theory in 2+1 dimensions are investigated. It is shown that (i) the zero-mass black hole has two exact supersymmetries; (ii) extreme lM=||J|| black holes with M≠0 have only one; and (iii) generic black holes do not have any. It is also argued that the zero-mass hole is the ground state of (1,1) adS supergravity with periodic (``Ramond'') boundary conditions on the spinor fields.
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https://arxiv.org/pdf/hep-th/9310194.pdf
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{'Supersymmetry of the 2+1 black holes': "Olivier Coussaert and Marc Henneaux ∗ Facult'e des Sciences, Universit'e Libre de Bruxelles, Campus Plaine C.P. 231, B-1050 Bruxelles, Belgium", 'Abstract': "The supersymmetry properties of the asymptotically anti-de Sitter black holes of Einstein theory in 2+1 dimensions are investigated. It is shown that (i) the zero mass black hole has two exact supersymmetries; (ii) extreme lM = | J | black holes with M /negationslash = 0 have only one; and (iii) generic black holes do not have any. It is also argued that the zero mass hole is the ground state of (1,1)-adS supergravity with periodic ('Ramond') boundary conditions on the spinor fields. \nAmong the black hole solutions of (2+1)-Einstein theory discovered recently [1], the one with zero mass and zero angular momentum stands apart. (i) It is the solution with smallest mass. (ii) It has zero temperature. (iii) It has zero entropy. We show in this letter that it enjoys also remarkable supersymmetry properties [2]-[10]. Namely, it is the black hole solution with the maximum number of exact supersymmetries. We shall first establish the result and shall then discuss its implications [11]. \nThe 2+1 black hole metric is given by [1] \nds 2 = -N 2 dt 2 + N -2 dt 2 + r 2 ( N ϕ dt + dϕ ) 2 N 2 = ( r/l ) 2 -M +( J/ 2 r ) 2 (1) N ϕ = -J/ 2 r 2 \nwhere M and J are respectively the mass and angular momentum of the hole, and where -l 2 is the cosmological constant. It can be obtained by making appropriate identifications of the anti-de Sitter metric [12], which corresponds to (1) with M = -1 and J = 0, \nds 2 adS = -[( r/l ) 2 +1] dt 2 +[( r/l ) 2 +1] -1 dr 2 + r 2 dϕ 2 (2) \nThe metric (1) with M /negationslash = -1 has only two Killing vectors [12]. If regarded as a solution of the equations of motion of adS supergravity with zero gravitini,it may possess, in addition, exact supersymmetries. Exact supersymmetries are by definition supersymmetry transformations leaving the metric (1) (with zero gravitini) invariant. The spinor parameters of these transformations solve the 'Killing spinor equation' \nD λ ψ = /epsilon1 2 l γ λ ψ (3) \nwhere /epsilon1 = 1 or 1 depending on the representation of the γ -matrices. \nAs it is well known, there are two inequivalent two-dimensional irreductible representations of the γ -matrices in three spacetime dimensions. One may be taken to be γ (0) = iσ 2 , γ (1) = σ 1 and γ (2) = σ 3 , where the σ k are the Pauli matrices. The other is given by γ ' ( λ ) = -γ ( λ ) . We shall consider here the simplest supergravity model with negative cosmological constant involving both representations, namely (1 , 1) adS supergravity [13]. \n- \nThe anti-de Sitter metric (2) possesses four Killing spinors, two for each inequivalent representation of the γ -matrices. In the radial tetrad frame \nh (0) = -[( r/l ) 2 +1] 1 2 dt h (1) = [( r/l ) 2 +1] -1 2 dr (4) h (2) = rdϕ \nthe Killing spinors are given by \nψ = [( N ads +1 2 ) 1 2 + /epsilon1 ( N ads -1 2 ) 1 2 γ (1) ] (5) × ( cos 1 2 ( ϕ + /epsilon1t/l ) -sin 1 2 ( ϕ + /epsilon1t/l ) γ (0) ) A \nwhere A is a constant spinor. \nSince the black hole metric can be obtained from (2) by making appropriate identifications, it possesses locally as many Killing spinors as anti-de Sitter space. However, only a subset of these Killing spinors are, in general, compatible with the identifications, i.e., invariant under the transformations of the discrete group used in the identifications. So, whereas all the local integrability conditions for the Killing equations (3) are fullfilled [14], there may be no Killing spinor at all because of global reasons. \nIn order to discuss which Killing spinors are compatible with the identifications, let us make a choice of coordinates in which the identifications take a simple form. As shown in [12] , anti-de Sitter space can be rewritten as \nds 2 ads = -[( R/l ) 2 -1] dT 2 +[( R/l ) 2 -1] -1 dR 2 + R 2 d Φ 2 (6) \nin new coordinates ( T, R, Φ) where Φ is not an angle but runs over the entire real line. The Killing spinors are, in the frame \nh (0) = -[( R/l ) 2 -1] 1 2 dT h (1) = [( R/l ) 2 -1] -1 2 dR (7) h (2) = Rd Φ \ngiven by \nψ = 1 √ 2 [(( R/l ) + 1) 1 2 + /epsilon1 (( R/l ) -1) 1 2 γ (1) ] (8) × ( cosh 1 2 (Φ + /epsilon1T /l ) + sinh 1 2 (Φ + /epsilon1T /l ) γ (2) ) A \nThe identifications appropriate to a non extreme black hole with angular momentum J and mass M ( | J | < Ml ) have been shown in [12] to be \n( T, Φ) ∼ ( T + J, Φ+ M ) , | J | < Ml (9) \nSince the Killing spinors are not invariant (even up to a sign) under these identifications, they are not well defined in the quotient space. Therefore, a generic black hole has no Killing spinor. \nLet us now consider the extreme case | J | = Ml . The identifications appropriate to that case are more complicated to describe in the coordinate system where (6) holds [16]. For that reason, we shall directly proceed to the explicit integration of the Killing spinor equations in the metric (1), where ϕ is an angle. For definitess, we treat the case J = Ml . The case J = -Ml is treated similarly. One may take as local Lorentz frame \nh (0) = -Ndt h (1) = N -1 dr (10) h (2) = -Ml 2 r dt + rdϕ \nOne finds that the Killing spinors solutions of (3) with /epsilon1 = 1 are given by \nψ = 1 2 [( U 1 2 + U -1 2 ) + ( U 1 2 -U -1 2 ) γ (1) ] (11) [1 + 1 2 ( γ (2) -γ (0) )( ϕ + t/l )] A \nwhere A is a constant spinor and U is given by \nU = l/r (( r/l ) 2 -M/ 2) . (12) \nThe Killing spinors for the other representation of the γ -matrices are \nψ = 1 2 ( r/l ) 1 2 [ αcosh ( √ M/ 2( t/l -ϕ )) + (13) βsinh ( √ M/ 2( t/l -ϕ ))] ( 1 -1 ) + 1 2 ( r/l ) -1 2 √ M/ 2[ αsinh ( √ M/ 2( t/l -ϕ )) + βcosh ( √ M/ 2( t/l -ϕ )) ( 1 1 ) \nwhere we recall that γ (1) = σ (1) , and where α and β are constants. The Killing spinor (11) is compatible with the periodicity of ϕ if and only if the linearly growing term in ϕ disappears, i.e., if and only if A is an eigenstate of γ (1) with eigenvalue +1. In that case, (11) does not depend on ϕ and is thus manifestly periodic. The Killing spinor (13) is never periodic or antiperiodic. There is thus only one Killing spinor for the extreme black hole with non vanishing mass. \nIn the limit M → 0, one gets from each sign of J a ϕ -independent Killing spinor. These read explicitly \nψ 1 = 1 2 ( r/l ) 1 2 ( 1 1 ) (14) \nand \nThe zero mass state has thus two exact supersymmetries. \nψ 2 = 1 2 ( r/l ) 1 2 ( 1 -1 ) . (15) \nThe Killing spinors of the extreme black hole solutions have the same asymptotic growing in r as the Killing spinors of anti-de Sitter space. However, they are periodic in ϕ , while those of anti-de Sitter space are antiperiodic. \nThis feature has interesting implications. It has been established in [17] that a negative cosmological constant allows for rich asymptotics. Namely, there exist boundary conditions on the gravitational variables such that the asymptotic symmetry algebra of (2+1)-gravity with a negative cosmological constant is the conformal algebra in two dimensions, i.e., twice the Virasoro algebra. The mass and angular momentum are respectively given by M = l -1 ( K 0 + L 0 ) and J = K 0 -L 0 , where K n and L n are the right and left Virasoro generators. These boundary conditions include the black hole solutions of [1]. \nNow, in a spacetime with the black hole topology R 2 × S 1 [12], one can consider spinor fields that are either periodic or anti-periodic in ϕ in the above radial triad frames. These different behaviours define inequivalent spinor structures and lead to different asymptotic superalgebras for (3+1)supergravity with a negative cosmological constant. The periodic case yields the Ramond graded extension of the Virasoro algebra and will be referred to as the 'Ramond sector' for that reason. The anti-periodic case yields the Neveu-Schwarz extension and will be called the 'Neveu-Schwarz sector'. \nJust as in 3 + 1 dimensions [18, 19, 20], the asymptotic supersymmetry algebra implies bounds for the generators K 0 and L 0 . The stronger ones are \nK 0 = G 1 / 2 G -1 / 2 + G -1 / 2 G 1 / 2 -1 2 ≥ -1 2 , (16) \nL 0 = ¯ G 1 / 2 ¯ G -1 / 2 + ¯ G -1 / 2 ¯ G 1 / 2 -1 2 ≥ -1 2 (17) \nfor the Neveu-Schwarz case, and \nK 0 = F 2 0 ≥ 0 (18) \nL 0 = ¯ F 2 0 ≥ 0 (19) \nfor the Ramond case. The G k and F k are the asymptotic right supersymmetry generators, while the ¯ G k and ¯ F k are the asymptotic left supersymmetry generators. \nThe exact supersymmetries of anti-de Sitter space belong to the NeveuSchwarz sector and are generated by the right and left supersymmetry charges with 'frequency' 1 / 2 and -1 / 2 (the Killing spinors have that dependence on ϕ ). Hence, anti-de Sitter space is annihilated by G 1 / 2 and ¯ G 1 / 2 and saturates the bound for the Neveu-Schwarz case, in agreement with M = -1. Similarly, the zero mass hole is invariant under the two zero-mode supersymmetries generated by F 0 and ¯ F 0 and saturates the bounds (18) and (19) of the Ramond case, leading to M = 0. Accordingly, the zero mass hole appears as the ground state of the Ramond sector. The extreme black holes lM = J | with M = 0 saturate only one of the bounds (18) or (19). \n| \n| In the above analysis, we have set the electric charge equal to zero. The reason why we did not consider charged black holes is that these appear to possess somewhat unphysical properties in 2+1 dimensions. (i) They fail to fullfill the fall-off conditions given in [17] for asymptotically anti-de sitter spaces. (ii) The energy M is not bounded from either below or above when the charge is different from zero. Indeed given an arbitrarily negative mass, the solution given in [1] possess an event horizon hiding the singularity for Q big enough. The unboundedness of the energy renders the solutions unstable and should imply the absence of asymptotic Killing spinors -a fortiori of exact Killing spinors. \n/negationslash \nOnce one imposes the asymptotic behaviour of [17], one must take Q = 0. This forces the electromagnetic field to vanish and makes the vector \npotential locally pure gauge. The vector potential is not necessarily globally pure gauge, however, since the fundamental group of black hole solutions is non trivial. Because of the presence of non contractible loops, the black hole spacetimes can support non zero holonomies of A λ , given by A t = 0, A r = 0, A ϕ = Constant . This is somewhat reminiscent of (3+1)-black holes with axionic charge [21]. It is no accident, since in 2+1 dimensions, the electromagnetic field ( and not the Kalb-Ramond field) is dual to the axion field . \nIn four-dimensional Einstein-Maxwell theory with zero cosmological constant, the only black holes with exact supersymmetries are the extreme Reissner-Nordstrom black holes. These have the further remarkable property that one can construct static, extreme multi-black hole solutions [22, 23], in which the Coulomb potential exactly balance the gravitational attraction. Both properties have sometimes been related, so that it is natural to ask whether one can also construct static multi-black holes solutions in 2+1 dimensions. It turns out that this is not the case. As we shall show in detail in a separate publication where the general static solution of 2+1 Einstein theory with negative cosmological constant will be constructed, there is no static, supersymmetric, pure multi-black hole metric without additional (undesirable) naked branch point singularities. \nTo conclude, we have shown in this letter that the zero mass state enjoys remarkable supersymmetry properties. These indicate that the zero mass state is the ground state of the Ramond sector of (1,1) adS supergravity. Furthermore, exact supersymmetry is associated with the precise bounds guaranteing the absence of naked singularity (cosmic censorship), as in 3+1 dimensions [9]. The extreme bound lM = | J | yields one supersymmetry. The bound M = 0 yields a second supersymmetry. A detailed presentation of this work, covering the extended ( p, q ) adS supergravity models, will be reported elsewhere.", 'Acknowledgements': "One of us (M. H.) is grateful to Andrew Strominger for interesting questions on the supersymmetry properties of the charged 2 + 1-black holes and to Claudio Teitelboim for useful comments and discussions. He also gratefully acknowledges the hospitality of the Institute for Advanced Study where this work has been partly carried out. O. C. is 'Chercheur I.R.S.I.A'. This research has been supported in part by research funds from F.N.R.S. and by \na research contract with the Community of the European Communities.", 'References': "- [1] M. Ba˜nados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 69 (1992)1849.\n- [2] Exact supersymmetry properties of (3+1)-black holes have been must studied since the realization that extremal Reisner-Nordstrom black holes have some unbroken supersymmetries[3-6]. These properties have been shown recently to play a central role in nonrenormaliztion theorems for extreme black holes[7]. They have been generalized to dilaton gravity[8-10].\n- [3] P.C. Aichelburg and R. Guven, Phys. Rev. D24 (1981)2066; D27 (1983) 456; Phys. Rev. Lett. 51 (1983)1613.\n- [4] G.W. Gibbons, in Supersymmetry, supergravity and related topics , eds. F. del Aguila, J. de Azc'arraga and L. Ib'a˜nez (World Scientific, Singapore, 1985) p.147; G.W. Gibbons and C.M. Hull, Phys. Lett. B109 (1982)190.\n- [5] K.P. Tod, Phys. Lett. B121 (1983)241.\n- [6] L.J. Romans, Nucl. Phys. B383 (1992)395.\n- [7] R. Kallosh, Phys. Lett. B282 (1992)80.\n- [8] G.W. Gibbons, Nucl. Phys. B207 (1982)337; G.W. Gibbons and K. Maeda, Nucl. Phys. B298 (1988)741; D. Garfinkle, G.T. Horowitz and A. Strominger, Phys. Rev. D43 (1991)3140; A. Shapere, S. Trivedi and F. Wilczek, Mod. Phys. Lett. A6 (1991)2677.\n- [9] R. Kallosh, A. Linde, T. Ort'ın, A. Peet and A. van Proyen, Phys. Rev. D46 (1992)5278.\n- [10] T. Ort'ın, Phys. Rev. D47 (1993)3136. \n- [11] Even though its horizon has zero length, we shall refer to the zero mass solution as being a black hole since the singularity, which coincides with the horizon, is invisible to an observer at infinity.\n- [12] M. Ba˜nados, M. Henneaux, C. Teitelboim and J. Zanelli, Phys. Rev. D48 (1993)1506.\n- [13] A.Ach'ucarro and P.K. Townsend, Phys. Lett. 180B (1986) 89.\n- [14] The fulfilment of the integrability condition is quite obvious in terms of the Chern-Simons formulation of (2+1)-supergravity [13, 15]. Indeed, the Killing equations then read ∇ λ ψ = 0 where ∇ λ is the covariant derivative in the SO (2 , 2)-connection. Since this connection is flat, the equations are integrable.\n- [15] E. Witten, Nucl. Phys. B311 (1988/1989) 46.\n- [16] If one takes | J | = Ml in the identifications (9), one does not get an asymptotically anti-de Sitter black hole solution. Nevertheless, the resulting quotient space is quite interesting: it has no singularity, no closed time like curves, two Killing spinors and four Killing vectors ( the vector field used in making the identification is everywhere spacelike and turns out to be self-dual ( J = Ml ) or anti-self-dual ( J = -Ml )).\n- [17] J.D. Brown and M. Henneaux, Commun. Math. Phys. 104 (1986)207.\n- [18] C. Teitelboim, Phys. Lett 69B (1977)240; S. Deser and C. Teitelboim, Phys. Rev. Lett. 39 (1977)249.\n- [19] E. Witten, Commun. Math. Phys. 80 (1981)397\n- [20] L.F. Abbott and S. Deser, Nucl. Phys. B195 (1982)76.\n- [21] M.J. Bowick, S.B. Giddings, J.A. Harvey, G.T. Horowitz and A. Strominger, Phys. Rev. Lett. 61 (1988)2823.\n- [22] A. Papapetrou, Proc. R. Irish Acad. A51 (1947)191; S.D. Majumdar, Phys. Rev. 72 (1947)930.\n- [23] J.B. Hartle and S.W. Hawking, Commun. Math. Phys. 26 (1972)87."}
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2003PhLB..559...65V
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Generalization of the KKW analysis for black hole radiation
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2003-01-01
| 2 | 0.45 | 164 |
['-', '-']
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[]
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An extension of the Keski-Vakkuri, Kraus and Wilczek (KKW) analysis to black hole spacetimes which are not Schwarzschild-type is presented. Preserving the regularity at the horizon and stationarity of the metric in order to deal with the across-horizon physics, a more general coordinate transformation is introduced. In this analysis the Hawking radiation is viewed as a tunnelling process which emanates from the non-Schwarzschild-type black hole. Expressions for the temperature and entropy of these non-Schwarzschild-type black holes are extracted. As a paradigm, in the context of this generalization, we consider the Garfinkle-Horowitz-Strominger (GHS) black hole as a dynamical background and we derive the modified temperature and entropy of GHS black hole. Deviations are eliminated and corresponding standard results are recovered to the lowest order in the emitted shell of energy. The extremal GHS black hole is found to be non-“frozen” since it is characterized by a constant non-zero temperature. Furthermore, the modified extremality condition forbids naked singularities to form from the collapse of the GHS black hole.
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https://arxiv.org/pdf/hep-th/0209185.pdf
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{'Elias C. Vagenas 1': "Departament d'Estructura i Constituents de la Mat'eria and CER for Astrophysics, Particle Physics and Cosmology Universitat de Barcelona Av. Diagonal 647 E-08028 Barcelona Spain", 'Abstract': "An extension of the Keski-Vakuri, Kraus and Wilczek (KKW) analysis to black hole spacetimes which are not Schwarzschild-type is presented. Preserving the regularity at the horizon and stationarity of the metric in order to deal with the across-horizon physics, a more general coordinate transformation is introduced. In this analysis the Hawking radiation is viewed as a tunnelling process which emanates from the non-Schwarzschild-type black hole. Expressions for the temperature and entropy of these non-Schwarzschild-type black holes are extracted. As a paradigm, in the context of this generalization, we consider the Garfinkle-Horowitz-Strominger (GHS) black hole as a dynamical background and we derive the modified temperature and entropy of GHS black hole. Deviations are eliminated and corresponding standard results are recovered to the lowest order in the emitted shell of energy. The extremal GHS black hole is found to be non-'frozen' since it is characterized by a constant non-zero temperature. Furthermore, the modified extremality condition forbids naked singularities to form from the collapse of the GHS black hole.", 'Introduction': 'The idea of Keski-Vakkuri, Kraus and Wilczek [1, 2, 3] (KKW) has been applied till now to Schwarzschild-type black hole geometries. In this semiclassical analysis the energy conservation plays a dominant role since it leads us to a dynamical Schwarzschild-type black hole background which in turn leads to a more realistic description of the black hole radiance [4, 5]. The total Arnowitt-Deser-Misner mass [6] is fixed while the mass of the Schwarzschild-type black hole decreases due to the emitted radiation. A coordinate transformation is implemented so that the line element be nonsingular at the horizon. This permits the study of across-horizon physics such as the black hole radiation. Using this methodology, it is has been made possible to exactly evaluate the non-thermal spectrum of the black hole radiation (the non-thermal character of the black hole radiation was already known since the seminal work of Hawking in mid-seventies). A direct byproduct of this analysis is that the black hole temperature is not only a function of the characteristics of the black hole but also of the energy of the emitted shell of energy. Additionally the black hole entropy is not that given by the area formula of Bekenstein and Hawking for the corresponding Schwarzschild-type black hole. \nIn the seminal works of Keski-Vakkuri, Kraus and Wilczek [1, 2, 3], although the starting point was spherically symmetric geometries, the analysis was restricted to Schwarzschild-type black holes, i.e. no general expressions for temperature and entropy were extracted for the above-mentioned geometries. Here we introduce a more general coordinate transformation in order to apply the KKW analysis to non-Schwarzschild-type black hole spacetimes. This general transformation preserves two conditions: (a) the regularity at the horizon which ensures that we are able to study across-horizon physics, (b) the stationarity of the non-static metric which implies that the time direction is a Killing vector, which are crucial in order to generalize the KKW analysis. The methodology followed is an appropriate modification of that adapted in the Schwarzschild-type black holes. The effect of this generalization in our calculation leads to exact expressions for the temperature and \nthe entropy of the non-Schwarzschild-type black holes which are not anymore the Hawking temperature and the Bekenstein-Hawking entropy (given by the area formula), respectively. \nThe outline of this paper is as follows. Section 1 is devoted to the presentation of KKW analysis in Schwarzschild-type black hole geometries. In Section 2 we extend the analysis of Section 1 to the case of non-Schwarzschildtype black hole geometries. In the framework of this semiclassical analysis, we derive exact expressions for the temperature and entropy of these black hole spacetimes. In Section 3 we implement the above-mentioned expressions for the case of the GHS black hole. We derive the corresponding modified temperature and entropy of the GHS black hole and in the lowest order of the emitted shell of energy standard results, i.e. Hawking temperature and Bekenstein-Hawking entropy are reproduced, respectively, as a verification of the validity of our results. We consider the extremality condition which now will be shifted since the charge Q of the four-dimensional GHS black hole will be reached by the mass M earlier. The temperature of the extremal GHS black hole is shown to be non-zero and singularities are found to be always hidden behind the event horizon of the GHS black hole. Finally, in Section 4 we end up with a short summary and concluding remarks.', '1 KKW Analysis': "The idea of Keski-Vakkuri, Kraus and Wilczek (KKW) was firstly utilized for the case of the four-dimensional Schwarzschild black hole by Parikh and Wilczek [7]. From that time till now the KKW methodology was also applied to several black hole solutions such as (d+1)-dimensional Anti-de-Sitter black hole [8], AdS(2) black hole [9], two-dimensional charged black hole solution derived from the effective string theory at the low-energy limit [10], two-dimensional charged (and uncharged) dilatonic black holes [11] (dimensionally reduced from (2+1) spinning (and spinless) BTZ black holes), (2+1)dimensional charged BTZ black hole [12, 13] and lately to a Schwarzschild-de Sitter spacetime [14]. \nAll these black hole backgrounds belong to the same family of geometries \nsince their line elements were Schwarzschild-like, i.e. they were of the type \nds 2 = -A ( r ) dt 2 + A -1 ( r ) dr 2 + r 2 d Ω (1) \nwhere the metric function A ( r ) had at least one (outer) event horizon ( r + ), i.e. \nA ( r + ) = 0 . (2) \nSince the frame of the KKW methodology is the Hawking phenomenon which main contribution comes from the event horizon, the line element should be regular at the event horizon. Therefore a choice of suitable coordinates was enforced and the ansantz is to pick up the a l ' a Painlev' e [15] coordinate transformation \n√ A ( r ) dt = √ A ( r ) dτ -√ √ √ √ 1 -A ( r ) A ( r ) dr (3) \nwhere τ is the new time coordinate (Painlev' e coordinate). \nAfter squaring expression (3) and substituting into equation (1), the line element becomes \nds 2 = -A ( r ) dτ 2 +2 √ 1 -A ( r ) dτdr + dr 2 + r 2 d Ω . (4) \nIt is obvious that there is no singularity at the event horizon r H and these coordinates are stationary, but not static. \nThe radial null geodesics are given by \n˙ r ≡ dr dτ = ± 1 -√ 1 -A ( r ) (5) \nwhere the upper (lower) sign in the above equation corresponds (under the assumption that τ increases towards future) to the outgoing (ingoing) geodesics. \nAt this point we will take into consideration the self-gravitation effect by fixing the total Arnowitt-Deser-Misner mass ( M ADM ) of the black hole and letting the mass M of the black hole to vary. A shell of energy ω is now radiated by the black hole. It travels on the outgoing geodesics which now \nare due to the fluctuation of the mass M of the black hole, derived by the modified line element \nds 2 = -A ( r, M -ω ) dτ 2 +2 √ 1 -A ( r, M -ω ) dτdr + dr 2 + r 2 d Ω . (6) \nThe outgoing radial null geodesics followed by the shell of energy will also be modified as \nAt this point let us remind ourselves of the following statements \n˙ r = 1 -√ 1 -A ( r, M -ω ) . (7) \n- 1. It is known that the emission rate Γ for a radiating source [5] is given as \nΓ ≈ e -βω = e +∆ S bh (8) \nwhere β is the inverse temperature ( T bh ) of the black hole and ∆ S bh is the change in the entropy of the black hole before and after the emission of the shell of energy ω (outgoing massless particle) \n∆ S bh = S bh ( M -ω ) -S bh ( M ) . (9) \n- 2. A canonical Hamiltonian treatment gives a simple result for the total action of a system [1] \nI = ∫ dτ [ p τ + dr dτ p r ] (10) \nwhere τ and r are the Painlev' e coordinate while p τ and p r are the corresponding conjugate momenta. \n- 3. A semiclassical (WKB) approximation gives the following expression for the emission rate [3] \nΓ ≈ e -2 Im I (11) \nwhere only the second term in equation (10) contributes to the imaginary part of the action. \nWe will consider here only the s-wave emission of massless particles. Therefore, using the above mentioned statements to the KKW methodology the imaginary part of the action can be obtained \nIm I = Im ∫ dr dτ p r dτ (12) \n= Im ∫ r + ( M -ω ) r + ( M ) p r dr (13) \nwhere r + is the outer event horizon of the black hole as mentioned before. It is useful to apply the Hamilton's equation \n˙ r = dH dp r (14) \n= d ( M -ω ) dp r (15) \ndp r = d ( M -ω ) ˙ r (16) \nand thus \nEquation (13) can be written as \nIm I = Im ∫ r + ( M -ω ) r + ( M ) ∫ p r 0 dp ' r dr (17) \nand substituting equation (16) we get \nIm I = Im ∫ r + ( M -ω ) r + ( M ) ∫ + ω 0 d ( M -ω ' ) ˙ r dr . (18) \nThus if we use the modified outgoing geodesics (7) the imaginary part of the action will be written as \nIm I = Im ∫ r + ( M ) r + ( M -ω ) ∫ + ω 0 dω ' 1 -√ 1 -A ( r, M -ω ' ) dr . (19) \nIt is easily seen, using the first statement, that the temperature and the entropy of the black hole is not the Hawking temperature ( T H ) and the entropy given by the area formula of Bekenstein and Hawking ( S BH ), respectively. Both of them are modified due to the specific modelling of the self-gravitation \neffect. Thus, the modified temperature and entropy of the black hole is given, respectively, by \nT bh = ω 2 Im ∫ r + ( M ) r + ( M -ω ) ∫ + ω 0 dω ' 1 -√ 1 -A ( r, M -ω ' ) dr -1 (20) \nwhere the entropy of the black hole with mass M must be equal to that given by the area formula of Bekenstein and Hawking ( S BH ). Therefore, the expression for the modified entropy will be \n S bh ( M -ω ) = S bh ( M ) -2 Im ∫ r + ( M ) r + ( M -ω ) ∫ + ω 0 dω ' 1 -1 -A ( r, M -ω ' ) dr (21) \n√ \nS bh ( M -ω ) = S BH -2 Im ∫ r + ( M ) r + ( M -ω ) ∫ + ω 0 dω ' 1 -√ 1 -A ( r, M -ω ' ) dr . (22)", '2 Generalizing the KKW Analysis': "The KKW methodology was, till now, restricted to Schwarzschild-type black hole backgrounds (see equation (1)). In this section we extend the KKW analysis to more general black hole backgrounds of the form \nds 2 = -A ( r ) dt 2 + B -1 ( r ) dr 2 + r 2 d Ω (23) \nwhere A ( r ) and B ( r ) are functions satisfying the equation \nA ( r ) · B -1 ( r ) = 1 . (24) \n/negationslash \nThe metric function B ( r ) had at least one (outer) event horizon ( r + ), i.e. \nB ( r + ) = 0 (25) \nsince a event horizon appears at a spacetime point where g rr = 0 and for the line element in (23) \ng rr = B ( r ) . (26) \nIn order for the total Arnowitt-Deser-Misner mass ( M ADM ) to be well-defined we restrict the class of metrics to those which are asymptotically flat, i.e. \nA ( r ) → 1 as r → + ∞ B ( r ) 1 as r + . \n→ \n→ \n∞ \nIt is obvious that since we would like to deal with the radiation phenomenon of a black hole we need to keep the regularity at the event horizon and the stationarity of the metric which in turn implies that the time direction is a Killing vector [16, 17]. Therefore, we introduce the following a l ' a Painlev' e more general, compared to that applied before for the Schwarzschild-type black holes, coordinate transformation \n√ A ( r ) dt = √ A ( r ) dτ -√ B -1 ( r ) -1 dr (27) \nwhere τ is the new time coordinate (Painlev' e coordinate). Substituting expression (27) in equation (23) the line element becomes \nds 2 = -A ( r ) dτ 2 +2 √ √ √ √ A ( r ) B ( r ) ( 1 -B ( r ) ) dτdr + dr 2 + r 2 d Ω . (28) \nThe radial null geodesics are now given by \n˙ r = √ √ √ √ A ( r ) B ( r ) [ ± 1 -√ 1 -B ( r ) ] (29) \nwhere the upper (lower) sign in the above equation corresponds, as before, to the outgoing (ingoing) geodesics under the assumption that τ increases towards future. \nAt this point we fix the total Arnowitt-Deser-Misner mass ( M ADM ) of the black hole since we want to include the effect of self-gravitation. On the contrary, we let the mass M of the black hole to fluctuate. A shell of energy ω which constitutes of massless particles considering only the s-wave part of emission, is now radiated by the black hole. The massless particles travel on the outgoing geodesics which now are due to the varying mass M of the black hole, derived by the modified line element \nds 2 = -A ( r, M -ω ) dτ 2 +2 √ √ √ √ A ( r, M -ω ) B ( r, M -ω ) ( 1 -B ( r, M -ω ) ) dτdr + dr 2 + r 2 d Ω . (30) \nThe outgoing radial null geodesics followed by the massless particles, i.e. the shell of energy, will also be modified as follows \n˙ r = √ √ √ √ A ( r, M -ω ) B ( r, M -ω ) [ ± 1 -√ 1 -B ( r, M -ω ) ] . (31) \nWe adopt the previously mentioned three statements. We follow the same steps as before in order to write down an expression for the imaginary part of the action. Using the modified outgoing geodesics (31), the imaginary part of the action will now be given by \nIm I = Im ∫ r + ( M ) r + ( M -ω ) ∫ + ω 0 dω ' √ ˜ A ˜ B [ 1 -√ 1 -˜ B ] dr (32) \nA = A ( r, M -ω ' ) (33) \n˜ \nwhere ˜ A and ˜ B are defined as follows \n˜ B = B ( r, M -ω ' ) . (34) \n˜ Finally considering a more general black hole geometry than that of the Schwarzschild-type, the expression for the modified temperature is given as \nT bh = ω 2 Im ∫ r + ( M ) r + ( M -ω ) ∫ + ω 0 dω ' √ ˜ A B [ 1 -√ 1 -˜ B ] dr -1 (35) \n and the corresponding expression for the modified entropy is given as \n˜ \nS bh ( M -ω ) = S BH -2 Im ∫ r + ( M ) r + ( M -ω ) ∫ + ω 0 dω ' √ ˜ A ˜ B [ 1 -√ 1 -˜ B ] dr . (36) \n˜ It is obvious that in case where the metric functions A and B satisfy the condition \nA ( r ) · B -1 ( r ) = 1 , (37) \nequations (35) and (36) coincide with the respective expressions (20) and (22) for the Schwarzschild-type black hole.", '3 GHS Black Hole': "The starting point will be the four-dimensional low-energy action obtained from string theory which is written in terms of the string metric as [18] \nS = ∫ d 4 x √ -g e -2 φ [ -R -4 ( ∇ φ ) 2 + F 2 ] (38) \nand the charged black hole metric is \nds 2 string = -( 1 -2 Me φ 0 r ) ( 1 -Q 2 e 3 φ 0 Mr ) dt 2 + dr 2 ( 1 -2 Me φ 0 r ) ( 1 -Q 2 e 3 φ 0 Mr ) + r 2 d Ω . (39) \nThis metric describes a black hole with an event horizon at \nr + = 2 Me φ 0 (40) \nwhen Q 2 < 2 e -2 φ 0 M 2 . The Hawking temperature of the GHS black hole solution (39) easily evaluated by the use of the periodicity of the Euclidean section, is given by \nT H = 1 8 πMe φ 0 . (41) \nIt is obvious that the Hawking temperature of the GHS black hole is independent of the charge Q , for Q < √ 2 e -φ 0 M . \nAt extremality, i.e. when Q 2 = 2 e -φ 0 M 2 , the GHS black hole solution (39) becomes \nds 2 string = -dt 2 + ( 1 -2 Me φ 0 r ) -2 dr 2 + r 2 d Ω . (42) \nand the corresponding Hawking temperature of the extremal GHS black hole (42) is \nT ext H = 0 (43) \nsince the Euclidean section is smooth but without identifications. \nIn order to implement the methodology introduced in the previous section we firstly identify the metric functions A ( r ) and B ( r ) for the case of GHS black hole by comparing equation (23) with (39) and we get \nA ( r ) = ( 1 -2 Me φ 0 r ) ( 1 -Q 2 e 3 φ 0 Mr ) (44) \nB ( r ) = ( 1 -2 Me φ 0 r )( 1 -Q 2 e 3 φ 0 Mr ) . (45) \nAvoiding to make the complicated computation of the integral in expression (32) for the specific black hole background (39) we make the following \napproximation \nSubstituting expression (46) in equation (32) and using the expressions of A ( r ) and B ( r ) for the GHS black hole, i.e. equations (44) and (45), respectively, we get \n√ √ √ √ ˜ A B ( 1 -√ 1 -˜ B ) ≈ 1 2 √ ˜ A ˜ B . (46) \n˜ \nIm I ≈ 2 Im ∫ r + ( M ) r + ( M -ω ) ∫ + ω 0 dω ' dr √ ˜ A ˜ B = 2 Im ∫ r + ( M ) r + ( M -ω ' ) ∫ + ω 0 dω ' dr ( 1 -2( M -ω ) e φ 0 r ) . (47) \nWe firstly perform the ω -integration which involves a contour integration into the lower half of ω ' plane and we finally get \nIm I = π 2 e -φ 0 [ r 2 + ( M ) -r 2 + ( M -ω ) ] . (48) \nThus the modified temperature (35) for the case of the GHS black hole (39) is given as \nT bh ( M,φ 0 , ω ) = ω 4 πM 2 e φ 0 [ 1 -( 1 -ω M ) 2 ] -1 (49) \nand the corresponding expression for the modified entropy (36) is given as \nS bh ( M,φ 0 , ω ) = S BH -4 πM 2 e φ 0 [ 1 -( 1 -ω M ) 2 ] . (50) \nWe see that there are deviations from the standard results derived for a fixed background. The temperature of the GHS black hole is not the Hawking temperature (41) and its entropy is not given by the Bekenstein-Hawking area formula [19] \nS BH = 1 4 A H = πr 2 + = 4 πM 2 e 2 φ 0 . (51) \nA welcomed but not unexpected result is that the modified temperature (49) evaluated to first order in ω yields the Hawking temperature of the GHS black \nhole (41). Additionally the modified entropy of the GHS black hole (50) to zeroth order in ω yields the corresponding Bekenstein-Hawking entropy (51). In the framework of our analysis, the extremal GHS black hole will be created when \nQ 2 = 2 e -φ 0 ( M -ω ) 2 . (52) \nIt is obvious that the extremality condition ( r + = r -) is modified and the temperature of the extremal GHS black hole is no longer zero but it is given as \nT ext bh ( M,Q,φ 0 ) = 1 4 πMe -φ 0 ( 1 -Q √ 2 M e -φ 0 ) . (53) \nAs a byproduct of this modification to the extremality condition, since the emitted shell of energy ω has to be always positive \nω = M -Q √ 2 e φ 0 > 0 , (54) \nit is implied that \nQ < √ 2 Me -φ 0 . (55) \nThus the extremality condition indicates that a naked singularity will never form from the collapse of the GHS black hole.", '4 Conclusions': "We have introduced a new, more general, coordinate transformation in order to generalize the KKW analysis to non-Schwarzschild-type black holes. Exact expressions for the temperature and the entropy of these black holes have been derived. Due to the specific modelling of the self-gravitation effect - described by the KKW analysis - the black hole temperature is not the Hawking temperature but depends explicitly on the emitted masssless (since we have restricted our analysis to the s-wave emission) particle's energy. The black hole entropy is also different from the corresponding entropy given by the area formula of Bekenstein and Hawking. It is easily seen that the modified entropy is less than the Bekenstein-Hawking entropy. In the context of our generalized KKW analysis, we have implemented the aforesaid expressions \nfor the case of the static four-dimensional charged black holes in string theory which are the well-known Garfinkle-Horowitz-Strominger (GHS) black holes. The temperature of the GHS black hole is no more the corresponding Hawking temperature and the entropy of the GHS black hole is no longer the corresponding Bekenstein-Hawking entropy. The 'greybody factors' showing up declare explicitly the dependence on the emitted particle's energy. As a verification of the validity of our generalized KKW analysis introduced here, the modified temperature and entropy of GHS black hole in the lowest order of the emitted shell of energy reproduce the standard results. Finally, we have shown that the extremal GHS black hole is no more 'frozen' but it is characterized by a nonzero background temperature since the extremality condition is modified (due to the specific modelling of the backreaction effect). It should also be noted that because of the modified extremality condition naked singularities are forbidden in a natural way.", 'Acknowledgements': "The author would like to thank the anonymous referee of Physics Letters B for his useful comments and Ass. Professor T. Christodoulakis for reading the manuscript. This work has been supported by the European Research and Training Network 'EUROGRID-Discrete Random Geometries: from Solid State Physics to Quantum Gravity' (HPRN-CT-1999-00161).", 'References': "- [1] P. Kraus and F. Wilczek, Nucl. Phys. B 433 (1995) 403, gr-qc/9408003.\n- [2] P. Kraus and F. Wilczek, Nucl. Phys. B 437 (1995) 231, hep-th/9411219.\n- [3] E. Keski-Vakkuri and P. Kraus, Phys. Rev. D 54 (1996) 7407, hep-th/9604151.\n- [4] S.W. Hawking, Nature 248 (1974) 30.\n- [5] S.W. Hawking, Commun. Math. Phys. 43 (1975) 199. \n- [6] R. Arnowitt, S. Deser and C.W. Misner, in Gravitation: An Introduction to Current Research , ed. by L. Witten (Wiley, New York, 1962).\n- [7] M.K. Parikh and F. Wilczek, Phys. Rev. Lett. 85 (2000) 5042, hep-th/9907001.\n- [8] S. Hemming and E. Keski-Vakkuri, Phys. Rev. D 64 (2001) 044006, gr-qc/0005115.\n- [9] Y. Kwon, Il Nuovo Cimento B 115 (2000) 469.\n- [10] E.C. Vagenas, Phys. Lett. B 503 (2001) 399, hep-th/0012134.\n- [11] E.C. Vagenas, Mod. Phys. Lett. A 17 (2002) 609, hep-th/0108147.\n- [12] E.C. Vagenas, Phys. Lett. B 533 (2002) 302, hep-th/0109108.\n- [13] A.J.M. Medved, Class. Quant. Grav. 19 (2002) 589, hep-th/0110289.\n- [14] A.J.M. Medved, Phys. Rev. D 66 (2002) 124009, hep-th/0207247.\n- [15] P. Painlev' e , C.R. Acad. Sci. (Paris) 173 (1921) 677.\n- [16] P. Kraus and F. Wilczek, Mod. Phys. Lett. A 9 (1921) 3713, gr-qc/9406042.\n- [17] M. K. Parikh, Phys. Lett. B 546 (2002) 189, hep-th/0204107.\n- [18] D. Garfinkle, G.T. Horowitz and A. Strominger, Phys. Rev. D 43 (1991) 3140; Erratum-ibid: D 45 (1992) 3888.\n- [19] M. Cadoni and S. Mignemi, Nucl. Phys. B 427 (1994) 669, hep-th/9312171."}
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2003ApJ...592..767P
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Accretion of Low Angular Momentum Material onto Black Holes: Two-dimensional Magnetohydrodynamic Case
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2003-01-01
| 3 | 0.47 | 164 |
['accretion', 'accretion disks', 'black hole physics', 'galaxies active', 'galaxies nuclei', 'methods numerical', 'mhd', 'astrophysics']
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We report on the second phase of our study of slightly rotating accretion flows onto black holes. We consider magnetohydrodynamical (MHD) accretion flows with a spherically symmetric density distribution at the outer boundary but with spherical symmetry broken by the introduction of a small, latitude-dependent angular momentum and a weak radial magnetic field. We study accretion flows by means of numerical two-dimensional, axisymmetric, MHD simulations with and without resistive heating. Our main result is that the properties of the accretion flow depend mostly on an equatorial accretion torus that is made of the material that has too much angular momentum to be accreted directly. The torus accretes, however, because of the transport of angular momentum due to the magnetorotational instability (MRI). Initially, accretion is dominated by the polar funnel, as in the hydrodynamic inviscid case, where material has zero or very low angular momentum. At the later phase of the evolution, the torus thickens toward the poles and develops a corona or an outflow or both. Consequently, the mass accretion through the funnel is stopped. The accretion of rotating gas through the torus is significantly reduced compared with the accretion of nonrotating gas (i.e., the Bondi rate). It is also much smaller than the accretion rate in the inviscid, weakly rotating case. Our results do not change if we switch on or off resistive heating. Overall our simulations are very similar to those presented by Stone, Pringle, Hawley, and Balbus despite different initial and outer boundary conditions. Thus, we confirm that MRI is very robust and controls the nature of radiatively inefficient accretion flows. Although the time-averaged properties of our models approach a steady state, we find that the instantaneous mass-accretion rate in the latter stages of our simulations is highly time-dependent, with the inner flow displaying three generic flow patterns.
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https://arxiv.org/pdf/astro-ph/0303093.pdf
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{'Accretion of low angular momentum material onto black holes: 2D magnetohydrodynamical case.': 'Daniel Proga and Mitchell C. Begelman 1 \n- JILA, University of Colorado, Boulder, CO 80309-0440, USA; [email protected], [email protected]\n- 1 also Department of Astrophysical and Planetary Sciences, University of Colorado at Boulder', 'ABSTRACT': 'We report on the second phase of our study of slightly rotating accretion flows onto black holes. We consider magnetohydrodynamical (MHD) accretion flows with a spherically symmetric density distribution at the outer boundary, but with spherical symmetry broken by the introduction of a small, latitude-dependent angular momentum and a weak radial magnetic field. We study accretion flows by means of numerical 2D, axisymmetric, MHD simulations with and without resistive heating. Our main result is that the properties of the accretion flow depend mostly on an equatorial accretion torus which is made of the material that has too much angular momentum to be accreted directly. The torus accretes, however, because of the transport of angular momentum due to the magnetorotational instability (MRI). Initially, accretion is dominated by the polar funnel, as in the hydrodynamic inviscid case, where material has zero or very low angular momentum. At the later phase of the evolution, the torus thickens towards the poles and develops a corona or an outflow or both. Consequently, the mass accretion through the funnel is stopped. The accretion of rotating gas through the torus is significantly reduced compared to the accretion of non-rotating gas (i.e., the Bondi rate). It is also much smaller than the accretion rate in the inviscid, weakly rotating case. Our results do not change if we switch on or off resistive heating. Overall our simulations are very similar to those presented by Stone, Pringle, Hawley and Balbus despite different initial and outer boundary conditions. Thus, we confirm that MRI is very robust and controls the nature of radiatively inefficient accretion flows. \nSubject headings: accretion - magnetohydrodynamics - black hole physics - outflows - galaxies: active - methods: numerical', '1. Introduction': "Accretion onto supermassive black holes (SMBHs) very likely powers some of the most dramatic phenomena of astrophysics, such as quasars and powerful radio galaxies. However, SMBH accretion does not always result in high radiative output, as evidenced by SMBHs that appear to spend most of their time in a remarkably quiescent state (e.g., Di Matteo et al. 1999, 2000, 2001; Loewenstein et al. 2001; for Sgr A ∗ , see review by Melia & Falcke 2001). Dim SMBHs are not something one would expect, because \nthese black holes are embedded in the relatively dense environments of galactic nuclei. Therefore it is natural to suppose that the gravity due to an SMBH will draw in matter at high rates, leading to a high system luminosity. \nEstimates for the accretion luminosity, L , rely on assumptions about the mass accretion rate, ˙ M a , and the efficiency of transforming the gas energy into radiation, η (i.e., L = ηc 2 ˙ M a ). Both ˙ M a and η are uncertain and there is no generally accepted model which could explain low luminosity SMBHs by predicting low enough ˙ M a or η , or both. \nThe result that SMBHs are dimmer than they should be is primarily due to the fact that we estimate ˙ M a based on the density and temperature of the gas in which the SMBHs are embedded. It is customary to adopt the analytic formula due to Bondi (1952) to estimate the mass accretion rate. Bondi (1952) considered spherically symmetric accretion from a non-rotating polytropic gas with uniform density ρ ∞ and sound speed c ∞ at infinity. Under these assumptions, a steady state solution to the equations of mass and momentum conservation exists with a mass accretion rate of \n˙ M B = λ 4 πR 2 B ρ ∞ c ∞ , (1) \nwhere λ is a dimensionless parameter that, for the Newtonian potential, depends only on the adiabatic index. The Bondi radius, R B , is defined as \nR B = GM c 2 ∞ , (2) \nwhere G is the gravitational constant and M is the mass of the accretor. \nRelatively high ˙ M a predicted by the Bondi formula is partially responsible for generating a lot of interest in accretion flows with low η , that is, where transfer of the flow internal energy to radiation is very inefficient. Nonradiatve accretion flow solutions are possible because binding energy dissipated in the gas can be advected through the event horizon before being radiated (Ichimaru 1977; Rees et al. 1982; Narayan & Yi 1994, 1995; Abramowicz et al. 1995). \nNeglect of radiative cooling in accretion flows does not lead to just one family of solutions, however. In particular, the above-mentioned pure advection-dominated inflows constitute only one possible class of solutions. Once rotation is allowed, radiatively inefficient hydrodynamical (HD) flows become subject to strong convection (Begelman & Meier 1982; Narayan & Yi 1995), which can fundamentally change the flow pattern and its radiative properties (Igumenshchev & Abramowicz 1999; Blandford & Begelman 1999; Stone, Pringle & Begelman 1999; Quataert & Narayan 1999; Narayan, Igumenshchev & Abramowicz 2000; Quataert & Gruzinov 2000). Numerical and theoretical studies show that convection alters the steep ( ∝ r -3 / 2 ) density profile of advection-dominated flows into a much flatter ( ∝ r -1 / 2 ) profile, which can explain the faintness of many SMBHs because it predicts relatively low density close to the black hole (i.e., ˙ M a is low in eq. 1). Similar structural changes occur in the magnetohydrodynamical (MHD) limit (Stone & Pringle 2001, SP01 hereafter; Hawley, Balbus, & Stone 2001; Machida, Matsumoto & Mineshige 2001; Igumenshchev & Narayan 2002; Hawley & Balbus 2002), although here the turbulence is probably driven by magnetorotational instability (MRI) rather than thermal convection (Balbus & Hawley 2002; but see Abramowicz et al. 2002 and Narayan et al. 2002 for alternative views). \nThe turbulent character of both HD and MHD models does not settle the issue of what happens to the energy and angular momentum that must be transported away. There are two possibilities: (i) turbulent transport effectively shuts off the accretion flow, turning it into a closed circulation (Narayan et al. 2000; Quataert & Gruzinov 2000) or (ii) turbulent transport drives powerful outflows that can strongly modify \nthe black hole's environment (Narayan & Yi 1994, 1995; Blandford & Begelman 1999). Recent MHD simulations bring new insights that may help us to resolve this issue. For example, Hawley & Balbus's (2002) three-dimensional MHD simulations show that, with and without resistive heating, some mass and energy in nonradiative accretion flows are carried off by an outflow in keeping with the outline of the second possibility. \nAn important element of the problem of very low SMBH luminosity is the rate at which mass is captured into the accretion flow. If this rate is far lower than ˙ M B , then the problem of very low SMBH luminosity becomes less severe. SMBHs draw matter from an extended medium and most authors assume that the Bondi (1952) formula provides an adequate approximation for the rate of mass supply. Evolution of the flow already captured by an SMBH has been studied extensively, but evolution of the flow including regions beyond the domination of gravity has not been given its due. The Bondi formula has been derived under the assumption that this gas is non-rotating and only under the influence of the central gravity. Thus, for a given gravitational field, the gas internal energy determines the accretion rate. By relaxing this assumption, introducing additional forces or sources of energy, one may find that the mass supply rate is much lower than the one predicted by the Bondi formula. For example, the rate at which matter is captured by a black hole can be severely limited when the matter is heated by X-rays produced near the black hole (Ostriker et al. 1976) or by mass outflow from the central region (Di Matteo et al. 2003; Fabbiano et al. 2003). In these two cases, the gas internal energy is increased. Introducing kinetic energy to the gas at infinity may have a similar effect: although the flow outside the Bondi accretion radius often can be described as nonrotating, even a tiny amount of angular momentum, l - when followed inward could severely limit the rate at which matter is captured by the black hole (e.g., Proga & Begelman 2003, PB03 hereafter, and references therein). \nIn PB03, we reported on the first phase of our study of slightly rotating accretion flows onto black holes. We considered inviscid, hydrodynamic accretion flows with a spherically symmetric density distribution at the outer boundary, but with spherical symmetry broken by the introduction of a small, latitude-dependent angular momentum. Namely we assumed that at the outer radial boundary, the specific angular momentum, l , depends on the polar angle, θ , as \nl ( θ ) = l 0 f ( θ ) , (3) \nwhere f = 1 on the equator ( θ = 90 · ) and monotonically decreases to zero at the poles ( θ = 0 · and 180 · ). PB03's main result was that the properties of the accretion flow do not depend as much on the outer boundary conditions (i.e., the amount as well as distribution of the angular momentum parameterized by l 0 and the form of f ) as on the geometry of the non-accreting matter. The material that has too much specific angular momentum to be accreted ( l > 2 R S c , where R S = 2 GM/c 2 is the radius of a Schwarzschild black hole) forms a thick torus near the equator. Consequently, the geometry of the polar region, where material is accreted (the funnel), and the mass accretion rate through it are constrained by the size and shape of the torus. PB03's results showed one way in which the mass accretion rate of slightly rotating gas can be significantly reduced compared to the accretion of non-rotating gas (i.e., the Bondi rate), and set the stage for calculations that will take into account the transport of angular momentum and energy as presented here. \nWe report on the second phase of our study and assess the gross properties of rotating accretion flows onto black holes with the inclusion of MHD effects. We consider a classic Bondi accretion flow modified by the introduction of (i) a small, latitude-dependent angular momentum at infinity, (ii) a pseudo-Newtonian gravitational potential and (iii) weak magnetic field. The imposed angular momentum and magnetic field are weak enough to have initially a negligible effect on the density distribution at the outer boundary, which \nremains spherically symmetric. Contrary to PB03, we now consider the transport of energy and angular momentum by including magnetic field effects. Therefore our model allows for accretion of matter which initially has a specific angular momentum higher than 2 R S c . Yet we still consider a simple model of an accretion flow, simpler than those occurring in nature, as we neglect the gravitational field due to the host galaxy, radiative heating and cooling effects and 3D effects of MHD. \nOur work is complementary to some other previous studies. Several authors considered accretion onto black holes with a focus on the evolution of rotationally supported thick tori including the transport of angular momentum and energy (e.g., Igumenshchev & Abramowicz 1999; SP01; Machida et al. 2001; Hawley & Balbus 2002; McKinney & Gammie 2002; Igumenshchev, Narayan & Abramowicz 2003). The main difference between our work and these studies is that our simulations have one degree of freedom more than the previous studies. The other authors consider cases where if not for the transport of angular momentum due to internal stresses, there would be neither time evolution nor mass accretion. For example, for their initial conditions Stone et al. (1999), SP01, Hawley & Balbus (2002), and Igumenshchev et al. (2003) adopted a bounded torus in hydrostatic equilibrium with uniform angular momentum, embedded in zero angular momentum ambient gas which is also in hydrostatic equilibrium. The zerol ambient gas has a very low density and is unimportant dynamically. Thus previous studies consider accretion from a finite reservoir of gas that is not refilled during simulations. On the other hand, we allow time evolution and mass accretion, even without internal stresses. We start our simulations from a radial inflow and allow for gas with a constant density but a range of angular momenta to enter the computational domain during simulations. Thus, our flow is a complex convolution of rotating and non-rotating flows with similar densities that may be sub-Keplerian over a very large range of radii. We find that despite these differences the accretion through the torus, facilitated by MRI, dominates the inner flow. Thus, our results reinforce the findings of the previous studies. \nWe note that if the lowl material were allowed to accrete onto the black hole then it could significantly contribute to the total mass accretion rate. However, detailed simulations are required to check what will happen in such a situation. Here we report on quite intriguing results, especially when compared to the HD inviscid simulations: the lowl material that can be accreted in the inviscid case (with no need for any transport of l ) is not accreted in the MHD case, whereas the highl material that could not be accreted in the inviscid case, is accreted in the MHD case. This seemingly paradoxical behavior can be understood by the fact that the energy and angular momentum from the MHD torus are deposited outside the torus in the polar region (see, e.g., Stone et al. 1999; Blandford & Begelman 1999; Blandford & Begelman 2002a, 2002b; Hawley & Balbus 2002), effectively choking off the funnel. In PB03, we speculated that this energy and angular momentum might interfere with the inflow in the funnel. But it was unclear whether this would lead to a significant net reduction of ˙ M a , especially when the compensating inflow in the torus was taken into account. The simulations we present here suggest that the compensation is negligible, and that the total accretion rate is far lower than in the inviscid case with a similar angular momentum distribution at infinity. \nThe outline of this paper is as follows. We describe our calculations in Section 2. In Section 3, we present our results. We summarize our results and discuss them together with their limitations in Section 4.", '2.1. Equations': "To calculate the structure and evolution of an accreting flow, we solve the equations of magnetohydrodynamics \nDρ Dt + ρ ∇· v = 0 , (4) \nρ D v Dt = -∇ P -ρ ∇ Φ+ 1 4 π ( ∇× B ) × B , (5) \nρ D Dt ( e ρ ) = -P ∇· v + η r J 2 , (6) \n∂ B ∂t = ∇× ( v × B -η r J ) , (7) \nwhere ρ is the mass density, P is the gas pressure, v is the fluid velocity, e is the internal energy density, Φ is the gravitational potential, B is the magnetic field vector, J is the current density, and η r is an anomalous resistivity. We adopt an adiabatic equation of state P = ( γ -1) e , and consider models with γ = 5 / 3. Our calculations are performed in spherical polar coordinates ( r, θ, φ ). We assume axial symmetry about the rotational axis of the accretion flow ( θ = 0 · and 180 · ). \nWe present simulations using the pseudo-Newtonian potential Φ introduced by Paczy'nski & Wiita (1980) \nΦ = -GM r -R S . (8) \nThis potential approximates general relativistic effects in the inner regions, for a nonrotating black hole. In particular, the Paczy'nski-Wiita (WP) potential reproduces the last stable circular orbit at r = 3 R S as well as the marginally bound orbit at r = 2 R S . \nTo compute resistivity, we follow SP01: \nη r = Q (∆ x ) 2 | J | / √ ρ, (9) \nwhere ∆ x is the grid spacing and Q is a dimensionless constant. We also follow SP01 in adopting Q = 0 . 1.", '2.2. Initial conditions and boundary conditions': "For the initial conditions of the fluid variables we follow PB03 and adopt a Bondi accretion flow with zero angular momentum everywhere except for the outermost part of the flow. In particular, we adopt v θ = 0 while v r and ρ are computed using the Bernoulli function and mass accretion rate for spherically symmetric Bondi accretion with the PW potential. We set ρ ∞ = 1 and specify c ∞ through R ' S ≡ R S /R B (note that R ' S = 2 c 2 ∞ /c 2 ). We specify the initial conditions by adopting a non-zero specific angular momentum l for the outer subsonic part of the flow. \nWe consider a general case where the angular momentum at the outer radius r o depends on the polar angle via \nl ( r o , θ ) = l 0 f ( θ ) , (10) \nwith f = 1 on the equator ( θ = 90 · ) and f = 0 at the poles ( θ = 0 · and 180 · ). We express the angular momentum on the equator as \nl 0 = √ R ' C R B c ∞ , (11) \nwhere R ' C is the 'circularization radius' on the equator in units of R B for the Newtonian potential (i.e., GM/r 2 = v 2 φ /r at r = R ' C R B ). \nWe adopt one form for the function f ( θ ): \nf 3 ( θ ) = { 0 for θ < θ o and θ > 180 · -θ o l 0 for θ o ≤ θ ≤ 180 · -θ o . (12) \nWe call this function f 3 to be consistent with the nomenclature in PB03. \nWe generate the initial magnetic field using a vector potential, i.e., B = ∇ × A . We consider one straightforward initial magnetic configuration: a purely radial field defined by the potential A = ( A r = 0 , A θ = 0 , A φ = A cos θ/r sin θ ). We scale the magnitude of the magnetic field using a parameter, β o ≡ 8 πP B ( r o ) /B 2 defined as the plasma parameter β ≡ 8 πP/B 2 at the outer boundary, r o , so that \nA = sign(cos θ ) √ (8 πP B ( r o ) /β o ) r 2 o , (13) \nwhere P B is the gas pressure associated with the Bondi solution at r o . Note that the magnetic field changes sign across the equator. \nOur standard computational domain is defined to occupy the radial range r i = 1 . 5 R S ≤ r ≤ r o = 1 . 2 R B and the angular range 0 · ≤ θ ≤ 180 · . We consider models with R ' S = 10 -3 . The r -θ domain is discretized into zones with 140 zones in the r direction and 100 zones in the θ direction. We fix zone size ratios, dr k +1 /dr k = 1 . 05, and dθ l /dθ l +1 = 1 . 0 for 0 · ≤ θ ≤ 180 · . However, we have also performed some runs with dθ l /dθ l +1 = 1 . 02 for 0 · ≤ θ ≤ 90 · and dθ l +1 /dθ l = 1 . 02 for 90 · ≤ θ ≤ 180 · (i.e., the zone spacing is decreasing toward the equator). \nThe boundary conditions are specified as follows. At the poles, (i.e., θ = 0 · and 180 · ), we apply an axis-of-symmetry boundary condition. At both the inner and outer radial boundaries, we apply an outflow boundary condition for all dynamical variables except the magnetic field. For the magnetic field at the outer boundary, we apply an outflow condition, whereas at the inner boundary, we follow SP01 and use a negative stress condition (i.e., we enforce B r B φ ≤ 0 at r = r i ). We also ran some models using an outflow condition for the magnetic field at the inner boundary and found similar results. As in PB03, to represent steady conditions at the outer radial boundary, during the evolution of each model we continue to apply the constraints that in the last zone in the radial direction, v θ = 0, v φ = l 0 f ( θ ) /r sin θ , and the density is fixed at the Bondi value at all times. Note that we allow v r to float. Additionally, we fix the magnetic field at its initial radial configuration in the last zone in the radial direction (i.e., B r = 1 r sin θ ∂ ( A φ sin θ ) ∂θ , B θ = 0 and B φ is allowed to float). To reduce the problems caused by very high Alfv'enic velocities in regions of very low density (i.e., to prevent the time step from being prohibitively small), we set a lower limit to the density on the grid as ρ min ( r ) = √ r i /r and enforce it at all times in all models. \nTo solve eqs. (4)-(7) we use the ZEUS-2D code described by Stone & Norman (1992a, 1992b), modified to implement the PW potential and resistive heating.", '3. Results': "We specify our model by several parameters. We set the length scale in terms of the black hole radius in units of the Bondi radius, R ' S . Our second parameter is the adiabatic index, γ . The third parameter (or \na function rather) is the angular momentum at the outer radial boundary, l = l 0 f ( θ ). The last parameter is the plasma parameter at the outer boundary, β o . \nFor practical reasons, we consider a relatively large value of R ' S = 10 -3 (see PB03 and below for reasons why R ' S < 10 -3 is not suitable for our purposes). Our choice of R ' S allows us to run our models over a couple of dynamical time scales at large radii and therefore obtain solutions which have lost memory of the initial conditions. We consider flows with γ = 5 / 3. Note that we allow the entropy of gas to increase due to nonadiabatic heating caused by the artificial viscosity and the resistivity (The artificial viscosity is a compressible bulk viscosity, not a shear viscosity, and does not affect rotation of the flow [Stone & Norman 1992a]). We assume an angular momentum distribution at the outer radial boundary as described in Section 2. We focus our attention on accretion of matter with low angular momentum, i.e., where the corresponding centrifugal force is small compared to gravity for all θ at the Bondi radius (see PB03 for details). Finally, we consider weak magnetic fields only. Our choice for the magnetic field strength is dictated by the requirement that the flow be initially super-Alfv'enic in the entire computational domain: | v p | > | v Ap | , for all radii, where v p ≡ √ v 2 r + v 2 θ is the poloidal fluid velocity and v Ap ≡ √ ( B 2 r + B 2 θ ) / 4 πρ is the poloidal Alfv'en velocity. \nTable 1 summarizes the properties of the simulations we discuss here. Columns (2) through (7) give the numerical resolution in the radial direction; the black hole radius compared to the Bondi radius, R ' S ; the circularization radius compared to the Bondi radius, R ' C ; the specific angular momentum on the equator at r = r o , l 0 , in units of 2 R S c ; the width of the angular distribution for which l ≤ 2 R S c , θ o ; and the angular momentum dependence on the polar angle at the outer boundary, f ( θ ), respectively. Columns (8) and (9) give the plasma parameter at the outer boundary, β o , and the dimensionless constant of the anomalous resistivity, Q , respectively. Table 1 also presents the final time at which we stopped each simulation (all times here are in units of the Keplerian orbital time at r = R B ), the range within which the maximum specific angular momentum at the inner radial boundary varies at the end of the simulation, l max a , and the mass accretion rate through the inner radial boundary measured near the end of the simulation, in units of the corresponding Bondi accretion rate. The mass accretion rate is time-averaged over 0.2 orbits at the end of each simulation, except for run C where it is averaged for 0.05 orbits only. Finally, column (13) gives comments about runs different from the standard runs (e.g., higher spatial resolution in the θ direction near the equator). \nOur simulations show that for l 0 ≥ 2 R S c the accretion flow consists of an equatorial torus with MHD turbulence driven by MRI. The latter produces accretion through the torus. The MHD turbulent torus also determines the fate of the material in the polar funnel, where l < 2 R S c and could be accreted directly. For cases with a very weak magnetic field or more generally at the beginning of simulations, there is supersonic funnel accretion as in the inviscid HD case. However, as MRI in the torus grows, the torus thickens and an outflow from the torus develops. As a result, the torus or its wind closes the polar funnel and quenches the accretion through the funnel. We describe an example of such an accretion flow in some detail first (Section 3.1). This is followed by a limited parameter survey in which we focus on varying two key aspects of our models: the strength of the magnetic field and the angular distribution of angular momentum on the outer boundary. \nIn this section we describe the properties and behavior of our model in which R ' S = 10 -3 , β o = 10 6 , and a step function describes the angular distribution of angular momentum on the outer boundary (run D). We assume that for 45 · ≤ θ ≤ 135 · , the specific angular momentum on the equator at the outer boundary equals 2 R S c , whereas for θ < 45 · and θ > 135 · , l = 0. This angular distribution of l on the outer boundary uses the same θ o as the fiducial model in PB03 (i.e., model B04f1a) for which R ' C = 0 . 1 and f ( θ ) = 1 - | cos θ | . Thus we assume that the material that cannot be accreted onto the black hole, without angular momentum transport, is located (at the outer boundary) relatively close to the equator. Inversely, we consider a relatively wide polar funnel containing zerol material. Our run H is the HD inviscid counterpart of run D and serves as a reference run. \nFigure 1 presents a sequence of density and angular velocity contours, and the direction of the fluid velocity for run D. After a transient episode of infall, the gas with l = 2 R S c piles up outside the black hole and forms a thick torus bounded by a centrifugal barrier near the rotation axis. Soon after the torus forms (i.e., a couple of orbits at r = r i ), the magnetic field is amplified both by MRI and by shear. The torus starts evolving rapidly and accretes onto the black hole. Another important effect of magnetic fields is that the torus produces a magnetized corona and an outflow. By a corona we mean gas outside a torus, with a low β ( < ∼ 0 . 1), density lower than in the torus by one or more orders of magnitude, and with vigorous circulation (see the top panels in Figure 8 for a good illustration of the torus and corona). On the other hand, by an outflow we mean gas with a systematic poloidal velocity directed away from the equator and the black hole. Initially, the role of the corona and outflow is negligible. For example, at the early phase of the evolution, 0 . 1 < ∼ t < ∼ 0 . 4, the lowl material close to the axis can accrete almost steadily through a funnel despite the torus corona and outflow (see Fig. 4). The mass accretion rate during this phase is dominated by the lowl material and is similar to ˙ M a for a pure HD inviscid flow where accretion occurs only through the polar funnel. The accretion rate due to the torus is > ∼ 1 orders of magnitude lower than ˙ M a due to the polar funnel. But at the later phase, t > 0 . 5, the MHD turbulent corona of the torus expands toward the poles and the torus outflow becomes stronger. They both eventually shut off the polar funnel accretion. Note that the polar outflow ('jet') reaches the outer boundary by t ∼ 0 . 5. A shock wave, associated with a transient, propagates outward through the entire computational domain during the first dynamical time scale. After the shock wave passes through the outer boundary, the flow in the torus is subsonic, highly variable and is directed inward near the equator and outward close to the poles. We note that the flow with non-zero angular momentum keeps rotating on cylinders over the whole simulation regardless of how complex is the velocity field. We shall return to this point below. Also we note that at the end of the simulation, the angular distribution of captured mass differs significantly from the spherically symmetric Bondi accretion flow. Only gas which enters the computational domain near the equator reaches the inner part of the domain. We will show the inner part of the flow and discuss its nature below. However, it is clear even from Fig. 1 that some of the equatorial inflow does not make it into the black hole but rather turns around and leaves the computational domain through the outer boundary. \nTo provide some insight into the time dependence, Figures 2 and 3 show the time evolution of the mass accretion rate in units of the corresponding Bondi rate. Initially, ˙ M a drops from 1 to 0.03 at t = 0 . 1. Then it rises sharply by one order of magnitude and starts oscillating irregularly between ∼ 0 . 003 and ∼ 0 . 03 -0 . 1 ˙ M a / ˙ M B for the remainder of the simulation. Figure 3a shows that toward the end of simulation, ˙ M a ( t ) has settled into a pattern that can be characterized as follows: ˙ M a sharply increases, then decreases gradually by ∼ 1 order of magnitude to an almost steady level and then increases sharply again. These sharp increases and gradual decreases are quasi-periodic and reoccur about every 0.07 orbits. Figure 3a shows also that on the top of this quite regular pattern of 'bursts' and exponential 'declines' there are short-lived 'dips' and 'spikes'. In summary, the time-dependence and level of accretion via the torus \nappear to have reached a persistent state. \nTo show the complex structure of the flow at small radii and gain some insight into the very highly time-dependent evolution of the mass accretion rate, Figures 4-7 and Figure 8 compare the inner accretion flow at four different times when a) accretion occurs through the torus and polar funnel ( t = 0 . 22, Fig. 4 and the top left panel in Fig. 8); b) accretion occurs only through the torus ( t = 2 . 39, Fig. 5 and the top right panel in Fig. 8); c) accretion through the torus is quenched by the strong magnetic field which forms a magnetized polar cylinder around the black hole; ( t = 2 . 41, Fig. 6 and the bottom left panel of Fig. 8); and d) accretion of lowl material occurs through the polar region outside the torus ( t = 2 . 36, Fig. 7 and the bottom right panel of Fig. 8). Figures 4-7 plot snapshots of the density, entropy, S ≡ ln( P/ρ γ ), angular velocity, specific angular momentum, and direction of the fluid velocity (top panels from left to right) and the total pressure, P tot = P + B 2 / 8 π , magnetic pressure, P mag = B 2 / 8 π , plasma parameter, β , toroidal magnetic field, and direction of the magnetic field (bottom panels from left to right). To display the generic features of run D in its four states in a more copact form, Fig. 8 compares only snapshots of the density over-plotted by the direction of the fluid velocity. Note vertical arrows on Figure 2 and Figure 3a which mark the times for these four snapshots: arrows a, b, c, d correspond to Figs. 4, 5, 6 and 7 (and top left, top right, bottom left and bottom right panels in Fig. 8), respectively. \nFigure 4 shows the inner flow characteristic of the early phase of the evolution (0 . 1 < ∼ t < ∼ 0 . 4). The torus made of highl material accretes onto the black hole due to MRI. The torus produces a magnetized corona with vigorous circulation bounded by the lowl material accreting onto the black hole through the polar funnel. The flow at this stage is similar to its HD inviscid counterpart because the polar funnel accretion is dominant and its ˙ M a is determined by the shape of the torus. There are two qualitative differences between run D in this early phase and its HD inviscid counterpart. In run D, the torus also accretes but at much lower rate than the funnel. In the HD inviscid case there is no torus accretion at all. The shape of the funnel in run D is also affected by the torus corona, albeit slightly. However, as the torus corona and outflow grow, the above mentioned differences become important. \nFigure 5 shows the inner flow at a later phase of the evolution, when the corona and outflow have fully developed and can block the lowl material incoming in the polar funnel. At this point accretion is due only to the torus and ˙ M a is at an intermediate level. During this intermediate ˙ M a state, matter is accreted via a torus extending down to the inner boundary. There can also be some accretion through the polar funnel but its rate is very low and the actual value of the accretion rate via the funnel is determined by the density floor we imposed (see Section 1). For small radii, the polar funnel is 'empty' because the torus outflow extends to the poles at large radii and shuts off the mass supply. However, the polar funnel has a significant pressure due to the magnetic field. In fact, the total pressure distribution is close to spherical despite the huge change in the gas pressure as we go from the poles to the equator for a given radius. The torus is time-variable due to the MHD turbulence. In particular, the density of the accreted material fluctuates and once it becomes too low, the magnetic field in the polar funnel can expand toward the equator and reconnect. At that instant the torus is pushed outward by the magnetic field and ˙ M a drops until the gas in the torus piles up and squashes the magnetic field (compare Figure 5 and 6). Fig. 6 shows that at very small radii, the magnetic field is vertical in this state instead of being radial as during other states shown. Each 'dip' in the accretion rate is followed by a 'spike' during which the torus unloads the material piled up during the dip. \nThe strong time variability of the inner accretion flow affects also the outer accretion flow. For example, the torus corona and outflow struggle constantly with the lowl material in the polar region. This material has neither rotational nor pressure support and therefore it would accrete if not for the torus corona and \noutflow. However, the torus corona and outflow sometimes become too weak to prevent the lowl material from accreting. When this happens we observe a burst of accretion. Figure 7 illustrates such a burst. Note a lowl stream reaching the inner boundary below the equator. It takes a relatively long time ( ∼ 0 . 03 orbits) for the torus corona and outflow to push this stream of lowl material away. Therefore ˙ M a decreases gradually with time to the level determined by the accretion rate due to the torus alone. \nWe conclude that the time-dependent accretion in run D is due to unsteady accretion via the torus, the presence of a very strong magnetic field at small radii, and lowl material trying to accrete outside the torus. The tension of the magnetic field tries to stop the gas incoming from the torus. Once the torus accretes too slowly the magnetic tension causes to the field lines to straighten and seals off the black hole because a highly magnetized cylinder forms around it. At this stage, gas flowing through the torus is constrained to move roughly parallel to the polar axis. This continues until enough lowl gas builds up in the torus to overcome the tension of the magnetic cylinder. \nDespite the differences in the flow in these various states, there are many properties of the flow that stay unchanged. We allow the entropy of gas to increase due to nonadiabatic heating caused by the artificial viscosity in run D. However, we note that the entropy is increased only inside the polar flow and is nearly constant inside the torus. This property indicates that MRI, not convection, is responsible for the complex nature of the flow. In particular, the vigorous circulation inside the torus is driven by MRI. The same is true when resistive heating is switched on (e.g., run F). In the discussion of the flow at large radii (Figure 1), we noted that despite complex velocity and magnetic fields the flow rotates on cylinders. Figs. 4-7 show that this property holds even at small radii, as indicated by the contours of Ω parallel to the rotation axis. The contours of specific angular momentum reveal that most of the changes in l occur near the equator just outside the torus. Specifically, l decreases from 1 to 0 with decreasing radius near the equator. However, inside the torus, l remains nearly constant with clear indication of some regions with l > 1! The very fact that we observe regions inside the torus with l > 1 indicates the outward transport of l because our initial and boundary conditions do not introduce any material with l > 1. In passing, we note that we decided to consider a step function for the angular distribution of l because we wish to see as clearly as possible the evolution of l . \nWe start our simulation with a very weak magnetic field and very slow rotation. Although our flow rotates differentially, the initial rotation is far from Keplerian in its values as well as its dependence on radius. The latter scales like r -2 , typical for a constant angular momentum fluid, rather than like r -1 . 5 as for a Keplerian disk. In fact, our initial conditions are such that Keplerian rotation is reached only at 2 R S . Since MRI is driven by the free energy in differential rotation, we have analyzed our solution in great detail to check whether MRI is indeed responsible for the nature of our flow. In particular, we have checked that the flow (the torus, more precisely) is subject to MRI for the actual rotational profile and the strength of the magnetic field (see eq. 108 in Balbus & Hawley 1998) and that our numerical resolution is adequate to follow the growth and saturation of MRI. Our analysis shows that most of the inner torus is unstable and that we are able to resolve, although marginally, the fastest growing MRI mode inside the inner torus. The wavelength of the mode is λ c = 2 πv A / √ 3Ω. For example, in run D we find that over most of the central region of the torus λ c / ∆ x ∼ 4 but there are also regions with λ c / ∆ x ∼ 100 and with λ c / ∆ x < ∼ 1. \nTo investigate the nature of our solutions in more detail, we have performed some simulations with a higher resolution near the equator, where MRI is expected to be most important. We find that our test run E, with dθ l /dθ l +1 = 1 . 02 for 0 · ≤ θ ≤ 90 · (i.e., the zone spacing is decreasing toward the equator in this region) and dθ l +1 /dθ l = 1 . 02 for 90 · ≤ θ ≤ 180 · , gives very similar results to those from run D. Additional confirmation of our interpretation of the results can be found by comparing them with expected \nscaling relations and with other relevant published results. Therefore, we will now present our results in a form similar to that presented by SP01. As will be clear from the following figures, our results resemble the other results despite significantly different initial and outer boundary conditions. Generally, we find that once MRI starts to operate it totally determines the nature of the flow. \nFigure 9 shows the radial profiles of several quantities in run D, angle-averaged over a small wedge near the equator (between θ = 87 · and 93 · ), and time-averaged over 12 data files covering a period when accretion is nearly steady and occurs only through the torus (i.e., orbits 2 . 388 through 2 . 417; see Fig. 3a). We can compare this plot directly with Figure 6 in SP01 for their run F. We indicate the location of the last stable circular orbit by the vertical dotted line in each panel. \nAs in the MHD models of SP01, the profiles of each variable are not simple power-laws but are rather complex. In particular, for r ' < ∼ 0 . 01 the density is nearly constant whereas for larger radii it decreases with increasing radius almost as r -1 . However, we note that this power-law at large radii is a relict of the initial conditions, i.e., the subsonic part of the Bondi inflow with a PW potential. The gas pressure is higher than the magnetic pressure for large radii. However, for r ' < ∼ 0 . 004 the two pressures are comparable. For r ' > 0 . 004, the magnetic pressure decreases with increasing radius much faster than the gas pressure. The rotational velocity is always sub-Keplerian. However, it peaks at nearly the Keplerian value for r ' = 0 . 006. For r ' > 0 . 006, the rotational velocity scales as r '-2 as for the angular momentum conserving fluid. \nTo measure the Reynolds stress, we follow Hawley (2000, see also SP01) and compute the difference between the angular momentum flux and mass flux times the mean angular momentum: \n< ρv r δv φ > = < ρv r v φ > -< ρv r >< v φ >, (14) \nand then normalize it to the gas pressure: \nα gas = < ρv r δv φ > / < P > . (15) \nWe find that for r ' < 0 . 002 the normalized Reynolds stress is negative and decreases sharply with decreasing radius (note that Figure 9 shows only the magnitude of the normalized Reynolds stress). For r ' > 0 . 002, the normalized Reynolds stress is positive, peaks at 0 . 15 for r ' ∼ 0 . 0025 and decreases with increases radius, for r ' < ∼ 0 . 0045. Beyond r ' ∼ 0 . 0045, α gas stays positive but is very small. On the other hand, the Maxwell stress, normalized to the magnetic pressure, α mag ≡ < 2 B r B φ > / < B 2 > , is negative for all radii except for a small radial range around r ' = 0 . 027 (note a local minimum in Fig. 9). Comparing the actual stresses rather than the 'alphas' (i.e., < ρv r δv φ > vs. < B r B φ / 2 > ), we find that the Maxwell stress transports angular momentum outward and is stronger than the Reynolds stress. \nThe last panel in Figure 9 shows that the toroidal component of the magnetic field is dominant at all radii. However, for r ' < ∼ 0 . 004, the radial component increases with decreasing radius and becomes comparable with the toroidal field. \nOverall, we find that the properties of the inner flow of our run D are strikingly similar to those found by SP01 in their run F. The main differences occur for very small radii ( r ' < 0 . 004). For example, we find weaker advection of such quantities as angular momentum by the infalling gas. This is consistent with the fact that in SP01 simulations, the torus accretes almost steadily (at least at small radii) whereas we find a very strong poloidal magnetic field parallel to the rotation axis, which tends to interrupt the torus accretion at small radii. \nOur simulations with an initial magnetic field weaker than that for run D (i.e., β o > 10 6 ) show that the accretion flow is qualitatively insensitive to β o . The main difference is in the time it takes the torus corona and outflow to push away the lowl material inflowing in the polar funnel. In particular, our simulations for β o = 10 7 (run G) show that the polar funnel accretion is stopped only after ∼ 1 orbit (see Fig. 2). Before that time, the accretion flow is similar to its inviscid HD counterpart: accretion occurs through a polar funnel where l < 1. The shape of the funnel and ˙ M a are determined by the shape of a torus where there is little accretion ( ˙ M a = 0 . 2 but the torus contribution is less than 1%). Only when the torus corona and outflow become well-developed do they shut off accretion through the funnel. Other differences between run G and D are: (i) at the end of run G, the torus accretion rate is lower than for run D and (ii) in run G, the torus accretion is not interrupted by the magnetic field at small radii. The decrease of ˙ M a via the torus with increasing β has been found before (e.g., SP01). We attribute the disappearance of dips in ˙ M a to the fact that the polar funnel is more sensitive than the torus to the initial and outer boundary magnetic field. In particular, the magnetic field in the torus is amplified both by MRI and by shear and is less dependent on the initial and outer conditions than the magnetic field in the polar funnel, which is amplified by accretion of the initial field. \nFor β o < 10 6 , the initial magnetic field is too strong to be consistent with our initial requirement of weak magnetic fields (e.g., run C). The main inconsistency is due to the fact that for β o < 10 6 the torsional waves are faster than the flow. Consequently, the waves reach the inner boundary before the material with l > 0 and the inner flow is disrupted by the waves reflected from the inner boundary. We observe a train of persistent shocks propagating outward which prevent the non-zero l material from reaching small radii. \nWe conclude that the accretion flow in run D, which we described in detail in the previous section, is a representative solution for a range of magnetic fields in the weak field regime, and is not just applicable to one particular initial field strength. Next we discuss our results for various l 0 and demonstrate that the generic features of run D appear to be robust and apply to a wide range of conditions.", '3.3. Dependence of accretion flow properties on l 0': "Our choice of l 0 = 1 and a step function for the angular distribution of l in run D was motivated by a wish to see if even a minimal angular momentum can reduce the mass accretion rate. We note that by setting l 0 = 1 we created rather unfavorable conditions for MRI to grow because the rotational velocity is sub-Keplerian in the whole computational domain except at r = 2 R S . As we showed above, MRI is very robust and dominates the nature of the flow even for sub-Keplerian rotation. \nIn reality the angular momentum distribution beyond the Bondi radius is likely to be complex. In particular, we expect a large range of l . Numerical experiments, such as ours, try to isolate the key elements of the accretion flow. For example, Stone et al. (1999), SP01, Hawley, Balbus & Stone (2001), and Hawley & Balbus (2002) considered a constant angular momentum hydrostatic torus for their initial conditions. Contrary to us, they assumed a larger angular momentum so that the circularization radius is larger than the black hole radius by a factor of few or more. \nTo make more direct contact with those previous simulations, we ran a couple of simulations with l = 4 R S c (runs I and J). We expect that a higher l will result in the development of a Keplerian flow for large radii, as the circularization radius is 8 R S and MRI will be needed to transfer angular momentum. \nOur run J is an HD inviscid accretion flow and serves as a reference run. We find that run J is \nconsistent with our other inviscid HD accretion runs performed for PB03. Namely, we observe the formation of a torus with l > 2 R S c , via which there is no accretion but the shape of which determines the geometry and ˙ M a of a polar accretion funnel. The funnel accretion and its ˙ M a settle into a steady state after t ∼ 0 . 6. The torus exhibits subsonic circulation. The specific angular momentum in the funnel is practically zero, as expected. The torus rotates on cylinders and l increases from 2 R S c to 4 R S c with increasing radius for r ' < ∼ 0 . 1 at the end of our simulations. Beyond 0.1, l is constant. The gradual change of l inside the torus is caused by a mixing of the zero l and non-zero l material during the initial phase of the evolution. During the later phase, the material with 2 R S c ≤ l ≤ 4 R S c remains in the torus as it cannot be accreted. \nRun I is similar to run J with the exception that we added a magnetic field ( β o = 10 6 ). In the early stages of run I, the flow consists of an accretion funnel and a torus. As in run J, l = 0 in the funnel and the torus is made of the material with l > 2 R S c . However, contrary to its inviscid counterpart, the torus in run I accretes onto the black hole due to MRI. Qualitatively, run I is similar to runs D and G. The main difference is in the duration of the phase when both the torus and polar funnel accrete (i.e., for run I it lasts for ∼ 2 orbits while for run D the duration is ∼ 0 . 3 and for run G it is ∼ 1 . 3). \nFigure 10 presents the radial profiles of several quantities in run I angle-averaged over a small wedge near the equator (between θ = 87 · and 93 · ), and time-averaged over 20 data files covering a period when accretion is nearly steady and occurs only through the torus (i.e., orbits 2.35 through 2.40). We can compare this plot directly with our Figure 9 and with Figure 6 in SP01 for their run F. As expected, run I is even more similar to SP01's run F than it is to run D. In particular, we find that in run I the rotational velocity is close to Keplerian for quite a wide range of radii. This is a clear indication of the outward transport of angular momentum. We note that for small radii (0 . 003 < ∼ r ' < ∼ 0 . 02), the density profile in run I scales almost as r '-1 / 2 and the density is lower than in run D for the same radii. However, we would be very cautious about drawing any conclusions from this scaling because of the relatively small range of radii in our computational domain (see Section 4). \nFinally, we comment on our runs B and A with zerol material which are simply magnetized Bondi flows. We consider these runs as tests of our code. Because these runs are for weak magnetic fields ( β = 10 6 and 10 5 for run B and A, respectively), they should be very similar to the Bondi flow (see PB03). Our simulations show that the mass accretion rate is indeed equal to the Bondi rate and the flow is almost spherically symmetric. The departure from spherical symmetry is due to numerical resistivity (in both runs) and artificial resistivity in run A. However, the heating caused by magnetic field reconnection is limited to the region very close to the equator and is very low, i.e., the maximum increase of entropy is on the equator at the inner radius, r i , and is 0.3 per cent for run B, and 2 per cent for run A.", '4. Discussion': "This paper presents the second phase of our study of slightly rotating accretion flows onto black holes where, in contrast to our previous paper, we have included magnetic fields. By adding MHD effects, we can calculate turbulent stresses generated self-consistently by MRI and thus include the transport of energy and angular momentum outward as needed to accrete matter with a specific angular momentum higher than 2 R S c . Our simulations support strongly our hypothesis that even very slow rotation of gas at large radii may be sufficient to reduce the mass accretion rate to the level required by observations. In what follows we will summarize our results, briefly review the limitations of our work, and discuss how the physical effects neglected here may change the results. \nWe have performed numerical 2D, axisymmetric, MHD simulations of rotating accretion flows onto a black hole. Our simulations are complementary to previous MHD simulations which considered strongly rotating accretion flows. We consider slightly rotating flows and attempt to mimic the boundary conditions of classic Bondi accretion flows as modified by the introduction of a small, latitude-dependent angular momentum at the outer boundary, a pseudo-Newtonian gravitational potential and weak poloidal magnetic fields. A weak radial magnetic field and the distribution of l with latitude allow the density distribution at infinity to approach spherical symmetry. The main result of our simulations is that the nature of the accretion flow is totally controlled by magnetic fields when l > ∼ 2 R S c . As in the HD inviscid case, which we studied in PB03, the material with l > ∼ 2 R S c forms an equatorial torus. However, in the MHD case, the torus accretes onto the black hole because of MRI and produces both a corona and an outflow. We find that the latter two can be strong enough to prevent accretion of the lowl material through the polar regions, the source of accretion in the HD inviscid case. The net mass accretion rate through the torus is lower than the Bondi rate and also lower than the accretion rate in the HD inviscid case. In PB03, we found that in order to reduce the mass accretion rate in the HD inviscid case, the polar accretion funnel had to be much narrower than the funnels allowed by a hydrostatic l = 2 R S c torus. Here we find that the outflow and corona produced by the accreting torus are natural mechanisms to narrow or even totally close the polar funnel for the accretion of the lowl material. We conclude that the inclusion of even slow rotational motion of the MHD flow at large radii can significantly reduce ˙ M a compared to the Bondi rate. \nThe accreting torus is the crucial component of our accretion flow, therefore we describe its main properties in more detail. Overall, the inner flow consists of a turbulent, gas pressure-dominated MHD torus with an outflow or corona or both, bounded by a magnetic pressure-dominated polar flow. The torus accretes near the equator. The accretion through the torus can be supplemented, in a quasi-periodic manner, by the lowl material in the polar regions. In fact, ˙ M a due to this polar accretion can be one order of magnitude larger than that due to the torus. The 'off torus' accretion is a manifestation of the facts that our initial and outer boundary conditions allow the zerol material to approach the black hole and that the torus corona and outflow are not always strong enough to push it away. Accretion through the torus can also be interrupted for a short time by strong poloidal magnetic field that builds up during accretion. However, we observe that even during periods when the torus is truncated, there is inflow of material inside the torus and its mass and pressure build up. Consequently, the magnetic field is quickly pushed inward by the torus and the gas from the torus can again fall onto the black hole. Thus, our simulations show a strong buildup of the poloidal magnetic field as in Igumenshchev, Narayan & Abramowicz (2003) but here we find that it is a transient and not a final solution. \nHawley & Balbus (2002) identified three well-defined dynamical components in their simulations of 3D MHD accretion flows: (i) a hot, thick, rotationally dominated Keplerian disk, (ii) a surrounding magnetized corona with vigorous circulation and outflow, and (iii) a magnetically confined jet along the centrifugal funnel. We also observe these three components in cases where the circularization radius is well outside the black hole, i.e., in cases similar to those studied by Hawley & Balbus (2002; see also SP01). We conclude that the three-component accretion flow is robust at small radii, despite the fact that we also allow for lowl material at large radii. \nWe note that R ' S in our simulations is much larger than that in real systems, where it is 10 -5 and smaller. Additionally, our relatively large R ' S and high γ do not allow us to capture, over a great radial range, the asymptotic behavior of the Bondi accretion flow, i.e., the flow in free fall yielding a steep ( ∝ r -3 / 2 ) density profile. The free fall approximation is valid for radii small compared to the sonic point, x s . However, our choice of R ' S and γ yield x s = 0 . 027 (PB03). Thus the approximation that ρ ∝ r -3 / 2 \nis valid over less than one order of magnitude in radius. This radial range is additionally reduced at very small radii because the free fall velocity in the PW potential is higher than for the Newtonian potential, so that the density profile is flatter than ρ ∝ r -3 / 2 for very small radii. For slightly rotating flows as in real systems, we lack analytic predictions. Therefore, we treat our simulations only as first-order indications of what may be happening in real systems. In particular, we find that our accretion flows differ significantly from advection-dominated accretion flows (ADAF) solutions. For example, ADAF solutions predict nearly spherically symmetric inflows with a steep density profile, whereas we find a thick torus with a flat ( ∼ r -1 / 2 ) density profile. Our solutions differ also from convection-dominated accretion flows (CDAF) solutions as we find an MHD turbulent torus with an outflow instead of a convection-driven closed circulation. Our solutions appear to be MHD analogs of the HD viscid self-similar solutions of Blandford & Begelman (1999), where inflows as well as outflows of gas play a critical role in determining the properties of the accretion flow. \nOur simulations do not include some of the physical processes that may be important in accretion flows onto black holes. Additionally, we have performed our simulations in 2D instead of 3D. Thus, in our simulations, a poloidal magnetic field can eventually dissipate according to the antidynamo theorem (e.g., Moffatt 1978). Additionally, we can not simulate the toroidal field MRI and we may be emphasizing the 'channel' solution mode (Hawley & Balbus 1992) which produces coherent streaming in the disk plane instead of more generic MHD turbulence. Obviously, 3D MHD simulations are required. However, one should not underestimate the importance of the larger radial dynamic range and time scale afforded by 2D simulations if one wants to make direct contact with observations and with theoretical models which often yield self-similar solutions. Future work should also include studies of accretion flows with various initial configurations of the magnetic field. \nACKNOWLEDGMENTS: We thank J.M. Stone for useful discussions. DP acknowledges support from NASA under LTSA grant NAG5-11736 and support provided by NASA through grant AR-09532 from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. MB acknowledges support from NSF grant AST-9876887.", 'REFERENCES': "Abramowicz, M.A., Chen, X., Kato, S., Lasota, J.-P., Regev, O. 1995, ApJ, 438, L37 \nAbramowicz, M.A., Igumenshchev, I. V., Quataert, E., & Narayan, R. 2002, ApJ, 565, 1101 \nBalbus, S.A., & Hawley, J.F. 1998, Rev. Mod. Phys., 70, 1. \nBalbus, S.A., & Hawley, J.F. 2002, ApJ, 573, 749 \nBalbus, S.A., & Hawley, J.F. 2002, ApJ, 573, 749 \nBegelman, M.C., & Meier, D.L. 1982, ApJ, 253, 873 \nBlandford, R.D., & Begelman, M.C. 1999, MNRAS, 303, L1 \nBlandford, R.D., & Begelman, M.C. 2002a, in preparation \nBlandford, R.D., & Begelman, M.C. 2002a, in preparation \nBlandford, R.D., & Begelman, M.C. 2002b, in preparation \nBondi, H. 1952, MNRAS, 112, 195 \nDi Matteo, T., Allen, S.W., Fabian, A.C., Wilson, A.S., & Young, A.J. 2003, ApJ, 582, 133 \nDi Matteo, T., Carilli, C.L., & Fabian, A.C. 2001, ApJ, 547, 731 \nDi Matteo, T., Fabian, A.C., Rees, M.J., Carilli, C. L., & Ivison, R.J. 1999, MNRAS, 305, 492 \nDi Matteo, T., Quataert, E., Allen, S.W., Narayan, R., & Fabian, A. C. 2000 MNRAS, 311, 507 \nFabbiano, G., Elvis, M., Markoff, S., Siemiginowska, A., Pellegrini, S., Zezas, A., Nicastro, F., Trinchieri, G., & McDowell, J., 2003, ApJ, in press (astro-ph/0301297) \nHawley, J.F. 1992, ApJ, 528, 462 \nHawley, J.F., & Balbus, S.A. 1992, ApJ, 400, 595 \nHawley, J.F., & Balbus, S.A. 2002, ApJ, 573, 749 \nHawley, J.F., Balbus, S.A., & Stone, J.M. 2001, ApJ, 554, L49 \nIchimaru, S. 1977, ApJ, 214, 840. \nIgumenshchev, I.V., & Abramowicz, M.A. 1999, MNRAS, 303, 309 \nIgumenshchev, I.V. & Narayan, R. 2002, ApJ, 566, 137 \nIgumenshchev, I.V., Narayan, R., & Abramowicz M.A. 2003, ApJ, submitted (astro-ph/0301402) \nLoewenstein, M., Mushotzky, R.F., Angelini, L., Arnaud, K.A., & Quataert, E. 2001, ApJ, 555, L21 \nMachida, M., Matsumoto, R., & Mineshige, S. 2001, PASJ, 53, L1 \nMelia, F. & Falcke, H. 2001, ARA&A, 39, 309 \nMcKinney, J.C. & Gammie, C.F. 2002, ApJ, 573, 728 \nMoffatt, K. 1978, Magnetic Field Generation in Electrically Conducting Fluids (Cambridge: Cambridge Univ. Press) \nNarayan, R., Igumenshchev, I.V., & Abramowicz, M.A. 2000, ApJ, 539, 798 \n```\nNarayan, R., Quataert E., Igumenshchev, I.V., & Abramowicz, M.A. 2002, ApJ, 577, 295 Narayan, R., & Yi, I. 1994, ApJ, 428, L13 Narayan, R., & Yi, I. 1995, ApJ, 444, 231 Ostriker, J.P., McCray, R., Weaver, R., & Yahil, A. 1976, ApJ, 208, L61 Paczy'nski, B., & Wiita, J.P. 1980, A&A, 88, 23 Proga, D., & Begelman M.C. 2003, ApJ, 582, 69 (PB03) Quataert, E., & Gruzinov A. 2000, ApJ, 545, 842 Quataert, E., & Narayan R. 1999, ApJ, 520, 298 Rees, M.J., Begelman, M.C., Blandford, R.D., & Phinney, E.S. 1982, Nature, 295, 17 Stone, J.M., & Norman, M.L. 1992a, ApJS, 80, 753 Stone, J.M., & Norman, M.L. 1992b, ApJS, 80, 791 Stone, J.M., & Pringle, J.E. 2001, MNRAS, 322, 461 (SP01) Stone, J.M., Pringle, J.E., & Begelman, M.C. 1999, MNRAS, 310, 1002\n``` \nFig. 1. A sequence of logarithmic density and angular velocity contours (left and middle panels, respectively) and velocity direction plots (right panel) from run D at times 0.11, 0.34, 0.52, 1.45, and 2.54. The minimum and maximum of log ρ are 0.25 and 2. We use eight equally spaced contour levels for log ρ . The angular velocity is in units of 2 c/R S . The minimum of log Ω is -6 and seven contour levels are equally spaced at intervals of ∆logΩ = 0 . 25. We show the direction of the velocity field by unit vectors. \nFig. 2. The time evolution of the mass accretion rate in units of the Bondi rate, for run D (solid line) and run G (dashed line). Vertical arrows mark times for run D corresponding to the snapshots shown in Figures 4, 5, 6, and 7 (arrow a, b, c, and d respectively) and in Figure 8. \nFig. 3. Late time evolution of the mass accretion rate in units of the Bondi rate, for run D (top panel) and run G (bottom panel). This figure is very similar to Fig. 2 but it shows in more detail ˙ M a as a function of time toward the end of the simulations. Vertical arrows mark times corresponding to the snapshots shown in Figures 5, 6, and 7 (arrow b, c, and d respectively) and in Figure 8. \nFig. 4. Two-dimensional structure of various quantities from the fiducial model (run D) near the beginning of simulations at t = 0 . 22, marked by arrow a in Fig. 2. At this time accretion onto the black hole occurs through both the torus and the polar funnel. The top panels from left to right are snapshots of log ρ , S , log Ω, l , and direction of the fluid velocity. The bottom panels from left to right are snapshots of log ( P + B 2 / 8 π ), log B 2 / 8 π , log β , log | B φ | , and direction of the poloidal magnetic field. All contour levels are equally spaced. There are eight contours for the density, log ρ , between 0.25 and 2; nine for the entropy, S , between 40 and 44; four for the angular velocity, log Ω, between -2.5 and -1.75; eleven for the specific angular momentum, l , between 0.1 and 1.1 (the level for 1.1 is indicated by the dotted contour); six for the total pressure, log ( P + B 2 / 8 π ), between 20.5 and 23; ten for the magnetic pressure, log B 2 / 8 π , between 17.5 and 22; seven for the plasma parameter, log β , between -3 and 3 (the levels for log β < 0 are indicated by the dotted contours); and six for the toroidal field, log | B φ | , between 7.5 and 10. \nFig. 5. Two-dimensional structure of various quantities from the fiducial model (run D) at t = 2 . 39, marked by arrow b in Figs. 2 and 3a. The snapshots are for the same quantities as in Fig. 4. The contour levels are also as in Fig. 4. This figure presents an example of an inner flow where accretion occurs only through the torus (see also Fig. 3a). \nFig. 6. Two-dimensional structure of various quantities from the fiducial model (run D) at t = 2 . 41, marked by arrow c in Figs. 2 and 3a. The snapshots are for the same quantities as in Fig. 4. The contour levels are also as in Fig. 4. This figure presents an example of an inner flow where there is no torus accretion but only very weak accretion through a very low density magnetized polar cylinder. This state is very short-lived (see Fig. 3a). \nFig. 7. Two-dimensional structure of various quantities from the fiducial model (run D) at t = 2 . 36, marked by arrow d in Figs. 2 and 3a. The snapshots are for the same quantities as in Fig. 4. The contour levels are also as in Fig. 4. This figure presents an example of an inner flow where accretion is dominated by lowl material which managed to reach the inner boundary despite a blocking corona and outflow from the torus. The torus also accretes at this time but at a lower rate than in the lowl inflow. As the lowl inflow is gradually pushed away by the torus corona and outflow, ˙ M a slowly decreases to the level where accretion occurs only via the torus. \nFig. 8. Maps of logarithmic density overplotted by the direction of the poloidal velocity. This figure compares the inner flow in four different accretion states (see figs. 2 and 3a) shown in more detail in Figs. 4-7. \nFig. 9. Radial profiles of various quantities from run D, time-averaged from 2 . 388 through 2 . 417 orbits. During this period accretion occurs only through a torus, as illustrated in Fig. 5 (see also Fig. 3a). To construct each plot, we averaged the profiles over angle between θ = 87 · and 93 · . The top middle panel plots the gas pressure (solid line) and magnetic pressure. The top right panel plots the rotational, radial, Keplerian, and Alfv'en velocities (solid, dashed, dot-dashed, and dotted line, respectively), as well as the sound speed (triple-dot dashed line). The bottom middle panel plots the Maxwell stress, α mag = 2 B r B φ /B 2 /P mag , and the Reynolds stress, α gas = < ρv r δv φ > /P (solid and dashed line, respectively). We calculate the Reynolds stress using eq. 15 and show only its amplitude. The bottom right panel plots the radial, latitudinal and toroidal components of the magnetic field (dot-dashed, dashed, and solid line, respectively). \nFig. 10. As Fig. 8 but for run I. \nTable 1: Summary of parameter survey. \n| Run | Resolution | R ' S | R ' C | l 0 | θ o | f ( θ ) | β o | Q | t f | l max a | ˙ M a / ˙ M B | Comments |\n|-------|--------------|---------|----------------|-------|-------|---------------|-------|-----|-------|---------------|-----------------|------------------------|\n| A | 140 | 10 - 3 | 0 | 0 | 0 | -- | 10 5 | 0.1 | 0.28 | 0 . 0 | 1 | |\n| B | 140 | 10 - 3 | 0 | 0 | 0 | --- | 10 6 | 0.1 | 0.48 | 0 . 0 | 1 | |\n| C | 140 | 10 - 3 | 8 × 10 - 3 | 1 | 44 · | step function | 10 5 | 0.1 | 0.1 | 0 . 0 | 0.05 | |\n| D | 140 | 10 - 3 | 8 × 10 - 3 | 1 | 44 · | step function | 10 6 | 0 | 2.54 | 0 . 0 - 0 . 9 | 0.025 | fiducial run |\n| E | 140 | 10 - 3 | 8 × 10 - 3 | 1 | 44 · | step function | 10 6 | 0 | 0.57 | 0 . 0 - 0 . 9 | 0.058 | dθ l /dθ l +1 = 1 . 02 |\n| F | 140 | 10 - 3 | 8 × 10 - 3 | 1 | 44 · | step function | 10 6 | 0.1 | 0.32 | 0 . 0 - 0 . 9 | 0.053 | |\n| G | 140 | 10 - 3 | 8 × 10 - 3 | 1 | 44 · | step function | 10 7 | 0 | 2.51 | 0 . 0 - 0 . 9 | 0.001 | |\n| H | 140 | 10 - 3 | 8 × 10 - 3 | 1 | 44 · | step function | ∞ | 0 | 0.64 | 0 . 9 | 0.17 | |\n| I | 140 | 10 - 3 | 3 . 2 × 10 - 2 | 2 | 44 · | step function | 10 6 | 0.1 | 2.54 | 0 . 0 - 0 . 9 | 0.013 | |\n| J | 140 | 10 - 3 | 3 . 2 × 10 - 2 | 2 | 44 · | step function | ∞ | 0.1 | 0.78 | 0 . 0 - 0 . 9 | 0.27 | | \n<!-- image --> \nFig. 2.- \n<!-- image --> \n<!-- image --> \nFig. 4.- \n<!-- image --> \nFig. 5.- \n<!-- image --> \nFig. 6.- \n<!-- image --> \n<!-- image --> \nFig. 7.- \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 9.- \n<!-- image --> \nFig. 10.- \n<!-- image -->"}
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1995PhRvD..52.2254M
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Cosmological production of charged black hole pairs
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1995-01-01
| 8 | 0.45 | 164 |
['-', '-', '-', '-', '-', '-', '-', '-']
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[]
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We investigate the pair creation of charged black holes in a backgrond with a positive cosmological constant. We consider C metrics with a cosmological constant, and show that the conical singularities in the metric only disappear when it reduces to the Reissner-Nordström-de Sitter metric. We construct an instanton describing the pair production of extreme black holes and an instanton describing the pair production of nonextreme black holes from the Reissner-Nordström-de Sitter metric, and calculate their actions. There are a number of striking similarities between these instantons and the Ernst instantons, which describe pair production in a background electromagnetic field. We also observe that the type I instanton in the ordinary C metric with zero cosmological constant is actually the Reissner-Nordström solution.
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[]
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https://arxiv.org/pdf/gr-qc/9504015.pdf
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{'Cosmological production of charged black hole pairs': 'R.B. Mann a and Simon F. Ross b Department of Applied Mathematics and Theoretical Physics University of Cambridge, Silver St., Cambridge CB3 9EW a [email protected] b [email protected] \nNovember 26, 2024 DAMTP/R-95/9', 'Abstract': 'We investigate the pair creation of charged black holes in a background with a positive cosmological constant. We consider C metrics with a cosmological constant, and show that the conical singularities in the metric only disappear when it reduces to the Reissner-Nordstrom de Sitter metric. We construct an instanton describing the pair production of extreme black holes and an instanton describing the pair production of non-extreme black holes from the Reissner-Nordstrom de Sitter metric, and calculate their actions. There are a number of striking similarities between these instantons and the Ernst instantons, which describe pair production in a background electromagnetic field. We also observe that the type I instanton in the ordinary C metric with zero cosmological constant is actually the Reissner-Nordstrom solution.', '1 Introduction': 'There has been considerable interest recently in studying black hole pair creation by instanton methods, and a number of interesting results have been obtained [1, 2, 3]. The study of black hole pair creation has so far been mostly restricted to the pair creation of oppositely-charged black holes by an electromagnetic field, where pair creation is possible because the negative potential energy of the created pair of black holes in the background electromagnetic field balances their rest mass energy. Black holes can, however, also be pair created by a cosmological background, as a positive cosmological constant supplies the necessary negative potential energy. We will examine the pair creation of electrically or magnetically charged black holes in a cosmological background. There are two instantons, which describe pair production of non-extreme and extreme black holes respectively. There is another instanton, the charged generalisation of the instanton found in [4], but careful analysis suggests this latter instanton does not formally represent black hole pair creation. \nWe find that some of the most interesting results of the electromagnetic case can be reproduced in the cosmological case; in particular, the pair creation rate is still determined by the entropy of the solutions. Indeed, there is an interesting similarity between the instantons we find and the instantons found in [1, 2] for the pair creation of black holes in an electromagnetic field. We feel that these similarities encourage the idea that the results of these instanton calculations represent real quantum gravity effects, and will not be qualitatively modified by the inclusion of quantum corrections. In other words, since the main features are similar in these two disparate models, we think that they represent features that should also be present in the full quantum theory. \nWe will begin by considering the charged C metric, which can be interpreted as representing a pair of oppositely-charged black holes accelerating away from each other in a flat background spacetime. As there is no force to accelerate the black holes, this metric is not in general regular. The approach taken in [5, 1] was to add a background electromagnetic field to this metric by a Harrison transformation, giving the Ernst metric, which could be made regular by a suitable choice of field strength. The background field provides the necessary force to accelerate the black holes. We will consider adding a cosmological constant to the C metric instead, and see if we can obtain a non-singular solution by using the cosmological acceleration to accelerate the black holes. We find that we can, but that the C metric reduces \nto the Reissner-Nordstrom de Sitter solution whenever the acceleration of the black holes is matched to the cosmological acceleration in this way. \nThere are in fact two ways to obtain a non-singular solution in the charged C metric itself. One is to set the acceleration to zero, in which case the C metric reduces to the Reissner-Nordstrom metric. The other way to get rid of the conical singularities is to allow the black hole and acceleration horizons to coincide, which has been referred to as the Type I instanton [1]. We have found that the C metric in this special case has the same functional form as the Reissner-Nordstrom metric, but it is now what is usually regarded as the azimuthal coordinate which is playing the role of time. That is, one can obtain this metric by analytically continuing φ → i Φ in the Euclidean Reissner-Nordstrom metric. \nAs the only non-singular versions of the charged C metric with a cosmological constant reduce to the Reissner-Nordstrom de Sitter metric, we would like to know what instantons can be made from this metric. We find that there are four types of instantons, referred to as the lukewarm, cold, ultracold [6], and charged Nariai solutions. The Euclidean section of the lukewarm solution has topology S 2 × S 2 , and it can be thought of as representing pair creation of non-extreme black holes. The Euclidean section of the cold solution has topology S 2 × R 2 , and it can be thought of as representing pair creation of extreme black holes (by an extreme black hole, we mean one in which the inner and outer horizons coincide). The charged Nariai solution has topology S 2 × S 2 ; its interpretation is not clear. The ultracold solution is just a special case of the cold solution where all three horizons coincide, but the metric is quite different in this special case. There is an analogy between this set of instantons and the instantons constructed from the Ernst solution in [2]: there one had an extreme black hole instanton, a non-extreme black hole instanton, and the Type I instanton; here we have the cold, lukewarm and charged Nariai instantons. \nIn the instanton approach, the wavefunction is approximated by Ψ ≈ e -I , where I is the action of the instanton, so the partition function, which gives the pair creation rate, is approximated by Z ≈ e -2 I . The action for Einstein-Maxwell with a cosmological constant is \nI = -1 16 π ∫ d 4 x √ g ( R -2Λ -F µν F µν ) + 1 8 π ∫ Σ d 3 x √ hK, (1) \nwhere R is the Ricci scalar of the metric g µν , Λ is the cosmological constant (we assume Λ > 0), and F µν is the electromagnetic field tensor. The boundary of the manifold is Σ, which has metric h ij , and extrinsic curvature K ij . \nSince there is no asymptotic region for these solutions, this action is the microcanonical action [7], and thus the entropy is [8] \nS = log Z = -2 I. (2) \nIn section 5, we calculate the action for the instantons described in section 4. We find that the pair creation rate given by these instantons is smaller than the rate which de Sitter space gives to propagate from nothing to an S 3 . Thus the contribution from de Sitter space will dominate that of these instantons in the path integral, in agreement with experience. We also find that the entropy of these solutions is always equal to one quarter of the area of the horizons which appear on the Euclidean section, as expected. These instantons are thus very similar to the Ernst instantons. We summarise our results in section 6.', '2 Cosmological C metrics': "The well-known C metric solution of the Einstein-Maxwell equations, which describes a pair of charged black holes undergoing uniform acceleration, was found in [9]. There is a less well-known generalisation of this solution to include a cosmological constant [10]. In the usual C metric, it is not possible to eliminate the possible conical singularities at both poles in general. The C metric is interpreted as describing two oppositely-charged black holes undergoing uniform acceleration, and these singularities are interpreted as representing 'rods' or 'strings' which provide the force necessary to accelerate the black holes. \nIf we consider the C metric with a positive cosmological constant, one might think that the cosmological constant can provide the necessary force, and so we should be able to eliminate the conical singularities, and thus obtain a completely regular metric. What we find is that the elimination of the conical singularities, which we regard as fixing the acceleration parameter A , causes the metric to reduce to the Reissner-Nordstrom de Sitter metric. \nThe cosmological C metric can be written as \nds 2 = 1 A 2 ( x -y ) 2 [ H ( y ) dt 2 -H -1 ( y ) dy 2 + G -1 ( x ) dx 2 + G ( x ) dϕ 2 ] , (3) \nwhere \nG ( x ) = 1 -x 2 (1 + r + Ax )(1 + r -Ax ) -Λ 3 A 2 , (4) \nand \nH ( y ) = 1 -y 2 (1 + r + Ay )(1 + r -Ay ) . (5) \nThe cosmological constant is Λ, which we will assume is positive. The gauge field in the magnetic case is \nF = -qdx ∧ dϕ, (6) \nand the gauge field in the electric case is \nF = -qdt ∧ dy, (7) \nwhere q = √ r + r -. We will denote by x 1 , x 2 , x 3 , x 4 the roots of G ( x ) in ascending order, and by y 1 , y 2 , y 3 , y 4 the roots of H ( y ) in ascending order. We will restrict the parameters Λ , r + , r -and A so that all these roots are real. As with the usual C metric, y = y 1 is interpreted as the inner black hole horizon, y = y 2 is interpreted as the outer black hole horizon, and y = y 3 is interpreted as the acceleration or cosmological horizon. \nWe will first restrict x to x 3 ≤ x ≤ x 4 in order to obtain a metric of the appropriate signature. We will restrict y to -∞ < y ≤ x , as when y = x the conformal factor in the metric diverges, and so this corresponds to infinity. In order to have a regular solution, we must avoid having conical singularities at x = x 3 or x = x 4 , so we must demand that [1] \nG ' ( x 3 ) = -G ' ( x 4 ) , (8) \nand identify ϕ periodically with period ∆ ϕ = 4 π/ | G ' ( x 3 ) | . This assumes that x 3 is at a finite distance from x 4 , but if one took x 3 = x 2 , then x = x 3 would lie at infinite proper distance from any other point, so that there could be no conical singularity there. That is, the ( x, ϕ ) sections would no longer be compact. This is analogous to what happens with the Euclidean section of an extreme black hole [3]. However, for positive cosmological constant, H ( y ) will have just two real roots if x 2 = x 3 , so this contradicts our restriction of the parameters. Therefore we must satisfy (8), which corresponds to \n( x 3 -x 4 )( x 3 -x 2 )( x 3 -x 1 ) = ( x 3 -x 4 )( x 4 -x 2 )( x 4 -x 1 ) , (9) \nand can only be satisfied by taking x 3 = x 4 . \nThis seems to imply that the ( x, ϕ ) section shrinks to a point, but this is just due to a poor choice of coordinate system. In the limit that x 3 = x 4 , the proper distance between x 3 and x 4 remains finite, as can be seen by the \nfollowing coordinate transformation. 1 The limit x 3 → x 4 corresponds to the limit A → √ Λ / 3 from above. Let us write 1 -Λ / (3 A 2 ) = /epsilon1 2 , so that the appropriate limit is /epsilon1 → 0. With this parametrisation, \nG ( x ) = /epsilon1 2 -x 2 (1 + r + Ax )(1 + r -Ax ) , (10) \nso x 3 ≈ -/epsilon1 and x 4 ≈ /epsilon1 . Then set \nx = /epsilon1 cos θ, ϕ = φ /epsilon1 . (11) \nIn the limit /epsilon1 → 0, the metric becomes \nds 2 = 3 Λ y 2 [ H ( y ) dt 2 -H -1 ( y ) dy 2 + dθ 2 +sin 2 θdφ 2 ] , (12) \nand if we make the further coordinate transformation \nr = -√ 3 / Λ y , T = √ 3 / Λ t, (13) \nit becomes \nwhere \nds 2 = -V ( r ) dT 2 + dr 2 V ( r ) + r 2 ( dθ 2 +sin 2 θdφ 2 ) , (14) \nV ( r ) = ( 1 -r + r )( 1 -r -r ) -Λ r 2 3 . (15) \nNote also that as ϕ has period 4 π/ | G ' ( x 3 ) | , φ has period 2 π . The range of r is from 0 to ∞ , and θ runs from 0 to π . The gauge field is now \nF = q sin θdθ ∧ dφ (16) \nin the magnetic case, and \nF = -q r 2 dT ∧ dr (17) \nin the electric case. Thus, this can be identified as a Reissner-Nordstrom de Sitter solution with charge q = √ r + r -and 'mass' M = ( r + + r -) / 2. Note that if we let Λ = 0, the limit x 3 = x 4 is just the limit A → 0, and what \nwe have just done specialises to the familiar statement that the C metric reduces to the Reissner-Nordstrom metric in this limit [1]. \nWe have assumed earlier that -∞ < y < x to avoid divergence in the conformal factor, but we could equally well have taken x < y < ∞ . Then y = y 2 is interpreted as the cosmological horizon, y = y 3 is the outer black hole horizon, and y = y 4 is the inner black hole horizon. We now have to take x 1 x ≤ x 2 to get a metric of the right signature. \n≤ To make this metric regular, we must take ϕ to be periodic with period ∆ ϕ = 4 π/ | G ' ( x 1 ) | , and require \nG ' ( x 1 ) = -G ' ( x 2 ) . (18) \nThis implies \n( x 1 -x 2 )( x 1 -x 3 )( x 1 -x 4 ) = ( x 1 -x 2 )( x 2 -x 3 )( x 2 -x 4 ) , (19) \nand thus x 1 = x 2 . We have found that this occurs when A = A c , where \nA 2 c = Λ 3 -[3( r + + r -) + γ ] 2 [( r + + r --γ ) 2 -16( r + -r -) 2 ] 4096 r 3 + r 3 -, (20) \nand \nγ = 9( r 2 + + r 2 -) -14 r + r -. (21) \nif we let A 2 c /A 2 = 1 -/epsilon1 2 , and X = x -x 1 , then \n√ \nG ( x ) = /epsilon1 2 -X 2 16 r + r -( a + bAX + cA 2 X 2 ) , (22) \na = (3( r + + r -) + γ ) γ, (23) \nb = -8 r + r -( r -+ r + + γ ) , (24) \nc = 16 r 2 + r 2 -. (25) \nTherefore, if we make coordinate transformations \nX = √ 16 r + r -a /epsilon1 cos θ, (26) \nand \nϕ = √ 16 r + r -a φ//epsilon1, (27) \n≤ \nwhere \nand \nthe metric becomes, in the limit /epsilon1 → 0, \nds 2 = 1 A 2 c ( y -x 1 ) 2 [ H ( y ) dt 2 -H -1 ( y ) dy 2 + 16 r + r -a ( dθ 2 +sin 2 θdφ 2 ) ] . (28) \nIf we make the further coordinate transformations \nr = √ 16 r + r -a 1 A c ( y -x 1 ) , T = √ a 16 r + r -t A c , (29) \nthe metric will become \nds 2 = -V ( r ) dT 2 + dr 2 V ( r ) + r 2 ( dθ 2 +sin 2 θdφ 2 ) , (30) \nwhere \nand \nV ( r ) = 1 -2 M r + Q 2 r 2 -Λ 3 r 2 , (31) \nM = -√ 16 r + r -a b 2 a , Q 2 = 16 r + r -c a 2 . (32) \nThe gauge field becomes \nF = 16 r + r -a q sin θdθ ∧ dφ = Q sin θdθ ∧ dφ (33) \nin the magnetic case, and \nF = Q r 2 dT ∧ dr (34) \nin the electric case. Therefore, this can be identified as a Reissner-Nordstrom de Sitter metric as well. Note that although the equations are more complicated, M and Q are still just functions of r + and r -, and M is positive if r + and r -are positive. \nIf the cosmological constant is set to zero, we see that the C metric is again only non-singular when it reduces to the Reissner-Nordstrom metric, that is, when the acceleration of the black holes vanishes. However, in this case, this happens for non-zero A , as we can easily see from (20). This is just an indication that we shouldn't think of A as simply parametrising the acceleration in this case.", '3 The Type I instanton': "When the cosmological constant is set to zero, we have seen that the ordinary C metric has no conical singularities when the acceleration vanishes, as we should have expected. However, when Λ = 0, we can eliminate the conical singularities in another way, as was first observed by Dowker et al [1]. It is now possible to set x 3 = x 2 , as x 3 = y 3 = ξ 3 and x 2 = y 2 = ξ 2 . Again, the apparent degeneracy of the two roots is merely an artifact of our coordinate system. As explained in [1], we may make a transformation so that the metric remains regular when ξ 3 = ξ 2 . \nThe C metric is given by (3) with Λ = 0. However, for consistency with [1], we will adopt a slightly different coordinate system in this section, and write the C metric as \nds 2 = 1 A 2 ( x -y ) 2 [ G ( y ) dt 2 -G -1 ( y ) dy 2 + G -1 ( x ) dx 2 + G ( x ) dϕ 2 ] (35) \nwith \nG ( ξ ) = [1 -ξ 2 (1 + r + Aξ )](1 + r -Aξ ) . (36) \nThe transformation between the two forms of the C metric is discussed in [9]. The gauge field is still (6) or (7). We will refer to the roots of G ( ξ ) as ξ 1 , ξ 2 , ξ 3 , ξ 4 in ascending order. As before, x is restricted to ξ 3 ≤ x ≤ ξ 4 to obtain the appropriate signature. In fact, if ξ 2 = ξ 3 , the appropriate range is actually ξ 3 < x ≤ ξ 4 , and the ( x, ϕ ) section becomes topologically R 2 . We can then obtain a regular solution by identifying ϕ with period 4 π/ G ' ( ξ 4 ) | . \n| √ \ny = √ 3( -1 + /epsilon1 cos χ ) , ψ = √ 3 /epsilon1t (37) \n| The two roots ξ 3 and ξ 4 will coincide when A = A c = 2 / (3 3 r + ). Following [1], we let r + A = 2 / (3 √ 3) -/epsilon1 2 / √ 3, so that the limit of coincident roots is /epsilon1 → 0. If we make the coordinate transformation \nthe metric is, in the limit /epsilon1 → 0, \nds 2 = 1 A 2 c ( x + √ 3) 2 [ -α sin 2 χdψ 2 + α -1 dχ 2 + G -1 ( x ) dx 2 + G ( x ) dϕ 2 ] , (38) \nwhere now \nG ( x ) = -2 3 √ 3 ( x + √ 3) 2 ( x -√ 3 / 2) ( 1 + 2 r -x 3 √ 3 r + ) (39) \nand \nα = 1 -2 r -3 r + . (40) \nIf we analytically continue ψ → i Ψ, and identify Ψ with period 2 πα -1 , we obtain an instanton with topology S 2 × R 2 , referred to as the type I instanton. It was not initially clear how to interpret this instanton. However, our experience with the cosmological C metric above suggests that it is related to the Reissner-Nordstrom instanton. We are encouraged in this guess by the fact that the (Ψ , χ ) two-sphere sections are round. √ \nThe range of x in this solution is -3 < x ≤ √ 3 / 2. Let ˜ x = x + 3, and φ = α Ψ, so that φ has period 2 π , and the Euclideanised metric becomes \n√ \n√ \nds 2 = α -1 A 2 c ˜ x 2 [ dχ 2 +sin 2 χdφ 2 + α ( G -1 (˜ x ) d ˜ x 2 + G (˜ x ) dϕ 2 ) ] . (41) \nIf we make a further coordinate transformation, \nr = α -1 / 2 A c ˜ x , τ = α 1 / 2 ϕ A c , (42) \nthe metric becomes \nds 2 = r 2 ( dχ 2 +sin 2 χdφ 2 ) + V ( r ) dτ 2 + dr 2 V ( r ) , (43) \nwhere \nV ( r ) = ( 1 -˜ r + r )( 1 -˜ r -r ) . (44) \nThe new parameters are ˜ r + = r + α -1 / 2 and ˜ r -= -r -α -3 / 2 . We also note that r runs from ˜ r + to ∞ on the Euclidean section. If the gauge field is (6), it becomes \nF = qα -1 r 2 dr ∧ dτ = -iQ r 2 dτ ∧ dr, (45) \nand if it is (7), it becomes \nF = -iqα -1 sin χdχ ∧ d Ψ = Q sin χdχ ∧ dφ, (46) \nwhere Q 2 = ˜ r + ˜ r -. Therefore, this instanton is seen to be the Euclidean Reissner-Nordstrom solution with charge Q and mass M = 1 2 (˜ r + +˜ r -), with the magnetic instanton being identified with the electric Reissner-Nordstrom solution and vice-versa. Thus we see again that eliminating the conical singularities is only possible when the C metric reduces to Reissner-Nordstrom. \nHowever, the coordinate we analytically continued to obtain a Euclidean section has been identified with the azimuthal coordinate in the Euclidean Reissner-Nordstrom solution. This means that the problem of understanding the physical significance, if any, of the C metric in this special case is equivalent to giving a physical interpretation of the metric (43) with φ = i Φ. Because τ is not analytically continued, this metric has one compact direction. This leads us to suspect that the interpretation is similar to that given for the five-dimensional black holes in [11]; that is, that this instanton should be interpreted as representing the decay of a Kaluza-Klein vacuum, Mink (2 , 1) S 1 . \nThe analytically continued metric is \n× \nds 2 = -r 2 sin 2 χd Φ 2 + r 2 dχ 2 + V ( r ) dτ 2 + dr 2 V ( r ) , (47) \nwhere V ( r ) is given in (44). So long as the period ∆ τ around the compact direction is chosen appropriately, the coordinate singularity at r = ˜ r + is harmless. Let us ignore for the moment the factors of V ( r ), and consider just the three-dimensional space ( r, χ, Φ), that is, consider \nds 2 = -r 2 sin 2 χd Φ 2 + dr 2 + r 2 dχ 2 . (48) \nThis metric describes a portion of three-dimensional Minkowski space. In fact, if we make the coordinate transformations \nz = r sin χ cosh Φ , t = r sin χ sinh Φ , y = r cos χ (49) \nthe metric (48) becomes \nds 2 = -dt 2 + dz 2 + dy 2 , (50) \nthe usual metric on Minkowski space. Because χ is restricted to 0 ≤ χ ≤ π , the original coordinates only cover the part z ≥ 0 of the Minkowski space; however there is no obstruction to extending the solution to the whole of Minkowski space. We could also set \nz = R sin ψ, y = R cos ψ, (51) \nand rewrite this metric as \nds 2 = -dt 2 + dR 2 + R 2 dψ 2 . (52) \nNow let us consider the effect of the factors of V ( r ) in (47). The metric (47) approaches the product manifold Mink (2 , 1) × S 1 asymptotically, but the coordinate r is now restricted to ˜ r + ≤ r ≤ ∞ . We can still make the same coordinate transformations, (49,51). The resulting metric approaches (52) × S 1 asymptotically, but the restriction r ≥ ˜ r + implies R 2 -t 2 ≥ ˜ r 2 + . Note that it is not possible to continue the metric beyond r = ˜ r + , as the radius of the circle direction vanishes there. \nThe physical interpretation is exactly the same as in [11]: if we assume the radius of the circle direction is relatively small, putative observers living in this space who didn't go too close to r = ˜ r + would think (47) described three-dimensional Minkowski space with the interior of the hyperboloid R 2 -t 2 = ˜ r 2 + omitted. The type I instanton should be interpreted as representing tunnelling from the vacuum (52) cross a circle to (47); that is, it describes the decay of a three-dimensional Kaluza-Klein vacuum.", '4 Reissner-Nordstrom de Sitter instantons': "Since the only special cases of the cosmological C metrics for which the metric is regular reduce to the Reissner-Nordstrom de Sitter metrics, consideration of the pair creation of charged black holes in a background with a positive cosmological constant reduces to a consideration of the non-singular instantons that can be constructed from the Reissner-Nordstrom de Sitter metric. This question has of course been studied before, notably in [6, 12], and in the uncharged case in [4], but we hope to present a unified picture which makes the relations between the instantons clear. \nThe Reissner-Nordstrom de Sitter metric is \nds 2 = -V ( r ) dt 2 + dr 2 V ( r ) + r 2 ( dθ 2 +sin 2 θdφ 2 ) , (53) \nwhere \nV ( r ) = 1 -2 M r + Q 2 r 2 -1 3 Λ r 2 . (54) \nThe gauge field is \nF = -Q r 2 dt ∧ dr (55) \nfor an electrically-charged solution, and \nF = Q sin θdθ ∧ dφ (56) \nfor a magnetically-charged solution. For the sake of simplicity, we will not consider dyonic solutions. This solution has three independent parameters, the 'mass' M , charge Q and cosmological constant Λ, which we will assume are all positive. There are then four roots of V ( r ), which we designate by r 1 , r 2 , r 3 , r 4 in ascending order. The first root is negative, and therefore has no physical significance. The remaining roots are interpreted as various horizons; r = r 2 is the inner (Cauchy) black hole horizon, r = r 3 is the outer (Killing) black hole horizon, and r = r 4 is the cosmological (acceleration) horizon. In the Lorentzian section, 0 r < ∞ . \n≤ \n∞ We want to construct instantons from this metric by analytically continuing t → iτ . To obtain a positive-definite metric, we must restrict r to r 3 ≤ r ≤ r 4 . There is then potentially a conical singularity at r = r 3 and at r = r 4 . However, if r 3 = r 2 , the range of r in the Euclidean section will be r 3 < r ≤ r 4 , as the double root in V ( r ) implies that the proper distance from any other point to r = r 3 along spacelike directions is infinite. In this case, we may obtain a regular instanton by identifying τ periodically with period 2 π/κ 4 , where κ 4 is the surface gravity of the horizon r = r 4 . This instanton will be referred to as the cold instanton, following [6]. \nIf we do not have r 2 = r 3 , then we must have \nκ 3 = κ 4 , (57) \nand identify τ with the same period. There are two ways to satisfy this condition; one is r 3 = r 4 , which gives an instanton analogous to that constructed out of the Nariai metric in [4], which we shall refer to as the charged Nariai instanton. This is very similar to the way in which the Type I instanton is obtained. The other is to set Q = M , which implies (57) [12]; we shall refer to this as the lukewarm instanton, following [6]. There is also a special case, when r 2 = r 3 = r 4 , which we refer to as the ultracold instanton, again following [6]. \nThe form of all these instantons in terms of the metric (53) has been given in detail in [6]; we will briefly summarise that discussion here. If there is a double root ρ of V ( r ), we can write \nV d ( r ) = ( 1 -ρ r ) 2 ( 1 -1 3 Λ( r 2 +2 ρr +3 ρ 2 ) ) , (58) \nand the mass and charge are thus given by \nM = ρ ( 1 -2 3 Λ ρ 2 ) , (59) \nQ 2 = ρ 2 (1 -Λ ρ 2 ) . (60) \nFor positive Λ, the double root ρ must lie in 0 < ρ < Λ -1 / 2 . The other positive root of V d ( r ) is \nb = √ 3Λ -1 -2 ρ 2 -ρ. (61) \nFor 0 < ρ 2 < Λ -1 / 2, b > ρ , so r 2 = r 3 = ρ : this solution gives the cold instanton. For Λ -1 / 2 < ρ 2 < Λ -1 , b < ρ , so r 3 = r 4 = ρ : this solution gives the charged Nariai instanton. If ρ 2 = Λ -1 / 2, then b = ρ = r 2 = r 3 = r 4 : this solution gives the ultracold instanton. The function V d ( r ) can also be rewritten as \nV d ( r ) = -r 2 ( b 2 +2 ρb +3 ρ 2 ) ( 1 -ρ r ) 2 ( 1 -b r )( 1 + 2 ρ + b r ) , (62) \nand we could also write M,Q and Λ as functions of ρ and b (see [6] for details). \nIf V ( r ) does not have a double root, then it must have two roots r 3 and r 4 such that \nκ 3 = 1 2 | V ' ( r 3 ) | = κ 4 = 1 2 | V ' ( r 4 ) | . (63) \nThis fixes V ( r ) to have the form \nV l ( r ) = ( 1 -r 3 r 4 ( r 3 + r 4 ) r ) 2 -r 2 ( r 3 + r 4 ) 2 , (64) \nfrom which one sees immediately that Q = M . We will now comment briefly on the nature of each of these instantons in turn. \nFor the lukewarm instanton, the topology of the Euclidean section is S 2 × S 2 . The common temperature of the two horizons is [12] \nT = 1 2 π √ √ √ √ √ Λ 3 1 -4 M √ Λ 3 . (65) \nThe Lorentzian section describes two black holes in de Sitter space, so this instanton represents pair creation of non-extreme black holes in thermal equilibrium with the cosmological acceleration radiation. For the cold instantons, the horizon at r = r 3 is at infinite distance, so the Euclidean section has topology S 2 × R 2 . There is a boundary B ∞ at the internal infinity r = r 3 = ρ . In the calculation of the action, we will take the boundary \nto lie at r = ρ + /epsilon1 , and then take the limit as /epsilon1 → 0, that is, as the boundary approaches B ∞ . The relation between the charge and mass is given parametrically by (59,60), and is displayed on Figure 1. The temperature of the horizon at r = r 4 is [6] \nT = b 2 π ( b 2 +2 ρb +3 ρ 2 ) ( 1 -ρ b ) 2 ( 1 + ρ b ) . (66) \nThe Lorentzian section describes two extreme black holes in de Sitter space, so this instanton represents pair creation of extreme black holes in thermal equilibrium with the acceleration radiation. For the charged Nariai instantons, where r 3 = r 4 , it is necessary to make a coordinate transformation and rewrite the metric as [7] \nds 2 = 1 A ( dχ 2 +sin 2 χdψ 2 ) + 1 B ( dθ 2 +sin 2 θdφ 2 ) , (67) \nwhere A and B are constants, with B > A , χ and θ run from 0 to π , and ψ and φ are periodic coordinates with period 2 π . The gauge field becomes \nF = Q sin θdθ ∧ dφ (68) \nin the magnetic case, and \nF = -iQ B A sin χdχ ∧ dψ (69) \nin the electric case. The cosmological constant is Λ = 1 2 ( A + B ), while ρ 2 = 1 /B , and M and Q are given by (59,60). This instanton has topology S 2 × S 2 ; indeed, it is just the direct product of two round two-spheres with different radii. The relation between the mass and charge is still given parametrically by (59,60), and is also displayed on Figure 1. However, although this solution is just a special case of Reissner-Nordstrom de Sitter, we find that the singularity retreats to infinite proper distance when r 3 = r 4 , and so there is no longer a global event horizon, and the Lorentzian section is just the direct product of two-dimensional de Sitter space and a two-sphere of fixed radius, dS 2 × S 2 . Thus the instanton doesn't represent pair creation of black holes. However, as in [4], higher-order quantum corrections will break the degeneracy of the two roots, so the charged Nariai solution will revert to an ordinary Reissner-Nordstrom de Sitter spacetime once these effects are included. \nFigure 1: The values of Q and M for which instantons can be obtained in the cosmological case. The plot is of the dimensionless quantities Q √ Λ vs. M √ Λ. The curve DU represents the cold solutions, DC represents the lukewarm solutions, and NU represents the charged Nariai solutions. The point at D is de Sitter space, while U is the ultracold case, and N is the Nariai solution. \n<!-- image --> \nThe ultracold case deserves a more detailed description. In this case r 2 = r 3 = r 4 , which may be regarded as the limit b = ρ in the metric (53) with V ( r ) having the double root form (62). In this case, the mass and charge are given by (59,60) with ρ = 1 / √ 2Λ, that is, \nM = 2 3 √ 2Λ , Q 2 = 1 4Λ . (70) \nSuppose ρ = 1 / √ 2Λ -/epsilon1 , and b = 1 / √ 2Λ + /epsilon1 , and consider the limit /epsilon1 → 0. We can construct two different metrics in this limit. First, define a new coordinate R by \nr = 1 / √ 2Λ + /epsilon1 cos √ 4 /epsilon1 (2Λ) 3 / 2 3 R , (71) \nψ = 4(2Λ) 3 / 2 3 /epsilon1 2 τ (72) \nand take \nin (53), with t → iτ . Then \nV d ( r ) = 2(2Λ) 3 / 2 3 /epsilon1 3 sin 2 ( √ 4 /epsilon1 (2Λ) 3 / 2 3 R )[cos( √ 4 /epsilon1 (2Λ) 3 / 2 3 R ) + 1] , (73) \nand the metric in the limit /epsilon1 → 0 is \nds 2 = R 2 dψ 2 + dR 2 + 1 2Λ ( dθ 2 +sin 2 θdφ 2 ) . (74) \nThus this instanton has topology S 2 × R 2 . In fact, the metric is the direct product of a flat R 2 and a round two-sphere of radius 1 / √ 2Λ. The internal infinity B ∞ is now at R = ∞ . We will take R = R 0 in the calculations, and then take the limit R 0 → ∞ . The angle ψ is interpreted as the imaginary time. We could construct the same instanton from (67) in the limit A → 0, by taking χ = √ AR . The gauge field is \nF = 1 2 √ Λ sin θdθ ∧ dφ (75) \nin the magnetic case, and becomes \nF = -i √ Λ RdR ∧ dψ (76) \nin the electric case. The Lorentzian metric obtained by taking ψ → i Ψ represents Mink (1 , 1) × S 2 in Rindler coordinates; the horizon at R = 0 is the Rindler horizon. \nAlternatively, we could define x by \nr = 1 √ 2Λ + √ 2(2Λ) 3 / 2 3 /epsilon1 3 / 2 x, (77) \nand take \nγ = √ 2(2Λ) 3 / 2 3 /epsilon1 3 / 2 τ. (78) \nThen \nV d ( r ) = 2 /epsilon1 3 (2Λ) 3 / 2 3 1 + √ 2(2Λ) 3 / 2 3 /epsilon1 1 / 2 x 2 1 -√ 2(2Λ) 3 / 2 3 /epsilon1 1 / 2 x , (79) \nand the metric in the limit /epsilon1 → 0 is \nds 2 = dγ 2 + dx 2 + 1 2Λ ( dθ 2 +sin 2 θdφ 2 ) . (80) \nThis instanton also has topology S 2 × R 2 , but the internal infinity B ∞ now has two components, x = ±∞ . We will evaluate the action for a region bounded by x = ± x 0 , and then take x 0 → ∞ . It is γ that is interpreted as the analogue of imaginary time, and γ runs from -∞ < γ < ∞ , so this looks just like flat space. The gauge field in this case is \nF = 1 2 √ Λ sin θdθ ∧ dφ (81) \nin the magnetic case, and \nF = i √ Λ dγ ∧ dx (82) \nin the electric case. This solution describes the neighbourhood of a point, when both horizons have receded to infinity. That is, the Lorentzian section is just Mink (1 , 1) S 2 in the usual coordinates. \nIn summary: for sufficiently small mass, there are two solutions, the lukewarm and cold solutions, which correspond to pair creation of nonextreme and extreme black holes respectively. At given mass, the cold solution has higher charge than the lukewarm solution. Once the mass \n× \nFigure 2: The values of Q and M for which instantons can be obtained in the electromagnetic case. The plot is of the dimensionless quantities qA vs. mA . The curve OA represents the extreme solutions, OC represents the lukewarm solutions, and BA represents the Type I instantons. The point at O is the Melvin solution. \n<!-- image --> \nreaches M = 1 / (3 √ Λ), there is a third solution, the charged Nariai solution, which has lower charge than the other two. When the mass reaches M = 3 / (4 √ 3Λ), the lukewarm and charged Nariai solutions coincide, and there is no lukewarm solution with higher mass. The cold and charged Nariai solutions coincide, in the ultracold solution, when the mass reaches M = 2 / (3 √ 2Λ), and there are no regular solutions where the mass is larger than this. The ratio of charge to mass is also at its largest at this point, where Q/M = 3 / (2 √ 2). There is an interesting analogy between the situation here and the Ernst instantons; with the Ernst solution, there are also three ways in which a non-singular instanton can be achieved. These are the non-extreme (or type II) instanton, the extreme instanton, and the type I instanton [1, 2]. The lukewarm solution may be thought of as the analogue of the non-extreme instanton, the cold solution as the analogue of the extreme instanton, and the charged Nariai solution as the analogue of the Type I instanton. The plots of mass versus charge for the ReissnerNordstrom de Sitter and Ernst instantons are given in Figure 1 and Figure 2 respectively. While the numerical values are not the same, the qualitative features of these two plots are strikingly similar.", '5 Pair creation rate and entropy': "In the previous section, we presented the Euclidean solutions which provide the instantons for pair creation of charged black holes in a cosmological background. We will now review their interpretation as instantons, and use them to obtain approximate rates for these processes. \nThe pair creation of non-extreme charged black holes in a cosmological background is described by propagation from nothing to a surface Σ with topology S 2 × S 1 , since such a surface may be thought of as a Wheeler wormhole attached to an S 3 , and S 3 is the topology of the spatial sections of de Sitter space. The pair creation of extreme charged black holes is similarly described by propagation to a surface with topology S 2 × R 1 . In this case there is also a boundary component B ∞ , which represents an 'internal infinity'. We will use Σ to denote the whole boundary in this case, and Σ s to denote the part with topology S 2 × R 1 . The amplitude for these processes will, at least formally, be given by a path integral; \nΨ = ∫ d [ g ] d [ A ] e -I . (83) \nThe integral is over all metrics and gauge fields which agree with the given boundary data on Σ, and Ψ may thus be thought of as a functional of the boundary data. We assume that, if there is a Euclidean classical solution which interpolates within the given boundary, then the integral is dominated by the contribution from it. That is, if there is an appropriate instanton, Ψ will be approximately \nΨ ≈ e -I , (84) \nwhere I is the action of this instanton. The partition function, and thus the pair creation rate, is given by the square of this amplitude. \nA spatial section of the lukewarm solution has topology S 2 × S 1 , so half of the Euclidean section provides an instanton for the pair creation of nonextreme charged black holes. We have to use half of the solution, as we want the extrinsic curvature of Σ to vanish, so that Σ can be interpreted as the zero-momentum initial data for the Lorentzian extension. Similarly, half of the cold solution can be used as an instanton to describe the pair creation of extreme charged black holes. Note that these instantons only exist when the data on Σ specified in the path integral agrees with the data induced by the solutions. \nThe whole of the Euclidean section in each case provides a 'bounce' solution. In asymptotically-flat situations, one can deal with the bounces \nrather than the instantons, but in a cosmological situation this is no longer possible, as the boundary data on the surface Σ provide crucial information [7]. We will therefore be interested in the calculation of the action for the instantons. \nWe also need to consider what action we should use in the calculation of (84). We want to use the action for which it is natural to fix the boundary data on Σ specified in the path integral (83). That is, we want to use an action whose variation gives the Euclidean equations of motion when the variation fixes these boundary data on Σ [8]. If we consider the action (1), we can see that its variation will be \nδI = (terms giving the equations of motion) + (gravitational boundary terms) + 1 4 π ∫ Σ d 3 x √ hF µν n µ δA ν , (85) \nwhere n µ is the normal to Σ and h ij is the induced metric on Σ (see [8] for a more detailed discussion of the gravitational boundary terms). Thus, the variation of (1) will only give the equations of motion if the variation is at fixed gauge potential on the boundary, A i . Note that it is not necessary to fix the component A µ n µ normal to the boundary. \nFor the magnetic Reissner-Nordstrom solutions, fixing the gauge potential fixes the charge on each of the black holes, as the magnetic charge is just given by the integral of F ij over a two-sphere lying in the boundary. However, in the electric case, fixing the gauge potential A i can be regarded as fixing a chemical potential ω which is conjugate to the charge [7]. Holding the charge fixed in the electric case is equivalent to fixing n µ F µi on the boundary, as the electric charge is given by the integral of the dual of F over a two-sphere lying in the boundary. Therefore, the appropriate action is \nI el = I -1 4 π ∫ Σ d 3 x √ hF µν n µ A ν , (86) \nas its variation is \nδI el = (terms giving the equations of motion) + (gravitational boundary terms) -1 4 π ∫ Σ d 3 xδ ( √ hF µν n µ ) A ν , (87) \nand so it gives the equations of motion when √ hn µ F µi , and thus the electric charge, is held fixed. \nLet us consider the pair creation where the electric and magnetic charge are held fixed on Σ. Then the appropriate action will be (1) in the magnetic case, and (86) in the electric case. It is also worth pointing out that, since we identify Σ (Σ s in the cold case) with a surface of zero extrinsic curvature in the Euclidean section, the gravitational boundary term in the action (1) will make no contribution to the action. Thus the action (1) for the instanton will just be half that of the whole Euclidean section. \nWe will now consider each of the instantons derived in section 4, and calculate the actions (1) in the magnetic case and (86) in the electric case. For the lukewarm solution we have, in the magnetic case, F 2 = 2 Q 2 /r 4 , and thus the action (1) is [12] \nI L = -Λ V (4) 8 π + 1 16 π ∫ d 4 x √ -gF 2 = -β Λ r 3 4 -r 3 3 12 + Q 2 4 β ( 1 r 3 -1 r 4 ) = -3 π 2Λ + πM √ 3 Λ , (88) \nwhere V (4) is the four-volume of the instanton, and β is the period of τ . In the electric case, F 2 = -2 Q 2 /r 4 , and we find that (1) gives \nI L E = -β Λ r 3 4 -r 3 3 12 -Q 2 4 β ( 1 r 3 -1 r 4 ) = -3 π 2Λ . (89) \nTo calculate the additional boundary term in (86), we have to pick a gauge for the Maxwell field. To obtain a unique result, we have to constrain the gauge choice to be regular at both horizons. A suitable gauge choice for the lukewarm solution is \nA = -i Q r 2 τdr. (90) \nIt might seem that this gauge choice involves a discontinuity at the horizons, but in fact it does not. To consider whether there is a discontinuity at the horizon, we should look at the gauge potential in orthonormal coordinates. An orthonormal frame for the metric (53) is \ne 0 = V ( r ) 1 / 2 dt, e 1 = V ( r ) -1 / 2 dr, e 2 = rdθ, e 3 = r sin θdφ, (91) \nand the gauge potential (90) is \nA = -iV ( r ) 1 / 2 Q r 2 τ e 1 , (92) \nwhich vanishes at r = r 3 and r = r 4 . \nTo evaluate the additional boundary term in (86), we take a coordinate system such that the boundary is the surface τ = 0 , β/ 2 in the Euclidean section, and we take the integral in the r direction on the boundary to run from the black hole horizon to the acceleration horizon along τ = 0, and back along τ = β/ 2. The additional term is \n1 4 π ∫ Σ d 3 x √ hF µν n µ A ν = -Q 2 2 β ( 1 r 3 -1 r 4 ) = -πM √ 3 Λ (93) \nand thus the action (86) in the electric case is \nI L el = I L E -1 4 π ∫ Σ d 3 x √ hF µν n µ A ν = -3 π 2Λ + πM √ 3 Λ . (94) \n- \nThe relevant action is thus the same for the electric and magnetic lukewarm instantons. It lies in the range 3 π/ 2Λ ≤ I L 3 π/ 4Λ, as M < √ 3 / Λ / 4. \n≤ √ For the cold solution, F 2 = 2 Q 2 /r 4 in the magnetic solution, and the action (1) is \n≤ - \nI C = β Λ b 3 -ρ 3 12 + Q 2 4 β ( 1 ρ -1 b ) = -π 2 b 2 . (95) \nThere is also an extrinsic curvature boundary term at B ∞ , but this vanishes. In the electric case, F 2 = -2 Q 2 /r 4 , so the action (1) is \nI C E = β Λ b 3 -ρ 3 12 -Q 2 4 β ( 1 ρ -1 b ) = -π 2 b 2 -Q 2 2 β ( 1 ρ -1 b ) . (96) \nA suitable gauge potential, which is regular everywhere on the instanton, is \nA = -iQ ( 1 r -1 b ) dτ. (97) \nThe integral over Σ now consists of two parts; there is an integral over the S 2 × R 1 factor, which we take to be from r = ρ + /epsilon1 to the acceleration horizon at r = b along τ = 0, and back along τ = β/ 2, and an integral over the internal infinity r = ρ + /epsilon1 , which is in the direction of decreasing τ . The additional boundary term in (86) is \n1 4 π ∫ Σ d 3 x √ hF µν n µ A ν = -Q 2 2 β ( 1 ρ -1 b ) , (98) \nand thus (86) is \nI C el = -π 2 b 2 . (99) \nAgain, the action is the same in the electric and magnetic cases. It lies in the range 3 π/ 2Λ ≤ I C π/ 4Λ. \n≤ The actions in the charged Nariai case have already been computed in [7]. In the magnetic case, F 2 = 2 Q 2 /B 2 , and the action (1) is \nI CN = -π B . (100) \nIn the electric case, F 2 = -2 Q 2 /B 2 , so the action (1) is \nI CN E = -π A . (101) \nA suitable gauge potential is \nA = i QB A sin( χ ) ψdχ. (102) \nWe take the boundary to be the surface ψ = 0 , ψ = π , and integrate from the black hole horizon ( χ = π ) to the acceleration horizon ( χ = 0) along ψ = 0, and back along ψ = π , so the additional boundary term in (86) is \n1 4 π ∫ Σ d 3 x √ hF µν n µ A ν = -1 4 π Q 2 B A ∫ ψ sin χ sin θdχdθdφ = -2 πQ 2 B A , (103) \nand thus (86) is \n- \n≤ - \nI CN el = -π A + 2 πQ 2 B A = -π B . (104) \nThe relevant action is the same in the electric and magnetic cases, and it lies in the range -π/ Λ I CN ≤ -π/ 2Λ. \n-≤ -For both metrics which can be constructed in the ultracold case, F 2 = 2Λ in the magnetic solution, so the volume contribution to the action (1) vanishes. Let us consider first the metric (74). Then \n≤ \nI UC 1 = -1 8 π ∫ B ∞ √ hK = -π 4Λ . (105) \nIn the electric solution, F 2 = -2Λ, so (1) gives \nI UC 1 E = -Λ V (4) 4 π -1 8 π ∫ B ∞ √ hK = -πR 2 0 / 4 -π 4Λ (106) \n(the boundary B ∞ in this case is the surface R = R 0 ). One could take the electric gauge potential to be \nA = -i 2 √ Λ R 2 dψ. (107) \nWe define Σ to be the surfaces ψ = 0 , ψ = π , together with the semi-circle at R = R 0 lying between them, and take the integral around the boundary to be from R = R 0 to R = 0 along ψ = 0, back along ψ = π , and around R = R 0 in the direction of decreasing ψ . The additional boundary term in (86) is \nso (86) is \n1 4 π ∫ Σ d 3 x √ hF µν n µ A ν = -πR 2 0 / 4 , (108) \nI UC 1 el = -π 4Λ . (109) \nThis action agrees with the limit of the action of the cold solution as it approaches ultracold. \nIf we consider instead the metric (80), the action vanishes in the magnetic case, I UC 2 = 0, as the extrinsic curvature surface term at x = ± x 0 vanishes as well. In the electric case, F 2 = -2Λ, so if we consider the action for the region between two surfaces γ = ± γ 0 , (1) gives \nI UC 2 E = -Λ V (4) 4 π = -2 x 0 γ 0 . (110) \nOne could take the electric gauge potential to be \nA = -i √ Λ xdγ. (111) \nThe additional boundary term in (86) is then \n1 4 π ∫ Σ d 3 x √ hF µν n µ A ν = -2 x 0 γ 0 , (112) \nso (86) vanishes as well, I UC 2 el = 0. \nOne can use the action of an instanton to approximate the wavefunction for the propagation to some final surface, so the action gives the approximate amplitude at which this process occurs. One can square this to get the rate. The actions we have just calculated for the cold and lukewarm instantons give the rate for pair creation of black holes in a cosmological background by these instantons. The pair creation rate is approximately \nΓ = Ψ 2 ≈ e -2 I , (113) \nin each case, where I is the relevant action. Since de Sitter space has action I de Sitter = -3 π/ 2Λ, which is the lower bound of the action for the cold and \nFigure 3: The action for the various instantons in the cosmological case. The action as a fraction of the action for de Sitter space, I/I de Sitter , is plotted against the dimensionless mass M √ Λ. The curve DU1 represents the cold solutions, DC represents the lukewarm solutions, and NU represents the charged Nariai solutions. The point at D is de Sitter space, N is the Nariai solution, and U1 and U2 represent the actions of the first and second type of ultracold solutions. Note that U does not correspond to one of the ultracold solutions. \n<!-- image --> \nlukewarm instantons, the rate at which black hole pairs are created relative to the rate at which de Sitter space is itself created is less than one. That is, the pair creation of black holes is suppressed. The situation is illustrated in Figure 3. \nSince the instantons are all cosmological solutions, there is no asymptotic region in the Euclidean section (that is, there is no 'point at infinity'). This can be interpreted as meaning that these solutions are closed systems, and thus necessarily have fixed energy. They should therefore be interpreted as a contribution to the microcanonical ensemble, as this is the thermodynamical ensemble at fixed energy. The partition function Z = Ψ 2 should therefore be interpreted as the density of states, and thus the entropy will be just the ln of this partition function, S = ln Z [8, 7]. The contribution to the entropy from the instantons is thus just \nS = -2 I. (114) \nAs we might expect, the entropy turns out to be just a quarter of the total area of the horizons which appear in the instanton (which we denote by A ). In the lukewarm case, there are two horizons, at r = r 3 and r = r 4 , so A / 4 = πr 2 3 + πr 2 4 , but \nso \nand thus \nr 3 , 4 = 1 2 √ 3 Λ ± √ 3 Λ -4 M √ 3 Λ , (115) \nA / 4 = 3 π Λ -2 πM √ 3 Λ , (116) \nS L = -2 I L = A / 4 . (117) \nIn the cold case, only the acceleration horizon at r = b is part of the instanton, and it has area A = 4 πb 2 , so \nS C = -2 I C = πb 2 = A / 4 . (118) \nIn the charged Nariai case, there are again two horizons, which both have area 4 π/B . Thus A / 4 = 2 π/B , and \nS CN = -2 I CN = 2 π/B = A / 4 . (119) \nFor the first type of ultracold solution, (74), the surface R = 0 is interpreted as a Rindler horizon, which has area A = 2 π/ Λ. Thus, \nS UC 1 = -2 I UC 1 = π/ 2Λ = A / 4 . (120) \nFor the other type of ultracold solution, there are no horizons, and the entropy vanishes, S UC 2 = -2 I UC 2 = 0, as we expect. Thus we see that horizons contribute to the gravitational entropy only if they are in the instanton; in particular, extreme black hole horizons make no contribution to the entropy, even if they have non-zero area, as discovered in [3]. \nThus, the usual relation between the entropy and the area of the horizons extends to all these solutions. Just as for the Ernst instantons, the pair creation of extreme black holes is suppressed relative to the pair creation of non-extreme black holes by a factor of e S bh , where S bh is the entropy associated with the black hole horizon.", '6 Conclusions': 'The pair creation of charged black holes by a strong electromagnetic field has been a subject of considerable recent interest. We have seen that charged black holes can also be pair created in a background with a positive cosmological constant. The nature of the instantons describing this pair creation is very similar to that of the instantons describing pair creation in an electromagnetic field. There is a non-extreme and an extreme instanton. As in the electromagnetic case, the pair creation of extreme black holes is suppressed relative to that of non-extreme black holes by a factor of e S bh , where S bh is the entropy associated with the black hole horizon. This is further evidence that e S bh should be regarded as the number of internal states of the black holes. \nThe pair creation rate obtained from the instantons is in fact just e S , where S is the total gravitational entropy of the instanton, that is, a quarter the area of the black hole and the cosmological horizons. Seen from this point of view, black hole pair creation in de Sitter space is suppressed simply because de Sitter space has a higher entropy; that is, the single horizon of de Sitter space has an area larger than the combined area of the horizons in the instantons. This is also similar to the electromagnetic case, where the pair creation rate was e ∆ A / 4+ A bh / 4 in the non-extreme case, with the suppression being due to the fact that the difference in acceleration horizon area ∆ A was negative. \nBecause the conical singularities in the charged C metric with a cosmological constant can only be eliminated when it reduces to the ReissnerNordstrom de Sitter metric, we are fairly confident that the instantons we have described in this paper are the only ones which can be interpreted as representing pair creation of charged black holes in a cosmological background. It might be interesting to see if these solutions could be extended to dilaton gravity with some kind of effective cosmological constant, as the presence of the dilaton field might allow some more possibilities. \nWe have also observed that the Type I instanton discovered in [1], where the conical singularities in the ordinary charged C metric are eliminated by allowing ξ 2 and ξ 3 to coincide, is in fact the Reissner-Nordstrom metric.', '7 Acknowledgements': 'R.B.M. is grateful for the hospitality of D.A.M.T.P., where this work was performed, and for the support of the Natural Sciences and Engineering Research Council of Canada. S.F.R. thanks Raphael Bousso for illuminating discussions, and the Association of Commonwealth Universities and the Natural Sciences and Engineering Research Council of Canada for financial support.', 'References': "- [1] H.F. Dowker, J.P. Gauntlett, D.A. Kastor and J. Traschen, Phys. Rev. D 49 , 2909 (1994).\n- [2] H.F. Dowker, J.P. Gauntlett, S.B. Giddings and G.T. Horowitz, Phys. Rev. D 50 , 2662 (1994).\n- [3] S.W. Hawking, G.T. Horowitz and S.F. Ross, Phys. Rev. D 51 , 4302 (1995).\n- [4] P. Ginsparg and M.J. Perry, Nucl. Phys. B222 , 245 (1983).\n- [5] F. J. Ernst, J. Math. Phys. 17 , 515 (1976).\n- [6] I.J. Romans, Nucl. Phys. B383 , 395 (1992).\n- [7] S.W. Hawking and S.F. Ross 'Duality between electric and magnetic black holes', DAMTP/R-95/8, hep-th/9504019. \n- [8] J.D. Brown and J.W. York, Phys. Rev. D 47 , 1420 (1993); J.D. Brown, J. Creighton and R.B. Mann, Phys. Rev. D 50 6394 (1994).\n- [9] W. Kinnersley and M. Walker, Phys. Rev. D 2 , 1359 (1970).\n- [10] J.F. Plebanski and M. Demianski, Ann. Phys. 98 , 98 (1976).\n- [11] E. Witten, Nucl. Phys. B195 , 481 (1982).\n- [12] F. Mellor and I. Moss, Phys. Lett. B222 , 361 (1989), ibid , Class. Quant. Grav. 8 , 1379 (1989)."}
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1997MNRAS.284..318S
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Capture of stellar mass compact objects by massive black holes in galactic cusps
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1997-01-01
| 6 | 0.46 | 164 |
['black hole physics', 'galaxies nuclei', 'astrophysics']
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[]
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A significant fraction of the stellar population in the cusps around central black holes of galaxies consists of compact remnants of evolved stars, such as white dwarfs, neutron stars and stellar mass black holes. We estimate the rate of capture of compact objects by massive central black holes, assuming that most spiral galaxies have a central black hole of modest mass (~10^6Msolar), and a cuspy spheroid. It is likely that the total capture rate is dominated by nucleated spirals. We estimate the flux of gravitational wave radiation from such coalescences, and the estimated detectable source count for proposed space-based gravitational wave observatories such as LISA. About one event per year should be detectable within 1Gpc, given very conservative estimates of the black hole masses and central galactic densities. We expect 10^2-10^3 detectable sources at lower frequencies (10^-4Hz) `en route' to capture. If stellar mass black holes are ubiquitous, the signal may be dominated by stellar mass black holes coalescing with massive black holes. The rate of white dwarf-white dwarf mergers in the cores of nucleated spirals is estimated at ~10^-6 per year per galaxy.
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| 2 |
https://arxiv.org/pdf/astro-ph/9608093.pdf
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{'No Header': '<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image -->', 'S/. Sigurdsson and M/.J/. Rees /1': 'Institute of Astr onomy/, Madingley R o ad/, Cambridge CB/3 /0HA /1 Institute for A dvanc e d Study/, Princ eton/, NJ /0/8/5/4/0', 'ABSTRA CT': 'A signi/ can t fraction of the stellar p opulation in the cusp around cen tral blac k holes of galaxies consists of compact remnan ts of ev olv ed stars/, suc h as white dw arfs/, neutron stars and stellar mass blac k holes/. W e estimate the rate of capture of compact ob jects b y massiv e cen tral blac k holes/, assuming most spiral galaxies ha v e a cen tral blac k hole of mo dest mass /( / /1/0 /6 M / /)/, and a cusp y spheroid/. It is lik ely that the total capture rate is dominated b y n ucleated spirals/. W e estimate the / ux of gra vitational w a v e radiation from suc h coalescences/, and the estimated detectable source coun t for prop osed space/{based gra vitational w a v e observ atories suc h as LISA/. Ab out one ev en t p er y ear should b e detectable within /1 Gp c/, giv en v ery conserv ativ e estimates of the blac k hole masses and cen tral galactic densities/. W e exp ect /1/0 /2 /{/1/0 /3 detectable sources at lo w er frequencies /(/1/0 /BnZr /4 Hz/) /\\en route/" to capture/. If stellar mass blac k holes are ubiquitous/, the signal ma y b e dominated b y stellar mass blac k holes coalescing with massiv e blac k holes/. The rate of white dw arf/{white dw arf mergers in the cores of n ucleated spirals is estimated at / /1/0 /BnZr /6 p er y ear p er galaxy /. \nKey w ords/: galaxies/{n uclei/, blac k holes/: stellar/{mergers', '/1 INTR ODUCTION': "There is increasingly strong evidence that massiv e blac k holes are found in the cen ters of galaxies /(see eg/. Kormendy /& Ric hstone /1/9/9/5/, T remaine /1/9/9/5/, Rees /1/9/9/0/)/. Observ a/tional evidence/, and theoretical considerations/, indicate that the masses/, M h /, range from under /1/0 /6 M / to o v er /1/0 /9 M / /. Blac k holes are though t to form during/, or p ossibly b efore the formation of the host galaxy /. Early infall of lo w angu/lar momen tum material/, and angular momen tum transp ort in the disk of gas surrounding the proto/{blac k hole fuels the early stages of blac k hole gro wth/, with accretion rates /> / /1 M / y /BnZr /1 inferred /(see eg/. Rees /1/9/9/0/)/. The lifetime of the resultan t quasar is uncertain/, it is p ossible that less than /1/% of galaxies host quasars/, with lifetimes /> / /1/0 /9 y /, but more lik ely that most galaxies undergo shorter p erio ds of activit y /(see eg/. Haehnelt and Rees /1/9/9/3/)/. \nIf the QSO dut y cycle is lo w/, then most galaxies m ust ha v e undergone mo derately short p erio ds of accretion on to a cen tral blac k hole/, in order to accoun t for the total QSO n um b ers/, and a t ypical galaxy has a M h /> / /1/0 /7 M / cen/tral blac k hole/. If the QSO//A GN dut y cycle is high/, then remnan ts in activ e galaxies are mostly sup er/{massiv e and as man y as /> / /9/9/% of /(non/{activ e/) galaxies will ha v e mo der/ate mass /( M h / /1/0 /6 M / /) blac k holes that did not undergo prolonged strong accretion episo des/. It is in teresting to note that for our o wn galaxy and the nearb y M/3/2 dw arf elliptical /, the b est estimates of the mass of an y cen tral blac k hole is /< / /3 /-/1/0 /6 M / /(Bender et al /1/9/9/6/, v an der Marel et al /1/9/9/6/, \nEc k art and Genzel /1/9/9/6/)/. The true blac k hole mass function probably lies b et w een the extremes men tioned/, with a tail of v ery high mass blac k holes in a few galaxies/, and some unkno wn distribution of masses in normal galaxies/. A useful guess for the blac k hole mass function assumes the mass of a cen tral blac k hole is some fraction of the mass of the stellar spheroid/, M h /= f g L /, where f g / /1/0 /BnZr /2 /: /5 M / /= L / /, and L is the galaxy luminosit y /(T remaine /1/9/9/5/)/. W e assume for no w that M h // L and a Sc hecter /(/1/9/7/6/) luminosit y function \n/ /( L /) /= / /0 / L L / / / e /BnZr /( L/=L / /) dL L / /; /(/1/) \nwhere L / /= /1 /: /8 /-/1/0 /1/0 L / /, / /0 /= /0 /: /0/0/8 Mp c /BnZr /3 and / /= /BnZr /1 /: /0/7 /(T remaine /1/9/9/5/, Efstathiou et al/. /1/9/8/8/)/. \nIn general a dense cusp of stars /(and p ossibly gas/) forms around an y cen tral mass in a galaxy /(P eebles /1/9/7/2/, Y oung /1/9/8/0/)/. Stars in these cusps will o ccasionally come close enough to the blac k hole to collide with eac h other or b e sw allo w ed b y the blac k hole /(see eg F rank and Rees /1/9/7/6/, Hills /1/9/7/5/)/. F or blac k holes with masses near the lo w end of the range exp ected /( M h / /1/0 /6 /BnZr /7 M / /) stars and /(sub/)gian ts are tidally disrupted if they come close to the blac k hole/; more massiv e blac k holes ma y sw allo w stars whole/. W e ex/p ect / /1/0/% of stars in an ev olv ed p opulation/, suc h as is observ ed in elliptical galaxies and the bulges of spiral galax/ies to b e white dw arfs/(WDs/)/, another /< / /1/% of stars ma y b e neutron stars/(NSs/) or ev en lo w mass /(/7 /BnZr /1/0/0 M / /) blac k holes/(LMBHs/)/, dep ending on the initial mass function and fate of ev olv ed massiv e stars/, whic h is uncertain/. These \n/2 \nS/. Sigur dsson /& M/.J/. R e es \nev olv ed stars will also encoun ter the cen tral blac k hole/, but w on/'t b e disrupted/, and ma y b e promising sources for gra v/itational w a v es detectable b y prop osed detectors suc h as LISA /(Danzmann et al/. /1/9/9/3/, Hough et al/. /1/9/9/5/)/.", '/1/./1 Stellar encoun ters with cen tral blac k holes': 'A star/, radius r / /, mass m / /, w ould b e tidally disrupted b y a blac k hole mass M h within a radius r T / /2 /-/( M h /=m / /) /1 /= /3 r / /. /(Rees /1/9/8/8/, Ev ans and Ko c hanek /1/9/8/9/, F rolo v et al/. /1/9/9/4/)/. F or a main sequence star of solar mass/, encoun tering a /1/0 /6 M /6 M / blac k hole/, r T / /1 /: /4 /-/1/0 /1/3 M /1 /= /3 /6 cm/( / /5/0 r S M /BnZr /2 /= /3 /6 /. In general the cen tral region of a galaxy will con tain b oth main sequence and ev olv ed stars/, with a range of masses and radii/. W e need to assume some global initial mass function for the stars/, here tak en to b e the Salp eter mass function/, dN / /=dm / /= m /BnZr /1 /BnZr x / / /, where x / /= /1 /: /3/5/. A t ypical galaxy will con tain an ev olv ed p opulation /, with a turno/ mass for the main sequence of /0 /: /7 /BnZr /1 /: /0M / /, plus p ossibly an additional y ounger p opula/tion/. The ev olv ed p opulation will t ypically consist of ab out /0/./2/% neutron stars of /1 /: /4M / /, ab out /7/% white dw arfs/, with masses from /0 /: /5/{/1 /: /3M / /, and p ossibly ab out /0/./0/3/% few M / blac k holes/. As w e discuss later/, ho w ev er/, dynamical e/ ects ma y alter these prop ortions within the cen tral cusp/. \nA degenerate/, dark/, compact remnan t of the same mass as a main sequence star has a m uc h smaller radius/. F or white dw arfs/, r W D / /0 /: /0/1 R / /, for neutron stars/, r N S / /1/0 /6 cm/, m uc h less than r / /. Consequen tly the white dw arf tidal ra/dius is smaller than that for a main sequence star of the same mass b y appro ximately r W D /=r / /. Since the Sc h w arzsc hild radius of a blac k hole/, r S /= /3 /-/1/0 /1/1 M h /= /(/1/0 /6 M / /)cm/, in/creases linearly with M h /, w e see that for a blac k hole with M h /> / /1/0 /6 M / /, r S /> r T for white dw arfs/. F or the v alues of M h considered here/, neutron stars are not sub ject to tidal disruption/; nor/, ob viously /, are an y lo w mass blac k holes/, whic h ma y also encoun ter the cen tral blac k hole/. Compact stellar remnan ts can orbit around galactic blac k holes inside the tidal disruption radius for stars/. \nThe orbital p erio d for a solar t yp e star around blac k hole at r T is P T /= /1/0 /4 s/, indep enden t of M h /. The c harac/teristic frequency /, f c /, of gra vitational radiation asso ciated with the orbit of compact remnan ts ab out the blac k hole inside r T is then /1/0 /BnZr /4 /{/1/0 /BnZr /2 Hz/. Gra vitational radiation of this frequency is not readily detectable b y Earth b ound de/tectors but is w ell matc hed to the exp ected sensitivit y of space based detectors /(Thorne /1/9/8/7/, /1/9/9/5/, Danzmann et al/. /1/9/9/3/, Hough et al/. /1/9/9/5/, Haehnelt /1/9/9/4/)/. The sensitivit y of these detectors cuts o/ b elo w / /1/0 /BnZr /3 Hz/, and the gra vita/tional radiation rate b y a star in the innermost stable orbit around a massiv e blac k hole scales as M /BnZr /2 h /. F or this reason w e are primarily in terested in mo derate mass cen tral blac k holes/, /0 /: /5 /< / M /6 /< / /5/. \n/ /6 / The corresp onding c haracteristic amplitude of the gra v/itational w a v es/, h c /, for a source at distance d Gpc /= d/= /1 Gp c /, M h / m / /, is \nh c /= /3 /: /7 /-/1/0 /BnZr /2/4 /1 d Gpc / m / M / / M /2 /= /3 /6 / /1/0 /4 P / /2 /= /3 g /1 /= /2 /; /(/2/) \nwhere P is the orbital p erio d in seconds and g is a geometric factor of order unit y /(Thorne /1/9/8/7/)/. F or d /> / /1 Gp c a cor/- \nafer \nrection for cosmological curv ature is necessary /. The exact w a v eforms for eccen tric orbits and resultan t detectabilit y of the radiation are more complicated /(see for example Cutler and Flannagan /1/9/9/5/, Ry an /1/9/9/5/, Shibata /1/9/9/4/, Junk er and Sc h/ /1/9/9/2/)/. \nWhat/, then is the lik ely rate of suc h ev en ts/? Hils and Bender /(/1/9/9/5/) recen tly considered the rate of capture in M/3/2/{lik e systems with isothermal cusps due to the di/ u/sion of compact stars in to the blac k hole/, the di/ usion b eing dominated in teraction with main sequence stars on more lo osely b ound orbits/. This w ork extends the consideration to non/{isothermal systems and includes the con tribution of b oth di/ usion and large/{angle scattering to the capture rate/, and the m utual in teraction of compact stars in tigh tly b ound orbits/.', '/2 CUSPS AND SW ALLO WING RA TES': 'Consider a massiv e blac k hole in the cen ter of some galaxy /. In general/, the blac k hole will b e surrounded b y some /(ev olv ed/) stellar p opulation with a range of masses corre/sp onding to some initial mass function and an asso ciated p opulation of ev olv ed remnan ts stars/, white dw arfs/, neu/tron stars and lo w mass blac k holes/. The stars will ha v e some underlying densit y pro/ le/, with the densit y / /( r /) t ypi/cally w ell appro ximated b y a /(brok en/) p o w er la w /(Kormendy and Ric hstone /1/9/9/5/) with some /\\break radius/"/, r b /. F or the galaxies whic h are the b est candidates for harb ouring blac k holes of mass M h / M /6 /, r b is observ ed to b e a few parsecs/. W e assume for no w that the stellar distribution is spherical and the v elo cit y distribution isotropic/. \nW e de/ ne a radius of in/ uence of the blac k hole/, r h /= GM h /=/ /2 c /, where / c is the one dimensional disp ersion for r /> / r h /. F or most systems of in terest the disp ersion is ap/pro ximately constan t for radii larger than but comparable to r h /, and r h is w ell de/ ned/. F or the systems w e will b e in terested in/, M h / M /6 and / c / /1/5/0 km s /BnZr /1 /. De/ ning / /1/6/6 /= / c /= /1/6/6 km s /BnZr /1 /, it is useful to scale \nr h /= /1 M /6 / /2 /1/6/6 p c /: /(/3/) \nW e de/ ne a c haracteristic dynamical time scale/, t dy n /= r h /=/ c /, the time scale for a star with c haracteristic /1/{D v elo cit y to cross r h /; for systems considered here t dy n /= /6/0/0/0/( r h /= p c /) /=/ /1/6/6 /) y ears/. Inside r h the densit y pro/ le is mo di/ ed b y the presence of the blac k hole /(P eebles /1/9/7/2/, Bahcall and W olf /1/9/7/6/, /1/9/7/7/, Cohn and Kulsrud /1/9/7/8/, Y oung /1/9/8/0/, Shapiro /1/9/8/5/, Murph y et al/. /1/9/9/1/, Quinlan et al/. /1/9/9/5/)/. \nA stellar p opulation /\\relaxes/" on some c haracteristic time scale/, t R /= p /2 / /3 c /=/ G /2 m / / log /(/1 /= /2 N / /)/, where N / is the n um b er of stars /(see eg/. Binney and T remaine /1/9/8/7/)/. Scaling to a normalised densit y / /6 /= / /= /1/0 /6 p c /BnZr /3 /, w e / nd t R /= /1/0 /1/0 / /3 /1/6/6 /=/ /6 y ears/. Note that t R /< / t H /(/= /1 /: /5 /-/1/0 /1/0 y ears/) is p ossible for reasonable v alues of / c /, / /. \nc If the lifetime of the stellar p opulation in the cen ter is longer than few times t R /, then the stellar p opulation is relaxed/; in particular mass segregation o ccurs due to dy/namical friction/. The lifetime of the p opulation of stars at the cen ter of the galaxy is ob viously /< / t H /, indeed it ma y b e / t H if there has b een recen t star formation with asso ci/ated mass loss/, or if the blac k hole arriv ed or formed in the', '/2/./1 Cusps': "A self/{consisten t cusp of stars forms around a cen tral blac k hole with some densit y pro/ le / /( r /< r h /) // r /BnZr C /(P eebles /1/9/7/2/, Bahcall and W olf /1/9/7/6/, /1/9/7/7/, Y oung /1/9/8/0/, Shapiro /1/9/8/5/, Quinlan et al/. /1/9/9/5/)/. In general C is a function of r /, with di/ eren t pro cesses mo difying the lo cal slop e at di/ eren t radii and densities/. The blac k hole ma y also induce a small anisotrop y in the v elo cit y distributio n /(Go o dman and Bin/ney /1/9/8/4/, Quinlan et al/. /1/9/9/5/)/. The cusp extends to some inner radius/, r in /, where the star are e/ ectiv ely destro y ed or sw allo w ed/. \nIt is useful to parametrise the cusp slop e as C /( r /) /= /3 /= /2 /+ p /( r /)/. With t R // / /( r /) /3 / /BnZr /1 inside r h /, / /( r /) // r /BnZr /1 /= /2 and hence t R // r p /. A blac k hole gro wing adiabatical ly in a / at isothermal core/, with t R / t H /, induces a cusp with p /( r /< r h /) /= /0 /(Y oung /1/9/8/0/, Quinlan /1/9/9/5/)/. This has the in teresting prop ert y that t R is indep enden t of radius inside r h /. F or ph ysical cores C / /2 /: /5 and hence p / /1 for plausible cusps whether relaxed or not /(see Quinlan et al/. /1/9/9/5/, Sigurdsson et al/. /1/9/9/5/, for discussion/)/. F or a relaxed p opulation of equal mass stars in the cen tral region /( t R /( r h /) / min f t H /; t f or m g /)/, p /( r /) /= /1 /= /4 and the relaxation time decreases slo wly with r /, while for a relaxed cusp with a range of stellar masses/, / p / /0 /: /3/, with steep er cusps for the more massiv e stars /(Bahcall and W olf /1/9/7/6/, /1/9/7/7/, Murph y et al/. /1/9/9/1/)/. \nTw o further quan tities are fundamen tal in determin/ing the structure of the cusp around the blac k hole/. The collisi on radius/, r coll /= /7 /-/1/0 /1/6 M /6 cm for main sequence stars /(smaller b y r W D /=r / for white dw arfs/, e/ ectiv ely zero for our purp oses for neutron stars and stellar mass blac k holes/)/, is the radius at whic h star/{star encoun ters cannot lead to large angle elastic scattering as the c haracteristic encoun ter v elo cities exceed the surface escap e v elo cities of the stars /(F rank and Rees /1/9/7/6/)/. Inside r coll t w o b o dy re/laxation is b y de/ nition ine/ ectiv e and t R b ecomes large for the relev an t p opulation/. Note that r coll / r T for b oth stars and white dw arfs for the blac k hole masses of in terest here/. Another quan tit y is the blac k hole w andering radius/, r w /, whic h measures the ro ot mean square displacemen t of the blac k hole from the galaxy/'s cen ter of mass due to the discreteness of the p oten tial of the galaxy /. F or a blac k hole in an isothermal stellar core r w / p m / /= M h r b /(Bahcall and W olf /1/9/7/6/)/. F or other core pro/ les the w andering radius is less w ell de/ ned /(Quinlan priv ate comm unication/)/. As/suming the graininess due to the stellar mass distribution dominates r w /, then for the densit y pro/ les and blac k hole masses of in terest here/, r h / r w / r T /. Consequen tly /, de/pletion of the main sequence stellar p opulation due to tidal disruption is not limited b y loss/{cone di/ usion in to r T as the blac k hole w anders on dynamical time scales short compared to t R /. The short p erio d random w alk of the cen tral blac k hole do es not a/ ect our rate estimates b ecause the rate es/timates are sto c hastic/, a v eraged o v er time and the ensem ble of blac k holes/. The short time scale / uctuations that mo v e the blac k hole a w a y from a compact remnan t star that w ould otherwise ha v e en tered the gra vitational w a v e loss/{cone/, are balanced on a v erage b y cen tral blac k holes random w alking in to the path of a compact remnan t star that w ould other/- \nwise not ha v e en tered the gra vitational radiation loss/{cone/. \nAnother critical radius is set b y the radius at whic h mergers of main sequence stars with eac h other b ecome im/p ortan t/, r m /. The rate for star/{star mergers p er star is sim/ply \nR cc /= / r /2 / / /( r /) / /( r /) m / /; /(/5/) \nfor r m / r h /, assuming / /> / v esc /, where v esc is the es/cap e v elo cit y at the surface of the star /(true for r / r h /)/. F or white dw arf/{white dw arf and white dw arf/{neutron star merger gra vitational fo cusing increases R cc b y a factor /(/1 /+ /2 Gm / /=r / / /( r /) /2 /)/. In our units/, \nR cc /= /1/0 /BnZr /3 / /6 / /1/6/6 / r / R / / /2 / M / m / / / r h r / /BnZr /3 /BnZr p p er t H /: /(/6/) \nBy de/ nition/, depletion of stars through merger b ecomes imp ortan t when R cc / t /BnZr /1 H /, pro vided t R /> / t H /, else deple/tion only b ecomes e/ ectiv e at r coll /. F or solar mass main sequence stars/, r m / /1/0 /BnZr /1 r h / r coll /, as exp ected/. In/side min f r m /; r coll g the cusp induced b y the blac k hole is / attened b y the depletion of main sequence stars due to mergers/, with the main sequence densit y pro/ le / attening to / /( r /< r m /) // r /BnZr D /, where D / /0 /{ /1 /= /2 /( p /= /BnZr /1 /{ /BnZr /3 /= /2/)/. /(Ligh tman and Shapiro /1/9/7/7/, Murph y et al/. /1/9/9/1/, Rauc h /1/9/9/5/)/. \nWhile the main sequence stellar densit y inside r m is / at/, this radius has no sp ecial signi/ cance for the compact remnan ts whic h w ould main tain a steep pro/ le inside r m /(see eg Murph y et al/. /1/9/9/1/)/, with \n/ cr /( r /< r m /) /= f cr / /( r m /) / r m r / /BnZr C /; /(/7/) \nwhere f cr is the fractional densit y of white dw arfs /(or neu/tron stars or stellar mass blac k holes/) at r m /, corrected for mass segregation if applicable/. Compact remnan ts stars also merge with main sequence stars/, as w ell as eac h other/, the rate for merger with main sequence stars is smaller than the star/{star merger rate b y a factor f cr /, and sim ulation s of suc h high sp eed mergers suggest in most cases the compact star will remain after the merger /(Shara and Regev /1/9/8/6/, Benz et al/. /1/9/8/9/, Ru/ ert /1/9/9/2/)/. Inside r coll /( M S /) the t w o b o dy re/laxation time scale for white dw arfs increases sharply as the e/ ectiv e densit y of b o dies a v ailable for large angle scattering decreases b y f cr /.", '/2/./2/./1 The in/ uenc e of the c entr al black hole': 'Consider no w the dynamics of the stars in the cusp/. The stellar orbits are dominated b y the gra vit y of the cen tral blac k hole/, but are p erturb ed b y the m utual in teractions of the stars/. Stars whic h v en ture to o close to the blac k hole are either tidally disrupted/, in the case of main sequence stars/, or sw allo w ed whole/, in the case of compact stellar remnan ts/. In the latter case/, the / ux of compact remnan ts in to the blac k hole is due to stars in the cusp scattered in to orbits that either plunge them directly in to the blac k hole/, or suc h that the p erib othron is small enough for gra vitational radiation to shrink the orbit more rapidly than in teractions with other ob jects either scatter the star a w a y from the blac k', '/4 S/. Sigur dsson /& M/.J/. R e es': 'hole again/, or put it on an orbit that plunges straigh t in to the blac k hole/. \nT o / rst non/{zero order/, gra vitational radiation leads to a decrease in orbital energy /, E /, and angular momen tum/, L /, with \ndE dt /= /BnZr /3/2 /5 G /4 c /5 M /3 h m /2 / r /5 p f /0 /( e /) /; /(/8/) \nand \ndL dt /= /3/2 /5 G /4 c /5 M /3 h m /2 / r /5 p g /0 /( e /) /; /(/9/) \nwhere r p /= a /(/1 /BnZr e /) is the p erib othron of an orbit with semi/{ ma jor axis a /, eccen tricit y e /, f /0 /( e /) /= /(/1 /BnZr e /2 /) /3 /= /2 /(/1 /+ /7/3 /= /2/4 e /2 /+ /3/7 /= /9/6 e /4 /) /= /(/1 /+ e /) /5 and g /0 /( e /) /= /(/1 /+ /7 /= /8 e /2 /)/(/1 /BnZr e /) /3 /= /(/1 /+ e /) /2 /. Note that f /0 is less sensitiv e than g /0 to the eccen tricit y for e / /1 while b oth E and L are equally sensitiv e to r p /. \np F rank and Rees /(/1/9/7/6/) considered the di/ usion of main sequence stars in cusps around cen tral blac k holes in galaxies with isothermal cores/. W e follo w their argumen t for the scat/tering of compact stellar remnan ts for the full range of adi/abatic and relaxed stellar cusps around cen tral blac k holes/. W e consider stars inside the cusp at radii r S / r /< / r h /. In order for the stars to b e sw allo w ed b y the cen tral blac k hole/, their orbits m ust b e within some critical radius r c /( r /)/. It is useful to de/ ne a /\\loss/{cone/"/, / /( r /) /= p /2 r min /= /3 r /, where r min is the p erib othron distance for the star at r /. As the stars orbit ab out the blac k hole/, the orbits are scattered b y the inhomogenous p oten tial they mo v e in/; it is useful to con/sider t w o regimes/; where the scattering angle is small com/pared to / /, whic h w e refer to as /\\di/ usion/"/, and scattering where the scattering angle is large compared to / /, whic h w e refer to as /\\kic ks/"/. The scattering in the resp ectiv e regimes can b e though t of as b eing due to the /\\P oisson noise/" in the p oten tial due to the discrete n um b er of stars for di/ usion/, and as elastic scattering o/ individ ual stars in the case of /\\kic ks/"/. In the absence of a cen tral blac k hole/, the scattering in and out of the loss/{cone w ould b e symmetric/, with the / ux in to the loss/{cone balanced b y the / ux out/. In the pres/ence of a blac k hole there is an additional source of orbital ev olution/: the secular deca y of the lo w angular momen tum orbits due to gra vitational radiation/. Hence there is a net loss of stars to the blac k hole/. \nThe ratios of the /\\scattering time scale/"/, t scat and the time scale for deca y through gra vitational radiation/, t GW /, to the orbital time scale/, t or b /( r /)/, can b e written \nt scat t or b /= / M h m / / /2 /1 N / /( r /) /: /(/1/0/) \nwhere N / /( r /) // r /3 /= /2 /BnZr p /( r /) is the n um b er of stars in terior to r /. The time scale for deca y through gra vitational radiation is set b y the energy radiation rate/, giv en b y \nt GW /= /6/4 /5 G /3 M /2 h m / c /5 a /4 f /( e /) /; /(/1/1/) \nwhere a is the semi/{ma jor axis of the stars orbit ab out the blac k hole/, and f /( e /) /= /(/1 /+ /7/3 /= /2/4 e /2 /+ /3/7 /= /9/6 e /4 /)/(/1 /+ e /) /BnZr /7 /= /2 /(/1 /BnZr e /) /BnZr /7 /= /2 /. F or the highly radial orbits w e/\'re in terested in/, e / /1/, giv en / /= p /2 /= /3 p /(/1 /BnZr e /) /= w e / nd \nt GW t or b /= /2/4 p /2 /8/5 / / /3 /2 / /7 /= /2 M h m / / r r S / /5 /= /2 / /7 /: /(/1/2/) \nt scat is the time scale for scattering out due to /\\kic ks/"/; the corresp onding di/ usion time scale is shorter b y a fac/tor r min /=r /= /3 / /2 /( r /) /= /2/. In order for a star to b e scattered to a small enough p erib othron that it will b e sw allo w ed due to gra vitational radiation/, w e require t scat / t GW /, or that / / / cr it /. Hence for /\\kic ks/"/, w e / nd \n/ cr it /= r /3 /2 / /8/5 / /2/4 p /2 / /1 /= /7 / M h m / N / /( r /) / /1 /= /7 / r r S / /BnZr /5 /= /1/4 /; /(/1/3/) \nand for di/ usion \n/ cr it /= r /3 /2 / /8/5 / /2/4 p /2 / /1 /= /5 / M h m / N / /( r /) / /1 /= /5 / r r S / /BnZr /1 /= /2 /: /(/1/4/) \nF or radii suc h that / cr it /( r /) / /1 the p o w er/{la w cusp of stars is no longer presen t as stars are sw allo w ed b y the cen tral blac k hole/. \nThe rate/, R s /, at whic h the stars are sw allo w ed/, either b y scattering straigh t in to the blac k hole/, or b y gradual shrink/age of their orbits is giv en b y \nR s /= N / /( r /) / /2 cr it t scat /; /(/1/5/) \nand w e can solv e for R s for a c hoice of cusp parameters and sum o v er the galaxy and blac k hole mass function for an estimate of the total rate/. Substituting for / cr it \nR s /( / /) /= /3 /2 C /2 /=/ / N / /( r /) /2 /BnZr /2 /=/ / m / M h / /2 /BnZr /2 /=/ / r r S / /BnZr /5 /=/ /1 P /; /(/1/6/) \nwhere / /= /5 for stars di/ using in the loss/{cone/, and / /= /7 for stars undergoing large angle scattering out of the loss/{ cone/, and C /2 /=/ / /= /3 /= /2/(/8/5 / /= /2/4 p /2/) /2 /= /7 / /2 /: /7 for / /= /7/, and C /2 /=/ / /= /3 /= /2/(/8/5 / /= /2/4 p /2 /) /2 /= /5 / /3 /: /4 for / /= /5/.', '/2/./2/./2 Single p ower law cusps': 'If the cusp densit y is a constan t p o w er la w/, the n um b er of stars/, N / /( r /)/, is giv en simply b y \nN / /( r /) /= /4 / Z r /0 / /( r /0 /) r /0 /2 dr /0 /; /(/1/7/) \nnormalisin g to / /( r h /) /= / /6 /= /1/0 /6 M / p c /BnZr /3 w e / nd \nN / /( r /) /= /1 /: /2 /-/1/0 /7 /3 /= /2 /BnZr p / /6 m / /= M / / r h /1 p c / /3 / r r h / /3 /= /2 /BnZr p /: /(/1/8/) \nAssume w e ha v e a single p o w er la w cusp/, that p is inde/p enden t of r for r /< r b /. It is useful to de/ ne r cr it /, where / cr it /( r cr it /) /= /1/, then \nr cr it /= C /1 /4/+ p / / M h m / / /1 /4/+ p /-/ /3 /= /2 /BnZr p /1 /: /2 /-/1/0 /7 / /6 / /1 /4/+ p / r h r S / /3 /BnZr /2 p /8/+/2 p r S /: /(/1/9/) \nA plot of r cr it vs/. p is sho wn in / gure /1/, for M /6 /; / /6 /= /1/, and r h /= /0 /: /5 /; /1 /: /0 and /5 /: /0 p c resp ectiv ely /. The curv es are sho wn for / /= /7/; r cr it is /2/5/% larger for / /= /5/. \nFigure /1/. r cr it vs p for single p o w er la w cusps for di/ eren t r h /. \nThen r cr it / /6/0/0 r S for p /= /0/, and r cr it / r S for p /= /1/, m uc h less than r m /. Large angle scattering b ecomes ine/ ec/tiv e for white dw arfs at r coll / /2/0/0/0 r S /> r cr it /, but neutron stars and stellar mass blac k holes ma y undergo large angle scattering to r / r S /. \nS W e consider the rate of capture for / /= /5 /; /7 separately /, still assuming a single p o w er la w cusp comp osed purely of compact remnan ts/. Substituting in equation /1/7/, w e can write \nR s /( / /; r /) /= F /( p/; M h /) / r r S / / /(/2/0/) \nwhere / /= /3 /BnZr /4 p /2 /BnZr /8 /BnZr /2 p / /; /(/2/1/) \nand \nF /( p/; M h /) /= /1/0 /BnZr /4 C /2 /=/ / A /( p /) /2 /BnZr /2 /=/ M /BnZr /3 /= /2/+/2 /=/ /6 /-/ r S r T / /BnZr /3 /= /2 / r S r h / /(/3 /BnZr /2 p /)/(/1 /BnZr /1 /=/ /) /; /(/2/2/) \nand \nA /( p /) /= /1 /: /2 /-/1/0 /7 /3 /= /2 /BnZr p / /6 m / /= M / / r h /1 p c / /3 /: /(/2/3/) \nT able /1 sho ws the resultan t r cr it and capture rates for di/ eren t slop e cusps/. \nThe di/ usion rate decreases with r for all p /. The large angle scattering rate increases with r for p /< /5 /= /2/4/. Ho w/ev er r cr it is m uc h smaller than r coll for white dw arfs for the steep er cusps while for / at cusps the di/ usion rate is rela/tiv ely insensitiv e to r /=r cr it /. Note the rates sho wn in the last column are not the true rates as r cr it /< r coll in all cases and large angle scattering is ine/ ectiv e at these radii for white dw arfs/. Figure /2 sho ws the capture rates for single p o w er la w cusps comp osed purely of compact remnan ts/, for some di/ eren t p /, neglecting r coll /. \ncoll The fraction of stars that plunges directly in to the blac k hole without gradual inspiral through gra vitational radia/tion is R s /( / /= /5/) /= /( R s /( / /= /5/) /+ R s /( / /= /7/)/)/. F or single p o w er la w cusps/, this can b e a small fraction/; for more re/alistic cusps w e exp ect /1 /= /3/{/1 /= /2 the stars to plunge rapidly \nT able /1/. Scaling of scattering rates/. Assuming / /( r h /) /= /1/0 /6 M / p c /BnZr /3 /, M h /= M /6 and / /= /1/6/6 km s /BnZr /1 /, r coll /=r S /= /2/0/0/0/, R s /( r /) // /( r /=r S /) / /, / /= /(/3 /BnZr /4 p /) /= /2 /BnZr /(/8 /BnZr /2 p /) /=/ /.Figure /2/. Plot of R s /( r /) for di/ eren t p /, assuming single p o w er la w cusps comp osed en tirely of compact remnan ts and ignoring existence of r coll /. The solid line is for / /= /7 /(large angle scatter/ing/) and the dotted line is for / /= /5 /(di/ usion/)/. \n| r cr it | p | / | t R /( r cr it /) /= y | R s /( r cr it /) y /BnZr /1 |\n|-----------|------------|--------------------|--------------------------|--------------------------------|\n| / /= /5 | | | | |\n| /6/0/7 | /0 | /BnZr /1 /= /1/0 | /1/0 /1/0 | /7 /: /3 / /1/0 /BnZr /1/4 |\n| /1/5/4 | /1 /= /4 | /BnZr /1 /= /2 | /6 / /1/0 /8 | /5 /: /2 / /1/0 /BnZr /1/1 |\n| /4/5 | /1 /= /2 | /BnZr /9 /= /1/0 | /2 / /1/0 /7 | /1 /: /1 / /1/0 /BnZr /7 |\n| /5 /: /4 | /1 | /BnZr /1/7 /= /1/0 | /5 / /1/0 /3 | /7 |\n| / /= /7 | | | | |\n| /6/0/7 | /0 | /+/5 /= /1/4 | /1/0 /1/0 | /1 /: /4 / /1/0 /BnZr /1/2 |\n| /1/9/2 | /5 /= /2/4 | /0 | /1/0 /9 | /1 /: /6 / /1/0 /BnZr /1/0 |\n| /1/5/4 | /1 /= /4 | /BnZr /1 /= /1/4 | /6 / /1/0 /8 | /4 /: /5 / /1/0 /BnZr /1/0 |\n| /4/5 | /1 /= /2 | /BnZr /1 /= /2 | /2 / /1/0 /7 | /4 /: /8 / /1/0 /BnZr /7 |\n| /1/5 | /0 /: /7/5 | /BnZr /1/3 /= /1/4 | /4 / /1/0 /5 | /1 /: /5 / /1/0 /BnZr /3 |\n| /5 /: /4 | /1 | /BnZr /1/9 /= /1/4 | /5 / /1/0 /3 | /1/4 | \nin to the blac k hole/, with the remainder undergoing a more gradual inspiral with the orbit eccen tricit y decreasing/. Most stars plunge from r / r m / r cr it and en ter r /< r cr it with eccen tricities of / /0 /: /9/9/9/. \nF or detection of gra vitational radiation/, the time the compact ob jects sp end at orbital p erio ds P /< / P /4 /(/= P /= /1/0 /4 s/) is critical/. The time to deca y for highly eccen tric orbits/, using the quadrup ole appro ximation/, is w ell appro ximated b y \n/ GW /= /1/0 /5 /(/1 /BnZr e /2 /) /7 /= /2 P /8 /= /3 /4 y /(/2/4/) \nfor M /6 /= /1/, m / /= M / /, note also / GW // M /BnZr /2 /= /3 h /; m /BnZr /1 / /(see eg/. Ra jagopal and Romani/, /1/9/9/5/)/. F or e /= /0 /: /9/9 at r /= /2/0/0/0 r S the time to inspiral is / /1/0 /4 y ears/, for e /= /0 /: /9/9/9 and r /= /4/0/0/0 r S the time to inspiral is only /3/0 y ears and the orbit has no time to circularise substan tiall y b efore reac hing \nS/. Sigur dsson /& M/.J/. R e es \np erio ds /< / P /4 /. F or steep /(large p /) cusps/, where the rate is dominated b y stars near r coll /, di/ usion will random w alk e to somewhat lo w er v alues for a signi/ can t fraction of the stars/, as discussed ab o v e/. The eccen tricit y /, e f /, as the star approac hes capture is smaller with/, /(/1 /BnZr e f /) / /4 /-/(/1 /BnZr e i /)/, where e i is the initial orbital eccen tricit y after scattering/, and the system ma y sp end times of order /1/0 /3 y ears at p e/rio ds / P /4 /. As discussed in a recen t pap er b y Rauc h and T remaine /(/1/9/9/6/)/, resonan t e/ ects ma y enhance the relax/ation rate of angular momen tum in Keplerian p oten tials/; ho w ev er/, this do es not app ear to b e e/ ectiv e for the orbital parameters of concern here/, b ecause the relativisti c preces/sion of the relev an t orbits /(with small p erib othrons/) is fast enough to destro y the resonances/. Eccen tricit y ev olution inside the loss/{cone is therefore small/. \nStellar mass blac k holes ma y con tribute strongly to the total observ able rate/. They spiral in an order of magnitude faster than white dw arfs and neutron stars and are observ/able for corresp ondingl y shorter times at these frequencies/. On the other hand/, h c is an order of magnitude larger for the lo w mass blac k holes than for white dw arfs/, and the v olume for detection is / /1/0 /3 times larger/. If stellar mass blac k holes form in p opulation I I in signi/ can t n um b ers/, /( f B H / /1/0 /BnZr /4 /)/, they will pro vide a strong c haracteristic signal for LISA /(see also P olnarev /& Rees /1/9/9/4/)/. \nIn practice cusps do not ha v e simple single p o w er la w densit y pro/ les as w e discussed ab o v e/, nor are they com/p osed solely from compact remnan ts/. W e no w consider the exp ected capture rates in real cusps/.', '/3 REAL GALAXIES': 'There are t w o classes of galaxies whose cen tres are go o d candidates for harb ouring cen tral blac k holes in the appro/priate mass range/, whic h ma y b e capturing compact ob/jects at an in teresting rate/: n ucleated spiral bulges/, suc h as that of our o wn galaxy/; and the cores of compact dw arf elliptic als lik e M/3/2/. Both ha v e lo w disp ersion/, steep high densit y cen tral cusps/, and probably con tain cen tral blac k holes of / /1/0 /6 M / /. M/3/2 w as discussed in detail b y Hils and Bender /(/1/9/9/5/)/, assuming di/ usion dominated the capture rate and that the cen tral cusp w as unrelaxed and isother/mal /( p /= /0/) It is lik ely that the core of M/3/2 is in fact re/laxed/, with p /= /0 /: /2/5/, although observ ation at radii / /0 /; /1 r h are consisten t with p /= /0/. Systems lik e M/3/2 are particu/larly promising sources b ecause of their large cen tral densi/ties and lo w cen tral disp ersion/. W e infer a capture rate of / /3 /-/1/0 /BnZr /8 p er y ear/, dominated b y large angle scattering from r / /5 /-/1/0 /3 r S /. A somewhat smaller di/ usion//col l is ion rate of /1 /: /8 /-/1/0 /BnZr /8 w as estimated b y Hils and Bender /(/1/9/9/5/)/. Unfortunately the space densit y of dw arf galaxies lik e M/3/2 app ears to b e v ery small/, probably as lo w as /1/0 /BnZr /5 Mp c /BnZr /3 /(Kormendy priv ate comm unication/, see also Gebhardt et al/. /1/9/9/6/)/, the total capture rate out to /1 Gp c is th us only ab out /1/0 /BnZr /2 y /BnZr /1 /.', '/3/./1 Bulges of spirals': 'The space densit y of n ucleated spirals with the appropriate bulge mass is higher / /1/0 /BnZr /2 /: /5 Mp c /BnZr /3 /, but the dynamics of their cen tral regions are more complicated/. A t r h t yp/ical densit y pro/ les corresp ond to p / /0 /: /5 /BnZr /0 /: /8/. F or the \ndensities and disp ersions seen in n uclei of spirals/, the cusp of main sequence stars is / attened due to stellar mergers at r m / /0 /: /1 r h as discussed ab o v e/. Inside r m the main sequence densit y pro/ le is / at/, / M S /( r /) /= / /0 /( r /=r m /) /BnZr /1 /= /2 /, / /0 /= / /6 /( r h /=r m /) /3 /= /2/+ p / /1/0 /8 M / p c /BnZr /3 /. The relaxation time at r m /, is t R /( r m /) / /3 /-/1/0 /9 y ears/. The total n um b er of main sequence stars inside r m is th us N M S / /5 /-/1/0 /5 /. The white dw arf/{main sequence merger rate is smaller b y a factor f cr /, and it is lik ely at the encoun ter v elo cities seen in these cores that a white dw arf w ould emerge relativ ely unscathed from an y suc h merger/. Th us w e exp ect the white dw arf /(and neu/tron star and lo w mass blac k hole/) densit y pro/ le to remain steep/. \nIncluding gra vitational fo cusing/, the cross/{section for WD/{WD mergers is R cc /( W D /) /= /6 /-/1/0 /BnZr /5 /( r h /=r /) /BnZr /2 /BnZr p t /BnZr /1 H /. Solving for r m /( W D /)/, and requiring R cc /( W D /) /= /5 as white dw arfs in terior to r m /( M S /) are replenished on t R /( r m /)/, w e / nd r m /( W D /) / /1/0 /BnZr /2 r h /< r coll /( M S /)/. The relaxation time scale for white dw arfs increases around r coll /( M S /) as WD/{MS scattering b ecomes ine/ ectiv e for t w o b o dy relax/ation/, but then decreases lik e r p inside r coll /( M S /)/, with r R /( W D /) /< / t H at r m /( W D /)/. Consequen tly /, relaxation main/tains the white dw arf densit y pro/ le at p / /0 /: /3 do wn to r coll /( W D /) / /5 /-/1/0 /BnZr /3 r h / /2 /-/1/0 /3 r S /. Inside r coll /( W D /) w e exp ect the densit y pro/ le of the white dw arf p opulation to / atten out to / W D /( r /) // /( r /=r m /( W D /)/) /BnZr /1 /= /2 /. \nW D The n um b er of white dw arfs in terior to r m /( M S /) is ap/pro ximately N W D /= /1/0 /4 /. The densit y of neutron stars and lo w mass blac k hole do es not / atten due to mergers/, but will lev el o/ due to relaxation inside r m /( W D /) to p /= /1 /= /4 pro/ le/. Note that with the densit y pro/ le / attened the relaxation time no longer decreases inside r coll /( W D /) Inside r coll /( W D /) w e th us / nd t R / t H again and a strong /( p /> / /0 /: /5/) neutron star and lo w mass blac k hole cusp ma y p ersist all the w a y to r cr it /. \ncr it The densit y pro/ le of the neutron star and lo w mass blac k hole p opulation follo ws that of the main sequence stars to r m /( M S /)/, / N S/=B H /= f N S/=B H / /6 /( r /=r h /) /( /BnZr /3 /= /2 /BnZr p /) /. A t r m /( M S /) the pro/ le ma y / atten to p /= /0 /: /2/5 /BnZr /0 /: /3 as relax/ation b ecomes e/ ectiv e/. The larger v alue is appropriate if the main sequence p opulation is ev olv ed/; the lo w er v alue is appropriate for a y ounger main sequence p opulation /(as ap/p ears to b e the case in our o wn galaxy/)/. The relaxed cusp pro/ le p ersists to r coll /( W D /) at whic h p oin t the t R is large and the pro/ le ma y steep en to p / /0 /: /5 again/. \nW e calculated n umerical large angle scattering rates and di/ usion rates for piecewise p o w er la w cusps/, where w e al/lo w ed for the c hanges in the densit y pro/ le inside r h due to stellar mergers/, c hanges in lo cal relaxation time scales/, and ine/ ectiv eness of collisio nal relaxation inside r coll /. Figure /3 sho ws the capture rates inferred as a function of radius for some c haracteristic pro/ les exp ected in the cen tres of n ucleated spirals/. T ypical inferred WD/{MBH capture rate/, through large angle scattering from r / /2 r coll /( W D /) is /1/0 /BnZr /7 p er y ear/, for n ucleated cusps with structural parameters lik e the Milky W a y /. Large angle scattering dominates the cap/ture rate/, and the total rate is dominated b y white dw arfs at radii / r m /( W D /) / /3 /-/1/0 /3 r S /. With / /1/0 /4 WDs inside r m the white dw arf p opulation is not signi/ can tly depleted b y mergers on time scales of t H /, and can b e replenished b y relaxation or stellar ev olution as implicitly assumed in the \nFigure /3/. The rate of capture of white dw arfs b y a /1/0 /6 M / cen tral blac k hole in a canonical n ucleated spiral galaxy /, as a function of radius/. The rate p eaks strongly at r / /2 r coll /( W D /) at / /1/0 /BnZr /7 p er y ear/. \nderiv ation/. \nderiv ation/. If the space densit y of these bulges is /0 /: /0/0/3 Mp c /BnZr /3 /, w e exp ect / /1 captures p er y ear within /1 Gp c/. A t an y one time w e ma y exp ect /1/0 /2 systems in the pro cess of capture within /1 Gp c/, or /0 /: /1 /BnZr /1 system within /1/0/0 Mp c/. Suc h systems w ould b e easily detectable b y LISA at /1/0/0 Mp c and migh t b e de/tected near /1/0 /BnZr /3 Hz out to /1 Gp c/. The optimal signal is ex/p ected from M /6 / /3/: for smaller M h the gra vitational w a v e amplitude b ecomes to o small/, for larger M h the frequencies b ecome to o lo w ev en for orbits close to r S /. Lo w cen tral dis/p ersion leads to higher capture rates/, fa v ouring systems with in trinsic densit y cusps similar to Hernquist pro/ les/, lo w cen/tral disp ersion and p /= /0 /: /8 inside r h /. Whether suc h systems are prev alen t in nature is an op en question/. \nIf the IMF is / at and the neutron star fraction is higher than the /0/./2/% exp ected from a Salp eter slop e mass func/tion/, then neutron star captures ma y b e comp etitiv e with the white dw arf capture rate due to the steep densit y pro// le exp ected for the neutron stars at all radii/. If WD/{WD mergers lead to accretion induced collapse and neutron star formation the exp ected capture rate ma y b e dominated b y neutron stars formed in the cusp/. R s /( N S /) /> / /1/0 /BnZr /7 are p ossi/ble for f N S / /1/%/, in whic h case the neutron stars con tribute signi/ can tl y to the total GW rate/. Ho w ev er/, neutron stars are exp ected to b e b orn with natal kic k v elo cities greater than / /1/6/6 and the cen tral p oten tial ma y not b e deep enough for man y neutron stars to remain in the cen ter after for/mation/, depressing the e/ ectiv e f cr /. Lo w mass blac k holes are detectable to larger distances/, but presumably o ccur in m uc h smaller n um b ers/. LMBHs are/, ho w ev er/, presumed not to receiv e natal kic ks/. \nIf there is a substan tial p opulation of stellar mass blac k holes/, the capture rate in spirals n uclei ma y b e as high as /1/0 /BnZr /5 p er y ear p er galaxy for /1/0 M / blac k holes/, with a global n um b er fraction of /2 /-/1/0 /BnZr /4 for the blac k holes /(see / gure /4/)/. With less than /1/0 /3 blac k holes in the cen tral regions/, suc h a rate is clearly not sustainable/, but w ould b e p ossible follo wing a n uclear star burst/, for a p erio d of /1/0 /8 y ears or so/. There is some evidence that the Milky W a y underw en t a n uclear starburst in the last /1/0 /9 y ears/, if this is t ypical of n ucleated spirals/, the p opulation a v eraged rate for stellar \nmass blac k hole mergers within /1 Gp c migh t b e as high as /1/0 /BnZr /6 p er y ear p er galaxy /, in whic h case w e migh t detect / /1/0 suc h system coalescing p er y ear with LISA/. The exp ected initial eccen tricities of the lo w mass blac k holes are / /0 /: /9/9/9/, and inspiral is rapid/.', '/3/./1/./1 Binaries/, anisotr opy and triaxiality': 'There are lik ely to b e some stellar binaries in galactic bulges/. Ho w ev er/, inside r h the densit y and disp ersion are large and only binaries with semi/{ma jor axis / r / are hard enough not to b e brok en up b y encoun ters with other stars/. Suc h binaries cannot lead to a larger capture rate as the semi/{ ma jor axis is /< / r cr it and encoun ters with the binaries are no more e/ ectiv e in scattering them in to the loss/{cone than comparable single star scatterings/. A signi/ can t fraction/, f b /, of white dw arf/{white dw arf binaries with semi/{ma jor axis / R / w ould lead to an enhanced WD/{WD merger rate/, b y a ratio of a W D /BnZr W D /=R W D / /1/0/0 f b /. \nW D /BnZr W D W D b The blac k hole also p olarizes the stellar distribution inducing a / nite tangen tial anisotrop y at small radii /(see Quinlan et al/. /1/9/9/5 for discussion/)/. There is consequen tly some bias to w ards circular orbits in the vicinit y of the blac k hole whic h will lo w er the estimated capture rate b y / /1/0/%/. As other pro cesses can induce comparably mild ra/dial anisotropies /(in particular ejection of single stars from the inner cusp b y lo w mass blac k holes b ound transien tly to the cen tral blac k hole/)/, this do es not c hange our estimate of the capture rate/. \nIn the inner cusp the stellar distribution is forced to/w ards sphericit y b y the blac k hole p oten tial /(there migh t in general b e some mo dest rotational / attening/)/. On larger scales the spheroid is most lik ely triaxial/, p ossibly strongly triaxial/. Orbit di/ usion in triaxial p oten tials ma y b e a strong factor in replenishi ng the loss/{cones of cen tral mas/siv e blac k holes/; this e/ ect is irrelev an t/, as the blac k hole mass is lo w enough that its /\\w andering/" ensures the loss/{ cone remains / lled indep enden t of the shap e of the stellar distributio n/. \nAs discussed b y /, for instance/, Sy er/, Clark e and Rees /(/1/9/9/1/) a main sequence star on an eccen tric orbit with p eri/b othron somewhat larger than r T can cum ulativ ely lose en/ergy b y impact on an accretion disk/, so that it ends up on a tigh tly b ound circular orbit/. This pro cess dep ends on the geometric cross section of the star and w ould b e corresp ond/ingly less e/ ectiv e for a compact ob ject/. The presence of a mo dest mass gas accretion disk around the cen tral blac k hole will therefor not a/ ect our presen t estimates of the capture rate for compact ob jects/.', '/3/./2 Stellar mergers': 'The rate of WD/{WD mergers is high within the blac k hole induced cusp/. The merger rate p er white dw arf at r m /( W D /)/, R cc /( W D /) /= /3 /-/1/0 /BnZr /1/0 p er y ear/, b y de/ nition/. There are /1/0 /4 white dw arfs in terior to r m /( W D /) giv en the assumed cusp parameters/. The in tegrated merger rate inside r m /( W D /)/, is N c /( W D /) / /1/0 /BnZr /6 y /BnZr /1 /, enough to destro y ev ery white dw arf in the cusp in a Hubble time/. As the cusp is relaxed at r m /( W D /) this is not a concern and the white dw arf p opula/tion can b e replenished b oth through lo cal stellar ev olution and replenishmen t from outside r m /( W D /)/. The exp ected \n/7 \n/8 \nS/. Sigur dsson /& M/.J/. R e es \nFigure /4/. The rate of capture of LMBHs b y a /1/0 /6 M / cen tral blac k hole in a canonical n ucleated spiral galaxy /, as a function of radius/. The rate p eaks strongly at r / /1/0 /3 r S at / /1/0 /BnZr /5 p er y ear/. \nn um b er of WD/{WD mergers within /1/0/0 Mp c is /1/0 p er y ear/, with a NS/-WD merger exp ected once ev ery few y ears within the same v olume/, dep ending on f N S /. As noted ab o v e/, a substan tial fraction of WD/{WD binaries could increase the WD/{WD merger rate b y an order of magnitude/. \nIf WD/{WD mergers lead to accretion induced collapse/, rather than total disruption of the white dw arfs/, then WD/{ WD mergers ma y lead to a substan tial neutron star p opula/tion in the cen tres of galaxies/, pro ducing up to /1/0 /4 neutron stars inside r m in a Hubble time/. This w ould on on a v erage double the estimated coalescence rates of white dw arfs and neutron stars with the cen tral blac k hole/. \nneutron stars with the cen tral blac k hole/. The white dw arf/{main sequence merger rate is / /1/0 /BnZr /5 p er y ear/, and w e assume a compact remnan t remains after the collision /. If the white dw arf is temp orarily b ound to the main sequence star after merger/, a common en v elop e phase ma y ensue/, pro ducing a large/, luminous stellar ob ject with a white dw arf core/. If suc h ob jects last /1/0 /7 y ears/, w e w ould exp ect / /1/0 /2 to b e observ ed at an y one time in the inner /0 /: /1 p c of a t ypical n ucleated bulge/.', '/4 CONCLUSIONS': 'W e estimate the capture rate of compact stellar remnan ts b y massiv e blac k holes in the cen ters of p o w er la w bulges of n ucleated spirals and compact dw arf elliptical s/. \nThe capture rate in M/3/2/{lik e elliptical s is high/, and lik ely to b e dominated b y large angle scattering b y other stars in the tigh tly/{b oun d cusp/. The net rate for M/3/2 lik e systems is lik ely to b e of order /1/0 /BnZr /8 p er y ear/, but the total rate in the galactic neigh b ourho o d is probably small due to the dearth of suc h systems/. \nThe total rate is lik ely dominated b y the p o w er la w bulges of /\\ordinary/" spiral galaxies suc h as our o wn Milky W a y /. The total exp ected rate is somewhat lo w er than naiv e estimates due to the / attening of the white dw arf densit y pro/ le through WD/{WD mergers/, with lik ely ev en t rates of order /1/0 /BnZr /8 p er y ear p er galaxy /. Ho w ev er/, due to their rel/ativ ely high space densit y suc h systems dominate the total observ able rate out to /1 Gp c/. \nW e conserv ativ ely estimate a minim um of /0 /: /1 captures p er y ear out to /1 Gp c/, with p erhaps /1/0 /2 systems observ able at lo w frequencies in the early stages of capture at an y one time/. Both the burst and p erio dic signals should b e de/tectable b y prop osed gra vitational radiation observ atories suc h as LISA/, out to few h undred Mp c/, with a c haracter/istic signal from the high eccen tricit y orbits/. If lo w mass blac k holes are presen t in signi/ can t n um b ers/, in the cen/ters of galaxies/, then the signal from LMBHs captured on to the cen tral blac k hole through large angle scattering ma y b e an order of magnitude larger still/, a v eraged o v er the lo cal p opulation of galaxies/. \np opulation of galaxies/. The WD/{MS merger rate is estimated at /1/0 /BnZr /7 p er y ear/, with O /(/1/0 /2 /) luminous stellar ob jects descended from suc h mergers observ able in the inner n ucleus at an y one time/. The WD/{WD merger rate is estimated at /1/0 /BnZr /6 p er y ear p er galaxy /, with sev eral p er y ear exp ected in the lo cal sup er/cluster/. Suc h ev en ts ma y b e detectable through X/{ra y or UV / aring b y curren t space based observ atories/. Chemical con tamination of the bulge from suc h ev en ts ma y also b e signi/ can t /(Khokhlo v and No vik o v in preparation/)/.', 'A CKNO WLEDGEMENTS': 'MJR gratefully ac kno wledges the supp ort of the Ro y al So/ciet y /. SS thanks the PP AR C and EU DGXI I for supp ort/.', 'REFERENCES': 'Bahcall J/.N/./, W olf R/.A/./, /1/9/7/6/, ApJ/, /2/0/9/, /2/1/4 Bahcall J/.N/./, W olf R/.A/./, /1/9/7/7/, ApJ/, /2/1/6/, /8/8/3 Begelman M/.C/./, Blandford/, R/.D/./, Rees M/.J/./, /1/9/8/0/, Nature/, /2/8/7/, /3/0/7 Bender/, R/./, Kormendy /, J/./, Dehnen/. W/./, /1/9/9/6/, Ap jL/, /4/6/4/, L/1/2/3 Benz/, W/./, Hills/, J/.G/./, Thieleman/, F/./{K/./, /1/9/8/9/, ApJ/, /3/4/2/, /9/8/6 Binney J/. J/./, T remaine S/./, /1/9/8/7/, in Galactic Dynamics/. Princeton/, p/./5/1/4 Cohn/, H/./, Kulsrud/, R/.M/./, /1/9/7/8/, ApJ/, /2/2/6/, /1/0/8/7 Cutler/, C/./, Kenne/ c k/, D/./, P oisson/, E/./, /1/9/9/4/, Ph ys/. Rev/. D/, /5/0/, /3/8/1/6 Cutler/, C/./, Flannagan/, / E/. E/./, /1/9/9/4/, Ph ys/. Rev/. 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Klu w er Academic/, p/. /1/5/3 T remaine/, S/./, /1/9/9/5/, in /\\Unsolv ed Problems in Astroph ysics/"/, ed/. J/. Bahcall and J/.P /. Ostrik er/, Princeton Univ ersit y Press /(Princeton/)/, in press v an der Marel/, R/.P /./, Quinlan/, G/.D/./, Sigurdsson/, S/./, de Zeeu w/, T/./, Hernquist/, L/./, /1/9/9/6/, in preparatio n Y oung/, P /.J/./, /1/9/8/0/, ApJ/, /2/4/2/, /1/2/3/2 \nCaptur e \nof c omp act stars by \nblack holes \nFigure /1/. r cr it vs p for single p o w er la w cusps for di/ eren t r h /. \nh Figure /2/. Plot of R s /( r /) for di/ eren t p /, assuming single p o w er la w cusps comp osed en tirely of compact remnan ts and ignoring existence of r coll /. The solid line is for / /= /7 /(large angle scattering/) and the dotted line is for / /= /5 /(di/ usion/)/. \nFigure /3/. The rate of capture of white dw arfs b y a /1/0 /6 M / cen tral blac k hole in a canonical n ucleated spiral galaxy /, as a function of radius/. The rate p eaks strongly at r / /2 r coll /( W D /) at / /1/0 /BnZr /7 p er y ear/. \nr / /2 r coll /( W D /) at / /1/0 p er y ear/. Figure /4/. The rate of capture of LMBHs b y a /1/0 /6 M / cen tral blac k hole in a canonical n ucleated spiral galaxy /, as a function of radius/. The rate p eaks strongly at r / /1/0 /3 r S at / /1/0 /BnZr /5 p er y ear/. \n<!-- image -->'}
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2011MNRAS.418.2292M
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Low-frequency oscillations in black holes: a spectral-timing approach to the case of GX 339-4
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2011-01-01
| 38 | 0.53 | 164 |
['accretion', 'accretion disks', 'stars binaries close', '-', 'astronomy x rays', '-']
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[]
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We analysed Rossi X-ray Timing Explorer (RXTE)/PCA and HEXTE data of the transient black hole binary GX 339-4, collected over a time-span of 8 years. We studied the properties and the behaviour of low-frequency quasi-periodic oscillations (QPOs) as a function of the integrated broad-band variability and the spectral parameters during four outbursts (2002, 2004, 2007 and 2010). Most of the QPOs could be classified following the ABC classification which has been proposed before. Our results show that the ABC classification can be extended to include spectral dependencies and that the three QPO types have indeed intrinsically different properties. In terms of the relation between QPO frequency and power-law flux, types A and C QPOs may follow the same relation, whereas the type B QPOs trace out a very different relation. Type B QPO frequencies clearly correlate with the power-law flux and are connected to local increases of the count rate. The frequencies of all QPOs observed in the rising phase of the 2002, 2007 and 2010 outbursts correlate with the disc flux. Our results can be interpreted within the framework of the recently proposed QPO models involving Lense-Thirring precession. We suggest that types C and A QPOs might be connected and could be interpreted as being the result of the same phenomenon observed at different stages of the outburst evolution, while a different physical process produces type B QPOs.
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[]
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https://arxiv.org/pdf/1108.0540.pdf
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{'S. Motta 1 , 2 , T. Mu˜noz-Darias 1 , P. Casella 3 , T. Belloni 1 , J. Homan 4': "1 INAF-Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807 Merate (LC), Italy \n2 Universit'a dell'Insubria, Via Valleggio 11, I-22100 Como, Italy \n3 School of Physics and Astronomy, University of Southampton, Southampton, Hampshire, SO17 1BJ, UK \n4 MIT Kavli Institute for Astrophysics and Space Research, 70 Vassar Street, Cambridge, MA 02139; \n26 October 2018", 'ABSTRACT': 'We analyzed RXTE/PCA and HEXTE data of the transient black hole binary GX 339-4, collected over a time span of eight years. We studied the properties and the behavior of low frequency quasi periodic oscillations (QPOs) as a function of the integrated broad-band variability and the spectral parameters during four outbursts (2002, 2004, 2007, 2010). Most of the QPOs could be classified following the ABC classification that has been proposed before. Our results show that the ABC classification can be extended to include spectral dependencies and that the three QPO types have indeed intrinsically different properties. In terms of the relation between QPO frequency and power-law flux, type-A and -C QPOs may follow the same relation, whereas the type-B QPOs trace out a very different relation. Type-B QPO frequencies clearly correlate with the powerlaw-flux and are connected to local increases of the count rate. The frequency of all QPOs observed in the rising phase of the 2002, 2007 and 2010 outburst correlate with the disk flux. Our results can be interpreted within the framework of recently proposed QPO models involving Lense-Thirring precession. We suggest that typeC and -A QPOs might be connected and could be interpreted as being the result of the same phenomenon observed at different stages of the outburst evolution, while a different physical process produces type-B QPOs. \nKey words: accretion disks - binaries: close - stars: individual: GX 339-4 - X-rays: stars', '1 INTRODUCTION': 'Quasi-periodic oscillations (QPOs) have been discovered in many systems and are thought to originate in the innermost regions of the accretion flows around stellar-mass black holes. Low-frequency QPOs (LFQPOs) with frequencies ranging from a few mHz to ∼ 10 Hz are a common feature in almost all black hole X-ray binaries (BHB) and were already found in several sources with Ginga and divided into different classes (see e.g. Miyamoto & Kitamoto 1991 for the case of GX 339-4 and Takizawa et al. 1997 for the case of GS 1124-68). Observations performed with the Rossi Xray Timing Explorer (RXTE) have led to an extraordinary progress in our knowledge on properties of the variability in BHBs (see van der Klis 2006, Remillard & McClintock 2006, Belloni 2010 for recent reviews) and it was only after RXTE was launched that LFQPOs were detected in most observed BHBs (see van der Klis 2004, Remillard & McClintock 2006). Three main types of LFQPOs, dubbed types A, B, and C, originally identified in the Power Density Spectra (PDS) of XTE J1550-564 (see Wijnands et al. 1999; Homan et al. 2001; Remillard et al. 2002), have been seen in several sources. \nThe systematic variations in the energy spectra of transient BHBs can be identified in terms of a pattern described in an \nX-ray hardness-intensity diagram (HID) (see Homan et al. 2001; Homan & Belloni 2005; Belloni et al. 2005, Belloni 2010). In many black hole candidates (BHC), different states are found to correspond to different branches/areas of a q-shaped HID pattern. Four main bright states (in addition to the quiescent state) have been identified in these sources, based on their spectral and timing properties (for a review Homan & Belloni 2005; McClintock & Remillard 2006; Belloni 2010). In particular, the analysis of the fast timing variations observed in the PDS plays a fundamental role in the state classification (see Homan et al. 2005, Belloni 2010). \nEven though the general evolution and the main transitions become apparent in the HID, providing a general description of the BHB evolution, it is not enough for detailed studies. Many observed properties change smoothly throughout the basic diagrams (HID, see e.g. Homan et al. 2001; rms vs. hardness diagram, see e.g. Belloni 2010; rms-intensity diagram, see Mu˜noz-Darias et al. 2011), but some do not. It is the inspection of the fast-variability properties which indicates the presence of abrupt variations that can be taken as landmarks to separate different states. In proximity of the HIMS/SIMS transition timing properties (in particular the appearence of different types of QPOs in the PDS) constitute the sole \nway to distinguish between HIMS/SIMS/HSS given the absence of differences in the spectral shape. \nThe different types of QPOs are currently identified on the basis of their intrinsic properties (mainly centroid frequency and width, but energy dependence and phase lags as well), of the underlying broad-band noise components (noise shape and total variability level) and of the relations among these quantities. Despite LFQPOs being known for several decades, their origin is still not understood and there is no consensus about their physical nature. However, the study of LFQPOs provides an indirect way to explore the accretion flow around black holes (and neutron stars). In particular, their association with specific spectral states and their phenomenology suggests that they could be a key ingredient in understanding the physical conditions that give origin to the different states. \nSeveral models have been proposed to explain the origin and the evolution of LFQPO in X-ray binaries. The geometry described in Esin et al. (1997) and Done et al. (2007) allow to explain the spectral evolution seen in BHBs and forms the basis for the LFQPO model proposed by Ingram et al. (2009), Ingram & Done (2010) and Ingram & Done (2011), which invokes Lense-Thirring precession. We shall refer to this model as the precession model . For the case of the neutron star source 4U 1728-34, Titarchuk & Osherovich (1999) show that LFQPO frequency is associated with radial oscillations in the boundary layer and the break frequency associated to the broad band noise in the PDS is determined by the characteristic diffusion time of the inward motion of the matter in the accretion flow. Tagger & Pellat (1999) associate the existence of LFQPO in X-ray binaries to an instability occurring in the inner part of disks threaded by a moderately strong vertical poloidal magnetic field. \nIn this paper, we examine the QPO parameters and compare them with the results of a complete spectral analysis. Since the presence and the properties of QPOs in BHBs are related to the spectral characteristics of the source, our goal is to identify the link between the spectral and fast timing variability properties and to highlight possible physical differences between the three types. The precession model explains both the type-C QPO frequencies and the presence of the associated broad-band variability (see also Tagger & Pellat 1999 for an alternative in the case of BHs) and most of spectral properties. For this reason, given the success of this model to explain many of the observed properties we will interpret our results in the context of the precession model and we will attempt to extend its predictions to the three types of QPOs.', '1.1 GX 339-4': 'GX 339-4 is a Low Mass X-Ray Binary (LMXB) harboring a > 6 M /circledot accreting black hole (Hynes et al. 2003; Mu˜noz-Darias et al. 2008). Since its discovery (Markert et al. 1973), the system has undergone several outbursts, becoming one of the most studied X-ray transients. The intense monitoring carried out by RXTE during the 2002, 2004, 2007 and 2010 outburst has yielded detailed studies on the evolution of black hole states throughout the outburst (see e.g. Belloni et al. 2005), making the source an ideal candidate for an extensive study on the spectral-timing properties of BHBs. Here, we use this rich data set to study the relation between the spectral and variability properties of GX 339-4 across the different outbursts. \nFigure 1. Top panel: HID for the 2002, 2004, 2007 and 2010 outburst of GX 339-4. Each point represents a single RXTE observation. Blue stars mark Type-C QPOs, red squares mark Type-B QPOs and green triangles mark Type-A QPOs. Black dots mark all the other RXTE observations of GX 339-4 that do not show low-frequency QPOs. Bottom panel: corresponding Rms-hardness diagram. Empty triangles stand for upper limits. \n<!-- image --> \nFigure 2. RID for the 2002, 2004, 2007 and 2010 outburst of GX 339-4. The symbols follow the same criteria as in Fig. 1 \n<!-- image -->', '2 OBSERVATIONS AND DATA ANALYSIS': 'We examined 1007 RXTE public archival observations of GX 3394 from 2002, 2004, 2007 and 2010 outburst and selected for our analysis only observations where somewhat narrow (i.e. Q /greaterorequalslant 2) feature was identifiable on top of peaked or power-law shaped noise components. The selection has been done by eye, then we fitted the PDS (see below) and non-significant (below 3 σ ) detections were excluded from the subsequent analysis. A total of 115 observations and 117 oscillations have been considered. \nThe RXTE data were obtained in several simultaneous modes. \nSTANDARD 2 and STANDARD modes for the PCA and HEXTE instruments respectively were used to create background and dead time corrected spectra. We extracted energy spectra from PCA and HEXTE for each observation using the standard RXTE software within HEASOFT V. 6.9. Only Proportional Counter Unit 2 from the PCA was used since only this unit was on during all the observations. A systematic error of 0 . 6% was added to the PCA spectra to account for residual uncertainties in the instrument calibration 1 . For 2002, 2004 and 2007, we used only data coming from HEXTE/Cluster B, which was correctly working in that period. Since HEXTE/Cluster B encountered technical problems at the end of 2009 2 , we decided to use data coming from HEXTE/Cluster A to analyze observations taken subsequently. We followed the standard procedure described in the RXTE cookbook 3 to produce source and background spectra, using data coming from HEXTE/Cluster B to produce a preliminary background from which we derived a background spectrum to be used together with HEXTE/Cluster A spectrum. \nWe accumulated background corrected PCU2 rates in the Standard 2 channel bands A = 4 - 44 (3.3 - 20.20 keV), B = 4 - 10 (3.3 - 6.1 keV) and C = 11 - 20 (6.1 - 10.2 keV). A is the total count rate, while the hardness was defined as H = C/B (Homan & Belloni 2005). \nPCA+HEXTE spectra were fitted with XSPEC V. 11 in the energy range 4 - 40 keV 4 and 20-200 keV respectively for data taken in 2002, 2004 and 2007 outbursts. For HEXTE spectra produced from data collected during 2010 we considered only the 20 - 150 keV spectral band. The reason for this was to exclude the harder part of the spectra that in some cases was affected by systematic problems due to an incorrect estimation of the background. For the same reason we ignored the energy range 50 - 80 keV, where line-like residuals can be found in the spectral fits 5 . \nWe calculated unabsorbed fluxes for different spectral components from the best fit to the energy spectra (see Sec. 2.2). We measured the total flux between 2.0 and 20.0 keV and the disk flux between 2 and 20 keV. We also measured the power-law flux in the 6 - 20 keV energy band. Since the spectral deconvolution can be problematic, we choose this energy interval to avoid contamination due to the confusion between disk and power-law component at lower energies. All the fluxes were normalized by the Crab flux in the respective energy bands in order to correct the fluctuations due to the variations in the instrument properties. For each outburst we used a Crab spectrum coming from an observation as close as possible to the central part of the outburst. The fluxes used for the correction are summerized in Table 1. \nFor our timing analysis, we used GOODXENON, EVENT and SINGLE BIT data modes. We used custom software under IDL and for each observation we produced power density spectra (PDS) from stretches 16 seconds long in the channel band 0-35 (2-15 keV). We averaged the PDS and subtracted the contribution due to Poissonian noise (see Zhang et al. 1995). The PDS were normalized according to Leahy et al. (1983) and converted to square \nTable 1. Crab fluxes used for the flux correction described in Sec. 2. Columns are: Crab observation ID, flux measured in the 2 - 20 keV band, flux measured in the 6 - 20 keV band, outburst the correction was applied to. \n| Observation ID | 2 - 20 keV flux erg/s/cm 2 | 6 - 20 keV flux erg/s/cm 2 | Outburst Outburst |\n|------------------|------------------------------|------------------------------|---------------------|\n| 70018-01-03-00 | 3.51e-08 | 1.72e-08 | 2002 |\n| 90129-02-01-00 | 3.44e-08 | 1.69e-08 | 2004 |\n| 92802-03-06-01 | 3.4e-08 | 1.65e-08 | 2007/2010 | \nFigure 3. Examples of type A, B and C QPOs from our GX 339-4 observations. The centroid peak is indicated. Upper panel: Obs. 92085-01-02-06. Middle panel: Obs. 95409-01-15-06. Bottom panel: Obs. 70109-04-01-01). The Poisson noise was not subtracted. \n<!-- image --> \nfractional rms (Belloni & Hasinger 1990). The integrated fractional rms was calculated over the 0.1 - 64 Hz band. PDS fitting was carried out with the standard XSPEC fitting package by using a oneto-one energy-frequency conversion and a unit response. Following Belloni et al. (2002), we fitted the noise components with three broad Lorentzian shapes, one zero-centered and other two centered at a few Hz. The QPOs were fitted with one Lorentzian each, only occasionally needing the addition of a Gaussian component to better approximate the shape of the narrow peaks and to reach values of reduced χ 2 close to 1. We examined PDS in the form of dynamical power density spectra (DPDS), computing Fast Fourier Transforms of windows of data 16s long. In some cases we used shorter time windows to better follow the evolution of a narrow feature. Where transitions between different power spectral shapes were seen, we separated different time intervals in order to obtain average power spectra for each shape.', '2.1 The QPO classification': 'We have classified the QPOs following Casella et al. (2004). The properties that allow one to classify the QPOs are the quality factor ( Q = ν centroid /FWHM ) and the shape of the noise associated with the oscillation. Frequency does not allow a classification, as the characteristic frequency intervals where the three types of QPOs appear largely overlap, nor does the rms. Casella et al. (2004) and Mu˜noz-Darias et al. (2011) quantify the noise level associated to the QPOs in terms of the total 6 fractional rms (0.1-64). We could classify ∼ 98% of the QPOs of GX 339-4 (see 3.1). In Table 3 we summarize the QPOs classification and in Fig. 3 we report one example for each type of QPO. \nIn Figure 1 we show the HID of GX 339-4 (top panel) and the rms versus hardness diagram (bottom panel, see Belloni 2010) including all the RXTE observations collected during 2002, 2004, 2007 and 2010 outbursts. Type-A, -B, -C QPOs are marked with green triangles, red squares and blue circles respectively. The HID is tracked counterclockwise, starting from the bottom right corner of the track. The upper and lower horizontal branches in the HID roughly correspond to the HIMS and SIMS, while the right and left vertical branches correspond respectively to the LHS and HSS. We will refer to the first part of the loop (from the right vertical branch to the left end of the top horizontal branch) as softening phase of the outburst and to the last part (from the left end of the bottom horizontal branch back to the right vertical branch) as to the hardening phase . \nIn the following we summarize the results of the ABC classification applied to our sample. \nType-C QPO In the early stages of all the four outbursts of GX339-4 (late LHS and HIMS, see Tab. 3 for details), two main components can be identified in the PDS: a strong flat-top noise and one or more QPO peaks. When more than one peak is observed, the peaks are harmonically related. The strongest and narrowest peak, which is usually the central one, is taken as the fundamental. When the identification of the fundamental remained difficult because of the presence of strong harmonic peaks, we followed the evolution of the PDS shape for the selection. In Tab. 4 we report parameters for the strongest peak in the PDS.The QPO is usually strong and narrow (Q /greaterorequalslant 6) and the centroid frequency varies in the 0.2-9 Hz range. Only in some cases, when the oscillations are weak and appear at very low frequencies, the Q-factor is slightly lower than 10. The addition of a Gaussian component to the multi-Lorentzian model is required in a few observations in order to better approximate the peak shape. Type-C QPOs are observed also in the late stages of all the outbursts. We refer to type-C QPOs observed in the lower branch as type-C ∗ . The PDS shows a noise component in the form of a broad Lorentzian and a QPO peak broader than at the beginning of an outburst. During the 2002 outburst, the typeC ∗ QPOs frequencies span the 4-9 Hz range, while in the 2004 only one Type-C ∗ QPO (at ∼ 3 Hz) is observed. In the 2007 outburst Type-C ∗ QPOs are seen in a slightly lower frequency interval (2-4 Hz). A second harmonic peak is sometimes present in the PDS. Even though the Type-C ∗ QPO centroid frequency ranges, rms properties and Q-values are different from the case of Type-C QPOs, it is possible to demonstrate that the properties of the two kinds of QPOs are continuously connected when ordered for increasing QPO frequency (see Sec. 3; see also Casella et al. 2005). \nType-A QPO Type-A QPOs are observed in 2002 and 2004 during the SIMS, when the flux of the source is close to its maximum. The PDS show a broad QPO (Q /lessorequalslant 3) with centroid frequency between 7.1 and 8.1 Hz associated to a weak power-law noise. Neither a subharmonic nor a second harmonic is observed. The PDS showing a Type-A QPOs have the lowest total fractional rms values of the sample. \nType-B QPO They are observed in the SIMS and the oscillations that appear in the PDS (Q /greaterorequalslant 6) are observable in the frequency ranges 0.8-6.4 Hz. All the type-B QPOs seen at low frequencies (i.e. below ∼ 3 Hz) belongs to the lower branch in the HID, while all the QPOs at higher frequencies are observed in the upper branch in the HID. The noise seen in the PDS is weak and the QPO peak shape is often more similar to a Gaussian rather than a Lorentzian, therefore we had to combine both components (Gaussian + Lorentzian) to obtain better fits. A weak second harmonic is often present in the PDS. Sometimes the hint of a sub-harmonic appears.', '2.2 Spectral Analysis': 'In order to obtain good fits and acceptable physical parameters, a model consisting of an exponentially cut off power law spectrum reflected from neutral material (Magdziarz & Zdziarski 1995) was used ( pexrav in XSPEC). The reflection parameter was left free to vary, while the inclination angle was fixed at 30 degrees (notice that the results only weakly depend on the inclination value assumed). A multi-color disk-blackbody ( diskbb ) was added to the model and a Gaussian emission line with centroid allowed to vary between 6.4 and 6.7 keV was further needed in order to obtain acceptable fits. The line width was constrained between 0.1 and 1.0 keV to prevent artificial broadening due to the response of XTE/PCA at 6.4 keV. The hydrogen column density ( wabs ), was frozen to 0 . 5 × 10 22 cm -2 (Zdziarski et al. 2004). The addition of an iron edge, justified by the presence of the iron line, does not improve the fits significantly. In Table 4 we show the relevant spectral parameters for the best fits. \nWhere Type-C QPOs are detected, the photon index is seen to rise from ∼ 1.5 to ∼ 2.8 and back to ∼ 1.5 as a function of time, consistently with what previously observed (see e.g. Motta et al. 2009). Following the loop in the HID the source becomes soft while approaching the HSS and subsequently becomes hard again going back to the LHS. As a consequence the photon index increases, remains almost constant (between ∼ 2.6 and ∼ 2.8) for a while and then decreases. When Type-A and B QPOs (during the softening phase) are seen in the PDS, the photon index is at its maximum. When type-B QPOs are observed in the hardening phase, they are associated to lower values of the photon index. Indeed, the spectra from SIMS observations in the hardening phase always show systematically lower photon indices. For a detailed analysis of the transitions between soft and hard state in GX 339-4 during 2010 outburst, see Stiele et al. (Submitted). The photon index also correlates with the LFQPO frequency, as was already noticed by Vignarca et al. (2003) in the cases of GRS 1915+105, GRO 1655-40, XTE J1550-564, XTE J1748-288 and 4U 1630-47. The same behavior was observed in H1743-322 by McClintock et al. (2009). All type-C QPOs follow the same relation rather than several branches depending on the outburst. Type-C and type-B QPOs also overlap quite well covering the same photon index interval. \nThe parameters associated to the iron line and to the reflection components do not show any clear correlations with the presence of the different types of QPOs and/or particular states. \nAs one can see from Table 4, not all the components of the model are present in all the observations. In all the observations where a Type-B or Type-A QPO is detected, a disk component ( diskbb ) is visible in the spectrum. When the Type-C QPOs are detected at hardness larger than 0.6 no disk component is observed. The disk appears at hardness 0.2 - 0.6 during the softening phase. During the hardening phase a disk component was needed only in some of the spectra from 2002. All the other spectral components (i.e. iron line, power-law and reflection components) are always necessary to obtain good fits. \nExamining the parameters related to the disk-blackbody component, it is clear that our constrains on the disc parameters are usually poor. Even when a soft component is clearly present in the spectra and is required in order to obtain good fits, it is often only marginally significant. This is expected since the working range of PCA (3-40 keV) allows to see only the high energy part of the disc black body component, above the Wien peak. It is also known that, even if the diskbb model provides a good description of the thermal component, the derived spectral parameters should not be interpreted literally (see e.g. Merloni et al. 2000, Remillard & McClintock 2006). However, when the thermal component is dominant, the parameters can be taken as reliable. \nDisentangling the different spectral components could be problematic when using spectra from RXTE. If one assumes that the hard x-ray emission comes from Comptonization of the soft disk photons on hot electrons, it is known that a simple power-law (or a power-law-like component, such as pexrav ) is not appropriate for the description of the hard Comptonization tail of the spectrum at lower energies (i.e. where the hard component overlaps with the soft emission from the disk blackbody). A simple power-law does not have the low-energy cutoff that is typical of a proper comptonization model (i.e. eqpair or compTT in XSPEC ) and therefore could affect the real contribution of the disk. However, when using RXTE data the adoption of a simple power-law (or power-lawlike) component is justified because the energy range where PCA spectra can be analyzed (above 3 keV) does not cover this problematic overlapping zone (just below the Wien peak of the multicolor disk-blackbody). Therefore, a simple power-law or a power-lawlike model such as pexrav is appropriated for the description of PCA spectra (see Mu˜noz-Darias et al. 011a and Stiele et al. Submitted). Despite the poor constraints on the disk parameters, the energy spectra are well fitted by the model used and the measures of the disk-fluxes reported in this work are to be considered reliable. This is supported by the fact that the disk flux correlates with the disk temperature.', '3.1 Rms-frequency relation': 'Once the QPOs of our sample were classified according to the ABC scheme, following Casella et al. (2004), we plotted the integrated fractional rms of each PDS versus the centroid frequency (see Fig. 4) to probe the link between the main QPO property (the frequency) and the total variability of the source. Several groups of points, associated to the Type-A, -B, -C QPOs, can be identified. \n- · Type-C QPOs cover the frequency range 0.1 and 9 Hz and the rms range 10-35%. Type-C QPO frequency is clearly anticorrelated with total fractional rms.\n- · Type-A QPOs form a group at frequencies in the range ∼ 7 - 8 Hz and rms of ∼ 3%. \nFigure 4. QPO centroid frequency vs. 0.1-64 Hz fractional rms. Each point corresponds to a different observation. Symbols correspond to QPO types: circles are Type-C QPOs, triangles type-A QPOs and squares type-B QPOs. The solid lines join for each outburst type-C QPOs during the outburst softening, while dashed lines join type-C QPOs during the hardening phase. Different colors mark different outburst: black 2002, blue 2004, green 2007 and red 2010. \n<!-- image -->', '· Type-B QPOs are located at a slightly higher rms ( ∼ 5-10%) in the 1-7 Hz range.': 'As one can see from Figure 4, the softening phase (solid lines) for each outburst shows lower rms than the hardening (dashed lines). This property has already been observed in other sources (see e.g. MAXI J1659-152, Mu˜noz-Darias et al. 011a) and can probably be understood in terms of a lower disk contribution to the emission. Only two outliers (i.e. the squares above and below the 5-10% of rms in Fig. 4) can be identified in Fig. 4 (Obs. #10, #15). Obs. #10 shows the typical PDS shape of a type-B QPO, even though with higher rms, while Obs. #15 shows a much more noisy PDS. Both the points lay far both from the type-C and -B region in Fig. 4. However, since they follow a relation similar to typeB QPOs in a flux-frequency plot (see 3.3), we tentatively classify those QPOs as Type-B. We notice that both the observations are taken in the decay phase of the outburst (2004 and 2007 respectively).', '3.2 Frequency-Power-law flux relation': "Since LFQPOs are known to be usually associated to the hard tail of the spectrum (see Churazov et al. 2001 and Sobolewska & ˙ Zycki 2006, but also Rodriguez et al. 2004 and Rodriguez et al. 2008, who showed tha LFQPO spectra display a moving high energy cutoff), we started investigating the relations between LFQPO frequencies and the power-law fluxes. The frequency-power-law flux relation is shown in Fig. 5. We refer to power-law flux as the Crab corrected flux from the power-law component in the 6 - 20 keV energy band (see Sec. 2.2) 7 . Different symbols represent the Type of QPOs and colors differentiate the four outbursts. For each outburst, a solid line connects in time all the QPOs from the first type-C to the first type-B QPO. The typical time interval between the last type-C \nFigure 5. QPO centroid frequency vs. 6-20 keV powerlaw flux, normalized to the Crab flux. Colors indicate the outburst: black stands for 2002, blue for 2004, green 2007 and red 2010. Symbols correspond to QPO types: stars are Type-C QPOs, triangles type-A QPOs and squares type-B QPOs. The solid lines join for each outburst type-C QPOs and the first type-B QPO detected after the disappearing of the Type-C QPO. The empty circles mark the type-C QPOs observed immediately before the appearance of a type-B QPO. Each point in the plot represents an entire RXTE observation in which a QPO was detected, apart from the cases in which a switch between two different types of QPOs was observed (Obs. #1/#26, Obs. #4/#27-#28). \n<!-- image --> \nand first type-B QPO is ∼ 1 day, even though sometimes other types of PDS are observed in between, especially if the transition period is not densely observed. When this happens, it might be possible to miss the appearance of a transient type-B QPO and see a typeA QPO after a type-C QPO. In Figure 5, the different QPO types follow clear and separate relations as function of the hard flux. \n- · Type-C QPOs (stars) lie on the right part of the diagram. The points trace well-defined tracks at different flux levels for each outburst (see the solid lines in the plot). The whole 0.1 - 9 Hz frequency range is spanned. Each track roughly correspond to the late LHS and HIMS observed in the softening phase of each outburst.\n- · Type-C ∗ QPOs follow tracks on the left part of the diagram. Those QPOs were observed in all the outbursts. Differently from what happens in the right part of the diagram, points belonging to different outburst span different and smaller frequency ranges. However, the maximum frequency reached is consistent with the softening tracks. We ascribe the fact that no QPO appear at lower frequencies to the count rate being very low.\n- · Type-A QPOs (triangles) cluster on a quite narrow frequency and flux range, close in frequency to the last Type-C QPOs seen before the transition to the SIMS, but at slightly lower fluxes (see Fig. 4). Type-A QPOs always appear in time after the detection of a type-B QPO. No Type-A QPO is observed in the left part of the plot, i.e. during the hardening phase at the end of the outburst (see also Fig. 1). 4.\n- · Type-B QPOs (squares) are sharply correlated with the power-law flux and the relation between frequency and powerlawflux is well described by a power-law of the form y = Ax B + C \n(where A = 19.4(8) , B = 0.18(6), C= -6.1(2), see Fig. 6). This correlation holds for a large range of flux, showing that these oscillations frequencies depend directly on the hard X-ray flux. However, unlike type-C QPOs, the points do not follow a clear path as a function of time. No oscillations are seen in a given flux range in the middle of the plot. As happens for type-C and type-C ∗ QPOs, this is due to the fact that also the SIMS (where type B QPOs are observed) is crossed two times, at either high or low fluxes. \nSince type-B QPO are found in a small hardness range, it might be argued that they behave like type-C QPOs when observed in a small hardness range. For this reason we checked whether typeC QPOs show a behavior similar to type-B QPOs once grouped in subsamples selected as a function of the hardness. We divided typeC QPOs in six subsamples and for each group we plotted the QPO centroid frequency as a function of the power-law flux. The result is shown in Fig. 7. \nType-C QPOs' frequencies only show a weak anti-correlation with the power-law flux, especially at high frequencies. When typeCQPOs' range overlaps the hardness range where type-B QPOs are found (red points in Fig. 7 in the hardness range 0.2-0.3) the (weak) correlation that they follow is exactly opposite to the one shown by type-B QPOs. This fact further strengthens the difference between type-C and B QPOs.", '3.3 Frequency-Disk flux relation': 'In Fig. 8 we show the relations between frequency and disk flux. Symbols and colors follow the same criteria of Fig. 5. We re- \nFigure 6. Type-B QPOs frequency as the function of the power-law flux. The correlation is well described by a power-law of the form y = Ax B + C , where A = 19.4(8) , B = 0.18(6), C= -6.1(2) \n<!-- image --> \nFigure 7. Type-C QPO centroid frequency versus power-law flux. Different colors marks the different hardness ranges in which the QPOs where divided. From the blue to the red the ranges are: 0.7-0.8, 0.6-0.7, 0.5-0.6, 0.4-0.5, 0.3-0.4, 0.2-0.3. \n<!-- image --> \nPOWERLAW FLUX/ CRAB FLUX (6 - 20 keV) \nfer to disk flux as the Crab corrected flux coming from the diskblackbody component in the 2 - 20 keV energy range. Since we could associate a measure of the disk flux only to a subsample of type-C QPOs, not the all of them are present in this plot. \nMost of the points trace out a well-defined track and the frequency clearly correlate with the disk flux. Those points correspond to all the QPOs seen during the upper branch of the four outbursts. Notice that for three of the four outburst (2002, 2007 and 2010 outburst) the upper branch in the HID loop is the same, while during the 2004 the upper branch is observed at a lower flux level. Note \nFigure 8. QPO centroid frequency versus soft flux. The symbols follow the same criteria as in Fig. 5. \n<!-- image --> \nthat this track includes type-C as well as type-A and B QPOs. TypeA QPOs are located in correspondence of the highest disk fluxes, while type-B QPOs cover approximately the same flux range of type-C QPOs. \nWe identify also a number of outliers, which correspond to two branches at different flux levels, forming other two tentative correlations. The clearest of the two is formed by all the type-C QPOs observed during the hardening phase (black points in the left upper corner of Fig. 8). Only during 2002 outburst it was possible to measure the disk flux during the lower branch. We ascribe this to the disk being fainter and/or colder during other outbursts than in 2002.', '3.4 Association of Type-B QPOs with local peaks in the light curve': 'In Fig. 9 we plot sections of the light curves (PCU data, 2-20 keV) of the four outburst of GX 339-4. From this figure it is clear that most of the type-B QPOs are found at times of local peaks in the light curve. In all the cases where different types of QPOs are observed within a short time interval, they follow a precise count rate segregation: type-B QPOs at highest count rates, Type-C QPOs immediately below and type-A QPOs are found at lower count rates. A relation QPO-type/count rate therefore seems to exist, although it is different from what Casella et al. (2005) observed in XTE J1859+226, where type-A QPOs are seen at highest count rates and type-B and -C appears below, even though they are not clearly separated in count rate. We notice that in all the outbursts, there is always a B at lower flux than some C in the same outburst, therefore the segregation in count rate in not absolutely true all along the outburst, but only for certain intervals (see Fig. 9). We also notice that outbursts 2002 and 2004 had a different initial evolution in comparison to 2007 and 2010, where the count-rate peak was reached after a monotonic rise. \nDespite a difference segregation in count rate, also in XTE J1859+226 type-B QPOs are always found at hardness higher than that of type-A QPOs and lower than that of type-C QPOs. In the case of XTE J1859+226 a certain overlap was observed, while there is no overlapping in the case of GX 339-4 during a single outburst. A similar correlation between QPO types and hardness was found in XTE J1550-564 (Homan et al. (2001)). \nFigure 9. Light curves of the 2002, 2004, 2007, 2010 outburst of GX339-4. Each point represents one entire RXTE observation. The symbols mark the different kinds of QPOs following the criteria used in Fig. 1. The Type-B QPOs appear associated with count-rate peaks. \n<!-- image -->', '3.5 Timing and spectral evolution': 'In four observations of our sample the PDS show rapid transitions between different shapes. In all cases the transitions involve type-B QPOs. \nIn Fig. 10 and 11 we show two examples of different behaviors, for Obs. #1/#35 and #4/#36-#37 respectively. In the case of Obs. #1/#35 (Obs. ID 70109-01-07-00, Fig. 10), a type-A QPO ( ∼ 7 Hz) is present in the first part of the observation, when the observed count rate was low. In the second part the light curve shows a net increase in count rate and simultaneously the onset of a type-B QPO ( ∼ 6 Hz) is observed (see Nespoli et al. 2003 for details). At the same time, an increase of the hard flux (from ∼ 11% to ∼ 13% of the total flux) is observed, as the variation in hardness suggests (see Tab. 3). \nDuring Obs. #4/#36-#37 (Obs. ID 70108-03-02-00, second PCA orbit, Fig. 11) a similar situation can be observed. In the first orbit the PDS shows a type-A QPO ( ∼ 7 Hz). In the second orbit the source count rate dropped abruptly from ∼ 2200 counts/s to ∼ 2100 \nFigure 10. Upper panel: Light curve for Obs. #1/#35. The red line marks the light curve interval where a type-A QPO was detected. The black line marks the interval where a type-B QPO was visible. Lower panel: PDS for the two time intervals Nespoli et al. 2003. \n<!-- image --> \ncounts/s 8 in few seconds. A type-B QPO ( ∼ 5.6 Hz), that was visible in the first part of the observation disappears leaving place to a type-A QPO ( ∼ 7 Hz), observable until the end of the interval and during the complete third orbit. Analogous to the previous case, a variation in flux takes place. When the type-B QPO disappears, the power-law flux is seen to decrease abruptly (from ∼ 15% to ∼ 11% of the total flux in ∼ 1s). The frequency of the type-A QPO before the appearance of the type-B QPO and and after its disappearance in the light curve is the same. \nTwo other cases are found in Obs. #5 and #30 (Obs. ID 7011001-47-00 and 95409-01-19-00), where, in correspondence of a rise in the count rate, a type-B QPO takes the place of power-lawshaped noise (i.e. no type-A or C QPO is observable before the onset of the type-B QPO). In all the mentioned cases it is clear that spectral differences can be very subtle, much more than the timing changes. \nSimilar fast transitions between different types of QPOs have already been observed in GX 339-4 (Miyamoto & Kitamoto 1991) and in other sources, such as XTE J1859+226 (Casella et al. 2004) and GS 1124-68 (Takizawa et al. 1997). For XTE J1859+226, the type-B QPO seems to be associated to a flaring behavior. However, also in this source the type-A QPO is always seen at slightly higher frequencies than the type-B QPOs. \nAs it happens for type-A/B QPOs, direct switching from/to type-C/type-B can be observed in GX339-4 as in other sources (see e.g. Miyamoto et al. 1991, Takizawa et al. 1997). However, the switch between type-B and -C QPOs is not as sharp as in the case of type-B and type-A QPOs: the transition between the two types usually comes with a complex behavior and Type-B QPOs appears in correspondence to peaks in the light curve (occurring at timescales of few seconds), consistently with what described in Sec. 3.4, and type-C QPOs are seen where the count rate drops. For this reason, transitions between/from type-C and type-B QPOs are not easily detectable and are worthy of a more detailed analysis that is beyond the scope of this work. For a more detailed study, see Homan et al. (in prep). \nFigure 11. Upper panel: Light curve for Obs. #4/#36-#37 (second orbit). The red line marks the light curve interval where a type-A QPO was detected. The black line marks the interval where a type-B QPO was visible. Lower panel: PDS for the two intervals. \n<!-- image --> \nTable 2. Summary of type-A, -B and -C QPOs properties in GX 339-4. (*) The bracket values correspond to the hardening phases. \n| | A | B | C |\n|-------|----------------------|---------------------------------------------------|---------------------------------------------------|\n| ν | 6.5-8 Hz | 0.8-6.4 Hz | 0.2-9 Hz |\n| Q | 1-3 | /greaterorequalslant 6( /greaterorequalslant 2) ∗ | /greaterorequalslant 6( /greaterorequalslant 2) ∗ |\n| rms | /lessorequalslant 5% | 5 - 10% | /greaterorequalslant 10% |\n| noise | weak red | weak red | strong flat-top |', '4 DISCUSSION': 'We analyzed RXTE/PCA and HEXTE data collected over eight years of observations of the transient BHB GX 339-4 to study the properties and the behavior of LFQPOs. 115 out of 117 oscillations could be classified into the three main types (A, B, C). The coherent scenario we constructed can be compared to that of other systems (such as XTE 1859+226 and XTE J1550-564) for which the ABC classification has been performed. Different properties and relations emerge from the analysis, allowing a better characterization of the different types of QPOs. Our results confirm that the ABC classification can be extended to include spectral dependencies. The main parameters of the different types of LFQPOs observed in GX 339-4 are summarized in Table 2. \nType-C QPO: this oscillation is found in LHS and HIMS (see Belloni 2010 and Homan et al 2011 in prep.). It is observed in both the softening and hardening outburst intervals, even though during the hardening it results weaker. It appears at hardness values in the 0.2 - 0.8 range and over a large frequency range (0.2 - 9 Hz). Its centroid frequency rises as function of the time as the source undergoes the softening phase and decreases as the source hardens during the decay phase. All type-C QPOs are spread above a variability level below which only type-B and type-A QPOs are observed (see Casella et al. 2004 and Mu˜noz-Darias et al. 2011). The observed 0.1-64 Hz rms has values between 10 and 35%. Those are positively correlated with the hardness ratio (see e.g. Belloni 2010) and negatively correlated to the frequency \n(see Fig. 4). Type-C QPOs form a clear but complex pattern in a frequency versus power-law-flux plane. The frequency type-C QPOs correlates with the disk flux and form different branches in a frequency versus disk-flux plane, corresponding to the softening and hardening phases. Type-C QPO frequencies correlates well with the hardness, stressing a clear dependence on the spectral shape. \nType-A QPO: this QPO is usually observed in the SIMS, which is indeed defined on the basis of the appearance of type-A and -B QPOs. It is found in a narrow hardness (0.20 - 0.22), frequency (6.5 - 8.0 Hz) and rms ( ∼ 2 - 3%) range. The frequency at which they are found is always very close to the frequency of the last type-C QPOs observed before the transition to the SIMS. We observed type-A QPO only during the softening phase (i.e. along the upper horizontal branch in HID). The lack of this type of QPO during the hardening phase can however be ascribed to the lower statistics, as the feature is weak and broad. This is the QPO type that is found to be associated to the lowest total fractional rms in our sample. In a frequency versus power-law-flux plot, type-A QPOs appear to be grouped close to the high-frequency end of the tracks defined by the type-C QPOs and are found around the same frequency of the type-C QPOs observed close to the transition to the SIMS. \nType-B QPO: its presence defines the SIMS. The frequency range where it is observed is 0.8 - 6.4 Hz. In addition, all type-B QPOs are observed at lower frequencies with respect to the type-C QPOs observed just before the transition to the SIMS. As for typeC QPOs, these oscillations are seen in both the softening and hardening outburst phase. Total fractional rms and hardness values are lower than in the case of type-C, but higher than for type-A QPOs, ranging in the intervals 5 - 10% and 0.2 - 0.3 respectively. It is noticeable that between 5 and 10% broad band total fractional rms only type-B QPOs are observed (see Mu˜noz-Darias et al. 2011). This type of QPO follows a sharp frequency-flux correlation for both disk and powerlaw flux (Fig. 5 and 8). \nFrom Fig. 5 one can see that type-B QPOs follow a clearly different path compared to the other classes, suggesting the presence of an intrinsic difference. \nIt is interesting to compare some of the properties reported here for GX 339-4 with those observed in other sources. McClintock et al. 2009 performed a detailed spectral analysis and a supporting timing analysis on the BHC H1743-322 and compared their results with the BHC XTE J1550-564, while Sobczak et al. (2000) compared the properties of XTE J1550-564 with GRO J1655-40. These three sources showed a behavior similar to GX 339-4 in the frequency-disk flux plane, spanning a larger flux range. Sobczak et al. (2000) also investigated the relations between frequency and powerlaw flux, finding opposite relations for GRO J1655-40 and XTE J1550-564. The frequency/powerlaw-flux correlation in GX 339-4 is similar to that of GRO J1655-40, but the frequency range covered by the two sources do not overlap, making impossible a direct comparison. \nThe relation between QPO frequency and power-law photon index is the same for GX 339-4 and H1743-322, XTE J1550-564 and GRO J1655-40 (Sobczak et al. 2000; McClintock et al. 2009), following the original correlation observed by Vignarca et al. (2003). As for the frequency-total rms plane, which was originally presented for XTE J1859+226 (Casella et al. (2005)), the \nsame relation is present also in H1743-322 and XTE J1550-564 (McClintock et al. 2009) .', '4.1 Similarities and differences: a common origin for QPO-types?': 'The different types of QPOs often show similar/compatible properties (eg. centroid frequency range, QPO profile, quality factor values), some of which suggest that there could be a common origin for the different classes. However, there are systematic differences that cannot be ignored. \nAs is clear from Fig. 4, a stringent relation between the total (0.1 - 64 Hz) fractional rms values and QPO-type exists and the different types of QPOs correspond to different and well separated rms ranges (see Casella et al. 2004 and Mu˜noz-Darias et al. 2011). At the same frequency two or sometimes even three different types of QPOs can be seen (not simultaneously) depending on the variability level at which the system is observed. \nDespite the clear separation in rms, type-C and type-A QPOs follow a similar hard-flux/frequency relation. In addition, type-A QPOs and the type-C QPOs observed just before the HIMS/SIMS transition show very similar frequencies (see 4 and Tab. 3), while type-B QPOs are systematically found at lower frequencies. Unfortunately type-A QPOs constitute only a small part of our entire sample of QPOs (8 out of 117), therefore it is not possible to completely exclude a bias in the frequency association due to the small number of detections. \nIt is clear that Type-A and -C QPOs are significantly different timing features, in particular for what concerns the broad band noise associated to the two types of oscillations (strong broad band noise in the case of type-C QPOs and weak power-law or weak peaked noise in the case of type-A QPOs). However, this is not enough to rule out the possibility that type-A and type-C QPOs share a common physical origin. \nHere we discuss the evolution of the LFQPOs in the framework of the model proposed by Done et al. (2007), that also suggest possible explanation of the simultaneous spectral transition seen during an outburst. In the geometry that these authors assume, the outer accretion flow takes the form of a cool, geometrically thin, optically thick accretion disk truncated at some radius, which is larger than the last stable orbit (in the HIMS). The inner accretion flow, instead, forms a hot, geometrically thick, optically thin configuration. The inward movement of the truncation radius with increasing mass accretion rates within the hard states gives a physical basis for the hard-soft transition when the disk finally reaches the last stable orbit. The inner disk radius evolution gives also a possible origin for the LFQPOs and to the associated noise observed in the power density spectra. In this scenario LFQPOs (and in particular type-C QPOs) arise from the vertical Lense-Thirring precession (Stella & Vietri 1998) of the misaligned inner hot flow, while the broad band noise arises from propagation of Magneto Rotational Instability (MRI, Balbus & Hawley 1991 9 ) fluctuations of the same hot flow (see Ingram et al. 2009, Ingram & Done 2010, Ingram & Done 2011 for details) 10 . The QPO frequency depends \non the truncation radius and also on the optical depth of the inner hot flow that produces them, as well on the inclination of the system (see also Homan et al. 2005) and to the optical depth of the precessing inner hot flow. In this scenario, the fact that typeA QPOs and the type-C QPOs observed close to the transition are seen at the same frequency, suggest either that they are produced in a region at similar radii or in a medium at the same physical conditions (i.e. same optical depth). This second possibility is supported by the clear difference in broad-band noise level in the PDS, that is thought to be related to MRI. The type-C QPO width sets the typical timescale for the QPO signal to remain coherent. Since the QPO is not always present in the light curve, there also exists an excitation timescale responsible for the QPO appearance (Lachowicz & Done 2010). The QPO can be broadened if the excitation timescale becomes shorter than the QPO timescale, as the excitation presumably interrupts the coherent QPO light curve by phase randomizing it. In the framework set by the global LenseThirring precession of the hot flow model (Fragile et al. 2007; Ingram et al. 2009; Ingram & Done 2011) the QPO could be triggered by turbulence, resulting in a vertical kick applied by the bending waves propagating in the outer accretion flow to the precessing inner hot flow at random phase. If more than one kick occur within the QPO timescale, the QPO coherent light curve is interrupted by a random phase shift. As a result, the type-C QPO would evolve in a broader and fainter feature (Chris Done, private communication). Thus, type-A and -C QPOs could be the result of the same physical process, i.e. Lense-Thirring precession. However, given the undeniable link between the type-A QPOs and very weak noise, it remains a fact that whatever process broadens and weakens the QPOs, should be also responsible for the collapse of the noise.', '4.2 The peculiar case of the type-B QPO': 'Type-B QPOs show properties that differentiate them from the other two classes. Analyzing the frequency-hard flux relation and the frequency-rms relation, it is evident that there is a clear discontinuity within the pattern defined by type-B QPOs and other types of QPOs. While all the QPO frequencies seem to correlate with the soft flux, only type-B QPOs show a sharp correlations with the hard flux. This is remarkable because when type-B and type-A/-C QPOs are seen at similar hardness (type-C QPOs observed just before the transition and all type-A QPOs), there are no differences in the spectral shape, as one might deduce from the HID, but only in flux (see below). In addition, type-B QPOs are systematically found at lower frequencies with respect to the last type-C QPOs and type-A QPOs (see 4 and 5). \nType-B QPOs can be transient (i.e appear/disappear in few seconds and are observable only for short periods, see also Takizawa et al. 1997) and vary significantly around their centroid frequency, with a characteristic time scale of ∼ 10s (see Nespoli et al. 2003). Also type-C QPOs can appear and disappear in few seconds, but they remain observable for long periods and can be easily followed in their frequency evolution during the hardto-soft or soft-to-hard transition. Because of its intrinsic faintness, type-A QPOs cannot be followed as can be done for type-B and -C QPOs, (see Nespoli et al. 2003). \nA noticeable peculiarity of type-B QPOs is the association to flux peaks. This is particularly evident for a direct switch from or to a type-B QPO (see Sec. 3.5). The association of type-B QPOs with increases in the count rate can be seen both in the total light curve (i.e. peaks in the count rate observed in the total light curve of the source, see also Fender et al. 2009) and on shorter timescales \n(i.e. when sudden increases in count rate take place during a single RXTE pointing). Under the assumption that the count rate tracks the accretion rate, type-B QPOs would be related to increases of the local accretion rate. Alternatively, the increase in the count rate associated to those QPOs could be related to the presence of a jet component that would contribute to the hard emission. A third possibility is that type-B QPOs might occur simultaneously to sudden changes in the geometry or radiative efficiency, which would possibly cause variations in the relative contribution of the emitting component to the spectrum and in the flux. \nStarting from the precession model and following a reasoning similar to that described above, we argue that type-B QPOs are either produced in a different region located at larger radii (with respect to the region where the type-C QPOs close to the transition and type-A QPOs might originate, to match the lower frequencies observed) or coming from a modulation operated by a medium with different physical properties. In the first case, it is necessary to find a process different from the vertical Lense-Thirring precession, that would be able to produce modulations at larger radii. This hypothesis is supported by the fact that when fast switches from/to typeA/-B QPOs are observed, the type-A QPOs is consistent with being still present when the type-B QPOs appears (see also Nespoli et al. 2003). This is also valid for type-A and -B detected in separated observations. Such a property suggests that two different, eventually simultaneous mechanisms, might be responsible of the production of type-C/A QPOs and type-B QPOs. For the second case, a process able to trigger fast transitions in the physical properties of the plasma would be needed. This hypothesis is supported, as for typeA QPOs, by the fact that the transition to type-B QPOs is associated to the significant difference in the PDS broad band noise level. There is also a third possibility: type-B QPOs could be produced in the same region where type-C QPOs come from, but thanks to a different pecession mode. In the precession model the QPO arises from the surface density-weighed Lense-Thirring-precession over the inner hot flow. A sudden change in the surface density profile - for example due to a jet ejection from the very inner regions of the accretion flow - would result in a different weighing and also a different precession frequency.', '5 CONCLUSIONS': "The large amount of RXTE observations of GX 339-4 in the past eight years allowed us to analyze the spectral and temporal behavior of the source over four outbursts. We considered all the observations where a low frequency oscillation was observed and performed a complete spectral and timing analysis. Almost all the oscillations observed in the PDS could be classified following the ABC classification and the three types of QPOs display different dependences on the spectral and timing parameters, further strengthening their intrinsic differences. \nWe conclude that type-B QPOs show properties that clearly differentiate them from other types of QPOs. Their frequencies clearly correlate with the powerlaw flux, tracing a complete different pattern with the respect to type-A and -C QPOs. Type-B QPOs follow a different behavior also in a rms vs rms plane. In addition, they show a peculiar association to increases in count rate that could reflect changes in accretion rate and/or geometry in the system. \nAll the types of QPOs can be explained through the precession model (Ingram et al. 2009, Ingram & Done 2010, Ingram & Done 2011) as the result of the vertical Lense-Thirring precession of a optical translucent inner hot flow in a truncated disk geometry. \nHowever, the characteristic properties of type-B QPOs suggest that they could be the effect of a physical phenomenon different from the Lense-Thirring precession and possibly somehow related to the transition/jet ejection mechanism. \nSM acknowledges Chris Done for hospitality during her visit at University of Durham and for useful discussions that led to significant improvement of this work and the referee Jerome Rodriguez for useful comments and suggestions. SM and TB acknowledge support from grant ASI-INAF I/009/10/. TMD acknowledges Univeristy of Amsterdam and Southampton for hospitality during his visits. The research leading to these results has received funding from PRIN INAF 2007 and from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement number ITN 215212 'Black Hole Universe'. PC acknowledges support from a EU Marie Curie Intra-European Fellowship within the 7th European Community Framework Programme, under contract no. 2009-237722. This work has been partially funded by the Spanish MEC under the Consolider-Ingenio 2010 Program grant CSD2006-00070: First Science with the GTC (http://www.iac.es/consolider-ingenio-gtc/).", '12 S. Motta et al.': 'Table 3: Power-spectral classification and variability parameters. Only observations with evidence of low frequency QPOs are listed. \n| # | Obs. ID | MJD | Outburst | Hardness ratio | Total fractional rms | QPO centroid Frequency | QPO type | State |\n|-------|-------------------------------|-------------------------|------------|-------------------------------------|-----------------------------------|---------------------------------|------------|-----------|\n| 1 | 70109-01-07-00b | 52411 . 601 | 2002 | 0 . 228 ± 0 . 001 | 7 . 9 ± 0 . 1 | 5 . 8 ± 0 . 1 | B | SIMS |\n| 2 | 70110-01-14-00 | 52416 . 596 | 2002 | 0 . 252 ± 0 . 001 | 9 . 2 ± 0 . 1 | 6 . 4 ± 0 . 1 | B | SIMS |\n| 3 | 70110-01-15-00 | 52419 . 238 | 2002 | 0 . 251 ± 0 . 001 | 9 . 0 ± 0 . 1 | 5 . 7 ± 0 . 03 | B | SIMS |\n| 4 | 70108-03-02-00b | 52419 . 432 | 2002 | 0 . 209 ± 0 . 001 | 9 . 3 ± 0 . 2 | 5 . 6 ± 0 . 1 | B | SIMS |\n| 5 | 70110-01-47-00b | 52532 . 749 | 2002 | 0 . 222 ± 0 . 001 | 7 . 2 ± 0 . 2 | 6 . 2 ± 0 . 1 | B | SIMS |\n| 6 | 70110-01-89-00 | 52707 . 915 | 2002 | 0 . 256 ± 0 . 004 | 9 . 7 ± 0 . 7 | 0 . 9 ± 0 . 05 | B | SIMS |\n| 7 | 90110-02-01-03 | 53232 . 993 | 2004 | 0 . 257 ± 0 . 001 | 9 . 4 ± 0 . 2 | 4 . 1 ± 0 . 04 | B | SIMS |\n| 8 | 90704-01-02-00 | 53233 . 389 | 2004 | 0 . 271 ± 0 . 001 | 9 . 1 ± 0 . 2 | 4 . 4 ± 0 . 2 | B | SIMS |\n| 9 | 60705-01-84-02 | 53333 . 899 | 2004 | 0 . 254 ± 0 . 001 | 8 . 8 ± 0 . 1 | 5 . 2 ± 0 . 1 | B | SIMS |\n| | 91105-04-10-00 | 53466 753 | 2004 | ± 003 | ± | | | |\n| 10 | | . | | 0 . 293 0 . | 12 . 8 0 . 7 | 3 . 4 ± 0 . 1 | B? | SIMS |\n| 11 | 92035-01-04-00 | 54147 . 011 | 2007 | 0 . 262 ± 0 . 001 | 8 . 4 ± 0 . 1 8 . 6 ± 0 . 1 | 6 . 7 ± 0 . 2 | B | SIMS |\n| 12 | 92085-01-03-01 | 54162 . 665 | 2007 | 0 . 259 ± 0 . 001 | 6 ± 1 | 6 . 4 ± 0 . 1 | B | SIMS |\n| 13 | 92704-03-10-00 | 54231 . 604 | 2007 | 0 . 264 ± 0 . 002 | | 1 . 0 ± 0 . 1 | B | SIMS |\n| 14 15 | 92704-03-10-11 | 54232 . 658 54233 565 | 2007 | 0 . 276 ± 0 . 002 ± | 10 . 0 ± 0 . 5 13 7 ± 0 4 | 1 . 7 ± 0 . 1 ± | B | SIMS |\n| | 92704-03-10-12 | . | 2007 | 0 . 335 0 . 002 0 ± 001 | . . ± | 1 . 8 0 . 2 ± | B? | SIMS |\n| 16 | 95409-01-15-02 | 55304 . 714 | 2010 | . 256 0 . | 8 . 3 0 . 2 | 5 . 6 0 . 1 | B | SIMS |\n| 17 | 95409-01-15-06 | 55308 . 983 | 2010 | 0 . 236 ± 0 . 001 | 6 . 7 ± 0 . 1 | 5 . 9 ± 0 . 2 | B | SIMS |\n| 18 19 | 95409-01-16-05 95409-01-17-00 | 55315 . 695 55316 . 114 | 2010 2010 | 0 . 262 ± 0 . 001 0 . 262 ± 0 . 001 | 9 . 3 ± 0 . 1 9 . 3 ± 0 . 1 | 6 . 1 ± 0 . 2 5 . 9 ± 0 . 1 | B B | SIMS SIMS |\n| 20 | 95409-01-17-05 | 55321 . 718 | 2010 | 0 . 243 ± 0 . 001 | 7 . 5 ± 0 . 1 | 5 . 3 ± 0 . 1 | B | SIMS |\n| 21 | 95409-01-17-06 | 55322 . 230 | 2010 | 0 . 238 ± 0 . 001 | 7 . 5 ± 0 . 2 | 5 . 2 ± 0 . 1 | B | SIMS |\n| 22 | 95409-01-18-00 | 55323 . 210 | 2010 | 0 . 259 ± 0 . 001 | 8 . 6 ± 0 . 1 | 5 . 5 ± 0 . 1 | B | SIMS |\n| 23 | 95335-01-01-07 | 55324 . 189 | 2010 | 0 . 255 ± 0 . 001 | 8 . 7 ± 0 . 2 | 5 . 3 ± 0 . 04 | B | SIMS |\n| 24 | 95335-01-01-00 | 55324 . 254 | 2010 | 0 . 247 ± 0 . 001 | 8 . 5 ± 0 . 1 | 5 . 3 ± 0 . 03 | B | SIMS |\n| 25 | 95335-01-01-01 | 55324 . 393 | 2010 | 0 . 240 ± 0 . 001 | 7 . 4 ± 0 . 1 | 5 . 1 ± 0 . 02 | B | SIMS |\n| 26 | 95335-01-01-05 | 55326 . 175 | 2010 | 0 . 228 ± 0 . 001 | 7 . 1 ± 0 . 2 | 4 . 9 ± 0 . 03 | B | SIMS |\n| 27 | 95335-01-01-06 | 55326 . 280 | 2010 | 0 . 223 ± 0 . 001 | 6 . 3 ± 0 . 2 | 4 . 9 ± 0 . 03 | B | SIMS |\n| 28 | 95409-01-18-04 | 55327 . 041 | 2010 | 0 . 235 ± 0 . 002 | 8 . 5 ± 0 . 4 | 4 . 8 ± 0 . 1 | B | SIMS |\n| 29 | 95409-01-18-05 | 55327 . 262 | 2010 | 0 . 234 ± 0 . 001 | 7 . 8 ± 0 . 3 | 4 . 9 ± 0 . 04 | B | SIMS |\n| 30 | 95409-01-19-00 | 55330 . 300 | 2010 | 0 . 223 ± 0 . 001 | 6 . 5 ± 0 . 6 | 4 . 7 ± 0 . 05 | B | SIMS |\n| 31 | 96409-01-04-04 | 55585 . 947 | 2010 | 0 . 282 ± 0 . 003 | 10 . 0 ± 0 . 6 | 2 . 0 ± 0 . 1 | B | SIMS |\n| 32 | 96409-01-04-05 | 55586 . 896 | 2010 | 0 . 269 ± 0 . 004 | 9 ± 2 | 0 . 9 ± 0 . 1 | B | SIMS |\n| 33 | 96409-01-05-01 | 55591 . 615 | 2010 | 0 . 292 ± 0 . 003 | 10 ± 1 | 1 . 7 ± 0 . 1 | B | SIMS |\n| 34 | 96409-01-05-02 | 55593 . 502 | 2010 | 0 . 299 ± 0 . 003 0 . 213 ± 0 . 001 | 8 ± 3 2 6 ± 0 1 | 1 . 8 ± 0 . 1 | B | SIMS |\n| 35 | 70109-01-07-00a | 52411 . 601 | 2002 | | . . | 7 . 0 ± 0 . 5 | A | SIMS |\n| 36 | 70108-03-02-00a | 52419 . 432 | 2002 | 0 . 214 ± 0 . 001 | 2 . 8 ± 0 . 2 | 6 . 7 ± 0 . 5 | A | SIMS |\n| 37 | 70108-03-02-00a2 | 52419 . 432 | 2002 | 0 . 210 ± 0 . 001 | 2 . 8 ± 0 . 2 | 6 . 8 ± 0 . 4 | A | SIMS |\n| | | | | ± | | 7 2 ± 0 6 | A | |\n| 38 | 70110-01-45-00 | 52524 . 948 | 2002 | 0 . 222 0 . 001 | 2 . 2 ± 0 . 2 | . . | A | SIMS |\n| 39 | 70109-01-23-00 | 52529 . 580 | 2002 | 0 . 209 ± 0 . 001 | 2 . 9 ± 0 . 2 | 7 . 4 ± 1 . 3 | | SIMS |\n| 40 | 70109-01-24-00 | 52536 . 358 | 2002 | 0 . 204 ± 0 . 001 | 2 . 4 ± 0 . 2 | 8 . 0 ± 0 . 9 | A | SIMS |\n| 41 | 92085-01-02-06 | 54160 . 896 | 2007 | 0 . 217 ± 0 . 001 | 2 . 6 ± 0 . 2 | 7 . 8 ± 0 . 7 | A | SIMS |\n| 42 | 92085-01-03-04 | 54165 . 527 | 2007 | 0 . 210 ± 0 . 001 | 2 . 7 ± 0 . 1 | 7 . 7 ± 0 . 7 | A | SIMS |\n| 43 | 40031-03-02-05 | 52388 . 054 52391 318 | 2002 2002 | 0 . 766 ± 0 . 003 0 . 763 ± 0 . 003 | 29 . 8 ± 0 . 3 29 . 7 ± 0 . 2 | 0 . 20 ± 0 . 01 0 . 22 ± 0 . 02 | C C | LHS HIMS |\n| 44 | 70109-01-05-01G | . | | 0 . 697 ± 0 . 002 | 22 . 2 ± 0 . 1 | 1 . 26 ± 0 . 01 | C | HIMS |\n| 45 | 70109-01-06-00 | 52400 . 83 | 2002 | | ± | | | |\n| 46 | 70108-03-01-00 | 52400 . 853 | 2002 | 0 . 694 ± 0 . 002 | 22 . 0 0 . 1 | 1 . 30 ± 0 . 01 | C | HIMS |\n| 47 | 70110-01-10-00 70109-04-01-00 | 52402 . 492 52405 . 58 | 2002 2002 | 0 . 562 ± 0 . 002 0 . 354 ± 0 . 001 | 20 . 1 ± 0 . 1 | 4 . 20 ± 0 . 08 | C | HIMS |\n| 48 | 70109-04-01-01 | 52405 . 713 | 2002 | 0 . 356 ± 0 . 001 | 15 . 45 ± 0 . 04 15 . 44 ± 0 . 02 | 5 . 46 ± 0 . 01 5 45 ± 0 01 | C | HIMS HIMS |\n| 49 | 70109-04-01-02 | 52406 . 07 | 2002 | 0 . 360 ± 0 . 001 | 15 . 6 ± 0 . 1 | . . 5 . 34 ± 0 . 02 | C C | HIMS |\n| 50 51 | 70110-01-11-00 | 52406 . 701 | 2002 | 0 . 342 ± 0 . 001 | 15 . 0 ± 0 . 1 | 5 . 82 ± 0 . 02 | C | HIMS |\n| 52 | 70110-01-12-00 | 52410 . 528 | 2002 | 0 . 266 ± 0 . 001 | 11 . 5 ± 0 . 1 | 8 . 1 ± 0 . 2 | C | HIMS |\n| 53 | 70109-01-37-00 | 52694 . 922 | 2002 | 0 . 237 ± 0 . 002 | 10 . 7 ± 0 . 4 | 8 . 6 ± 0 . 2 | C | HIMS |\n| 54 | 70128-02-02-00 | 52696 . 355 | 2002 | 0 . 272 ± 0 . 001 | 12 . 1 ± 0 . 1 | 8 . 02 ± 0 . 04 ± | C | HIMS |\n| 55 | 50117-01-03-01 | 52706 . 767 | 2002 | 0 . 364 ± 0 . 003 | 19 . 2 ± 0 . 3 | 6 . 7 0 . 1 | C | HIMS |\n| 56 | 50117-01-03-00 | 52706 . 84 | 2002 | 0 . 363 ± 0 . 002 | 18 . 7 ± 0 . 2 | 6 . 77 ± 0 . 02 | C | HIMS |', 'Low Frequency Quasi Periodic Oscillations in GX 339-4 13': 'Table 3 - continued from previous page \n| # | Obs. ID | MJD | Outburst | Hardness ratio | Total fractional rms | QPO centroid Frequency | QPO type | State |\n|---------|-------------------------------|-------------------------|------------|-------------------------------------|-------------------------------|---------------------------------|------------|-----------|\n| 57 | 70109-02-01-00 | 52709 . 859 | 2002 | 0 . 293 ± 0 . 002 | 14 . 2 ± 0 . 2 | 8 . 0 ± 0 . 1 | C | HIMS |\n| 58 | 70109-02-01-01 | 52709 . 991 | 2002 | 0 . 289 ± 0 . 002 | 14 . 3 ± 0 . 4 | 8 . 1 ± 0 . 1 | C | HIMS |\n| 59 | 60705-01-56-00 | 52710 . 715 | 2002 | 0 . 306 ± 0 . 003 | 15 . 1 ± 0 . 4 | 7 . 8 ± 0 . 1 | C | HIMS |\n| 60 | 70110-01-94-00 | 52724 . 226 | 2002 | 0 . 395 ± 0 . 004 | 18 . 2 ± 0 . 4 | 6 . 1 ± 0 . 1 | C | HIMS |\n| 61 | 70110-01-95-00 | 52727 . 252 | 2002 | 0 . 480 ± 0 . 005 | 22 . 8 ± 0 . 5 | 4 . 7 ± 0 . 1 | C | HIMS |\n| 62 | 60705-01-59-00 | 52731 . 562 | 2002 | 0 . 586 ± 0 . 003 | 25 . 2 ± 0 . 2 | 2 . 9 ± 0 . 1 | C | HIMS |\n| 63 | 60705-01-68-00 | 53218 . 11 | 2004 | 0 . 763 ± 0 . 005 | 30 . 8 ± 0 . 3 | 0 . 5 ± 0 . 0 | C | LHS |\n| 64 | 60705-01-68-01 | 53222 . 24 | 2004 | 0 . 728 ± 0 . 004 | 26 . 9 ± 0 . 2 | 1 . 03 ± 0 . 04 | C | HIMS |\n| 65 | 60705-01-69-00 | 53225 . 40 | 2004 | 0 . 714 ± 0 . 003 | 24 . 9 ± 0 . 2 | 1 . 3 ± 0 . 0 | C | HIMS |\n| 66 | 90704-01-01-00 | 53226 . 43 | 2004 | 0 . 705 ± 0 . 003 | 24 . 9 ± 0 . 1 | 2 . 0 ± 0 . 1 | C | HIMS |\n| | | | | ± | ± | ± | | |\n| 67 | 60705-01-69-01 | 53228 . 99 | 2004 | 0 . 646 0 . 003 | 23 . 7 0 . 2 | 2 . 9 0 . 2 | C | HIMS |\n| 68 | 60705-01-70-00 | 53230 . 96 | 2004 | 0 . 437 ± 0 . 002 | 18 . 3 ± 0 . 2 17 . 5 ± 0 . 1 | 4 . 3 ± 0 . 1 | C | HIMS |\n| 69 | 90110-02-01-02 | 53232 . 34 | 2004 | 0 . 386 ± 0 . 002 | 16 . 35 ± 0 . 04 | 5 . 2 ± 0 . 1 | C | HIMS |\n| 70 71 | 90110-02-01-00 90704-01-11-00 | 53232 . 40 53472 . 33 | 2004 2004 | 0 . 361 ± 0 . 001 0 . 584 ± 0 . 004 | 24 . 9 ± 0 . 3 | 5 . 8 ± 0 . 1 2 . 7 ± 0 . 1 | C C | HIMS HIMS |\n| 72 | 92035-01-02-01 | 54133 . 922 | 2007 | 0 . 771 ± 0 . 003 | 30 . 8 ± 0 . 2 | 0 . 28 ± 0 . 01 | C | LHS |\n| 73 | 92035-01-02-02 | 54135 . 033 | 2007 | 0 . 771 ± 0 . 003 | 30 . 8 ± 0 . 2 | | | |\n| | | | 2007 | 0 . 766 ± 0 . 003 | 30 . 1 ± 0 . 2 | 0 . 30 ± 0 . 01 | C | LHS |\n| 74 | 92035-01-02-03 | 54136 . 015 54136 997 | 2007 | 0 . 759 ± 0 . 003 | 29 . 8 ± 0 . 2 | 0 . 37 ± 0 . 01 ± | C | LHS |\n| 75 76 | 92035-01-02-04 92035-01-02-08 | . 54137 . 851 | 2007 | 0 . 748 ± 0 . 003 | 27 . 5 ± 0 . 3 | 0 . 43 0 . 01 0 . 55 ± 0 . 02 | C C | LHS HIMS |\n| 77 | 92035-01-02-07 | 54138 . 83 | 2007 | 0 . 731 ± 0 . 002 | 25 . 8 ± 0 . 2 | 0 . 90 ± 0 . 01 | C | HIMS |\n| 78 | 92035-01-02-06 | 54139 . 942 | 2007 | 0 . 686 ± 0 . 002 | 22 . 4 ± 0 . 1 | 0 . 99 ± 0 . 01 | C | HIMS |\n| 79 | 92035-01-03-00 | 54140 . 204 | 2007 | 0 . 670 ± 0 . 002 | 21 . 7 ± 0 . 1 | 1 . 13 ± 0 . 01 | C | HIMS |\n| 80 | 92035-01-03-01 | 54141 . 055 | 2007 | 0 . 621 ± 0 . 002 | 20 . 7 ± 0 . 1 | 1 . 68 ± 0 . 01 | C | HIMS |\n| 81 | 92035-01-03-02 | 54142 . 036 | 2007 | 0 . 547 ± 0 . 002 | 19 . 6 ± 0 . 1 | 2 . 45 ± 0 . 01 | C | HIMS |\n| 82 | 92035-01-03-03 | 54143 . 019 | 2007 | 0 . 461 ± 0 . 002 | 18 . 3 ± 0 . 1 | 3 . 52 ± 0 . 01 | C | HIMS |\n| 83 | 92428-01-04-00 | 54143 . 870 | 2007 | 0 . 411 ± 0 . 001 | 17 . 1 ± 0 . 1 | 4 . 34 ± 0 . 02 | C | HIMS |\n| 84 | 92428-01-04-01 | 54143 . 951 | 2007 | 0 . 419 ± 0 . 001 | 17 . 2 ± 0 . 1 | 4 . 23 ± 0 . 02 | C | HIMS |\n| 85 | 92428-01-04-02 | 54144 . 086 | 2007 | 0 . 424 ± 0 . 002 | 17 . 4 ± 0 . 1 | 4 . 13 ± 0 . 03 | C | HIMS |\n| 86 | 92428-01-04-03 | 54144 . 871 | 2007 | 0 . 380 ± 0 . 001 | 16 . 4 ± 0 . 1 | 4 . 99 ± 0 . 03 | C | HIMS |\n| 87 | 92035-01-03-05 | 54145 . 114 | 2007 | 0 . 343 ± 0 . 001 | 14 . 9 ± 0 . 1 | 5 . 80 ± 0 . 03 | C | HIMS |\n| 88 | 92085-01-03-00 | 54161 . 669 | 2007 | 0 . 295 ± 0 . 001 | 12 . 9 ± 0 . 1 | 7 . 1 ± 0 . 1 | C | HIMS |\n| 89 | 92085-01-03-02 | 54163 . 698 | 2007 | 0 . 288 ± 0 . 001 | 12 . 5 ± 0 . 1 | 7 . 3 ± 0 . 2 | C | HIMS |\n| 90 | 92085-01-03-03 | 54164 . 557 54234 839 | 2007 | 0 . 296 ± 0 . 001 ± | 12 . 4 ± 0 . 1 | 7 . 0 ± 0 . 2 | C | HIMS |\n| 91 | 92704-03-11-00 | . | 2007 | 0 . 493 0 . 004 | 21 . 3 ± 0 . 4 | 4 . 0 ± 0 . 1 | C | HIMS |\n| 92 | 92704-03-11-01 | 54235 . 791 | 2007 | 0 . 549 ± 0 . 005 | 22 . 1 ± 0 . 4 | 3 . 3 ± 0 . 1 | C | HIMS |\n| 93 | 92704-04-01-01 | 54236 . 446 | 2007 | 0 . 577 ± 0 . 004 | 22 . 8 ± 0 . 6 | 3 . 0 ± 0 . 2 | C | HIMS |\n| 94 | 92704-04-01-02 | 54236 . 513 | 2007 | 0 . 587 ± 0 . 004 | 24 . 8 ± 0 . 4 | 2 . 7 ± 0 . 1 | C | HIMS |\n| 95 | 92704-03-12-00 | 54236 . 591 | 2007 | 0 . 598 ± 0 . 005 | 25 . 5 ± 1 . 1 | 2 . 7 ± 0 . 4 | C | HIMS |\n| 96 | 92704-04-01-04 | 54237 . 356 | 2007 | 0 . 593 ± 0 . 004 | 25 . 2 ± 0 . 5 | 2 . 8 ± 0 . 1 | C | HIMS |\n| 97 | 92704-04-01-05 | 54237 . 421 | 2007 | 0 . 584 ± 0 . 004 | 24 . 5 ± 0 . 4 | 3 . 1 ± 0 . 3 | C | HIMS |\n| 98 | 92704-03-12-01 | 54237 . 488 | 2007 | 0 . 596 ± 0 . 004 | 24 . 8 ± 0 . 3 | 2 . 7 ± 0 . 1 | C | HIMS |\n| 99 | 95409-01-12-04 | 55286 . 727 | 2010 | 0 . 787 ± 0 . 003 | 32 . 4 ± 0 . 4 | 0 . 22 ± 0 . 01 | C | LHS |\n| 100 | 95409-01-13-03 | 55288 . 367 | 2010 | 0 . 783 ± 0 . 003 | 31 . 5 ± 0 . 3 | 0 . 2 ± 0 . 1 | C | LHS |\n| 101 | 95409-01-13-00 | 55289 . 618 | 2010 | 0 . 777 ± 0 . 003 | 31 . 0 ± 0 . 3 | 0 . 26 ± 0 . 01 | C | LHS |\n| 102 | 95409-01-13-04 | 55290 . 722 | 2010 | 0 . 781 ± 0 . 003 | 31 . 7 ± 0 . 2 | 0 . 29 ± 0 . 01 | C | LHS |\n| 103 | 95409-01-13-02 | 55291 . 649 | 2010 | 0 . 775 ± 0 . 003 | 31 . 6 ± 0 . 3 | 0 . 32 ± 0 . 01 | C | LHS |\n| 104 | 95409-01-13-05 | 55292 . 779 55293 . 088 | 2010 | 0 . 777 ± 0 . 003 0 . 772 ± 0 . 003 | 30 . 9 ± 0 . 4 | 0 . 38 ± 0 . 02 | C | LHS |\n| 105 | 95409-01-13-01 95409-01-13-06 | 55294 . 124 | 2010 | 0 . 770 ± 0 . 003 | 30 . 6 ± 0 . 4 ± | 0 . 38 ± 0 . 05 | C | LHS |\n| 106 | 95409-01-14-01 | | 2010 | 0 . 734 ± 0 . 003 | 30 . 1 0 . 4 26 . 7 ± 0 . 3 | 0 . 47 ± 0 . 02 1 . 04 ± 0 . 01 | C C | LHS HIMS |\n| 107 | 95409-01-14-02 | 55296 . 248 55297 . 87 | 2010 2010 | 0 . 672 ± 0 . 002 | 21 . 9 ± 0 . 1 | 1 . 25 ± 0 . 01 | C | HIMS |\n| 108 | 95409-01-14-03 | 55298 . 70 | 2010 | 0 . 648 ± 0 . 003 | ± | 1 . 59 ± 0 . 01 | C | |\n| 109 110 | 95409-01-14-06 | 55299 . 766 | 2010 | 0 . 564 ± 0 . 002 | 21 . 6 0 . 2 20 . 0 ± 0 . 1 | 2 . 43 ± 0 . 01 | C | HIMS HIMS |\n| 111 | 95409-01-14-04 | 55300 . 336 | 2010 | 0 . 563 ± 0 . 002 | 20 . 0 ± 0 . 2 | 2 . 38 ± 0 . 01 | C | HIMS |\n| 112 | 95409-01-14-07 | 55300 . 923 | 2010 | 0 . 515 ± 0 . 002 | 19 . 4 ± 0 . 1 | 2 . 92 ± 0 . 0 | C | HIMS |\n| 113 | 95409-01-14-05 | 55301 . 789 | 2010 | 0 . 454 ± 0 . 002 | 17 . 7 ± 0 . 2 | 3 . 64 ± 0 . 02 | C | HIMS |\n| 114 | 95409-01-15-00 | 55302 . 196 | 2010 | 0 . 425 ± 0 . 002 | 17 . 5 ± 0 . 2 | 4 . 15 ± 0 . 03 | C | HIMS |\n| 115 | 95409-01-15-01 | 55303 . 604 | 2010 | 0 . 346 ± 0 . 001 | 14 . 6 ± 0 . 1 | 5 . 65 ± 0 . 04 | C | HIMS |', '14 S. Motta et al.': 'Table 3 - continued from previous page \n| # | Obs. ID | MJD | Outburst | Hardness ratio | Total fractional rms | QPO centroid Frequency | QPO type | State |\n|-----|----------------|-------------|------------|-------------------|------------------------|--------------------------|------------|---------|\n| 116 | 95409-01-17-02 | 55318 . 441 | 2010 | 0 . 308 ± 0 . 001 | 13 . 7 ± 0 . 1 | 6 . 67 ± 0 . 19 | C | HIMS |\n| 117 | 96409-01-06-01 | 55598 . 700 | 2010 | 0 . 491 ± 0 . 004 | 26 . 7 ± 1 . 5 | 4 . 52 ± 0 . 30 | C | HIMS | \nTable 4: Columns are: observation number, reduced χ 2 ,inner disc temperature (kT), inner disc radius R (assuming a distance of 10 kpc and an inclination of 30 o ), photon index Γ , fold Energy E fold (corresponding to high energy cutoff), total flux, hard flux and disk flux calculated in the 2 - 20 keV band and expressed in units of Crab flux. \n| # | reduced χ 2 | T Innradius | R Inn Disk | Γ | E fold | F tot /F Crab | F hard /F Crab | F disk /F Crab | instruments |\n|-------|---------------|---------------------------------------------------|-----------------|-------------------------------|-----------------|-----------------|------------------|------------------|-------------------------|\n| 1 | 0.92 | 0 . 94 +0 . 02 - 0 . 04 | 43 +3 - 2 | 2 . 6 +0 . 1 - 0 . 1 | - | 0.501 | 0.063 | 0.321 | PCA+HEXTE |\n| 2 | 1.01 | 0 . 93 +0 . 04 - 0 . 04 | 41 +5 - 4 | 2 . 6 +0 . 2 - 0 . 1 | - | 0.462 | 0.070 | 0.270 | PCA+HEXTE |\n| 3 | 1.11 | 0 . 93 +0 . 05 - 0 . 04 | 38 +5 - 4 | 2 . 8 +0 . 2 - 0 . 2 | - | 0.434 | 0.066 | 0.235 | PCA+HEXTE |\n| 4 | 0.81 | 0 . 90 +0 . 04 | 43 +6 - 4 | 2 . 6 +0 . 2 - 0 . 1 | - | 0.395 | 0.048 | 0.259 | PCA+HEXTE |\n| 5 | 0.96 | - 0 . 03 0 . 91 +0 . 04 - 0 . 03 | 45 +4 - 4 | 3 . 0 +0 . 2 - 0 . 2 | - | 0.494 | 0.060 | 0.297 | PCA+HEXTE |\n| 6 | 0.81 | 0 . 61 +0 . 06 - 0 . 15 | 33 +31 - 7 | 3 . 8 +0 . 2 - 0 . 3 | - | 0.027 | 0.004 | 0.016 | PCA+HEXTE |\n| 7 | 0.85 | 0 . 80 +0 . 06 - 0 . 03 | 37 +4 - 6 | 2 . 7 +0 . 2 - 0 . | - | 0.178 | 0.027 | 0.104 | PCA+HEXTE |\n| 8 | 0.84 | 0 . 85 +0 . 04 - 0 . 03 | 31 +3 - 3 | 4 2 . 6 +0 . 1 - 0 . 1 | - | 0.175 | 0.029 | 0.099 | PCA+HEXTE |\n| 9 | 1.07 | 0 . 90 +0 . 05 - 0 . 03 | 38 +3 | 2 . 5 +0 . 1 1 | - | | | | PCA+HEXTE |\n| 10 | 0.93 | 0 . 81 +0 . 11 - 0 . 07 | - 4 19 +5 | - 0 . 2 . 4 +0 . 4 | - | 0.314 | 0.047 | 0.195 | PCA+HEXTE |\n| | | +0 . 04 | - 4 +3 | - 0 . 4 +0 . 1 03 | | 0.056 | 0.011 | 0.030 | |\n| 11 | 0.92 | 0 . 93 - 0 . 02 | 46 - 4 | 2 . 7 - 0 . | - | 0.653 | 0.099 | 0.367 | PCA+HEXTE |\n| 12 | 1.08 0.89 | 0 . 92 +0 . 03 - 0 . 03 0 . 76 +0 . 05 | 44 +4 - 3 25 +4 | 2 . 7 +0 . 1 - 0 . 1 +0 . | - | 0.536 | 0.080 | 0.307 0.035 | PCA+HEXTE |\n| 13 14 | 0.99 | - 0 . 04 0 . 78 +0 . 04 | - 4 22 +5 | 2 . 2 3 - 0 . 3 2 . 1 +0 . 3 | - - | 0.054 | 0.008 0.009 | 0.033 | PCA+HEXTE |\n| 15 | 1.04 | - 0 . 05 0 . 8 +0 . 1 - 0 . 1 | - 3 16 +3 - 3 | - 0 . 2 2 . 1 +0 . | - | 0.053 | 0.010 | 0.025 | PCA+HEXTE |\n| | | 0 +0 . 03 | +4 | 3 - 0 . 2 4 | | 0.046 | | | PCA+HEXTE |\n| 16 | 1.26 | . 96 - 0 . 04 | 40 - 3 2 +4 | . +0 . 04 - 0 . 04 | - | 0.496 | 0.071 | 0.318 | PCA+HEXTE * |\n| 17 | 1.18 | 0 . 90 +0 . 02 - 0 . 03 | 48 - 3 2 38 +4 | . 5 +0 . 1 . 05 | - | 0.492 | 0.062 | 0.329 | PCA+HEXTE * |\n| 18 | 1.00 | 0 . 94 +0 . 03 - 0 . 04 +0 . 04 | - 3 43 +3 - 4 | - 0 2 . 5 +0 . 1 - 0 . 1 +0 . | - | 0.438 | 0.068 | 0.261 | PCA+HEXTE * |\n| 19 20 | 1.25 1.23 | 0 . 91 - 0 . 02 0 . 93 +0 . 04 | 2 39 +3 - 4 | . 3 1 - 0 . 04 2 . 4 +0 . 1 | - - | 0.432 0.382 | 0.064 | 0.275 | PCA+HEXTE * PCA+HEXTE * |\n| 21 | 1.54 | - 0 . 03 0 . 89 +0 . 04 - | 2 . | - 0 . 1 36 +0 . 04 | - | 0.383 | 0.052 | 0.252 | PCA+HEXTE * |\n| 22 | 1.74 | 0 . 04 44 +0 . 04 38 | +4 | - 0 . 04 04 | - | 0.384 | 0.049 | 0.263 | |\n| 23 | 1.07 | 0 . 93 - 0 . 03 - . 03 41 | +4 - 5 2 . | 38 +0 . - 0 . 04 | - | 0.391 | 0.057 | 0.242 | PCA+HEXTE * |\n| | | 0 . 89 +0 - 0 . 03 . 04 | 4 +5 4 | . 7 +0 . 4 - 0 +0 | | | 0.056 | 0.225 | PCA+HEXTE * |\n| 24 | 1.07 | - 0 . 87 +0 - 0 . 01 43 +4 - | 2 4 2 | . 4 . 7 . 2 - 0 . 2 | - | 0.385 | 0.054 | 0.226 | PCA+HEXTE * |\n| 25 | 1.12 | 0 . 86 +0 . 04 03 45 | 2 | . 7 +0 . 2 - 0 . | - | 0.380 | 0.051 | 0.231 | PCA+HEXTE * |\n| 26 | 1.03 | - 0 . 0 . 84 +0 . 04 03 47 +5 - | +3 - 4 | 2 . 9 +0 . 2 | - | 0.371 | 0.046 | 0.219 | PCA+HEXTE * |\n| 27 | 1.28 | - 0 . 0 . 85 +0 . 04 - 0 . 03 46 +5 - | 5 2 5 2 | - 0 . 4 . 7 +0 . 3 - 0 . 3 | - | 0.357 | 0.043 | 0.230 | PCA+HEXTE * |\n| 28 | 1.26 | 0 . 89 +0 . 05 - 0 . 04 41 +5 - | 5 2 | . 3 +0 . 1 - 0 . 1 | - | 0.327 | 0.042 | 0.230 | PCA+HEXTE * |\n| 29 | 1.50 | 0 . 90 +0 . 05 - 0 . 05 40 - 04 +5 | +7 4 2 . | 4 +0 . 1 - 0 . 03 +0 . 1 | - | 0.342 | 0.045 | 0.232 | PCA+HEXTE * |\n| 30 31 | 1.21 1.27 | 0 . 86 +0 . - 0 . 03 46 - 4 0 . 71 +0 . 06 31 +14 | 2 . 2 | 4 - 0 . 1 . 0 +0 . 2 | - - | 0.330 0.055 | 0.039 0.009 | 0.232 0.038 | PCA+HEXTE * PCA+HEXTE * |\n| 33 | 0.94 | - 0 . 03 0 . 77 +0 . 06 - 0 . 07 22 - 4 06 +2 | - 3 +4 1 | - 0 . 1 . 8 +0 . 1 - 0 . +0 | - | 0.042 | 0.007 | 0.028 | PCA+HEXTE * |\n| 34 | 1.31 | 0 . 85 +0 . 15 | 1 . | 1 . 1 . | | | 0.007 | 0.023 | |\n| | | - 0 . 04 - 2 02 +6 | | 8 - 0 | - | 0.035 | | | PCA+HEXTE * |\n| 35 | 1.09 | 0 . 94 +0 . - 0 . 44 - 2 | 2 | 1 . 6 +0 . 1 0 1 | - | 0.488 | 0.055 | 0.336 | PCA+HEXTE |\n| 36 | 1.31 | 04 0 . 90 +0 . 02 01 44 +3 - 3 | 2 | - . +0 . 1 1 | - | | | | |\n| 37 | 1.31 | - 0 . 0 90 +0 . 02 44 +3 | 2 | . 6 - 0 . +0 . 1 | - | 0.394 | 0.045 | 0.271 | PCA+HEXTE |\n| 38 | 0.97 | . - 0 . 02 - +0 . 04 44 +5 | 3 | . 6 - 0 . 1 +0 . 1 | - | 0.394 | 0.045 | 0.271 | PCA+HEXTE |\n| | | 0 . 95 - 0 . 04 - 02 +3 | 4 2 | . 7 - 0 . | | 0.531 | 0.065 | 0.342 | PCA+HEXTE |\n| 39 | 0.85 | 0 . 95 +0 . - 0 . 03 43 - | 2 2 | 2 . 6 +0 . 1 - 0 . 1 | - | 0.480 | 0.053 | 0.335 | PCA+HEXTE |\n| 40 | 0.89 | 0 . 90 +0 . 03 - 0 . 02 48 +3 - +0 . 04 | 4 2 +2 | . 6 +0 . 1 - 0 . 1 +0 1 | - | 0.448 | 0.047 | 0.311 | PCA+HEXTE |\n| 41 | 0.86 | 0 . 90 - 0 . 02 +0 . 04 | 51 - 4 +3 | 2 . 7 . - 0 . 1 | - | 0.535 | 0.060 | 0.357 | PCA+HEXTE |\n| 42 | 1.08 | 0 . 88 - 0 . 02 49 | - 4 | 2 . 8 +0 . 1 - 0 . 2 | - | 0.461 | 0.050 | 0.306 | PCA+HEXTE |\n| 43 | 1.19 | - | - 1 . | 74 +0 . 02 - 0 02 | 119 +8 - 7 | 0.401 | 0.236 | 0.000 | PCA+HEXTE |\n| 44 | 0.88 | - | - 1 . | . 78 +0 . 02 - 0 . 02 +0 . 02 | 133 +10 - 9 +11 | 0.407 | 0.238 | 0.000 | PCA+HEXTE |\n| 45 | 0.91 | - | - 1 . | 96 - 0 . 02 +0 . 02 | 122 - 10 129 +9 | 0.430 | 0.230 | 0.000 | PCA+HEXTE |\n| 46 47 | 1.50 0.84 | - - | - 1 . 2 . | 98 - 0 . 02 45 +0 . 03 | - 8 - | 0.428 0.410 | 0.227 | 0.000 | PCA+HEXTE |\n| 48 | 0.80 | 1 . 01 +0 . 05 0 . 05 | - 26 +3 2 . | - 0 . 03 59 +0 . 07 | - | | 0.173 0.122 | 0.000 0.164 | PCA+HEXTE PCA+HEXTE |\n| 49 | 1.12 | - 0 . 99 +0 . 04 - 0 . 04 | - 2 27 +3 - 2 . | - 0 . 22 56 +0 . 06 - | - | 0.494 0.499 | 0.123 | 0.171 | PCA+HEXTE |\n| 50 | | 1 . 0 . 1 0 . 1 | 2 24 | 0 . 15 2 . 7 +0 . 1 0 . | | | | 0.140 | PCA+HEXTE |\n| 51 | 0.85 | +0 - 1 . 03 +0 . 06 | +3 - 4 28 +3 | - 1 2 . 2 +0 . 4 | - | 0.471 | 0.119 | 0.215 | PCA+HEXTE |\n| 52 | 0.84 | - 0 . 06 . 04 0 . 03 | - 3 | . 2 . 1 | - | 0.460 | 0.106 | | |\n| | 0.94 | 0 . 94 +0 - | 39 +4 - 3 | 2 . 5 | - | 0.467 | | 0.259 | PCA+HEXTE |\n| 53 54 | 0.89 | 0 . 69 +0 . 05 | 30 +8 - | - 0 +0 - 0 . 2 2 . 4 +0 . 6 | - | 0.046 | 0.078 0.006 | | PCA+HEXTE |\n| | 1.36 | - 0 . 05 0 . +0 . 03 - 0 . 03 | 5 23 +3 - 3 | - 0 . 2 2 . 6 +0 . 1 0 . 1 | - | 0.041 | 0.007 | 0.030 0.022 | PCA+HEXTE |\n| 55 | | 73 +0 . 1 | +6 2 | - +0 . 2 - | | | | | |\n| | 0.67 | 0 . 9 - 0 . 1 | 11 - | 2 . 5 0 . 4 | - | 0.034 | 0.009 | 0.011 | PCA+HEXTE | \n| # | reduced χ 2 | T Innradius | R Inn Disk | Γ | E fold | F tot /F Crab | F hard /F Crab | F disk /F Crab | instruments |\n|---------|---------------|----------------------------------------------|------------------------------|----------------------------------------------|-------------------|-----------------|------------------|------------------|-------------------------|\n| 56 | 1.11 | 0 78 +0 . 07 - 0 . 06 | 15 +4 - 3 | 2 . 5 +0 . 2 - 0 . 1 | - | 0.035 | 0.009 | 0.013 | PCA+HEXTE |\n| 57 | 0.97 | . 0 . 75 +0 . 04 - 0 . 04 | 19 +3 - 2 | 2 . 2 +0 . 2 - 0 . 2 | - | 0.032 | 0.006 | 0.019 | PCA+HEXTE |\n| 58 | 0.74 | 0 . 7 +0 . 1 - 0 . 1 | 27 +8 5 | 2 . 4 +0 . 3 - 0 . 4 | - | 0.035 | 0.006 | 0.021 | PCA+HEXTE |\n| 59 | 1.01 | 0 . 7 +0 . 1 - 0 . 1 | - 21 +6 | 2 . 8 +0 . 4 | | | | 0.013 | PCA+HEXTE |\n| 60 | 1.29 | - | - 6 - | - 0 . 4 3 . 2 +0 . 1 | - - | 0.026 0.035 | 0.005 0.010 | 0.000 | PCA+HEXTE |\n| 61 | 1.03 | - | - | - 0 . 1 2 . 9 +0 . 1 - 0 . 1 | - | 0.034 | 0.012 | 0.000 | PCA+HEXTE |\n| 62 | 1.17 | - | - | 2 . 48 +0 . 03 - 0 . 06 | - | 0.033 | 0.015 | 0.000 | PCA+HEXTE |\n| 63 | 1.02 | - | - | 1 . 83 +0 . 03 - 0 . 03 | - | 0.102 | 0.061 | 0.000 | PCA+HEXTE |\n| 64 | 1.15 | - | - | 1 . 98 +0 . 03 - 0 . 03 | - | 0.126 | 0.070 | 0.000 | PCA+HEXTE |\n| | | | | 1 97 +0 . 03 | | | | | |\n| 65 | 1.20 | - | - | . - 0 . 01 | - | 0.132 | 0.073 | 0.000 | PCA+HEXTE |\n| 66 | 1.40 | - | - | 2 . 01 +0 . 03 - 0 . 02 | - | 0.136 | 0.074 | 0.000 | PCA+HEXTE |\n| 67 68 | 0.90 0.94 | - 1 2 +0 . 1 | - 8 +6 | 2 . 23 +0 . 03 - 0 . 03 2 . 3 +0 . 1 | - - | 0.134 0.124 | 0.066 0.040 | 0.000 0.038 | PCA+HEXTE PCA+HEXTE |\n| 69 | 1.09 | . - 0 . 1 1 . 17 +0 . 09 | - 1 11 +2 | - 0 . 1 1 . 7 +0 . 7 | 64 +13 | | | | |\n| 70 | 0.88 | - 0 . 09 1 . 13 +0 . 05 . | - 1 12 +1 | - 0 . 1 1 . 9 +0 . 4 | - 17 102 +11 | 0.127 0.127 | 0.035 0.033 | 0.065 0.064 | PCA+HEXTE PCA+HEXTE |\n| 71 | 1.14 | - 0 08 - | - 1 - | - 0 . 2 2 . 4 +0 . 1 - 0 . 1 | - 38 - | 0.046 | 0.021 | 0.000 | PCA+HEXTE |\n| 72 | 1.30 | - | - | 1 . 82 +0 . 02 - 0 . 02 | 163 +15 - 14 | 0.367 | 0.214 | 0.000 | PCA+HEXTE |\n| | | | | +0 . 02 | +11 | | | | |\n| 73 | 1.19 | - | - | 1 . 80 - 0 . 02 | 139 - 9 | 0.371 | 0.216 | 0.000 | PCA+HEXTE |\n| 74 75 | 1.52 1.25 | - - | - | 1 . 83 +0 . 02 - 0 . 02 +0 . 02 | 146 +13 - 10 +11 | 0.386 | 0.222 | 0.000 | PCA+HEXTE PCA+HEXTE |\n| 76 | 1.17 | - | - - | 1 . 84 - 0 . 02 1 . 86 +0 . 02 | 137 - 9 131 +16 | 0.413 0.424 | 0.236 0.239 | 0.000 0.000 | PCA+HEXTE |\n| 77 | 1.13 | - | - | - 0 . 02 . 92 +0 . 02 | - 12 140 +13 | 0.433 | 0.238 | 0.000 | PCA+HEXTE |\n| 78 | 1.03 | - | - | 1 - 0 . 02 2 . 05 +0 . 02 | - 11 142 +14 | 0.434 | 0.223 | 0.000 | PCA+HEXTE |\n| 79 | 1.05 | - | - | - 0 . 02 2 . 09 +0 . 02 - 0 . 02 | - 11 164 +19 - 16 | 0.436 | 0.219 | 0.000 | PCA+HEXTE |\n| 80 | 0.81 | - | - | 2 . 25 +0 . 02 - 0 . 02 | - | 0.443 | 0.206 | 0.000 | PCA+HEXTE |\n| 81 | 0.92 | - | - | 2 . 539 +0 . 004 - 0 . 004 | - | 0.475 | 0.191 | 0.000 | PCA+HEXTE |\n| 82 | 1.00 | 1 . 0 +0 . 0 - 0 . 1 +0 . 0 | 19 +6 - 3 | 2 . 46 +0 . 14 - 0 . 22 | - | 0.501 | 0.166 | 0.085 | PCA+HEXTE PCA+HEXTE |\n| 83 | 1.00 | 0 . 980 - 0 . 004 +0 . 0 | 23 . 4 +0 . 2 - 0 . 2 | 2 . 589 +0 . 003 - 0 . 003 | - | 0.533 | 0.154 | 0.121 | |\n| 84 | 0.96 | 1 . 035 - 0 . 005 | 18 . 8 +0 . 2 - 0 . 2 | 2 . 637 +0 . 004 - 0 . 004 | - | 0.529 | 0.157 | 0.103 | PCA+HEXTE |\n| 85 | 0.85 | 1 . 02 +0 . 0 - 0 . 01 +0 . 0 | 19 . 0 +0 . 2 - 0 . 2 +0 . 2 | 2 . 638 +0 . 004 - 0 . 004 | - | 0.522 | 0.157 | 0.098 | PCA+HEXTE |\n| 86 | 0.88 | 1 . 024 - 0 . 004 +0 07 | 22 . 6 - 0 . 2 +3 | 2 . 699 +0 . 004 - 0 . 004 +0 05 | - | 0.538 | 0.141 | 0.142 | PCA+HEXTE |\n| 87 88 | 1.10 1.05 | 0 . 98 . - 0 . 02 0 . 929 +0 . 00 - 0 . 002 | 30 - 4 38 . 2 +0 . 2 | 2 . 63 . - 0 . 05 2 . 532 +0 . 003 - 0 . 003 | - - | 0.554 0.488 | 0.127 | 0.203 0.246 | PCA+HEXTE |\n| 89 | 1.26 | 0 . 922 +0 . 00 | - 0 . 2 38 . 5 +0 . 2 | 2 . 525 +0 . 003 0 003 | - | 0.462 | 0.092 0.085 | 0.240 | PCA+HEXTE PCA+HEXTE |\n| 90 | 0.97 | - 0 . 002 0 . 898 +0 . 00 . 002 | - 0 . 2 39 . 6 +0 . 2 . 2 | - . 2 . 641 +0 . 003 | - | 0.465 | 0.088 | 0.222 | PCA+HEXTE |\n| 91 | 0.98 | - 0 - | - 0 - | - 0 . 003 2 . 88 +0 . 02 - 0 . 01 | - | 0.048 | 0.018 | 0.000 | PCA+HEXTE |\n| 92 | 1.32 | - | - | 2 . 53 +0 . 02 - 0 . 01 | - | 0.046 | 0.019 | 0.000 | PCA+HEXTE |\n| | | | | +0 . 01 0 | | | | | |\n| 93 | 1.23 | - | - | 2 . 46 - . 01 | - | 0.045 | 0.020 | 0.000 | PCA+HEXTE |\n| 94 | 1.28 | - | - | 2 . 45 +0 . 01 - 0 . 01 | - | 0.045 | 0.021 | 0.000 | PCA+HEXTE |\n| 95 | 1.30 | - | - | 2 . 38 +0 . 02 - 0 . 01 | - | 0.045 | 0.021 | 0.000 | PCA+HEXTE |\n| 96 | 1.16 | - | - | 2 . 45 +0 . 01 - 0 . 01 | - | 0.045 | 0.021 | 0.000 | PCA+HEXTE |\n| 97 | 1.25 | - | - | 2 . 52 +0 . 01 - 0 . 01 | - | 0.044 | 0.020 | 0.000 | PCA+HEXTE |\n| 98 | 1.97 | - | - | 2 . 45 +0 . 06 - 0 . 05 | - | 0.043 | 0.020 | 0.000 | PCA+HEXTE |\n| 99 | 1.47 | - | - | 1 . 81 +0 . 02 - 0 . 02 | - | 0.285 | 0.168 | 0.000 | PCA+HEXTE * |\n| 100 | 1.77 | - | - | 1 . 80 +0 . 02 - 0 . 02 | - | 0.293 | 0.172 | 0.000 | PCA+HEXTE * |\n| 101 | 1.55 | - | - | 1 . 82 +0 . 02 - 0 . 03 | - | 0.300 | 0.175 | 0.000 | PCA+HEXTE * |\n| 102 | 1.86 1.73 | - - | - - | 1 . 82 - 0 . 03 1 . 85 +0 . 02 - 0 . 03 | - - | 0.307 0.317 | 0.179 | 0.000 | PCA+HEXTE * |\n| 103 104 | 1.68 | - | - | 1 . 86 +0 . 02 - 0 . 04 | - | 0.317 | 0.183 | 0.000 | PCA+HEXTE * PCA+HEXTE * |\n| 105 | 1.24 | - | - | 1 . 81 +0 . 03 | 171 . 893 +24 | 0.335 | 0.183 0.194 | 0.000 | |\n| 106 | 1.36 | - | - | - 0 . 03 1 . 85 +0 . 03 | - 18 - | 0.340 | 0.194 | 0.000 0.000 | PCA+HEXTE * |\n| 107 | 1.08 | - | - | - 0 . 02 1 . 92 +0 . 03 | 191 . 624 +37 | | | | PCA+HEXTE * PCA+HEXTE * |\n| 108 | 1.25 | - | | - 0 . 04 +0 . 03 | - 31 - | 0.347 | 0.190 | 0.000 | |\n| 109 | 1.50 | - | - | 2 . 10 - 0 . 03 | | 0.367 | 0.184 | 0.000 | PCA+HEXTE * |\n| 110 | | +0 . | - +330 | 2 . 20 +0 . 00 - 0 . 00 | - | 0.375 | 0.179 | 0.000 | PCA+HEXTE * |\n| | 1.17 | 0 . 7 57 - 0 . 3 | 39 - 10 +188 | 2 . 29 +0 . 09 - 0 . 16 | - | 0.405 | | 0.043 | PCA+HEXTE * |\n| 111 112 | 1.10 1.55 | 0 . 8 +0 . 5 - 0 . 4 0 . 98 +0 . 22 - 0 . 07 | 22 - 9 20 +10 - 5 | 2 . 2 +0 . 2 - 0 . 2 2 . 1 +0 . 2 - 0 . 1 | - 15 +150 | 0.392 | 0.160 0.158 | 0.046 | PCA+HEXTE * PCA+HEXTE * |\n| | | | | | 153 . - 52 | 0.392 | 0.144 | 0.083 | |\n| 113 | | 027 +0 . 004 | 20 . 9 +0 . 2 | 2 . 185 +0 . - 0 . | | | | | |\n| | 1.19 | 1 . - 0 . 004 | - 0 . 2 | 004 004 | - | 0.405 | 0.129 | 0.122 | PCA+HEXTE * |', 'Low Frequency Quasi Periodic Oscillations in GX 339-4 17': 'Table 4 - continued from previous page \n| # | reduced χ 2 | T Innradius | R Inn Disk | Γ | E fold | F tot /F Crab | F hard /F Crab | F disk /F Crab | instruments |\n|-----|---------------|----------------------------|------------------------------|--------------------------------|----------|-----------------|------------------|------------------|---------------|\n| 114 | 1.19 | 0 . 9 +0 . 3 - 0 . 1 | 30 +15 - 7 | 2 . 4 +0 . 2 - 0 . 4 | - | 0.429 | 0.132 | 0.137 | PCA+HEXTE * |\n| 115 | 1.4 | 1 . 002 +0 . 003 - 0 . 003 | 28 . 9 +0 . 1 - 0 . 1 +0 . 2 | 2 . 274 +0 . 004 - 0 . 004 004 | - | 0.454 | 0.1 | 0.214 | PCA+HEXTE * |\n| 116 | 1.11 | 0 . 853 +0 . 002 - 0 . 002 | 42 . 8 - 0 . 2 | 2 . 457 +0 . - 0 . 004 | - | 0.387 | 0.074 | 0.197 | PCA+HEXTE * |\n| 117 | 1.24 | - | - | 2 . 95316 +0 . 10 - 0 . 09 | - | 0.032 | 0.012 | 0 | PCA+HEXTE * |', 'REFERENCES': "| Balbus S. 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2010PhRvL.105c1302E
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Black Hole Entropy and SU(2) Chern-Simons Theory
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2010-01-01
| 9 | 0.45 | 164 |
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Black holes (BH’s) in equilibrium can be defined locally in terms of the so-called isolated horizon boundary condition given on a null surface representing the event horizon. We show that this boundary condition can be treated in a manifestly SU(2) invariant manner. Upon quantization, state counting is expressed in terms of the dimension of Chern-Simons Hilbert spaces on a sphere with punctures. Remarkably, when considering an ensemble of fixed horizon area a<SUB>H</SUB>, the counting can be mapped to simply counting the number of SU(2) intertwiners compatible with the spins labeling the punctures. The resulting BH entropy is proportional to a<SUB>H</SUB> with logarithmic corrections ΔS=-(3)/(2)loga<SUB>H</SUB>. Our treatment from first principles settles previous controversies concerning the counting of states.
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https://arxiv.org/pdf/0905.3168.pdf
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{'Black hole entropy and SU (2) Chern-Simons theory': "Jonathan Engle 1 , Karim Noui 2 , and Alejandro Perez 1 1 Centre de Physique Th'eorique ∗ , Campus de Luminy, 13288 Marseille, France. and 2 Laboratoire de Math'ematique et Physique Th'eorique † , 37200 Tours, France. (Dated: September 12, 2018) \nBlack holes in equilibrium can be defined locally in terms of the so-called isolated horizon boundary condition given on a null surface representing the event horizon. We show that this boundary condition can be treated in a manifestly SU (2) invariant manner. Upon quantization, state counting is expressed in terms of the dimension of Chern-Simons Hilbert spaces on a sphere with punctures. Remarkably, when considering an ensemble of fixed horizon area a H , the counting can be mapped to simply counting the number of SU (2) intertwiners compatible with the spins labelling the punctures. The resulting BH entropy is proportional to a H with logarithmic corrections ∆ S = -3 2 log a H . Our treatment from first principles settles previous controversies concerning the counting of states. \nPACS numbers: \nBlack holes are intriguing solutions of general relativity describing the physics of gravitational collapse. These fascinating systems-whose existence in our universe is supported by a great amount of observational evidenceare remarkably simple. However, in the interior of the event horizon, the predictive power of classical general relativity breaks down due to the unavoidable appearance of un-physical divergences of the gravitational field ( singularities ). Dimensional arguments imply that quantum effects cannot be neglected near singularities . In this precise sense, black holes (BH) provide the most tantalizing theoretical evidence for the need of a more fundamental (quantum) description of the gravitational field. \nQuantum effects are also important outside the horizon. Indeed the semiclassical calculations of Hawking [1] show that BH's radiate as perfect black bodies at temperature proportional to their surface gravity and have an entropy S = a H / 4 /lscript 2 p , where /lscript 2 p = G /planckover2pi1 /c 3 is the Planck area. This entropy is expected to arise from the huge number of microstates of the underlying fundamental quantum theory describing the BH, and therefore its computation from basic principles is an important test of any candidate quantum theory of gravity. This letter proposes a new and more fundamental framework for the computation of BH entropy in loop quantum gravity (LQG) and establishes a precise relationship between SU (2) Chern-Simons (CS) theory and quantum black hole physics as first explored in [3]. \nOur treatment clarifies the description of both the classical as well as the quantum theory of black holes in LQG making the full picture more transparent. We show that, in contrast with prior results [2], the gauge symmetry of LQG need not be reduced from SU (2) to U (1) at the horizon. Even when the U (1) reduction is perfectly viable at the classical level, it leads to imposition of certain \ncomponents of the quantum constraints only in a weak sense. Our SU (2) invariant formulation-equivalent to the U (1) at the classical level-avoids this issue and allows the imposition of the constraints strongly in the Dirac sense. This leads to a drastic simplification of the quantum theory in which states of a black hole are now in one-to-one correspondence with the fundamental basic volume excitations of LQG given by single intertwiner states. This settles certain controversies concerning the relevant quantum numbers to be considered in the counting of states. The main quantitative result of our work is the correction of the value of the BH entropy. \nThe standard definition of a BH as a spacetime region of no escape is a global definition. This notion of BH requires a complete knowledge of a spacetime geometry and is therefore not suitable for describing local physics. This is solved by using instead the notion of Isolated horizons (IH): defined by extracting from the definition of a Killing horizon the minimum conditions necessary for the laws of BH mechanics to hold [4]. They may be thought of as 'apparent horizons in equilibrium'. Even though IH are very general, allowing rotation and distortion, for simplicity here we concentrate on the case in which the horizon geometry is spherically symmetric. In the vacuum case, the latter are easy to visualize in terms of the characteristic formulation of general relativity with initial data given on null surfaces: spacetimes with such IH are solutions to Einstein's equations where Schwarzschild data are given on the horizon and suitable free radiation is given at a transversal null surface [6] (see Fig. 1). \nThe calculation of black hole entropy in LQG is done by quantizing the sector of the phase space of general relativity corresponding to solutions having an IH. At the technical level this sector is defined by postulating the existence of a null boundary ∆ ⊂ M with topology S 2 × R with the pull-back of the gravitational field to ∆ satisfying the isolated horizon boundary conditions. \nIt is well known that the initial value formulation of general relativity can be characterized in terms of a triad field e i a through Σ = e ∧ e -encoding the intrinsic spatial metric of M as q ab = e i a e j b δ ij -and certain components of \nthe extrinsic curvature K ab of M defined by K i a = K ab e b i . It can be shown that the symplectic structure of gravity \nΩ M ( δ 1 , δ 2 ) = 1 8 πG ∫ M [ δ 1 Σ i ∧ δ 2 K i -δ 2 Σ i ∧ δ 1 K i ] (1) \nis preserved in the presence of an IH. More precisely in the shaded space-time region in Fig. 1 one has \nΩ M 2 ( δ 1 , δ 2 ) = Ω M 1 ( δ 1 , δ 2 ) . (2) \nThat is, the symplectic flux across the isolated horizon ∆ vanishes due to the isolated horizon boundary condition [4, 5]. One also has that, on shell, phase space tangent vectors δ α , δ v of the form \nδ α Σ = [ α, Σ] , δ α K = [ α, K ]; δ v Σ = L v Σ , δ v K = L v K \nfor α : M → su (2) and v ∈ Vect( M ) tangent to the horizon, are degenerate directions of Ω M from which one concludes that SU (2) triad rotations and diffeomorphisms are gauge symmetries [7]. Hence, the IH boundary condition breaks neither the symmetry under these diffeomorphisms nor the SU (2) internal gauge symmetry introduced by the use of triad variables. \nFIG. 1: The characteristic data for a (vacuum) spherically symmetric isolated horizon corresponds to Schwarzschild data on ∆, and free radiation data on the transversal null surface. \n<!-- image --> \nAshtekar-Barbero connection variables are necessary for the quantization ' a la LQG. When there is no boundary the SU (2) connection \nA i a = Γ i + βK i a (3) \nis canonically conjugate to /epsilon1 abc β -1 Σ i bc / 2 where β is the so-called Immirzi parameter. As shown below, in the presence of a boundary the situation is more subtle: the symplectic structure acquires a boundary term Ω H . Due to the fact that, at the horizon, phase space tangent vectors δ are linear combinations of SU (2) gauge transformations and diffeomorphisms tangent to the horizon H = M ∩ ∆, the boundary term Ω H is completely fixed by the requirement of gauge invariance, i.e., the condition that local SU (2) transformations, now taking the form \nδ α Σ = [ α, Σ] , δ α A = -d A α (4) \nas well as diffeomorphisms preserving H , continue to be degenerate directions of the symplectic structure. The symplectic structure, in the new variables, becomes 1 \nκ Ω M = ∫ M 2 δ [1 Σ i ∧ δ 2] A i -a H π (1 -β 2 ) ∫ H δ 1 A i ∧ δ 2 A i , (5) \nwhere κ = 8 πGβ , a H is the horizon area, and we have used the IH boundary condition which in terms of Ashtekar-Barbero variables is found [5] to take the form \nΣ i + a H π (1 -β 2 ) F i ( A ) = 0 . (6) \nNote that the boundary contribution to the symplectic structure is given by an SU (2) CS symplectic form. One can also show directly that the boundary term contribution is necessary for time evolution to preserve the symplectic form. \nAnother consequence of the fact that SU (2) transformations and diffeomorphisms preserving H are gauge is that (in the canonical formulation) they are Hamiltonian vector fields generated by first class constraints. More precisely one has that \nΩ( δ α , δ ) + δG [ α, A, Σ] = 0 , Ω( δ v , δ ) + δV [ v, A, Σ] = 0 , (7) \nwhere G and V are the Gauss and Diffeo constraints respectively. They take the form \nG [ α, A, Σ] = ∫ M α i ( d A Σ i / ( κβ )) + ∫ H α i [ a H πκβ (1 -β 2 ) F i + 1 κβ Σ i ] ≈ 0 , \nfor all α : M → su (2), and \nV [ v, A, Σ] = ∫ M 1 κβ [ Σ i ∧ v /rightanglese F i -v /rightanglese A i d A Σ i ] -∫ H v /rightanglese A i [ a H πκβ (1 -β 2 ) F i + 1 κβ Σ i ] ≈ 0 , \nfor all v ∈ Vect( M ) that is tangent to H at the horizon. Notice that the previous constraints have the usual Gauss and diffeo constraint bulk-terms, plus boundary-terms \n1 This follows firstly from Ω M ( δ α , δ ) = 0 implying \n-κ Ω H = ∫ M δ α Σ i ∧ δA i -δ Σ i ∧ δ α A i = ∫ M [ α, Σ] i ∧ δA i + δ Σ i ∧ d A α i = ∫ M d ( α i δ Σ i ) -α i δ ( d A Σ i )= -a H π (1 -β 2 ) ∫ H α i δF i ( A ) = a H π (1 -β 2 ) ∫ H δ α A i ∧ δA i . \nwhere we used the Gauss law δ ( d A Σ) = 0, condition (6), and that boundary terms at infinity vanish. A similar calculation for diffeos completes the proof (see [5] for all details). \ngiven by smearings of (6) on H . Their Poisson algebra is \n{ G [ α, A, Σ] , G [ β, A, Σ] } = G ([ α, β ] , A, Σ) { G [ α, A, Σ] , V [ v, A, Σ] } = G ( L v α, A, Σ) { V [ v, A, Σ] , V [ w, A, Σ] } = V ([ v, w ] , A, Σ) , (8) \nwhere we have ignored the Poisson brackets involving the scalar constraint as its smearing must vanish on H , i.e., it does not affect the first class nature of the previous constraints. Thus (6) are first class constraints and can be implemented 'a la Dirac in the quantum theory. \nThis fact and the form of the symplectic structure motivates one to handle the quantization of the bulk and horizon degrees of freedom (d.o.f.) separately. As in standard LQG [8] one first considers (bulk) Hilbert spaces H B γ defined on a graph γ ⊂ M with end points on H , denoted γ ∩ H . The quantum operator associated with Σ in (6) is \n/epsilon1 ab ˆ Σ i ab ( x ) = 8 πGβ ∑ p ∈ γ ∩ H δ ( x, x p ) ˆ J i ( p ) (9) \nwhere [ ˆ J i ( p ) , ˆ J j ( p )] = /epsilon1 ij k ˆ J k ( p ) at each p ∈ γ ∩ H . Consider a basis of H B γ of eigen-states of both J p · J p as well as J 3 p for all p ∈ γ ∩ H with eigenvalues /planckover2pi1 2 j p ( j p +1) and /planckover2pi1 m p respectively. These states are spin network states, here denoted |{ j p , m p } n 1 ; ··· 〉 , where j p and m p are the spins and magnetic numbers labeling n edges puncturing the horizon at points x p (other labels are left implicit). They are also eigenstates of the horizon area operator ˆ a H \nˆ a H |{ j p , m p } n 1 ; ··· 〉 = 8 πβ/lscript 2 p n ∑ p =1 √ j p ( j p +1) |{ j p , m p } n 1 ; ··· 〉 . \nWe can decompose H B γ according to \nH B γ = ⊕ { j p } p ∈ γ ∩ H H B γ ( { j p } ) (10) \nfor spaces H B γ ( { j p } ) spanned by states |{ j p , m p } n 1 ; ··· 〉 for a given n-tuple j p } . \n{ \n} Substituting the expression (9) into (6) we get \n-a H π (1 -β 2 ) /epsilon1 ab ˆ F i ab = 8 πGβ ∑ p ∈ γ ∩ H δ ( x, x p ) ˆ J i ( p ) (11) \nThis equation tells us that the surface Hilbert space, H H γ ∩ H is precisely the one corresponding to (the well studied [8]) SU (2) CS theory in the presence of particles with CS level k = a H / (2 πβ (1 -β 2 ) /lscript 2 p ). The curvature of the (quantum) CS connection vanishes everywhere on H except at the position of the defects where we find conical singularities of strength encoded in the quantum operators ˆ J i p . \nThe solutions of (11) restricted to the graph γ are found to be elements of the Hilbert space [5] \nH γ = ⊕ { j p } p ∈ γ ∩ H H inv γ ( { j p } ) ⊗ H CS k ( { j p } ) , (12) \nwhere H inv γ ( { j p } ) is a proper subspace of H B γ ( { j p } ) spanned by area eigenstates, and H CS k ( { j p } ) are the CS Hilbert spaces which turn out to be completely determined by the total spin of punctures { j p } [8]. The full Hilbert space of solutions of (11) is obtained as the projective limit of the spaces H γ . The IH boundary condition implies that lapse must be zero at the horizon so that the scalar constraint is only imposed in the bulk. \nThe entropy of the IH is computed by the formula S = tr( ρ IH log ρ IH ) where the density matrix ρ IH is obtained by tracing over the bulk d.o.f., while restricting to horizon states that are compatible with the macroscopic area parameter a H . Assuming that there exist at least one solution of the bulk constraints for every admissible state on the boundary, the entropy is given by S = log( N ) where N is the number of horizon states compatible with the given macroscopic horizon area a H . After a moment of reflection one sees that \nN = ∑ n ;( j ) n 1 dim[ H CS k ( j 1 ··· j n )] , (13) \nwhere the labels j 1 · · · j p of the punctures are constrained by the condition \na H -/epsilon1 ≤ 8 πβ/lscript 2 p n ∑ p =1 √ j p ( j p +1) ≤ a H + /epsilon1. (14) \nSimilar formulae were first used in [9]. \nIt turns out that due to (14) we can compute the entropy for a H >> β/lscript 2 p (not necessarily infinite). The reason is that the representation theory of U q ( SU (2))describing H CS k for finite k -implies \ndim[ H CS k ( j 1 ··· j n )] = dim[Inv( ⊗ p j p )] , (15) \nas long as all the j p as well as the interwining internal spins are less than k/ 2. But for Immirzi parameter in the range | β | ≤ √ 3 this is precisely granted by (14) [5] . All this simplifies the entropy formula considerably. The previous dimension corresponds to the number of independent states one has if one models the black hole by a single SU (2) intertwiner! \nLet us conclude with a few remarks. \nWe have shown that the spherically symmetric isolated horizon is described by a symplectic form Ω M that, when written in the (connection) variables suitable for quantization, acquires a horizon contribution corresponding to an SU(2) CS theory. Our derivation of the (conserved) symplectic structure is straightforward. We first observe that SU (2) and diffeomorphism gauge invariance is not broken by the IH boundary condition: they continue to be degenerate directions of Ω M on shell. This by itself is then sufficient for deriving the boundary term that arises when writing the symplectic structure in terms of Ashtekar-Barbero connection variables (see also [5]). \nNote that no d.o.f. is available at the horizon in the classical theory as the IH boundary condition completely fixes the geometry at ∆ (the IH condition allows a single \n(characteristic) initial data once a H is fixed (see fig. 1)). Nevertheless, non trivial d.o.f. arise as would be gauge d.o.f. upon quantization. These are described by SU (2) CS theory with (an arbitrary number of) defects which couple to gauge d.o.f. through the dimensionless parameter 16 π 2 β (1 -β 2 ) /lscript 2 p √ j ( j +1) /a H , i.e., the ratio of a basic quantum of area carried by the defect to the total area of the horizon. These would be gauge excitations are entirely responsible for the entropy. \nWe obtain a remarkably simple formula for the horizon entropy: the number of states of the horizon is simply given in terms of the (well studied) dimension of the Hilbert spaces of CS theory with punctures labeled by spins, which-due to the area constraint (14), and for the range | β | ≤ √ 3 including the physical value of β [13]-is just the dimension of the singlet component in the tensor product of the representations carried by punctures. The black hole density matrix ρ IH is the identity on Inv( ⊗ p j p ) for admissible j p . Similar counting formulae have been proposed in the literature [10] by means of heuristic arguments. Our derivation from first principles in particular clarifies previous proposals. \nGeneral arguments and simple estimates indicate that the entropy will turn out to be S BH = β 0 a H / (4 β/lscript 2 p ), where β 0 is a constant to be determined. The new counting techniques of [12] are expected to be very useful for this. Thus the result to leading order remains unchanged. However, subleading corrections will have the form ∆ S = -3 2 log a H (instead of ∆ S = -1 2 log a H in the U (1) treatment) matching other approaches [11]. This is due to the full SU (2) nature of the IH quantum constraints imposed here, and this is a clear-cut indication that the U (1) treatment overcounts states. The value β 0 and the log-correction has been recently computed for | β | < √ 3 [13]. The range | β | ≥ √ 3 is unphysical as the quantum group structure imposes additional constraints driving the entropy below the physical value a H / 4. \nIn ref. [4] the classical description of the IH was first \n- [1] S. W. Hawking, Commun. Math. Phys. 43 (1975) 199.\n- [2] A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, Phys. Rev. Lett. 80 (1998) 904. A. Ashtekar, J.C. Baez and K. Krasnov, Adv. Theor. Math. Phys. 4 (2000) 1.\n- [3] K. Krasnov, Gen. Rel. Grav. 30 (1998) 53 [arXiv:gr-qc/9605047]. L. Smolin, J. Math. Phys. 36 (1995) 6417.\n- [4] A. Ashtekar, A. Corichi and K. Krasnov, Adv. Theor. Math. Phys. 3 (2000) 419.\n- [5] J. Engle, K. Noui, A. Perez and D. Pranzetti, arXiv:1006.0634 [gr-qc].\n- [6] J. Lewandowski, Class. Quant. Grav. 17 (2000) L53.\n- [7] C. Crnkovic and E. Witten, in 'Three hundred years of gravitation'; ed. S. Hawking, W. Israel. J. Lee and R.M. Wald, J. Math. Phys. 31 (1990) 725. A. Ashtekar, L. Bombelli and O. Reula, in '200 Years After Lagrange', Ed. by M. Francaviglia, D. Holm. \ndone in terms of the null tetrad formalism. In this case the null surface defining the Horizon provides the natural structure for a partial gauge fixing from the internal gauge SL (2 , C ) to U (1). In this setting one fixes an internal direction r i ∈ su (2) and the IH boundary condition (6) becomes \ndV + 2 π a H Σ i r i = 0 , Σ i x i = 0 , Σ i y i = 0 , (16) \nwhere x i , y i ∈ su (2) are arbitrary vectors completing an internal triad. In the quantum theory [2] only the first of the previous constraints is imposed strongly, whiledue to the non-commutativity of Σ i in LQG-the other two can only be imposed weakly, namely in [2] one has 〈 Σ i x i 〉 = 〈 Σ i y i 〉 = 0. However, this leads to a larger set of admissible states (over counting). To solve this problem, within the U (1) model, one would have to solve the two constraints Σ i x i = 0 = Σ i y i at the classical level first, implementing the reduction also on the pull back of two forms Σ i on H . However, this would introduce formidable complications for the quantization of the bulk degrees of freedom in terms of LQG techniques. Our SU (2) treatment resolves this problem as now the three components of (6) are first class constraints. Dirac implementation leads to a smaller subset of admissible surface states that are relevant in the entropy calculation. \nWe thank A. Ashtekar, M. Knecht, M. Montesinos, D. Pranzetti, M. Reisenberger, and C. Rovelli for discussions, and an anonymous referee for exchanges that have considerably improved the presentation of our results. This work was supported in part by the Agence Nationale de la Recherche; grant ANR-06-BLAN-0050. J.E. was supported by NSF grant OISE-0601844, and thanks Thomas Thiemann and the Albert-Einstein-Institut for hospitality. A.P. is a member l'Institut Universitaire de France . \n- [8] E. Witten, Commun. Math. Phys. 121 (1989) 351.\n- [9] R.K. Kaul and P. Majumdar, Phys. Lett. B 439 (1998) 267. R.K. Kaul and P. Majumdar, Phys. Rev. Lett. 84 (2000) 5255.\n- [10] C. Rovelli, Phys. Rev. Lett. 77 (1996) 3288. E. Livine and D. Terno, Nucl. Phs. B 741 (2006) 131.\n- [11] A. Ghosh and P. Mitra, Phys. Rev. D 71 (2005) 027502. G. Gour, Phys. Rev. D 66 (2002) 104022. S. Carlip, Class. Quant. Grav. 17 (2000) 4175.\n- [12] I. Agullo, J.F. Barbero, J. Diaz-Polo, E. FernandezBorja and E.J.S. Villasenor, Phys. Rev. Lett. 100 (2008) 211301. J.F. Barbero G. and E.J.S. Villasenor, Phys. Rev. D 77 (2008) 121502. Class. Quant. Grav. 26 (2009) 035017\n- [13] I. Agullo, J.F. Barbero., E.F. Borja, J. Diaz-Polo and E.J.S. Villasenor, arXiv:0906.4529 [gr-qc]."}
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2010CQGra..27k4006L
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Remnant masses, spins and recoils from the merger of generic black hole binaries
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2010-01-01
| 27 | 0.48 | 164 |
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We obtain empirical formulae for the final remnant black hole mass, spin, and recoil velocity from merging black hole binaries (BHBs) with arbitrary mass ratios and spins. Our formulae are based on the mass ratio and spin dependence of the post-Newtonian expressions for the instantaneous radiated energy, linear momentum, and angular momentum, as well as the ISCO binding energy and angular momentum. The relative weight between the different terms is fixed by amplitude parameters chosen through a least-squares fit of recently available fully nonlinear numerical simulations. These formulae can be used for statistical studies of N-body simulations of galaxy cores and clusters, and the cosmological growth of supermassive black holes. As an example, we use these formulae to obtain a universal spin magnitude distribution of merged black holes and recoil velocity distributions for dry and hot/cold wet mergers. We also revisit the long-term orbital precession and resonances and discuss how they affect spin distributions before the merging regime.
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https://arxiv.org/pdf/0904.3541.pdf
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{'Carlos O. Lousto, Manuela Campanelli, Yosef Zlochower, and Hiroyuki Nakano': 'Center for Computational Relativity and Gravitation, and School of Mathematical Sciences, Rochester Institute of Technology, 85 Lomb Memorial Drive, Rochester, New York 14623 \nAbstract. We obtain empirical formulae for the final remnant black hole mass, spin, and recoil velocity from merging black-hole binaries with arbitrary mass ratios and spins. Our formulae are based on the mass ratio and spin dependence of the postNewtonian expressions for the instantaneous radiated energy, linear momentum, and angular momentum, as well as the ISCO binding energy and angular momentum. The relative weight between the different terms is fixed by amplitude parameters chosen through a least-squares fit of recently available fully nonlinear numerical simulations. These formulae can be used for statistical studies of N-body simulations of galaxy cores and clusters, and the cosmological growth of supermassive black holes. As an example, we use these formulae to obtain a universal spin magnitude distribution of merged black holes and recoil velocity distributions for dry and hot/cold wet mergers. We also revisit the long term orbital precession and resonances and discuss how they affect spin distributions before the merging regime.', '1. Introduction': "Black holes at the centers of galaxies and globular clusters significantly impact the dynamical evolution of these astronomical structures. Of particular importance to the dynamics are the black-hole (BH) mass, spin, and location (if off-center); properties that can significantly change following a BH merger. When two galaxies merge, an event that is expected to occur several times during a galaxy's evolution, the supermassive BH at their centers form a black-hole binary (BHB) that eventually inspirals and merges. Similarly, intermediate-mass BHs in globular clusters can form tight BHBs that inspiral and merge. Consequently, accurate models for the mass, spin, and gravitational recoil of the merger remnants of BHBs are of great astrophysical interest. However, these models require accurate simulations of merging BHs, a problem in the highly-nonlinear regime, that only recently became feasible due to breakthroughs in Numerical Relativity [1, 2, 3]. \nThe first attempts at modeling the remnant BH of BHB mergers using fully nonlinear simulations utilized the 'Lazarus method' [4], which combined short-term numerical simulations of BHBs, just prior to merger, with perturbative calculations. With the advent of the 'moving punctures' [2, 3] and generalized harmonic [1] approaches, it became possible to accurately model merging BHBs from inspiral through merger and ringdown using fully-nonlinear numerical simulations. As a result of these breakthroughs, NR groups from around the world have been able to develop heuristic models for the properties of remnant BHs as a function of the orbital and intrinsic BH parameters of the binary (at-least for part of the parameter space). \nThe initial attempts at modeling the properties of remnant BH focused on the mass and spin using ad hoc interpolation formulae. In [5, 6, 7] we studied equal-mass, spinning BHBs, where the individual BH spins were aligned and counter-aligned with the orbital angular momentum, using fully nonlinear numerical calculations. We found a simple quadratic polynomial relating the final mass and spin of the remnant with the spins of the individual BHs. This scenario was later revisited in [8, 9], and in [10] the authors generalized the formula for the remnant spin (by assuming that the angular momentum is only radiated along the orbital axis, and neglecting the energy loss) in order to model arbitrary BH configurations (these assumptions were relaxed in a followup paper [11]). A generic formulae for the final spin was proposed in [12] based on simulations with aligned and non-aligned spins. A more comprehensive approach, using a generic Taylor expansion consistent with the the physical symmetries of the problem, and with parameters chosen by a least-squares fit to many simulations, was developed in [13]. All of these models used low-order polynomial interpolation functions to predict the remnant mass and spin as a function of the individual BH masses and spins. On the other hand, in [14], a different approach, based on approximate analytic models for the merger, was used. Here the authors extended the particle limit approximation for the radiated mass and angular momentum to the comparable-mass regime; ignoring effects of post-ISCO (Innermost Stable Circular Orbit) gravitational radiation. This approach was further improved in [15] by taking binding energies into account. All of \nthese approaches show a certain degree of agreement with the remnant masses and spins obtained in the few dozen fully nonlinear numerical simulations available, but significant uncertainties concerning accuracy outside this range of the parameter space remain. Here we propose a set of formulae that incorporate the benefits of both approaches and regimes in a unified way; using analytic techniques to develop empirical models with free parameters determined by numerical results. \nDue to its significant impact on astrophysics, the modeling of the remnant recoil followed an independent path, particularly since the discovery [16, 17] that the spins of the black holes play a crucial role in producing recoils of up to 4000 km s -1 . The realization that the merger of BHBs can produce recoil velocities that allow the remnant to escape from major galaxies led to numerous theoretical and observational efforts to find traces of this phenomenon. Several studies made predictions of specific observational features of recoiling supermassive black holes in the cores of galaxies in the electromagnetic spectrum [18, 19, 20, 21, 22, 23, 24]. Notably, there began to appear observations indicating the possibility of detection of such effects [25, 26, 27], and although alternative explanations are possible [28, 29, 30], there is still the exciting possibility that these observations can lead to the first confirmation of a prediction of General Relativity in the highly-dynamical, strong-field regime. \nThis paper is organized as follows. In Sec. 2 we describe our empirical formula for the remnant gravitational recoil and provide the leading coefficients for this formula. In Sec. 3 we describe our formula for the final remnant mass, while in Sec. 4 we describe the formula for the final remnant spins. We provide fits to the constants in the remnant mass and spin formula in Sec. 5. In Sec. 6 we revisit the gravitational alignment and antialignment mechanisms for long term inspiral orbits, and discuss the consequences and applications of our formulae in Sec. 7.", '2. Remnant Recoil Velocities': "In our approach to the recoil problem [16, 17] we used post-Newtonian (PN) theory as a guide to model the recoil dependence on the physical parameters of the progenitor BHB (See Eqs. (3.31) in [31]), while arguing that only full numerical simulations can produce the correct amplitude of the effect (see Eq. (3) below). For example, in the instantaneous radiated linear momentum, there are terms of the form \nd glyph[vector] P dt = · · · + η 2 1 + q [ glyph[vector] F ( glyph[vector]r , glyph[vector]v ) · ( glyph[vector] α 2 -qglyph[vector]α 1 ) ] ˆ glyph[vector] L, (1) \nwhere ˆ glyph[vector] L is the unit vector pointing along the instantaneous orbital angular momentum, glyph[vector] F ( glyph[vector]r , glyph[vector]v ) is a vector in the orbital plane that is only a function of the orbital position and its time derivative, q = m 1 /m 2 ≤ 1 is the mass ratio, η = q/ (1 + q ) 2 is the symmetric mass ratio, and glyph[vector] α i = glyph[vector] S i /m 2 i is the intrinsic spin on black hole i . We incorporate this term by adding a term \nglyph[vector] V recoil = · · · + ( K η 2 (1 + q ) [∣ ∣ α ⊥ 2 -qα ⊥ 1 ∣ ∣ cos(Θ ∆ -Θ 0 ) ] ) ˆ glyph[vector] L (2) \nto our fitting formula for the recoil velocity (see Eq. (3) below), where the fitting constants K and Θ 0 approximate the net effect of the dynamics of this term during the last few M of the rapid plunge (where most of the recoil is generated) and Θ ∆ is the angle that glyph[vector] ∆ = M 2 ( glyph[vector] α 2 -qglyph[vector]α 1 ) / (1 + q ) where M = m 1 + m 2 , makes with the infall direction at merger. Our heuristic formula describing the recoil velocity of BHB remnants was theoretically verified in several ways. In [17] we confirmed the sinusoidal dependence [cos Θ ∆ in Eq. (3)] of the recoil on the direction of the in-plane spin for the so-called 'superkick' configurations, a result that was reproduced in [32] for binaries with different initial separations. While in [33] the authors verified the decomposition of the spin-dependence of the recoil into spin components perpendicular and parallel to the orbital plane. Similarly, in [34] the authors determined that the quadratic-in-spin corrections to the in-plane recoil velocity are less than 5% of the total recoil. Recently in [35] we confirmed the leading η 2 (where η is symmetric mass ratio) dependence of the large recoils out of the orbital plane (see also [36]). \nHere we augment our original empirical formula with subleading terms, higher order in the mass ratio, and include a new term linear in the total spin, motivated by higher order post-Newtonian computations [37], \nglyph[vector] V recoil ( q, glyph[vector]α ) = v m ˆ e 1 + v ⊥ (cos ξ ˆ e 1 +sin ξ ˆ e 2 ) + v ‖ ˆ n ‖ , \nv m = A η 2 (1 -q ) (1 + q ) [1 + Bη ] , v ⊥ = H η 2 (1 + q ) [ (1 + B H η ) ( α ‖ 2 -qα ‖ 1 ) + H S (1 -q ) (1 + q ) 2 ( α ‖ 2 + q 2 α ‖ 1 ) ] , v ‖ = K η 2 (1 + q ) [ (1 + B K η ) ∣ ∣ α ⊥ 2 -qα ⊥ 1 ∣ ∣ cos(Θ ∆ -Θ 0 ) + K S (1 -q ) (1 + q ) 2 ∣ ∣ α ⊥ 2 + q 2 α ⊥ 1 ∣ ∣ cos(Θ S -Θ 1 ) ] , (3) \nwhere the index ⊥ and ‖ refer to perpendicular and parallel to the orbital angular momentum respectively and ˆ n ‖ = ˆ glyph[vector] L . ˆ e 1 , ˆ e 2 are orthogonal unit vectors in the orbital plane, and ξ measures the angle between the unequal mass and spin contribution to the recoil velocity in the orbital plane. The new constants H S and K S can be determined from new generic BHB simulations as the data become available. The angles, Θ ∆ and Θ S , are the angles between the in-plane component of glyph[vector] ∆ = M ( glyph[vector] S 2 /m 2 -glyph[vector] S 1 /m 1 ) or glyph[vector] S = glyph[vector] S 1 + glyph[vector] S 2 and the infall direction at merger. Phases Θ 0 and Θ 1 depend on the initial separation of the holes for quasicircular orbits. A crucial observation is that the dominant contribution to the recoil is generated near the time of formation of the common horizon of the merging black holes (See, for instance Fig. 6 in [38]). The formula (3) above describing the recoil applies at this moment (or averaged coefficients around this maximum generation of recoil), and has proven to represent the distribution of velocities with sufficient accuracy for astrophysical applications. \nThe most recent estimates for the above parameters can be found in [35] and references therein. The current best estimates are: A = 1 . 2 × 10 4 km s -1 , B = -0 . 93, \nH = (6 . 9 ± 0 . 5) × 10 3 km s -1 , K = (6 . 0 ± 0 . 1) × 10 4 km s -1 , and ξ ∼ 145 · . Note that we can use the data from [35] to obtain K = (6 . 072 ± 0 . 065) × 10 4 kms -1 , if we assume that B K and K S are negligible. Finally, if we fit the data to find K and K S simultaneously we obtain K = (6 . 20 ± 0 . 12) × 10 4 kms -1 and K S = -0 . 056 ± 0 . 041, where we made the additional assumption that Θ 0 = Θ 1 (since glyph[vector] S = glyph[vector] ∆ for these runs). An attempt to fit K , K S , B K simultaneously does not produce robust results with currently available data (one of the reasons for this is that different values of K and B K produce very similar predicted recoil velocities over the range 0 . 1 ≤ q ≤ 1). Note that the values for the dominant K term are reasonably insensitive to the different choices for the fits, while finding the subleading terms require additional runs and higher accuracy. \nThe above equation (3) contains all the expected linear terms in the spin, and includes ten fitting parameters. Based on the works [37] one could add quadratic terms, and this will be published elsewhere by the authors. \nFrom a practical point of view, for statistical simulations of BHB mergers, where the directions of the spins and infall direction is not known, one should take a uniform distribution for the in plane-components of ˆ α 1 and ˆ α 2 over all possible angles, define Θ S and Θ ∆ with respect to a fixed arbitrary in-plane vector (say ˆ x ), and take Θ 0 = 0. The resulting distribution of recoil velocities will be independent of the choice of the arbitrary in-plane vector (but will depend weakly on Θ 1 ). If we ignore the subleading K S correction, then Θ 1 will not enter the recoil calculation. It's effects can be incorporated by including the K S term and averaging over all possible values of Θ 1 .", '3. Remnant Mass': 'Motivated by the success of the empirical formula for the recoil, we propose a new empirical formula for the total radiated energy based on the post-Newtonian equations for the instantaneous radiated energy (see Eqs. (3.25) in [31], and for the quadratic terms in the spin see Eq. (5.4) in [37]): \nδM/M = η ˜ E ISCO + E 2 η 2 + E 3 η 3 + η 2 (1 + q ) 2 { E S ( α ‖ 2 + q 2 α ‖ 1 ) + E ∆ (1 -q ) ( α ‖ 2 -q α ‖ 1 ) + E A | glyph[vector] α 2 + qglyph[vector]α 1 | 2 + E B | α ⊥ 2 + qα ⊥ 1 | 2 ( cos 2 (Θ + -Θ 2 ) + E C ) + E D | glyph[vector] α 2 -qglyph[vector]α 1 | 2 + E E | α ⊥ 2 -qα ⊥ 1 | 2 ( cos 2 (Θ --Θ 3 ) + E F ) } , (4) \nwhere Θ ± are the angles that glyph[vector] ∆ ± = M ( glyph[vector] S 2 /m 2 ± glyph[vector] S 1 /m 1 ) make with the radial direction during the final plunge and merger (for comparable-mass BHs, a sizable fraction of the radiation is emitted during this final plunge, see for instance Fig. 6 in [38]). Phases Θ 2 , 3 are parameters that give the angle of maximum radiation for these terms, and depend on the initial separation and parameters of the binary at the beginning of the numerical simulation. \nIn addition to the terms arising from the instantaneous radiated energy, which gives \n12 fitting parameters, we also included terms associated with the secular loss of energy in the inspiral period from essentially infinite separation down to the plunge. In order to model this contribution we adopted the effective one body form [39] supplemented by the η 2 effects from self force calculations [40] and 2PN effects of the spins (see Eq. (4.6) in [31]), to obtain \n˜ E ISCO ≈ (1 -√ 8 / 3) + 0 . 103803 η + 1 36 √ 3(1 + q ) 2 [ q (1 + 2 q ) α ‖ 1 +(2 + q ) α ‖ 2 ] -5 324 √ 2(1 + q ) 2 [ glyph[vector] α 2 2 -3( α ‖ 2 ) 2 -2 q ( glyph[vector] α 1 · glyph[vector] α 2 -3 α ‖ 1 α ‖ 2 ) + q 2 ( glyph[vector] α 2 1 -3( α ‖ 1 ) 2 ) ] + O ( α 3 ) . (5) \nThe above expression only includes quadratic-in-spin terms for compactness, hence it is expected to produce reliable results for intrinsic spin magnitudes α i < 0 . 8 (because the binding energy is a very steep function of α for α > 0 . 8 and the quadratic expressions above are no longer appropriate). Note that we used the full expressions from [39] to obtain our fitting parameters. Here we fit the leading-order parameters using available data, and as new data become available, we expect to be able fit the remaining parameters.', '4. Remnant Spin': 'In an analogous way, we propose an empirical formula for the final remnant spin based on the post-Newtonian equations for the radiated angular momentum and the angular momentum of a circular binary at close separations (see Eqs. (3.28) and (4.7) in [31]), \nglyph[vector] α final = (1 -δM/M ) -2 { η ˜ glyph[vector] J ISCO + ( J 2 η 2 + J 3 η 3 ) ˆ n ‖ + η 2 (1 + q ) 2 ([ J A ( α ‖ 2 + q 2 α ‖ 1 ) + J B (1 -q ) ( α ‖ 2 -q α ‖ 1 ) ] ˆ n ‖ +(1 -q ) | glyph[vector] α ⊥ 2 -q glyph[vector]α ⊥ 1 | √ J ∆ cos[2(Θ ∆ -Θ 4 )] + J M ∆ ˆ n ⊥ + | glyph[vector] α ⊥ 2 + q 2 glyph[vector] α ⊥ 1 | √ J S cos[2(Θ S -Θ 5 )] + J MS ˆ n ⊥ )} . (6) \nNote that, even at linear order, there are important contributions of generic spinning black holes producing radiation in directions off the orbital axis that do not vanish in the equal-mass or zero-total-spin cases. The above formula can be augmented by quadraticin-the-spins terms [37, 41] of a form similar to the terms added to the radiated energy formula (4). However, these terms are less significant for modeling the final spin (see, for instance Fig. 21 of [7]). \nAgain, we use the effective one body resummation form [39], supplemented with the η 2 effects from self force calculations [40] and the 2PN effects of the spins (see Eq. (4.7) in [31]), to obtain \n˜ glyph[vector] J ISCO ≈ { 2 √ 3 -1 . 5255862 η -1 9 √ 2(1 + q ) 2 [ q (7 + 8 q ) α ‖ 1 +(8 + 7 q ) α ‖ 2 ] \nRemnant masses, Spins and recoils from the Merger of Generic Black-Hole Binaries 7 \n+ 2 9 √ 3(1 + q ) 2 [ glyph[vector] α 2 2 -3( α ‖ 2 ) 2 -2 q ( glyph[vector] α 1 · glyph[vector] α 2 -3 α ‖ 1 α ‖ 2 ) + q 2 ( glyph[vector] α 2 1 -3( α ‖ 1 ) 2 ) ] } ˆ n ‖ -1 9 √ 2(1 + q ) 2 [ q (1 + 4 q ) glyph[vector] α 1 +(4 + q ) glyph[vector] α 2 ] + 1 η ( glyph[vector] α 2 + q 2 glyph[vector] α 1 ) (1 + q ) 2 + O ( α 3 ) . (7) \nThis expression represents a quadratic expansion in the spin-dependence, hence we expect to produce reliable results for intrinsic spin magnitudes α i < 0 . 8 (hence α final < 0 . 9).', '5. Determination of fitting parameters': "Here we show how results from current full numerical simulations can be used to determine the fitting constants in the equations for the final remnant mass and spins of a BHB merger. This procedure can be repeated and extended as we have access to new runs and can also help in designing new simulations to optimally determine all fitting constants and better cover the 7-dimensional physical parameter space of BHBs. We used Mathematica's LinearRegression and NonLinearRegress functions to find the fitting parameters and estimate the errors in the parameters. Our method for finding the fitting parameters was to first fit to simulations with symmetries that caused most terms to vanish in order to fit to as few parameters at a time as possible. Then, after fixing the parameters we found in earlier fits, we fit to simulations with less symmetry to obtain other parameters. For example, we first find E 2 and E 3 , and then using these values, fit additional data to obtain E S , etc. \nEnergy radiated: For the non-spinning case, we fit the data from 8 simulations found in [42, 43] (see also [44]). Here we fit E Rad versus η , where E Rad is the total radiated energy for a given configuration minus the binding energy of the initial configuration (where the binding energy is negative). We calculate the binding energy using the 3PN accurate expressions given in [45]. A fit of the resulting data gives E 2 = 0 . 341 ± 0 . 014 and E 3 = 0 . 522 ± 0 . 062. In order to estimate E S , E ∆ , E A , and E D , We use the remnant masses from 13 simulations for spinning BHBs with spins aligned with the orbital angular momentum given in [46, 47] (see also [5]), and find E S = 0 . 673 ± 0 . 035, E ∆ = -0 . 36 ± 0 . 37, E A = -0 . 014 ± 0 . 021, and E D = 0 . 26 ± 0 . 44. These large uncertainties in the fitting parameters are due to the effect of correcting for the binding energy in these simulations. Finally, fits from the final remnant masses from 5 simulations [17] yields E E = 0 . 09594 ± 0 . 00045 and fits from 5 equal-mass configurations in [35] yield E B = 0 . 045 ± 0 . 010. An accurate fit to E D is not possible with the configurations available in [35]. Note that our fits for E ∆ , E A , and E D are consistent with the parameters set to zero. This is due to the fact that the errors introduced in renormalizing the data are of the same order as the effects of these subleading terms. \nAngular momentum radiated: For the non-spinning case, we fit the data from 8 simulations in [42, 43], and find J 2 = -2 . 81 ± 0 . 11 and J 3 = 1 . 69 ± 0 . 51. A fit to J A and J B from 13 simulations in [46, 47] yields J A = -2 . 97 ± 0 . 26 and J B = -1 . 73 ± 0 . 80. However, we determined that the uncertainty in J A and J B is actually closer to 1 . 0 by \nconsidering fits to the independent datasets in [46]. \nFrom the combined fit, we find that 2 . 42% < δM/M < 9 . 45% and 0 . 34 < α final < 0 . 92 for the equal-mass, aligned spin scenario, in the region where the fit is valid ( | α final | < 0 . 9). \nFinally, we note that much of the errors in the fitting parameters are due to differences in the normalizations between the various runs. Some authors choose normalize their simulations such that m 1 + m 2 = 1, which approximates a binary that inspiraled from infinity with an initial mass of 1, while others choose to normalize their simulations such that the initial ADM mass is 1. In this latter case we attempted to renormalize the results using the 3PN expression for the binding energy. However, the errors introduced by renormalizing data, or assuming that the ADM mass at infinite separation is 1, introduces uncertainties in our fitting parameters. This affects both δM/M directly and glyph[vector] α final indirectly through δM/M . Ideally we would use a set of simulations with the same normalization and all starting from the same initial orbital frequency. \nFrom a practical point of view, for statistical simulations of BHB mergers, where the infall direction and the directions of the spins at merger are not known, one should take a uniform distribution for the in plane-components of ˆ α 1 and ˆ α 2 over all possible angles, define the angles Θ S , Θ ∆ , and Θ + (note Θ -= Θ ∆ ) with respect to a fixed arbitrary in-plane vector (say ˆ x ), and take a uniform distribution for the unknown angles Θ 1 , 3 , 5 . The angles Θ 0 , 2 , 4 can be set to zero, since the final distributions will be independent of this choice (the distribution will only be a weak function of the relative angles Θ 0 -Θ 1 , etc.). The resulting distributions will be independent of the choice of the arbitrary inplane vector (but will depend weakly on Θ 1 , 3 , 5 ). However, the angles Θ 1 , 3 , 5 only appear in subleading expressions and the uncertainties in the final distributions of the spins, masses, and recoils should not be significant for astrophysical applications.", '6. Inspiral phase': "One of the important application of our formulae is to study statistical distributions of the final mass, spin and recoil of the remnant merged black hole given an initial distribution of individual spins and mass ratios. This kind of studies have been performed lately, see for instance [48], assuming initial random distribution of individual spin directions and magnitudes as well as mass ratios. This choice was supported by the post-Newtonian studies [48] that in the (dry) inspiral phase, preliminary to the final merger we have modeled in this paper, there is not an strong alignment of the spins with the orbital angular momentum, as there would be if, for instance, we would have large accretion of gas in the system (wet mergers). \nThe simulations in [48] actually found that, gravitational radiation induced precession of the orbital plane during the inspiral phase leads to an small bias of the spins towards counter-alignment. These results were the product of integrating 3.5PN equations of motion form separations r = 50 M down to the merger regime around \nr = 5 M . It was point out by studying averaged PN equations in the quasicircular orbits regime [49, 50, 51], that on longer time scales there are resonances that might affect the distributions of spin directions by the time of merger. Since these studies are complementary to those presented in [48], we will investigate this issue analytically at a lower PN order than in the numerical studies of Ref. [48], but retaining the radiation reaction effects on the orbital plane for consistency with the integration of the PN equations of motion in the Hamiltonian formalism. \nIn terms of the notation and approach of Ref. [48] we consider \n( ˆ glyph[vector] L · ˆ glyph[vector] S i ) · = ˙ glyph[vector] L · glyph[vector] S i | glyph[vector] L || glyph[vector] S i | + glyph[vector] L · ˙ glyph[vector] S i | glyph[vector] L || glyph[vector] S i | -glyph[vector] L · glyph[vector] S i | glyph[vector] L | · | glyph[vector] L | 2 | glyph[vector] S i | -glyph[vector] L · glyph[vector] S i | glyph[vector] S i | · | glyph[vector] L || glyph[vector] S i | 2 , (8) \nwhere we can set | glyph[vector] S i | · = 0 at this order of approximation. In [48] ˙ glyph[vector] S i and the conservative part of ˙ glyph[vector] L terms did not contribute due to the nature of the statistical studies performed in that paper. We hence focus on the dissipative part. \n( ˆ glyph[vector] L · ˆ glyph[vector] S i ) · dis = ˙ glyph[vector] L dis · glyph[vector] S i | glyph[vector] L || glyph[vector] S i | -glyph[vector] L · glyph[vector] S i | glyph[vector] L | · dis | glyph[vector] L | 2 | glyph[vector] S i | , (9) \nWith the PN techniques described [48] we find \n( ˆ glyph[vector] L · ˆ glyph[vector] S 1 ) · dis = -8 15 v 11 ω M q (1 + q ) 4 1 | glyph[vector] α 1 | × { q (61 q +48)( ˆ glyph[vector] P · glyph[vector] α 1 ) 2 +(61 + 48 q ) ( ˆ glyph[vector] P · glyph[vector] α 1 )( ˆ glyph[vector] P · glyph[vector] α 2 ) } , (10) ( ˆ glyph[vector] L · ˆ glyph[vector] S 2 ) · dis = -8 15 v 11 ω M q (1 + q ) 4 1 | glyph[vector] α 2 | × { q (61 q +48)( ˆ glyph[vector] P · glyph[vector] α 1 )( ˆ glyph[vector] P · glyph[vector] α 2 ) + (61 + 48 q ) ( ˆ glyph[vector] P · glyph[vector] α 2 ) 2 } . (11) \nNote that this expressions are defined negative when averaged over spin directions with only the squared terms ( ˆ glyph[vector] P · glyph[vector] α i ) 2 contributing. By integrating them over time, we obtain similar results to the expression for ( ˆ glyph[vector] L · ˆ glyph[vector] S ) · dis in Eq (18) of [48], that lead us to the conclusion that distributions of spins show some bias towards counter-alignment with respect to the orbital angular momentum. Note that the instantaneous counteralignment mechanism acts at every radius, with increasing strength for small separations, where the orbital velocity v ω is large. \nTo investigate the small mass ratio limit, i.e. q → 0, we compute the time integral (roughly speaking, multiply by 1/q) of Eq. (11), for instance. We can then see that if the larger black hole's spin, glyph[vector] S 2 , is initially randomly distributed in the limit q → 0 it ends up with some counteralignment. On the other hand, the smaller black hole's spin, glyph[vector] S 1 , would remain to be random oriented as seen from the vanishing of the right hand side of Eq. (10). \nNote that the above equations do not use orbital averages since the effect is particularly strong in the latest part of the inspiral, when averages are not a good approximation. In the alternative regime, when the inspiral motion is very slow, \nresonance orbits have been found using orbit averaged descriptions [49, 50, 51, 52]. These resonance orbits lead to alignment or antialignment of spins if one starts from an initial aligned or antialigned large hole and allow random orientations for the less massive one. Note that if both spins are allowed to be chosen at random initially, as we assumed in our computations, then the resulting evolution leads to still random distributed spins. \nThe resonance mechanism is complementary to the mechanism we studied in [48]. The former takes place on very long time scales compared to precession, while the later mechanism is quadratic in the spins (as seen in Eqs. (10) and (11), hence higher order.) In order to quantify which of them is the predominant mechanism long term numerical integration of the (non averaged) equations of motion is required.", '7. Discussion': 'In this paper, we provided a framework to describe the bulk properties of the remnant of a BHB merger. Our framework is based on PN scaling and fitting the results of full numerical simulations. The new formulae are physically motivated, incorporate the correct mass ratio dependence, and account for the radiation of angular momentum both parallel and perpendicular to the orbital angular momentum. These formulae have a symmetric dependence on the mass ratio and spins, while still including the correct particle limit. We also extended the successful recoil formula (3) by adding nonleading terms that include all the linear dependence in the spins, as well as higher mass ratio powers. \nUnlike in the formula for the remnant recoil case, the energy lost by the binary during the inspiral phase is a non-trivial fraction of the total radiated energy (and is, in fact, the dominant contribution in the small mass ratio limit). We thus included both the instantaneous radiative terms in (4) and the binding energy at the ISCO Eq. (5) (to take into account the secular loss of energy from very large distances down into the merger and plunge regime). Similarly, in order to model the final remnant spin, we need to take into account both the angular momentum of the system near the ISCO, see Eq. (7), and the subsequent loss of angular momentum in the final plunge (which is particularly important in comparable-mass mergers). \nUsing the fitted coefficients in the above formulae, we find that for equal-mass, non-spinning binaries, the net energy radiated is 5% of the total mass and the final spin is α ≈ 0 . 69, both in good agreement with the most accurate full numerical runs [53]. For maximally spinning BHBs with spin aligned and counteraligned we estimate that quadratic corrections lead to radiated energies between 10% and 3% respectively. As for the magnitude of the remnant spin, the linear estimates are between 0 . 97 and 0 . 41 respectively, with quadratic corrections slightly reducing those values. These results show that the cosmic censorship hypothesis is obeyed (i.e. no naked singularities are formed) and are in good agreement with earlier estimates [7]. \nThe set of formulae (4) and (6) with the fitting constants determined as in the Sec. 5 \nTable 1. The following parameters give the current best estimates for the constants in Eqs. (4) and (6). These parameters were used to generate the spin-magnitude distribution in Fig. 1. \n| E 2 | 0 . 341 | E 3 | 0 . 522 | E ∆ | - 0 . 3689 | E A | - 0 . 0136 |\n|-------|-----------|-------|-----------|-------|--------------|-------|--------------|\n| E B | 0 . 045 | E C | 0 | E D | 0 . 2611 | E E | 0 . 0959 |\n| E F | 0 | | | | | | |\n| J 2 | - 2 . 81 | J 3 | 1 . 69 | J A | - 2 . 9667 | J B | - 1 . 7296 |\n| J ∆ | 0 | J M | 0 | J S | 0 | J MS | 0 | \ncan be used to describe the final stage of BHB mergers in theoretical, N-body, statistical studies in astrophysics and cosmology [48, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66] by choosing a distribution of the initial intrinsic physical parameters of the binaries ( q, glyph[vector] S 1 , glyph[vector] S 2 ) and mapping them to the final distribution of recoil velocities, spins and masses after the mergers. As an example of such an application of the above formulae, we calculate the expected distribution of spin magnitudes of astrophysical supermassive and intermediate-mass BHs (which are expected to have undergone several mergers). To do this, we first consider a set of 10 6 binaries with uniform distributions of mass ratio (from 0 to 1), uniform orientations of the spin directions, and uniform distributions of spin magnitudes. We then use our formulae (see Table 1 for the values of the constants that we used) to predict the spin-magnitude distribution of the merger remnants and repeat the calculation, again with uniform distributions in mass ratio and spin directions, but with this new spin-magnitude distribution (see also [66, 12]). The resulting spin-distribution, after each subsequent set of mergers, approaches a fixed distribution. The spin distribution that results after ten generations of mergers is shown in Fig. 1. The final results are insensitive to the initial distribution and quickly converge, in a few generations of mergers, to the displayed curve, which represents a universal distribution of the intrinsic spin magnitudes (with a maximum near 0.7 and mean in the range (0 . 5 , 0 . 8)) of the remnant BHs of dry BHB mergers (when neglecting accretion). In order to provide a simple analytical model for this distribution, we fit it to the Kumaraswamy functional form [67] f ( x ; a, b ) = abx a -1 (1 -x a ) b -1 , and find a = 5 . 91 ± 0 . 04, b = 5 . 33 ± 0 . 07. We choose this functional form because it allows for a skewed distribution (and fits the results better than a beta distribution), however, the fit underestimates the probability for producing small spins. \nWe have also considered the effect of wet mergers on the final spin and recoil velocities. A first account of accretion effects during the long inspiral phase of binary black holes have been given in [68] using the smoothed particle hydrodynamics approximation (SPH). To evaluate the accretion effects on the statistical distributions, according to [68], we have considered distributions that at merger have 0 . 3 ≤ α i ≤ 0 . 9 and orientations within 10 deg and 30 deg for cold and hot accretion disks respectively. We have also assumed a flat distribution in mass ratios in the region 0 ≤ q ≤ 1. The \nFigure 1. The spin magnitude distribution for dry mergers. We plot the distributions of spins α = S/m 2 of the final remnant after many mergers. This distribution does not change significantly following additional mergers and peaks at α ≈ 0 . 73. We also display the distributions representing wet mergers for hot and cold accretion disks. They are highly peaked distributions at around α ≈ 0 . 88 and α ≈ 0 . 9 respectively. \n<!-- image --> \nresults for the final spin distributions are displayed in Fig. 1 and show the dramatic change in the spin distributions due to accretion. Note that this accretion effects on spin will be less important on black holes with masses larger than 10 7 M glyph[circledot] [69]. \nThe same statistical analysis can be made with the magnitude of the recoil velocity of the remnant final black hole when we consider a set of 10 6 binaries with uniform distributions of mass ratios in the range 0 ≤ q ≤ 1. For dry mergers we consider uniform orientations of the spin directions, and uniform distributions of spin magnitudes. We evaluate Eq. (3) each time and obtain the distribution with the extended tail beyond 1000 kms -1 in Fig. 2. The other two distributions correspond to the wet mergers with 0 . 3 ≤ α i ≤ 0 . 9 and orientations within 10 deg and 30 deg for cold and hot accretion disks respectively according to Ref. [68]. The results show a tighter distribution around low recoil velocities for cold than for hot accretion disks around the merging black holes. \nFinally, another use of the remnant formulae can be found in modeling waveforms in the intermediate and small mass ratio limits using the techniques of Ref. [70] by providing an accurate a priori estimation for the background black-hole mass and spin.', 'acknowledgments': 'We thank E. Berti for discussion on the resonances of PN evolutions and M. Volonteri on describing the SPH results in detail. We gratefully acknowledge NSF for financial support from grant PHY-0722315, PHY-0653303, PHY 0714388, and PHY 0722703; and NASA for financial support from grant NASA 07-ATFP07-0158 and HST-AR-11763. \nFigure 2. The the recoil magnitude distribution for dry mergers displaying a tail extending beyond 1000 km s -1 . We also display the distributions representing recoils for wet mergers for hot and cold accretion disks. The cold disk leads to a recoil velocity distribution highly peaked at around v ≈ 80 kms -1 while the hot accretion disk extends the magnitude of the recoil to several hundred km s -1 . \n<!-- image -->', 'References': "- [1] Pretorius F 2005 Phys. Rev. Lett. 95 121101 ( Preprint gr-qc/0507014 )\n- [2] Campanelli M, Lousto C O, Marronetti P and Zlochower Y 2006 Phys. Rev. Lett. 96 111101 ( Preprint gr-qc/0511048 )\n- [3] Baker J G, Centrella J, Choi D I, Koppitz M and van Meter J 2006 Phys. Rev. Lett. 96 111102 ( Preprint gr-qc/0511103 )\n- [4] Baker J G, Campanelli M, Lousto C O and Takahashi R 2004 Phys. Rev. D69 027505 ( Preprint astro-ph/0305287 )\n- [5] Campanelli M, Lousto C O and Zlochower Y 2006 Phys. Rev. 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2008Sci...321.1060B
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Star Formation Around Supermassive Black Holes
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2008-01-01
| 23 | 0.51 | 164 |
['-', 'astrophysics']
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[]
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The presence of young massive stars orbiting on eccentric rings within a few tenths of a parsec of the supermassive black hole in the galactic center is challenging for theories of star formation. The high tidal shear from the black hole should tear apart the molecular clouds that form stars elsewhere in the Galaxy, and transport of stars to the galactic center also appears unlikely during their lifetimes. We conducted numerical simulations of the infall of a giant molecular cloud that interacts with the black hole. The transfer of energy during closest approach allows part of the cloud to become bound to the black hole, forming an eccentric disk that quickly fragments to form stars. Compressional heating due to the black hole raises the temperature of the gas up to several hundred to several thousand kelvin, ensuring that the fragmentation produces relatively high stellar masses. These stars retain the eccentricity of the disk and, for a sufficiently massive initial cloud, produce an extremely top-heavy distribution of stellar masses. This potentially repetitive process may explain the presence of multiple eccentric rings of young stars in the presence of a supermassive black hole.
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[]
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https://arxiv.org/pdf/0810.2723.pdf
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{'Star Formation Around Supermassive Black Holes': 'I.A. Bonnell 1 , W.K.M. Rice 2 \n1 Scottish Universities Physics Alliance, University of St Andrews, Physics & Astronomy, North Haugh, St Andrews, Fife KY16 9SS, UK. \n2 Scottish Universities Physics Alliance, Institute for Astronomy, University of Edinburgh, Blackford Hill, Edinburgh EH9 3HJ, UK. \nThe presence of young massive stars orbiting on eccentric rings within a few tenths of a parsec of the supermassive black hole in the Galactic centre is challenging for theories of star formation. The high tidal shear from the black hole should tear apart the molecular clouds that form stars elsewhere in the Galaxy, while transporting the stars to the Galactic centre also appears unlikely during their stellar lifetimes. We present numerical simulations of the infall of a giant molecular cloud that interacts with the black hole. The transfer of energy during closest approach allows part of the cloud to become bound to the black hole, forming an eccentric disc that quickly fragments to form stars. Compressional heating due to the black hole raises the temperature of the gas to 100-1000K, ensuring that the fragmentation produces relatively high stellar masses. These stars retain the eccentricity of the disc and, for a sufficiently massive initial cloud, produce an extremely top-heavy distribution of stellar masses. This potentially repetitive process can therefore explain the presence of multiple eccentric rings of young stars in the presence of a supermassive black hole. \nBy tracking the motions of young massive stars, two teams of astronomers have uncovered the existence of a 3.6 × 10 6 Mo (solar masses) supermassive black hole (SMBH) in the center of our Galaxy (1-3). In addition, one, possibly two, eccentric rings of young massive stars orbit slightly further out, near ~0.1 pc (4), while lowmass stars are sparse in the region (5), making our Galactic center the best example of an atypical distribution of stellar masses (or stellar initial mass function, IMF). \nThe presence of young massive stars in the vicinity of the Galactic centre is difficult to reconcile with current models of star formation where turbulent molecular clouds produce a mostly clustered population of stars with a remarkably constant distribution of stellar masses, covering stars with masses from less than a tenth to greater than 100 times the mass of the sun (6). The tidal pull from the supermassive back hole should disrupt anything as tenuous as a molecular cloud, thereby destroying it before it can even form and thus removing the necessary conditions for star formation (7). This leaves the possibilities that either the stars formed elsewhere and migrated to the Galactic center, or that the stars formed in situ by an unusual mechanism, such as the fragmentation of a gaseous disc rotating around the supermassive black hole (8,9). \nA stellar cluster could migrate to the Galactic center by losing energy through its gravitational disturbance of the background stars in the Galaxy. However, this appears to take too long to explain the existence of the young stars found there (10). Star formation in an accretion disk around the supermassive black hole is possible if the disc is sufficiently massive (11), and if it is able to cool sufficiently rapidly (12,13). A priori a disc should fragment producing circular orbits, not the eccentric rings recently detected. Dynamical relaxation of a top-heavy IMF could increase the \neccentricities of individual stars but is unlikely to reach the values observed suggesting that the stars are likely to have formed from an initially eccentric disc (14,15). Here, we explore how the accretion disc forms around the black hole, and in particular how it could form with an initial eccentricity, by simulating the evolution of a giant molecular cloud (GMC) falling towards a supermassive black hole (16). The cloud is presumed to be the result of a collision at a distance of several pc from the Galactic centre, sending it on this plunging orbit towards the black hole. \nWe carried out the simulations using Smooth Particle Hydrodynamics (SPH), a Lagrangian hydrodynamics formalism (17). We considered two situations, a 10 4 Mo molecular cloud falling towards a 10 6 Mo black hole, and a 10 5 Mo molecular cloud falling towards a 3 × 10 6 Mo black hole. The clouds were initially placed 3 pc from the black hole and on an orbit with an impact parameter of ~ 0.1 pc. The clouds were turbulently supported and had a minimum temperature of 100 K due to the background radiation field. We included an approximate radiative transfer formalism with compressional heating balanced by cooling rates derived from estimated optical depths (18). Dense protostellar fragments were replaced by sink-particles (19) that accreted all bound gas particles that fell within 200 AU. Additional information as to the details of the simulation are available in the supplementary online material. Our computations were performed using the U.K. Astrophysical Fluids Facility (UKAFF) and our local SUPA HPC facility. \nThe early evolution of both clouds was broadly similar over the ~ 2 x 10 4 years required to reach the Galactic center. As the 10 4 Mo molecular cloud fell towards the 10 6 Mo black hole (Fig. 1), it became tidally distorted, and unbound, due to the strong \ngravitational field. The turbulence in the cloud formed local structures some of which collapsed to form stars before the cloud reached the black hole. Stars formed before closest approach remained unbound and escaped the system. In contrast, 10 per cent of the gas cloud became bound due to the combination of gas dissipation and tidal torques at closest approach and remained in orbit within a few tenths of a pc. The tidal disruption of the self-gravitating cloud formed coherent spiral structures that transferred orbital angular momentum outwards while shocks from infalling streams, which passed either side of the black hole, also removed orbital angular momentum from the gas, allowing it to form an eccentric disc-like structure. \nStructure in the infalling gas ensured that the disk of bound gas was very clumpy, forming spiral patterns that grew due to the diskÕs self-gravity and fragmented to form individual stars. The fragmentation occurred very quickly (on an orbital timescale) to form 498 stars with eccentricities between e = 0.6 and e = 0.76 and semi-major axes between a = 0.11 pc and a = 0.19 pc. The larger 10 5 Mo cloud (the final state is shown in Fig. 2) formed 198 stars with semi-major axes between a = 0.02 pc and a = 0.13 pc and eccentricities between e = 0 and e = 0.53. The larger mass and size of the cloud resulted in more angular momentum being removed by the tidal torques and by direct shocks on the incoming streams such that the eccentric disc and stars were formed closer in to the black hole. The stellar discs were fairly thin with an initial H/R varying from 0.1 to 0.2. Dynamical relaxation of these disks would increase the H/R slightly over several million years. \nThe resulting masses of the stars formed depends crucially on the balance between compressional heating of the gas as it infalls towards the black hole and the radiative \ncooling given by the gas temperature and the local optical depth. Near the black hole, the heating dominated, increasing the gas temperature to between several hundred and several thousand degrees K. This resulted in a Jeans mass, the minimum fragment mass, of order 0.5 Mo for the 10 4 Mo cloud and up to 10 to 50 Mo for the 10 5 Mo cloud. The higher mass cloud produced higher temperatures and hence higher Jeans masses as more gas was captured closer in to the black hole. This increased the compressional heating and decreased the cooling rate due to the higher the optical depths. Stars that formed further out, near 0.1 pc, had masses of several Mo as in the lower-mass cloud. \nThe stellar mass functions from the two simulations (Fig. 3 and Fig. 4) are very different: the lower-mass cloud formed a typical stellar mass distribution peaking at 0.8 Mo and following a Salpeter-type power law at higher-masses [d N (log m ) = m -Γ d(log m ),with Γ ~ 1.35]. The higher-mass cloud produced a bimodal mass function: a population of very massive stars with masses between ~ 10 and ~100 Mo, and a population of lower mass stars. The higher-mass stars formed in the inner ring (a~0.02 pc) while the lower-mass stars formed further out (a~ 0.05-0.1 pc) due to the different gas temperatures produced (see online material). As additional gas remained bound at larger radii, it is possible that more lower-mass stars would eventually form if the simulation was followed further in time. \nIn addition to forming the stars, 10 to 30 % of the infalling gas clouds were accreted onto the black hole. This accretion implies only that the material is bound within the size of the sink-particleÕs accretion radius of 4000 AU, and in fact this material had sufficient angular momentum to form a disk at radii of 1000-4000 AU around the black hole. \nThe actual size and evolution of this inner disk is not determined by our simulations and could involve further star formation or self-gravity driven accretion. Assuming that all the gas would accrete directly onto the black hole on a viscous timescale of order 10 7 years yields an average mass accretion rate of order 10 -4 Mo / yr for the10 4 Mo cloud and 10 -3 Mo / yr for the 10 5 Mo cloud. This gives a maximum accretion luminosity of order 10 43 ergs/s, or about 1 % of the Eddingtom luminosity. This additional source of radiation, and that from the newly formed stars could increase the gas temperature in the disk and thus the fragment masses. \nOur simulations show that an infalling molecular cloud can indeed form an eccentric disc around a supermassive black hole and that although the tidal force of the black hole will disrupt the cloud, it does not destroy the small-scale structures that seed the disk fragmentation. Furthermore, the compressional heating of the infalling gas results in the formation of a population of stars biased towards higher masses. The stellar masses depend crucially on the mass of the infalling cloud and on its impact parameter allowing for a variety of final outcomes. The initial disc eccentricity also means that the stars can form with initial eccentricities and, if the molecular cloud is sufficiently massive, the stars that form may be extremely massive. This is, therefore, a viable mechanism for forming the rings of young, massive stars within ~ 0.1 pc of the galactic centre. What is still unclear however, is the origin of the infalling cloud and the probability of the small impact parameter that is required. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig 1. The evolution of a 10 4 M molecular cloud is shown falling toward a 10 6 M GLYPH<0> black hole. The upper left image shows the region within 1.5 pc of the black hole and the colors illustrate the column density on a logarithmic scale between 0.01 g cm -2 and 100 g cm -2 . The upper right image is at a later time and shows the region within 1 pc of the SMBH with the color scale from 0.025 g cm -2 and 250 g cm -2 . The lower two images are a later time ands show the region within 0.5 pc of the black hole and the colors illustrate the column density on a logarithmic scale between 0.1 g cm -2 and 1000 g cm -2 . Although the cloud was tidally disrupted by the black hole, some of the material is captured by the black hole to form an eccentric disc that quickly fragments to form stars. These are illustrated by the white dots and have eccentricities between e = 0.6 and e = 0.76 and semi-major axes between a = 0.11 pc and a = 0.19 pc. A small population of stars also formed quite early, are visible in the top right panel, and can be seen being ejected from the system in the bottom right panel. \n<!-- image --> \nFig 2. The final state of the simulation of a 10 5 Mo molecular cloud falling towards a 3 × 10 6 Mo SMBH. The image shows the region within 0.25 pc of the SMBH located at the centre with the colors illustrating column densities between 0.75 g cm -2 and 7500 g cm -2 . A portion of the cloud has formed a disc around the SMBH, while Ð at the stage shown here Ð most of the mass is still outside the region shown. The disc fragments very quickly producing 198 stars with semi-major axes between a = 0.04 pc and a = 0.13 pc and eccentricities between e = 0 and e = 0.53. \n<!-- image --> \nFig 3. Mass function of the stars formed in the simulation illustrated in Fig 1. The stars form with masses close to 0.1 Mo, but grow quickly through gas accretion. The mass function there fore has a peak at ~ 0.8 Mo, above which it has a power-law form with a slope comparable to that of the salpeter slope, illustrated by the diagonal line. \n<!-- image --> \nFig 4 . Mass function of the stars formed in the simulation illustrated in Fig 2. The mass function is extremely top-heavy and appears to have two populations of stars, a population of massive stars with masses from 10 Mo to 100 Mo and another with masses between 1 Mo and 10 Mo . \n<!-- image -->', 'References and notes': '- 1. R. Genzel, et al. , Mon. Not. R. Astron. Soc. , 291 , 219 (1997)\n- 2. R. Schodel, et al. , Nature , 416 , 694 (2002)\n- 3. A. Ghez, et al. , Astrophys. J. , 620 , 744 (2005)\n- 4. T. Paumard, et al. , Astrophys. J. , 643 , 1011 (2006)\n- 5. S. Nayakshin, R. Sunyaev, Mon. Not. R. Astron. Soc. , 364 , L23 (2005)\n- 6. B. Elmegreen, Astrophys J. , 648 , 572, (2006)\n- 7. E.S. Phinney, 1989 , in The Center of the Galaxy: Proceedings of the 136th symposium of the IAU, ed. M. Morris (Dordrecht: Kluwer), 543\n- 8. Y. Levin, A.M. Beloborodov, Astrophys. J. , 590 , L33 (2003)\n- 9. S. Nayakshin, J. Cuadra, Aston. Astrophys. , 437 , 437 (2005)\n- 10. M.A. GŸrkan, F.A. Rasio, Astrophys. J. , 628 , 236 (2005)\n- 11. S. Nayakshin, J. Cuadra, V. Springel, Mon. Not. R. Astron. Soc. , 379 , 21 (2007)\n- 12. C.F. Gammie, Astrophys. J. , 553 , 174\n- 13. W.K.M. Rice, et al. , Mon. Not. R. Astron. Soc. , 339 , 1025 (2003)\n- 14. R.D. Alexander, M.C. Begelman, P.J. Armitage, Astrophys. J. , 654 , 907 (2007)\n- 15. R.D. Alexander, P.J. Armitage, J. Cuadra, M.C. Begelman, Astrophys. J. , 674 , 927 (2008)\n- 16. F. Yusef-Zadeh, J. Braatz, M. Wardle, D. Roberts, Astrophys. J., in press (arXiv:0805.3274)\n- 17. J.J. Monaghan, Ann. Rev. Astron. Astrophys. , 30 , 543 (1992)\n- 18. D. Stamatellos, A.P. Whitworth, T. Bisbas, S. Goodwin, Astron. Astrophys. , 475 , 37 (2007) \n19. M.R. Bate, I.A. Bonnell, N.M. Price, Mon. Not. R. Astron. Soc. , 277 , 362 (1995)', 'Supplemental on-line information': 'We carried out the simulations using Smooth Particle Hydrodynamics (SPH), a Lagrangian hydrodynamics formalism (17). The two clouds considered are 1) a 10 4 Mo molecular cloud with a 0.5 pc radius, falling towards a 10 6 Mo SMBH, and 2) a 10 5 Mo molecular cloud with a radius of 1 pc falling towards a 3 × 10 6 Mo SMBH. The clouds were initially placed 3 pc from the SMBH, infalling at 40 km/s with a tangential velocicty of 8 km/s, yielding an impact parameter of ~ 0.1 pc. The clouds were represented by 4.5 × 10 6 particles, had an initial temperature of 100 K to account for the background radiation field, and Jeans masses of ~ 10 Mo. The minimum resolvable protostellar masses were therefore ~ 0.1 Mo for the ~ 10 4 Mo cloud and 1 Mo for the 10 5 Mo (20). We also performed a higher-resolution run of the second case where regions of interest had an increased mass resolution down to 0.1 Mo., Gas particles that were accreted onto sink-particles (or turned into sink-particles) in the lower resolution run, were split into 9 lower-mass. This mass resolution was always lower than the Jeans mass at the point of fragmentation, ensuring the simulation was adequately resolved. Furthermore, the high-resolution run produced qualitatively and quantitatively similar results. Dense protostellar fragments were replaced by sink-particles (19), which interacted with the rest of the simulation only through gravity and accretion of the gas. \nThe sink-particles accreted any bound gas particles that fell within 200 AU, while any gas particles that came within 40 AU was accreted, regardless of its properties. Sink-particles were created if the gas particles had a smoothing length smaller than the accretion radius, were bound with a viral parameter <1/2 and collapsing. Creation densities were ≥ 10 14 Mo /pc ( ≥ 10 -8 g cm -3 ), well above the critical tidal densities. Only one particle per timestep could be turned into a sink-particle, which immediately accreted all its neighbours ensuring that spurious fragmentation did not occur. The central black hole was modelled by a sink particle with a larger accretion radius of 4000 AU. for bound gas. Encounters between sink-particles had their gravitational forces smoothed by the SPH kernel within 200 AU. \nThe simulations evolved under a radiative transfer formalism that is described in detail by Stamatellos et al (18). In this formalism, the column density for each particle was determined from its density and gravitational potential energy, the latter reflecting the overall distribution of mass. Rosseland mean opacities (21) were then used to determine the optical depth and consequently the cooling rate, which, together with the hydrodynamic heating rate gave the equilibrium temperature and a thermalisation timescale. The internal energy was updated using an implicit scheme. The equation of state took into account the different chemical states of hydrogen and helium, and the rotational and vibrational degrees of freedom of H2 (22). We assumed that the background radiation field would prevent the cloud from cooling below 100 K. Our computations were performed using the U.K. Astrophysical Fluids Facility (UKAFF) and our local SUPA HPC facility. \nThe clouds were initially turbulent, with the turbulence modelled by a divergence-free Gaussian velocity field with power spectrum P ( k ) α k -4 , where k is the wavenumber of the velocity perturbations (23). The velocities were normalized such that the kinetic energy was equal to the absolute magnitude of the potential energy. Including the thermal energy, the clouds were initially marginally unbound but the dissipation of kinetic energy due to shocks ensured the cloud would become globally bound in isolation. Such isolated clouds would dissipate their turbulent energies on the free-fall timescale of tff ~ 10 5 years. Previous simulations of isolated clouds show that the first stars form at 0.5 tff, or 5 x 10 4 years and continues for 2 tff (24). \nThe lower-mass cloud was evolved for 5 x 10 4 years by which point the cloud had passed the point of closest approach with the black hole. The higher-mass cloud was evolved for 3 x 10 4 years by which time the infalling gas had formed an eccentric disc that fragmented. The bulk of the latter cloud was still infalling at this point leaving open the possibility of further star formation in this simulation. The largely unbound nature of both infalling clouds, due to the tidal forces from the central black hole, ensured that few stars formed directly from the collapsing cloud. The stars that did form on the way in did not lose significant kinetic energy to become closely bound to the black hole. Instead, they retained their large initial semi-major axis from the black hole. The dissipative nature of the gas was necessary to bind material in close orbits around the black hole. The accretion onto the central black hole in the 10 5 Mo cloud is also likely to increase from 10, to nearer the 30 % as in the longer evolution of the lower-mass cloud. The unresolved inner disk could also conceivably form additional stars. \nThe fragmentation of the eccentric disk implies that the Toomre Q-value was Q<1. The infalling material is initially tidally unbound and the Q-value varies between 100 and 10 5 . The rapid increase in the surface density in the eccentric disk forced Q to drop to Q~ 0.1 at the point of fragmentation, in contrast to models where the disk is built up more slowly (10). Surface densities of the disks were of order 10 7 to 10 8 Mo pc -2 . The stars formed at near their Jeans masses (see below) with the majority of their mass being accreted on a timescale of a few thousand years. A total of 6 % of the lower-mass cloud and 3 % of the higher-mass cloud was turned into stars. The average mass accretion rate onto the stars was ~10 -4 Mo /yr for the lower-mass cloud, 10 -3 Mo /yr for the higher mass cloud with a peak accretion rate of ~ 4 x 10 -2 Mo /yr. These accretion rates would in reality be onto the accretion discs around the forming stars that would modulate the accretion rate over longer time periods. Fragmentation of these disks could conceivably reduce the stellar masses while producing binary systems. It should also be noted that longer-term accretion as the bulk of the highermass cloud falls in could also significantly increase the stellar masses. \nThe resultant stellar masses depend crucially on the gas thermodynamics. Fig 5 plots the temperature distribution of the gas in the higher-mass cloud. In the figure we have plotted every tenth particle plus all particles (red squares) that have densities high enough to prevent them from being tidally disrupted by the supermassive black hole. The compressional heating as the gas fell in and formed the disk dominated over the cooling and increased the temperature to several thousand K. These high temperatures give an initial Jeans mass that vary from several to ~ 50 Mo, resulting in high fragment masses and thus a population of massive stars that form quickly in the disk. This is shown in Fig 6 which plots the Jeans mass of every tenth particle plus the \nJeans mass of the neighbour sphere of every particle that has a density high enough to prevent it from being tidally disrupted by the supermassive black hole. The lower mass stars form later once the gas has cooled down to near 100 K. It should be noted, however, that we do not include any feedback from these massive stars and this, if included, could inhibit the later formation of the lower mass stars. This is shown in \nThe final state of the high-mass simulation is summarised in Fig. 7 which shows the mass distribution of the stars and gas still in orbit around the black hole. The masses of the individual stars and the cumulative mass of the gas is plotted against the virial radius of each sink or gas particle. The highest mass stars are all located in the inner 0.02-0.03 pc with intermediate mass stars found between 0.03 and 0.15 pc. Several stars that formed in the infalling cloud are located at > 1 pc.', 'References': '- 20. M.R. Bate, A. Burkert, Mon. Not. R. Astron. Soc. , 288 , 1060 (1997)\n- 21. K.R. Bell, D.N.C. Lin, Astrophys. J. , 427 , 987 (1994)\n- 22. A.C. Boley, T.W. Hartquist, R.H. Durisen, S. Michael, Astrophys. J. , 656 , L89 (2007)\n- 23. E.C. Ostriker, J . M. Stone, C.F. Gammie, Astrophys. J. , 546 , 980 (2001)\n- 24. I.A. Bonnell, M.R. Bate, S.G. Vine, Mon. Not. R. Astron. Soc., 343 , 413 (2003) \nFig 5. Scatter plot showing the particle temperatures plotted against distance from the SMBH. We plot every tenth particle, plus all those particles whose densities are high enough such that they are unlikely to be tidally disrupted by the SMBH. The relatively high temperatures (~ 1000 K) means that the initial Jeans mass is ~ 1 Mo to 50 Mo, and those stars that form early are able to grow quickly and become relatively massive. \n<!-- image --> \nFig 6. Scatter plot showing the Jeans mass plotted against distance from the SMBH. We plot every tenth particle, plus the Jeans mass of the neighbor sphere of all those particles whose densities are high enough such that they are unlikely to be tidally disrupted by the SMBH. The relatively high temperatures (~ 1000 K) means that the initial Jeans mass is ~ 1 Mo to 50 Mo, and those stars that form early are able to grow quickly and become relatively massive. \n<!-- image --> \nFig 7. The cumulative mass distribution of gas (dashed line) and of gas and stars (solid line) plotted against semi-major axis. The individual stellar masses are also plotted (red crosses) at the location of their individual semi-major axis. \n<!-- image -->'}
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2005PhRvD..72b4021Z
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Accurate black hole evolutions by fourth-order numerical relativity
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2005-01-01
| 6 | 0.45 | 164 |
['-', '-', '-', '-', 'methods numerical', '-', 'perturbation theory', '-', 'waves', '-', '-']
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We present techniques for successfully performing numerical relativity simulations of binary black holes with fourth-order accuracy. Our simulations are based on a new coding framework which currently supports higher-order finite differencing for the Baumgarte-Shapiro-Shibata-Nakamura formulation of Einstein’s equations, but which is designed to be readily applicable to a broad class of formulations. We apply our techniques to a standard set of numerical relativity test problems, demonstrating the fourth-order accuracy of the solutions. Finally we apply our approach to binary black hole head-on collisions, calculating the waveforms of gravitational radiation generated and demonstrating significant improvements in waveform accuracy over second-order methods with typically achievable numerical resolution.
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https://arxiv.org/pdf/gr-qc/0505055.pdf
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{'Accurate black hole evolutions by fourth-order numerical relativity': "Y. Zlochower, 1 J. G. Baker, 2 M. Campanelli, 1 and C. O. Lousto 1 \n1 Department of Physics and Astronomy, and Center for Gravitational Wave Astronomy, \nThe University of Texas at Brownsville, Brownsville, Texas 78520 \n2 Gravitational Astrophysics Laboratory, NASA Goddard Space Flight Center, Greenbelt, Maryland 20771 (Dated: November 26, 2024) \nWe present techniques for successfully performing numerical relativity simulations of binary black holes with fourth-order accuracy. Our simulations are based on a new coding framework which currently supports higher order finite differencing for the BSSN formulation of Einstein's equations, but which is designed to be readily applicable to a broad class of formulations. We apply our techniques to a standard set of numerical relativity test problems, demonstrating the fourth-order accuracy of the solutions. Finally we apply our approach to binary black hole head-on collisions, calculating the waveforms of gravitational radiation generated and demonstrating significant improvements in waveform accuracy over second-order methods with typically achievable numerical resolution. \nPACS numbers: 04.25.Dm, 04.25.Nx, 04.30.Db, 04.70.Bw", 'I. INTRODUCTION': "Gravitational wave astronomy will soon provide astrophysicists with a new tool to observe and analyze some of the most energetic phenomena in our universe. Collisions of compact objects, such as neutron stars, stellarmass black holes, and super-massive black holes, should produce characteristic gravitational wave signals that are observable to cosmological distances. In particular binary black hole systems are among the most promising sources of gravitational waves for both the current generation of ground-based detectors, such as LIGO [1], and for the next generation of space-based detectors, such as LISA [2, 3]. While early ground-based detectors will need to use templates from theoretically derived waveforms in order to extract the weak signal from the much larger noise, space-based detectors will require accurate models of the gravitational wave sources to extract the characteristic information (e.g., mass, spin) from the detected gravitational waves. Thus there is a critical need for accurate gravitational waveforms from realistic simulations of merging black holes. \nNumerical studies of Einstein's equations play an essential role in studying highly dynamic systems, such as binary black holes, that have little or no symmetry. These are not only computationally very demanding, requiring high-performance massive parallel computers, but also mathematically and numerically very challenging. Despite these difficulties a great deal of progress has been made in the last few years [4], and it is now possible to follow binary black hole evolutions during the last moments of the merger phase [5, 6, 7] and perhaps even through a complete orbit [8]. The calculation of the gravitational radiation emission from merging binary black holes has also been made possible through the use of the Lazarus approach, which bridges numerical relativity and perturbative techniques to extract approximate gravitational waveforms [9, 10, 11, 12, 13]. These calculations are in good agreement with recent numerical relativity evolutions [7]. \nAt present, one of the most serious limiting factor in numerical relativity has been the inability to extract accurate and reliable results from stable three-dimensional numerical simulations of coalescing binary black hole spacetimes. Considerable resources have been dedicated to finding numerically stable formulations of the Einstein evolution equations [6, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]. This effort has led to stable evolutions of single black hole spacetimes [25, 26, 27, 28]. However, there is a growing realization that solving a well-posed formulation of the Einstein equations with second-order accurate finite differencing is not sufficient to produce accurate simulations of binary black hole spacetimes. Unfortunately, black hole simulations require large domains with high resolution near the horizons, and the unigrid second-order accurate codes developed in the past are not sufficient, given the foreseeable constraints on memory and CPU speed, to produce accurate waveforms from merging black holes [29]. The Numerical Relativity community is currently pursuing several different approaches to produce accurate evolutions. Among them are the use of finite elements, spectral methods [30, 31], adaptive mesh finite-difference codes [32, 33, 34, 35, 36], and higher order finite-difference codes [37]. In addition, fourth-order accurate algorithms have recently been designed to evolve perturbations of nonrotating [38] and rotating black holes [39]. Ideally, the next generation of finite-difference codes should combine higher order finitedifferencing in combination with mesh refinement techniques. However, considerable progress toward highly accurate evolutions can be achieved using a unigrid higherorder finite-difference code along with a coordinate system, such as 'Fisheye' [10], that concentrates the gridpoints in the central region containing the black holes. In this paper we present a successful approach to performing fourth-order accurate numerical relativity simulations. \nOur simulations are based on a new numerical relativity evolution code framework, LazEv , to solve the full non-linear Einstein's equations in 3D. The LazEv framework is designed to be modular. The code consists of \na 'Method of Lines' time integrator along with several Mathematica scripts [40] that convert tensorial equations into finite-difference algorithms (of arbitrary order) in C. This modularity allows us to quickly implement new formulations and gauge choices as needed. We have currently implemented the ADM formulation as well as several 'flavors' of the BSSN formulation [14, 15, 17]. In Sec. II we describe the LazEv framework in detail. In Sec. III we enumerate many of the basic techniques utilized in numerical relativity simulations which we have implemented within LazEv , providing the foundation for detailed testing and application in black hole simulations. \nIn Sec. IV we test our fourth-order techniques on standard numerical relativity testbed problems [41] demonstrating fourth-order accuracy. Finally, in Sec V we apply these methods to study head-on binary black hole collisions. Although head-on collisions have limited astrophysical relevance, there has been recent interest in using these spacetimes to develop and test numerical techniques for evolving more generic binary black hole spacetimes [33, 34, 42]. In this paper we demonstrate that our fourth-order techniques can realize significant improvements in gravitational radiation waveform accuracies at typically achievable numerical resolutions.", 'II. THE LAZEV EVOLUTION FRAMEWORK': "The LazEv evolution framework is a foundation for rapid implementation of numerical relativity evolution system routines, supporting higher order finite differencing techniques with 'method of lines' (MoL) integration [43]. Our numerical code uses the Cactus Computational Toolkit [44] for parallelization and IO. We have constructed the code so that it is compatible with the initial data and analysis thorns provided with Cactus. Specific evolution routines are produced using Mathematica scripts [40] that convert tensorial equations into finite difference algorithms in C. These scripts provide several types of consistency checks that prevent many types of potential coding errors in setting up the tensorial equations. The Mathematica scripts support generic centered spatial finite differencing, with optional upwinded differencing (and other off-center differencing) provided by external macros. Currently we use second and fourth-order spatial finite differencing. \nThe LazEv framework applies the MoL technique for solving first-order-in-time, hyperbolic PDE's, in which the PDE's are converted into coupled ODE's for all variables at every gridpoint. This is achieved by choosing a discrete numerical grid and finite difference stencils for the spatial derivatives (the finite different stencils couple the fields at neighboring gridpoints), and then integrating the time derivatives of the fields at all gridpoints. This time integration can be carried out by standard ODE integrators. \nThe LazEv MoL integrator provides a generic framework for integrating hyperbolic PDE's using Runge- \nKutta or ICN style time integrations. The integrator itself has no knowledge of the system that is being evolved. The MoL integrator provides internal timebins pre-ministep , ministep , dissipation , and post-ministep , which are called during each step of the Runge-Kutta or ICN integration (there are several ministeps in each full timestep). The evolution system is chosen by registering routines with MoL to be evaluated during each of these bins. For example, in the BSSN system (see Sec. III A) we register routines to calculate the time derivative ∂ t of all the BSSN variables in the ministep timebin, a routine to rescale the BSSN conformal metric ˜ γ ij to unit determinant in post-ministep , and a routine to subtract off the numerical trace of the BSSN trace-free, conformal extrinsic curvature ˜ A ij in post-ministep . In the ADM system, on the other hand, we register routines only in the ministep bin. Additional evolution systems may also register routines during these steps. For example, the gauge evolution systems, which are independent of the main evolution systems, also register routines in ministep and post-ministep . \nCurrently we have implemented ICN (second-order) and Runge-Kutta (second, third, and fourth-order). In some cases we use Kreiss-Oliger [45] dissipation of the form \n∂ t F → RHS +( -1) r/ 2 /epsilon1 ∑ i h i r +1 D i + r/ 2+1 D i -r/ 2+1 F, (2.1) \nwhere ∂ t F = RHS is one of the evolution equations, h i is the gridspacing in the i th direction, D i + and D i -are the forward and backwards differencing operators (in the i th direction), and r is the order of the finite differencing used to evaluate RHS . \nWe use several different styles of finite difference stencils, depending on the order of accuracy desired and the type of derivative considered. Our standard choices for fourth-order accurate evolutions are: \n∂ x F i,j,k = 1 12 dx ( F i -2 ,j,k -8 F i -1 ,j,k + 8 F i +1 ,j,k -F i +2 ,j,k ) , (2.2) ∂ xx F i,j,k = 1 12 dx 2 ( -F i +2 ,j,k +16 F i +1 ,j,k -30 F i,j,k +16 F i -1 ,j,k -F i -2 ,j,k ) , (2.3) ∂ xy F i,j,k = 1 144 dxdy [ F i -2 ,j -2 ,k -8 F i -1 ,j -2 ,k + 8 F i +1 ,j -2 ,k -F i +2 ,j -2 ,k -8( F i -2 ,j -1 ,k -8 F i -1 ,j -1 ,k + 8 F i +1 ,j -1 ,k -F i +2 ,j -1 ,k ) + 8( F i -2 ,j +1 ,k -8 F i -1 ,j +1 ,k + 8 F i +1 ,j +1 ,k -F i +2 ,j +1 ,k ) -( F i -2 ,j +2 ,k -8 F i -1 ,j +2 ,k + 8 F i +1 ,j +2 ,k -F i +2 ,j +2 ,k )] (2.4) \nNote that the mixed xy derivative is obtained by applying \nthe x-derivative and y-derivatives sequentially (the order is irrelevant). \nWe do not use standard centered differencing for the advection terms (i.e. terms of form β i ∂ i F ). For these terms we use the following upwinded stencils: \n∂ x F i,j,k = 1 12 dx ( -F i -3 ,j,k +6 F i -2 ,j,k -18 F i -1 ,j,k + 10 F i,j,k +3 F i +1 ,j,k ) for β x < 0 (2.5) \n∂ x F i,j,k = 1 12 dx ( F i +3 ,j,k -6 F i +2 ,j,k +18 F i +1 ,j,k -10 F i,j,k -3 F i -1 ,j,k ) for β x > 0 (2.6) \nIn addition we use lower-order centered differencing on the planes adjacent to the boundaries. We have two choices for how the lower-order stencils are constructed. We can choose to use second-order centered differencing at all points on these planes and for all directions, or we can use second-order centered differencing only in the direction perpendicular to the boundary. When we use this latter choice we construct mixed second derivatives by applying the fourth-order accurate centered first derivative operator (in a direction tangent to the boundary) to the second-order centered derivative operator (perpendicular to the boundary). \nAlthough we plan to implement excision boundary conditions soon, we currently use the puncture approach (see Ref. [42] for a comparison of methods), along with singularity avoiding slicings, to evolve black hole space-times. When using punctures, we found that we needed to either use second-order stencils (but still fourth-order RungeKutta time integration) in regions inside the apparent horizons, or use a second-order upwinded stencil for the advection terms over the entire computational domain. The former method produces significantly better waveforms. See Sec. V C for further details.", 'III. FORMULATION': 'Within our LazEv framework we have implemented support for several standard options in formulating numerical relativity modeling problems. Such a formulation includes selection of an Einstein evolution system and a choice of gauge for the evolving spacetime. In this section we discuss the formulation options presently available within our LazEv framework. Of these we have specific realizations which are appropriate for the fourthorder applications discussed in the subsequent sections.', 'A. Evolution Systems': "Many alternative formulations of Einstein's evolution equations have been considered so far (see Ref. [46] and references therein). Within the most popular Cauchy \nor 3 + 1 approach, one can currently choose among the first order symmetric hyperbolic formulations of the evolution equations which are mathematically very attractive [19, 20, 22, 23], and various flavors of spatially second-order hyperbolic formulations which are numerically more tractable [47, 48]. While the former have the advantage of possessing a well-posed continuum limit in the presence of maximally dissipative boundary conditions, it is not yet clear how one can handle dozens of free parameters [19, 20] and dynamical gauge choices for the lapse and shift. On the other hand, some second-order hyperbolic formulations have proven to be empirically much more robust than others [15, 17, 49, 50]. In this paper we will focus on the Baumgarte-Shapiro-ShibataNakamura (BSSN) [14, 15, 17] system, a strongly hyperbolic system [51, 52] that has been shown to have some attractive stability properties [51, 52, 53]. \nThe BSSN system is an extension of the standard ADM [54] system with better numerical properties than the original system. In the ADM system the Einstein Equations are split into six evolution equations for the metric (along with six auxiliary evolution equations for the extrinsic curvature) \n∂ 0 γ ij = -2 αK ij , (3.1) \n+ α ( R ij + KK ij -2 K ik K k j ) , (3.2) \n-D i D j α, \nand four constraints equations \nH ≡ R + K 2 -K ij K ij = 0 , (3.3) \nM i ≡ D j ( K ij -γ ij K ) = 0 . (3.4) \nHere ∂ 0 is the operator ∂ t -L β , L β is the Lie derivative with respect to the shift vector β i , D i is the covariant derivative associated with the 3-metric γ ij , R ij is the three-dimensional Ricci tensor, R the Ricci scalar, and K is the trace of K ij . \nThe BSSN system of equations is obtained from the standard ADM equations by the following substitutions, \nγ ij e 4 φ ˜ γ ij , (3.5) \nK ij → e 4 φ ( ˜ A ij + 1 3 ˜ γ ij K ) , (3.6) \n→ \nwhere ˜ γ = det ˜ γ ij = 1 and ˜ A i i = 0. Three additional variables \n˜ Γ i = -∂ j ˜ γ ij (3.7) \nare also introduced. The BSSN variables (˜ γ ij , ˜ A ij , K , φ , and ˜ Γ i ) obey the following evolution equations [55] \n∂ 0 ˜ γ ij = -2 α ˜ A ij , (3.8) \n∂ 0 φ = -1 6 αK, (3.9) \n∂ 0 ˜ A ij = e -4 φ ( -D i D j α + αR ij ) TF + \n∂ 0 K ij \n= \nα K ˜ A ij -2 ˜ A ik ˜ A k j ) , (3.10) \n( \n∂ t ˜ Γ i = ˜ γ jk ∂ j ∂ k β i + 1 3 ˜ γ ij ∂ j ∂ k β k + β j ∂ j ˜ Γ i -˜ Γ j ∂ j β i + 2 3 ˜ Γ i ∂ j β j -2 ˜ A ij ∂ j α + 2 α ( ˜ Γ i jk ˜ A jk +6 ˜ A ij ∂ j φ -2 3 ˜ γ ij ∂ j K ) , (3.12) \n) ∂ 0 K = -D i D i α + α ( ˜ A ij ˜ A ij + 1 3 K 2 ) , (3.11) \nwhere TF indicates that only the trace-free part of the tensor is used and R ij = ˜ R ij + R φ ij is given by \nR φ ij = -2 ˜ D i ˜ D j φ -2˜ γ ij ˜ D k ˜ D k φ +4 ˜ D i φ ˜ D j φ -4˜ γ ij ˜ D k φ ˜ D k φ, (3.13) ˜ R ij = 1 ˜ lm ∂ l ∂ m ˜ γ ij + ˜ γ k ( i ∂ j ) ˜ Γ k + ˜ Γ k ˜ Γ ( ij ) k + ˜ γ lm 2 ˜ Γ k l ( i ˜ Γ j ) km + ˜ Γ k im ˜ Γ klj ) , (3.14) \n-2 γ γ ( \nand ˜ D i is the covariant derivative with respect to ˜ γ ij . ˜ Γ i is replaced by -∂ j ˜ γ ij in Eq's (3.8) - (3.14) wherever it is not differentiated. Note that Eq. (3.12) gives the ∂ t derivative of ˜ Γ i rather than the ∂ 0 derivative. The Lie derivatives of the non-tensorial quantities ( φ , ˜ γ ij , and ˜ A ij ) are given by \nL β φ = β k ∂ k φ + 1 6 ∂ k β k , (3.15) L β ˜ γ ij = β k ∂ k ˜ γ ij + ˜ γ ik ∂ j β k + ˜ γ jk ∂ i β k -2 3 ˜ γ ij ∂ k β k , (3.16) L β ˜ A ij = β k ∂ k ˜ A ij + ˜ A ik ∂ j β k + ˜ A jk ∂ i β k -2 3 ˜ A ij ∂ k β k . (3.17) \nIn addition to the evolution equations, the BSSN variables must also obey the following seven differential constraint equations \nH = R -˜ A ij ˜ A ij + 2 3 K 2 = 0 , (3.18) e 4 φ M i = ∂ j ˜ A ij + ˜ Γ i jk ˜ A jk +6 ˜ A ij ∂ j φ -2 3 ˜ γ ij ∂ j K = 0 , (3.19) G i = ˜ Γ i + ∂ j ˜ γ ij = 0 , (3.20) \nand the following two algebraic constraint equations \n˜ γ = 1 , (3.21) \n˜ A i i = 0 . (3.22) \nWe monitor the differential constraints but do not enforce them. However, we enforce the algebraic constraints by rescaling the evolved ˜ γ ij and subtracting off the trace of the evolved ˜ A ij at every timestep. \nand \n∂ t β i = B i -ζ β i , ∂ t B i = F ∂ t ˜ Γ i -ηB i , (3.27) \nwhere η and ζ are some prescribed functions over space, and F is a function of α and ψ . Unless otherwise specified, we take ζ = 0. \nWe can evaluate ( R ij ) TF = R ij -1 / 3 γ ij R in two ways. We could either calculate R ij and find its trace R numerically and subtract off the numerical trace, or we can use the Hamiltonian constraint Eq. (3.18) to calculate R and subtract that from R ij . Our tests showed very little difference between the two approaches even when the Hamiltonian constraint violations were relatively large. Unless otherwise specified we use the latter method.", 'B. Gauge Choices': "We have implemented three basic types of lapse conditions: (i) maximal slicing, (ii) 'Bona-Mass'o' type lapses [25, 55, 56, 57], (ii) and densitized lapses. The maximal slicing condition has the form \n∆ α = β i ∂ i K + αK ij K ij , (3.23) \nand is implemented using a modified version of the Cactus 'Maximal' thorn [44] in combination with a fourth-order accurate version of Bernd Brugmann's 'BAM Elliptic' [44, 58] thorn. This fourth-order accurate version of 'BAM Elliptic' was developed at UTB by Mark Hannam. The 'K-driver' lapses have the form \n∂ t α = -α 2 f ( α ) A, (3.24) \nwhere \n∂ t A = ∂ t K -ξA, (3.25) \nand ξ is some specified function (usually zero). We have also implemented modifications to this general form by replacing ∂ t α with ∂ 0 α , adding D i β i to the right-handside of Eq. (3.24), and multiplying A by factors of ψ n in Eq. (3.24), where ψ is the conformal factor of the puncture data. Equation (3.24) includes harmonic slicing ( f ( α ) = 1), 1+log slicing ( f ( α ) = 2 /α ), and 'shockavoiding' slicing ( f ( α ) = 8 3 ( α (3 -α )) -1 ) [59]. \n-We have implemented 'Gamma-driver' shift conditions [55] as well as static corotating shifts. The two versions of the 'Gamma-driver' shifts that we implemented are \n∂ t β i = F B i -ζ β i , ∂ t B i = ∂ t ˜ Γ i -ηB i , (3.26)", 'C. Boundaries': 'Thus far we have implemented fairly simple boundary conditions. For fourth-order upwinded stencils we alternatively use centered finite differencing or second-order upwinding at the second point from the boundary, and second-order centered differencing at the first point from the boundary. The boundary points are filled using radiative boundary conditions [55]. However, when the shift is zero, or sufficiently small, we evolve ˜ γ ij on the boundary using the zero-shift form of Eq. (3.8). We have observed that the boundary algorithm has very little effect on the convergence of the /lscript = 2 component of the waveforms in Sec. V. \nThese boundary conditions are not known to be well posed and do lead to incoming constraint violating modes. Our future work will involve imposing constraintpreserving boundary conditions [48, 60, 61].', 'IV. APPLES WITH APPLES TESTS': "We apply the LazEv framework to a standard set of numerical relativity tests known as 'Apples with Apples' tests [41]. These tests consists of evolving spacetimes with R × T 3 topology with the advantage that there are no boundaries. The results from these testbeds indicate that LazEv is stable and fourth-order convergent with our choices of fourth-order stencils.", 'A. Gauge wave test': "For this test we evolve the metric \nds 2 = -Hdt 2 + Hdx 2 + dy 2 + dz 2 , (4.1) \nwhere \nH = 1 + A sin ( 2 π ( x -t ) d ) , \nand d is the wavelength. The ADM variables for this metric have the form \nγ xx = 1 + A sin ( 2 π ( x -t ) d ) , (4.2) \nγ yy = γ zz = 1 , (4.3) \n( \nK xx = Aπ d cos ( 2 π ( x -t ) d ) √ 1 + A sin 2 π ( x -t ) d ) , (4.4) \n) α = √ 1 + A sin ( 2 π ( x -t ) d ) , (4.5) \nwhere all remaining ADM variables (including β i ) are zero. Note that α obeys the harmonic slicing condition \n∂ t α = -α 2 K. (4.6) \nFIG. 1: The L 2 norm of δγ xx = γ xx -γ analytic xx , rescaled by ρ 4 / 16, for the one-dimensional 'Gauge Wave' test with A = 0 . 01. Note the near perfect overlap for 630 crossing times and that the larger ρ runs are convergent longer. The runs are acceptably accurate when the (non-rescaled) norm of the error is smaller than 10 -4 (i.e. 1% of A ). \n<!-- image --> \nIn our simulations α is 'live' and we evolve it using the harmonic slicing condition. A two-dimensional test is obtained by the coordinate transformation \nx → 1 √ 2 ( x + y ) , y → 1 √ 2 ( y -x ) , \nproducing non-trivial γ xx , γ yy , K xx , and K yy . \nIn order to obtain stable runs we needed to include dissipation of the form given in Eq. (2.1), with dissipation coefficient /epsilon1 = 0 . 125. \nThese one and two-dimensional problems were evolved using a three-dimensional grid with periodic boundary conditions in all directions. We chose a wavelength ( d ) of 1 and constructed the grid so that it contained 6+1 /h points (6 points for ghost-zones) in the non-trivial directions and 9 points in the trivial directions (the stencil requires 3 ghost-zones), where h is the gridsize and 1 /h is an integer. \nOur first test is a weak gauge wave with amplitude A = 0 . 01 and resolutions h = 0 . 02 /ρ , where ρ = 2, 4, 8. Figure 1 shows the L 2 norm of the error in γ xx versus time (rescaled by a factor of ρ 4 / 16). Note that fourth-order convergence (as evident by the overlap of the rescaled curves) implies that the error for the ρ = 8 case is 256 times smaller than the ρ = 2 case. This relationship breaks down near 630 crossing times, though the higher resolution runs remain in the convergence regime longer. Using the criterion that the numerical solution is sufficiently accurate if the norm of the error is smaller than 1% of the amplitude A , we find that the fourth-order \nFIG. 2: The L 2 norm of H , rescaled by ρ 4 / 16, for the onedimensional 'Gauge Wave' test with A = 0 . 01. Note the near perfect overlap for 800 crossing times and that the larger ρ runs are convergent longer. The runs are acceptably accurate when the (non-rescaled) norm of the Hamiltonian constraint is smaller than 10 -4 (i.e. 1% of A ) \n<!-- image --> \ncode produces accurate results to t = 308, t = 720, and t = 865 for the ρ = 2, ρ = 4, and ρ = 8 runs respectively. Figure 2 shows the L 2 norm of H (rescaled by a factor of ρ 4 / 16). Again, the higher resolution runs remain in the convergence regime longer. Using the criterion that the numerical solution is sufficiently accurate if the norm of the Hamiltonian constraint is smaller than 1% of the amplitude A , we find that the fourth-order code produces accurate results to t = 130, t = 495, and t = 638 for the ρ = 2, ρ = 4, and ρ = 8 runs respectively. The Hamiltonian constrain violation is a stricter measure of the quality of the results than the error in γ xx because the Hamiltonian constraint involves second derivatives of the metric (which are harder to model). \nOur next test is a stronger gauge wave. Figure 3 shows the L 2 norm of H versus time (rescaled by a factor of ρ 4 / 16) for the gauge-wave test with amplitude A = 0 . 1 and resolutions h = 0 . 02 /ρ , where ρ = 2, 4, 8. Note that in this higher amplitude case the runs remain convergent for 80 crossing times (approximately one-tenth as long as the A = 0 . 01 runs). Using the criterion that the numerical solution is sufficiently accurate if the norm of the Hamiltonian constraint is smaller than 1% of the amplitude A , we find that the fourth-order code produces accurate results to t = 19 . 5, t = 46, and t = 60 for the ρ = 2, ρ = 4, and ρ = 8 runs respectively. Hence these A = 0 . 1 runs are accurate for roughly one-tenth the time that the A = 0 . 01 are accurate. \nThough we have applied our fully three-dimensional code to the last two test problems, the dynamics in these cases are non-trivial only in one (numerical grid) direction. Our last gauge wave test is non-trivial in two \nFIG. 3: The L 2 norm of H , rescaled by ρ 4 / 16, for the onedimensional 'Gauge Wave' test with A = 0 . 1. Note the near perfect overlap for 80 crossing times and that the larger ρ runs are convergent longer. The runs are acceptably accurate when the (non-rescaled) norm of the Hamiltonian constraint is smaller than 10 -3 (i.e. 1% of A ). \n<!-- image --> \nFIG. 4: The L 2 norm of H , rescaled by ρ 4 / 16, for the twodimensional 'Gauge Wave' test with A = 0 . 1. Note the near perfect overlap for 55 crossing times and that the larger ρ runs are convergent longer. The runs are acceptably accurate when the (non-rescaled) norm of the Hamiltonian constraint is smaller than 10 -3 (i.e. 1% of A ). \n<!-- image --> \ngrid directions. Figure 4 shows the L 2 norm of H versus time (rescaled by a factor of ρ 4 / 16) for the twodimensional gauge-wave test with amplitude A = 0 . 1 and resolutions h = 0 . 02 /ρ , where ρ = 2, 4, 8. These two-dimensional runs are less stable than the corresponding one-dimensional run, crashing before 100 crossing \ntimes. Convergence is, nonetheless, maintained for about 55 crossing times, longer for the higher resolution runs. Using the criterion that the numerical solution is sufficiently accurate if the norm of the Hamiltonian constraint is smaller than 1% of the amplitude A , we find that the fourth-order code produces accurate results to t = 10, t = 30, and t = 40 for the ρ = 2, ρ = 4, and ρ = 8 runs respectively. Note that these two-dimensional A = 0 . 1 runs are accurate roughly two-thirds as long as the corresponding one-dimensional A = 0 . 1 runs.", 'B. Gowdy wave test': "In this section we present results for the 'Polarized Gowdy Wave' test. The 'Polarized Gowdy Wave' metric is given by \nds 2 = t -1 / 2 e λ/ 2 ( -dt 2 + dz 2 )+ t ( e P dx 2 + e -P dy 2 ) , (4.7) \nwhere \nP = J 0 (2 πt ) cos(2 πz ) , (4.8) λ = -2 πtJ 0 (2 πt ) J 1 (2 πt ) cos 2 (2 πz ) + 2 π 2 z 2 [ J 2 0 (2 πt ) + J 2 1 (2 πt ) ] -1 2 ((2 π ) 2 [ J 2 0 (2 π ) + J 2 1 (2 π )] -2 πJ 0 (2 π ) J 1 (2 π )) . (4.9) \nWe use this metric to obtain initial data and evolve backwards in time using the harmonic slicing condition. The above metric only varies as a function of z and t but we can obtain a two-dimensional problem by introducing new coordinates x = ˜ x , y = ˜ y , z = ˜ z + ˜ y . \nHere again we use a full three-dimensional grid to solve one and two-dimensional problems. We use periodic boundary conditions in all directions, and our grid consists of 1 /h points (plus 6 for ghost-zones) in the non-trivial directions, and 9 points in the trivial directions. Additionally, we needed to add dissipation of the form given in Eq. (2.1), with dissipation coefficient /epsilon1 = 0 . 000625, to stabilize the runs. \nFigures 5 and 6 show the fourth-order convergence of the Hamiltonian constraint and the γ zz component of the metric of the one-dimensional Gowdy Wave test for 1000 crossing times. Note that these convergence plots imply that the errors in γ zz and H in the ρ = 8 run are 256 times smaller than these errors in the ρ = 2 run. Unlike the previous gauge-wave test, here the amplitude of the metric functions are damped. So our criterion for an acceptable value of the Hamiltonian constrain violation is time dependent. In this case we will use the criterion that the error norm, divided by the norm of γ zz , must be smaller than 1%. At the end of the run the L 2 of γ zz (the function, not the error) is of order 1. Both the Hamiltonian constraint and the error in γ zz are smaller than this quality criterion for all resolutions. \nFigure 7 shows the fourth-order convergence of the Hamiltonian constraint for the two-dimensional Gowdy \nFIG. 5: The L 2 norm of H , rescaled by ρ 4 / 16, for the onedimensional 'Gowdy Wave' test. Note the good agreement between the curves for 1000 crossing times and that the evolution is backwards in time \n<!-- image --> \nFIG. 6: The L 2 norm of the error in γ zz , rescaled by the L 2 norm of γ zz and by ρ 4 / 16, for the one-dimensional 'Gowdy Wave' test. Note the good agreement between the curves for 1000 crossing times and that the relative error in γ zz is less than 2 · 10 -4 at 1000 crossing times. The evolution is backwards in time. \n<!-- image --> \nWave test. The early time lack of convergence is due to roundoff effects, and, although the ρ = 2 curve does not lie on the ρ = 4 and ρ = 8 curves, the overlap of the ρ = 4 and ρ = 8 curves confirm that the code is fourth-order convergent with sufficient resolution. \nFigure 8 shows the fourth-order convergence of γ yy for the two-dimensional Gowdy Wave test. Unlike H , γ yy clearly demonstrates fourth-order convergence at the low \nFIG. 7: The L 2 norm of H , rescaled by ρ 4 / 16, for the twodimensional 'Gowdy Wave' test. The test was limited to 50 crossing times due to limited resources. The early time lack of convergence is due to low amplitude high frequency noise in the ρ = 8 results. Although the two higher resolution curves do not lie on top of ρ = 2 curve, the overlap of the two high resolution curves indicates that the Hamiltonian constraint is converging to fourth-order. Note that the evolution is backwards in time. \n<!-- image --> \n( ρ = 2) resolution. \nWe conclude from the gauge wave and Gowdy wave results that our BSSN code is stable, accurate, and convergent.", 'V. HEAD-ON BINARY BLACK-HOLE COLLISIONS': "In this section we present results for head-on collisions of two equal-mass Misner-Wheeler-Brill-Lindquist (MWBL) black holes [62, 63]. These results extend our previous tests of LazEv to more interesting nonlinear spacetimes containing binary black holes. \nThe MWBL data represent conformally flat slices of multiple black hole space-times with n punctures. The data are parametrized by n puncture masses m i and n puncture positions /vectorr i (in the conformal space). The ADM mass is given by the sum of m i , and the ADM linear momentum and angular momentum are zero. The data have the form \nγ ij = ψ 4 δ ij , (5.1) \nK ij = 0 , (5.2) \nψ = 1 + n ∑ i =1 m i 2 r i , (5.3) \nwhere \nand r i = | /vectorr -/vectorr i | . \nFIG. 8: The L 2 norm of the error in γ yy , rescaled by ρ 4 / 16, for the two-dimensional 'Gowdy Wave' test. The test was limited to 50 crossing times due to limited resources. The excellent overlap of all three curves indicates that the code is converging to fourth-order. Note that the evolution is backwards in time. \n<!-- image -->", 'A. Setup': "We used MWBL octant-symmetric data consisting of two equal mass black holes aligned along the z-axis, allowing us to evolve the data using the { x > 0 , y > 0 , z > 0 } octant. In addition, we use a 'Transition Fisheye' transformation [10] to enlarge the physical domain without sacrificing resolution near the punctures. The 'Transition Fisheye' transformation is a smooth radial transformation from an inner resolution fixed by the Cactus gridspacing ( h ) to an outer resolution that is generally lower than the inner resolution. This transformation is parametrized by an outer 'de-resolution parameter' a (the effective grid-spacing at the edge of the grid is ah ), a transition width parameter ' s ', and the center of the transition region ' r 0 '. The transformation has the form \nr physical = r ( a +(1 -a ) R ( r )) , (5.4) \nwhere r physical is the physical radius corresponding to the coordinate radius r and R ( r ) is given by \nR ( r ) = s 2 r tanh r 0 s ln ( cosh r + r 0 s cosh r -r 0 s ) . (5.5) \nFor these runs we set the mass parameter of the two holes to 0 . 5 M , and the puncture positions to (0 , 0 , -1 . 1515 M ) and (0 , 0 , 1 . 1515 M ). For most of the runs, the computational grid extended to 12 . 3 M , which corresponds to a physical size of 26 M . We also performed some short runs where the computational grid extended to 24 . 6 M , which corresponds to a physical size of 63 M . These latter runs were used to determine the effect of the boundaries on the waveforms, constraint violations, \nand horizon mass. The Fisheye parameters used in these runs were a = 3, s = 1 . 2 M , r 0 = 5 . 5 M . We performed runs at resolutions of h = 1 . 1515 M/ (9 ρ ) with gridsizes of (96 ρ ) 3 gridpoints along one octant, where ρ = 1 , 2 , 4. The gridsizes were chosen so that the punctures lie halfway between gridpoints. For the runs with the boundary at 63 M we used the same resolutions but chose gridsizes of (192 ρ ) 3 gridpoint along one octant, where ρ = 1 , 2. \nWe evolved the data with the standard 1+log lapse, where the initial value of A (see Eq. 3.25) is zero, and a modified 1+log lapse, where the initial value of A is \nA ( t = 0) = c exp -( ψ -1) -2 /σ ) . (5.6) \n( \n) We choose 1 for the initial value of the lapse. When using the modified lapse we set c = 0 . 5, and σ = 0 . 8. This modified lapse, unlike the original 1+log lapse, collapses at the puncture in the continuum limit. Analytically the standard 1+log lapse should retain its initial value at the puncture throughout the entire evolution. Since we start with a lapse of one, the lapse should not collapse at the puncture. However, the lapse will collapse near the puncture, and a lack of resolution drives the lapse to zero at the puncture as well. However, as the grid is refined this artificial collapse at the puncture is delayed, leading to short wavelength features and blow-ups near the puncture. The initial value of A in Eq. (5.6) forces the lapse to collapse near the puncture in the continuum limit. Unfortunately this modification introduces its own problems as described below (see Fig. 21). \nWe used the second form of the 'Gamma-Driver' shift Eq. (3.27) to evolve the shift. The initial value of the shift and its time derivative were zero. We used both a constant η ( η = 2 . 8 /M ) and a spatially varying η equal to 2 . 8 /M at infinity and 5 . 6 /M at the punctures, as suggested by Diener [64]. The spatially varying η was of the form [42] \nη = η p -η p -η ∞ ( ψ -1) 2 +1 , (5.7) \nwhere η p and η ∞ are parameters specifying η at the puncture and infinity respectively, and ψ is the puncture data conformal factor. The function F in the 'Gamma-Driver' shift was set to F = 3 4 α/ψ 4 . \nPure fourth-order runs proved to be very unstable near the punctures both with and without upwinded stencils, and Kreiss-Oliger dissipation further destabilized these runs (near the punctures). We had success in stabilizing these runs with two techniques: (i) using second-order upwinding of advection terms over the entire grid, and (ii) a localized (within the apparent horizon) order reduction (LOR) to all second-order accurate spatial differentiation (including second-order upwinding of advection terms). This latter method provides significantly more accurate waveforms. When using the LOR method we found that lower-order accuracy is needed both at the punctures and at the origin. However, we only reduce the order of accuracy near the origin after a common apparent horizon \nforms (typically 5 M after the common apparent horizon forms). We found that Kreiss-Oliger dissipation can be used to remove late time instabilities if the dissipation coefficient is set to zero in the LOR region. However, we did not use dissipation in the runs presented below. \nWe implement LOR in the following way. First we calculate the time derivatives of all variables using the standard fourth-order stencils in Eq's (2.2) - (2.6). Then we overwrite all points within some given coordinateellipsoid with the time derivatives calculated using the standard second-order stencils (with second-order upwinded advection terms). The dissipation operator inside the LOR region can be either the same one chosen for the rest of the grid or a lower-order operator (in either case the dissipation coefficient inside the LOR region can be different from the dissipation coefficient used for the rest of the grid). Only the spatial finite differencing is changed; the time integrator is the same for all gridpoints. The size of the ellipsoid is chosen at runtime and up to eleven ellipsoids may be used. The user is free to choose when each ellipsoid is activated. For example, in the head-on binary black hole collision case, we specify a small ellipsoid for each of the individual apparent horizons and a larger ellipsoid for the common apparent horizon, where the larger ellipsoid is activated only after the common apparent horizons forms. The sizes of these ellipsoids can be determined by evolving with secondorder accuracy over the entire grid and finding the sizes of the apparent horizons versus time. \nFor the head-on-collision runs the LOR regions consisted of spheres of radius 0 . 512 M centered on the punctures activated at t = 2 M , as well as an ellipsoid with semi-axes { 0 . 512 M, 0 . 512 M, 1 . 66 M } activated at t = 11 . 6 M (the punctures were located on the z-axis). For comparison, the common apparent horizon has a minimum radius of 1 . 56 M and a maximum radius of 1 . 88 M at t = 11 . 4 M . The individual apparent horizons are approximately spherical with radii 0 . 61 M at t = 1 . 98 M . \nWe used radiative boundary conditions for all variables except β i (which were evolved via ∂ t β i = B i on the boundary). Table I gives the radiative boundary condition parameters for all variables. In Table I ' a ' is the Fisheye de-resolution parameter. We enforce the algebraic constraints Eq's (3.21),(3.22) prior to applying the boundary conditions. Hence, the boundary points do not satisfy these constraints exactly. \nWe updated the Zorro thorn of the Lazarus Toolkit [10] to compute the Weyl scalars to fourth-order accuracy and to make it compatible with octant, bitant, and quadrant symmetries in any direction as well as π rotation symmetry about the z-axis. In addition we added a spherical harmonic decomposition routine to generate the /lscript = 2 and /lscript = 4 modes shown below. \nTABLE I: Radiative Boundary Condition Parameters \n| Variables | Asymptote | speed |\n|------------------|-----------------|---------|\n| ˜ g ii | 1 | 1 /a |\n| ˜ g ij ( i = j ) | 0 | 1 /a |\n| ˜ A ij | 0 | 1 /a |\n| ˜ Γ i | 0 | 1 /a |\n| φ | (1 / 2) ln( a ) | √ 2 /a |\n| K | 0 | √ 2 /a |\n| α | 0 | √ 2 /a |\n| B i | 0 | 1 /a |\n| A | 0 | 1 /a |", 'B. Second-Order Accurate Results': "We first confirmed that our code can reproduce the second-order accurate waveforms published in [55]. For these runs we used standard 1+log lapse (i.e. A (0) = 0) and Gamma-Driver shift with η = 2 . 8. We ran with gridsizes of (96 ρ ) 3 gridpoints and grid resolutions of h = 1 . 1515 / (9 ρ ) for ρ = 1 , 2 , 4. We calculated ψ 4 as well as its ( /lscript = 2 , m = 0) and ( /lscript = 4 , m = 0) components using Zorro [10]. Note that the ( /lscript = 2 , m = 0) and ( /lscript = 4 , m = 0) modes of ψ 4 are purely real for this test. \nFigure 9 shows the ( /lscript = 2 , m = 0) mode of ψ 4 at r = 5 M for these three resolutions along with the Richardson extrapolation of these data. In addition Fig. 9 shows the differences 4( ψ 4 | ρ =2 -ψ 4 | ρ =4 ) and ψ 4 | ρ =1 -ψ 4 | ρ =2 . These two difference curves overlap reasonably well indicating that the waveforms are second-order accurate. However, the phase drift between the two curves makes an evaluation of the exact order of convergence difficult. Figure 10 shows the convergence rate for this mode given by \nν conv = log 2 ψ 4 | ρ =1 -ψ 4 | ρ =2 ψ 4 | ρ =2 -ψ 4 | ρ =4 . \nThe measured convergence rate oscillates wildly but averages to about 2. \nStrictly speaking r = 5 M is not in the radiation zone so ψ 4 given above does not represent the asymptotic waveform. However, the code's performance in calculating the ( /lscript = 2 , m = 0) mode of ψ 4 at r = 5 M is indicative of its performance in calculating this mode at large r . Additionally, boundary errors contaminate the /lscript = 4 modes at large r (see Sec. V C). Thus by extracting at r = 5 M we can maximize the amount of non-trivial, physically correct quasinormal oscillations in the waveforms. \nFigure 11 shows the L 2 norm of the Hamiltonian constraint for these three runs. Note that the constraints have not been rescaled. From the figure we can see that the constraints tend to get bigger as the resolution is increased. The source of these constraint violations is located between the puncture and origin. High frequency features develop there, leading to strong constraint violations. However, as seen in Fig. 12 these large constraint violations do not leak out of the apparent horizon. Thus, \nFIG. 9: The top plot shows the ( /lscript = 2 , m = 0) mode of ψ 4 at r = 5 M for the second-order evolution of the MWBL data for gridsizes h = 1 . 1515 / (9 ρ ), as well as the Richardson extrapolated value. The bottom plot shows the differences ( ψ 4 | ρ =1 -ψ 4 | ρ =2 ) and 4( ψ 4 | ρ =2 -ψ 4 | ρ =4 ) between these waveforms for this mode. Note that the latter difference has been rescaled by a factor of four. Second-order convergence is demonstrated by the reasonable overlap between these two differences. \n<!-- image --> \nalthough there are significant problems inside the horizon, the region outside the horizon remains uncontaminated. \nOur calculations of the gravitational waveforms from the Newman-Penrose scalar ψ 4 , which contains higherorder spatial derivatives of the metric, do not reveal any additional degree of noise and distortion with respect the waveforms obtained from the Zerilli-Moncrief formalism [34, 42]. \nFIG. 10: Convergence rate of the ( /lscript = 2 , m = 0) mode of ψ 4 at r = 5 M for a purely second-order evolution of MWBL head-on collision data. \n<!-- image --> \nFIG. 11: The L 2 norm of the Hamiltonian constraint violation versus time for the second-order accurate head-on collision runs. Note that the constraint violation increases with resolution. \n<!-- image -->", 'C. Fourth-Order Accurate Results': "Purely fourth-order runs of puncture data proved to be unstable with more than one black hole. We found that we could stabilize the evolution by reducing the spatial discretization to second-order inside the apparent horizons, and we successfully ran the MWBL head-on collision data with fourth-order discretization, fourth-order Runge Kutta time integration, and second-order LOR regions inside the apparent horizons. For these runs we used the spatially varying η form of the Gamma-driver shift given by Eq. (3.27) and Eq. (5.7). Figure 13 demon- \nFIG. 12: Hamiltonian constraint violation (absolute value) versus z along the z-axis at time t = 80 M for the ρ = 4 secondorder head-on collision run. Note that the large constraint violation inside the apparent horizon ( r = 2 . 8 M ) has not leaked out into the exterior spacetime. \n<!-- image --> \ntrates the convergence of the ( /lscript = 2 , m = 0) component of ψ 4 at r = 5 M . Note that the system is not strictly fourth-order accurate. There appears to be an additional phase discrepancy between the two differences. However, the amplitude of the differences appears to be falling at a rate consistent with better than fourth-order accuracy. The ρ = 4 run crashed at 47 . 5 M due to an instability near the origin (see Fig. 19). This unstable mode was triggered by advection terms (the collapse of the lapse near the origin ensures that these are the only terms which can lead to a blowup). The fields which blew up most strongly were the ˜ Γ i . This blow-up does not appear to be related to the quadratic blow-up in φ at the puncture discussed later. That term led to a blow up proportional to e t 2 (in H ) near the puncture, which masked the unstable behavior near the origin. Both blow-ups can be modified by various choices in the gauge conditions and it is likely that a better choice of gauge will lead to longer fourth-order evolutions. \nA fit of the ( /lscript = 2 , m = 0) mode of ψ 4 to the quasinormal form f ∼ e -at sin( ωt ) gave an exponential damping factor of a = 0 . 084 /M and frequency of ω = 0 . 373 /M . The exponential damping factor agrees to within 6% of the expected value of 0 . 0889625 /M , and the frequency agrees to within 0 . 2% of the expected value of 0 . 373672 /M [65]. \nIn Fig. 14 we show the Richardson extrapolated value of the ( /lscript = 2 , m = 0) mode of ψ 4 , calculated using the second-order accurate results, along with the values obtained from second and fourth-order evolutions with LOR. Note that the medium resolution fourth-order run outperforms the high resolution second-order run. Fig- \nFIG. 13: The top plot shows the ( /lscript = 2 , m = 0) mode of ψ 4 at r = 5 M , produced using fourth-order evolution with LOR, for 3 resolutions, along with the Richardson extrapolation curve of these data. The ρ = 2 and ρ = 4 curves are indistinguishable from the extrapolated curve. The bottom plot shows rescaled differences ψ 4 | ρ =1 -ψ 4 | ρ =2 and 16( ψ 4 | ρ =2 -ψ 4 | ρ =4 ) (note the factor of 16). The convergence rate for the fourthorder runs using LOR inside the apparent horizons is not strictly fourth-order due to the phase discrepancy between the two curves, but average waveform differences between resolutions is consistent with fourth-order accuracy. Note that the amplitude of the dotted curve is less than the amplitude of the solid curve indicating better than fourth-order reduction in the amplitude of the errors. \n<!-- image --> \nre 15 shows a magnified view of Fig. 14 (note that the legends in the two figures are different). \nFigure 16 shows the convergence of the ( /lscript = 4 , m = 0) mode of ψ 4 at r = 5 M calculated using fourthorder accuracy with LOR. The plot shows the differences ψ 4 | ρ =1 -ψ 4 | ρ =2 and 16( ψ 4 | ρ =2 -ψ 4 | ρ =4 ), as well the mode itself for ρ = 4. Note that the latter rescaled difference is smaller than the former. This indicates that the \nFIG. 14: A comparison of the ( /lscript = 2 , m = 0) mode of ψ 4 at r = 5 M for second and fourth-order evolutions with LOR. Note that the ρ = 1 fourth-order waveform is better than the ρ = 2 second-order waveform. Similarly the ρ = 2 fourthorder waveform is better than the ρ = 4 second-order waveforms. The Richardson extrapolated waveform (based on the second-order waveforms) is indistinguishable from the ρ = 2 and ρ = 4 fourth-order waveforms. \n<!-- image --> \n( /lscript = 4 , m = 0) component of the waveform is converging faster than fourth-order. In addition, note that the error in the ρ = 1 waveform is comparable to the amplitude of the waveform. \nFigure 17 shows a magnified view of the ( /lscript = 4 , m = 0) mode of ψ 4 at r = 5 M for the ρ = 1, ρ = 2, and ρ = 4 fourth-order runs as well as the ρ = 4 second-order run. Note that the ρ = 4 second-order results are again inferior to the ρ = 2 fourth-order (with LOR) results. \nThe /lscript = 4 modes, unlike the /lscript = 2 modes, contain significant contamination from the boundary at late times. For example, when extracting at r = 13 M we find that the amplitude of the ( /lscript = 4 , m = 4) mode of ψ 4 (which is non-zero due to boundary errors) is approximately 50% that of the ( /lscript = 4 , m = 0) mode at t = 45 M . This contamination is stronger when the extraction sphere is further out. However, the waveform extracted at 5 M shows significantly less contamination. The reason for this is that the incoming error is delayed by about 8 M while the outgoing mode is advanced by 8 M (for a net gain of 16 M in reliable data). In addition the outgoing mode, which has a 1 /r falloff, is correspondingly larger at this smaller radius. This boundary contamination problem can be mitigated by pushing the outer boundary further out. This can be achieved within the existing LazEv framework using a stronger fisheye transformation or adding more gridpoints. \nAlthough the waveforms from the fourth-order runs converge as expected the Hamiltonian constraint does not. High frequency features near the puncture and ori- \nFIG. 15: A comparison of the ( /lscript = 2 , m = 0) mode of ψ 4 at r = 5 M for second and fourth-order evolutions with LOR. Note that the ρ = 2 fourth-order waveform is better than the ρ = 4 second-order waveforms. The second-order ρ = 4 curve lies above the extrapolated curve while the fourth-order ρ = 1 curve lies below. \n<!-- image --> \ngin lead to large Hamiltonian constraint violation. At the points surrounding the puncture these high frequency features induce a quadratic blow-up in φ for the ρ = 4 run. This quadratic behavior, which is confined to the nearest neighbor points to the puncture, leads to a blowup of e t 2 in the Hamiltonian constraint due to terms proportional to e 4 φ . This blow-up in H is entirely localized to the puncture and does not affect points outside. In the following figures we demonstrate that the large constraint violations inside the apparent horizon do not leak out and that the constraint violations outside the apparent horizon converge to zero when boundary effects are removed. \nFigure 18 shows the Hamiltonian constraint along the z-axis (the points surrounding the puncture have been removed) at t = 47 . 2 M for the second and fourth-order runs for ρ = 4. The Hamiltonian constraint inside the horizon is as much as 10 4 times larger for the fourthorder run. However, outside the horizon the constraint violations are very similar. Note that we do not expect the constraints to converge to zero in this case because our boundary conditions are not constraint preserving and boundary constraint violations have contaminated the solution at this time (the boundaries were at 26 M in physical coordinates). The ρ = 4 fourth-order run with LOR crashed due to an instability near the origin. Figure 19 shows the unstable mode in . \nIn Fig. 20 we show the L 2 norms of the Hamiltonian constraint H , momentum constraint M i , and the BSSN constraints G i (note that the x and y components of the both momentum and BSSN constraints are equal). We restricted these norms to the region outside \nH \nFIG. 16: The ( /lscript = 4 , m = 0) mode of ψ 4 for ρ = 4 as well as the differences between the ρ = 1 and ρ = 2 waveforms and the ρ = 2 and ρ = 4 waveforms (the latter rescaled by 16). The convergence rate of the ( /lscript = 4 , m = 0) mode of ψ 4 at r = 5 M for the fourth-order runs with LOR is better than fourthorder. Note that the error in the ρ = 1 waveform (as evident by the difference between the ρ = 1 and ρ = 2 waveforms) is approximately 50% of the amplitude of the waveform at 20 M and larger than the waveform beyond 30 M . \n<!-- image --> \nr physical = 3 M (the horizon is at r physical ∼ 2 M ) and inside r physical = 26 M , where r physical is the physical radius. The outer boundaries were located at 63 M in physical coordinates. These runs required double the number of gridpoints (along each direction) as the standard runs for the same resolutions. The reasonable overlap of the ρ = 1 and rescaled ρ = 2 curves for 43 M of evolution indicates that the L 2 norms are approximately fourthorder convergent (the convergence is lost near t = 40 M due to constraint violating modes propagating in from the boundary). In addition, the amplitude of the constraint violations is ∼ 10 -5 , which is nine order of magnitude smaller than the maximum Hamiltonian violation inside the horizon. We conclude from this figure that the second-order errors introduced by the LOR differencing, as well as the extremely large Hamiltonian constraint violation inside the apparent horizon, do not propagate outside of the apparent horizon. \nThe extreme Hamiltonian violation near the puncture can be removed using the modified 1+log lapse. However, this modification tends to destabilize the runs at late times. Figure 21 shows the L 2 norm of the Hamiltonian constraint violation for the ρ = 4 fourth-order run with LOR. The modified lapse produces a constraint violation smaller than 1 for 25 M but also causes the run to crash at 26 . 5 M . The standard 1+log lapse produces a 'stable' evolution to 47 . 5 M (see above) but has a constraint violation proportional to e t 2 at the puncture. If short term evolutions, like the ones required for \nFIG. 17: The ( /lscript = 4 , m = 0) mode of ψ 4 at r = 5 M for fourth-order evolutions with LOR as well as second-order evolutions. Note that the ρ = 1 fourth-order waveform has an order 100% error and that the ρ = 2 fourth-order waveform has a substantially smaller error than the ρ = 4 second-order waveform. \n<!-- image --> \nthe Lazarus techniques, are required, then the modified 1+log is preferred. \nFigure 22 shows the horizon mass versus time, as calculated by the AHFinderDirect thorn [66], for the ρ = 2 and ρ = 4 runs with LOR. The plot also shows the horizon mass of a ρ = 2 run with double the standard number of gridpoints in each direction and outer boundary at 63 M in physical coordinates. The relatively sharp increase in the horizon mass at t = 32 M is due to contamination from the boundary. When the boundary is moved out to 63 M (from 26 M ) this increase is delayed by roughly ∆ t = 37 M . Also note that the slope of the increase is reduced when the boundary is further out. The measured convergence rate of the horizon mass (based on runs with the boundary at 26 M ) exceeded 3.8 for 30 M of evolution. \nThe horizon-mass versus time plot shows that the late time boundary contamination contains significant erroneous gravitational radiation. To determine when the calculated gravitational waveforms are good approximations to the true waveforms, we need to confirm that (i) the waveforms converge and do not change significantly when the resolution is further increased, (ii) the waveforms do not change significantly when the boundary effects are removed (which we achieved by causally disconnecting the boundaries from the observer), and (iii) the Hamiltonian constraint violation (inside the extraction region) is much smaller than the waveform amplitude. We have shown that the waveforms converge, and that the Hamiltonian constraint violation converges in the extraction region (see Fig. 20). Here we show the effects of the boundary on the Hamiltonian constraint and the waveforms. Figure 23 shows Hamiltonian constraint vio- \nFIG. 18: The absolute value of the Hamiltonian constraint along the z-axis at t = 47 . 2 M for the second and fourth-order runs. The points near the puncture have been removed (the violation at the puncture was 10 117 ) from the fourth-order data. Note that the constraint violation is 10 4 times bigger inside the horizon for the 4th order run, and that the extreme constraint violation does not leak out of the horizon (located at z = 2 . 8 M ). \n<!-- image --> \nFIG. 19: The unstable mode in H for the ρ = 4 fourth-order runs. \n<!-- image --> \nlation along the z axis at time t = 40 . 8 M for the standard ρ = 2 run and a run with twice the standard number of gridpoints. The boundary in this latter case was at 63 M (note that the z -axis plots the fisheye coordinate and that z fisheye = 12 . 3 M corresponds to the physical coordinate z physical = 26 M and z fisheye = 24 . 6 M corresponds to z physical = 63 M ). The figure shows the Hamiltonian constraint outside the apparent horizon. Note that at this \nFIG. 20: The L 2 norm, restricted to the region 3 M < r physical < 26 M , of all constraint violations. The boundaries were located at 63 M . All norms have been multiplied by 10 5 and the ρ = 2 norms have been rescaled by an additional factor of 16. In each panel the solid curves correspond to ρ = 1 and the dashed curves to ρ = 2. The reasonable overlap of the dashed and solid curves indicate that the constraint violations are converging to zero to fourth-order. The constraints no longer converge to zero after t ∼ 43 M due to boundary effects. \n<!-- image --> \nFIG. 21: The L 2 norm of the Hamiltonian constraint violation for the ρ = 4 fourth-order evolution with LOR for the standard and modified 1+log lapses. Note the e t 2 blow-up in the run using the standard 1+log lapse and that the run using the modified 1+log lapse crashes at 26 . 5 M . \n<!-- image --> \nFIG. 22: The horizon mass versus time for the fourth-order runs with LOR. The thick solid line and the dotted line show the horizon mass versus time for the ρ = 2 resolution with the physical boundary at 63 M and 26 M respectively. Note that the sharp increase in mass at later times is due to boundary effects. The dashed line shows the horizon mass for the ρ = 4 run with the boundaries at 26 M . The error in the horizon mass at t = 63 . 5 M is 2 . 6% for the ρ = 2 run with boundary at 26 M . \n<!-- image --> \ntime the boundary errors for the larger run have just reached z fisheye = 12 M . The Hamiltonian constraint, in the range z fisheye = 5 M to z fisheye = 12 M , is 100 times smaller for the run with the larger boundary. Even at t = 70 M , when the boundary errors have contaminated the entire grid, the Hamiltonian constraint is 20 times smaller for the larger run in this range. \nAs seen in Fig. 24, the effect of reducing the boundary noise is not readily apparent in the ( /lscript = 2 , m = 0) mode of ψ 4 . However, as seen in Fig. 25, the effect is readily observable in the ( /lscript = 4 , m = 0) mode. We conclude, based on these two figures, that the ( /lscript = 2 , m = 0) mode is represented accurately to 55 M when the boundaries are located at 26 M , while the ( /lscript = 4 , m = 0) mode is represented accurately only to 33 M for the same boundary location. The ( /lscript = 2 , m = 0) mode is more accurate because the boundary contamination contains relatively high frequencies which are filtered more effectively by the ( /lscript = 2 , m = 0) angular integration. Note that we placed the observer at r = 5 M and that the waveforms become inaccurate sooner when the extraction sphere is placed at larger radii. \nWe conclude this section by showing the waveforms produced with fourth-order centered spatial finite differencing for all terms except the advection terms, for which we used second-order upwinded differencing. The waveforms produced by this method are inferior to those produced with LOR techniques but superior to those produced by the standard second-order technique. The time \nFIG. 23: The Hamiltonian constraint along the z axis at t = 40 . 8 M for the fourth-order (with LOR) runs with gridspacing h = 1 . 1515 / 18 (i.e. ρ = 2). z fisheye = 12 . 3 M corresponds to 26 M in physical coordinates and z fisheye = 24 . 6 M corresponds to 63 M in physical coordinates. Boundary contamination for the larger run has just reached z fisheye = 12 M . Note that in the range 5 M < z fisheye < 12 M the Hamiltonian constraint is 100 time smaller when the boundary is pushed out to 63 M . \n<!-- image --> \nFIG. 24: The ( /lscript = 2 , m = 0) mode of ψ 4 (observer at r = 5 M ) for the fourth-order (with LOR) ρ = 2 run with the physical boundary at 26 M (standard) and 63 M . The effect of the boundary is not significant until t = 55 M . \n<!-- image --> \nintegration was carried out with the standard fourthorder Runge Kutta integrator. Figure 26 shows the differences ψ 4 | ρ =1 -ψ 4 | ρ =2 and ψ 4 | ρ =2 -ψ 4 | ρ =4 with the latter rescaled by 4 (to indicate second-order convergence) for the ( /lscript = 2 , m = 0) mode of ψ 4 at r = 5 M . The two curves do not overlap exactly due to phase drift. Despite \nFIG. 25: The ( /lscript = 4 , m = 0) mode of ψ 4 (observer at r = 5 M ) for the fourth-order (with LOR) ρ = 2 run with the physical boundary at 26 M (standard) and 63 M . The effect of the boundary is significant at t > 33 M . \n<!-- image --> \nthe second-order accuracy of the algorithm, the waveforms produced with this technique are more accurate than those produced by the purely second-order spatial differencing. Figure 27 shows the ( /lscript = 2 , m = 0) mode of ψ 4 at r = 5 M for the standard second-order evolution as well as the mixed fourth-order with second-order upwinding evolution. Note that the ρ = 2 waveform from the latter technique is of a similar quality to the ρ = 4 waveform from the purely second-order technique. The ρ = 4 run crashed at 47 . 5 M with the same mode that killed the ρ = 4 LOR run. Both techniques crashed at the same time and in the same way because the instability near the origin is driven by advection terms, and both techniques use the same upwinded advection stencil near the origin. Figure 19 shows the unstable mode (along the z-axis) that crashes the ρ = 4 fourth-order runs (with LOR and with second-order upwinding). \nWe computed the energy radiated from the head-on collision by several different methods. We first estimated it by computing the difference between the final horizon mass and the total ADM mass, and obtained a radiated energy in the range (7 -8) × 10 -4 M . We also used the Lazarus method to extract Cauchy data for the Teukolsky equation at relatively early evolution times T ≤ 20 M , and obtained E radiated = (8 ± 1) × 10 -4 M . The direct integration of the /lscript = 2 and /lscript = 4 waveforms that we present here, extracted at numerical radii between 5 M and 10 M , produce energy estimates of the order ∼ 6 . 6 × 10 -4 M . \nFIG. 26: Convergence plot for the mixed fourth-ordered centered differencing with second-order upwinded differencing runs. Note that the difference ψ 4 | ρ =2 -ψ 4 | ρ =4 has been multiplied by 4, indicating approximate second-order convergence. \n<!-- image --> \nFIG. 27: A comparison of the ( /lscript = 2 , m = 0) mode of ψ 4 at r = 5 M produced using the standard second-order evolution and a mixed fourth-order with second-order upwinding evolution. Note that the ρ = 2 waveform from the mixed fourth-order/second-order upwinding is of similar quality to the ρ = 4 second-order waveform. The extrapolated values used in this plot are based on the waveforms from the fourthorder runs with LOR. \n<!-- image -->", 'VI. CONCLUSION': "We developed a new framework, LazEv , for evolving the Einstein equations using 3+1 decompositions. LazEv is capable of evolving with arbitrary-order finite difference stencils along with second, third, and fourth-order time integrators. The overall LazEv design has a few \nnovel features which will improve significantly upon previous setups which have been used in the Lazarus approach [9, 10, 11, 12, 67]. LazEv is a flexible and modular evolution package, well-suited for rapid development and experimentation with well-posed hyperbolic systems of evolution equations, new 'live' gauge conditions, and sophisticated boundary conditions using arbitrary order finite differencing. \nWe implemented the BSSN formulation using this new framework and demonstrated that this new code passes the Apples with Apples testsuite and that the code reproduces the second-order accurate head-on binary black hole collisions waveforms published in Ref. [55]. We found that fourth-order accurate evolutions of the same data were not stable without some reduction of order and that the most accurate waveforms were obtained by evolving the data with second-order accurate stencils inside the apparent horizons and fourth-order stencils outside the horizon. In that case we found that the waveforms were fourth-order convergent and that the fourth-order accurate 192 3 gridpoint runs outperformed the second-order-accurate 384 3 gridpoint runs. This means that our fourth-order evolutions give better quality waveforms with over an order-of-magnitude smaller computational expense when compared to the second-order evolutions. \nWe also found that using a second-order upwinded stencil for the advection terms was sufficient to stabilize the runs. However, this introduces a second-order error in the waveform. Nevertheless, the waveforms produced by this latter method are superior to those produced by ordinary second-order accurate evolutions. \nIn this paper we demonstrated that the LazEv framework can be used to evolve binary black hole spacetimes. We plan to extend this work to include orbiting black holes starting from initial data based on the conformal thin-sandwich formulation [68] as well as study recoil velocities from unequal mass head-on collisions [69]. We also plan to extend the framework to include excision, fixed mesh refinement, constraint damping [70, 71], and constraint preserving boundary conditions.", 'Acknowledgments': "We thank Peter Diener for providing the Gamma driver shift parameters needed to stabilize the fourthorder BBH runs. We thank Manuel Tiglio and Peter Diener for helpful discussions on fourth-order evolutions. \nWe thank Peter Diener, Mark Hannam, and Bernard Kelly for helpful discussions and for carefully reading this manuscript. \nWe gratefully acknowledge the support of the NASA Center for Gravitational Wave Astronomy at University of Texas at Brownsville (NAG5-13396) and the NSF for financial support from grants PHY-0140326 and PHY0354867. 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Lett. 87 , 271103 (2001), gr-qc/0012079.\n- [47] G. Nagy, O. E. Ortiz, and O. A. Reula, Phys. Rev. D 70 , 044012 (2004), gr-qc/0402123.\n- [48] C. Gundlach and J. M. Martin-Garcia, Phys. Rev. D 70 , 044032 (2004), gr-qc/0403019.\n- [49] F. Pretorius, Class. Quant. Grav. 22 , 425 (2005), grqc/0407110.\n- [50] H.-O. Kreiss and O. E. Ortiz, Lect. Notes Phys. 604 , 359 ((2002)), gr-qc/0106085.\n- [51] O. Sarbach, G. Calabrese, J. Pullin, and M. Tiglio, Phys. Rev. D 66 , 064002 (2002).\n- [52] H. Beyer and O. Sarbach, Phys. Rev. D 70 , 104004 (2004), gr-qc/0406003.\n- [53] M. Alcubierre, G. Allen, B. Brugmann, E. Seidel, and W.-M. Suen, Phys. Rev. D 62 , 124011 (2000), gr-q \n/9908079. \n- [54] R. Arnowitt, S. Deser, and C. W. Misner, in Gravitation: An Introduction to Current Research , edited by L. Witten (John Wiley, New York, 1962), pp. 227-265, gr-qc/0405109.\n- [55] M. Alcubierre, B. Brugmann, P. Diener, M. Koppitz, D. Pollney, E. Seidel, and R. Takahashi, Phys. Rev. D 67 , 084023 (2003), gr-qc/0206072.\n- [56] C. Bona, J. Mass'o, E. Seidel, and J. Stela, Phys. Rev. Lett. 75 , 600 (1995), gr-qc/9412071.\n- [57] J. Balakrishna, G. Daues, E. Seidel, W.-M. Suen, M. Tobias, and E. Wang, Class. Quantum Grav. 13 , L135 (1996).\n- [58] S. Brandt and B. Brugmann, Phys. Rev. Lett. 78 , 3606 (1997), gr-qc/9703066.\n- [59] M. Alcubierre, Class. Quantum Grav. 20 , 607 (2003), gr-qc/0210050.\n- [60] S. Frittelli and R. G'omez, Phys. Rev. D 70 , 064008 (2004), gr-qc/0404070.\n- [61] G. Calabrese, L. Lehner, and M. Tiglio, Phys. Rev. D \n- 65 , 104031 (2002), gr-qc/0111003.\n- [62] D. Brill and R. Lindquist, Phys. Rev. 131 , 471 (1963).\n- [63] C. Misner and J. Wheeler, Ann. Phys. (N.Y.) 2 , 525 (1957).\n- [64] P. Diener, private Communication.\n- [65] H.-P. Nollert and B. G. Schmidt, Phys. Rev. D 45 , 2617 (1992).\n- [66] J. Thornburg, Class. Quantum Grav. 21 , 743 (2004), grqc/0306056, URL http://stacks.iop.org/0264-9381/ 21/743 .\n- [67] J. Baker, B. Brugmann, M. Campanelli, and C. O. Lousto, Class. Quantum Grav. 17 , L149 (2000), grqc/0003027.\n- [68] M. D. Hannam, conformal thin-sandwich puncture black holes: quasi-circular orbits (in preparation).\n- [69] P. Anninos and S. Brandt, Phys. Rev. Lett. 81 , 508 (1998), gr-qc/9806031.\n- [70] P. Marronetti (2005), preprint gr-qc/0501043.\n- [71] B. K. Berger (2004), gr-qc/0410058."}
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2004ApJ...601..428K
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The Three Spectral Regimes Found in the Stellar Black Hole XTE J1550-564 in Its High/Soft State
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2004-01-01
| 9 | 0.5 | 164 |
['accretion', 'accretion disks', 'black hole physics', '-', 'astronomy x rays', 'astronomy x rays', 'astrophysics']
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The present paper describes the analysis of multiple RXTE PCA data of the black hole binary with superluminal jet, XTE J1550-564, acquired during its 1998-1999 outburst. The X-ray spectra show features typical of the high/soft spectral state and can approximately be described by an optically thick disk spectrum plus a power-law tail. Three distinct spectral regimes, which we call the ``standard regime,'' the ``anomalous regime,'' and the ``apparently standard regime,'' have been found from the entire set of the observed spectra. When the X-ray luminosity is well below ~6×10<SUP>38</SUP> ergs s<SUP>-1</SUP> (assuming a distance of 5 kpc), XTE J1550-564 resides in the standard regime, in which the soft spectral component dominates the power-law component and the observed disk inner radius is kept constant. When the luminosity exceeds the critical luminosity, the apparently standard regime is realized, in which the luminosity of the optically thick disk rises less steeply with the temperature, and the spectral shape is moderately distorted from that of the standard accretion disk. In this regime, the radial temperature gradient of the disk has been found to be flatter than that of the standard accretion disk. The results of the apparently standard regime suggest a slim disk, which is a solution predicted for a high mass accretion rate. In the intermediate anomalous regime (or very high state in the literature), the spectrum becomes much harder, and the disk inner radius derived using a simple disk model spectrum apparently varies significantly with time. These properties can be explained as a result of significant thermal inverse Comptonization of the disk photons, as was found from GRO J1655-40 in its anomalous regime by Kubota and coworkers.
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https://arxiv.org/pdf/astro-ph/0310085.pdf
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{'The three spectral regimes found in the stellar black hole XTE J1550-564 in its high/soft state': 'Aya Kubota \nInstitute of Space and Astronautical Science, 3-1-1 Yoshinodai, Sagamihara, Kanagawa 229-8510, Japan \[email protected] \nKazuo Makishima 1 \nDepartment of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan', 'ABSTRACT': 'The present paper describes the analysis of multiple RXTE /PCA data of the black hole binary with superluminal jet, XTE J1550 -564, acquired during its 1999-2000 outburst. The X-ray spectra show features typical of the high/soft spectral state, and can approximately be described by an optically thick disk spectrum plus a power-law tail. Three distinct spectral regimes, named standard regime, anomalous regime, and apparently standard regime, have been found from the entire set of the observed spectra. When the X-ray luminosity is well below ∼ 6 × 10 38 erg s -1 (assuming a distance of 5 kpc), XTE J1550 -564 resides in the standard regime , where the soft spectral component dominates the power-law component and the observed disk inner radius is kept constant. When the luminosity exceeds the critical luminosity, the apparently standard regime is realized, where luminosity of the optically thick disk rises less steeply with the temperature, and the spectral shape is moderately distorted from that of the standard accretion disk. In this regime, radial temperature gradient of the disk has been found to be flatter than that of the standard accretion disk. The results of the apparently standard regime are suggestive of a slim disk (e.g., Abramowicz et al. 1988, Watarai et al. 2000) which is a solution predicted under high mass accretion rate. In the intermediate anomalous regime , the spectrum becomes much harder, and the disk inner radius derived using a simple disk model spectrum apparently varies significantly with time. These properties can \nbe explained as a result of significant thermal inverse Comptonization of the disk photons, as was found from GRO J1655 -40 in its anomalous regime by Kubota, Makishima and Ebisawa (2001). \nSubject headings: accretion, accretion disks- black hole physics-stars:individual (XTE J1550 -564) -X-rays:stars', '1. Introduction': "In a close binary consisting of a mass accreting stellar-mass black hole and a mass donating normal star, the accreting matter releases its gravitational energy as X-ray radiation. When the mass accretion rate ˙ M is high, such a black hole binary is usually found in a so-called high/soft state, of which X-ray spectrum is characterized by a very soft component accompanied by a power-law tail. As described, e.g., by Makishima et al. (1986), the soft spectral component is interpreted as thermal emission from an optically thick accretion disk around the black hole, as it can be well reproduced by a multi-color disk model (MCD model; Mitsuda et al. 1984). This model approximates a spectrum from a standard accretion disk (Shakura & Sunyaev 1973). The MCD model has two spectral parameters; the maximum disk color temperature, T in , and the apparent disk inner radius, r in ; the latter can be related to the true inner radius, R in , via a correction for spectral hardening (Shimura & Takahara 1997) and a boundary condition (Kubota et al. 1998). As the disk luminosity changes significantly, the value of R in is usually observed to remain constant at the innermost Keplerian orbit for the black hole, 6 R g where R g = GM/c 2 is the gravitational radius (e.g., Ebisawa et al. 1993). \nAlthough this 'standard picture' is successful for many of the high/soft-state black hole binaries (e.g., Tanaka & Lewin 1995; McClintock & Remillard 2003), it has been pointed out theoretically that the standard disk can be stable only over a limited range of ˙ M . Actually, a series of new solutions to the accretion flow have been discovered, including a slim disk solution, which takes advective cooling into account (Abramowicz et al. 1988; Watarai et al. 2000). In addition, a 'very high state' has been observationally found as a derivative state from the soft state (Miyamoto et al. 1991; van der Klis 1994). Characterized by an enhanced hard component and significant variations in both T in and r in , the very high state is regarded as a possible violation of the simple-minded standard-disk picture. \nA clue to this problem has been recently given by Kubota, Makishima, & Ebisawa (2001; hereafter Paper I) from an analysis of the multiple RXTE /PCA data of the black hole transient with superluminal jet, GRO J1655 -40. While the source behavior was \ndescribed adequately by the standard-disk picture over some period (called standard regime ) of the entire PCA data span, the other period was characterized by the previously suggested deviation from such a standard behavior (called anomalous regime ). The enhanced hard X-ray spectrum in the anomalous regime has been interpreted successfully as a result of significant inverse Compton scattering of the disk photons by some high energy electrons. The inner radius of the underlying optically thick disk has been found to be kept constant, when the effect of the Comptonization is taken into account. This result has been reinforced by Kobayashi et al. (2003). \nIn order to reinforce the view obtained from GRO J1655 -40, and to deepen our understanding of the physics of accretion under high values of ˙ M , the RXTE data of the Xray transient XTE J1550 -564 is analyzed in this paper. This transient source was discovered on 1998 September 7 by RXTE /ASM and CGRO /BATSE (Wilson et al. 1998, Smith 1998), and now confirmed as a superluminal jet source (Hannikainen et al. 2001). Figure 1 shows a lightcurve of this source, obtained with the RXTE /ASM. As indicated with down-arrows in this figure, this outburst was continuously monitored by the RXTE pointing observations through 1999 May 20. Optical observations have established that the system consists of a late type sub giant (G8IV to K4III) and a black hole, the latter having a dynamical mass of M X = 8 . 4-11.2 M /circledot (Orosz et al. 2002). The binary inclination angle and the distance to the source are estimated to be i = 67 · -75 · and D = 2 . 8-7.6 kpc, respectively. In this paper, i = 70 · is used as a crude estimate, and its distance is denoted as D = D 5 · 5 kpc. \nIn § 2, the observation and data reduction are briefly described. In § 3, the PCA spectra are analyzed using the canonical MCD plus power-law model, leading to the identification of three characteristic regimes; standard regime , anomalous regime , and apparently standard regime , in the increasing order of the luminosity. It is confirmed in § 4 that the spectra in the anomalous regime can be well explained by the strong inverse Compton scattering, as found in Paper I. In § 5, the apparently standard regime spectra are characterized in terms of the radial temperature gradient of the optically-thick disk, with a conclusion that a slim disk is probably realized.", '2. Observation and data reduction': "As indicated with down-arrows in Fig. 1, the entire outburst was covered by 184 pointing observations with RXTE ; the obtained data was analyzed in a standard way by Sobczak et al. (2000). By analyzing the outburst rise phase ( ∼ 40 days in Fig .1) covered by 14 pointing observations, Wilson & Done (2001) reported that this source experienced a significant spectral evolution. Figure 1 in their paper shows that the power-law photon index \ndescribing the PCA spectra below ∼ 20 keV switched from 1.4-1.7 to 2-3, which are typical of the low/hard state (e.g., Tanaka 1997), and the high/soft or very high state (e.g., Grove et al. 1998), respectively. Meanwhile, the 20-100 keV portion of the spectrum maintained a convex shape. Therefore, the source is inferred to have made a spectral transition from the low/hard state into the very high state (Wilson & Done 2001), or into the anomalous regime defined in Paper I, rather than into the canonical high/soft state wherein the spectrum above 20 keV would not exhibit any cutoff. \nThese published results indicate that the spectra acquired during the first ∼ 40 days of the outburst is possibly related to development of the optically thick accretion disk; in response to a sudden increase in ˙ M , an optically thick accretion disk developed inward and reached the last stable orbit. Also the data after day 225 (1999 April 20) shows characteristics of the low/hard state. Therefore, the present paper focuses on the data taken from 128 PCA pointing observations between day 43 (1998 October 20) and 224 (1999 April 19), which are thought to represent relatively steady states with high ˙ M . For the systematic analyses of all the data set, the HEXTE data is not analyzed here because of poor statistics in some pointings. \nFollowing the standard procedure for bright sources, good PCA data was selected and processed. The data was excluded when the target elevation angle was less than 10 · above the Earth's Limb, or when the actual pointing direction was more than 1 ' . 2 away from the pointed direction. In particular, the data was discarded if it was acquired within 30 minutes after the spacecraft passage through South Atlantic Anomaly. The selected data from the individual proportional counter units was co-added and used for spectral analyses. The standard dead time correction procedure was applied to the data. The PCA background was estimated for each observation, using the software package pcabackest (version 2.1e), supplied by the RXTE Guest Observer's Facility at NASA/GSFC. Sometimes, the use of pcabackest resulted in an overestimate of the PCA background up to ∼ 10%. Such a systematic over-estimate was corrected in the same way as in Paper I. That is, the on-source spectra were compared to the predicted model background spectra in the hardest energy band ( > 80 keV), where the signal flux is usually negligible. If necessary, the normalization factor of the background spectrum was changed. The PCA response matrix was made for each observation by utilizing the software package pcarsp (version 7.10). In order to take into account the calibration uncertainties, 1% systematic errors are added to the data. Over the 20-35 keV range, the systematic errors are increased to 10%, to cope with the response uncertainties associated with the Xe-K edge at ∼ 30 keV. Although this could be an overestimate, the 20-35 keV data is utilized only in § 4, and the results remain essentially unchanged even if it is reduced to 2 %.", '3.1. Characterization of the observed spectra': 'The 3-20 keV PCA spectra of XTE J1550 -564 were analyzed by employing the canonical MCD plus power-law model. The two constituent continuum components were subjected to common photoelectric absorption, with the column fixed at N H = 1 × 10 22 cm -2 , which is reasonable for the source distance of 5 kpc and its Galactic position of ( l, b ) = (325 · . 9 , -1 · . 83). The following results are not affected by changing the value of N H to 5 × 10 21 cm -2 . The data requires additionally an absorption edge to the power-law component around 7-9 keV in terms of a smeared edge model (Ebisawa et al. 1994) and a narrow Gaussian line around 6.5-6.7 keV for the Fe-K line. \nFigure 2 shows time histories of the best-fit model parameters, including the disk bolometric luminosity L disk (Mitsuda et al. 1984; Makishima et al. 1986), the 1-100 keV power-law luminosity L pow calculated assuming an isotropic emission and their sum, L tot ≡ L disk + L pow . Along the time history, the entire observational span can be divided into the following eight periods (Periods 1-8 denoted in Fig. 2a) of relatively distinct properties, by mainly referring to the behavior of L pow and L disk . \n- · Periods 1 and 7 are characterized by dominance of L pow and significant reductions in r in .\n- · In Periods 2 and 8, L tot decreases keeping L pow rather low, while r in remains rather large and constant.\n- · In Period 3, L disk gradually increases, but L pow remains negligible. The power-law component is too weak to constrain its photon index Γ. It is therefore fixed at Γ = 2 . 0.\n- · In Periods 4 and 5, L pow is negligible (hence Γ being still fixed at 2.0), while L tot saturates at ∼ 6 × 10 38 · D 5 2 erg s -1 ; this is about ∼ 40 · D 5 2 % of the Eddington limit, L E , for a black hole of 10 M /circledot .\n- · Period 6 may be intermediate between Periods 5 and 7. \nTypical spectra representing these periods are shown in Fig. 3, and their best-fit parameters are given in Table 1. The fits are acceptable for the data in Periods 1, 2, and 6-8, but sometimes unacceptable in Periods 3-5 (Fig. 2e). This problem is considered again in § 5.', '3.2. Observed three spectral regimes': "To examine each Period for the validity of the standard picture, Fig. 4 shows several scatter plots between the spectral parameters. The data points of Periods 2, 3, and 8 are thus confirmed to satisfy the standard picture, because r in is therein kept constant at ∼ 60 · D 5 km while T in changes significantly, ∼ 0.5-1 keV (Fig. 2, Fig. 4a). By taking into account a correction factor for the boundary condition of ξ = 0 . 412 (Kubota et al. 1998) and a color hardening factor of κ = 1 . 7 (Shimura & Takahara 1997), the true inner radius is estimated as R in = r in · ξ · κ 2 ∼ 71 · D 5 km. This value seems to be slightly smaller than 6 R g for a 10 M /circledot black hole, 90 km, though the source distance is not well constrained. \nThe data points in Period 1 and 7 clearly violate the standard picture. Even though the fits are acceptable (Fig. 2e), r in is observed to change significantly over 30-50 · D 5 km (Fig. 2c), and T in shows strong positive deviation from their long-term trend (Fig. 2b). In these Periods, the hard component dominates the MCD component (Fig. 2a), and values of Γ are largest among these eight Periods (Fig. 2b). All these properties make Periods 1 and 7 reminiscent of the anomalous regime of GRO J1655 -40 (Paper I). \nThe classification of Period 4-6 is somewhat ambiguous. The hard emission is negligible (Fig. 2a), and the spectral shape is similar to those in the standard regime (Fig. 3). However, as is clear from Fig. 2a, the data suffers from a strong saturation in L disk . In addition, a slight increase of T in under the saturation in L disk gives rise to a slight decrease in r in (Fig. 4a). In this sense, Periods 4-6 are called the apparently standard regime in this paper. Thus, XTE J1550 -564 exhibits three characteristic spectral regimes, the standard (Periods 2, 3, 8), apparently standard (4, 5, 6), and anomalous (1, 7) regimes . \nThe L disk -L pow diagram, presented in Fig. 4c, can be used to understand the spectral evolution. The entire span of this source is found to show a clockwise loop; starting from a highestL pow point (Period 1; anomalous regime ), the source moves to the left along a lowL disk branch (Period 2; standard regime ), then increases in L disk with L pow kept low (Period 3; standard regime ), and reaches a ceiling at L disk , ∼ 6 × 10 38 · D 5 2 erg s -1 . It then moves to the right (Periods 4, 5 and 6; apparently standard regime ). Finally, in Period 7 ( anomalous regime ), it returns to nearly the same position as Period 1, with Period 8 ( standard regime ) being a simple repetition of Period 2. Thus, the source fortunately exhibited a complete one cycle. \nFigure 5a shows a L disk -T in diagram, which is useful to examine the validity of the standard picture against luminosity. A simple relation of L disk ∝ T in 4 means the constancy of r in (see also Fig 4a), and hence the goodness of the standard picture. In this diagram, the data points in the standard regime indeed satisfy this relation, while those in the anomalous \nregime deviate significantly. The data points in the apparently standard regime , clustered at the uppermost end of the diagram, deviate weakly from the standard L disk ∝ T in 4 relation, exhibiting a flatter dependence of L disk on T in as L disk ∝ T in 2 . \nIt is useful to note here that the overall spectral shape of the MCD model well describes emission from a standard disk, even though it ignores the inner boundary condition of the disk. As long as the PCA spectra of T in = 0 . 5-2 keV are concerned, the values of T in and r in (with the corrections mentioned before) agree with 4-5 % with those obtained by a more accurate model, e.g., a diskpn model (Gierli'nski et al. 1999; Gierli'nski & Done 2003) in xspec . Therefore, the characteristics shown in Fig. 4-5, are not due to incompleteness of the MCD model but mean that the anomalous regime and the apparently standard regime are intrinsically different from the standard regime .", '4. Reanalyses of the anomalous regime data - confirmation of the strong inverse Compton scattering -': 'Now that Period 1 and 7 have been inferred to be the anomalous regime , these data can be reanalyzed employing the concept of the strong disk Comptonization which successfully explains the anomalous regime of GRO J1655 -40 (Paper I). In the case of GRO J1655 -40, the hard component in the anomalous regime has been found to be different in behavior from the power-law tail of the standard regime , in several points, including negative correlation between L pow and L disk , and systematically higher values of Γ. It is hence argued in Paper I that in the anomalous regime the spectrum is contributed significantly by a third spectral component, which is harder than the MCD emission but softer than the power-law tail in the standard regime . The strong anti-correlation between L pow and L disk has been taken for evidence that this third component strongly and negatively correlates with the MCD component. It is therefore natural to assume that a fraction of the photons from the optically thick accretion disk are converted into the third spectral component, most probably through inverse Compton scattering by high energy electrons which may reside somewhere around the disk. \nFollowing Paper I, the PCA spectra in the anomalous regime of XTE J1550 -564 are re-fitted with a three-component model, obtained by adding a Comptonized component to the original two component model. In this paper, a thermal Comptonization model ( thcomp ; Zycki, Done, & Smith 1999 ) is utilized to reproduce the Comptonized component, instead of the Comptonized blackbody model ( compbb ; Nishimura, Mitsuda, & Itoh 1986) which was used in Paper I. The thcomp model is based on a solution of the Kompaneets equation (Lightman & Zdziarski 1987). It well describes thermal cutoff around the electron \ntemperature of the plasma, T e , and allows the MCD model to be used as a seed photon spectrum. In contrast, the compbb model is useable only energies below kT e , and it assumes a single temperature black body emission as a seed photon spectrum. The thcomp model has actually improved the fit goodness compared to the compbb model because of difference in the seed photon spectra. The main results of the present paper are however independent of such modelings, and same results were obtained by using the compbb model. In this subsection, the 3-50 keV data is used to better constrain the wide-band spectral shape. \nFor the spectral fitting, many of the model parameters were fixed to default values after Paper I. Namely, the maximum color temperature of the seed photons was tied to T in , to reproduce the situation whereby a part of the original MCD photons are up-scattered. Furthermore, T e was fixed at a representative value of 20 keV since it could not be constrained and is consistent with that obtained during the first 14 observations by Wilson & Done (2001). Neither a reflection component nor relativistic smearing was added to the model. Hence the free parameters of the thcomp model are two; thcomp photon index Γ thc which expresses the spectral shape below kT e , and its normalization. Moreover, a value of Γ of the original power-law component was fixed at 2.0. As a result, the additional number of the free parameters for spectral fittings is reduced to only one. \nFigure 6 shows the same anomalous-regime spectrum as presented in Fig. 3a, but fitted with the three-component model over the expanded energy range. A solid line represents the additional thcomp component. The fit is acceptable, and the result implies that the dominant hard spectral component is mostly produced by the strong disk Comptonization. The re-estimated values of T in for all the data in the anomalous regime are plotted again in Fig. 2b with open triangles. By considering the disk Comptonization, the highly deviated data points in terms of T in have thus settled back to a smooth long-term trend as was already found in GRO J1655 -40 (Paper I). The luminosity is also re-estimated in Fig. 5b as L disk + L thc , where L thc is the estimated 0.01-100 keV thcomp luminosity, assuming an isotropic emission. Thus, L disk + L thc plotted against the revised T in approximately recovers the standard L disk ∝ T in 4 relation for optically-thick accretion disks. \nThe value of r in is difficult to be estimated precisely under the strong Comptonization, because of uncertainties of both geometry and optical depth, τ es , of the cloud. In the present paper, it is approximately calculated by referring to observed photon flux and a re-estimated value of T in , on an assumption that few photons are scattered back into the optically thick disk and hence the number of the observed photons is conserved through inverse Compton scatterings. Details of this procedure are described in Appendix-A. The re-estimated values of r in , plotted in Fig. 2c with open triangles, now appear to remain almost stable at ∼ 60 · D 5 km. Consequently, the optically thick accretion disk can also be considered to remain \nrelatively stable, even when a significant fraction of the MCD photons are Comptonized. Therefore, the picture of disk Comptonization is reconfirmed in the anomalous regime , as suggested in GRO J1655 -40 (Paper I; Kobayashi et al. 2003). \nTable 2 summarizes the best-fit parameters associated with the exemplified spectra in the anomalous regime . The value of Γ thc ∼ 2 . 8 implies a y -parameter of ∼ 0 . 2, or τ es ∼ 1 . 8 calculated via a following formula (e.g., Sunyaev & Titarchuk 1980) assuming T e = 20 keV; \nτ es = √ 2 . 25 + 3 ( T e / 511keV) · { (Γ thc +0 . 5) 2 -2 . 25 } -1 . 5 . (1) \nThe derived parameters are similar to those of GRO J1655 -40. The smallness of y and τ es is consistent with the comparatively small fraction of L thc relative to L disk + L thc , typically < 0 . 5. It is also consistent with the assumption made in re-evaluating r in in Appendix-A and Fig. 5b, that the mean number of scattering is not too large and the fractional energy change is small. \nFrom these results, the anomalous behavior of XTE J1550 -564 observed in Periods 1 and 7 can be identified with that of GRO J1655 -40 in the anomalous regime : the optically-thick standard accretion disk is present, but the Comptonization converts a significant fraction of its emission into the hard component.', '5.1. Properties of the apparently standard regime': 'As shown in § 3, the apparently standard regime corresponds to the most luminous phase of the outburst, and the data in this regime occupies the upper-right region of the L disk -T in diagram (Fig. 5b). In this regime, T in gradually changed keeping L disk almost constant at 6 × 10 38 · D 5 2 erg s -1 ( ∼ 0 . 4 L E ). As a result, the data points deviate from the standard L disk ∝ T in 4 relation, as L disk ∝ T in 2 . The obtained values of r in are not constant but become smaller than those in the standard regime , exhibiting a weak correlation with T in as r in ∝ T in -1 . Figure 4a clearly shows this behavior. \nAs seen in Fig. 3, the spectra in the apparently standard regime show a dominant soft component accompanied by a very weak hard tail. Although these properties are similar to that of the standard soft state, the apparently standard regime is something different from the standard regime because of the inconstancy of r in (or moderate saturation of L disk ), and the absence of the Fe-K line feature that is usually found in the standard regime . Moreover, the canonical spectral model often failed to give acceptable fits to the data (Fig. 2e). As are \nclearly seen in the residuals of Fig. 3d-e, the discrepancy between the data and the best-fit canonical model appears as a low-energy excess and a spectral hump around 13 keV. This could be a result of fixing Γ of the hard tail component at 2.0; accordingly, the fits were repeated by leaving Γ free to vary. Then, the fits became acceptable, but they required so large values of Γ (see Fig. 2d, Table 1), that the spectrum below ∼ 5 keV is mostly accounted for by the power-law component rather than the MCD component; the observed low-energy excess is filled up artificially by the steep power-law. Such a fit could be physically inappropriate. \nThe above results on the apparently standard regime suggest a subtle difference in the accretion disk configuration from the standard disk, in such a way that the softest end of the observed spectrum is more enhanced than is described by the MCD model. Such a change, in turn, may arise if, e.g., the radiative efficiency of the inner disk region becomes reduced and the radial temperature gradient flattens. In order to quantify this idea, in § 5.2 a p -free disk model is constructed as a mathematically generalized function of the MCD model. The data in the apparently standard regime is then examined in § 5.3 by utilizing this model function.', '5.2. Formalism of the p -free disk model': 'The concept of the standard accretion disk assumes that the energy released by accretion is half stored in the Keplerian kinetic energy, and half radiated away as local blackbody emission. As a result, the spectrum of the standard disk can be described as a geometricallyweighted sum of multi-temperature blackbody components of which the local temperature depends as T ( r ) ∝ r -3 / 4 on the distance, r , from the central black hole. Therefore, any departure of the physical condition assumed by the standard disk picture will make the radial temperature gradient deviate from the canonical value of -3 / 4. Such a deviation will in turn cause a slight deformation of the radially-integrated X-ray spectra that are observed. In order to quantify this idea, the MCD model is generalized as seen below, after initial attempts by Mineshige et al. (1994, for GS 1124 -68) and Hirano et al. (1995, for Cyg X-2). \nThe main assumptions of the model developed here are that a disk local temperature is described by T ( r ) = T in · ( r/r in ) -p , and that the disk locally emits a blackbody spectrum. Here, p is a dimension-less positive parameter introduced to generalize the MCD formalism, with p = 3 / 4 implying the MCD model. The spectrum from this model function can be written as \nf p ( E ) = 2 π cos i · r 2 in p · d 2 ∫ T in T out ( T T in ) -2 p -1 B ( E,T ) dT T in , (2) \nwith B ( E,T ) being a blackbody spectrum of temperature T . For convenience, hereafter \nthis mathematical model function is called p -free disk model. As p decreases, the spectrum becomes softer than the MCD spectrum of the same T in , because the radiation from outer parts of the accretion disk is emphasized as seen in equation (2).', '5.3. Spectral fitting with the p -free disk model': "The PCA spectra of Periods 3-5 (one segment of standard regime and two of apparently standard regime ) have been re-fitted by replacing the MCD model component with the p -free disk model. Although an accurate determination of p is difficult when the power-law component is strong, the PCA spectra in Period 3-5 are fortunately free from this obstacle. The condition of spectral fitting is otherwise the same as in § 3. That is, the absorption column N H and Γ of the power-law component are fixed at 1 × 10 22 cm -2 and 2.0, respectively. The Gaussian line was not included. \nThe time histories of the obtained p -free disk model parameters are given in Fig. 7. In Period 4 and 5 ( apparently standard regime ), the fit goodness has been significantly improved by allowing p free. Figure 8 shows the same apparently standard regime spectrum originally presented in Fig. 2e, fitted this time with the p -free disk model, and Table 3 shows the examples of the p -free model parameters. \nIn Fig. 9, the best fit values of p are plotted against T in ; here, instead of the values of T in obtained by the p -free disk model, those by the MCD model are employed, in order to avoid any systematic coupling between p and T in . The dependence of p on T in thus changes at ∼ 1 keV. As a function of T in , p increases up to ∼ 1 keV (Period 3, the standard regime ), beyond which it decreases abruptly and the correlation turns negative (Period 4-5, the apparently standard regime ). To examine these results for various systematic effects, the p -free disk fits have been extensively repeated by changing the fitting conditions; fixing N H to 5 × 10 21 cm -2 instead of 1 × 10 22 cm -2 , fixing Γ to 2.2 instead of 2.0, including a Gaussian line of which the central energy is constrained to 6.2-6.9 keV. These different conditions slightly affected the absolute values of p , but did not affect the characteristic p vs. T in behavior of Fig. 9a. \nThus, the value of p has been found to deviate from 3/4 as the source enters deep into the apparently standard regime . However, a still larger excursion of p is observed during the standard regime , where p = 3 / 4 should be obtained. Therefore, the changes of p in Fig. 9 could partially or entirely be due to some artifacts, and p could deviate from 3/4, even if the standard accretion disk is realized. This could actually happens, because the MCD model is a mere approximation of the exact standard-disk solution; the actual temperature gradient of a standard accretion disk must be flatter than -3 / 4 near the innermost disk edge, where \nthe temperature will approach zero. As T in decreases, the limited PCA band pass will sample preferentially the emission from inner disk regions, thus making p different from 3/4. \nIn order to address the above issue, a theoretical approach was first attempted. That is, a number of 3-20 keV PCA spectra were simulated, using two theoretical model spectra in the xspec which are known to be more accurate than the MCD model. One is the diskpn model, which is based on the Shakura-Sunyaev solution in a pseudo-Newtonian potential; it properly takes the inner boundary condition into account. The other is so-called a GRAD model (Hanawa 1989; Ebisawa, Mitsuda, & Hanawa 1991), which considers full general relativistic effects for a Schwarzshild metric. Then, the simulated spectra were fitted with the p -free disk model. As a result, both diskpn and GRAD models gave p = 0 . 72-0.75 as long as T in is in the range 1.2-1.5 keV. Furthermore, p was found to change (over 0.51.0) for lower input disk temperature of T in = 0 . 8-1.5 keV. However, the p -T in relation turned out to be quite different between the two input models, and to depend significantly on the inclination angle when the GRAD model is used. In addition to these theoretical complications, neither the diskpn model nor the GRAD model can be considered accurate yet, since they (as well as the MCD model) neglect possible spectral deviation from a pure local blackbody. Accordingly, this approach has been concluded unrealistic. \nIn this paper, instead, the PCA-determined values of p for a standard disk has been calibrated empirically, by using actual spectral data of a prototypical black hole binary, LMC X-3, in the standard regime . This black hole binary perfectly satisfies the standard picture up to ∼ L E for a black hole of 5-7.2 M /circledot (Paper I, Wilms et al. 2001), and its system inclination angle, 65 · -69 · , is similar to that of XTE J1550 -564. The p -free disk model was applied to the PCA spectra of LMC X-3 obtained by 128 pointed RXTE observations from 1996 February to 1999 January (Kubota 2001). The best fit values of p from LMC X3 are plotted against T in in Fig. 9b. As expected, a positive correlation between p and T in artificially appears even for this prototypical 'standard-disk' object. Therefore, the behavior of XTE J1550 -564 (in Fig. 9a) for T in ≤ 1keV (Period 3, the standard regime ) can be understood as an artifact. In contrast, the behavior of XTE J1550 -564 for T in ≥ 1 keV is clearly distinct from that of LMC X-3. The conclusion is that the temperature gradient of XTE J1550 -564 in Period 4 and 5 becomes intrinsically smaller than in the standard regime , and hence the accretion disk in the apparently standard regime in reality deviates from the standard picture.", '6.1. Overall picture from the observation': 'Through the detailed analysis of the PCA data of XTE J1550 -564, the three spectral regimes have been identified. Their relation on the L disk -T in plane is illustrated schematically in Fig. 10, while their properties can be summarized as follows. \n- 1. When L disk is well below a certain critical upper-limit luminosity, L c ∼ 6 × 10 38 · D 5 2 erg s -1 ( ∼ 0 . 4 · D 5 2 L E ), the spectral behavior can be explained by the standarddisk picture. This is the standard regime .\n- 2. When L disk hits L c , it is moderately saturated as L disk ∝ T in 2 , and hence r in shows a weak correlation to T in as r in ∝ T in -1 . Although the spectrum, consisting of a dominant soft component and a weak hard tail, resembles that in the standard regime , the radial temperature gradient in the disk (represented by p ) becomes flatter than in the standard disk. This is the apparently standard regime ( § 5).\n- 3. At an intermediate case ( L disk ∼ L c ), the spectral hard component dominates, and the apparent inner-disk radius is no longer constant. This is the anomalous regime ( § 4). These effects can be explained by a sudden increase of the disk Comptonization, while the underlying disk itself remains in the standard state and r in is kept constant.\n- 4. The source evolves from the standard to apparently standard regimes , then to the anomalous regime , and returns again to the standard regime . \nThe anomalous regime is naturally identified with that found in GRO J1655 -40, and the scenario of Comptonization suggested in Paper I successfully applies to XTE J1550 -564 as well. It has been confirmed that the violent variation in r in in the anomalous regime is apparently caused by strong disk Comptonization, with the underlying optically thick disk extending down to the last stable orbit like in the standard regime . The apparently standard regime is possibly the same as Period 1 of GRO J1655 -40. Because intensity variation in Period 1 of the source was small, Paper I did not discuss the source behavior in that period. However, GRO J1655 -40 resides in this period in the upper right corner on the L disk -T in plane (see Paper I), and its spectra consist of a dominant disk component and a very weak hard tail. These properties are basically the same as those of XTE J1550 -564 in the apparently standard regime .', '6.2. Comparison with theoretical predictions': "As is well known, theoretical solutions to the steady-state accretion flow form an S -shaped locus on the plane of ˙ M vs. the surface density of the disk (e.g., Abramowicz et al. 1988, Chen & Taam 1993, Kato, Fukue, & Mineshige 1998). The locus involves to thermally stable branches; the standard Shakura-Sunyaev accretion disk solution, and the slim-disk solution, realized when ˙ M is relatively low and very high, respectively. Evidently, the standard regime can be identified with the standard Shakura-Sunyaev solution. \nThe slim disk solution takes into account the effect of the advection, in addition to the viscous heating and the radiative cooling. This effect becomes important when ˙ M is high and hence the luminosity is close to L E . Under this condition, any increase in ˙ M would be balanced by an increase in the advective transport, accompanied by little increase in L disk ; this agrees with the observed mild saturation in L disk observed in the apparently standard regime . Watarai et al. (2000) simulated many slim disk spectra, fitted them with the MCD model, and derived an empirical relation between r in and the apparent T in as r in ∝ T in -1 (or L disk ∝ T in 2 ). This is exactly what has been observed in Fig. 5a. Furthermore, Watarai et al. (2000) showed that the temperature gradient becomes flatter as the advective cooling becomes important because of a progressive suppression of disk emissivity. In the extreme case, p can reduce to 0.5. Thus, the overall source behavior in the apparently standard regime agrees very well with the prediction by the slim disk model. Of course, this state assignment is still tentative, because the slim disk solution still neglects important effects such as general relativity, magnetic field, and photon trapping (Ohsuga et al. 2002). Other solutions would have to be considered as well. \nIn addition to the two stable branchs, a thermally and secularly unstable branch is known to exist between them. This branch is recognized as a negative slope on the S -shape sequence. By comparing the obtained L disk -T in diagram to the S -shape sequence, the anomalous regime may have some relation to the unstable branch of the sequence. In other words, the instability of the standard disk may cause the anomalous regime . Interestingly, the quasi-periodic oscillations (QPOs) are observed preferentially in the anomalous regime , in GRO J1655 -40 (Remillard et al. 1999) and XTE J1550 -564 (Remillard et al. 2002). Therefore, the QPO is likely to relate to the existence of the Compton cloud. \nThe observed three distinctive regimes are likely to reflect the change of the accretion disk structure from the standard accretion disk to other solutions, as the radiative cooling becomes progressively inefficient. These results hence provide, at least potentially, one of the first observational accounts of the long predicted S -shaped sequence. \nThe authors would like to thank Hajime Inoue, Shin Mineshige and Chris Done, for their valuable comments. They also thank Kazuhiro Nakazawa, Tsunefumi Mizuno and Ken Ebisawa for their helpful discussions. Thanks are also due to Piotr Zycki and Marek Gierli'nski for their help with the thcomp and the diskpn models. The authors are grateful to Dave Willis for his reading of this paper, and to the anonymous referee for his/her useful comments. A. K. is supported by Japan Society for the Promotion of Science Postdoctoral Fellowship for Young Scientists.", 'A. A formula to re-estimate r in under Comptonization': "Under the presence of strong disk Comptonization, the apparent disk inner radius r in may be calculated as \nF p disk + F p thc · 2 cos i = 0 . 0165 · ( r 2 in · cos i ( D/ 10kpc) 2 ) · ( T in 1 keV ) 3 photons s -1 cm -2 , (A1) \nwhere F p disk and F p thc are 0.01-100 keV photon flux from the direct disk component and the Comptonized component, respectively. Here, T in refers to the disk temperature obtained by considering the inverse Compton scattering as is done in § 4. The first parentheses is just the same as the normalization factor of the MCD model in the xspec . This formula is based on an assumption that there are few photons which are injected again into the optically thick cool disk (i.e., optical depth of the cloud is not very large). It can be derived through the following steps. \n- 1. 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A. 1999, MNRAS, 309, 561 \nTable 1: The best-fit parameters of XTE J1550 -564 with the MCD fit \n| Per. | date a | T in [keV] | r in [km] b | Γ | L disk c | L pow | E c [keV] | eqw [eV] | χ 2 /ν |\n|--------|----------|----------------------------|----------------|-------------------------|------------|---------|----------------------|-------------|----------|\n| 1 | 43 | 1 . 05 ± 0 . 04 | 36 +3 - 2 | 2 . 47 ± 0 . 05 | 2.07 | 2.18 | 6 . 6 +0 . 3 - 0 . 4 | 45 ± 22 | 15.3/37 |\n| 2 | 71 | 0 . 67 ± 0 . 01 | 60 ± 3 | 1 . 97 +0 . 07 - 0 . 06 | 0.92 | 0.062 | 6 . 5 +0 . 2 - 0 . 3 | 84 ± 28 | 21.2/37 |\n| 3 | 91 | 0 . 682 ± 0 . 008 | 60 ± 2 | (2.0 fixed) | 1.01 | 0.027 | 6 . 3 +0 . 3 - 0 . 2 | 80 ± 30 | 24.9/38 |\n| | | 0 . 686 +0 . 008 - 0 . 011 | 52 +2 - 3 | 2 . 2 +0 . 1 - 0 . 2 | 1.02 | 0.028 | 6 . 2 ± 0 . 3 | 68 +37 - 30 | 23.1/37 |\n| 4 | 125 | 1 . 117 ± 0 . 004 | 53 . 7 ± 0 . 6 | (2.0 fixed) | 5.8 | 0.022 | - | < 10 . 3 d | 94.0/38 |\n| | | 1 . 105 +0 . 006 - 0 . 004 | 54 ± 1 | 3 . 7 +0 . 5 - 0 . 3 | 5.73 | 0.369 | 6 . 1 +0 . 5 - 0 . 4 | 22 ± 17 | 18.1/37 |\n| 5 | 143 | 1 . 141 ± 0 . 004 | 52 . 1 ± 0 . 6 | (2.0 fixed) | 5.93 | 0.031 | - | < 9 . 2 d | 92.8/38 |\n| | | 1 . 133 +0 . 007 - 0 . 005 | 51 ± 1 | 3 . 9 ± 0 . 3 | 5.61 | 0.834 | 6 . 1 +0 . 5 - 0 . 6 | 18 +20 - 15 | 21.4/37 |\n| 6 | 172 | 1 . 06 ± 0 . 02 | 57 ± 2 | 2 . 21 +0 . 11 - 0 . 08 | 5.27 | 0.75 | 6 . 7 ± 0 . 3 | 46 +27 - 11 | 9.91/37 |\n| 7 | 182 | 1 . 10 ± 0 . 05 | 31 ± 2 | 2 . 55 +0 . 08 - 0 . 04 | 1.9 | 2.85 | 6 . 6 +0 . 4 - 0 . 3 | 35 ± 19 | 7.5/37 |\n| 8 | 192 | 0 . 78 ± 0 . 02 | 60 ± 3 | 2 . 01 +0 . 11 - 0 . 05 | 1.68 | 0.19 | 6 . 6 ± 0 . 2 | 63 ± 24 | 17.9/37 | \nNOTE.-Errors represent 90% confidence limits. The PCA 3-20 keV spectra are used. For all the spectral fitting, \nthe smedge and a Gaussian with fixed σ = 0 . 1 keV are included, and N H is fixed at 1 × 10 22 cm -2 . \n- a Days since 1998 September 7.\n- b Apparent inner radii under assumptions of i = 70 · and D = 5 kpc. \n- d Upper limit of Gaussian line with fixed E c = 6 . 4 keV. \nTable 2: The best-fit parameters with the thcomp fit \n| Per. | date | T in | [keV] | r in [km] a | Γ thc L | disk b L pow | L thc b | χ 2 /ν |\n|--------|--------|-------------------------|------------|---------------|----------------------|----------------|-----------|----------|\n| 1 | 43 | 0 . 96 - 0 . | +0 . 04 06 | 53 +5 - 3 | 2 . 8 +0 . 2 - 0 . 1 | 2.29 0.70 | 1.27 | 33.6/74 |\n| 7 | 182 | 0 . 96 +0 . 05 - 0 . 06 | 55 +5 - | 4 | 2 . 8 ± 0 . 1 2.37 | 0.69 | 1.73 | 34.5/74 | \nNOTE.-The PCA 3-50 keV spectra are used. \nTable 3: The best-fit parameters with the p -free disk fit \n| Per. | date | T in [keV] | p | E c | eqw [keV] | χ 2 /ν |\n|--------|--------|-----------------------------------------------------------------|----------------------------------------------|--------------------------|-------------|-----------------|\n| 3 | 91 | 0 . 76 +0 . 06 - 0 . 02 | 0 . 41 +0 . 04 - 0 . 05 | - | - | 25.4/39 |\n| 4 | 125 | 0 . 73 ± 0 . 05 1 . 16 +0 . 01 - 0 . 02 1 . 17 +0 . 02 - 0 . 01 | 0 . 5 +0 . 2 - 0 . 1 0 . 59 +0 . 04 - 0 . 02 | 6 . 3 +0 . 3 < - 0 . 3 - | 48 - | 22.0/37 54.8/39 |\n| 5 | 143 | 1 . 18 +0 . 02 - 0 . 01 | 0 . 56 ± 0 . 03 0 . 60 ± 0 . 03 | 5 . 9 +0 . 2 - 0 . 3 - | 45 - | 38.9/37 57.6/39 |\n| 5 | | 1 . 20 ± 0 . 02 | 0 . 57 ± 0 . 03 | 6 . 0 +0 . 2 - 0 . 3 | 31 | 47.7/37 | \nNOTE.-The PCA 3-20 keV spectra are used. The power-law photon index is fixed at 2.0. \nFig. 1.- The 1.5-12 keV lightcurve of XTE J1550 -564, obtained with the RXTE /ASM. The pointing observations are indicated with down-arrows. The horizontal arrow represents the period studied in the present paper. \n<!-- image --> \nFig. 2.- Evolution of the spectral parameters of XTE J1550 -564. (a) Time histories of L disk (filled circles), L pow (open circles), and L tot (a solid line), all in the unit of 10 38 · D 5 2 erg s -1 . As for the disk luminosity, i = 70 · is assumed. The eight characteristic periods are indicated at the top. (b)-(e) Those of T in , r in , Γ, and χ 2 / dof, respectively. Open triangles in panel (b), (c), and (e) are obtained by incorporating the disk-Comptonization. Crosses in panel (d) and (e) for Period 3-5 show the results from the free-Γ fitting. \n<!-- image --> \nFig. 3.- Typical PCA spectra of XTE J1550 -564 in Periods 1-8, compared with the predictions of the best-fit MCD plus power-law model. Background has been subtracted, but the instrumental response is not removed. The lower panel of each spectrum shows fit residuals. \n<!-- image --> \nFig. 4.- Several scatter plots among the spectral parameters. Panels (a), (b) and (c) shows r in against T in , Γ against L pow , and L disk against L pow , respectively. In panels (a) and (c), Period 1-8 are specified by eight kinds of symbols presented in panel (a); data points of standard , anomalous and apparently standard regimes are shown with filled symbols, crosses, and grey open ones, respectively. In panel (b), the data for Periods 3-5 are excluded because of the weakness of the power-law tail. \n<!-- image --> \nFig. 5.- (a) The calculated L disk plotted against the observed T in . Periods 1-8 are specified by the same eight kinds of symbols as Fig. 4. The solid and dotted lines represent the L disk ∝ T in 4 and L disk ∝ T in 2 relations, respectively. The source distance and inclination are assumed to be 5 kpc and 60 · , respectively. (b) Same as panel (a), but the data points in Period 1 and 7 ( anomalous regime ) are re-calculated considering the Comptonized component as L disk + L thc . \n<!-- image --> \nFig. 6.- The PCA spectrum in the anomalous regime (the same as Fig. 3a), fitted with the three-component model incorporating the Comptonized component (the medium hardness one shown with a solid line). \n<!-- image --> \nFig. 7.- Time histories of the best-fit parameters of the p -free disk model (diamond). For comparison, the MCD parameters are also plotted with filled circles in the top and bottom panels. In the middle panel, p = 0 . 75 is indicated as a dotted line. \n<!-- image --> \nFig. 8.- The PCA spectrum in the apparently standard regime (the same as Fig. 2e), fitted with the p -free disk model plus power-law (Γ = 2 . 0 fixed). \n<!-- image --> \nFig. 9.- The best fit values of p , plotted against T in determined with the MCD model. The PCA results for XTE J1550 -564 in Period 3-5 (panel a) are compared with those of LMC X-3 (panel b). Symbols for XTE J1550 -564 are the same as in Fig. 3. In panel (b), data points with large error bars (∆ p > 0 . 5) are not shown. As obtained in the MCD fit to the data of LMC X-3 (Kubota 2001), the values of N H and Γ are fixed at 7 . 5 × 10 20 cm -2 and Γ = 2 . 5, respectively. \n<!-- image --> \nFig. 10.- A schematic classification of the three spectral regimes on the L disk -T in diagram. Thick solid and dashed lines show the source behavior obtained under the MCD plus powerlaw fit. \n<!-- image --> \nlog T in"}
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2004PhRvD..70j7501A
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Severe constraints on the loop-quantum-gravity energy-momentum dispersion relation from the black-hole area-entropy law
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2004-01-01
| 6 | 0.45 | 164 |
['-', '-', '-', '-', '-']
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[]
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We explore a possible connection between two aspects of loop quantum gravity which have been extensively studied in the recent literature: the black-hole area-entropy law and the energy-momentum dispersion relation. We observe that the original Bekenstein argument for the area-entropy law implicitly requires information on the energy-momentum dispersion relation and on the position-momentum uncertainty relation. Recent results show that in first approximation black-hole entropy in loop quantum gravity depends linearly on the area, with small correction terms which have logarithmic or inverse-power dependence on the area. And it has been argued that in loop quantum gravity the dispersion relation should include terms that depend linearly on the Planck length, while no evidence of modification of the position-momentum uncertainty relation has been found. We observe that this scenario with Planck-length-linear modification of the dispersion relation and unmodified position-momentum uncertainty relation is incompatible with the black-hole-entropy results, since it would give rise to a term in the entropy formula going like the square root of the area.
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[]
| 3 |
https://arxiv.org/pdf/gr-qc/0405084.pdf
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{'Giovanni AMELINO-CAMELIA b , Michele ARZANO b and Andrea PROCACCINI a': "a Dipartimento di Fisica, Universit'a di Roma 'La Sapienza' and INFN Sez. Roma1, P.le Moro 2, 00185 Roma, Italy b Institute of Field Physics, Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599-3255, USA \nWe explore a possible connection between two aspects of Loop Quantum Gravity which have been extensively studied in the recent literature: the black-hole area-entropy law and the energymomentum dispersion relation. We observe that the original Bekenstein argument for the areaentropy law implicitly requires information on the energy-momentum dispersion relation. Recent results show that in first approximation black-hole entropy in Loop Quantum Gravity depends linearly on the area, with small correction terms which have logarithmic or inverse-power dependence on the area. Preliminary studies of the Loop-Quantum-Gravity dispersion relation reported some evidence of the presence of terms that depend linearly on the Planck length, but we observe that this possibility is excluded since it would require, for consistency, a contribution to black-hole entropy going like the square root of the area.", 'I. INTRODUCTION': "The intuition that the entropy of a black hole should be proportional to its (horizon-surface) area, up to corrections that can be neglected when the area A is much larger than the square of the Planck length L p , has provided an important element of guidance for quantum-gravity research. It is noteworthy that, as shown by Bekenstein [1], this contribution to black hole entropy can be obtained from very simple ingredients. One starts from the general-relativity result [2] that the minimum increase of area when the black hole absorbs a classical particle of energy E and size s is ∆ A /similarequal 8 πL 2 p Es (in 'natural units' with ¯ h = c = 1). Taking into account the quantum properties of particles one can estimate s as roughly given by the position uncertainty δx , and, since a particle with position uncertainty δx should at least [3] have energy E ∼ 1 /δx , this leads to the conclusion [1,4] that the minimum change in the black-hole area must be of order L 2 p , independently of the size of the area. Then using the fact that, also independently of the size of the area, this minimum increase of area should correspond to the minimum ('one bit') change of entropy one easily obtains [1] the proportionality between black-hole entropy and area. \nIt is remarkable that, in spite of the humble ingredients of this Bekenstein analysis, the entropy-area relation introduced such a valuable constraint for quantum-gravity research. And a rather challenging constraint, since attempts to reproduce the entropy-area-linearity result using directly some quantum properties of black holes were unsuccessful for nearly three decades. But over the last few years both in String Theory and in Loop Quantum Gravity the needed techniques for the analysis of entropy on the basis of quantum properties of black holes were developed. These results [5-8] now go even beyond the entropy-area-proportionality contribution: they establish that the leading correction should be of log-area type, so that one expects (for A /greatermuch L 2 p ) an entropy-area relation for black holes of the type \nS = A 4 L 2 p + ρ ln A L 2 p + O ( L 2 p A ) . (I.1) \nFor the case of Loop Quantum Gravity, which is here of interest, there is still no consensus on the coefficient of the logarithmic correction, ρ , but it is established [6-8] that there are no correction terms with stronger-than-logarithimic dependence on the area. \nWe observe that the availability of results on the log-area correction might provide motivation for reversing the Bekenstein argument: the knowledge of black-hole entropy up to the leading log correction can be used to establish the Planck-scale modifications of the ingredients of the Bekenstein analysis. \nIn particular, the mentioned role of the relation E ≥ 1 /δx in the Bekenstein analysis appears to provide an opportunity to put under scrutiny some scenarios for the energy-momentum dispersion relation in Loop Quantum Gravity. Several recent studies have tentatively argued that the Loop-Quantum-Gravity dispersion relation might involve a term with a linear dependence on the Planck length, and, as we observe in Section II, this in turn requires a Planck-length modification of the relation E ≥ 1 /δx between the energy and position uncertainty of a particle. However, as we show in Section III, the resulting modification of the E ≥ 1 /δx relation would in turn lead, following the Bekenstein argument, to a contribution to black-hole entropy that goes like the square root of the area. Since such a square-root contribution is, as mentioned, excluded by direct analysis of black-hole entropy in Loop Quantum Gravity, we conclude that the presence in the energy-momentum dispersion relation of a term with linear dependence on the Planck length is also excluded.", 'II. LOOP-QUANTUM-GRAVITY DISPERSION RELATION AND ITS IMPLICATIONS FOR THE E ≥ 1 /δX RELATION': "The possibility of Planck-scale modifications of the dispersion relation has been considered extensively in the recent quantum-gravity literature [9-11] and in particular in Loop Quantum Gravity [12-15]. \nSome calculations in Loop Quantum Gravity [12,13] provide support for the idea of an energy-momentum dispersion relation that for a particle of high energy would take the approximate form \nE /similarequal p + m 2 2 p + αL p E 2 , (II.1) \nwhere α is a coefficient of order 1. However, these results must be viewed as preliminary [14,15] since they essentially consider perturbations of 'weave states' [12,13], rather than perturbations of the ground state of the theory. It is not surprising (and therefore not necessarily insightful) that there would be some states of the theory whose excitations have a modified spectrum. If instead a relation of the type (II.1) was applicable to excitations of the ground state of the theory this would provide a striking characteristic of the Loop-Quantum-Gravity approach. \nSeveral papers have been devoted to the derivation of tighter an tighter experimental limits on coefficients of the α type for Loop Quantum Gravity (see, e.g. , Ref. [16] and references therein). As announced we intend to show here that the linear-inL p term can be excluded already on theoretical grounds, because of an inconsistency with the black-hole-entropy results. \nIn this section we start by observing that a modified dispersion relation implies a modification of the relation E ≥ 1 /δx between the energy of a particle and its position uncertainty. We can see this by simply following the familiar derivation [3] of the relation E ≥ 1 /δx , substituting, where applicable, the standard special-relativistic dispersion relation with the Planck-scale modified dispersion relation. It is convenient to focus first [3] on the case of a particle of mass M at rest, whose position is being measured by a procedure involving a collision with a photon of energy E γ and momentum p γ . In order to measure the particle position with precision δx one should use a photon with momentum uncertainty δp γ ≥ 1 /δx . Following the standard argument [3], one takes this δp γ ≥ 1 /δx relation and converts it into the relation δE γ ≥ 1 /δx , using the special-relativistic dispersion relation, and then the relation δE γ ≥ 1 /δx is converted into the relation M ≥ 1 /δx because the measurement procedure requires 1 M ≥ δE γ . If indeed Loop Quantum Gravity hosts a Planck-scale-modified dispersion relation of the form (II.1), it is easy to see that, following the same reasoning, one would obtain from δp γ 1 /δx the requirement M ≥ (1 /δx )[1 + 2 α ( L p /δx )]. \n≥ \n≥ These results strictly apply only to the measurement of the position of a particle at rest, but they can be straightforwardly generalized [3] (simply using a boost) to the case of measurement of the position of a particle of energy E . In the case of the standard dispersion relation (without Planck-scale modification) one obtains the familiar E ≥ 1 /δx . In the case of (II.1) one instead easily finds that \nE ≥ 1 δx ( 1 + 2 α L p δx ) . (II.2)", 'III. A REQUIREMENT OF CONSISTENCY WITH THE BLACK-HOLE ENTROPY ANALYSIS': "We now intend to show that the linear-inL p modification of the relation between the energy of a particle and its position uncertainty, which follows from the corresponding modification of the energy-momentum dispersion relation, should be disallowed in Loop Quantum Gravity since it leads to a contribution to the black-hole entropy-area relation which has already been excluded in direct black-hole-entropy analyses. \nWe do this by following the original Bekenstein argument [1]. As done in Ref. [1] we take as starting point the general-relativistic result which establishes that the area of a black hole changes according to ∆ A ≥ 8 πEs when a classical particle of energy E and size s is absorbed. In order to describe the absorption of a quantum particle one must describe the size of the particle in terms of the uncertainty in its position [1,4], s ∼ δx , and take into account a 'calibration 2 factor' [17-19] (ln 2) / 2 π that connects the ∆ A ≥ 8 πEs classical-particle result with the quantumparticle estimate ∆ A ≥ 4(ln 2) L 2 p Eδx . Following the original Bekenstein argument [1] one then enforces the relation E ≥ 1 /δx (and this leads to ∆ A ≥ 4(ln 2) L 2 p ), but we must take into account the Planck-length modification in (II.2), obtaining \n∆ A ≥ 4(ln 2) [ L 2 p +2 αL 3 p δx ] /similarequal 4(ln 2) [ L 2 p +2 αL 3 p R S ] /similarequal 4(ln 2) [ L 2 p + α 4 √ πL 3 p √ A ] , \nwhere we also used the fact that in falling in the black hole the particle acquires [18,21,22] position uncertainty δx R S , where R S is the Schwarzschild radius (and of course A = 4 πR 2 S ). \nNext, following again Bekenstein [1], one assumes that the entropy depends only on the area of the black hole, and one uses the fact that according to information theory the minimum increase of entropy should be ln 2, independently of the value of the area: \ndS dA /similarequal min (∆ S ) min (∆ A ) /similarequal ln 2 4(ln 2) L 2 p [ 1 + α 4 √ π L p √ A ] /similarequal ( 1 4 L 2 p -α √ π L p √ A ) . (III.1) \n∼ \nFrom this one easily obtains (up to an irrelevant constant contribution to entropy): \nS /similarequal A 4 L 2 p -2 α √ π √ A L p . (III.2) \nWe therefore conclude that when a quantum-gravity theory predicts the presence of a linear-inL p contribution to the energy-momentum dispersion relation it should correspondingly predict the presence of √ A contribution to black-hole entropy. Since in Loop Quantum Gravity such a √ A contribution to black-hole entropy has already been excluded [6-8] in direct black-hole entropy studies, we conclude that in Loop Quantum Gravity the presence of linear-inL p contributions to the energy-momentum dispersion relation is excluded. \nIt is instead plausible that Loop Quantum Gravity might host a dispersion relation of the type \nE /similarequal p + m 2 2 p + ˜ αL 2 p E 3 , (III.3) \nwith a quadratic-inL p contribution. In fact, the careful reader can easily adapt our analysis to the case of the dispersion relation (III.3), finding that the quadratic-inL p contribution to the dispersion relation ultimately leads to a leading correction to the black-hole-entropy formula which is of log-area type, consistently with the indications obtained in direct black-hole entropy studies [6-8].", 'ACKNOWLEDGMENTS': "G. A.-C. gratefully acknowledges conversations with O. Dryer, D. Oriti, C. Rovelli and L. Smolin. The work of M. A. was supported by a Fellowship from The Graduate School of The University of North Carolina. M. A. also thanks the Department of Physics of the University of Rome for hospitality. \n- [1] J. D. Bekenstein, Phys. Rev. D7 (1973) 2333.\n- [2] D. Christodoulou, Phys. Rev. Lett. 25 (1970) 1596; D. Christodoulou and R. Ruffini, Phys. Rev. D4 (1971) 3552.\n- [3] E. M. Lifshitz, L. P. Pitaevskii and V. B. Berestetskii, 'Landau-Lifshitz Course of Theoretical Physics, Volume 4: Quantum Electrodynamics' (Reed Educational and Professional Publishing, 1982).\n- [4] S. Hod, Phys. Rev. Lett. 81 (1998) 4293.\n- [5] A. Strominger and C. Vafa, Phys. Lett. B379 (1996) 99; S. N. Solodukhin, Phys. Rev. D57 (1998) 2410.\n- [6] C. Rovelli, Phys. Rev. Lett. 77 (1996) 3288.\n- [7] A. Ashtekar, Phys. Rev. Lett. 80 (1998) 904.\n- [8] R.K. Kaul and P. Majumdar, Phys. Rev. Lett. 84 (2000) 5255.\n- [9] G. Amelino-Camelia, J. Ellis, N.E. Mavromatos, D.V. Nanopoulos and S. Sarkar, Nature 393 (1998) 763.\n- [10] L.J. Garay, Phys. Rev. Lett. 80 (1998) 2508.\n- [11] G. Amelino-Camelia, gr-qc/0012051, Int. J. Mod. Phys. D11 (2002) 35; J. Magueijo and L. Smolin, gr-qc/0207085, Phys.Rev. D67 (2003) 044017; Jerzy Kowalski-Glikman and S. Nowak, hep-th/0204245, Int. J. Mod. Phys. D12 (2003) 299; G. Amelino-Camelia, gr-qc/0207049, Nature 418 (2002) 34.\n- [12] R. Gambini and J. Pullin, Phys. Rev. D59 (1999) 124021.\n- [13] J. Alfaro, H.A. Morales-Tecotl and L.F. Urrutia, Phys. Rev. Lett. 84 (2000) 2318.\n- [14] L. Smolin, hep-th/0209079.\n- [15] G. Amelino-Camelia, L. Smolin and A. Starodubtsev, hep-th/0306134 (Class. Quant. Grav., in press).\n- [16] D. Sudarsky, L. Urrutia, H. Vucetich Phys. Rev. Lett. 89 (2002) 231301.\n- [17] P. Chen and R.J. Adler, Nucl. Phys. Proc. Suppl. 124 (2003) 103.\n- [18] P. S. Custodio and J.E. Horvath, Class. Quant. Grav. 20 (2003) L197.\n- [19] T. Damour, hep-th/0401160; M. Cavaglia, S. Das and R. Maartens, Class. Quant. Grav. 20 (2003) L205.\n- [20] S.W. Hawking, Nature 248 (1974) 30.\n- [21] M.Yu. Kuchiev and V.V. Flambaum, gr-qc/0312065.\n- [22] R.J. Adler and T.K. Das, Phys. Rev. D14 (1976) 2472; R.S. Hanni and R. Ruffini, 'Lines of Force of a point Charge Near a Schwarzschild Black Hole', in Black Holes , eds. C. DeWitt and B.S. DeWitt (Gordon Breach 1973)."}
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2014NanoL..14.6424W
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Black Phosphorus Radio-Frequency Transistors
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2014-01-01
| 19 | 0.45 | 164 |
['-']
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Few-layer and thin film forms of layered black phosphorus (BP) have recently emerged as a promising material for applications in high performance nanoelectronics and infrared optoelectronics. Layered BP thin film offers a moderate bandgap of around 0.3 eV and high carrier mobility, leading to transistors with decent on-off ratio and high on-state current density. Here, we demonstrate the gigahertz frequency operation of black phosphorus field-effect transistors for the first time. The BP transistors demonstrated here show excellent current saturation with an on-off ratio exceeding 2000. We achieved a current density in excess of 270 mA/mm and DC transconductance above 180 mS/mm for hole conduction. Using standard high frequency characterization techniques, we measured a short-circuit current-gain cut-off frequency fT of 12 GHz and a maximum oscillation frequency fmax of 20 GHz in 300 nm channel length devices. BP devices may offer advantages over graphene transistors for high frequency electronics in terms of voltage and power gain due to the good current saturation properties arising from their finite bandgap, thus enabling the future ubiquitous transistor technology that can operate in the multi-GHz frequency range and beyond.
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| 9 |
https://arxiv.org/pdf/1410.3880.pdf
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{'Black Phosphorus Radio-Frequency Transistors': 'Han Wang 1, *, Xiaomu Wang 2 , Fengnian Xia 2, *, Luhao Wang 1 , Hao Jiang 3 , \nQiangfei Xia 3 , Matthew L. Chin 4 , Madan Dubey 4 , Shu-jen Han 5 \n1 Ming Hsieh Department of Electrical Engineering, University of Southern California, Los Angeles, CA 90089 \n- 2 Department of Electrical Engineering, Yale University, New Haven, CT 06511 \n3 Department of Electrical & Computer Engineering, University of Massachusetts, \nAmherst, MA 01003 \n4 Sensors and Electron Devices Directorate, US Army Research Laboratory, \nAdelphi, MD 20723 \n5 IBM T. J. Watson Research Center, Yorktown Heights, NY 10598 \n* Email: [email protected], [email protected]', 'Introduction': "Typical channel materials in thin film electronics include amorphous or polycrystalline silicon 1-3 , organic compounds 4,5 , and oxides 6,7 . These materials usually exhibit a sizable bandgap but compromised carrier mobility, making them undesirable for radio-frequency (RF) electronics. Early interests in using two-dimensional materials for RF thin film electronics focused mainly on graphene. Since the first demonstration of high frequency current gain in graphene field-effect transistors (FETs) in 2007 8 , graphene has inspired great interests for RF applications due to its high mobility, high carrier velocity and long mean free path 9-14 . However, several years of intensive research has revealed that graphene transistors might suffer from a few fundamental limitations that will restrict its high frequency performance. The key shortcoming of graphene RF transistors is their lack of current saturation as a result of graphene's zero-bandgap nature, that can lead to reduced voltage and power gains 15,16 , and to a lesser extent, the current gain 17 . As a result, most graphene RF FETs may have relatively high short-circuit current-gain cut-off frequency ( f T), but significantly lower maximum oscillation frequency ( f max), which benchmarks the power gain of the transistor. On the other hand, transistors based on transition metal dichalcogenides, such as molybdenum disulfide (MoS2) and tungsten diselenide (WSe2), only showed limited potential for high frequency applications because of their relatively low mobility 18 , 19 . Recently, layered black phosphorus (BP) thin film has emerged as a promising candidate for high performance thin film electronics due to its moderate bandgap of 0.3 eV and high carrier mobility. The recent experiment has shown hole mobility exceeding 650 cm 2 V -1 s -1 at room temperature and above 1,000 cm 2 V -1 s -1 at 120 K along the light effective mass (x) \ndirection 20 . While in bulk black phosphorus, mobilities exceeding 1,000 cm 2 V -1 s -1 at room temperature for both electrons and holes, and exceeding 50,000 cm 2 V -1 s -1 at 30 K have been demonstrated 21 . In this work, we demonstrate for the first time the operation of black phosphorus FETs in the gigahertz frequency range. Future ubiquitous transistor technologies using this novel layered material with high mobility and highly desirable current saturation property may revolutionize the electronic systems for many civilian and defense applications.", 'Device fabrication': 'Orthorhombic bulk black phosphorus is an elemental layered material with D2h point group symmetry as shown in Fig. 1a. It is the most stable allotrope of phosphorus and the electronic properties of bulk BP have been studied several decades ago 22-26 . Recently, BP in its few-layer and thin-film form has inspired renewed interests among physicists and engineers due to its promising potential for application in thin film electronics and infrared optoelectronics 20,27-29 . The layered nature of BP crystals allows single- and multi-layer atomic crystals to be obtained through mechanical exfoliation techniques. In our experiments, multi-layer BP was first exfoliated using the standard micromechanical cleavage method from a bulk BP crystal, then transferred onto 300 nm thick silicon dioxide thermally grown on highly resistive silicon (>10 4 Ω·cm). Atomic force microscope (AFM) can be used to determine the number of atomic layers in a BP flake, which has a layer-to-layer spacing of 0.53 nm 20,21 . The x-, y- and z-directions of the crystal lattice are indicated in Fig. 1a. In a series of recent publications, the mobility \nand the on-off current ratio of BP FETs have been studied with respect to the thickness and layer number in the BP channel 20,27,28 . In this work, BP thin films with thicknesses around 6-10 nm were selected for the best balanced mobility and on-off properties for RF applications. Transistors using thinner BP films exhibit an on-off current ratio as high as 2×10 5 but low mobility of below 100 cm 2 V -1 s -1 due to the scattering from the external environment. Utilization of excessively thick BP films can lead to higher mobility but a reduced on-off ratio and less pronounced current saturation. The inset of Fig. 1b shows the optical micrograph of a typical BP flake used for device fabrication. AFM measurements indicate a thickness of 8.5 nm in this flake (Fig. 1b). To enhance the channel mobility, crystal orientations of the flakes were first identified using either Raman spectroscopy or infrared spectroscopy techniques and the transistors were built along the light effective mass (x-) direction of the BP lattice. Fig. 1c shows the Raman spectrum of the BP flake with the excitation laser polarized along the x-direction. The characteristic peaks at 470, 440, and 365 cm -1 correspond to 2 g A , B2g, and 1 g A modes, respectively 20,30 . The 2 g A mode has higher intensity compared to B2g, and 1 g A modes with this particular excitation laser polarization. Fig. 1d shows the polarization-resolved infrared spectra of the flake and the x-direction can be clearly identified as the direction with the highest optical conductivity around the band edge 20,21 . The source and drain electrodes were formed with 1 nm Ti/20 nm Pd/ 30 nm Au metal stack, which favors ptype carrier injection into the channel owing to the large work-function (~5.22 eV) of Pd. The gate dielectric was made with 21 nm of HfO2 deposited by atomic layer deposition (ALD) technique at a temperature of 150 C. The typical dielectric constant is around 13, determined by ellipsometric measurement on a control sample. Finally, the gate electrode \nwas defined by electron-beam lithography to form transistors with sub-micrometer channel lengths. The fabrication process is described in more detail in Supporting Information. Fig. 2b shows the optical micrograph of the full layout of the device. The standard ground-signal-ground (GSG) pads were fabricated to realize signal transition from microwave coax cables to on-chip coplanar waveguide electrodes.', 'DC Characterization': 'Fig. 2a shows the schematic of the transistor structure. Fig. 2c and 2d display the DC characteristics of a top-gated BP FET with a 300 nm channel length (LG). The device was built along the x-direction of the 8.5 nm thick flake shown in Fig. 1. Fig. 2c shows the measured drain current ( I DS) as a function of drain-source bias voltage ( V DS) at gate bias ( V GS) from 0 V to -2 V in steps of -0.5 V. The current saturation in the BP transistor is clearly visible and significantly improved from that in most graphene FETs due to the finite bandgap in BP thin film. This is typical for a BP device with ~8.5 nm channel thickness and agrees well with previous demonstrations 20 . Good current saturation characteristics will lead to low output conductance defined as 𝑔 0 = d𝐼 DS d𝑉 DS | 𝑉 GS =constant , which is the differential drain current change with respect to the variation in drain voltage bias. Low output conductance is critical for improving voltage and power gains, and to a lesser extent, the current gain in RF transistors. As reported in our previous work 20 , typical Hall mobility along x-direction of BP flakes with thickness around 8 nm is above 400 cm 2 V -1 s -1 at room temperature. Fig. 2d shows I DS as a function of V GS at V DS=-2 V, where onoff current ratio over 2×10 3 is achieved. The as-fabricated device shows p-type conduction with threshold voltage around -0.7 V. A key factor influencing the high- \nll signal response of a transistor is its transconductance ( 𝑔 m ), defined as the first derivative of the transfer characteristics, i.e. 𝑔 m = d𝐼 DS d𝑉 GS | 𝑉 DS =constant . The inset of Fig. 2d shows the measured 𝑔 m as a function of the gate voltage at V DS=-2 V. The peak value of this extrinsic 𝑔 m exceeds 180 mS/mm at V G=-1.75 V while the peak on-state current density measured exceeds 270 mA/mm at V DS=-2 V and V GS=-2.5 V. Fig. 2d also shows I DS as a function of top-gate voltage V GS at a drain bias of V DS=-2 V with the current plotted in logarithmic scale. The on-off current ratio of the device exceeds 2×10 3 . Hence, the BP transistor demonstrated here shows significant advantages over graphene transistors in terms of its current saturation properties and on-off current ratio. Typical monolayer graphene devices have on-off current ratio less than 10 at room temperature. In bilayer graphene devices, the on-off current ratio only reaches around 100 even with bandgap opening induced by a strong external electrical field 31 .', 'RF Characterization': "To characterize the high frequency performance of BP RF transistors, we used the standard S-parameter measurement with on-chip probing utilizing GSG probes and Agilent N5230 vector network analyzer up to 50 GHz. Key figures of merit for microwave transistors can then be obtained for the BP devices. The network analyzer and the entire testing fixture were first calibrated using standard open, short, and load calibrations. Standard open and short structures were then used to de-embed the signals of the parallel and series parasitics associated with the measurement pads and connections 15,32-34 . The measurement procedure used in this work followed strictly the standard calibration and de-embedding processes widely accepted in the semiconductor \nindustry, where the calibration step moves the reference plane to the tips of the GSG probes and the de-embedding step gives access to the performance of the active device region. \nFig. 3a and 3b plot the short-circuit current gain (h21), the unilateral power gain (U) and maximum stable gain (MSG)/maximum available gain (MAG) extracted from Sparameters before and after de-embedding, respectively, measured at V DS=-2.0 V and V GS=-1.7 V for the device with LG=300 nm. The plot of |h21| 2 follows the characteristic 1/ f relation with respect to frequency at a 20 dB/dec slope 16 . Fig. 3a and 3b show that the 300 nm channel length device has a f T of 7.8 GHz before de-embedding and 12 GHz after de-embedding, as extracted from the frequencies at which |h21| reaches unity. Gummel's method provides another way of extracting the cut-off frequency 35 where f T is extracted from the reciprocal of the initial slope in the imaginary part of 1/h21 vs. frequency plot. The values of f T obtained by both 1/ f extraction and Gummel's method match closely for both 300 nm (Fig. 3c and 3d) and 1 m (see Fig. S1 in the Supporting Information) channel length devices before and after de-embedding. \nWhile f T is an important figure of merit related to the intrinsic speed of the BP transistor, another key figure of merit for analog application is the maximum oscillation frequency. It is the highest possible operating frequency at which a transistor can still amplify power. f max can be extracted from U or MSG/MAG of the device 16,36 . The unilateral power gain, U, also known more generally as Mason ' s U invariant, is a key parameter for any general two-port network. It carries great significance as an invariant parameter of the system under linear, lossless and reciprocal transformations. In Mason ' s classic work 37,38 , the rich physical meaning of U was interpreted in three different ways \nas a maximum power gain, as a device activity measure, and as an invariant under a class of bilinear Möbius transformations. In transistor characterizations, U is the power gain under the condition of (1) Unilateralization , and (2) Conjugate matched load for maximum power transfer . In Fig. 3a and 3b, the plots of U follow a 20 dB/dec slope, and both U and MSG/MAG plots give similar f max of 12 GHz before de-embedding and 20 GHz after de-embedding, respectively. \nThe 300 nm channel length device has an extrinsic f T·LG product of 3.6 GHz m. Using a slightly different design of the open pattern where the gate electrode in the 'open' structure extends into the spacing between the source and drain electrodes 15 , we can eliminate most of the gate-source and gate-drain parasitic capacitances C gs and C gd. This will allow the extraction of a new f T value that reflects the more intrinsic property of the BP channel, and an intrinsic f T value ( f T,int) close to 51 GHz is obtained for the same device, which corresponds to the average saturation velocity in the channel approximately equal to v sat=2 f T· L G ~ 9.6×10 6 cm/s. However, we would like to emphasize that f T,int only represents the upper limit of the possible frequency spectrum for this transistor. In any practical applications, the C gs and C gd of the device always significantly affect the device performance. As a result, we report f T=12 GHz and f max=20 GHz in Fig. 3 as the practically operable cut-off frequencies of the active device region 15,33 , which are extracted based on standard characterization techniques commonly used for the characterization in silicon and III-V high frequency transistors. The intrinsic cut-off frequency of 51 GHz is extracted only as a way to approximately estimate the saturation velocity and it may not be appropriate for technology benchmarking. The RF characteristics for a device with 1 m channel length are also reported in the Supporting \nInformation. The 1 m channel length device has peak f T=2.8 GHz and f max=5.1 GHz before de-embedding and f T=3.3 GHz and f max=5.6 GHz after de-embedding (see Fig. S1 in the Supporting Information). \nThe intrinsic limit of the cut-off frequency can be estimated using 𝑓 𝑇 = 𝑔 𝑚 2𝜋𝐶 𝑔𝑠 . Assuming the same gm and gate dielectrics property from the measured data, an f T above 100 GHz may be reached if the channel length reduces to 30 nm. On the other hand, f max depends on both the current and voltage gains. It is related to f T, following 𝑓 𝑚𝑎𝑥 = 𝑓 𝑇 2 √ 𝑟 0 𝑅 𝑔 +𝑅 𝑖 . We can see that a high f T will enhance f max and good current saturation, i.e. high output resistance ( r 0), low gate resistance ( R g) and low input resistance ( R i) are critical for improving f max of the device. \nFig. 4 shows the magnitude of the small-signal open-circuit voltage gain |z21/z11| as a function of frequency before and after de-embedding. An open-circuit voltage gain refers to the voltage gain subject to infinitely high load impedance. Here, the voltage gain is obtained from the ratio between the open circuit forward transfer impedance z21 and the open circuit input impedance z11. Since 𝑧 11 = 𝑣 1 𝑖 1 | 𝑖 2 =0 and 𝑧 21 = 𝑣 2 𝑖 1 | 𝑖 2 =0 , the ratio z 21/ z 11 is equal to 𝑣 2 𝑣 1 | 𝑖 2 =0 , i.e. the voltage gain with the output port in open condition. v 1 and v 2 refer to the voltages at the input and output ports, respectively. In transistors, v 1 is the small-signal input voltage at the gate and the v 2 is the small-signal output voltage at the drain. Based on the definition of g m and g 0 discussed earlier, we can see that the voltage gain |z21/z11| is closely related to the ratio gm/g0. Devices with good current saturation and high transcondutance are hence expected to have high voltage gains. The BP transistors show good voltage gain characteristics. As shown in Fig. 4, the before-de-embedding \nvoltage gain stays above unity (0 dB) up to 13 GHz. The after-de-embedding voltage gain is close to 20 dB at 2 GHz and stays above unity (0 dB) for the entire frequency range measured up to 50 GHz. For a longer channel length (1 m) device (see Fig. S2 in the Supporting Information), the before-de-embedding voltage gain is above unity (0 dB) up to 10 GHz. The after-de-embedding voltage gain is above 15 dB at 1 GHz and stays above unity (0 dB) up to 30 GHz.", 'Summary': 'In this work, we investigated the high frequency characteristics of black phosphorus field effect transistors, whose channels were fabricated along the light effective mass (x-) direction. The device operates well into the GHz frequency range of the radio frequency spectrum. We carried out standard S-parameter measurements to characterize the high frequency response of these top-gated BP transistors. The shortcircuit current gain, maximum stable gain/maximum available gain, unilateral power gain and voltage gain of the devices were carefully extracted. The short-circuit current gain of BP transistors shows the 20 dB/dec 1/ f frequency dependence at high frequency. We measured a peak short-circuit current gain cutoff frequency f T of 12 GHz and maximum oscillation frequency f max of 20 GHz for a 300 nm channel length BP transistor, demonstrating the GHz operation of BP devices for the first time. These results clearly reveal the potentials of BP transistors to function as power and voltage amplifiers in multi-GHz frequency analogue and digital electronics demanded by many emerging civilian and military applications.', 'References': "- 1. Powell, M. J. The physics of amorphous-silicon thin-film transistors. IEEE Trans. Electron Devices , 36 , 2753-2763 (1989).\n- 2. Street, R. A. Thin-Film Transistors. Adv. Mater., 21 , 2007-2022 (2009).\n- 3. Hawkins, W. G. Polycrystalline-silicon device technology for large-area electronics. IEEE Trans. Electron Devices , 33 , 477-481 (1986).\n- 4. Forrest, S. R. The path to ubiquitous and low-cost organic electronic appliances on plastic. Nature , 428 , 911-918 (2004).\n- 5. Dimitrakopoulos, C. D., Mascaro, D. J. Organic thin-film transistors: A review of recent advances. IBM J. Res. Dev. , 45 , 11-27 (2004).\n- 6. Nomura, K. et al. Amorphous Oxide Semiconductors for High-Performance Flexible Thin-Film Transistors. Japanese Journal of Applied Physics , 45 , 4303 (2006).\n- 7. Carcia, P. F., McLean, R. S., Reilly, M. H. & Nunes, G. Jr., Transparent ZnO thin-film transistor fabricated by RF magnetron sputtering. Appl. Phys. Lett. , 82 , 1117 (2003);\n- 8. Meric, I., Baklitskaya, N., Kim, P. & Shepard, K. L. RF performance of top-gated, zero-bandgap graphene field-effect transistors. IEEE IEDM Tech. Digest , 2008.\n- 9. Lin, Y.-M., Dimitrakopoulos, C. D., Jenkins, K. A., Farmer, D. B., Chiu, H.-Y., Grill, A. & Avouris, P. 100-GHz Transistors from Wafer-Scale Epitaxial Graphene. Science , 327 , 662 (2010).\n- 10. Wu, Y., Lin, Y., Bol, A. A., Jenkins, K. A., Xia, F., Farmer, D. B., Zhu, Y., & Avouris, P. High-frequency, scaled graphene transistors on diamond-like carbon. Nature , 472 , 74-78 (2011).\n- 11. Liao, L. et al. High-speed graphene transistors with a self-aligned nanowire gate. Nature , 467 , 305-308 (2010).\n- 12. Wang, H., Nezich, D., Kong, J. & Palacios, T. Graphene frequency multipliers. IEEE Elec. Dev. Lett. , 30 , 547-549 (2009).\n- 13. Wang, H., Hsu, A., Wu, J., Kong, J. & Palacios, T. Graphene-based ambipolar RF mixers. IEEE Elec. Dev. Lett. , 31 , 906-908 (2010). \n- 14. Han, S.-J., Valdes-Garcia, A., Oida, S., Jenkins, K. A. & Haensch, W. Graphene radio frequency receiver integrated circuit. Nat. Commun. , 5 :3086 doi:10.1038/ncomms4086 (2014).\n- 15. Wang, H. Chapter 4. In Two-Dimensional Materials for Electronic Applications . Ph.\n- D. Thesis, Massachusetts Institute of Technology, Cambridge, MA, USA, 2014. http://dspace.mit.edu/handle/1721.1/84899\n- 16. Schwierz, F. & Liou, J. J. Modern Microwave Transistors: Theory, Design, and Performance , Wiley-Interscience, 1st edition, 2002.\n- 17. Tasker, P. J. & Hughes, B. Importance of source and drain resistance to the maximum f T of millimeter-wave MODFETs. IEEE Elec. Dev. Lett. , 10 , 291-293 (1989).\n- 18. Zhu, W., Low, T., Lee, Y.-H., Wang, H., Farmer, D. B., Kong, J., Xia, F. & Avouris, \nP. Electronic transport and device prospects of monolayer molybdenum disulphide grown \nby chemical vapour deposition. Nat. Commun. , 5: 3087 doi:10.1038/ncomms4087 (2014). \n- 19. Huang, J.-K. et al. Large-Area Synthesis of Highly Crystalline WSe2 Monolayers and Device Applications, ACS Nano , 8 , 923-930 (2014).\n- 20. Xia, F., Wang, H. & Jia, Y. Rediscovering black phosphorus as an anisotropic layered material for optoelectronics and electronics. Nat. Commun. , 5 :4458 doi:10.1038/ncomms5458 (2014).\n- 21. Morita, A. Semiconducting black phosphorus. Appl. Phys. A , 39 , 227-242, 1986.\n- 22. Keyes, R. The electrical properties of black phosphorus. Phys. Rev. 92 , 580-584 (1953).\n- 23. Warschauer, D. Electrical and optical properties of crystalline black phosphorus. J. Appl. Phys. 34 , 1853-1860 (1963).\n- 24. Jamieson, J. Crystal structures adopted by black phosphorus at high pressures. Science 139 , 1291-1292 (1963).\n- 25. Wittig, J. & Matthias, B. T. Superconducting phosphorus. Science 160 , 994-995 (1968).\n- 26. Maruyama, Y., Suzuki, S., Kobayashi, K. & Tanuma, S. Synthesis and some properties of black phosphorus single crystals. Physica , 105B , 99-102 (1981). \n- 27. Li, L. et al. Black phosphorus field-effect transistors. Nat. Nanotechnol. , 9 , 372-377 (2014).\n- 28. Liu, H. et al. Phosphorene: an unexplored 2D semiconductor with a high hole mobility. ACS Nano , 8 , 4033-4041 (2014).\n- 29. Koenig, S., Doganov, R., Schmidt, H., Castro Neto, A. & Ozyilmaz, B. Electric field effect in ultrathin black phosphorus. Appl. Phys. Lett. , 104 , 103106 (2014).\n- 30. Akahama, Y., Kobayashi, M. & Kawamura, H. Raman study of black phosphorus up to 13 GPa. Solid State Comm. 104 , 311-315 (1997).\n- 31. Xia, F., Farmer, D. B., Lin, Y.-M., Avouris, P. Graphene field-effect transistors with high on/off current ratio and large transport band gap at room temperature. Nano Lett. , 10 , 715-718 (2010).\n- 32. R. J. Collier and A. D. Skinner, Microwave Measurements, 3rd Edition, the Institution of Engineering and Technology, 2007.\n- 33. Koolen, M. C. A., Geelen, J. A. & Versleijen, M. P. J. An improved de-embedding technique for on-wafer high-frequency characterization. Proceedings of the Bipolar Circuits and Technology Meeting 1991 , 188-191 (1991).\n- 34. Kim, J.-Y., Choi, M.-K. & Lee, S.-H. A 'Thru-Short-Open' De-embedding Method for Accurate On-Wafer RF Measurements of Nano-Scale MOSFETs. JSTS:Journal of Semiconductor Technology and Science , 12 , 53-58 (2012).\n- 35. Gummel, H. K. On the definition of the cutoff frequency f T. Proceedings of the IEEE , 57 , 12, 2159-2159 (1969).\n- 36. S-Parameter Techniques, HP Application Note 95-1 , \nhttp://www.hparchive.com/Application\\_Notes/HP-AN-95-1.pdf. \n- 37. Mason, S. J. Power gain in feedback amplifiers. Trans. IRE Professional Group on Circuit Theory, CT-1 , 20-25 (1954). (This work was previously reported in Tech. Rep. No. 257, Research Laboratory of Electronics, MIT, Cambridge, MA, 1953.\n- 38. Gupta, M. S., Power Gain in Feedback Amplifiers, a Classic Revisited, IEEE Trans. Microwave Theory and Techniques, 40 , 5 (1992).", 'Competing financial interests': 'Authors declare no competing financial interests.', 'Supporting Information': 'Information on the device fabrication process and characterization method, additional high frequency characterization data are included in the Supporting Information. This material is available free of charge via the Internet at http://pubs.acs.org.', 'Additional information': 'Correspondence and requests for materials should be addressed to H.W. \n([email protected]) and F. X. ([email protected])', 'Figure 1. Characterization of the black phosphorus (BP) thin film.': '(a) Layered crystal structure of black phosphorus. The spacing between adjacent layers is 5.3 Å. (b) atomic force microscope (AFM) data showing the thickness of a BP flake. The inset shows the optical micrograph of the BP flake with thickness around 8.5 nm. (c) Raman spectrum of the BP flake along x-direction. (d) Polarization-resolved infrared spectra of a BP flake.', 'Figure 2. DC characteristics of BP transistors.': '- (a) Schematic of the BP transistor device structure. (b) Optical micrograph of the fabricated device. (c) Output characteristics of the BP transistor. L G= 300 nm. (d) Transfer characteristics of the same BP transistor plotted in both linear and logarithmic \nscale. V DS=-2 V. The device has an on-off current ratio exceeding 2×10 3 . The inset shows the transconductance gm of the device.', 'Figure 3. Current and power gain in BP transistors at GHz frequency.': "(a) and (b) the short-circuit current gain h21, the maximum stable gain and maximum available gain MSG/MAG and the unilateral power gain U of the 300 nm channel length device before and after de-embedding, respectively. The device has f T=7.8 GHz, f max=12 GHz before de-embedding, and f T=12 GHz, f max=20 GHz after de-embedding. (c) and (d) the imaginary part of 1/h21 as a function of frequency before and after de-embedding, respectively. Based on Gummel's method, the initial slope of the curve is equal to 1/ f T.", 'Figure 4. Open-circuit voltage gain in BP transistors at GHz frequency': 'The open-circuit voltage gain (z21/z11) before and after de-embedding is shown as a function of the frequency. After de-embedding, the voltage gain stays close to 20 dB up to 2 GHz and is above unity (0 dB) in the entire measurement range up to 50 GHz. The grey dashed line is a guide to the eyes. \nFigure 1. Characterization of the black phosphorus (BP) thin film \n<!-- image --> \nFigure 2. DC characteristics of BP transistors \n<!-- image --> \n<!-- image --> \nb \n<!-- image --> \n<!-- image --> \nFigure 3. Current and power gain in BP transistors at GHz frequency \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 4. Open-circuit voltage gain in BP transistors at GHz frequency \n<!-- image -->', 'Methods': 'Top-gated transistor fabrication. The fabrication of our devices starts with the exfoliation of BP thin films from bulk BP crystals onto 300 nm SiO2 on a Si substrate, which has pre-patterned alignment grids, using the micro-mechanical cleavage technique. The thickness of the SiO2 was selected to provide the optimal optical contrast for locating BP flakes relative to the alignment grids. The thickness of BP layers was measured by atomic force microscopy (AFM). The next step was to pattern the resist layer for metallization using a Vistec 100 kV electron-beam lithography system based on poly (methyl methacrylate) (950k MW PMMA). PMMA A3 was spun on wafer at a speed of 3000 rpm for 1 minute and was then bakes at 175 degrees for 3 minutes. The dose for exposure is 1100 µC cm -2 . Development was performed in 1:3 MIBK: IPA (Methyl isobutyl ketone: Isopropanol) for 90 s. We then evaporated 1 nm Ti/20 nm Pd/30 nm Au \nfollowed by lift-off in acetone to form the contacts. The HfO2 gate dielectric is deposited using atomic layer deposition (ALD) at 150 C. The gate electrode is also patterned using Vistec 100 kV electron-beam lithography system. \nAFM. Atomic force microscopy (AFM) for identifying the thin film thickness was performed on a Digital Instruments/Veeco Dimension 3000 system. \nIR spectroscopy. Bruker Optics Fourier Transfer Infrared spectrometer (Vertex 70) integrated with a Hyperion 2000 microscope system was used to measure the infrared spectroscopy of the BP flakes in the 800 cm -1 to 4000 cm -1 range. The linear polarization of the incident light was achieved using an infrared polarizer. \nElectrical characterization. DC electrical characterizations were performed using an Agilent B1500 semiconductor parameter analyzer and a Lakeshore cryogenic probe station with micromanipulation probes. The high frequency characterizations were performed using an Agilent N5230A Vector Network Analyzer.', 'High frequency characterization of BP transistor with 1 m channel length': "The short-circuit current and power gain of a BP transistor with 1 m channel length are shown in Fig. S1. The open-circuit voltage gain z21/z11 both before and after de-embedding are shown in Fig. S2. The device has f T=2.8 GHz, f max=5.1 GHz before de-embedding, and f T=3.3 GHz, f max=5.6 GHz after de-embedding. After de-embedding, the voltage gain stays above 15 dB up around 1 GHz and is above unity (0 dB) up to 30 GHz. \nFig. S1 (a) and (b) the short-circuit current gain h21, the maximum stable gain and maximum available gain MSG/MAG and the unilateral power gain U of the device with 1 m channel length before and after deembedding, respectively. The device has f T=2.8 GHz, f max=5.1 GHz before de-embedding, and f T=3.3 GHz, f max=5.6 GHz after de-embedding. (c) and (d) the imaginary part of 1/h21 as a function of frequency before and after de-embedding, respectively. Based on Gummel's method, the initial slope of the curve is equal to 1/ f T. \n<!-- image --> \nFig. S2 the open-circuit voltage gain (z21/z11) before and after de-embedding is shown as a function of the frequency. After de-embedding, the voltage gain stays above 15 dB up to around 1 GHz and is above unity (0 dB) up to 30 GHz. \n<!-- image -->"}
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2017MNRAS.472.2422M
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The cosmic merger rate of stellar black hole binaries from the Illustris simulation
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2017-01-01
| 23 | 0.5 | 164 |
['black hole physics', 'gravitational waves', 'methods numerical', 'stars black holes', 'stars luminosity function;mass function', '-', '-', '-', '-']
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[]
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The cosmic merger rate density of black hole binaries (BHBs) can give us an essential clue to constraining the formation channels of BHBs, in light of current and forthcoming gravitational wave detections. Following a Monte Carlo approach, we couple new population-synthesis models of BHBs with the Illustris cosmological simulation, to study the cosmic history of BHB mergers. We explore six population-synthesis models, varying the prescriptions for supernovae, common envelope and natal kicks. In most considered models, the cosmic BHB merger rate follows the same trend as the cosmic star formation rate. The normalization of the cosmic BHB merger rate strongly depends on the treatment of common envelope and on the distribution of natal kicks. We find that most BHBs merging within LIGO's instrumental horizon come from relatively metal-poor progenitors (<0.2 Z<SUB>⊙</SUB>). The total masses of merging BHBs span a large range of values, from ∼6 to ∼82 M<SUB>⊙</SUB>. In our fiducial model, merging BHBs consistent with GW150914, GW151226 and GW170104 represent ∼6, 3 and 12 per cent of all BHBs merging within the LIGO horizon, respectively. The heavy systems, like GW150914, come from metal-poor progenitors (<0.15 Z<SUB>⊙</SUB>). Most GW150914-like systems merging in the local Universe appear to have formed at high redshift, with a long delay time. In contrast, GW151226-like systems form and merge all the way through the cosmic history, from progenitors with a broad range of metallicities. Future detections will be crucial to put constraints on common envelope, on natal kicks, and on the BHB mass function.
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[]
| 4 |
https://arxiv.org/pdf/1708.05722.pdf
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{'The cosmic merger rate of stellar black hole binaries from the Illustris simulation': "Michela Mapelli 1 , 2 , Nicola Giacobbo 1 , 3 , Emanuele Ripamonti 3 , Mario Spera 1 , 2 , 4 \n- 1 INAF-Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, I-35122, Padova, Italy, [email protected] 2 INFN, Milano Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy\n- 3 Physics and Astronomy Department Galileo Galilei, University of Padova, Vicolo dell'Osservatorio 3, I-35122, Padova, Italy\n- 4 Physics Department Giuseppe Occhialini, University of Milano Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy \n22 August 2017", 'ABSTRACT': "The cosmic merger rate density of black hole binaries (BHBs) can give us an essential clue to constraining the formation channels of BHBs, in light of current and forthcoming gravitational wave detections. Following a Monte Carlo approach, we couple new population-synthesis models of BHBs with the Illustris cosmological simulation, to study the cosmic history of BHB mergers. We explore six population-synthesis models, varying the prescriptions for supernovae, common envelope, and natal kicks. In most considered models, the cosmic BHB merger rate follows the same trend as the cosmic star formation rate. The normalization of the cosmic BHB merger rate strongly depends on the treatment of common envelope and on the distribution of natal kicks. We find that most BHBs merging within LIGO's instrumental horizon come from relatively metal-poor progenitors ( < 0 . 2 Z glyph[circledot] ). The total masses of merging BHBs span a large range of values, from ∼ 6 to ∼ 82 M glyph[circledot] . In our fiducial model, merging BHBs consistent with GW150914, GW151226 and GW170104 represent ∼ 6, 3, and 12 per cent of all BHBs merging within the LIGO horizon, respectively. The heavy systems, like GW150914, come from metal-poor progenitors ( < 0 . 15 Z glyph[circledot] ). Most GW150914-like systems merging in the local Universe appear to have formed at high redshift, with a long delay time. In contrast, GW151226-like systems form and merge all the way through the cosmic history, from progenitors with a broad range of metallicities. Future detections will be crucial to put constraints on common envelope, on natal kicks, and on the BHB mass function. \nKey words: stars: black holes - gravitational waves - methods: numerical - stars: mass-loss - black hole physics", '1 INTRODUCTION': "The first direct detection of gravitational waves (GWs, Abbott et al. 2016b) opens a new perspective for the study of compact object (CO) binaries. Black hole binaries (BHBs) have been predicted and studied for a long time (e.g. Tutukov & Yungelson 1973; Thorne 1987; Schutz 1989; Kulkarni et al. 1993; Sigurdsson & Hernquist 1993; Portegies Zwart & McMillan 2000; Colpi et al. 2003; Belczynski et al. 2004), but the three events observed by LIGO so far (GW150914, GW151226, and GW170104) are the first observational confirmation of their existence. \nMoreover, the two black holes (BHs) associated with GW150914 and one of the two BHs associated with GW170104 are surprisingly massive: 36 . 2 +5 . 2 -3 . 8 M glyph[circledot] , 29 . 1 +3 . 7 -4 . 4 M glyph[circledot] (Abbott et al. 2016c,a), and 31 . 2 +8 . 4 -6 . 0 M glyph[circledot] (Abbott et al. 2017), respectively. If they are the remnants of \nmassive stars, such massive BHs should have formed from relatively metal-poor ( Z ≤ 0 . 5 Z glyph[circledot] ) progenitors, which are expected to collapse directly to BHs (e.g. Mapelli et al. 2009, 2010, 2013; Belczynski et al. 2010; Spera et al. 2015). Dynamical processes (such as exchanges or runaway collisions in dense star clusters) might also contribute to enhancing the formation of massive BHBs similar to GW150914 and GW170104 (e.g. Ziosi et al. 2014; Chatterjee et al. 2017; Rodriguez et al. 2015, 2016; Mapelli 2016). Alternatively, GW150914 and GW170104 might be the result of primordial BHs born from gravitational collapse in the early Universe (e.g. Bird et al. 2016; Carr et al. 2016; Inomata et al. 2017). \nConstraining the formation epoch and the birthplace of BHBs is one of the key points to interpret the nature of GW events associated with BHB mergers. This requires to model the formation and evolution of BHBs in a cosmolog- \nical context. This task is currently a challenge, because of the huge dynamical range between the scale of cosmological structures (tens of Mpc) and the scale of binary evolution ( < ∼ few AU). Moreover, since the progenitor's metallicity appears to be crucial for the BH mass (Spera et al. 2015), any attempt to reconstruct the cosmic formation of BHBs should account for the local and global evolution of metallicity in the proper way. \nFor these reasons, only few authors attempted to put the formation of BHBs in a cosmological frame. Dominik et al. (2013) plant CO binaries into the cosmic history through a Monte Carlo-based algorithm. They generate a sample of galaxies based on a Press-Schechter like function (Fontana et al. 2006), adopt the average metallicity evolution described by Pei et al. (1999), and finally associate the CO binaries to a given redshift bin based on the cosmic star formation rate (SFR) evolution (Strolger et al. 2004). This gives a BHB merger rate density reaching its maximum at z ∼ 4 -5 and then slowly decreasing down to z = 0. \nSimilarly, Belczynski et al. (2016b) generate isolated BHBs and then distribute them as a function of redshift, adopting an updated version of the cosmic SFR density and of the average metallicity evolution (Madau & Dickinson 2014). This approach does not account for the massmetallicity relation observed in galaxies (Maiolino et al. 2008). The resulting merger rate density peaks at z ∼ 2. If only GW150914-like systems are considered, the distribution of the formation times of these systems is markedly bimodal with two peaks, one ∼ 11 -12 Gyr ago and the second one ∼ 2 -3 Gyr ago. \nIn contrast, Lamberts et al. (2016) account for the cosmological evolution through a Press-Schechter like formalism (Cole et al. 2008) with a redshift-dependent massmetallicity relation (Ma et al. 2016). This ensures that the metallicity of a galaxy depends on its mass, consistent with the observations (Maiolino et al. 2008). Lamberts et al. (2016) do not recover the strongly bimodal birth-time distribution of GW150914-like systems reported by Belczynski et al. (2016b). Their predicted BHB merger rate is ∼ 850 Gpc -3 yr -1 , significantly larger than inferred from LIGO observations ( ∼ 9 -240 Gpc -3 yr -1 , Abbott et al. 2016a). A conceptually similar approach was followed also by Dvorkin et al. (2016) and Elbert et al. (2017). \nThe formalism adopted by Dominik et al. (2013), Belczynski et al. (2016b), and even Lamberts et al. (2016) cannot give us detailed information on the evolution of the host galaxy of a CO binary. Thus, O'Shaughnessy et al. (2017b) follow a complementary approach: they start from a cosmological simulation and pick up four test galaxies, which they re-simulate at high resolution, by doing a 'zoom-in'. Then, they add BHBs to the location of star forming particles in the simulation. They find a significantly higher merger rate per unit mass in dwarf galaxies than in Milky-Way-like galaxies. \nRecently, Schneider et al. (2017) characterize the formation and coalescence sites of GW events, by coupling the metallicity-dependent binary population synthesis code SeBa (Portegies Zwart & Verbunt 1996; Mapelli et al. 2013) with a (4 Mpc) 3 simulation performed with the GAMESH pipeline (Graziani et al. 2015, 2017). GAMESH interfaces an N-body simulation with a semi-analytic model for galaxy formation, and a radiative-transfer code. With this ap- \nproach, Schneider et al. (2017) find that the observed GW events occur most likely in star forming galaxies with stellar mass > 10 10 M glyph[circledot] . \nIn this paper, we follow a new approach, complementary to previous work: we draw the cosmic history of the BHB merger rate by coupling up-to-date population synthesis simulations of BHBs with the public Illustris-1 cosmological simulation (Vogelsberger et al. 2014b). The Illustris-1 is the highest resolution hydrodynamical simulation run in the frame of the Illustris project (Vogelsberger et al. 2014a). In the following, we refer to the Illustris-1 simply as Illustris. The Illustris box (length = 106 . 5 Mpc comoving) is considerably larger than the one adopted by Schneider et al. (2017), ensuring that we are considering a less biased portion of the Universe, even if with lower resolution. \nWe plant our BHBs in the cosmological simulation through a Monte Carlo approach, based on the metallicity of star particles. Our BHBs were generated by evolving isolated stellar binaries with a new version of the BSE code (Hurley et al. 2002), which includes up-to-date recipes for stellar evolution and stellar winds (Vink et al. 2011; Chen et al. 2015). Moreover, we include the effect of pulsational pair-instability and pair-instability supernovae (Woosley 2017) which were neglected in most previous studies. This approach allows us to follow the merger history of BHBs, accounting for the evolution of their environment.", '2 METHODS': 'To reconstruct the cosmic history of BHB mergers, we couple the Illustris simulation with a large set of populationsynthesis simulations of isolated binaries. The main ingredients of our model are the following.', '2.1 The BHBs': "We simulate the evolution of isolated stellar binaries through an updated version of the BSE code (Hurley et al. 2000, 2002). The changes with respect to the original version of BSE are described in a companion paper by Giacobbo et al. (in prep.). Here we summarize the most important prescriptions. Stellar winds have been updated based on the equations described in Belczynski et al. (2010). Namely, a treatment of stellar winds following Vink et al. (2001) and Vink & de Koter (2005) is included for O-type and WolfRayet stars, respectively. In this model, mass loss by stellar winds depends on metallicity, both in the main sequence (MS) and in later evolutionary stages. \nWith respect to Belczynski et al. (2010), there is one crucial update: we take into account the dependence of the mass loss ˙ M on the electron-scattering Eddington ratio Γ (Grafener & Hamann 2008; Vink et al. 2011; Vink 2016). Following Chen et al. (2015), the mass loss scales as ˙ M ∝ Z α (where Z is the star metallicity) with α = 0 . 85 if the electron-scattering Eddington ratio of a star is Γ < 2 / 3, and α = 2 . 45 -2 . 4 Γ if Γ ≥ 2 / 3. This ensures that the dependence of mass loss on metallicity almost vanishes if the star is radiation-pressure dominated. With this relatively small change, we obtain a mass spectrum of BHs similar to the one published by Spera et al. (2015) and based on the \nPARSEC stellar evolution tracks (Bressan et al. 2012; Tang et al. 2014; Chen et al. 2015). \nOur new version of BSE also includes new fitting formulas for the core radii, as described in Hall & Tout (2014). This is a crucial ingredient for the study of BHBs, because the fate of a common envelope phase depends on the core radius. \nFurthermore, we included in BSE new recipes for corecollapse supernovae (SNe). In particular, we implemented both the rapid (R) and the delayed (D) models for SN explosion presented by Fryer et al. (2012). In Appendix A, we detail the prescriptions for the CO mass in the rapid and in the delayed SN model. Finally, we added a formalism to account for pair-instability and pulsational pair-instability SNe, following Spera & Mapelli (2017) (see also Belczynski et al. 2016c; Woosley 2017). \nWith the new code, we ran six sets of populationsynthesis simulations. The details of the six sets are given in Table 1. In particular, the simulation set labelled as 'R' adopts the rapid SN model, while all the others (labelled as 'D') adopt the delayed model for core-collapse SNe. Pulsational pair instability and pair instability SNe are included in all runs. \nFor the common envelope (CE) phase, we use the same formalism as described in Hurley et al. (2002), which depends on two free parameters, α and λ . According to this formalism, α is the fraction of binding energy converted into kinetic energy of the envelope, while λ describes the geometry of the envelope. In the formalism by Hurley et al. (2002), α and λ always appear as their product αλ . In simulations D, R, DHG, and DK we use α = 1, λ = 0 . 1. The latter choice of λ is quite well motivated for massive stars (e.g. Xu & Li 2010; Loveridge et al. 2011). In simulation D0.02 we assume α = 0 . 2, λ = 0 . 1, while in simulation D1.5 we use α = 3 and λ = 0 . 5. \nThe treatment of Hertzsprung gap (HG) donors was found to be critical in previous studies (e.g. Dominik et al. 2012). A HG star lacks a steep density gradient between the core and envelope. Thus, its response to a CE should be similar to that of a MS star (Ivanova & Taam 2004). In the standard version of BSE , MS donors entering a CE phase are forced to merge with the accretor, while HG donors are allowed to survive the CE phase. In our run DHG, we adopted the default setting of BSE allowing HG donors to survive a CE phase. In all the other simulations we modified BSE , by imposing that a HG donor merges with its companion if they enter a CE phase. \nFinally, the natal kick of the CO is another essential ingredient, because it can unbind a binary. There are no conclusive observational constraints on the natal kick of BHs, even if some recent studies indicate that high-velocity kicks are possible (Repetto et al. 2012, 2017; O'Shaughnessy et al. 2017a). Thus, we draw the natal kicks from the Maxwellian distribution described in Hobbs et al. (2005), with dispersion σ = 265 km s -1 . This distribution was obtained from the proper motions of 233 isolated Galactic pulsars. \nIn model DK, we assume that BH kicks follow the same distribution as neutron star (NS) kicks. In all the other models, we scale the velocities drawn from this distribution by the amount of fallback, according to: \nv BH = v NS (1 -f fb ) , (1) \nTable 1. Properties of the population-synthesis simulations. \n| Name | SN | α | λ | HG | Kick |\n|--------|---------|-----|-----|------|--------|\n| D | Delayed | 1 | 0.1 | new | F12 |\n| R | Rapid | 1 | 0.1 | new | F12 |\n| DHG | Delayed | 1 | 0.1 | BSE | F12 |\n| DK | Delayed | 1 | 0.1 | new | H05 |\n| D0.02 | Delayed | 0.2 | 0.1 | new | F12 |\n| D1.5 | Delayed | 3 | 0.5 | new | F12 | \nColumn 1: model name; column 2: SN model (delayed and rapid from Fryer et al. 2012); column 3: value of α ; column 4: value of λ ; column 5: treatment for HG stars (' BSE ' means same treatment as in BSE , 'new' means that we force all CE binaries with a HG donor to merge); column 6: model for the SN kick. H05 means that we use the distribution from Hobbs et al. (2005). F12 means that we rescale the natal kicks by the fallback, as described in Fryer et al. (2012). See also equation 1 and the text for details. \nwhere v BH is the natal kick for the BH, v NS is the natal kick for a NS (drawn from the distribution proposed by Hobbs et al. 2005), and f fb (ranging from 0 to 1) is the amount of fallback on the proto-NS (Fryer et al. 2012; Spera et al. 2015). The definition of f fb depends on the adopted corecollapse SN prescription. In Appendix A, we detail the values of f fb for the two considered core-collapse SN models. \nFor each of the six simulation sets described in Table 1, we simulate 12 sub-sets with metallicity Z = 0 . 0002 , 0.0004, 0.0008, 0.0012, 0.0016, 0.002, 0.004, 0.006, 0.008, 0.012, 0.016, and 0.02. Throughout the paper, we define solar metallicity as Z glyph[circledot] = 0 . 02. Thus, the 12 sub-sets correspond to metallicity Z = 0 . 01, 0.02, 0.04, 0.06, 0.08, 0.1, 0.2, 0.3, 0.4, 0.6, 0.8, and 1.0 Z glyph[circledot] . In each sub-set we simulate 10 7 stellar binaries. Thus, each of the six sets of simulations is composed of 1 . 2 × 10 8 massive binaries. The mass of the primary ( m p ) is randomly drawn from a Kroupa initial mass function (Kroupa 2001) ranging 1 from 5 to 150 M glyph[circledot] , and the mass of the secondary ( m s ) is sampled according to the distribution F ( q ) ∝ q -0 . 1 (where q = m s /m p ) in a range [0 . 1 -1] m p . The orbital period P and the eccentricity e are randomly extracted from the distribution F ( P ) ∝ (log 10 P ) -0 . 55 , with 0 . 15 ≤ log 10 ( P/ day) ≤ 5 . 5, and F ( e ) ∝ e -0 . 4 , with 0 ≤ e < 1, as suggested by Sana et al. (2012).", '2.2 The Illustris': "The Illustris simulation covers a comoving volume of (106 . 5 Mpc) 3 , and has an initial dark matter and baryonic matter mass resolution of 6 . 26 × 10 6 and 1 . 26 × 10 6 M glyph[circledot] , respectively (Vogelsberger et al. 2014b,a). At redshift zero the softening length is ∼ 710 pc, while the smallest hydrodynamical cells have a length of 48 pc. The large size of the \nIllustris' box ensures that we are modelling an unbiased portion of the Universe, satisfying the cosmological principle. The main drawback is that the population of dwarf galaxies is heavily under-resolved. In Appendix B, we estimate the impact of resolution on our main results, by comparing the Illustris-1 with the lower-resolution Illustris-3 simulation. Moreover, in a companion paper (Schneider et al. 2017), we follow a complementary approach: we combine our population-synthesis models with the GAMESH simulation, which has a box of only (4 Mpc) 3 , but much higher resolution, in order to quantify the contribution of dwarf galaxies to the BHB merger rate. \nThe Illustris was run with the moving mesh code AREPO , to solve the inviscid Euler equations (Springel 2010). The Illustris includes a treatment for sub-grid physics (cooling, star formation, SNe, super-massive BH formation, accretion and merger, AGN feedback, etc), as described in Vogelsberger et al. (2013). The model of sub-grid physics adopted in the Illustris is known to produce a massmetallicity relation (Genel et al. 2014; Genel 2016) which is sensibly steeper than the observed one (see the discussion in Vogelsberger et al. 2013 and Torrey et al. 2014). Moreover, the simulated mass-metallicity relation does not show the observed turnover at high stellar mass ( > ∼ 10 10 M glyph[circledot] ). In Appendix B we estimate that the impact of these differences between simulated and observed mass-metallicity relation on the BHB merger rate is ∼ 20 per cent. \nAs for the cosmology, the Illustris adopts WMAP-9 results for the cosmological parameters (Hinshaw et al. 2013), that is Ω M = 0 . 2726, Ω Λ = 0 . 7274, Ω b = 0 . 0456, and H 0 = 100 h km s -1 Mpc -1 , with h = 0 . 704. \nThrough the web-based interface (API) made available by the the Illustris project ( http://www.illustris-project.org/ ), we downloaded the stellar particles in each snapshot, including information on their formation time, initial mass, and metallicity. A total of 108 snapshots have stellar particles, from redshift z ∼ 16 to 0. More details on the Illustris can be found in the presentation papers (Vogelsberger et al. 2014b,a), in the release paper (Nelson et al. 2015) and on the aforementioned website.", '2.3 Planting BHBs into a cosmological simulation': "We wrote a Monte Carlo code to associate the simulated BHBs to the Illustris, performing the following operations. \nFor each of the simulation sets listed in Table 1, we extract information on those BHBs merging within a Hubble time. Namely, we store the BH masses and the delay time t delay between the formation of the progenitor stellar binary and the merger of the BHB. We also store information on the total initial stellar mass M BSE of each sub-set of simulations with the same metallicity (including binary systems which do not evolve into BHBs). \nWeread each stellar particle from the Illustris only once, when it first appears in the snapshots. We store information on its initial mass M Ill , formation redshift z Ill , and metallicity Z Ill . We then find the metallicity that best matches Z Ill among the 12 metallicities simulated with BSE 2 . \nWe then associate to each Illustris' particle a number n BHB of merging BHBs, randomly extracted from the subset with the best-matching metallicity, based on the following algorithm: \nn BHB = N BSE M Ill M BSE f corr f bin , (2) \nwhere M Ill is the initial stellar mass of the Illustris' particle and M BSE is the initial stellar mass in the BSE sub-set with the selected metallicity. In our calculations, M Ill < M BSE . N BSE is the number of merging BHBs within the simulated sub-set of initial stellar mass M BSE . f corr = 0 . 285 is a correction factor, accounting for the fact that we actually simulate only primaries with m p ≥ 5 M glyph[circledot] , neglecting lower mass stars. Finally, f bin accounts for the fact that we simulate only binary systems, whereas a fraction of stars are single. Here we assume that 50 per cent of stars are in binaries, thus f bin = 0 . 5. We note that f bin is only a scale factor and our results can be rescaled to a different f bin a posteriori. We notice that N BSE f corr f bin /M BSE is (by definition) the number of merging BHBs per unit stellar mass at a given metallicity. \nWith this procedure, we associate to each Illustris' particle a number n BHB of randomly selected merging BHBs whose progenitors have metallicity Z glyph[similarequal] Z Ill . \nWe then estimate the look-back time of the merger ( t merg ) of each BHB in the randomly selected sample as \nt merg = t form -t delay , (3) \nwhere t delay is the time between the formation of the progenitor stellar binary and the merger of the BHB, and t form is the look back time at which the Illustris' particle has formed, calculated as \nt form = 1 H 0 ∫ z Ill 0 1 (1 + z ) [Ω M (1 + z ) 3 +Ω Λ ] 1 / 2 d z, (4) \nwhere the cosmological parameters are set to WMAP-9 values (for consistency with the Illustris) and z Ill is the formation redshift of the Illustris' particle. \nAccording to this definition, t merg is also a look back time: it tells us how far away from us the BHB merged. For our analysis, we consider only BHBs with t merg ≥ 0, i.e. we do not consider BHBs that will merge in the future. \nWe repeat the same procedure for each of the six simulation sets in Table 1 and we obtain six different models of the cosmic BHB merger evolution.", '3.1 Merger rate': 'Figure 1 shows the cosmic BHB merger rate density ( R BHB ) in the comoving frame, derived from our simulations. To obtain the BHB merger rate shown in this Figure, we extracted the number of BHB mergers ( N BHB ) per time bin (each bin \nparticle a BSE set with Z Ill = 0 . 02 ( Z Ill = 0 . 0002), since the maximum (minimum) metallicity we simulated with BSE is 0.02 (0.0002). This procedure is particularly arbitrary for population III stars, whose binarity properties are not known. However, we show in Section 4 that population III stars do not significantly affect the rate of detectable BHB mergers. \nFigure 1. Left y -axis: cosmic merger rate density of BHBs ( R BHB ) in the comoving frame, as a function of the look-back time t lb (bottom x axis) and of the redshift z (top x axis) in our models. Red solid line: D (fiducial model); black dashed line: R; violet dash-dot line: DHG; orange dashed line: DK; blue dotted line: D0.02; green dash-dot line: D1.5. Green shaded area: BHB merger rate inferred from LIGO detections (Abbott et al. 2016a). Right y -axis: cosmic SFR density from the Illustris (grey thin solid line), as a function of the look-back time t lb (bottom x axis) and of the redshift z (top x axis). \n<!-- image --> \nTable 2. Comoving BHB merger-rate density at redshift z = 0 and z = 0 . 2 (corresponding to t lb = 0 and 2.43 Gyr, respectively). \n| Name | R BHB ( z = 0) [Gpc - 3 yr - 1 ] | R BHB ( z = 0 . 2) [Gpc - 3 yr - 1 ] |\n|--------|------------------------------------|----------------------------------------|\n| D | 125 | 181 |\n| R | 155 | 228 |\n| DHG | 572 | 772 |\n| DK | 20 | 29 |\n| D1.5 | 145 | 181 |\n| D0.02 | 278 | 279 | \nColumn 1: model name; column 2: present-time BHB merger rate density; column 3: BHB merger rate density at z = 0 . 2. \nspanned over ∆ t = 10 Myr from z ∼ 16 to z = 0) and then we did the following simple conversion: \nR BHB = N BHB ( l box Gpc ) -3 ( ∆ t yr ) -1 , (5) \nwhere l box = 106 . 5 Mpc is the size of the Illustris box (in the comoving frame) and ∆ t is the size of the time bin (∆ t =10 Myr). \nFrom Fig. 1 it is apparent that the overall behaviour of the merger rate is the same for all considered BHB models, with the partial exception of D0.02 (see Table 1 for details about the models). The behaviour of the merger rate density as a function of time depends only on the SFR (given by the Illustris and thus common to all BHB models) and on the delay between the formation time of a stellar binary and the \nmerger time of the BHB born from the stellar binary (which depends on the BHB model). \nThe shape of the merger rate density in Fig. 1 resembles the one of the cosmic SFR density (e.g. Madau & Dickinson 2014) with a peak at t lb = 11 . 29 Gyr (i.e. z ∼ 2 . 7). The decrease of the merger rate approaching z = 0 is more gentle than the decrease of the SFR density, because of BHBs that formed at high redshift but merge with a delay of several Gyr (see the next section). \nThe main difference between the considered BHB models is the normalization of the merger rate density, which depends on the BHB merger efficiency (see Table 2 for details). In particular, the present-time merger rate density ranges from R BHB ∼ 125 Gpc -3 yr -1 to R BHB ∼ 155 Gpc -3 yr -1 in models D, D1.5 and R. Model D0.02 results in a factor of two larger rate ( R BHB ∼ 280 Gpc -3 yr -1 ). Finally, the rate is much higher ( R BHB ∼ 570 Gpc -3 yr -1 ) in model DHG and much lower ( R BHB ∼ 20 Gpc -3 yr -1 ) in model DK (see Table 2). \nThe BHB merger rate density inferred from the first LIGO observations (O1 run) is R BHB = 9 -240 Gpc -3 yr -1 (Abbott et al. 2016a). While this paper was being reviewed, the inferred rate was updated to R BHB = 12 -213 Gpc -3 yr -1 , based on the first results of the O2 run (Abbott et al. 2017). Thus, the present-day merger rate density of models D, R, DK, and D1.5 are consistent with observations, while D0.02 is slightly above the observed range and DHG gives a much higher rate. Thus, models in which HG stars can survive a CE phase (DHG) are not consistent with the observed merger rate, unless natal kicks are much higher than assumed. \nNatal kicks have a strong impact on the BHB merger rate: R BHB is a factor of ∼ 6 lower in run DK than in run D, which differ only by the kick prescription. In run D the magnitude of the kick depends on the amount of fallback. In run DK all BHs receive a natal kick, drawn from the same distribution as Galactic single pulsars (Hobbs et al. 2005). The merger rate density of both run D and DK are consistent with current observations, but future detections might be able to discriminate between such models. \nRuns D, D0.02 and D1.5 differ by the choice of the α and λ CE parameters. Unlike the other considered effects (SN model, SN kicks and HG treatment), the choice of CE parameters affects not only the normalization but also the shape of the BHB merger rate density as a function of time. Since the SFR history is the same for all models, this difference indicates that models with different CE parameters have also different distributions for the delay time. \nThe effect of the choice of αλ is much more important at high redshift than at low redshift. At z < 0 . 3, the difference between runs D and D1.5 is negligible, while the difference between run D and D0.02 is about a factor of two. \nFinally, runs R and D have similar merger rates (within a factor of 1.3). This indicates that the choice of the corecollapse SN model (rapid or delayed) does not affect the BHB merger rate significantly. In the following, we will consider run D as our fiducial model. \nFigure 2 also shows that the stellar progenitors of BHBs have all possible metallicities ranging from Z ∼ 0 up to ∼ 0 . 7 Z glyph[circledot] . The most common metallicity of BHBs merging at low redshift is 0 . 05 < ∼ Z/ Z glyph[circledot] < ∼ 0 . 2, i.e. significantly sub-solar. This result comes from a combination of two factors. Firstly, relatively metal-poor stars form efficiently even at z = 0, as expected from the mass-metallicity relation. Secondly, many BHBs merging at z = 0 formed at highredshift, where low metallicity was more common. Mergers associated with solar or super-solar metallicity are strongly suppressed in our models, because stellar radii are larger at higher metallicity, causing early mergers of massive stars before they become BHBs. \n<!-- image --> \nFigure 2. Metallicity of progenitors of merging BHBs in the fiducial model (D). Upper panel: metallicity versus formation time of the stellar progenitors ( t form ). Lower panel: metallicity of the stellar progenitors versus merger time of the BHBs ( t merg ). Both t form and t merg are expressed as look-back time. The colour-coded map (in logarithmic scale) indicates the number of merging BHBs per cell. The black lines are isocontours enclosing a number of merging BHBs ranging from 5 × 10 3 to 5 × 10 6 (as indicated by the black labels). \n<!-- image -->', "3.2 Formation time, progenitor's metallicity and BHB masses": 'In this section, we discuss the main properties of merging BHBs in the Illustris simulation. We consider only our fiducial run D. \nFigure 2 maps the metallicity of the stellar progenitors of merging BHBs. In the upper panel the metallicity is plotted against the formation time of the stellar progenitors, while the lower panel shows the metallicity versus the merger time of BHBs. From the comparison between the two panels, it is apparent that a large fraction of metal-poor systems which formed at high redshift merge at relatively low redshift with a long delay time. For example, ∼ 2 × 10 6 BHBs with progenitor metallicity Z ∼ 0 . 1 Z glyph[circledot] merge at redshift z ∼ 0 in the simulation, but only ∼ 5 × 10 4 of them form at redshift z ∼ 0. This implies that a significant number \n<!-- image --> \nFigure 3. Upper (lower) panel: total mass (chirp mass) of merging BHBs as a function of t merg in the fiducial model. t merg is expressed as look-back time. The colour-coded map (in logarithmic scale) indicates the number of merging BHBs per cell. The black lines are isocontours enclosing a number of merging BHBs ranging from 5 × 10 3 to 5 × 10 5 (as indicated by the black labels). \n<!-- image --> \nof merging BHBs visible in the LIGO instrumental horizon were born in the high-redshift Universe and possibly in a metal-poor environment. \nAt higher redshift, the percentage of merging BHBs born in metal-poor environments increases, and the contribution of lower metallicities becomes more important. How- \nFigure 4. Distribution of the delay time ( t delay , estimated as the time elapsed between the formation of a stellar binary and the merger of the BHB formed from this stellar binary) for the simulated BHBs in runs D (red solid line), R (black dashed line), DK (orange dashed line), D1.5 (green dash-dot line) and D0.02 (blue dotted line). Only BHBs merging within the mass dependent instrumental horizon of LIGO are shown. Purple dotted line: d N/ d t ∝ t -1 . Note that (unlike t merg and t form ) t delay is not a look-back time. The value N merg on the y axis is the number of simulated BHBs per time bin (∆ t = 100 Myr). \n<!-- image --> \never, we stress that even at t merg > ∼ 11 Gyr there is a nonnegligible fraction of systems with metallicity > 0 . 1 Z glyph[circledot] . \nFigure 3 shows the behaviour of the total mass ( M = m p + m s , where m p and m s are the mass of the primary and secondary BH, respectively) and of the chirp mass ( m chirp = m 3 / 5 p m 3 / 5 s M -1 / 5 ) as a function of the merger time. The distribution of total masses (chirp masses) peaks at 20 ≤ M/ M glyph[circledot] ≤ 45 (8 ≤ m chirp / M glyph[circledot] ≤ 20), but we find merging systems with total masses (chirp masses) ranging from ∼ 6 to ∼ 82 M glyph[circledot] ( ∼ 3 to ∼ 35 M glyph[circledot] ). From this Figure it is apparent that there is nearly no dependence of the BHB mass on the merging time. This is primarily a consequence of the broad distribution of delay times (see next Section).', '3.3 BHBs merging within the LIGO horizon': 'In this section, we focus only on simulated BHBs that merge within the LIGO instrumental horizon, defined as the luminosity distance at which GWs from a face-on, equal-mass, overhead binary would be detected with signal-to-noise ratio of 8 (Abbott et al. 2016c). To account for the dependence of the instrumental horizon on the BHB mass, we use the curve reported in Fig. 4 of Abbott et al. (2016c) for the 2015-2016 LIGO sensitivity (left-hand panel). \nIn order to extract from our simulations all BHBs merging inside the LIGO instrumental horizon, we check whether the luminosity distance at the time of merger is smaller than the instrumental horizon for a BHB with the same total mass, as given in Fig. 4 of Abbott et al. (2016c). \nIn this section we also compare the properties of our fiducial model with the other runs. Figure 4 shows the delay time distribution for the BHB merging within the LIGO horizon. This distribution depends only on the BSE models and is not affected by the cosmological simulation. In run D, \nFigure 5. Red solid line: distribution of formation time t form for the simulated BHBs in the fiducial model (run D). Blue dashed line: distribution of merger time t merg for the simulated BHBs in the fiducial model. Only BHBs merging within the LIGO instrumental horizon are shown. Bottom x axis: t form and t merg expressed as look-back time. Top x axis: t form and t merg expressed as redshift. The value N merg on the y axis is the number of simulated BHBs per time bin (∆ t = 10 Myr). \n<!-- image --> \nthe delay distribution matches the behaviour d N/ d t ∝ t -1 found in previous studies (Belczynski et al. 2016b; Lamberts et al. 2016), but only for t delay ≤ 2 Gyr. For longer delay times, the distribution flattens considerably: it becomes nearly independent of time. This explains why a large fraction of metal-poor systems formed at high redshift merge within the LIGO horizon (see Fig. 2 and Section 3.2). Runs R (adopting the rapid SN model) and DK (assuming large BH kicks) behave exactly the same as run D. \nIn contrast, the delay time distributions of runs D1.5 and especially D0.02 (which differ from run D only for the CE parameters) are significantly different. They decrease more steeply than ∝ t -1 at short delay time. This produces a much lower number of BHB mergers with delay time 0 . 01 ≤ t delay / Gyr ≤ 4. For t delay > 4 Gyr, the number of BHB mergers in runs D1.5 and D0.02 becomes significantly higher than that of run D. This difference in the distribution of delay times explains why the BHB merger rate density in run D is higher (lower) at high (low) redshift than that of runs D1.5 and D0.02. \nFigure 5 shows the distribution of formation times t form and the merger times t merg of BHBs that merge within the LIGO horizon in our fiducial model. In Fig. 5, the merger time peaks at z ∼ 0 . 1 -0 . 2. This results from the convolution between the LIGO horizon (which depends on the BHB mass) and the increase of the merger rate as the redshift increases (see Fig. 1). Interestingly, the estimated redshift of GW150914 and GW151226 is z ∼ 0 . 1, while z ∼ 0 . 2 is the redshift of the candidate event (LVT151012) and of the most recent detection (GW170104). \nThe distribution of t form and t merg in the other models is similar to the one shown in Fig. 5. The main differences arise from the distribution of delay times. For this reason, the distribution of formation times in run D0.02 has a much higher peak at high redshift than that of run D. \nFigure 6 shows the distribution of total masses and chirp \nFigure 6 is significantly affected by the fact that the LIGO horizon depends on the BHB mass. More massive binaries can be observed also if they merge at higher redshift ( z ∼ 0 . 4 if M ∼ 50 M glyph[circledot] ). Thus, their contribution to Fig 6 is enhanced with respect to that of lighter BHBs. \n<!-- image --> \nFigure 6. Distribution of total masses (upper panel) and chirp masses (lower panel) of the simulated BHBs merging within the LIGO horizon. Red solid line: run D; black dashed line: R; orange dashed line: DK; green dash-dot line: D1.5; blue dotted line: D0.02. The value N merg on the y axis is the number of simulated BHBs per mass bin (∆ m = 2 M glyph[circledot] for both the chirp and the total mass). \n<!-- image --> \nmasses of the simulated BHBs that merge within the LIGO horizon. The chirp masses (total masses) range between 3 and 35 M glyph[circledot] (6 and 82 M glyph[circledot] ). \nRuns D1.5 and D0.02 are significantly different from the others, with a peak at lower masses ( M ∼ 20 -40 M glyph[circledot] , m chirp ∼ 8 -20 M glyph[circledot] ) and a dearth of massive systems. Runs with αλ = 0 . 1 are more similar between each other. For high total masses ( M > 20 M glyph[circledot] ) there is no appreciable difference between runs D and R (which differ for the SN model). At lower masses, run R (assuming a rapid SN model) has a peak at 11 ≤ M/ M glyph[circledot] ≤ 17, which is completely absent if the delayed SN model is assumed. Moreover, in run D and in the other runs assuming a delayed SN models, BHs with mass down to 3 M glyph[circledot] are allowed to form, while no BHs with mass < 5 M glyph[circledot] form in run R. \nFinally, the distribution of BHB masses in run DK is very similar to that of run D, except for one important detail: there is no peak for total BHB masses (chirp masses) 75 < M/ M glyph[circledot] < 82 (30 < m chirp / M glyph[circledot] < 35). This comes from the fact that in run DK all BHs receive a strong natal kick (regardless of their mass), while in the other runs the kick is modulated by the fallback. \nFigure 7. Distribution of metallicity ( Z ) of the stellar progenitors of the simulated BHBs that merge within the LIGO horizon. Red solid line: run D; black dashed line: R; orange dashed line: DK; green dash-dot line: D1.5; blue dotted line: D0.02. The value N merg on the y axis is the number of simulated BHBs per metallicity bin (∆ log Z = 0 . 01). The step-like features in the plot correspond to the metallicity groups in the BSE simulations. \n<!-- image --> \nFuture LIGO-Virgo detections will enable us to reconstruct total mass and chirp mass distributions. Hopefully, the observed distributions will be able to discriminate between different models, indicating which one captures the main physics of BHB evolution. \nFinally, Fig. 7 shows the distribution of metallicity of the stellar progenitors of the simulated BHBs merging within LIGO horizon. In runs D, DK and R, the metallicity of BHB progenitors peaks in the range 0 . 01 ≤ Z/ Z glyph[circledot] ≤ 0 . 2. The metallicity distribution drops for Z > ∼ 0 . 5 Z glyph[circledot] . \nAlso in this case, runs with different CE parameters (D1.5 and D0.02) have a different trend. In run D0.02 the metallicity of BHB progenitors peaks at 0 . 15 < Z/ Z glyph[circledot] < 0 . 4, significantly higher than in runs with αλ = 0 . 1. In contrast, in run D1.5 the most metal-poor systems are more efficient in producing merging BHBs than in the other runs. The step-like features clearly visible in Fig. 7 are model artifacts, due to the fact that we simulated BHBs only in 12 metallicity bins (see Section 2).', '3.4 GW150914, LVT151012, GW151226, and GW170104-like systems': "Finally, we focus on the properties of simulated systems that match the three observed GW events, GW150914 ( m p = 36 . 2 +5 . 2 -3 . 8 M glyph[circledot] , m s = 29 . 1 +3 . 7 -4 . 4 M glyph[circledot] , Abbott et al. 2016a), GW151226 ( m p = 14 . 2 +8 . 3 -3 . 7 M glyph[circledot] , m s = 7 . 5 +2 . 3 -2 . 3 M glyph[circledot] , Abbott et al. 2016a), and GW170104 ( m p = 31 . 2 +8 . 4 -6 . 0 M glyph[circledot] , m s = 19 . 4 +5 . 3 -5 . 9 M glyph[circledot] , Abbott et al. 2017), and the fourth possible signal, LVT151012 ( m p = 23 +18 -6 M glyph[circledot] , m s = 13 +4 -5 M glyph[circledot] , Abbott et al. 2016a). From our simulations, we extract all merging BHBs with masses consistent with those of the BHs \nFigure 8 shows the behaviour of the merger time and the formation time for simulated systems like GW150914, LVT151012, GW151226 and GW170104 in our fiducial model. It is apparent that the vast majority of GW150914like systems form at high redshifts ( z > 1), with a peak at t form ∼ 12 -12 . 3 Gyr (corresponding to z ∼ 3 . 6 -4 . 3 in the Illustris' cosmology). Mergers also peak at high redshift \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 8. Distribution of the formation time ( t form , red solid line) and of the merger time ( t merg , blue dashed line) for the simulated BHBs matching the mass of GW150914 (top left), LVT151012 (top right), GW151226 (bottom left) and GW170104 (bottom right) in the fiducial model (run D). Thin lines indicate all simulated systems matching the mass of GW150914, LVT151012, GW151226, and GW170104, while thick lines indicate only systems that merge within LIGO's horizon. Both t form and t merg are expressed in look-back time (bottom x axis) and redshift (top x axis). The value N merg on the y axis is the number of simulated BHBs per time bin (∆ t = 10 Myr for both the merger time and the formation time). \n<!-- image --> \nTable 3. Percentage of simulated BHBs merging in the LIGO horizon with mass consistent with the detected events. \n| Name | GW150914 | LVT151012 | GW151226 | GW170104 |\n|--------|------------|-------------|------------|------------|\n| D | 6.0% | 9.9% | 3.4% | 12.1% |\n| R | 5.3% | 8.9% | 3.3% | 10.8% |\n| DK | 9.8% | 11.0% | 2.5% | 12.5% |\n| D1.5 | 8.4% | 13.1% | 5.0% | 10.4% |\n| D0.02 | 2.8% | 12.9% | 4.9% | 5.2% | \nColumn 1: model name; columns 2, 3, 4, and 5: percentage of simulated BHBs with mass consistent with GW150914, \nLVT151012, GW151226 and GW170104, respectively, which merge within the LIGO horizon. This percentage is calculated over all simulated BHBs that merge within the LIGO horizon. \nassociated with GW150914, GW151226, GW170104, and possibly LVT151012 (within 90% credible intervals, as given by Abbott et al. 2016a). Table 3 shows the percentage of merging BHBs that have masses consistent with GW150914, \nLVT151012, GW151226 and GW170104, if we consider only BHBs that merge within the LIGO horizon. \nBHB mergers consistent with GW150914, LVT151012, GW151226, and GW170104 exist in all of our models. In particular, we expect ∼ 6 %, ∼ 10 %, ∼ 3 % and ∼ 12 % of events, inside the LIGO horizon, to have mass consistent with GW150914, LVT151012, GW151226, and GW170104 respectively, in run D (see Table 3 for the other runs). Interestingly, GW150914-like events appear to be more common than GW151226-like ones within the 2015-2016 LIGO's horizon. Also, LVT151012-like events seem to be the most common ones within LIGO's horizon, but this is mainly due to the fact that the mass of LVT151012 is much more uncertain than that of both GW150914 and GW151226, thus more systems fall within the allowed window. \nFigure 10 shows the metallicity distribution of the stellar progenitors of GW150914-like, GW151226-like, LVT151012-like, and GW170104-like systems in our fiducial model. As expected, GW150914-like systems form only at relatively low metallicity: Z < ∼ 0 . 15 Z glyph[circledot] . Metal-poor stars ( Z < ∼ 0 . 3 Z glyph[circledot] ) are the most common progenitors also for GW170104-like systems. In contrast, GW151226-like systems form nearly at all metallicities up to Z ∼ 0 . 7 Z glyph[circledot] . \n<!-- image --> \nFigure 9. Metallicity of progenitors of systems matching the mass of GW150914 in the fiducial model (D). Upper panel: metallicity versus formation time of the stellar progenitors ( t form ). Lower panel: metallicity of the stellar progenitors versus merger time of the BHBs ( t merg ). Both t form and t merg are expressed as look-back time. The colour-coded map (in logarithmic scale) indicates the number of merging BHBs per cell. The black lines are isocontours enclosing a number of merging BHBs ranging from 5 × 10 3 to 5 × 10 5 (as indicated by the black labels). \n<!-- image --> \n( t merg ∼ 12 Gyr). However, the long delay time between formation and merger for several systems causes t merg to decrease more gently than t form when approaching z = 0. \nIf we consider only GW150914-like systems that merge within the LIGO horizon, their formation time still peaks at high redshift ( z ∼ 2 -3), even if the slope of t form is much less steep. This is consistent with Schneider et al. (2017), who predict that GW150914-like events form preferentially at redshift 2 . 4 ≤ z ≤ 4 . 2. \nIn the nearby Universe, we expect a factor of > ∼ 10 (and up to ∼ 400) higher merger rate of GW150914-like systems than their formation rate. In run D, the birth rate of new GW150914-like systems at z ∼ 0 . 1 (the redshift associated with the LIGO detection) is only 0.4 Gpc -3 yr -1 , whereas the expected merger rate of GW150914-like systems at redshift z ∼ 0 . 1 is ∼ 5 . 2 Gpc -3 yr -1 (see Table 4). This result is perfectly consistent with the estimates from the LIGO-Virgo collaboration (3 . 4 +8 . 8 -2 . 8 Gpc -3 yr -1 , Abbott et al. 2016a). \nFig. 9 shows that this effect is linked to the metallicity of GW150914 progenitors. In fact, most GW150914-like sys- \nTable 4. Expected merger rate for GW150914, LVT151012, GW151226 and GW170104-like events. To estimate these values we consider only systems merging at redshift consistent with the LIGO detections. \n| Name | GW150914 [Gpc - 3 yr - 1 ] | LVT151012 [Gpc - 3 yr - 1 | GW151226 [Gpc - 3 yr - 1 ] | GW170104 [Gpc - 3 yr - 1 ] |\n|--------|------------------------------|-----------------------------|------------------------------|------------------------------|\n| D | 5.2 | 20.9 | 8.3 | 15.9 |\n| R | 5.2 | 20.7 | 9.3 | 16.0 |\n| DK | 1.5 | 3.5 | 1.0 | 2.9 |\n| D1.5 | 7.7 | 21.8 | 10.9 | 12.2 |\n| D0.02 | 5.1 | 37.4 | 18.7 | 10.5 |\n| A16 | 3 . 4 +8 . 8 - 2 . 8 | 9 . 1 +31 - 8 . 5 | 36 +95 - 30 | - | \nColumn 1: model name; columns 2, 3, 4, and 5: merger rate of simulated BHBs with mass and merger redshift consistent with GW150914, LVT151012, GW151226, and GW170104, respectively. The last line shows the rate inferred from LIGO observations (see Table II of Abbott et al. 2016a) for GW150914, LVT151012, and GW151226. \ntems merging at low redshift have 0 . 03 < Z/ Z glyph[circledot] < 0 . 05, but the number of progenitor systems forming at low redshift in this metallicity range is a factor of ∼ 100 lower than the number of merging GW150914-like systems. \nWe find a similar trend for GW170104-like systems. In run D, the birth rate of new GW170104-like systems at z ∼ 0 . 18 (the redshift associated with the LIGO detection) is only 0.4 Gpc -3 yr -1 , whereas the expected merger rate of GW170104-like systems at redshift z ∼ 0 . 18 is ∼ 12 . 2 Gpc -3 yr -1 (see Table 4). \nIn contrast, the shift between formation time and merger time is rather negligible for systems like GW151226. From our simulations, the expected merger rate of GW151226-like systems at redshift z ∼ 0 . 1 (the redshift associated with the LIGO detection) is ∼ 8 . 3 Gpc -3 yr -1 (see Table 4), similar to their birth rate ( ∼ 3 . 1 Gpc -3 yr -1 at z ∼ 0 . 1). This merger rate is consistent with the estimates from the LIGO-Virgo collaboration, even if close to the low tail (36 +95 -30 Gpc -3 yr -1 , Abbott et al. 2016a). \nLVT151012-like events behave in a similar way to GW151226, with a mild offset between their current merger time and their current formation time (Fig. 8). The merger rate of LVT151012-like events is ∼ 20 Gpc -3 yr -1 at redshift z ∼ 0 . 2 (i.e. the redshift associated with the possible LIGO signal), significantly higher than that of the other two events. This happens because the predicted BHB merger rate at z ∼ 0 . 2 is higher by a factor of ∼ 1 . 3 than that at z ∼ 0 . 1 and because the mass of LVT151012 is more uncertain than that of GW151226 and GW150914 (thus, more simulated systems are consistent with it). This merger rate is also consistent with the estimates from the LIGO-Virgo collaboration (9 . 1 +31 -8 . 5 Gpc -3 yr -1 , Abbott et al. 2016a). \nFigure 10. Metallicity distribution of simulated GW150914like (blues dashed line), LVT151012-like (green dash-dot line), GW151226-like (red solid line) and GW170104-like systems (violet dotted line) in the fiducial model (run D). The value N merg on the y axis is the number of simulated BHBs per metallicity bin (∆log Z = 0 . 01). The step-like features in the plot correspond to the metallicity groups in the BSE simulations. \n<!-- image --> \nLVT151012-like systems also form with a broad range of metallicities (up to Z ∼ 0 . 5 Z glyph[circledot] ).", '4 DISCUSSION: COMPARISON WITH PREVIOUS WORK AND CAVEATS': "The method we followed in this paper ensures that we account for the cosmic SFR and for the mass-metallicity relation, at least within the limitations of state-of-the-art cosmological simulations. This is a crucial point for understanding the cosmic evolution of BHB mergers, since the mass of BHs is expected to depend on the metallicity of the progenitor stars. \nThe cosmic BHB merger rate we obtain from our models (Fig. 1) approximately follows the same trend as the cosmic SFR, suggesting that the slope of the curve is primarily set by star formation. The peak of the BHB merger rate is at z ∼ 2 -3, corresponding to the peak of the cosmic SFR (Madau & Dickinson 2014). This is in reasonable agreement with previous results (e.g. Dominik et al. 2013; Belczynski et al. 2016b). \nThe normalization of the BHB merger rate in Fig. 1 strongly depends on the treatment of CE (especially for HG stars) and on the distribution of natal kicks. In particular, models in which HG donors can survive the CE phase disagree with the BHB merger rate estimated from LIGO observations, unless very large natal kicks are assumed for most BHs (in agreement with Belczynski et al. 2016b). Different distributions of the natal kicks result in a factor of > ∼ 6 different BHB merger rate, given the large uncertainties in the distribution of BH natal kicks. Future observations by LIGO-Virgo will likely allow us to put constraints on the natal kicks of BHs (Vitale et al. 2017a,b; Stevenson et al. 2017; Zevin et al. 2017). \nThe behaviour of the BHB merger rate differs significantly from the cosmic SFR only if the CE parameters are drastically different ( αλ = 0 . 02) from our fiducial val- \ns ( αλ = 0 . 1). According to the CE formalism (Webbink 1984), low values of αλ imply that is more difficult to eject the envelope. Thus, if αλ is very low, the closest binaries merge prematurely during a CE phase, before they become BHBs. This explains why the merger rate of D0.02 is much lower than that of D at high redshift ( z > 1). On the other hand, a very low value of αλ implies that the shrinking of the orbit of the two cores within the CE is more efficient. Thus, looser binaries going through a CE phase might shrink enough to produce tight BHBs, which merge in a Hubble time. This might explain why the merger rate density of D0.02 increases at low redshift with respect to that of run D: in run D0.02, there is a higher number of BHBs with long delay time (Fig. 4), which form at high redshift but merge at low redshift. \nIn contrast, if αλ is high, the CE is ejected easily and the orbit does not shrink efficiently. Thus, even if most binaries survive the CE phase, a lower number of BHBs become sufficiently close to merge in a Hubble time. These merging BHBs will have, on average, a longer delay time. This explains why less BHBs merge at high redshift ( z > 0 . 3) in run D1.5 with respect to run D, while at low redshift ( z < 0 . 3) the merger rate of D1.5 is slightly higher than that of D. \nAmong previous studies, Lamberts et al. (2016) and Belczynski et al. (2016b) contain several results that can be compared with ours quite straightforwardly. To produce their sample of BHBs, Belczynski et al. (2016b) use the startrack code (Belczynski et al. 2008), while Lamberts et al. (2016) use another updated version of the BSE code (Hurley et al. 2002) and focus only on the study of GW150914-like systems. The present-time BHB merger rates we obtain with αλ = 0 . 1 are consistent (within a factor of 2) with those shown by Belczynski et al. (2016b), who adopt a similar choice for the CE parameters. In contrast, Lamberts et al. (2016) obtain a much larger present-time BHB merger rate ( ∼ 850 Gpc -3 yr -1 ) than expected from LIGO detections. As to the origin of this difference, we note that Lamberts et al. (2016) do not update the recipes for core radii in BSE , and (most importantly) do not include pulsational pair instability SNe, while this paper and Belczynski et al. (2016b) do. \nIf we restrict our analysis to systems like GW150914, GW151226, and GW170104, we find that their formation history also mimics the cosmic BHB merger rate density, but with a substantial difference. The distribution of formation times for GW150914-like and GW170104-like binaries peaks at high redshift ( z ∼ 2 -4) and then drops much faster than the total BHB merger rate density, while GW151226-like systems follow the general trend as the BHB merger rate. As a consequence, present-day GW150914-like and GW170104like events are dominated by systems that formed at high redshift and have a long delay time. The reason is that GW150914-like and GW170104-like systems form mainly from metal-poor progenitors. We find a rather smooth distribution of the birth times for GW150914-like systems, consistently with Lamberts et al. (2016) and at odds with Belczynski et al. (2016b). This is not surprising, since both us and Lamberts et al. (2016) account for the mass-metallicity relation. Our main findings for GW150914, LVT151012 and GW151226 are also consistent with the results of Schneider et al. (2017), who predict that most GW150914 candidates form at high redshift (2 . 4 ≤ z ≤ 4 . 2), whereas both \nGW151226 and LVT151012 have a broad range of possible formation redshifts. \nWe now discuss the main caveats of the present work. First, we have not included the impact of stellar dynamics on the evolution of BHBs. This might be a serious issue for our work, because massive stars are known to form especially in dense massive star clusters (see e.g. Weidner & Kroupa 2006), which are active dynamical places. Dynamical exchanges might lead to the formation of additional BHBs, which are likely more massive and more eccentric than average (e.g. Downing et al. 2010; Ziosi et al. 2014; Askar et al. 2017; Rodriguez et al. 2016; Banerjee 2017). Very massive BHs or even intermediate-mass BHs might form from runaway collisions (Portegies Zwart et al. 2004; Giersz et al. 2015; Mapelli 2016). Furthermore, Kozai-Lidov resonances in triple systems might enhance the merger rate both in dense star clusters (e.g. Kimpson et al. 2016; Antonini & Rasio 2016) and in the field (e.g. Antonini et al. 2017). A study of the cosmic BHB merger rate including the effects of dynamics is still missing, because it poses a serious numerical challenge. \nMoreover, we neglect the contribution of very massive stars ( > 150 M glyph[circledot] ), because the current version of BSE does not include them. While these very massive objects are presumably very rare, they might significantly contribute to the merger rate of the most massive BHBs (e.g. Mapelli 2016). \nFurthermore, our calculations neglect the chemically homogeneous evolutionary formation channel, recently proposed by Mandel & de Mink (2016), which might account for R BHB ∼ 10 -20 Gpc -3 yr -1 (see also Marchant et al. 2016 and de Mink & Mandel 2016). \nIn our analysis, we simply assume that population III stars behave as 'normal' metal-poor stars (with Z = 0 . 0002). While this choice is partially motivated by the fact that stellar winds are already highly inefficient at Z ∼ 0 . 0002 (see e.g. figure 6 of Spera et al. 2015), we do not know whether the mass function and the binary fraction of population III stars were significantly different from those of population II stars. On the other hand, we have checked that the contribution of population III stars (defined as Illustris' star particles with metallicity Z ≤ 0 . 0001) to the cosmic BHB merger rate is negligible, especially within the LIGO horizon (see Fig. 11). We refer to other studies (e.g. Kinugawa et al. 2014, 2016; Belczynski et al. 2016a; Hartwig et al. 2016; Inayoshi et al. 2017) for a more accurate treatment of BHBs from population III stars. \nWe also stress that we keep using the original fitting formulas of Hurley et al. (2000) for the photospheric radius and luminosity of stars, while our knowledge of the evolution of massive stars changed significantly in the last decades (e.g. Martins & Palacios 2013; Chieffi & Limongi 2013; Chen et al. 2015). In forthcoming studies, we will account for more recent stellar evolution prescriptions in the frame of the new SEVN population-synthesis code (Spera et al., in preparation). \nFinally, the Illustris reproduces reasonably well many observational features (such as the cosmic SFR density and the galaxy luminosity function), but has several limitations which might affect the BHB merger rate. For example, it predicts a too mild decline in the cosmic SFR density at z < 1 (Pillepich et al. 2017). This might lead to an overestimate of the present-day merger rate by several per cents (see Ap- \nFigure 11. Upper panel: cosmic merger rate density of BHBs ( R BHB ) in run D, if we include (red solid line) or neglect (blue dashed line) population III stars (i.e. Illustris' stars with metallicity Z ≤ 0 . 0001). Lower panel: residuals of the model with population III stars with respect to the model without them. \n<!-- image --> \npendix B). Galaxies in the Illustris follow a mass-metallicity relation which is sensibly steeper than the observed relation (Torrey et al. 2014). As we discuss in Appendix B, this should affect the BHB merger rate density by ∼ 20 per cent. Thus, it is important to repeat the same exercise that we did in this paper also with other cosmological simulations and to check for discrepancies 3 .", '5 SUMMARY': "We reconstructed the cosmic BHB merger rate by planting BHBs into the Illustris cosmological box. The Illustris' box (length=106.5 Mpc) is large enough to guarantee that we are modelling an unbiased portion of the Universe, satisfying the cosmological principle. \nThe population of BHBs is estimated through population synthesis simulations of isolated binaries, performed with an updated version of BSE . In particular, our new version of BSE includes up-to-date prescriptions for stellar winds of massive stars, during and after the MS. We account not only for the dependence of stellar winds on metallicity, but also for the effect of the electron-scattering Eddington limit (Chen et al. 2015). Up-to-date recipes for core collapse SNe, pair-instability and pulsational pair-instability SNe are also included. \nWe perform six different sets of runs with BSE , changing the SN prescription, the CE efficiency, the treatment of HG stars, and the distribution of natal kicks (Table 1). Each of these six simulations produces a population of merging BHBs, depending on the metallicity of the progenitor stars. We then interface the population of merging BHBs with the \n3 An updated Illustris simulation will be available soon, with an improved algorithm for AGN feedback and galactic winds (Pillepich et al. 2017). \nIllustris simulation through a Monte Carlo model, to obtain the cosmic BHB merger rate density for each of the six BSE simulation sets. \nThe cosmic BHB merger rate follows the same trend as the cosmic SFR, with a peak at z ∼ 2 -3, in all simulation sets (Fig. 1). In contrast, the normalization of the BHB merger rate strongly depends on the specific BSE simulation set. In particular, the treatment of CE and the distribution of SN kicks appear to be the most important processes. \nModels in which a HG donor is allowed to survive the CE phase are not consistent with LIGO observations (unless very high natal kicks are assumed), because they give a too high BHB present-time merger rate density. The choice of the CE parameters (we study values of αλ ranging from 0.02 to 1.5, in addition to our fiducial value αλ = 0 . 1) significantly affects the merger rate at high redshift, while it induces only minor changes (a factor of 2) in the low-redshift merger rate ( z < 0 . 3). \nPopulation-synthesis simulations with large natal kicks (distributed according to Hobbs et al. 2005) result in a present-day BHB merger rate R BHB ∼ 20 Gpc -3 yr -1 , while population-synthesis simulations with lower kicks (accounting for the amount of fallback) give a merger rate R BHB ∼ 125 -155 Gpc -3 yr -1 . Both rates are still consistent with the constraints from LIGO observations (9 -240 Gpc yr -1 , Abbott et al. 2016a), but forthcoming detections might be able to distinguish between them. \nFrom our simulations, we can also trace the metallicity of the progenitors of merging BHBs. We find that most BHB mergers detectable by current GW interferometers come from relatively metal-poor progenitors, ranging from ∼ 0 . 015 Z glyph[circledot] to ∼ 0 . 2 Z glyph[circledot] (Figs 2 and 7). \nThe merging BHBs have chirp masses (total masses) ranging from ∼ 3 to ∼ 35 M glyph[circledot] ( ∼ 6 to ∼ 82 M glyph[circledot] ), with a slight dependence on the SN prescription and on CE parameters (Fig. 6). \nIf we focus on systems similar to GW150914, GW151226 and GW170104, we can obtain some useful hints on their progenitors. The formation of GW150914-like and GW170104-like systems is more efficient at high redshift and drops in the local Universe (Fig. 8). Most GW150914-like and GW170104-like systems merging in the local Universe appear to have formed at higher redshift with a long delay time. This happens because only genuinely metal-poor stars can produce GW150914-like and GW170104-like systems (with metallicity Z < 0 . 15 Z glyph[circledot] and Z < 0 . 3 Z glyph[circledot] , respectively; see Fig. 10). In contrast, GW151226-like systems form and merge all the way through the cosmic history (from z ∼ 9 to z = 0). The progenitors of GW151226-like systems can be either metal-rich or metal-poor stars with about the same probability. \nIn our fiducial model (run D) the percentage of GW150914-like systems merging within the LIGO instrumental horizon is higher ( ∼ 6 %) than the percentage of GW151226-like systems ( ∼ 3 %, see Table 3). This might suggest that LIGO and Virgo will observe more events like GW150914 than like GW151226. \nIn conclusion, this study provides several clues about merging BHBs and their progenitors. We show that the BHB merger rate density poses constraints on both the CE process and the natal kicks of BHs. Forthcoming GW detections will allow us to further strengthen these constraints. More- \nover, our results suggest that most BHBs merging within the LIGO horizon formed from relatively metal-poor progenitors ( Z < 0 . 2 Z glyph[circledot] ). The Illustris simulation also includes information on the properties of BHB host galaxies. In a forthcoming paper, we will explore the properties of the galaxies where BHBs form and merge. The same analysis will be done also for NS binary systems and NS-BH binary systems, providing a clue for the detection of electromagnetic counterparts.", 'ACKNOWLEDGMENTS': "We thank the anonymous referee for their critical reading of the manuscript. We thank Alessandro Bressan, Marica Branchesi, Raffaella Schneider, Luca Graziani, Elena D'Onghia and Roberto Maiolino for useful discussions. We warmly thank The Illustris team for making their simulations publicly available. Numerical calculations have been performed through a CINECA-INFN agreement and through a CINECA-INAF agreement, providing access to resources on GALILEO and MARCONI at CINECA. MM and MS acknowledge financial support from the Italian Ministry of Education, University and Research (MIUR) through grant FIRB 2012 RBFR12PM1F, and from INAF through grant PRIN-2014-14. MM acknowledges financial support from the MERAC Foundation.", 'REFERENCES': "Abbott B. P., et al., 2016a, Physical Review X, 6, 041015 Abbott B. P., et al., 2016b, Phys. Rev. Lett., 116, 061102 Abbott B. P., et al., 2016c, ApJ, 818, L22 Abbott B. P., et al., 2017, Physical Review Letters, 118, 221101 Antonini F., Rasio F. A., 2016, ApJ, 831, 187 Antonini F., Toonen S., Hamers A. S., 2017, ApJ, 841, 77 Askar A., Szkudlarek M., Gondek-Rosi'nska D., Giersz M., Bulik T., 2017, MNRAS, 464, L36 Banerjee S., 2017, MNRAS, 467, 524 Belczynski K., Sadowski A., Rasio F. A., 2004, ApJ, 611, 1068 Belczynski K., Kalogera V., Rasio F. 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E., Mandel I., 2016, MNRAS, 460, 3545 |", 'APPENDIX A: PRESCRIPTIONS FOR CORE-COLLAPSE SN AND FOR FALLBACK': 'We adopt two different models for core-collapse SNe: the rapid (R) and the delayed (D) model. Both models were introduced by Fryer et al. (2012) and they differ by the timescale over which the explosion occurs: < 250 ms after the bounce for the rapid model, > 250 ms for the delayed mechanism. \nFor the rapid SN mechanism, a fixed mass of the protocompact object, m proto = 1 . 0 M glyph[circledot] , is assumed. In this case, the fallback parameter f fb is defined as \nf fb = 0 . 2 ( m fin -m proto ) -1 m CO < 2 . 5 M glyph[circledot] 0 . 286 m CO -0 . 514 m fin -m proto 2 . 5 M glyph[circledot] ≤ m CO < 6 . 0 M glyph[circledot] 1 . 0 6 . 0 M glyph[circledot] ≤ m CO < 7 . 0 M glyph[circledot] α R m CO + β R 7 . 0 M glyph[circledot] ≤ m CO < 11 . 0 M glyph[circledot] 1 . 0 m CO ≥ 11 . 0 M glyph[circledot] (A1) \nwhere m fin is the final mass of the star, m proto is the mass of the proto-NS, m CO is the mass of the Carbon-Oxygen core, α R ≡ 0 . 25 -1 . 275 ( m fin -m proto ) -1 , and β R ≡ 1 -11 α R . \nFor all models, the amount of mass that falls onto the proto-NS is defined as m fb = f fb ( m fin -m proto ). The final mass of the compact object is given by M BH = m proto + m fb .', 'A0.2 Delayed SN model': 'For the delayed SN mechanism, the mass of the protocompact object is defined as \nm proto = 1 . 2 M glyph[circledot] m CO < 3 . 5 M glyph[circledot] 1 . 3 M glyph[circledot] 3 . 5 M glyph[circledot] ≤ m CO < 6 . 0 M glyph[circledot] 1 . 4 M glyph[circledot] 6 . 0 M glyph[circledot] ≤ m CO < 11 . 0 M glyph[circledot] 1 . 6 M glyph[circledot] m CO ≥ 11 . 0 M glyph[circledot] . (A2) \nThe amount of fallback is determined using the following relations \nf fb = 0 . 2 m fin -m proto m CO < 2 . 5 M glyph[circledot] 0 . 5 m CO -1 . 05 M glyph[circledot] m fin -m proto 2 . 5 M glyph[circledot] ≤ m CO < 3 . 5 M glyph[circledot] α D m CO + β D 3 . 5 M glyph[circledot] ≤ m CO < 11 . 0 M glyph[circledot] 1 . 0 m CO ≥ 11 . 0 M glyph[circledot] (A3) \nwhere α D ≡ 0 . 133 -0 . 093 ( m fin -m proto ) -1 and β D ≡ 1 -11 α D .', 'APPENDIX B: THE IMPACT OF RESOLUTION AND MASS-METALLICITY RELATION': 'In this section, we estimate the impact of the numerical resolution and of the simulated mass-metallicity relation on the main results of this paper. In Fig. B1 and Table B1, we compare the BHB merger rate density we obtain from the Illustris-1 simulation (model Il-1) with the BHB merger rate density we obtain from the Illustris-3 simulation (model Il3), which has a factor of ∼ 60 poorer resolution. The current BHB merger rate density in the Illustris-3 simulation is ∼ 40 % lower than in the Illustris-1 simulation, because of the lower resolution. Galaxies with stellar mass < 10 9 M glyph[circledot] are under-resolved (they consist of < 1000 star particles) even in the Illustris-1 simulation. It is reasonable to expect that accounting for these low-mass galaxies might further boost the merger rate. \nFigure B1. Left y -axis: cosmic merger rate density of BHBs ( R BHB ) in the comoving frame, as a function of the look-back time t lb (bottom x axis) and of the redshift z (top x axis). Red solid line: model D applied to the Illustris-1 simulation (same as in Fig. 1); violet dash-dot line: model D applied to the Illustris-3 simulation; blue dashed line: model D applied to the Illustris-3 simulation and adopting Maiolino et al. (2008; hereafter M08) fitting formulas for the mass-metallicity relation. Green shaded area: BHB merger rate inferred from LIGO detections (Abbott et al. 2016a). Right y -axis: cosmic SFR density from the Illustris1 (grey thin solid line) and from Madau & Dickinson 2014 (grey thin dashed line), as a function of the look-back time t lb (bottom x axis) and of the redshift z (top x axis). \n<!-- image --> \nTable B1. Comoving BHB merger-rate density at redshift z = 0 and z = 0 . 2. \n| Name | R BHB ( z = 0) [Gpc - 3 yr - 1 ] | R BHB ( z = 0 . 2) [Gpc - 3 yr - 1 ] |\n|--------------|------------------------------------|----------------------------------------|\n| Il-1 | 125 | 181 |\n| Il-3 | 78 | 114 |\n| Il-3 and M08 | 96 | 135 | \nSame as Table 1 but for the check runs. Column 1: model name; column 2: present-time BHB merger rate density; column 3: BHB merger rate density at z = 0 . 2. \nTable B2. Best fit parameters for the mass-metallicity relation in equation B1 at different redshift. The values of M 0 and K 0 at z = 3 . 5 come from Mannucci et al. (2009), while the other values come from Maiolino et al. (2008).Column 1: redshift; column 2 and 3: values of the parameters in equation B1 at different redshift. \n| z | log M 0 | K 0 |\n|------|-----------|-------|\n| 0.07 | 11.18 | 9.04 |\n| 0.7 | 11.57 | 9.04 |\n| 2.2 | 12.38 | 8.99 |\n| 3.5 | 12.28 | 8.69 |', '16 Mapelli et al.': "In a companion paper (Schneider et al. 2017), we follow a complementary approach by coupling our populationsynthesis models with the GAMESH pipeline (Graziani et al. 2015, 2017). GAMESH consists of a (4 Mpc) 3 box and reaches a much higher resolution: even galaxies with a stellar mass of ∼ 10 6 M glyph[circledot] are effectively resolved. Schneider et al. (2017) predict that all GW150914-like systems in the GAMESH box form in progenitor galaxies with a stellar mass < 5 × 10 6 M glyph[circledot] (see their Figure 2). Thus, the merger rate of GW150914-like systems is probably underestimated in the current study, due to the Illustris resolution. \nFig. B1 also shows the difference between the cosmic SFR density in the Illustris-1 simulation and the cosmic SFR density from Madau & Dickinson (2014). At redshift z < 0 . 3 the Illustris-1 simulation predicts a higher SFR density by a factor of ∼ 1 . 5. This suggests that the current BHB merger rate density might be slightly overestimated in our calculations. \nAs we discussed in Section 2.2, the model of sub-grid physics adopted in the Illustris produces a mass-metallicity relation which significantly differs from the observed one. In particular, the simulated mass-metallicity relation is sensibly steeper than the observed one and does not show the observed turnover at high stellar mass (Vogelsberger et al. 2013; Torrey et al. 2014). We quantify the impact of these differences between simulated and observed mass-metallicity relation through the following procedure. \nIn the Illustris-3 simulation, we override the metallicity of a given star particle with the metallicity we expect from the observed mass-metallicity relation. For the observed relation, we adopt the fitting formula by Maiolino et al. (2008) and Mannucci et al. (2009): \n12 + log [O / H] = -0 . 0864 (log M ∗ -log M 0 ) 2 + K 0 , (B1) \nwhere M ∗ is the total stellar mass of the host galaxy in solar masses, while M 0 and K 0 are given in Table B2. For intermediate redshifts between those in Table B2, we obtain the metallicity by linear interpolation. At redshift z < 0 . 07 ( z > 3 . 5) we simply use the same coefficients as for z = 0 . 07 ( z = 3 . 5). The new metallicity of each Illustris' star is randomly extracted from a Gaussian distribution with mean value given by equation B1 (where M ∗ is the total stellar mass of the sub-halo hosting the Illustris' star) and standard deviation σ = 0 . 5 dex (accounting for metallicity dispersion within galaxies). \nIn Fig. B1 and Table B1 we show the BHB merger rate density we obtain with this procedure, compared to the BHB merger rate density we derive using the simulated metallicity of each Illustris-3 particle. The maximum difference between the two curves is only ∼ 20 per cent. The same procedure can be applied to the Illustris-1 simulation, but is a factor of 60 more computationally expensive. We will show the results for the Illustris-1 in our follow-up paper."}
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2009PhRvA..80d3601M
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Black-hole radiation in Bose-Einstein condensates
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2009-01-01
| 10 | 0.45 | 164 |
['-', '-', '-', 'dynamical systems', '-', '-', '-', '-', '-']
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[]
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We study the phonon fluxes emitted when the condensate velocity crosses the speed of sound, i.e., in backgrounds which are analogous to that of a black hole. We focus on elongated one-dimensional condensates and on stationary flows. Our theoretical analysis and numerical results are based on the Bogoliubov-de Gennes equation without further approximation. The spectral properties of the fluxes and of the long distance density-density correlations are obtained, both with and without an initial temperature. In realistic conditions, we show that the condensate temperature dominates the fluxes and thus hides the presence of the spontaneous emission (the Hawking effect). We also explain why the temperature amplifies the long distance correlations which are intrinsic to this effect. This confirms that the correlation pattern offers a neat signature of the Hawking effect. Optimal conditions for observing the pattern are discussed, as well as correlation patterns associated with scattering of classical waves. Flows associated with white holes are also considered.
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https://arxiv.org/pdf/0905.3634.pdf
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{'Black-hole radiation in Bose-Einstein condensates': "Jean Macher ∗ and Renaud Parentani † Laboratoire de Physique Th'eorique, CNRS UMR 8627, Bˆat. 210, Universit'e Paris-Sud 11, 91405 Orsay Cedex, France (Dated: October 23, 2018) \nWe study the phonon fluxes emitted when the condensate velocity crosses the speed of sound, i.e., in backgrounds which are analogous to that of a black hole. We focus on elongated one dimensional condensates and on stationary flows. Our theoretical analysis and numerical results are based on the Bogoliubov-de Gennes equation without further approximation. The spectral properties of the fluxes and of the long distance density-density correlations are obtained, both with and without an initial temperature. In realistic conditions, we show that the condensate temperature dominates the fluxes and thus hides the presence of the spontaneous emission (the Hawking effect). We also explain why the temperature amplifies the long distance correlations which are intrinsic to this effect. This confirms that the correlation pattern offers a neat signature of the Hawking effect. Optimal conditions for observing the pattern are discussed, as well as correlation patterns associated with scattering of classical waves. Flows associated with white holes are also considered. \nPACS numbers: 03.75.Kk, 04.62.+v, 04.70.Dy", 'I. INTRODUCTION': "The analogy between sound propagation in nonhomogeneous media and light propagation in curved spacetimes has opened the possibility to detect the analogue of black hole radiation in the lab [1]. Indeed, when sound waves propagate in a flowing medium whose velocity crosses at some point the speed of sound, they experience the analogue of an event horizon. If the phonon state is stationary and regular across this sonic horizon, one expects to obtain a thermal flux of phonons with a temperature k B T = /planckover2pi1 κ/ 2 π , where κ is the gradient of the flow velocity evaluated at the sonic horizon. Since the analogy works perfectly in the hydrodynamical limit, the above result should be valid at least when κ is much smaller than the critical wave-vector characterizing the dispersion [2, 3, 4, 5]. \nFollowing the original work of Unruh, various setups were proposed, see [6] for a review. In Refs. [7, 8, 9, 10, 11, 12, 13, 14, 15] the particular case of sound waves in dilute Bose-Einstein condensates (BEC) was considered. These condensates have nice properties both from an experimental and a theoretical point of view. From the first, progress in the manipulation and control of their physical properties is rapid, and from the second, the equations for the condensate and the phonons are well understood, as well as the approximations they involve [16]. \nIn this work, we first aim to derive the properties of the phonon fluxes without making use of the gravitational analogy. In fact we also aim to determine the validity range of the analogy. To achieve these goals, our analytical and numerical analysis are both based on the (ex- \nact) Bogoliubov-de Gennes (BdG) equation. More specifically, we consider one dimensional stationary flows which contain one sonic horizon (black or white) surrounded by two infinite asymptotically homogeneous regions. In this case, at fixed conserved frequency | ω | , three types of asymptotic phonons exist, and a complete description of their scattering is given in terms of a 3 × 3 Bogoliubov transformation [17]. \nOur second aim is to provide quantitative estimates of the spectral properties of the fluxes, in order to guide or explain experimental projects. To this end, we have performed a systematic numerical analysis. \nOur third aim is to understand the links between local and nonlocal observables. In Gaussian states (e.g. vacuum and thermal states), the physical predictions encoded in a Bogoliubov transformation are all contained in two groups of expectation values: first occupation numbers (in the present case there are three of them, one for each type of outgoing phonons: ¯ n i ω = 〈 a out † ω, i a out ω, i 〉 , i = 1 , 2 , 3), and second, interference terms, such as 〈 a out ω, i a out ω, j 〉 with i /negationslash = j , which determine the long distance correlations [18, 19, 20]. In this study, we were motivated by the recent work [14] where a distinct pattern of density correlations was 'numerically observed' when a sonic horizon forms. In what follows, we shall explain both the properties of this pattern, and also why, when taking into account the condensate temperature, the initial distributions of phonons tend to hide the black hole radiation in the occupation numbers whereas, at the same time, they reinforce the correlation pattern without affecting its spatial properties. This confirms that the correlation pattern offers a neat signature of the Hawking effect [13]. \nIn Sec. II, we derive the mode equation for the phonon field in one dimensional stationary condensates. We obtain an explicit fourth order equation which is valid for all nonhomogeneous condensates. To be general, the sonic horizons we consider can be due either to the velocity flow v ( x ), or to a varying sound speed c ( x ), or to a com- \nbination of these two. We shall later see that some fluxes properties (such as the backscattering mixing left- and right-moving phonons) crucially depend on the particular combination which is realized. \nIn Sec. III, we analyze the mode equation, we identify the combinations of solutions which describe initial and final asymptotic phonons, and relate them by the aforementioned 3 × 3 Bogoliubov transformation. In the next section, we show how it governs both local and nonlocal observables. This is carried out twice, without and with an initial temperature. In appendices, we consider the scattering of coherent states, which links the previous analysis with hydrodynamical experiments [21], and white holes. \nIn Sec. V, we numerically solve the mode equation and obtain the spectral properties of the emitted phonons. We consider both weakly and strongly dispersive regimes. The transition from one to the other characterizes the validity range of the gravitational analogy.", 'A. Dilute gases': "We give here the basic ingredients which describe the condensates and their linearized perturbations. The reader is referred to the review [16] for more details. \nIn a second quantized formalism, the set of atoms is described by a field operator Ψ( t, x ) which annihilates an atom at t, x , and which obeys the equal time commutator \n[Ψ( t, x ) , Ψ † ( t, x ' )] = δ 3 ( x -x ' ) . (1) \nThe time evolution of Ψ is given by the Heisenberg equation \ni /planckover2pi1 ∂ t Ψ( t, x ) = [Ψ( t, x ) , H ] , (2) \nwhere the Hamiltonian is \nH = ∫ d 3 x { /planckover2pi1 2 2 m ∇ x Ψ † ∇ x Ψ+ V Ψ † Ψ+ g 2 Ψ † Ψ † ΨΨ } . (3) \nIn this expression, m is the mass of the atoms, V the external potential, and g the effective coupling constant which describes the scattering of atoms by a local term. \nAt sufficiently low temperature, of the order of 300 nK for 10 4 atoms, a large fraction of the atoms condense in a common state. To separate this state from its perturbations, the field operator is decomposed into a c -number wave describing the condensed atoms, Ψ 0 , and a fluctuation: \nΨ = Ψ 0 + ˜ φ. (4) \nIn the mean field approximation, Ψ 0 satisfies the GrossPitaevskii equation \ni /planckover2pi1 ∂ t Ψ 0 = [ T + V + gρ 0 ] Ψ 0 , (5) \nwhere the kinetic operator is T = -/planckover2pi1 2 ∇ 2 x / 2 m . This equation guarantees that ρ 0 = | Ψ 0 | 2 obeys the continuity equation: ∂ t ρ 0 +div( ρ 0 v ) = 0, where v is the condensate velocity.", 'B. Stationary condensates': "In the general case, V and g depend on both x and t . In the body of this work we shall only consider stationary cases. In Appendix A the time-dependent case is presented. Before proceeding, let us make clear that stationarity means here that there is a Galilean frame (that needs not coincide with the lab frame) in which V, g and therefore ρ 0 only depend on x . From now on we work in this 'preferred' frame. \nIn this frame, the condensate wave function has the form \nΨ 0 ( t, x ) = e -iµt/ /planckover2pi1 × √ ρ 0 ( x ) e iW 0 ( x ) , (6) \nwhere ρ 0 ( x ) gives the (mean) density of condensed atoms, µ is the chemical potential, and k 0 ( x ) = ∂ x W 0 is the condensate wave vector. \nIn the sequel, we shall work with one dimensional condensates. This means that the transverse excitations have energies much higher than the longitudinal ones, and than the interaction energy gρ 0 . The transverse excitations can therefore be ignored at sufficiently low energies. This simplifying hypothesis can be relaxed without significantly modifying the forthcoming treatment. For one dimensional stationary condensates, the continuity equation reduces to \nρ 0 ( x ) v ( x ) = cst . (7) \nwhere v ( x ) = /planckover2pi1 k 0 ( x ) /m is the velocity of the condensate, and where x is the longitudinal coordinate. Plugging Eq. (6) in Eq. (5) gives \nµ = [ mv 2 ( x ) 2 + ρ -1 / 2 0 Tρ 1 / 2 0 + V ( x ) + g ( x ) ρ 0 ( x ) ] . (8) \nBecause of Eq. (7), the nonhomogeneity of the background can be characterized by only two functions. We shall use v ( x ) and the x -dependent speed of sound \nc 2 ( x ) = g ( x ) ρ 0 ( x ) m , (9) \nbecause the equation for the relative fluctuations only depend on these functions.", 'C. Bogoliubov-de Gennes equation': "We show that the relative fluctuations obey, at the linear order, a fourth order equation which does not involve \nthe external potential. Inserting Eq. (4) in Eq. (2), and linearizing the equation in ˜ φ , one gets \ni /planckover2pi1 ∂ t ˜ φ = T + V +2 g | Ψ 0 | 2 ] ˜ φ + g Ψ 2 0 ˜ φ † . (10) \n[ \n] Given the structure of this equation and Eq. (8), we found that it is mathematically more convenient to work with the relative fluctuation φ defined by \nΨ = Ψ 0 (1 + φ ) . (11) \nThen using Eqs. (6,9), one gets \ni /planckover2pi1 ( ∂ t + v∂ x ) φ = T ρ φ + mc 2 φ + φ † ] , (12) \n[ \n] where we have introduced the 'dressed' kinetic operator \nT ρ = -/planckover2pi1 2 2 mρ 0 ∂ x ρ 0 ∂ x = -/planckover2pi1 2 v 2 m ∂ x 1 v ∂ x , (13) \nwhich takes into account the nonhomogeneous character of the condensate density. The last expression is obtained using Eq. (7), and is valid for stationary condensates only. \nIn this last case, the field operator can be written as a superposition of the form \nφ ω ( t, x ) = a ω e -iωt φ ω ( x ) + a † ω e -iωt ϕ ω ( x ) ) ∗ , (14) \n( \n) where a ω , a † ω are phonon annihilation and creation operators. Inserting Eq. (14) in Eq. (12), and taking the commutator with both a ω and a † ω yields \n[ /planckover2pi1 ( ω + iv ∂ x ) -T ρ -mc 2 ] φ ω = mc 2 ϕ ω , -/planckover2pi1 ( ω + iv ∂ x ) -T ρ -mc 2 ] ϕ ω = mc 2 φ ω . (15) \n[ \n] It is instructive to obtain a single equation for φ ω by eliminating ϕ ω . This can be done by dividing the first line by c 2 and acting on the resulting equation with the operator between brackets in the second line. After some manipulation, we obtain \n{ [ /planckover2pi1 ( ω + iv ∂ x ) + T ρ ] 1 c 2 [ -/planckover2pi1 ( ω + iv ∂ x ) + T ρ ] -/planckover2pi1 2 v∂ x 1 v ∂ x } φ ω = 0 . (16) \nThis equation (or equivalently Eq. (15)) is valid for all stationary 1D condensates. It contains no approximation besides those involved in the BdG equation [15, 16]. In Appendix B we analyze its properties as well as its relations with other dispersive models.", 'D. Near horizon trajectories and background profiles': "Asonic horizon can be obtained in two cases depending on whether it is c + v or c -v that crosses zero. Assuming it is c + v , the condensate flows to the left, i.e., v < 0. \nWe further require that c + v smoothly crosses zero, i.e., the following expansion is valid in a finite range: \nc + v = κx + O ( x 2 ) . (17) \nWe set to x = 0 the location of the sonic horizon, and we call the near horizon region , the range of x where the neglect of nonlinear terms in Eq. (17) is valid. \nTo verify that this profile gives rise to a black hole horizon, we can either refer to the gravitational analogy [6], or directly analyze the characteristics of Eq. (16). Adopting the second approach, we perform a long wavelength approximation to Eq. (16), i.e., we drop the T ρ terms, and consider the WKB solutions of the resulting equation. In this approximation, right ( u ) and left ( v ) moving solutions (with respect to the condensate) decouple and are governed by x -dependent momenta k u ω , and k v ω respectively. These are solutions to \nω -k ω v ( x ) = ± k ω c ( x ) , (18) \nwhere the + ( -) sign governs k u ω ( k v ω ), and where c ( x ) > 0. To get the space-time trajectories we consider the Hamilton-Jacobi equations ( dx/dt = ∂ω/∂k, dk/dt = -∂ω/∂x ). These trajectories give the locus of constructive interference when considering wave-packets. In the near horizon region, for the right movers, Eq. (18) gives \nk = k 0 e -κt , x = x 0 e κt . (19) \nBy definition, whenever such equations are obtained with κ > 0, one is dealing with a black hole horizon. When instead κ < 0, the structure of the trajectories is that of a white hole. The two cases are related by a time reversal symmetry, see Appendix D. \nIn both cases the relevant quantity is the 'decay rate' κ , given by the gradient \n∣ \nκ = d ( c + v ) dx ∣ ∣ ∣ horizon , (20) \n∣ evaluated at the sonic horizon c + v = 0. (In the general relativistic jargon it is called the 'surface gravity' (when multiplied by c ). It plays a crucial role in the laws of black hole thermodynamics [22, 23], and it fixes the temperature of black hole radiation when using relativistic fields [24].) One should also point out that the left movers, the v -modes, are hardly affected by the horizon since -c + v ∼ 2 v -κx is approximately constant in the near horizon region. \nIn the sequel, we shall work with \nc ( x ) + v ( x ) = c 0 D × sign( x ) tanh 1 n [( | κx | Dc 0 ) n ] . (21) \nThe parameter 1 > D > 0 determines the size of the near horizon region, namely | ∆ x | = Dc 0 /κ . As we shall see, it plays a critical role in the deviations with respect to the standard relativistic fluxes. The power n \nFigure 1: Upper plot: shape of the profile ( c + v ) /c 0 as a function of κx/c 0 , for D = 0 . 5 (solid line) and D = 1 (dashed line). Lower plot: separate profiles c ( x ) /c 0 and v ( x ) /c 0 , for D = 0 . 5 and n = 2. The solid lines correspond to q = 0 . 3 and the dashed lines to q = 0 . 7. Both pairs of profiles give rise to the same function c + v , represented by the solid line in the upper plot. \n<!-- image --> \ncontrols the sharpness of the transition from the near horizon behavior to the asymptotic flat regions on either side. For n → ∞ , the transition becomes sharp. As discussed in [5, 17], sharp transitions give rise to nonadiabatic effects which produce superimposed oscillations on the fluxes. In this paper, we shall work with n equal to 2, and we refer to [17] for more details about this aspect. \nGiven that the left-moving v -modes 'see' the combination c -v , and are coupled to the u -modes, we need to fix both v and c , and not only the combination of Eq. (21). To this end, we introduce a new parameter q which specifies how c + v is shared between c and v : \nc ( x ) = c 0 +(1 -q ) [ c ( x ) + v ( x )] , v ( x ) = -c 0 + q [ c ( x ) + v ( x )] . (22) \nFor q = 1 (resp. q = 0) the hole is purely due to the gradient of v (resp. c ). In [14] the analysis was carried out with q = 0, whereas in the recent experiment [25], one finds q /similarequal 0 . 7. In our numerical analysis we shall see that the value of q has an important impact on the spectrum of the Hawking radiation. The influence of D and q on the functions c ( x ) and v ( x ) is illustrated in Fig. 1.", 'III. THEORETICAL ANALYSIS': 'To prepare the numerical analysis of Eq. (15), it is worth studying the modes ( φ ω , ϕ ω ), and the Bogoliubov transformation relating the asymptotic in and out mode bases. \nWe remind the reader that for a relativistic 2D massless field, the Bogoliubov transformation induced by propa- \nFigure 2: Graphical resolution of the dispersion relation Eq. (23), for q = 1 (left plot), and q = 0 (rightplot). The straight lines represent ω -v ± k and the curves represent Ω ± ( k ). The solutions k ( ω ) are given by the abscissa of their intersections. \n<!-- image --> \ngating the field in Eq. (21) is particularly simple because left and right moving modes decouple, see e.g. [26]. When dealing with Eq. (16), one faces two novelties. First, at fixed ω , four independent solutions now exist, and secondly, the left-right decoupling is no longer exact (even in the dispersionless limit m → ∞ ). Similar features have been already confronted in [5, 17]. However, the present case is more general, since both v and c vary. It is also more complicated, because φ and Eq. (A6) are complex, whereas the field and the mode equation in the former works were both real.', 'A. Asymptotic solutions': 'In the above profiles, c and v are asymptotically constant in the regions | κx | /greatermuch Dc 0 . In these regions, the general solution of Eq. (16) is a superposition of plane waves e ikx with constant amplitudes. The asymptotic roots k ( ω ) are solutions of (see Eq. (B3)) \n( ω -v ± k ) 2 = c 2 ± k 2 + /planckover2pi1 2 k 4 4 m 2 = Ω 2 ± ( k ) . (23) \nFor ω > 0, in the subsonic region c + < | v + | , there is one real root k u ω > 0 which corresponds to a right mover and another real root k v ω < 0 which describes a left mover, see Fig. 2, where the two extreme cases q = 1 (left plot) and q = 0 (right plot) are represented. There is also a pair of complex conjugated roots (since the equation is real). The root with a negative (resp. positive) imaginary part corresponds to a growing (resp. decaying) mode to the right. \nIn the supersonic region c -> | v -| , for ω larger than a critical frequency ω max (we compute its value below) there are only two real roots, as in a subsonic flow. Instead for 0 < ω < ω max four real roots exist. This doubling of the number of real roots is illustrated in Fig. 3. Thus the pair of growing and decaying modes which existed in the subsonic flow is replaced by a couple of oscillatory modes. Such a replacement is a generic feature \nFigure 3: Graphical resolution of the dispersion relation in the supersonic region for three values of ω : 0 < ω < ω max , ω = ω max and ω > ω max . \n<!-- image --> \nof QFT in external fields, see [27]. In BEC it will occur at all horizons, for both black and white holes.', 'B. Maximal frequency ω max': "The maximal frequency ω max is the value of ω where the two extra real roots merge. It is thus reached when the straight line ω -v -k is tangent to -Ω -( k ), the negative root of Eq. (23) evaluated in the supersonic region. The corresponding value of k max is \nk max = 1 √ 2 m /planckover2pi1 √ v 2 --4 c 2 -+ | v -| √ v 2 -+8 c 2 -. (24) \nThen ω max is obtained by replacing k by k max in Eq. (23). Using Eq. (21) and Eq. (22), it is thus of the form \nω max = Λ × f ( D,q ) , (25) \nwhere the 'healing' frequency Λ is related to the healing length computed with c 0 \nξ 0 = /planckover2pi1 √ 2 mc 0 , (26) \nby Λ = √ 2 c 0 /ξ 0 . (The prefactor √ 2 has been added so that the quartic term in the dispersion relation Eq. (23) be equal to k 4 c 4 0 / Λ 2 .) In what follows, Λ (or the adimensional λ = Λ /κ ) is referred to as the dispersion scale and is used to characterize the importance of dispersion. \nThe contours of constant ω max /κ in the ( D,λ )-plane are shown in Fig. 4, for q = 0 and q = 1. One sees that q has only little influence on the value of ω max /κ when this latter is small, but the effect becomes significant for larger values. The physical consequences of this shall be seen in Secs. V C 2. Notice that for D /lessmuch 1, one has f ( D,q ) ∝ D 3 / 2 . This means that ω max can be much smaller than Λ. \nFigure 4: Contours of constant ω max /κ in the ( D,λ ) plane, for q = 0 (solid lines) and q = 1 (dashed lines). \n<!-- image -->", 'C. Mode orthonormality and mode completeness': "To proceed to the canonical quantization of φ , one needs a mode basis which is orthonormal and complete. The orthonormality is defined with respect to the conserved scalar product on the space of the solutions of Eq. (15). In terms of the doublet W i = ( φ i , ϕ i ) the scalar product takes the form [28] \n( W 1 | W 2 ) = ∫ ∞ -∞ dxρ 0 ( x ) [ φ ∗ 1 φ 2 -ϕ ∗ 1 ϕ 2 ] . (27) \nThe presence of ρ 0 in this product follows from the use of the rescaled fluctuations defined in Eq. (11). Similarly, the equal time commutator reads \n[ φ ( t, x ) , φ † ( t, x ' )] = 1 ρ 0 ( x ) δ ( x -x ' ) . (28) \nThis x -dependent measure induces no difficulty, and moreover one can get rid of it by using the non-Cartesian coordinate y defined by dy = dxρ 0 ( x ). \nWhen the condensate is homogeneous the quantization is rather straightforward and well known [16]. However, when the flow c + v possesses a horizon, the situation is more subtle. Therefore, before considering the inhomogeneous backgrounds of Eq. (21), let us quantize φ when c and v are constant both in space and time. The complete description we shall obtain transposes to a BEC that of Ref. [17].", '1. Homogeneous condensates, k -representation': "In homogeneous condensates, it is appropriate to express the field in terms of exponentials e ikx and creation/destruction operators labeled by the (real) wavevector k , \nˆ φ = ∫ ∞ -∞ dk [ ˆ a k φ k +ˆ a † k ϕ ∗ k ] , (29) \nwhere \nφ k ( t, x ) = e -iω k t + ikx √ 2 πρ 0 u k , ϕ k ( t, x ) = e -iω k t + ikx √ 2 πρ 0 v k , (30) \nand where ω k is the positive solution of Eq. (23). Using these expressions, and introducing ¯ W k = ( ϕ ∗ k , φ ∗ k ), the 'bar' doublet associated with W k = ( φ k , ϕ k ), one verifies that the orthonormality conditions \n( W k | W k ' ) = -( ¯ W k | ¯ W k ' ) = δ ( k -k ' ) , ( ¯ W k | W k ' ) = 0 , (31) \nare satisfied when the amplitudes u k , v k take their standard value [29], irrespectively of the (sub- or supersonic) value of the condensate velocity v . Explicitly, using Eq. (15) and Eq. (B3), one obtains \nv k = D k u k , | u k | 2 -| v k | 2 = 1 , D k = 1 mc 2 [ /planckover2pi1 √ c 2 k 2 + /planckover2pi1 2 k 4 4 m 2 -( mc 2 + /planckover2pi1 2 k 2 2 m ) ] . (32) \nThe fact that D k is independent of the condensate velocity v follows from Galilean invariance, the condensate being homogeneous. Indeed, in the coordinates t c , x c comoving with the fluid, related to the coordinates t, x by t c = t , x c = x -vt , the 2D wave vector of components ( ω k , k ) in the t, x system has comoving components (Ω k = ω k -vk, k ) where all reference to v drops out when using k to label modes, as can be seen from Eq. (23). \nThe operators ˆ a k , ˆ a † k obey the usual bosonic commutation relation [ˆ a k , ˆ a † k ' ] = δ ( k -k ' ). The relationships between mode doublets W k and these operators follow from \nˆ a k = ( W k | ˆ W ) , ˆ a † k = -( ¯ W k | ˆ W ) , (33) \nwhere the doublet operator is ˆ W = ( ˆ φ, ˆ φ † ). \nIn this k -representation, one can also easily verify that the mode basis is complete. That is, starting with the commutators [ˆ a k , ˆ a † k ' ] = δ ( k -k ' ) and the doublets W k , making use of the completeness (in the sense of Fourier analysis) of the exponentials e ikx with k real from [ -∞ , ∞ ], and using that D k = D -k (which is a necessary condition) one verifies that Eq. (28) is satisfied, again irrespectively of the value of the condensate velocity v . \nTherefore, when using ω instead of k to label modes and operators, one should discard the growing and the decaying solutions of Eq. (16). Unlike the oscillatory plane waves they cannot be reached in a limiting procedure, starting with square integrable functions.", '2. Nonhomogeneous condensates, ω -representation': "When considering nonhomogeneous but stationary flows, one should use the conserved frequency ω to label modes. Then one notices that the change of variable k → ω in Eq. (29) proceeds very differently according to the sub- or supersonic character of the flow. As a preliminary step, we consider separately sub- and supersonic homogeneous flows. A superscript u ( v ) shall be added to characterize right (left) movers with respect to the condensate. \nIn a homogeneous subsonic flow, only two real roots of Eq. (23) exist: k u ( ω ) > 0 and k v ( ω ) < 0. Given that dk/dω never crosses zero, one can re-express Eq. (29) as \nφ ( t, x ) = ∫ ∞ 0 dω [ e -iωt ˆ φ ω ( x ) + e + iωt ˆ ϕ † ω ( x ) ] , (34) \nwhere \nˆ φ ω ( x ) = ˆ a u ω φ u ω ( x ) + ˆ a v ω φ v ω ( x ) , ˆ ϕ ω ( x ) = ˆ a u ω ϕ u ω ( x ) + ˆ a v ω ϕ v ω ( x ) . (35) \nThe rescaled modes and operators are φ ω = φ k √ dk/dω , and ˆ a ω = ˆ a k √ dk/dω . One easily verifies that the factors √ dk/dω guarantee that all δ ( k -k ' ) obtained in the former subsection are consistently replaced by δ ( ω -ω ' ). That is, the operators ˆ a ω , ˆ a † ω obey [ˆ a ω , ˆ a † ω ' ] = δ ( ω -ω ' ). Similarly the modes φ u ω and φ v ω are orthogonal to each other, and possess unit positive norm (in the sense of a Dirac distribution δ ( ω ω ' )). \nWe now consider a homogeneous supersonic flow. Starting again from Eq. (29) one decomposes, as in Eq. (34), the field as an integral over ω of a sum of right- and left-moving modes. When considering the leftmoving sector in left-moving flows, dk/dω does not cross zero. Therefore, as in subsonic flows, all left-moving (positive norm) modes can still be monotonically labeled by ω belonging to [0 , ∞ ]. The same is no longer true for the right-moving sector. In fact, the integral ∫ ∞ 0 dk splits into an integral over ω belonging to [0 , ∞ ] plus another piece over negative frequencies belonging to [ -ω max , 0]. This last interval terminates at -ω max where the two new real roots k u ( ω ) > 0 merge. \n- \nThus, for ω > ω max , ˆ φ ω , ˆ ϕ ω read as in Eq. (35) since one has only one (positive norm) u -root. Instead, when 0 < ω < ω max , three real u -roots exist: the continuation (in ω ) of this positive norm one, plus two new roots with \nnegative Ω, see Fig. 3. In this case, one has \nˆ φ ω ( x ) ≡ ∫ dt 2 π e iωt φ ( t, x ) = ˆ a u ω φ u ω ( x ) + ˆ a v ω φ v ω ( x ) + ∑ l =1 , 2 ˆ a u † -ω,l [ ϕ u -ω,l ( x ) ] ∗ , (36) \nˆ ϕ † ω ( x ) ≡ ∫ dt 2 π e -iωt φ ( t, x ) = ˆ a u † ω [ ϕ u ω ( x )] ∗ +ˆ a v † ω [ ϕ v ω ( x )] ∗ + ∑ l =1 , 2 ˆ a u -ω,l φ u -ω,l ( x ) . (37) \nIn the first equation, a complex conjugate and a subscript -ω have been used to characterize the two new modes. This means that the two doublets W u -ω,l = ( φ u -ω,l , ϕ u -ω,l ) have a positive norm and obey Eq. (15) with a frequency i∂ t = -ω < 0. It should also be noticed that the above operators ˆ φ ω ( x ) , ˆ ϕ ω ( x ) contain both annihilation and creation sectors. This allows one to write φ ( t, x ) as in Eq. (34), i.e. as an integral over ω [0 , ∞ ]. \n∈ \n∞ In metrics which contain a transition from a subsonic to a supersonic flow, because of the scattering on v ( x ), the modes that are bounded on one side of the horizon are generally not bounded on the other side. However, given the linearity of Eq. (15), one can always construct bounded modes as linear combinations of the above modes, requiring that the coefficient of the growing mode be zero [17]. \nIn fact, when ω < ω max , three independent bounded combinations can be constructed and ˆ φ ω , instead of Eq. (36), now reads: \nˆ φ ω ( x ) = ˆ a u ω φ u ω ( x ) + ˆ a v ω φ v ω ( x ) + ˆ a u † -ω ϕ u -ω ( x ) ] ∗ , (38) \n[ \n] and similarly for ˆ ϕ ω ( x ). In this expression, φ u ω , φ u -ω and ϕ u -ω stand for yet unspecified, normalized bounded modes. The particular cases of the in and out bases shall be defined in the next section. For ω > ω max instead, there is one growing mode on each side of the horizon. Hence only two independent bounded linear combinations can be constructed, so that ˆ φ ω is still decomposed as in Eq. (35).", '1. Vacuum instability and spontaneous pair production': "Since we are dealing with a stationary situation, the energy of the state of φ is constant. Hence, one might have thought that no phonons could possibly be spontaneously emitted. This is not the case, because the vacuum is unstable against the production of phonon pairs when two conditions are met. First, negative energy excitations must exist. We saw in the last section that it is the case when the flow is supersonic, for ω < ω max . \nHowever, this is not enough, as can be understood by considering a homogeneous condensate propagating in a frictionless translation faster than sound in the lab frame. One also needs spatial gradients to define unambiguously the unique 'preferred' stationary frame, and to couple the so-defined negative energy excitations to positive energy ones. When these conditions are met, the Hamiltonian of φ has the following structure \nH = ∫ ω max 0 dω /planckover2pi1 ω ( ˆ a u † ω ˆ a u ω +ˆ a v † ω ˆ a v ω -ˆ a u † -ω ˆ a u -ω ) + ∫ ∞ ω max dω /planckover2pi1 ω ( ˆ a u † ω ˆ a u ω +ˆ a v † ω ˆ a v ω ) . (39) \nFrom this expression three conclusions can be drawn. First, phonons with ω > ω max cannot participate in the pair production since no partner with the corresponding negative frequency exists. Second, both u and v positive frequency phonons participate in the vacuum instability. Which of the two channels contributes most depends on the intensity of the coupling with the negative frequency excitations. Third, the diagonalization of H in the sector 0 < ω < ω max is ambiguous because by a unitary ( ω -diagonal) Bogoliubov transformation, the above quadratic form is left unchanged. \nTherefore, one needs an additional physical criterion to remove this ambiguity. In the general (time-dependent) case no such criterion exists, and the notion of phonon is inherently ambiguous [23, 30]. However, in the present case of stationary asymptotically homogeneous condensates, there is no ambiguity to define two complete sets of modes. The first set of modes (the 'in' modes) characterizes the initial phonons propagating towards the sonic horizon. The second set, the 'out' modes, characterizes the asymptotic particle content of the scattered field configurations. Having the two sets, we can compute how they are related and how this relationship governs the decay of the vacuum.", '2. Space-time structure of in and out wave-packets': "The procedure to identify the in and out modes is standard [26]. One should construct 'broad wave packets', i.e., superpositions of stationary modes, so as to extract the asymptotic temporal behavior by looking at the stationary phase condition ∂ ω S = 0. This equation is equivalent to Hamilton's equation since the mode phase S coincides with the Hamilton-Jacobi action in the WKB approximation. From this asymptotic behavior, one identifies the modes (in fact the doublets W ω, a = ( φ ω, a , ϕ ω, a )) that are associated with each initial (and final) onephonon state. It should also be pointed out that these modes acquire a precise physical meaning when considering coherent states, see App. C. \nIt is useful to visualize these modes. Let us first describe the in mode φ u,in ω , i.e., the particular solution of Eq. (16) which contains in the past only a right-moving \nFigure 5: Schematic space-time representation of a wavepacket made out of modes φ u,in ω , for ω > ω max . At early times, the packet is purely right-moving. It is then scattered into a transmitted right-moving part and a reflected left-moving one. The dashed line represents the initially unexcited 'ancestor' of the late-time left-moving packet. \n<!-- image --> \nFigure 6: Same as Fig. 5 for ω < ω max , when pair production occurs. \n<!-- image --> \npacket. We shall do it twice, both for ω > ω max , where there is only some elastic scattering (see Fig. 5), and for ω < ω max , in the presence of pair creation (see Fig. 6). In both cases, initially, one only has the incoming branch with unit norm. At late time, when ω > ω max , one has a transmitted u -mode with amplitude T ω , and a reflected v -mode with amplitude R ω . For ω < ω max , in addition to these modes there is the negative frequency mode ( ϕ u,out -ω ) ∗ . The description of the other in and out modes is obtained without difficulty.", '3. Bogoliubov transformation': "From Eq. (39) one sees that for ω > ω max one has only two modes with positive frequency/energy. Because the condensate is not translation invariant, there is some scattering. Therefore, in and out φ modes are related by \nφ u,in ω = T ω φ u,out ω + R ω φ v,out ω , φ v,in ω = -R ∗ ω φ u,out ω + T ∗ ω φ v,out ω . (40) \nThe other modes, the ϕ u/v, in/out ω , are related among themselves by the same relations. Since there is no mixing of φ ω with ϕ ∗ ω , the conservation of the norm trivially implies | T ω | 2 + | R ω | 2 = 1. One is thus dealing with an elastic scattering between u and v modes, and T ω , R ω are the transmitted and reflected amplitudes, respectively. This is a 'trivial' transformation in the sense that there is no spontaneous pair production. The vacuum of these high frequency modes is thus stable. \nIt should be pointed out that this elastic scattering is also found, for all modes, when the flow remains everywhere subsonic. In fact, because of dispersion, the high frequency modes with ω > ω max do not see/experience the presence of the sonic horizon. This can be verified by computing the trajectories followed by wave-packets centered around some ω > ω max . \nThe situation is radically different for ω < ω max . In this case the mode mixing is nontrivial, and three equations are needed to characterize the transformation \nφ u,in ω = α ω φ u,out ω + β -ω ( ϕ u,out -ω ) ∗ + ˜ A ω φ v,out ω , φ v,in ω = α v ω φ v,out ω + B ω ( ϕ u,out -ω ) ∗ + A ω φ u,out ω , (41) φ u,in -ω = α -ω φ u,out -ω + β ω ( ϕ u,out ω ) ∗ + ˜ B ω ( ϕ v,out ω ) ∗ . \nThe coefficients are given by the overlap of the corresponding (normalized) in and out doublets, e.g. \nβ -ω = -( ¯ W u,out -ω | W u,in ω ) . (42) \nThe normalization of the coefficients then immediately follows, e.g. the first equation (together with the corresponding one for ϕ u,in ω ) gives \n| α ω | 2 + | ˜ A ω | 2 -| β -ω | 2 = 1 . (43) \nIn this expression, the minus sign comes from negative norm doublets, see Eq. (31). \nIt should be noticed that the above enlarged Bogoliubov transformation governs the general case. It applies indeed to any stationary situation when there is one type of negative frequency modes (here the u -modes φ u -ω ). It should also be noticed that in situations with two sonic horizons, such as black hole white hole pairs [25, 31, 32], the Bogoliubov transformation will be more complicated than Eq. (41).", 'IV. FLUXES, DENSITY FLUCTUATIONS, AND CORRELATIONS': "Most of the work dedicated to the Hawking effect concentrates on the particle content, or the energy content, of the outgoing flux [23, 30]. These two observables are related to | β ω | 2 of Eq. (41). However, it was also noticed that there exist Einstein-Podolski-Rosen (EPR) correlations between the outgoing particles and their partners with negative frequency ω . Unlike the energy flux, these correlations are weighted by α ω β ∗ ω and originate from interfering terms (i.e., nondiagonal in the occupation number). In the case of gravitational black holes, little attention has been given to the late time correlations because they are hidden for the external observers since the partners are trapped inside the horizon [26]. Nevertheless these correlations have well-defined properties [33]. Moreover, they extend to the past and can be revealed by sending quanta to stimulate the emission process [34, 35, 36]. \nWith the advent of acoustic black holes, the situation completely changes because one has access to both regions, and can therefore probe the EPR correlations. In this respect, acoustic black holes are similar to what is found in the homogeneous time-dependent BEC [37, 38, 39, 40], and inflationary cosmology. In that case, the primordial fluctuations, seeds of the galaxy clusters and of the temperature anisotropies in the Cosmic Microwave Background, also result from pair creation [41]. Moreover, their correlations possess a welldefined space-time structure [20] which, on the one hand, affects today's observables because both members are within our Hubble patch, and on the other hand, is very similar to that found in time-dependent BEC [38, 42]. \nWe start our analysis with the 3 occupation numbers, with and without an initial temperature. The first case is similar to that of [17], but many new features arise because of the different structure of the wave equation and because both c and v vary. The second case is completely new, as are the next sections where we relate the coefficients of Eq. (41) to both local and nonlocal density fluctuations.", 'A. Occupation numbers in the initial vacuum': "Let us first consider the in vacuum | 0 in 〉 , i.e., the state annihilated by the destruction operators ˆ a u,in ω , ˆ a v,in ω and ˆ a u,in -ω , defined by the in modes through Eq. (33). For gravitational black holes, this is the physically relevant state (for outgoing u -configurations) after a few e -folding times ∆ t = 1 /κ . In fact, because of the exponential redshift effect associated with the near horizon propagation, see Eq. (19), the transient effects due to infalling quanta are exponentially rapidly washed out. Therefore, the choice of the initial distribution of quanta does not affect the stationary properties of the outgoing flux. This is no longer necessarily true for acoustic black holes, because \nthe dispersion limits the number of e -folding times during which Eq. (19) applies [43, 44]. Thus, one should analyze each case to see if the vacuum state provides a reliable approximation. \nAssuming this is the case, the mean occupation numbers are \n¯ n vac ω = 〈 0 in | ˆ a u, out † ω ˆ a u, out ω | 0 in 〉 = | β ω | 2 , ¯ n vac , v ω = 〈 0 in | ˆ a v, out † ω ˆ a v, out ω | 0 in 〉 = | ˜ B ω | 2 , (44) ¯ n vac -ω = 〈 0 in | ˆ a u, out † -ω ˆ a u, out -ω | 0 in 〉 = | β -ω | 2 + | B ω | 2 = ¯ n vac ω + ¯ n vac , v ω . \nThese expressions follow when using Eq. (41) and Eq. (33) to express the out operators in terms of in ones. \nWhen compared with the simpler case where u and v modes decouple [26] the main novelty is that v -quanta are also produced (to the left of the horizon, in the 'inside' region when using the gravitational analogy). Their occupation number is ¯ n vac , v ω . Because of this, ¯ n vac ω , the numbers of u phonons emitted to the right is always smaller than that emitted to the left: ¯ n vac -ω . In fact, ¯ n vac -ω = ¯ n vac ω +¯ n vac , v ω tells us that in addition to the usual 'Hawking' channel, there is a new channel where one of the partner is a v -quantum. When the latter is negligible, for | ˜ B ω | 2 /lessmuch | β ω | 2 , one recovers the simpler case where ¯ n vac -ω = ¯ n vac ω .", 'B. Occupation numbers from an initial thermal state': "In a BEC, in realistic situations, there will always be some residual temperature. The order of magnitude of this temperature is given by the chemical potential µ of Eq. (8), and thus inversely proportional to the healing length ξ . Moreover, the characteristic wavelength of the condensate inhomogeneity ∼ c/κ will generally be larger than ξ . Therefore, since Hawking temperature is k B T H = /planckover2pi1 κ/ 2 π , the initial distribution of phonons could well hide the Hawking effect. \nFor concreteness, to characterize the initial state, we assume that the three distributions have the same temperature. Using Ω in , the initial value of the comoving frequency, the three initial occupation numbers ¯ n in ω , ¯ n in, v ω , ¯ n in -ω are \n¯ n in,a ω = ( e β T Ω in,a ( ω ) -1 ) -1 , (45) \nwhere β T is related to the (initial) temperature by k B T in = /planckover2pi1 /β T . This choice means that, before the scattering in the near horizon region, the temperature measured in the frame comoving with the fluid is the same for all modes. \nHowever, because of the scattering, modes sharing the same constant frequency ω mix. Hence we must use ω to characterize the initial distributions. For each type of modes, we thus need to express Ω in in terms of ω . \nThis explains the presence of the index a in the comoving frequency in the above equation. Using Eq. (23), the values corresponding to the three in modes of Eq. (41) are \nΩ in,u ( ω ) = ω -v -k u ( ω ) , Ω in,v ( ω ) = ω -v + k v ( ω ) , Ω in,u ( -ω ) = -ω -v -k u ( -ω ) , (46) \nwhere the three roots k u ( ω ) > 0 , k v ( ω ) < 0 , k u ( -ω ) > 0, are clearly seen in Fig. 2. \nGiven the initial occupation numbers, Eq. (41) fixes the final ones to be \n¯ n fin ω = ¯ n in ω + | A ω | 2 (¯ n in, v ω -¯ n in ω ) + | β ω | 2 (1 + ¯ n in -ω + ¯ n in ω ) , ¯ n fin , v ω = ¯ n in, v ω + | ˜ A ω | 2 (¯ n in ω -¯ n in, v ω ) + | ˜ B ω | 2 (1 + ¯ n in -ω + ¯ n in, v ω ) , (47) ¯ n fin -ω = ¯ n in -ω + | β -ω | 2 (1 + ¯ n in -ω + ¯ n in ω ) + | B ω | 2 (1 + ¯ n in -ω + ¯ n in, v ω ) . \nThe interpretation of these equations is clear. The first term is the corresponding initial occupation number. Then for the first two equations, the second term is due to the elastic scattering between u and v modes which adds (or subtract) particles according to the strength of the scattering, whereas the last term is due to the induced emission which involves both the partner's initial occupation number ¯ n in -ω and that of the species itself. In the third line instead, one has two induced emission terms because there are two 'pair creation' channels and no 'elastic' channel.", 'C. How to get rid of thermal noises?': 'From the first equation in (47), one can easily imagine that the Hawking radiation (HR), i.e., the spontaneous creation of pairs weighted by | β ω | 2 given in the first line of Eq. (44), might be hidden by the presence of initial distributions. See also [15] for a discussion of the consequences of three-phonon interactions. \nThere is yet another difficulty which can complicate the detection of HR, namely the possibility to distinguish right- from left-moving phonons. In the case one can, detecting HR requires that | β ω | 2 be larger than, or at least of the same order as, both ¯ n in ω and | A ω | 2 ¯ n in, v ω . The first condition could be satisfied because the initial distribution of u modes can be significantly redshifted. By this we mean that one can have Ω in,u ( ± ω ) /greatermuch ω and thus possibly Ω in,u ( ± ω ) /T in /greatermuch ω/T H , although a priori T in /greatermuch T H . The second condition might be more problematic because the v modes are hardly redshifted. Nevertheless, the inequality | A ω | 2 ¯ n in, v ω /lessorsimilar | β ω | 2 could also be satisfied because, as we shall see, in certain cases the u -v mixing is very small. Therefore, when one is able to \ndistinguish left- from right-moving phonons, it could be possible to detect Hawking radiation, even when the initial temperature is larger than Hawking temperature. In Sec. V D, we study numerically realistic situations and confirm this possibility. In the case one cannot distinguish left- from right-moving phonons, the situation is much worse. The dominant noise term would be ¯ n in, v ω , and an initial temperature larger than Hawking temperature would hide Hawking radiation. \nBefore proceeding to the numerical analysis, we study density fluctuations and nonlocal density correlations, firstly because phonon occupation numbers are not directly measurable (see however [10]) whereas density fluctuations are, and secondly because nonlocal correlations are amplified by initial thermal distributions instead of being smeared by them. \nThe reader interested in the spectral properties can read Sec. V first and go back afterwards to the next section.', 'D. Density fluctuations': "Given that φ is a complex field, there are several ways to characterize the fluctuations in a BEC: either through the density correlation function which is governed by Re φ , see Eq. (B5), or through the phase correlations governed by Im φ , or even through the crossed phasedensity correlations. In what follows we only discuss the density-density correlations as the extension to the two other types is easily made. \nTo simplify the forthcoming expressions we introduce the field operator ˆ χ = φ + φ † . In stationary cases, it can be decomposed as in Eq. (34), and in terms of the same operators as those of Eq. (38). The only change is that the wave functions φ a ω , ϕ a ω are all replaced by \nχ a ω ( x ) = φ a ω ( x ) + ϕ a ω ( x ) . (48) \nSince this correspondence applies to both the in and out sets, and since both φ a ω and ϕ a ω obey transformation of Eq. (41), the three initial χ in, a ω are also related to the three final χ out, a ω by Eq. (41). Therefore, even though ˆ χ is not a canonical field, as it does not obey canonical commutators, it can be treated as a genuine quantum field when computing correlation functions. \nThe statistical properties of the density fluctuations encoded in a given state are characterized by the anticommutator \nG in ( t, x ; t ' , x ' ) = 1 2 Tr[ ˆ ρ in { ˆ χ ( t, x ) , ˆ χ ( t ' , x ' ) } ] = ∫ ∞ -∞ dω e -iω ( t -t ' ) G in ω ( x, x ' ) , (49) \nwhere ˆ ρ in is the initial density matrix, since we work in the Heisenberg representation. In the second line we passed to a Fourier transform since we assumed that both \nthe condensate and the state are stationary (in the 'preferred' frame). \nWhen the initial state ˆ ρ in is incoherent, using the in basis to express χ , only three terms having the same structure are obtained \nG in ω ( x, x ' ) = ( ¯ n in ω +1 / 2 ) χ in, u ω ( x ) [ χ in, u ω ( x ' ) ] ∗ + ( ¯ n in , v ω +1 / 2 ) χ in, v ω ( x ) [ χ in, v ω ( x ' ) ] ∗ + ¯ n in -ω +1 / 2 ) [ χ in, u -ω ( x ) ] ∗ χ in, v -ω ( x ' ) . (50) \n( \n) This expression is valid for ω > 0; for ω < 0, one has G in -ω ( x, x ' ) = [ G in ω ( x, x ' )] ∗ since G in ( t, x ; t ' , x ' ) is real. In the above equation, the initial occupation numbers are given by \nn in , i ω × δ ij = Tr[ˆ ρ in a in, i † ω a in, j ω ] . (51) \nDue to the incoherence of ˆ ρ in , only the diagonal terms remain. Additional terms would be obtained if the state ˆ ρ in contained correlations among the initial configurations. In what follows we assume it does not (see however App. C). \nBecause of the scattering near the sonic horizon, G in ω ( x, x ' ) has a rather complicated structure, as can be seen by decomposing the in modes into out ones. In fact G in ω ( x, x ' ) encodes both local observables related to the occupation numbers of Eq. (47), and non-local ones governed by correlators such as Tr[ˆ ρ in a out, i ω a out, j ω ] with i = j . \n/negationslash", 'E. Coincidence point limit': "Far from the horizon so that the scattering is completed, i.e., when | x | /greatermuch Dc 0 /κ , one should express the in modes in terms of their asymptotic plane wave content ∼ e ik a ω x . To ease the reading, we shall call these asymptotic contributions by the corresponding wave, χ in, a ω or χ out, a ω , for which the amplitude of this contribution is unity, and, to avoid any misinterpretation, we shall add an upper index 'as' to make clear that only the unit, plane wave contribution should be kept. In terms of these, using Eq. (47), in the asymptotic right region, one gets \nG in ω ( x, x ) → (¯ n fin , u ω +1 / 2) ×| χ out, u, as ω | 2 +(¯ n in , v ω +1 / 2) ×| χ in, v, as ω | 2 , (52) \nwhereas, on the left, the 'power' is asymptotically equal to \nG in ω ( x, x ) → (¯ n fin , v ω +1 / 2) ×| χ out, v, as ω | 2 +(¯ n fin , u -ω +1 / 2) ×| χ out, u, as -ω | 2 +(¯ n in , u ω +1 / 2) ×| χ in, u, as ω | 2 +(¯ n in , u -ω +1 / 2) ×| χ in, u, as -ω | 2 . (53) \nSince the point x = x ' lives in an asymptotic region where the condensate is homogeneous, the norm \nof the asymptotic modes χ as ω is x independent. Notice also that we discarded all oscillatory terms, such as χ out, u, as ω ( χ in, v, as ω ) ∗ ∼ e i ( k u ω -k v ω ) x , because they rapidly oscillate as x → ∞ , and thus drop out when averaging over ω . \nThe interpretation of the above equations is clear. When working in the state ˆ ρ in , the two asymptotic values of the correlator are the sum of the contributions of the waves that have been scattered, governed by the final occupation numbers, \nn fin , i ω = Tr[ˆ ρ in a out, i † ω a out, i ω ] , (54) \nand of the waves which have not propagated through the horizon region, and whose occupation numbers are the initial ones given in Eq. (51). From Eq. (52) one clearly sees that the second term (the v contribution) will hide the Hawking radiation whenever ¯ n in , v is much larger than ¯ n fin , u , which is the case in realistic situations, as we shall see in Sec. V D 3. \nIt is therefore also of interest to compute \nF ( t, x ; t ' , x ' ) = i 2 ρ 0 ( ∂ x ' -∂ x )Tr [ ˆ ρ in Ψ † ( t ' , x ' ) Ψ( t, x ) ] = ∫ ∞ -∞ dω e -iω ( t -t ' ) F ω ( x, x ' ) . (55) \nIn the coincidence point limit ( t, x ) = ( t ' , x ' ), /planckover2pi1 ρ 0 F /m is the atom flux at ( t, x ). It is the sum of ρ 0 v = ρ 0 /planckover2pi1 k 0 /m , the flux of the condensed atoms, and of the integral over ω of F ω ( x, x ). In the asymptotic right region, using Eq. (14), for ω > 0, one has F ω = F dr ω + F com ω where \nF dr ω = k 0 × [ ¯ n fin ω | φ u,out, as ω ( x ) | 2 + ¯ n in,v ω | φ v,in, as ω ( x ) | 2 ] , F com ω = ¯ n fin ω k u,out ω | φ u,out, as ω ( x ) | 2 + ¯ n in,v ω k v,in ω | φ v,in, as ω ( x ) | 2 . (56) \nFor ω < 0, one has the same expressions with opposite signs, with φ ω replaced by ϕ ω , and ¯ n ω replaced by ¯ n ω +1. F dr ω arises from the uncondensed atoms and is due to the dragging of the background, while F com ω is the atom flux measured in the frame comoving with the condensate. As in the case of the density fluctuations, the term arising from the initial distribution of v -phonons largely dominates. However, since k v,in ω < 0, F com ω is related to the difference of the terms appearing in G in ω ( x, x ), up to k -dependent factors (different for each term). Thus, if one has access to both the atom flux and the density fluctuations in the right asymptotic region, there is a greater hope to have access to ¯ n fin ω . \nNote that any other local observable constructed out of two fields, like the depletion, will have the same structure in the right asymptotic region and will thus suffer from the same limitations, namely it will be largely dominated by the contribution of n in,v ω . This reinforces the interest to consider nonlocal observables.", '1. Late time entanglement': "We now study the long distance correlations, for | x -x ' | /greatermuch Dc 0 /κ > ξ 0 . In this case G in ω ( x, x ' ) displays a rich structure. Considering x > 0, x ' < 0, one gets a priori eight terms, since one has two asymptotic modes χ as on the right, and four on the left, see Eq. (36). However, many of them drop out because they destructively interfere upon integrating over ω , see the discussion after Eq. (53). Hence, when the initial state ˆ ρ in is incoherent, no (long distance) correlations among in modes exist. Similarly, all terms mixing in and out χ as modes will destructively interfere. In fact, only out phonons are entangled by the scattering in the near horizon region. Hence only correlations among asymptotic out modes will contribute. Given that in the subsonic region, there is only one asymptotic out mode, χ out, u, as ω , and in the supersonic region, two such modes exist, for x > 0, x ' < 0, one has \nG in ω ( x, x ' ) = χ out, u, as ω ( x ) × { A ω [ χ out, v, as ω ( x ' ) ] ∗ + B ω χ out, u, as -ω ( x ' ) } . (57) \nWhen both x and x ' are taken in the subsonic region, no long distance correlations can develop since χ out, u, as ω is the only asymptotic out mode. On the contrary, when both x and x ' are negative, in the supersonic region, the two asymptotic modes are entangled and this will show up in \nG in ω ( x, x ' ) = C ω χ out, v, as ω ( x ) χ out, u, as -ω ( x ' ) . (58) \nGiven the 3 × 3 character of Eq. (41), three types of late time correlations could have been expected since three different couples of out modes can be formed. \nTo compute A ω , B ω and C ω one can either express the r.h.s. of Eq. (50) in terms of out modes and identify the coefficients multipying the corresponding couple of χ out, as ω , or equally start with Eq. (49) and decompose the field χ using the out basis. Adopting the second, more rapid method, we get \nA ω = Tr[ˆ ρ in a out, u ω a out, v † ω ] , B ω = Tr[ˆ ρ in a out, u ω a out, u -ω ] , C ω = Tr[ˆ ρ in a out, v ω a out, u -ω ] , (59) \nwhere we have used the fact that the a out operators commute in each product. Unlike the coincidence point limit of G in ω ( x, x ' ) which is governed by diagonal terms, see Eqs. (52, 53), the long distance correlations are governed by terms which are nondiagonal in occupation number. This is exactly as in homogeneous situations [20, 42] and in fact will always be found when the parametric amplification (or the scattering) conserves a quantity, the \nfrequency ω here, the spatial wave-vector k in homogeneous cases. A straightforward calculation gives \nA ω = ¯ n in ω α ω ˜ A ∗ ω + n in, v ω A ω α v ∗ ω +(¯ n in -ω +1) β ∗ ω ˜ B ω , B ω = ¯ n in ω α ω β ∗ -ω + ¯ n in, v ω A ω B ∗ ω +(¯ n in -ω +1) β ∗ ω α -ω , (60) C ω = ¯ n in ω ˜ A ω ˜ β ∗ -ω + ¯ n in, v ω α v ω B ∗ ω +(¯ n in -ω +1) ˜ B ∗ ω α -ω . \nIn the vacuum, the residual correlations all involve the negative frequency modes because these enter in both pair creation channels. \nAt this point, an important observation should be made. Because the initial distributions ¯ n in, a only appear in factors multiplying terms already present in the in vacuum, the long distance correlations will not be erased by the presence of initial quanta. In fact an initial temperature will in general amplify the long distance correlations induced by interactions. 1 It was numerically observed in [14], as was the fact that the A ω coefficient, arising from the product a u ω a v † ω , is nonzero even in a subsonic flow. 2 \nWe finally remind the reader that the above correlations were obtained using the BdG equation (15) which neglects phonon interactions [15]. One might therefore worry that the entanglement is reduced upon taking into account such interactions. We refer to Refs. [45, 46, 47] for an analysis of this point in a cosmological context. In brief, the weakness of the nonlinearities guarantees that the entanglement is hardly reduced.", '2. Spatial structure of long distance correlations': "At fixed ω no spatial structure emerges from Eqs. (57, 58). To get the spatial properties of the correlations, one needs to take the inverse Fourier transform. Then, as it is the case when considering wave-packets, constructive interferences will develop along the characteristics of the mode equation. To ease the reading of the forthcoming expression we found convenient to return to Eq. (49) and to re-introduce t and t ' . \nAs usual, the constructive interference condition gives the stationary phase condition ∂ ω S = 0. In the present case, using a WKB approximation for the modes χ a ω ( x ) ∼ exp( i ∫ x dyk a ω ( y )), the three phases of the terms weighted \nby A ω , B ω and C ω are respectively \nS A ( t, t ' , x, x ' ; ω ) = -ω ( t -t ' ) + ∫ x z dyk u ω ( y ) -∫ x ' z dyk v ω ( y ) + arg(ln A ω ) , S B ( t, t ' , x, x ' ; ω ) = -ω ( t -t ' ) + ∫ x z dyk u ω ( y ) + ∫ x ' z dyk u -ω ( y ) + arg(ln B ω ) , (61) S C ( t, t ' , x, x ' ; ω ) = -ω ( t -t ' ) + ∫ x z dyk v ω ( y ) + ∫ x ' z dyk u -ω ( y ) + arg(ln C ω ) , \nwhere z is an arbitrary location where the absolute (unobservable) phase of the χ a ω is fixed. The stationary phase condition gives \n( t -t ' ) = ∫ x z dy ∂ ω k u ω ( y ) -∫ x ' z dy ∂ ω k v ω ( y ) + ∂ ω arg(ln A ω ) , (62) \nfor the first line, and similar equations for the second and third lines. It seems a priori that the choice of z matters. However this is not the case because z enters in A ω in such a way that a change of z leaves the r.h.s. of the equation unchanged. (This is because the coefficients of Eq. (59) contain the operators a † , a which are 'contravariant' with respect to a phase shift of the corresponding mode.) The physically meaningful phase that comes out of these expressions has the role of fixing the location where the interactions occur. [19] \nThis result becomes exact in the limit where x and x ' are taken far away from the scattering zone, and it amounts to put z = 0 and to to drop the last term in Eq. (62), and similarly for the equations involving B ω and C ω . There could be some finite phase shift with respect to these WKB estimates, but these do not change with x and x ' , and hence give subdominant effects in the large x limit. In this limit, the properties of the correlation pattern derived from Eq. (62) are thus independent of both the norm and the phase of the coefficient A ω . Hence the same long distance space-time pattern will be found both in the limit κ → 0 and κ →∞ , i.e. in regimes which do not give the standard Hawking radiation. Therefore if Hawking radiation implies the pattern, the converse is not true (when defining Hawking radiation as the near thermal radiation associated with a finite surface gravity κ ). \nIn the large distance limit, when putting t = t ' , the stationary phase condition applied to Eq. (61) gives, for the A ω , B ω and C ω terms respectively, \n∆ t HJ, u ω ( x ) = ∆ t HJ,v ω ( x ' ) , ∆ t HJ, u ω ( x ) = ∆ t HJ,u -ω ( x ' ) , ∆ t HJ, v ω ( x ) = ∆ t HJ,u -ω ( x ' ) , (63) \nwhere \n∆ t HJ,a ω ( x ) = ∫ x 0 dy ∂ ω k a ω ( y ) (64) \nis the time it takes the a -type phonon of frequency ω to propagate from x = 0 to x , in virtue of the HamiltonJacobi equation determining the group velocity \nv a gr ( ω ) = ( ∂ ω k a ω ) -1 . (65) \nFor each type of correlations, we see that the locus of constructive interference of, say, x ' given x , is given by the value of x ' reached at the same lapse time it takes the partner to reach x , both phonons starting their journey near the horizon x = 0. In the large x limit, the dominant contribution comes from the uniform motion outside the horizon region. Thus the above three constructive interferences occur at locations x, x ' related by \nx v a gr ( ω ) = x ' v b gr ( ω ) . (66) \nUsing Eq. (23), the asymptotic group velocities are, for x → + ∞ , \nv u gr ( ω ) = ∂ k Ω + + v + = c 2 + k 1 + k 2 ξ 2 + Ω + ( k ) + v + , (67) \nfor the right movers in the subsonic region, the 'Hawking quanta'; and, for x →-∞ , \nv v gr ( ω ) = -∂ k Ω -+ v -= -c 2 -k 1 + k 2 ξ 2 -Ω -( k ) + v -, v u gr ( -ω ) = ∂ k Ω -+ v -= c 2 -k 1 + k 2 ξ 2 -Ω -( k ) + v -, (68) \nfor the v quanta and the u partners in the supersonic region. In the dispersionless regime, for kξ ± /lessmuch 1, these group velocities are independent of ω and respectively equal to c + + v + in Eq. (67), and -c -+ v -, c -+ v -for Eq. (68). Therefore, in this regime, the correlation pattern is independent of ω , and will show up in G in ( t, x ; t ' = t, x ' ) of Eq. (49). \nThese three types of correlations have been (numerically) observed in [14]. They have been also correctly interpreted save for two aspects. First, the C ω branch has been attributed to the 'partial elastic scattering' of right movers of positive frequency. If this explanation applies to the first term, it does not to the last two which involve pair creation B ω coefficients mixing χ v ω and χ u -ω . Second, the A ω is claimed to 'originate from thermal effects'. From the first line of Eq. (60) we see that it is indeed amplified by thermal effects, but, because of the third term, it is already present in the vacuum. Being quadratic in frequency mixing coefficients, it is too weak to be easily seen in the numerical simulations. However using a black and white print of the arXiv version of [14], these correlations are quite visible in the last two \nplots in Figure 2, see Figure 6 for their orientation in the x, x ' plane. Upon contacting the authors, they agree that these correlations are indeed present at zero temperature. When preparing the revised version of this paper, we became aware of [48] which agrees with this point, as with the rest of our analysis, and which also contains additional interesting results. \nConcerning the relative amplitude of A ω , B ω and C ω it should be noticed that, in general, there is no clear ordering. Instead, when the u -v mixing is weak, the Bogoliubov coefficients A ω , B ω are much smaller than β ω (see Sec. V C 4) and therefore, B ω is the largest. In this regime, one recovers the properties of the B ω branch relating the u modes across the horizon that have been known for a while. In the context of gravitational black holes, using relativistic fields, they can be found in [26, 33]. As of acoustic black holes, it was understood in [4] that the late time behavior of these correlations is essentially unaffected by dispersive effects, 3 see also [43, 44]. Finally, that the B ω correlations determine (in the hydrodynamical limit) the long distance density correlations in BECs was stressed in [13]. \nIn App. C we present an alternative and simple way to characterize the correlation pattern. It consists in sending classical waves -described by highly excited coherent states- towards the horizon. This approach is worth considering because it allows to relate the above study of Eq. (49) both to the experiments in hydrodynamics described in [21], and to the wave packet analysis of [4].", 'A. Numerical procedure': 'The numerical procedure we used was elaborated from that of [17]. The wave equation of that reference was replaced by the BdG equation Eq. (15), where the velocity v and the sound speed c both vary, see Eq. (22). The extraction of the Bogoliubov coefficients from the numerical solutions to Eq. (15) took into account the specific normalization of the modes ( φ ω , ϕ ω ). For a presentation of the procedure itself, we refer to [17].', 'B. Value of the parameters': 'Let us determine the number of free parameters and their realistic ranges. 4 A typical value for the average sound speed c 0 is \nc 0 = 0 . 15 cm · s -1 . (69) \nAssuming that the condensate is made out of 85 Rb, the mass of the atoms is \nm = 1 . 5 × 10 -25 kg . (70) \nThis yields the healing length, \nξ 0 = /planckover2pi1 √ 2 mc 0 = √ 2 c 0 Λ /similarequal 3 . 3 × 10 -5 cm . (71) \nThe distance over which the variation of the sound speed and flow velocity takes place, that is, the distance separating the asymptotic regions where these speeds are constant, cannot be smaller than a few healing lengths. To reduce the number of free parameters, we assume in this work that it is of the order of 10 ξ 0 (see however Sec. V C 6). Then, given our parameterization Eq. (21), the gradient at the horizon is \nκ = c 0 D 5 ξ 0 . (72) \nWith Eq. (71), this yields a relationship between D and λ = Λ /κ : \nλ /similarequal 7 D . (73) \nas well as an expression for the Hawking temperature: \nT H = /planckover2pi1 κ 2 πk B /similarequal D × 1 . 1 nK , (74) \nwhere we have used Eq. (72) and the numerical values above. We choose as free parameters D and q . Thus Eq. (73) fixes λ . \nFairly large relative variations around the average sound speed c 0 can be achieved experimentally, for instance by means of a Feshbach resonance that modifies the coupling constant g along the flowing BEC. Hence we shall consider values of D from 0 . 1 to 0 . 7 (higher values are not excluded experimentally, but proved difficult to reach with our code). The corresponding values of λ go from 70 to 10, while the order of magnitude of Hawking temperature is \nT H /similarequal 0 . 1 -0 . 8 nK . (75) \nIn general there will be a variation of both the sound speed and the flow velocity. Which one varies most depends on the experimental conditions. In [14], q was taken to be zero. In the following, the typical value of q will be 0 . 3. In Sec. V C 3, we shall nevertheless explore the whole range 0 < q < 1. \nIn realistic conditions a condensate has an effective temperature approximately fixed by the chemical potential µ /similarequal mc 2 0 . To be able to quantitatively compare Hawking temperature with this effect, we define \nT ξ 0 = mc 2 0 k B = /planckover2pi1 c 0 ξ 0 k B /similarequal 30 nK , (76) \nand in the following we consider initial temperatures (entering Eq. (45)) \nT in = τ T ξ 0 , (77) \nwith τ ranging from 1 / 3 to 1, so that T in goes from 10 nK to 30 nK. Equation (76) together with (71) yields \nT in T H = τ × √ 2 πλ. (78) \nIf λ is constrained by Eq. (73), this can be rewritten as \nT in T H /similarequal 30 τ D . (79) \nThe fact that T in is about two orders of magnitude higher than the Hawking temperature will surely complicate measuring the spectral properties of the Hawking radiation. \nIn the following sections, we start by studying these spectral properties assuming the condensate has zero temperature. The analysis when there is an initial temperature is then performed in Sec. V D. Our main goal is to understand how the adimensional parameters D,λ,q affect the fluxes. To this end it proves very convenient to work with the rescaled frequency ω/κ and rescaled energy flux f ω defined below.', 'C. Spectral properties of HR at zero temperature': 'Let us first study the energy flux of positive frequency u quanta on the right of the horizon, i.e., what corresponds to Hawking radiation. We denote by F the total energy flux, and we define the energy flux density as: \nf ω = 2 π k B T H dF dω = 2 π ω κ | β ω | 2 . (80) \nThe factor 2 π/k B T H is here for convenience, so that f ω be dimensionless and normalized to 1 at ω = 0 when the occupation number n ω = | β ω | 2 is the standard, Planckian one with temperature T H . To characterize the deviations from the standard flux, it is also convenient to use the effective temperature T ω defined as \nn ω = 1 exp( /planckover2pi1 ω/k B T ω ) -1 . (81)', '1. Typical spectra': 'In Fig. 7 f ω and T ω /T H are represented versus ω/κ for D = 0 . 1, 0 . 4 and 0 . 7. The corresponding λ are fixed by Eq. (73) and are respectively λ = 70, 18 and 10. The parameter q specifying the relative contribution of c and v to the gradient at the horizon, see Eq. (22), is fixed to 0 . 3. The three values of ω max /κ are respectively 1 . 17, 2 . 2 and 2 . 54. The energy flux and T ω quickly drop to zero when approaching ω max , as expected. Until short before ω max , T ω is nearly constant. \nNote nevertheless that this constant temperature can differ from T H : for D = 0 . 1, the asymptotic temperature when ω → 0 is equal to T 0 = 1 . 08 T H . For D = 0 . 4 and D = 0 . 7, we found T 0 = 0 . 996 T H and T 0 = 1 . 0004 T H respectively, so T 0 differs from T H only by a fraction of a percent when D is large enough. Note also that the scale separation condition λ /greatermuch 1 is not the relevant criterion to predict the importance of the deviation with respect to the standard spectrum, since λ is much larger for D = 0 . 1 than for the other two values. Instead, it is the ratio ω max /κ that controls the deviation from the standard temperature. This is confirmed in Sec. V C 2. \nThe thermality of the spectra can be characterized more precisely. To this end we define \n∆ T 0 = T 0 -T ω = k B T 0 / /planckover2pi1 T 0 , (82) \nthat measures the running of the temperature. For the three spectra of Fig. 7, we found | ∆ T 0 | /similarequal 0 . 1%. This establishes that, to a very good approximation, the energy spectra are Planckian, but truncated very close to ω max . At this point it should be stressed that to reach this result, nowhere have we used the gravitational analogy. In fact the analogy is validated by our results since it would have predicted a thermal spectrum at the standard Hawking temperature, missing however the truncation near ω max and the small running | ∆ T 0 | ).', '2. ω max /κ controls the deviation from the standard spectrum': 'In this section, in order to show that ω max /κ is the main quantity that controls the modifications with respect to the standard Planckian spectrum with temperature T H , we relax the constraint (73), and allow for arbitrary values of λ for a given D . We characterize the deviation with respect to the standard result by the relative difference \n∆ H = f ω -f H ω f H ω ∣ ∣ ∣ ω = ω H , (83) \n∣ \nas a function of ω max /κ , where f H ω denotes the Planckian energy flux density with temperature T H and where ω H = k B T H / /planckover2pi1 . \n<!-- image --> \nFigure 7: Energy flux density f ω (left plot) and effective temperature T ω /T H (right plot) versus ω/κ , for ( D,λ ) = (0 . 1 , 70), (0 . 4 , 18) and (0 . 7 , 10). q is fixed to 0 . 3. \n<!-- image --> \nFigure 8: ∆ H versus ω max /κ for D = 0 . 1 and D = 0 . 7. q is fixed to 0 . 3. \n<!-- image --> \nIn Fig. 8, for q = 0 . 3, the curves associated with D = 0 . 1 and D = 0 . 7 have similar shapes, with a maximum around ω max /κ = 0 . 6, with height 0 . 35 for D = 0 . 1 and 0 . 18 for D = 0 . 7. The deviation then decreases and in both cases becomes less than a percent as soon as ω max /κ > 2. This confirms what was found (in a different setup) in Ref. [17]. This is not the end of the story however, since as we now show, there can be significant deviations from thermality in a BEC, even when ω max /κ is large, depending on the value of the parameter q .', '3. Effect of inhomogeneity: deviations from thermality': 'In the preceding sections, we fixed q = 0 . 3, thus restricting attention to experimental cases where the variation of c + v is mainly due to the sound speed. This would be the case for instance if one used a Feshbach resonance \nFigure 9: Effective temperature as a function of ω/κ for D = 1, q = 0 . 7 and λ = 6. This situation describes approximately the experimental realization of Ref. [25]. \n<!-- image --> \nto vary the coupling constant g across the BEC. In [25], the authors report to have created a WH/BH horizon pair in a BEC using a different technique, where a local increase in the flow velocity leads to a decrease in the speed of sound. Their profile c + v is not symmetric with respect to the horizon, contrary to our parameterization, and has two horizons. Ignoring the WH horizon we can approximately describe their experimental realization within our setup. From their experimental values one gets λ /similarequal 6, D /similarequal 1, and q /similarequal 0 . 7, which yield ω max /κ /similarequal 3. Given the results of the preceding section, one expects a robust spectrum of phonon radiation. This is however not the case, as shown in Fig. 9 where the effective temperature T ω is shown as a function of the frequency. The temperature becomes equal to T H only for frequencies ω /similarequal κ . In the low frequency part the power is suppressed. From \nFigure 10: Effective temperature as a function of ω/κ for D = 0 . 4 and ( q, λ ) = (0 , 19 . 54), (0 . 5 , 17 . 15), (1 , 15 . 47). ω max /κ is the same for all curves, equal to 2 . 2. \n<!-- image --> \nthis case we learn that the precise mixture of c and v used to form the horizon significantly affects the properties of the Hawking flux and that the radiation produced in the setup of Ref. [25] should not have a thermal spectrum. \nThe effect of q is investigated more systematically in Fig. 10, for a lower value of ω max /κ so as to ensure better numerical control. In that figure, T ω /T H is plotted versus ω/κ for D = 0 . 4, and q = 0 (homogeneous BEC where only c varies), q = 0 . 5 and q = 1 (only v varies). For each value of q , λ is tuned so that we work at fixed ω max /κ to facilitate the comparison. For q = 0 and q = 1, the effective temperature varies significantly and the radiation is thus not thermal. For q = 0 . 5, as in the previous plots with q = 0 . 3, the radiation is (almost) thermal. We verified that the same results hold with other values of D and λ , with the same particular role of q = 0 . 5. \nThe strong effect of q on the Hawking flux can be understood as follows. Contrary to the setup studied in [17] where the u and v sectors were completely decoupled in the dispersionless limit, there is no such decoupling in the present settings. The creation of left-moving quanta and the elastic scattering of u quanta into v quanta will thus be large, even when the dispersion plays no role. The above results thus suggest that for q close to 0.5, the u -v mixing is small and gets larger for extreme values of q near 0 and 1. To confirm this, we now turn to the properties of the flux of left-moving quanta, ¯ n v ω .', '4. Mixing of u and v quanta': 'Figure 11 shows the particle flux of v quanta ¯ n v ω and the elastic scattering coefficient | A ω | 2 as a function of ω/κ for the same parameters as in Fig. 10. \nAt low frequencies, | A ω | 2 is nearly constant. It is two orders of magnitude smaller for q = 0 . 5 than for the other two values. It should also be noticed that | A ω | 2 does not vanish for ω → ω max since both φ u ω and φ v ω remain welldefined above ω max so that | A ω | 2 connects smoothly to \nFigure 11: | A ω | 2 (upper plot) and n v ω versus ω/κ . The parameters are identical to those of Fig. 10. \n<!-- image -->', 'R ω | 2 at ω = ω max .': '| Near ω max , ¯ n v ω goes to zero in the three cases. Instead, the low frequency behavior of ¯ n v ω changes dramatically depending on the value of q . First, the curves corresponding to q = 0 and q = 1 join at low frequencies, and are proportional there to 1 /ω . This behavior in 1 /ω means that the energy flux /planckover2pi1 ω ¯ n v ω carried by the v quanta is constant and nonvanishing at low frequencies. Secondly, for q = 0 . 5, ¯ n v ω stays everywhere below 10 -3 and is proportional to ω at low frequencies. This proportionality with ω was also obtained in the different setup of Ref. [17], and seems to indicate that the case q = 0 . 5 (and not q = 1 as one would have thought naively) is effectively similar to the setup of that reference. Even though this particular point is most probably an artefact of Eq. (22) where v and c follow the same function, the qualitative conclusion that the u -v mixing is lower when the horizon is formed by a variation of both v and c rather than of only one of them, should hold experimentally. We emphasize this point because a small u -v mixing guarantees that the Hawking B type of correlations is the largest, as explained at the end of Sec. IV F 2. \nFigure 12 shows the influence of D on the u -v mixing coefficients ¯ n v ω and | A ω | 2 , for q = 0 . 3. Both the creation of v quanta and the elastic scattering coefficient A ω are significantly affected by D and increase by more than one order of magnitude between D = 0 . 1 and D = 0 . 7. This will have important consequences when taking into account an initial temperature.', '5. Dispersionless elastic scattering': 'It is instructive to look at the behavior of | A ω H | 2 evaluated for /planckover2pi1 ω H = k B T H as a function of ω max /κ , and for fixed values of D and q . It is shown in Fig. 13. When ω max /κ is greater than about 2, | A ω H | 2 becomes nearly constant, with a value that depends on D and q . This \nFigure 12: | A ω | 2 (upper plot) and ¯ n v ω versus ω/κ , for various values of D . The legend applies to both plots. The values of λ are identical to those of Fig. 7 and q is fixed to 0 . 3. \n<!-- image --> \nFigure 13: | A ω | 2 evaluated for /planckover2pi1 ω = k B T H , versus ω max /κ for D = 0 . 1, D = 0 . 4 and D = 0 . 7. q is fixed to 0 . 3 in the upper plot and 0 . 5 in the lower one. \n<!-- image --> \nmeans that for large ω max /κ , the dispersion no longer plays any role. This is to be opposed to what was found in [17], where | A ω H | 2 scaled approximately as λ -4 . The reason is that in a BEC, the wave equation Eq. (16) does not factorize into a u and a v part in the dispersionless limit λ →∞ . Thus, there is always some elastic scattering between u and v modes. What Fig. 13 shows is that the dispersionless value of | A ω H | 2 is quickly reached, or equivalently that the λ -dependent contributions vanish rapidly for increasing λ . This λ -independent elastic scattering exists also for q = 0 . 5, whereas we have seen that in this case ¯ n v ω has a behavior similar to the corresponding one in [17]. This is somewhat surprising. \nNote finally that, since | A ω | 2 is nearly constant in ω , see Fig. 11 and Fig. 12, the value | A ω H | 2 for some ( D,q ) \nFigure 14: Effective temperature T ω divided by Hawking temperature as a function of ω/κ for q = 0 . 3 and ( D,λ ) = (0 . 2 , 3 . 5) (solid line) and (0 . 4 , 1 . 8) (dashed line) when the variation of c + v occurs on one healing length. \n<!-- image --> \nin the regime where λ plays no role actually gives | A ω | 2 for all ω .', '6. Strongly dispersive regimes': 'In Fig. 8, we saw that, starting in the robust regime ω max /κ > 2 and reducing ω max /κ , ∆ H first increases, which indicates also an increasing asymptotic temperature T 0 , and then monotonically decreases for ω max /κ /lessorsimilar 0 . 6. We refer to the latter regime as the strongly dispersive regime. It is reached when the variation of c + v occurs on much smaller distances than what is assumed in Eq. (73). \nIn Fig. 14, the variation is assumed to occur on one healing length. This amounts to change the numerical factor in Eq. (73) so that now Λ /κ = 0 . 7 /D . The effective temperature T ω is shown for q = 0 . 3 and ( D,λ ) = (0 . 2 , 3 . 5) and (0 . 4 , 1 . 8). The corresponding values of ω max /κ are 0 . 16 and 0 . 22, that is, much smaller than in Fig. 7. The shape of the two curves is remarkably similar to what was obtained in the right plot of Fig. 7: as soon as the frequency is less than about 0 . 2 × ω max , T ω is nearly constant (the running ∆ T 0 is equal to 7% and 5%) and then drops quickly to zero when approaching ω max . However, T 0 , the asymptotic value of T ω , significantly differs from the Hawking temperature: it is respectively equal to 0 . 6 T H and 0 . 75 T H . From this we conclude that when dispersion is strong, the low-frequency spectrum remains in 1 /ω as for a Planck spectrum. However since the cutoff frequency ω max is only /planckover2pi1 ω max / ( k B T 0 ) = 1 . 7 and 1 . 9 respectively, the spectrum is no longer Planck- \nFigure 15: Effective temperature T ω divided by the healing temperature of Eq. (76), as a function of ω/ω max for q = 0 . 7 and D = 1 in extremely dispersive cases, when κ /greatermuch c 0 /ξ 0 . \n<!-- image --> \ny notice that T 0 /T H becomes smaller and smaller as ω max /κ decreases. \nIt is thus interesting to consider extremely dispersive cases where c 0 κ -1 is much smaller than ξ 0 . In Fig. 15 we represented T ω /T ξ 0 as a function of ω/ω max , with D = 1, q = 0 . 3 and small values of λ from 0 . 1 to 10 -3 , corresponding to ω max /κ ranging from 4 × 10 -2 to 4 × 10 -4 . T ω still tends to a constant value T 0 at low frequencies and the low-frequency part of the spectrum still behaves as ω -1 . The ratio /planckover2pi1 ω max / ( k B T 0 ) is nearly constant, equal to 1 . 6 for all three curves. The running is large, of the order of 12%. An important result is that the ratio T 0 /T ξ 0 saturates at a constant value as ω max decreases. 6 This demonstrates that κ becomes irrelevant and that T 0 is fixed only by ξ 0 . We have verified that this result also holds for other values of D and q , but with different asymptotic values for T 0 /T ξ 0 .', '1. Energy spectrum': "As pointed out in Secs. IV B and V B, the residual temperature of a condensate is expected to be about two orders of magnitude higher than T H . The spectra \nof the previous sections are thus unlikely to be observed as such and it is necessary to include the effects of an initial temperature. This is easily done using Eq. (47) and the numerical values of the coefficients of the Bogoliubov transformation. Assuming that the three initial occupation numbers are characterized by a common comoving temperature, they are given by Eq. (45). Their calculation reduces to the computation of the functions Ω in ( ω ) for the three types of modes, which is easily done by solving Eq. (23). Since Ω in, a are three nontrivial functions of ω , the initial distributions are not Planckian in ω . \nThe energy spectrum emitted to the right of the horizon, with a nonzero initial temperature, is defined as \nf fin ω = 2 π ω κ n fin ω . (84) \nIt is represented in Fig. 16 for the same set of parameters as in Fig. 7. Two values of the initial temperature are considered: a conservative one, T in = 30 nK, and an optimistic one, T in = 10nK. With the notations and the constraints of Sec. V B, they correspond to τ = 1 and τ = 0 . 3 respectively. For D = 0 . 1, D = 0 . 4 and D = 0 . 7 the ratio T in /T H is respectively 300, 75 and 43 when τ = 1, and 90, 23 and 13, when τ = 0 . 3. \nThe spectra differ greatly from those at zero temperature, and have a nontrivial behavior. Without surprise, they no longer vanish when approaching ω max because neither ¯ n in nor | A ω | 2 do. The most interesting point is that for D = 0 . 4 and 0 . 7 and the lower value of τ , the spectra follow relatively closely the zero-temperature ones until ω /similarequal κ . Thus, for frequencies below κ , large values of D , and low initial temperatures, the measure of the phonon energy spectrum to the right gives a good estimate of the zero-temperature flux (i.e., Hawking radiation). \nTo better understand these spectra, we show in Fig. 17 the contributions to the energy flux of each of the three terms in Eq. (47), along with the full spectrum, in the two extreme cases D = 0 . 1 with τ = 1, and D = 0 . 7 with τ = 0 . 3. \nIn both cases, at high frequencies, the initial distribution ¯ n in ω largely dominates the spectrum, and all we see is just the flux of the initial quanta. At low frequencies instead, the main contribution comes from the spontaneous plus stimulated emission term | β ω | 2 (1+¯ n in ω +¯ n in -ω ), called 'creation term' in Fig. 17. For D = 0 . 1, the u -v scattering term is small and never dominates, but it is never completely negligible. The contribution from | A ω | 2 ¯ n in, v ω quickly becomes comparable to the exponentially decreasing stimulated emission. \nFor D = 0 . 7, this contribution is not much smaller than the creation term at low frequencies, as could be expected from the results of Fig. 12. On the other hand, the contribution from ¯ n in ω long remains very small compared to the other two, until about ω = κ . Thus the creation+stimulated emission term is actually dominated by the Hawking pair creation effect, since ¯ n in -ω is close to ¯ n in ω . This explains the relative similarity between the spectra with τ = 0 . 3 and τ = 1 in Fig. 16, when D = 0 . 7. \nFigure 16: Energy flux emitted to the right of the horizon. q is fixed to 0 . 3. The dotted spectra are those of Fig. 7. The values τ = 1 (solid lines) and τ = 0 . 3 (dashed lines) correspond respectively to an initial temperature T in = 30 nK and T in = 10nK, see Eq. (77). \n<!-- image --> \nFigure 17: Different contributions to the energy flux to the right of the horizon. Left plot: D = 0 . 1, τ = 1. Right plot: D = 0 . 7, τ = 0 . 3. \n<!-- image --> \nIn brief, the main lesson from these plots is that, in general, there is no clear hierarchy between the various contributions in ¯ n fin ω , which makes the interpretation of a measurement highly nontrivial. Nevertheless, let us try to identify what the optimal conditions are for such a detection.", '2. Optimal experimental conditions': 'The comoving frequency Ω of the right-moving quanta is strongly redshifted in the black hole geometry, which implies that the initial Ω u,in ( ω ) is much larger than ω . Since the integrated redshift increases with λ , one could ask whether lowering the Hawking temperature with respect to the initial temperature T in , with a fixed value of D , could not improve the results above. Indeed, if the increase of the redshift were such that Ω in /T in would grow with λ , the initial occupation numbers n in ω and n in -ω would decrease exponentially, and one could hope to have the stimulated plus creation term dominate over the other two on a larger interval of frequency ω . This is not the case however, as we now explain. \nThe initial proper frequency Ω in ( ω ) /κ for the three types of modes is shown in Fig. 18, for D = 0 . 4, q = 0 . 3, and different values of λ . As expected, at a given ω , Ω in,u /κ is an increasing function of λ . However it scales only as λ , and since the ratio T in /T H also is proportional to λ , Ω in,u /T in hardly changes. On the other hand, the left movers suffer almost no redshift, and Ω in,v has almost no dependence on λ . Thus, Ω in,v /T in decreases as λ -1 and the initial occupation number of the v modes increases as λ for a given ω . \nThese remarks are summarized in Fig. 19, where the occupation numbers ¯ n in ω of Eq. (45) are represented for the same parameters as in Fig. 18. ¯ n in ω and ¯ n in -ω are almost equal and, more importantly, do not change with \n<!-- image --> \n<!-- image --> \nFigure 18: Initial proper frequency Ω in /κ as a function of ω/κ for the u, ω (left plot), u, -ω (middle plot) and v, ω modes (right plot), for D = 0 . 4, q = 0 . 3 and τ = 0 . 3, and different values of λ . The legend applies to all plots. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 19: Initial occupation numbers n in ω (left plot), n in -ω (middle plot) and n in,v ω (right plot) versus ω/κ , for the same values of the parameters as in Fig. 18. \n<!-- image --> \nλ at low frequencies. ¯ n in,v ω on the other hand explodes when λ increases. Since | A ω | 2 does not depend on λ in first approximation, the consequence is that the scattering contribution in ¯ n fin ω dominates at low frequencies when λ increases, since the other contributions remain unchanged. In conclusion, there is no gain in lowering T H . \nIn fact, as can be seen from the figures of the previous section, the optimal conditions to be able to measure directly the zero temperature flux f ω are reached when there is a large relative variation of c + v on the smallest possible distance, that is the largest possible D with the smallest possible λ . Remember also that c and v should contribute nearly equally to the variation for | A ω | 2 to be as small as possible.', '3. Density fluctuations and atom flux': "As explained in Sec. IV C, the flux f ω studied in the previous sections is observable only if one can distinguish experimentally between left- and right-moving phonons. It is thus important to address the question whether the easily accessible density fluctuations also contain some signature of the particle creation process. The function ρ 0 vG in ω of Eq. (52), evaluated in the coincidence point limit in the right asymptotic region, is represented in the upper plot of Fig. 20 for D = 0 . 7, q = 0 . 3 and τ = 0 . 3. This quantity is adimensional, does not depend explicitly on ρ 0 and c 0 and is proportional to the power spectrum of the density fluctuations. In the lower plot, F com ω , defined in Eq. (56) and adimensionalized by multiplication with ρ 0 vc 0 /κ , is represented, for the same set of parameters. \nFigure 20: Upper plot: adimensionalized (by multiplication by ρ 0 v ) power spectrum of density fluctuations as a function of ω/κ . Lower plot: adimensionalized comoving atom flux as a function of ω/κ . In both plots, D = 0 . 7, q = 0 . 3 and τ = 0 . 3, and G in ω and F com ω are evaluated in the right asymptotic region, in the coincidence point limit. 'HR related' refers to the contribution weighted by n fin ω and 'initial excitations' to the one weighted by n v,in ω . \n<!-- image --> \nIn both cases, the contributions governed respectively by n fin ω (referred to as 'HR related' in the figure) and by n in,v ω have been plotted separately. They are of the same order of magnitude but for ω /lessmuch κ , that is, in the region where n fin ω is mainly due to Hawking radiation, see Fig. 16, the contribution from the initial excitations is greater by a factor of about 2 in the density fluctuations, and of up to 6 in the atom flux. This confirms the remark made in the theoretical analysis that no clear signature of Hawking radiation can be observed in these local observables. However, as pointed out in Sec. IV E, one could hope to combine both observables to extract n fin ω .", 'VI. CONCLUSIONS': "In this work, we presented a complete description of the scattering of the phonon modes propagating in stationary, one dimensional condensates that possess a sonic horizon. Our description is based on the BdG equation, and the scattering in the horizon region is expressed in terms of 3 × 3 Bogoliubov transformations relating asymptotic one phonon modes. \nWe have shown that this scattering affects two types of observables, local ones such as the density fluctuations, and the long distance correlations. The former are governed by expectation values which are diagonal in the occupation number, see Eq. (52), whereas the latter are determined by interference terms, Eq. (59), which reveal the entangled nature of the final state. \nWhen taking into account the condensate temperature, \nthe local observables are in general dominated by the initial distributions, whereas nonlocal correlations are amplified by these initial distributions without having their pattern modified (when the initial state is uncorrelated). Therefore the latter probably constitute the clearest indication that the Hawking effect is taking place. However, it is worth noting that the spatial structure of the pattern is not specific to HR strictly speaking, i.e. to a nearly Planckian spectrum, since it is determined before all by the structure of the 3 × 3 transformation of Eq. (41) rather than by the value of its coefficients. In fact, similar patterns are found when considering the scattering of classical waves, as explained in App. C, and in the limiting cases κ →∞ and κ = 0 (with nonvanishing higher derivatives of c + v at x = 0 for the latter) where the spectrum is no longer approximately Planckian (as we verified). \nIn the last part, we numerically integrated the BdG equation, and obtained the occupation numbers of the three kinds of phonons, both with and without an initial temperature. The main results are the following. \nFirstly, in the initial vacuum and when the u -v mixing is small ( q /similarequal 0 . 5), the spectrum of the outgoing (right-moving) phonons is Planckian, with a temperature determined by the gradient κ of Eq. (20), as soon as ω max /κ /greaterorsimilar 2 and for frequencies ω < ω max , see Fig. 7. This result, derived directly from the BdG equation (without further approximation), makes precise the domain of validity of the analogy between relativistic fields in black hole metrics and a phonon field in the corresponding 'dumb hole'. When ω max /κ < 2, the analogy fails even for the lowest frequencies, see Fig. 14, which invalidates the condition usually found in the literature that the analogy is applicable for wavelengths greater than the healing length. When the u -v mixing is not negligible, the spectrum deviates from the Planck spectrum, see Figs. 10 and 11. This is due to grey body factors (that can also be computed to a good approximation using the gravitational analogy [49]). \nSecondly, as for the detection of the analog HR in a BEC, if one can distinguish left- from right-moving phonons, the Hawking flux could be observable when the relative variation of c + v is large and happens on a small number of healing lengths. Instead, if they cannot be distinguished, local observables are completely dominated by the initial temperature of the condensate. One must then resort to nonlocal observables such as densitydensity correlations [13, 14]. \nThirdly, in App. D, we established the relationship between the fluxes emitted by white and black hole flows. Because of the blueshift effect in WH flows, the correlation pattern is more amplified by the presence of initial quanta than in BH flows and should therefore be easier to detect. This could well be the optimal case for observing HR in the laboratory through the correlation pattern.", 'Acknowledgments': "R.P. is grateful to Roberto Balbinot for many stimulating discussions over the last years, as well as his comments on an early version of this paper. We both would like to thank the participants of the workshop 'Towards the observation of Hawking radiation in condensed matter systems' (http://www.uv.es/workshopEHR/) held at IFIC in Valencia in February 2009 for interesting remarks following the presentation of results contained in this work. We are also grateful to Jeff Steinhauer for comments about his work [25] and BEC in general, and to Iacopo Carusotto for comments on App. C.", 'Appendix A: NONSTATIONARY CASE': 'In this appendix we consider nonstationary condensates. As we shall see, little is modified when compared with the stationary case studied in the body of the paper. To ease the comparison we still work with the one dimensional case. The extention to the three dimensional case is trivial. \nIn the mean field approximation, the condensate is now described by \nΨ 0 ( t, x ) = ρ 0 ( x, t ) e iW 0 ( t,x ) . (A1) \nThis wave satifies Eq. (5) where V , and g depend both on x and t . (As pointed out in [14], some tuning of V and g might be necessary not to produce time-dependent effects that might hide the Hawking radiation.) The conservation of the number of atoms follows from Eq. (5), and gives the continuity equation \n√ \n∂ t ρ 0 + ∂ x ( ρ 0 v ) = 0 , (A2) \nwhere v ( t, x ) = /planckover2pi1 k 0 ( t, x ) /m is the velocity of the condensate, and k 0 ( t, x ) = ∂ x W 0 ( t, x ) its wave vector. Plugging Eq. (A1) into Eq. (5) and using Eq. (A2) still gives Eq. (8) where the chemical potential is replaced by /planckover2pi1 ω 0 ( t, x ) = -/planckover2pi1 ∂ t W 0 . Then, because of Eq. (A2), the condensate is still characterized by v ( t, x ) and the speed of sound \nc 2 ( t, x ) = g ( t, x ) ρ 0 ( t, x ) m . (A3) \nTo describe the phonon modes, as in Sec. II, it is convenient to work with the relative fluctuation φ defined in Eq. (11). Then, using Eqs. (A1, A3), one still gets Eq. (12). To get the c -number modes which shall be used to proceed to the second quantization, we look for a complete orthonormal family of doublets W j = ( φ j , ϕ j ), where the index j now does not correspond to a conserved quantum number. The scalar product with respect to which orthonormality is defined is still given by Eq. (27), with ρ 0 ( x ) replaced by ρ 0 ( x, t 0 ). (The choice of the time t 0 when the product is evaluated does not \nchange its value since it is conserved.) This family enters φ as in Eq. (14): \nφ ( t, x ) = ∑ j [ ˆ a j φ j ( t, x ) + ˆ a † j ( ϕ j ( t, x )) ∗ ] , (A4) \nwhere ˆ a j and ˆ a † j are annihilation and destruction operators, with which one can construct the phonon Fock space. However, contrary to the case studied in the text, there is in general no clear interpretation of the vectors in this Fock space in terms of particle content, and the notion of phonons is inherently ambiguous. The notion of phonon can be recovered if the condensate is stationary in the asymptotic past and the asymptotic future. In that case, as in the text, one can define two Fock spaces, the in one and the out one. \nInserting Eq. (A4) in Eq. (12) and taking the commutator with ˆ a j and ˆ a † j yields: \n[ i /planckover2pi1 ( ∂ t + v∂ x ) -T ρ -mc 2 ] φ j = mc 2 ϕ j , [ -i /planckover2pi1 ( ∂ t + iv∂ x ) -T ρ -mc 2 ϕ j = mc 2 φ j . (A5) \n[ As in the body of the paper, one can eliminate ϕ j and obtain \n] \n{ [ /planckover2pi1 ( ∂ t + v∂ x ) -iT ρ ] 1 c 2 [ /planckover2pi1 ( ∂ t + v∂ x ) + iT ρ ] +2 mT ρ } φ j = 0 . (A6) \nWe notice that all kinetic terms are of the form of Eq. (13). This is related to the fact that T ρ is self-adjoint when using the scalar product of Eq. (27).', '1. Eikonal approximation and importance of ordering': 'It is worth exploring the properties of Eq. (16), the stationary version of Eq. (A6). We first notice that \nc 2 [ /planckover2pi1 ( ω + iv ∂ x ) + T ρ ] 1 c 2 = [ /planckover2pi1 ( ω + iv ∂ x ) + T ρ ] + c 2 [ ( i /planckover2pi1 v ∂ x + T ρ ) , 1 c 2 ] . (B1) \nThis makes explicit that the spatial gradient of c 2 affects Eq. (16) only through a commutator. We also notice that ϕ ω , the other mode of Eq. (14) obeys \n{ [ -/planckover2pi1 ( ω + iv ∂ x ) + T ρ ] 1 c 2 [ /planckover2pi1 ( ω + iv ∂ x ) + T ρ ] -/planckover2pi1 2 v∂ x 1 v ∂ x } ϕ ω = 0 . (B2) \nThe only differences between the equations for φ ω and for ϕ ω come from the sign of the commutator between \nv ∂ x and c -2 . When this commutator is neglected, as it is in the WKB approximation, φ ω and ϕ ω thus obey the same equation. \nWhen working to leading order in a WKB (or eikonal) approximation, i.e., inserting φ ω ∼ e i ∫ dxk ω in Eq. (16) (or inserting ϕ ω ∼ e i ∫ dxk ω in Eq. (B2)) gives the dispersion relation in a moving fluid of velocity v , \n( ω -kv ) 2 = Ω 2 = k 2 c 2 + /planckover2pi1 2 k 4 4 m 2 = c 2 k 2 ( 1 + ξ 2 k 2 2 ) , (B3) \nwhere ξ = /planckover2pi1 / 2 mc is the healing length. We recover the quartic dispersion relation between the frequency in the comoving frame Ω = ( ω -kv ), and the wave vector k . It should be noticed that in a stationary nonhomogeneous flow, Ω and k depend on x through v ( x ) and c ( x ), whereas the frequency ω is a globally defined constant. Remember also that ω is not necessarily the lab frequency, because it is defined in the frame in which the condensate quantities only depend on x . \n√ \nReturning to Eq. (16), it is clear that its role is to fix the exact properties of the ODE obeyed by φ ω . These properties could not have been inferred starting from Eq. (B3), and applying the substitution k → -i∂ x , because this naive rule could neither predict the ordering of T , v ( x ) and c 2 ( x ) found in Eq. (16), nor the different one found in Eq. (B2).', '2. Hydrodynamical limit and Euler mode equation': 'In the hydrodynamical (dispersionless) limit, for long wavelengths with respect to the healing length ξ , one can drop the two operators T ρ in Eq. (16). Then Eq. (16) reduces to the Eulerian mode equation which describes sound waves in a moving fluid, and this with the correct nontrivial ordering of T , v ( x ) and c 2 ( x ). Let us verify this. \nThe Eulerian action is usually written in terms of the velocity potential ψ , related to the velocity fluctuation by δv = ∂ x ψ , see e.g. [43]. To make contact between ψ and the φ field, one should compare the fluctuations of Ψ described as in Eq. (11) to those written as \nδ Ψ = δ ( ρ 1 / 2 e iW ) = Ψ 0 ( δρ 2 ρ 0 + iδW ) = Ψ 0 φ. (B4) \nUsing δv = /planckover2pi1 m ∂ x δW = ∂ x ψ , one obtains \nφ + φ † = δρ ρ 0 , φ -φ † 2 i = θ = δW = m /planckover2pi1 ψ. (B5) \nUsing the phase fluctuation θ = mψ/ /planckover2pi1 rather than ψ itself, the Euler action is \nS E = /planckover2pi1 2 2 m ∫ dtdxρ 0 { 1 c 2 [( ∂ t + v∂ x ) θ ] 2 -( ∂ x θ ) 2 } . (B6) \nThen, using the continuity equation, the mode equation reads \n[ ( ∂ t + v∂ x ) 1 c 2 ( ∂ t + v∂ x ) -1 ρ 0 ∂ x ρ 0 ∂ x ] θ = 0 . (B7) \nWhen working at fixed ω = i∂ t , and using vρ 0 = cst . , as announced, one recovers the dispersionless limit of Eq. (16) obtained by taking the limit T ρ → 0. \n→ Notice that Eq. (B7) is a generalization of the (dispersionless limit of the) mode equation which has been generally studied in the literature, see Refs. [3, 4, 5, 17, 43]. In those references, the equation also followed from Eq. (B6), but with the extra hypothesis that both c and ρ 0 can be approximated by constants, in which case one recovers the massless relativistic 2D mode equation.', "3. Link with Unruh's dispersive models": 'It is worth noticing that the dispersive properties of the phonons come through the two operators T ρ in Eq. (16). This is not what we would have obtained had we used the rules of [3] with a quartic superluminal dispersion. Indeed, the Eulerian action supplemented by a quartic term ( k 4 / Λ 2 ) is \nS Λ = /planckover2pi1 2 2 m ∫ dtdxρ 0 { 1 c 2 [( ∂ t + v∂ x ) θ ] 2 -( ∂ x θ ) 2 + 1 Λ 2 ( ∂ 2 x θ ) 2 } , (B8) \nand the corresponding mode equation reads, in a stationary condensate and at fixed ω , \n[ ( ω + iv∂ x ) 1 c 2 ( ω + iv∂ x ) -v ∂ x 1 v ∂ x + v Λ 2 ∂ 2 x 1 v ∂ 2 x ] θ ω = 0 . (B9) \nIn nonhomogeneous situations, the quartic term encoding the dispersion differs from that of Eq. (16), which means that θ ω ( x ) will differ from φ ω ( x ) in any nontrivial background. \nFinally, it is also worth noticing that Eq. (16) can be obtained from an action for a single field with a quadratic kinetic term, and which generalizes the Euler action: \nS φ = -/planckover2pi1 2 2 m ∫ dtdxρ 0 { 1 c 2 [( ∂ t + v∂ x + i T ρ /planckover2pi1 ) φ ] 2 -( ∂ x φ ) 2 } . (B10) \nThis action can be obtained from S = ∫ dt d 3 x ( i /planckover2pi1 Ψ ∗ ∂ t Ψ -H ), where H is given in Eq. (3), using Eq. (11), keeping all quadratic terms in φ, φ ∗ , and using Eq. (12) to eliminate φ ∗ in favor of φ . The conjugate momentum that enters the equal time commutator \n[ φ ( x ) , π ( y )] = i /planckover2pi1 δ ( x -y ) is rather unusual \nπ = -/planckover2pi1 2 m ρ 0 c 2 ( ∂ t + v∂ x + i T ρ /planckover2pi1 ) φ = i /planckover2pi1 ρ 0 φ + φ † ) = i /planckover2pi1 δρ, (B11) \n( \n) where we used Eq. (B5). It thus obeys π = -π † . Using Eq. (A4), π decomposes as \nπ ( t, x ) = ∑ j [ ˆ a j π j ( t, x ) + ˆ a † j (¯ π j ( t, x )) ∗ ] , (B12) \nwhere π j = i /planckover2pi1 ρ 0 ( φ j + ϕ j ) = -¯ π j . The conserved scalar product (which generalizes the standard Klein-Gordon one [23] and which agrees with Eq. (27)) is \n( φ 2 | φ 1 ) = i /planckover2pi1 ∫ dx ( ϕ ∗ 2 π 1 -¯ π ∗ 2 φ 1 ) . (B13) \nIt is not clear whether this alternative way of describing the phonon field presents any advantage over the original version of Eq. (15). It could nevertheless be useful to study how dispersion affects the analogy with gravitational systems. [39, 40, 50]', 'Appendix C: CORRELATION PATTERNS IN THE CLASSICAL LIMIT': "Rather than using the vacuum or a thermal state as in the text, we assume that the initial state also contains a highly excited coherent state. 7 This state can be obtained by making use of the displacement operator \nˆ D w = exp w ˆ a † ω -w ∗ ˆ a ω ) . (C1) \n( \n) Any of the three initial operators a in,j ω appearing in Eq. (51) can be used to construct the corresponding initial wave. One can also consider wave packets engendered by \nˆ D d, w = exp [∫ dω ( wd ω ˆ a † ω -w ∗ d ∗ ω ˆ a ω ) ] . (C2) \n| When ˆ ρ in , the initial state without the coherent state, is such that Tr[ˆ ρ in ˆ χ ( t, x )] = 0, the anticommutator in the presence of the coherent state separates as \nWhen imposing the normalization ∫ dω | d ω | 2 = 1, the mean occupation number of initial quanta added by ˆ D d, w is w | 2 , as it is for ˆ D w in Eq. (C1). \n| \nG ( t, x ; t ' , x ' ) = 1 2 Tr [( ˆ D d, w ˆ ρ in ˆ D † d, w ) { ˆ χ ( t, x ) , ˆ χ ( t ' , x ' ) } ] , = ¯ χ d ( t, x ) ¯ χ d ( t, x ) + G in ( t, x ; t ' , x ' ) , (C3) \nwhere G in is given in Eq. (49), and where the mean value of the field operator is \n¯ χ d ( t, x ) = Tr [( ˆ D d, w ˆ ρ in ˆ D † d, w ) ˆ χ ( t, x ) ] = 〈 0 in | ˆ D † d, w ˆ χ ( t, x ) ˆ D d, w | 0 in 〉 , (C4) \nas if the initial state was the in vacuum. (To get these equations, we have used Tr[ˆ ρ in ˆ χ ( t, x )] = 0 and the relation ˆ D † d, w ˆ a ω ˆ D d, w = ˆ a ω + wd ω , see e.g. [51].) From the decomposition Eq. (C3), we see that the correlation pattern is the sum of the pattern encoded in the state ˆ ρ in we studied in the former sections, plus the new pattern encoded in the real wave packet ¯ χ d ( t, x ) associated with the coherent state. \nAssuming that this wave packet is initially made only with ˆ a in, u ω , it is given by \n¯ χ d ( t, x ) = ∫ ∞ 0 dω [ wd ω e -iωt χ in, u ω ( x ) + c.c. ] . (C5) \nAt early times, before it enters the near horizon region, it describes a single wave packet propagating against the flow in the region where the flow is supersonic, see Fig. 6. As shown in that figure, at late time, it splits into three wave-packets, \n¯ χ d ( t, x ) = ∫ ∞ 0 dω [ wd ω e -iωt α ω χ out, u ω ( x ) + c.c. ] + ∫ ∞ 0 dω [ wd ω e -iωt ˜ A ω χ out, v ω ( x ) + c.c. ] + ∫ ω max 0 dω { wd ω e -iωt β -ω [ χ out, u -ω ( x ) ] ∗ + c.c. } . (C6) \nThe first one describes the transmitted wave. It is amplified with respect to the initial wave if ∫ dω | α ω d ω | 2 > 1 which needs not be always the case, as can be seen from Eq. (39). The second wave describes the left moving packet obtained by elastic scattering. The third wave is due to stimulated pair creation process, see the minus sign in Eq. (39). It describes the partner's wave, and is present only if d ω has non-vanishing components for ω < ω max since β -ω = 0 for ω > ω max . Notice that the Fourier component multiplying the mode e iωt χ out, u -ω is the complex conjugate of β -ω wd ω . The fact that d ∗ ω appears guarantees that when replacing d ω by d ω e iωt 0 , the partner wave with negative ω stays synchronized, i.e. it is shifted by the same lapse of time as the two other wave packets with positive ω . Notice also that the presence of the coefficient β -ω implies that the partner wave function cannot be localized in a region smaller than c/κ because β -ω ∼ e -πω/κ for ω /greatermuch κ . In this we recover the width of the correlations to the partner which is found when using the correlation function [13, 35]. \nWe can now relate the space-time pattern encoded in G in , the second term in Eq. (C3), to that encoded in ¯ χ d . Since these two terms have a completely different origin, one might a priori think that the two patterns will be radically different. However, this is not the case. \nTwo conditions must be met for the patterns to be approximately the same. The first concerns the out modes, whereas the second concerns the Bogoliubov coefficients appearing in Eq. (C6). At a given late time t , i.e. after a lapse of time ∆ t , the chosen wave-packet ¯ χ d has been scattered, the pattern encoded in ¯ χ d is given by the three values of x where the waves constructively interfere. Working far from the horizon, i.e. κ ∆ t /greatermuch 1 (see also the discussion in the paragraph after Eq. (62)), and using Eq. (64), these are the solutions of \n∆ t HJ,a ω d ( x ) = ∆ t, (C7) \nwhere ω d = ∫ dωω | d ω | 2 is the mean frequency of the wave packet. When the dispersive effects are weak at late time, as it is the case for ω d /lessmuch c/ξ 0 , see the discussion after Eq. (68), the solutions of Eq. (C7) are essentially independent of ω d . Therefore, in the large x limit, one finds the same pattern of correlations whether one considers the two-point function at some given time t as a function of x ' given x , see Eq. (63), or whether one looks for the two partner waves when using a packet which contains a branch that arrives at x at time t . \nIn the above reasoning, we have neglected the ω dependence of the Bogoliubov coefficients. If their relative phase would vary rapidly, and therefore that of z ω = β -ω /α ω as well, Eq. (C7) could receive significant corrections that might depend on the value of ω d , thereby giving different patterns for different wave packets. This is not the case. When using a massless two dimensional relativistic field, one finds that the pattern is universal as arg z ω is truly independent of ω , see e.g. Eq. (3.49) in [26]. Upon considering dispersive fields, when both ω d /lessmuch c/ξ 0 and κ /lessmuch c/ξ 0 are satisfied, one finds [4] that the late time pattern is unmodified even though the early pattern is completely different and depends on both the value of Λ = c/ξ 0 and the sub- or superluminal character of dispersion [43, 44]. These results were obtained using saddle point and WKB approximations which are both reliable in the present regime. The analysis was performed with a scalar dispersive field, but it also applies to a phonon field in a BEC because the differences between the various dispersive models, see Appendix B, are not relevant in a WKB regime. \nIn brief, we saw that the late time pattern obtained from the scattering of classical waves is (to leading order in ω/ Λ) the same as that obtained from amplifying vacuum (or thermal) fluctuations. This correspondence follows from the fact that both patterns are obtained from the (common) decomposition of in modes into out modes. Given this we can address a question which has been raised by several of our colleagues: to what extent observing the scattering of classical waves could be considered as a signature of the Hawking effect? \nA convincing signature [21] would consist in measuring the contribution of the negative frequency modes which anomalously contribute to Eq. (39), once sent a wavepacket composed only of positive ω , as it is the case in Eq. (C6). The experimental difficulty is different for BH \nand WH flows. In BH flows, one has to be sure that the initial packet contains no negative modes otherwise their final contribution will be dominated by the initial one, thereby preventing a neat measure of β -ω . The values of ω with β -ω /negationslash = 0 are such that | ω | < ω max . Given Eq. (23), they correspond to an interval of initial values for k ( ω ): \nk max = k ( -ω max ) < k ( ω ) < ¯ k max = k ( ω max ) , (C8) \nwhere \n¯ k max = k max + √ k 2 max + 4 m 2 /planckover2pi1 2 ω 2 max k 2 max . (C9) \nThe restriction to modes with positive frequency imposes k > k 0 = k (0) = 2 m /planckover2pi1 ( v 2 --c 2 -) 1 / 2 . By a numerical analysis, we found that the ratio ( ¯ k max -k 0 ) /k 0 is of the order of 20%, independently of m (or ξ ), and weakly varying with D and q . Thus in principle, one could selectively excite (e.g. by Bragg spectroscopy techniques [52, 53]) certain values of k in that interval. However in experiments, there will be a tension between the limited lifetime of the condensate and the narrowness of the packets in k space which gives broad packets in real space with long traveling times. Orders of magnitude taken from [52] and [25] indicate that there could be one wave packet satisfying all desiderata. \nFor WH flows, because of the time reversal symmetry, the situation is the opposite in that there is no difficulty to prepare an initial wave-packet containing only positive frequencies, whereas the final positive and negative frequency packets might be difficult to distinguish as they will follow similar trajectories [21].", 'Appendix D: WHITE HOLES': "So far we only considered condensate flows that possess a sonic horizon which is analogous to that of a black hole. However sonic horizons which act as a white hole are closely related to the black hole ones. Indeed, the transformation \nv ( x ) →-v ( x ) , c ( x ) → c ( x ) , (D1) \ntransforms the profile studied in the text, Eq. (21), into a white hole profile, in which the characteristics focus forward in time, see Eq. (19) with κ < 0. (This equation now applies to the left-moving modes, so that the quanta mainly produced are left-moving ones.) \nThe equivalent of Eq. (A6) in this WH profile reads: \n{ [ /planckover2pi1 ( ∂ t -v∂ x ) -iT ρ ] 1 c 2 [ /planckover2pi1 ( ∂ t -v∂ x ) + iT ρ ] +2 mT ρ } φ j = 0 . (D2) \nFigure 21: Occupation number of the v quanta for a BH (¯ n v ω , solid line) and of the u quanta for the corresponding WH under the transformation (D1) (¯ n WH , u ω , dashed line), as functions of ω/κ for q = 0 . 3 and D = 0 . 4 (upper plot) and D = 0 . 1 (lower plot). \n<!-- image --> \nFigure 22: Relative difference ˛ ˛ | β ω | 2 -| β -ω | 2 ˛ ˛ / | β ω | 2 as a function of ω . Same parameters as in Fig. 21. \n<!-- image --> \nUnder a time-reversal t → -t , Eq. (D2) becomes the complex conjugate of Eq. (A6). This proves that there is a one-to-one mapping given by: \nT : φ BH j ( t, x ) ↦→ φ WH j ( t, x ) = [ φ BH j ( -t, x ) ] ∗ . (D3) \nThe same relation holds between ϕ BH j et ϕ WH j . The doublets W WH j and W BH j related by Eq. (D3) thus have the same norm for the scalar product (27). \nWith these remarks, it is manifest that in a stationary \nWHflow, when working at fixed frequency ω , the modes: \nφ v,in, WH ω ( t, x ) = e -iωt [ φ u,out, BH ω ( x ) ] ∗ , φ v,in, WH -ω ( t, x ) = e + iωt [ φ u,out, BH -ω ( x ) ] ∗ , φ u,in, WH ω ( t, x ) = e -iωt φ v,out, BH ω ( x ) ] ∗ , (D4) \n[ \n] (and similarly for ϕ i,in, WH ω ) form a complete in basis. Notice that the in/out character and the u / v character of the modes is interchanged between the BH and the WH. Indeed, because of the time-reversal symmetry, the phase and group velocities of the modes change sign. The Bogoliubov transformation relating the in to the out WH modes is thus the inverse of (41). This implies that when dealing with a WH flow in the initial vacuum, instead of Eq. (44), the mean occupation numbers of emitted quanta are \n¯ n WH ω = | β -ω | 2 , ¯ n u, WH ω = | B ω | 2 , ¯ n WH -ω = ¯ n WH ω + ¯ n WH , u ω = ¯ n BH -ω . (D5) \nThe third line shows that the total number of produced pairs is equal to that for the corresponding BH. However since | β ω | 2 /negationslash = | β -ω | 2 , the repartition of the quanta with positive ω into left and right movers differs. \nBefore considering the implication of these relations, it is worth noting that only one quantity governs all differences between the occupation numbers and the elastic scattering in the WH and BH cases. Indeed, Eq. (D5) (or the third equation in (44)) shows that \n| β ω | 2 + | ˜ B ω | 2 = | β -ω | 2 + | B ω | 2 , (D6) \nwhile the conservation of the norm of φ u, out , BH ω and of φ u, in , BH ω gives: \n| α ω | 2 + | ˜ A ω | 2 -| β -ω | 2 = | α ω | 2 + | A ω | 2 -| β ω | 2 (= 1) . (D7) \nThus, \n| β ω | 2 -| β -ω | 2 = | B ω | 2 -| ˜ B ω | 2 = | A ω | 2 -| ˜ A ω | 2 . (D8) \nThe relative difference between the occupation numbers in the BH and WH cases is small whenever the u -v mixing coefficients are small, see for instance Sec. V C 3 in [17]. In the case of a BEC, however, the u -v mixing coefficients are not necessarily small and can even grow as 1 /ω at low frequencies, see Sec. V C 4. \nTo further analyze the difference between WH and BH fluxes, in Fig. 21, we have represented | B ω | 2 and | ˜ B ω | 2 for q = 0 . 3 and two values of ( D,λ ). Contrary to | ˜ B ω | 2 that grows as 1 /ω , | B ω | 2 is nearly constant at low frequencies, so that their difference grows like 1 /ω . The consequence of this is that the relative difference ( | β ω | 2 -| β -ω | 2 ) / | β ω | 2 tends to a constant at low frequencies, as is verified in Fig. 22, where this difference is shown for the same parameters as the previous figure. This constant value can \nbe significant. For D = 0 . 4 it is slightly greater than 1%; for D = 0 . 1, it is very small, of the order of 0 . 1%. For both values of D , the relative difference becomes important near ω max , but both | β ω | 2 and | β -ω | 2 vanish when reaching this frequency. \nFrom Eq. (D5) several important lessons can be drawn. Firstly, we established that, when starting with the vacuum state, the fluxes of phonons emitted by a WH flow are directly related to those of the corresponding BH flow. \nSecondly, from this correspondance, the WH flows appear to be as stable as the corresponding BH flows. It should be stressed however that this conclusion is reached when using real frequencies ω , and not taking into account the modes that are not asymptotically bounded. Whether this implies that WH flows are stable is a moot point which deserves further study, see [54]. \nThirdly, when expressed in the asymptotic regions in terms of the comoving frequency Ω a ( ω ) (or the wave number k a ( ω )) the spectral properties of the WH fluxes are radically different from those of the corresponding BH. This is because, when considering a WH flow, the relationships between the comoving Ω a and the conserved frequency ω are time reversed with respect to those for a BH. Hence, unlike what was found for a BH flow, \n- [1] W. G. Unruh, Phys. Rev. Lett. 46 , 1351 (1981).\n- [2] T. Jacobson, Phys. Rev. D 44 , 1731 (1991).\n- [3] W. G. Unruh, Phys. Rev. D 51 , 2827 (1995).\n- [4] R. Brout, S. Massar, R. Parentani, and P. Spindel, Phys. Rev. D 52 , 4559 (1995), hep-th/9506121.\n- [5] S. Corley and T. Jacobson, Phys. Rev. D 54 , 1568 (1996), hep-th/9601073.\n- [6] C. Barcelo, S. Liberati, and M. Visser, Living Rev. Rel. 8 , 12 (2005), gr-qc/0505065.\n- [7] L. J. 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Parentani, talk given at the workshop 'Towards the observation of Hawking radiation in condensed matter \nEq. (46) now determines the final values of the Ω a which are thus blueshifted. In the ideal case of an initial vacuum, this would be a great advantage since the detection of higher frequency phonons is easier. When taking into account the nonvanishing temperature of the condensate, the initial occupation numbers are now given by n a, in , WH ( ω ) = ( e Ω a, out , BH ( ω ) /T in -1 ) -1 , hence they are much larger than those in the corresponding BH case since Ω a, out , BH /similarequal ω . It implies that the initial occupation numbers in WH flows will always be larger than the number of quanta spontaneously created. From this, two conclusions can be drawn. First, the detection of the analog Hawking radiation, based on the measurement of final occupation numbers, will be more difficult than in the corresponding BH. Second, on the contrary, the observation of the long-distance correlations of Eqs. (57) and (58) will be greatly facilitated as the larger initial occupation numbers increase their amplitude. \nFourthly, WH might also be more appropriate than BH to manipulate coherent states because at early time, as explained in App. C, positive and negative ω wave packets follow very different trajectories, whereas for BH they essentially follow the same trajectory. \n- systems' (2009), URL http://www.uv.es/workshopEHR .\n- [19] D. Campo and R. Parentani, Phys. Rev. D 67 , 103522 (2003), gr-qc/0301044.\n- [20] D. Campo and R. Parentani, Phys. Rev. D 70 , 105020 (2004), gr-qc/0312055.\n- [21] G. Rousseaux, C. Mathis, P. Ma¨ıssa, T. Philbin, and U. Leonhardt, New J. Phys. 10 , 053015 (2008).\n- [22] J. M. Bardeen, B. Carter, and S. W. Hawking, Commun. Math. Phys. 31 , 161 (1973).\n- [23] R. M. 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Philbin, in Quantum Analogues: From Phase Transitions to Black Holes and Cosmology (Springer, Berlin, 2007), 0803.0669.\n- [33] S. Massar and R. Parentani, Phys. Rev. D 54 , 7444 (1996).\n- [34] R. M. Wald, Phys. Rev. D 13 , 3176 (1976).\n- [35] R. D. Carlitz and R. S. Willey, Phys. Rev. D 36 , 2327 (1987).\n- [36] R. D. Carlitz and R. S. Willey, Phys. Rev. D 36 , 2336 (1987).\n- [37] C. Barcelo, S. Liberati, and M. Visser, Int. J. Mod. Phys. D 12 , 1641 (2003), gr-qc/0305061.\n- [38] P. O. Fedichev and U. R. Fischer, Phys. Rev. A 69 , 033602 (2004), cond-mat/0303063.\n- [39] P. Jain, S. Weinfurtner, M. Visser, and C. W. Gardiner, Phys. Rev. A 76 , 033616 (2007), 0705.2077.\n- [40] S. Weinfurtner, P. Jain, M. Visser, and C. W. Gardiner, Class. Quant. Grav. 26 , 065012 (2009), 0801.2673.\n- [41] V. F. Mukhanov, H. A. Feldman, and R. H. Brandenberger, Phys. Rept. 215 , 203 (1992).\n- [42] I. Carusotto, R. Balbinot, A. Fabbri, and A. Recati (2009), 0907.2314.\n- [43] R. Balbinot, A. Fabbri, S. Fagnocchi, and R. Parentani, Riv. Nuovo Cim. 28 , 1 (2005), gr-qc/0601079.\n- [44] T. Jacobson and R. Parentani, Phys. Rev. D 76 , 024006 (2007), hep-th/0703233. \n- [45] D. Campo and R. Parentani, Phys. Rev. D 72 , 045015 (2005), astro-ph/0505379.\n- [46] D. Campo and R. Parentani, Phys. Rev. D 78 , 065044 (2008), 0805.0548.\n- [47] D. Campo and R. Parentani, Phys. Rev. D 78 , 065045 (2008), 0805.0424.\n- [48] A. Recati, N. Pavloff, and I. Carusotto (2009), 0907.4305.\n- [49] P. R. Anderson, talk given at the workshop 'Towards the observation of Hawking radiation in condensed matter systems' (2009), URL http://www.uv.es/workshopEHR .\n- [50] R. Balbinot, S. Fagnocchi, and A. Fabbri, Physical Review A 75 , 043622 (2007).\n- [51] U. Leonhardt, Measuring the quantum state of light (Cambridge University Press, 1997).\n- [52] J. Steinhauer, N. Katz, R. Ozeri, N. Davidson, C. Tozzo, and F. Dalfovo, Phys. Rev. Lett. 90 , 060404 (2003).\n- [53] R. Ozeri, N. Katz, J. Steinhauer, and N. Davidson, Rev. Mod. Phys. 77 , 187 (2005).\n- [54] C. Barcelo, A. Cano, L. J. Garay, and G. Jannes, Phys. Rev. D 74 , 024008 (2006), gr-qc/0603089."}
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2019JHEP...05..205Y
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The quantum gravity dynamics of near extremal black holes
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2019-01-01
| 35 | 0.45 | 164 |
['-', '-', 'black hole physics', '-', '-', '-']
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[]
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We study the quantum effects of Near-Extremal black holes near their horizons. The gravitational dynamics in such backgrounds are closely connected to a particle in AdS <SUB>2</SUB> with constant electric field. We use this picture to solve the theory exactly. We will give a formula to calculate all correlation functions with quantum gravity backreactions as well as the exact Wheeler-DeWitt wavefunction. Using the WdW wavefunction, we investigate the complexity growth in quantum gravity.
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[]
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https://arxiv.org/pdf/1809.08647.pdf
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{'Zhenbin Yang': 'Jadwin Hall, Princeton University, Princeton, NJ 08540, USA', 'Abstract': 'We study the quantum effects of Near-Extremal black holes near their horizons. The gravitational dynamics in such backgrounds are closely connected to a particle in AdS 2 with constant electric field. We use this picture to solve the theory exactly. We will give a formula to calculate all correlation functions with quantum gravity backreactions as well as the exact Wheeler-DeWitt wavefunction. Using the WdW wavefunction, we investigate the complexity growth in quantum gravity.', 'Contents': '| 1 | Introduction | 2 |\n|---------------------------------------|------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|\n| 2 | Classical Solutions | 3 |\n| 3 | Charged Particle in AdS 2 | 4 |\n| 4 | Solving the Quantum Mechanical Problem | 7 |\n| 4.1 . . . | The Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 9 |\n| 4.2 . . | Partition Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 9 |\n| 5 | Quantum Gravity at Schwarzian Limit | 12 |\n| 5.1 | The Propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 12 |\n| 5.2 | Wheeler-DeWitt Wavefunction . . . . . . . . . . . . . . . . . . . . . . . . | 15 |\n| 6 | Correlation Functions in Quantum Gravity 21 Gravitational Feynman Diagram . . . . . . . . . . . . . . . . . . . . . . . . 21 Two Point Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 ETH and the KMS condition . . . . . . . . . . . . . . . . . . . . . . . . . 24 Three Point Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Einstein-Rosen Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 28 29 B Wavefunction with matter 30 | Correlation Functions in Quantum Gravity 21 Gravitational Feynman Diagram . . . . . . . . . . . . . . . . . . . . . . . . 21 Two Point Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 ETH and the KMS condition . . . . . . . . . . . . . . . . . . . . . . . . . 24 Three Point Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Einstein-Rosen Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 28 29 B Wavefunction with matter 30 |\n| | 6.1 6.2 | |\n| | 6.5 | |\n| 7 | Conclusion | |\n| | A Gauge Fix SL(2,R) | |\n| C Boundary effective action from CFT | | 31 |\n| | | 32 |\n| | D Connection with the relativistic particle and Pair Production | |\n| F Trajectories in Real Magnetic Field | E Details on the Schwarzian limit of Propagator | 34 34 |', '1 Introduction': "Near-Extremal black holes have a universal structure near their horizons: there is an AdS 2 throat with a slowly varying internal space. Its low energy gravitational dynamics is captured universally by the following effective action in two dimensions [1]: \nI = -φ 0 2 (∫ R +2 ∫ ∂ M K ) ︸ ︷︷ ︸ Einstein-Hilbert Action -1 2 (∫ M φ ( R +2) + 2 ∫ ∂M φ b K ) ︸ ︷︷ ︸ Jackiw-Teitelboim action + S matter ( g, ψ ) , (1.1) \nwhere the dilaton field φ + φ 0 represents the size of internal space. We have separated the size of internal space into two parts: φ 0 is its value at extremality. It sets the value of the extremal entropy which comes from the first term in (1.1). φ is the deviaton from this value. We have also added matter that only couples to the metric. This is a reasonable assumption when matter comes from Kaluza Klein reduction, where the coupling to the dilaton would involve φ/φ 0 glyph[lessmuch] 1. \nThe action ∫ φ ( R +2)+2 ∫ φ b K is the so-called Jackiw-Teitelboim action [2, 3], and will be the main focus of our paper. This action is one of the simplest nontrivial gravitational actions in two dimensions. 1 . It is simple because the bulk geometry is a rigid AdS 2 space fixed by the equation of motion of the dilaton field. Its nontriviality arises from the remaining boundary action. Schematically, the gravitational action is reduced to the following form: \nI = -2 πφ 0 χ ( M ) -φ b ∫ ∂M K + S matter ( g, ψ ) . (1.2) \nAnd the motion of the boundary is controlled by its extrinsic curvature. Our goal will be to quantize this action and to provide expression for the full quantum gravity correlators of (1.1). This problem was considered before in [4, 5, 6, 7, 8] from various points of view. Here we will add one other point of view where we reduce the problem to the motion of a relativistic particle in an electric field, building on a suggestion in Kitaev's talk at IAS [9]. More precisely, one can consider a relativistic particle in a Lorentzian AdS 2 target space moving under the influence of an electric field. The coupling to electric field can also be viewed as a coupling to a spin connection so that it becomes a particle with spin as suggested by Kitaev. Alternatively we can start from a non-relativistic particle moving in hyperbolic space, H 2 , under the influence of a magnetic field b . After analytic continuation in b to imaginary values we get the problem of interest. \nUsing this point of view one can think of the full quantum gravity problem as the combination of two problems. First we consider quantum fields propagating in AdS 2 (or H 2 in the Euclidean case) and then we add the 'gravitational particle' which couples to the quantum fields by changing their boundary location in AdS 2 . The discussion of quantum fields will be standard and depends on the particular model one interested in, therefore \nwe will mainly focus on solving the second problem. Generically, solving the gravitational problem is challenging and is not exactly equivalent to a quantum mechanical particle. One needs to worry about what functional space one will integrate over. For example, in path integrals, one usually integrates over all trajectories including those with self-intersections. However self-intersecting boundaries in gravitational system have no obvious meaning. On that account, more precisely the gravitational problem is equal to a self-avioding particle. Nevertheless, it turns out that one can take a particular limit of this model, namely large φ b , to avoid this issue and a treatment of the boundary theory as an ordinary particle is justified. It is also true that the JT gravity can be rewritten as a Schwarzian action only in this limit. We call this the Schwarzian limit and will only focus on solving the JT action in the Schwarzian limit. Solving the model away from Schwarzian limit was considered recently by Kitaev and Suh [8]. \nOur result can be summarized as follows: \nFirst, we will give a formula to calculate all correlation functions with quantum gravity backreaction (formula 6.74). Second, we will give the exact Wheeler-DeWitt wavefunction in the Schwarzian limit, which has been analyzed classically by Harlow and Jafferis [10]. Last, we consider the recent proposed conjecture about complexity growth in this exact Wheeler-DeWitt wavefunction and show that the complexity maintains linear growth after taking quantum gravity effects into account. This, to our knowledge, is the first test of the gravitational conjecture made by Susskind [11] that the size of ERB grows linearly for as long as quantum mechanics allows. \nThis paper is organized as follows: in the first section, we will review the classical calculation of this model and introduce notations; in the second section, we will make the dictionary between the JT model and a particle in a magnetic field; in the third and fourth sections, we will solve the quantum mechanical problem and derive the propagator and WdW wavefunction in the Schwarzian limit; in the last section, we will talk about gravitaional backreaction on correlators as well as complexity growth. As a useful notion in our calculation, we introduce a notation called gravitational Feyman diagrams.", '2 Classical Solutions': 'Let us first consider the classical solutions of the Jackiw-Teitelboim model (1.1). See [12] for further discussion. The equation of motion of the dilaton field imposes R = -2 and fixes the geometry to be AdS 2 , or H 2 in the Euclidean case. This is also true if we have additional matter coupled with metric only, as in (1.1). The equations for the metric constrain the dilaton \n( ∇ µ ∇ ν φ -g µν ∇ 2 φ + g µν φ ) + T M µν = 0 (2.3) \nThese equations are compatible with each other thanks to the conservation of the matter stress tensor. They do not allow any propagating mode. In fact, setting T M µν = 0, and using a high momentum approximation we can write (2.3) as ( k µ k ν -g µν k 2 ) φ ( glyph[vector] k ) = 0, which then implies φ ( glyph[vector] k ) = 0, for large k . \nMore precisely, after introducing the Euclidean AdS 2 coordinates ds 2 = dρ 2 +sinh 2 ρdϕ 2 , we can solve (2.3) in AdS 2 with no matter. Up to an SL(2) transformation the solution is \nφ = φ h cosh ρ, (2.4) \nwhere φ h is a constant that is fixed by the boundary conditions. At the boundary we fix the metric along the boundary and the value of the dilaton field \nds ‖ = du φ r glyph[epsilon1] , φ = φ b = φ r glyph[epsilon1] (2.5) \nwhere we think of u as the time of the boundary theory. It is simply a rescaled version of proper time. Similarly φ r is a rescaled value of the dilaton. We will be interested in taking glyph[epsilon1] → 0. With these rescalings the value of φ h in the interior remains fixed as we take glyph[epsilon1] → 0 as φ h = 2 π/β where β is the inverse temperature, u ∼ u + β . Notice that due to the factor of φ r in the first expression in (2.5) we are measuring time in units of the constant φ r , which has dimensions of length. We did this for convenience. A nice feature that appears after taking the glyph[epsilon1] → 0 limit is that the action (1.2) can be written as the Schwarzian action for the boundary curve labeled by ϕ ( u ) [13]: \nI = -∫ duSch (tan ϕ ( u ) 2 , u ) (2.6) \nThe fluctuation of the boundary shape can be understood as the fluctuation of the dilaton distribution in the bulk. A bit more explicitly we can say that the dilaton boundary condition fixes the location of the boundary at ρ b given by φ b = φ h cosh ρ b , and the metric at that location relates the time ϕ to u by φ r du = glyph[epsilon1] sinh ρ b dϕ . We get the above formulas noticing that the period of ϕ is 2 π while that of u is β , which fixes glyph[epsilon1] sinh ρ b .', '3 Charged Particle in AdS 2': "Despite the absence of a bulk propagating mode there is still a non-trivial dynamical gravitational degree of freedom. There are various ways to describe it. Here we will think of it as arising from the motion of the physical boundary of AdS 2 inside a rigid AdS 2 space. This picture is most clear for finite glyph[epsilon1] in (2.5), but it is true even as glyph[epsilon1] → 0. The dynamics of the boundary is SL(2) invariant. This SL(2) invariance is a gauge symmetry since it simply reflects the freedom we have for cutting out a piece of AdS 2 space that we will call the 'inside'. It is important that the dilaton field we discussed above is produced after we put in the boundary and it moves together with the boundary under this SL(2) gauge transformation. It is a bit like the Mach principle, the location in AdS 2 is only defined after we fix the boundary (or distant 'stars'). \nWe can make this picture of a dynamical boundary more manifest as follows. Since the bulk Jackiw-Teitelboim action (1.1) is linear in φ , we can integrate out the dilaton field \nwhich sets the metric to that of AdS 2 and removes the bulk term in the action, leaving only the term involving the extrinsic curvature \nI = -φ r glyph[epsilon1] ∫ du √ gK (3.7) \nThis action, however, is divergent as we take glyph[epsilon1] to zero. This divergence is simply proportional to the length of the boundary and can be interpreted as a contribution to the ground state energy of the system. So we introduce a counterterm proportional to the length of the boundary to cancel it. This is just a common shift of the energies of all states. It is also convenient to use the Gauss-Bonnet theorem to relate the extrinsic curvature to an integral over the bulk \n∫ ∂ M du √ gK = 2 πχ ( M ) -1 2 ∫ M R (3.8) \nSince the curvature is a constant, the bulk integral is actually proportional to the total area A of our space. That is we have the regularized action: \nI = -φ r glyph[epsilon1] ∫ ∂M du √ g ( K -1 ︸︷︷︸ counterterm ) = -φ r glyph[epsilon1] ( 2 πχ ( M ) -1 2 ∫ M R -∫ ∂M du √ g ) = -2 πqχ ( M ) -qA + qL, q ≡ φ r glyph[epsilon1] , L = βφ r glyph[epsilon1] (3.9) \nWe now define an external gauge field a µ as \na ϕ = cosh ρ -1 , a ρ = 0 , f ρϕ = sinh ρ = √ g, (3.10) \nand write the action as follows \nI = -2 πq + qL -q ∫ a (3.11) \nwhere we used that χ ( M ) is a topological invariant equal to one, in our case, where the topology is that of a disk. The term qL is just the length of the boundary. So this action has a form somewhat similar to the action of a relativistic charged particle moving in AdS 2 in the presence of a constant electric field. There are a couple of important differences. First we are summing only over trajectories of fixed proper length set by the inverse temperature β . Second, in the JT theory we are treating the SL (2) symmetry as a gauge symmetry. And finally, in the JT theory we identify the proper length with the boundary time, viewing configurations which differ only by a shift in proper time as inequivalent. In fact, all these changes simplify the problem: we can actually think of the problem as a non-relativistic particle moving on H 2 in an electric field. In appendix D we discuss in more detail the connection to the relativistic particle. \nIn fact, precisely the problem we are interested in has been discussed by Polyakov in [14], Chapter 9, as an an intermediate step for the sum over paths. Now we would also like \nto point out that we can directly get to the final formula by using the discussion there, where he explicitly shows that for a particle in flat space the sum over paths of fixed proper length that stretch between two points glyph[vector]x and glyph[vector]x ' gives \n∫ D glyph[vector]xe -m 0 ˜ τ δ ( ˙ glyph[vector]x 2 -1) = e -1 2 µ 2 τ 〈 x ' | e -τH | x 〉 = e -1 2 µ 2 τ ∫ D x exp ( -∫ τ 0 dτ ' 1 2 ˙ glyph[vector]x 2 ) (3.12) \nµ 2 is the regularized mass and ˜ τ is related to τ by a multiplicative renormalization. The JT model consists precisely of a functional integral of this form, where we fix the proper length along the boundary. There are two simple modifications, first the particle is in a curved H 2 space and second we have the coupling to the electric field. These are minor modifications, but the arguments leading to (3.12) continue to be valid so that the partition function of the JT model can be written directly: \n∫ D glyph[vector]xe 2 πq -m 0 ˜ τ + q ∫ a δ ( ˙ x 2 + ˙ y 2 y 2 -q 2 ) = e 2 πq -1 2 µ 2 τ Tr e -τH = e 2 πq -1 2 µ 2 τ ∫ D x exp ( -∫ τ 0 dτ ' 1 2 ˙ x 2 + ˙ y 2 y 2 -q ˙ x y ) (3.13) \nThe delta function implements the first condition in (2.5) at each point along the path. The last path integral can be done exactly by doing canonical quantization of the action (section 4) and by comparing the result with the one from the Schwarzian action [6] we can determine that τ is the inverse temperature β . \nIn the above discussion we have been fixing the time along the boundary. Instead we can fix the energy at the boundary, where the energy is the variable conjugate to time. This can be done by simply integrating (3.13) times e βE over β along the imaginary axis. This fixes the energy of the non-relativistic problem by generating a δ ( H -E ). More precisely, we will argue that after doing a spectral decomposition we can write the propagator at coincident points as \nZ JT ( β ) = ∫ ∞ 0 ρ ( E ) e -βE dE -→ ρ ( E ) = ∫ i ∞ -i ∞ dβ i e Eβ Z JT ( β ) (3.14) \nwhere the function ρ ( E ) can then be interpreted as a 'density of states' in the microcannonical ensemble. We will give its explicit form in section (4.2). For now, we only want to contrast this integral with a superficially similar one that appears when we compute the relativistic propagator \ne -2 πq ∫ ∞ 0 e Eβ Z JT ( β ) = 〈 φ ( x ) φ ( x ) 〉 (3.15) \nwhich gives the relativistic propagator of a massive particle in an electric field at coincident points (we can also compute this at non-coincident points to get a finite answer). The total mass of the particle is \nm = q -E q (3.16) \nFor large q this is above threshold for pair creation 2 . The pair creation interpretation is appropriate for the problem in (3.15), but not for (3.14). In both problems we have a \nclassical approximation to the dynamics that corresponds to a particle describing a big circular trajectory in hyperbolic space at radius ρ c : \ntanh ρ c = m q (3.17) \nFor the problem in (3.15), fluctuations around this circle lead to an instability, with a single negative mode and an imaginary part in the partition function (3.15). This single negative mode corresponds to small fluctuations of the overall size of the circular trajectory around (3.17). On the other hand in (3.14) we are integrating the same mode along a different contour, along the imaginary axis, where we get a real and finite answer. Furthermore, the imaginary part in the partition function (3.15) comes precisely from the trajectory describing pair creation, which is also the type of contribution captured in (3.14). \nFinally, in the relativistic particle problem, we expect that the pair creation amplitude should be exponentially suppressed for large q , while the partition function for the JT model is not. In fact, for large q the exponential suppression factor for pair creation goes as e -2 πq , which is precisely cancelled by a similar factor in (D.113), to obtain something finite in the large q limit.", '4 Solving the Quantum Mechanical Problem': "As we explained above the solution of the JT theory is equivalent to considering a nonrelativistic particle in AdS 2 or H 2 . We first consider the Euclidean problem, of a particle moving in H 2 . An ordinary magnetic field in H 2 leads to an Euclidean action of the form \nS = ∫ du 1 2 ˙ x 2 + ˙ y 2 y 2 + ib ∫ du ˙ x y -1 2 ( b 2 + 1 4 ) ∫ du , b = iq (4.18) \nIf b is real we will call it a magnetic field, when q is real we will call it an 'electric' field. The last term is a constant we introduced for convenience. Its only effect will be to shift the ground state energy. It is interesting to compute the classical solutions and the corresponding action for (4.18). These solutions are simplest in the ρ and ϕ coordinates, using the SL(2) symmetry we find that the trajectories are given by ( t = -iu ): \n1 2 sinh 2 ρ ( dϕ dt ) 2 + q 2 2 -1 8 = E, cosh ρ = qβ 2 π , dϕ du = 2 π β . (4.19) \nIn this classical limit we get the following relations for the action and the temperature: \nβ 2 π = 1 √ 2 E + 1 4 -S = 2 π 2 β + β 8 -2 πq (4.20) \nWhen b is real, this system is fairly conventional and it was solved in [15] . Its detailed spectrum depends on b . For very large b we have a series of Landau levels and also a continuous spectrum. In fact, already the classical problem contains closed circular orbits, related to the discrete Landau levels, as well as orbits that go all the way to infinity. 3 The number of discrete Landau levels decreases as we decrease the magnetic field and for 0 < b < 1 2 we only get a continuous spectrum. The system has a SL (2) symmetry and the spectrum organizes into SL(2) representations, which are all in the continuous series for 0 < b < 1 / 2. For real q we also find a continuous spectrum which we can view as the analytic continuation of the one for this last range of b . \nThe canonical momenta of the action (4.18) are: \np x = ˙ x y 2 + iq y ; p y = ˙ y y 2 . (4.21) \nAnd the Hamiltonian conjugate to τ L is thus: \nH = ˙ x 2 + ˙ y 2 2 y 2 + q 2 2 = y 2 2 [( p x -i q y ) 2 + p 2 y ] + q 2 2 -1 8 (4.22) \nNote that the Hamiltonian is not Hermitian. However, it is PT-symmetric (here parity reflects x and p x ) and for that reason the spectrum is still real, see [16]. The action is invariant under SL (2 , R ) transformations generated by \nL 0 = xp x + yp y ; L -1 = p x ; L 1 = ( y 2 -x 2 ) p x -2 xyp y -2 iqy (4.23) \nNotice the extra q dependent term in L 1 that arises due to the presence of a magnetic field. Up to a simple additive constant, the Hamiltonian is proportional to the Casimir operator \nH = 1 2 ( L 2 0 + 1 2 L -1 L 1 + 1 2 L 1 L -1 ) + q 2 2 -1 8 (4.24) \nAs is common practice, let us label the states by quantum numbers j = 1 2 + is and k , so that H | j, k 〉 = j (1 -j ) | j, k 〉 and L -1 | j, k 〉 = k | j, k 〉 . We can find the eigenfunctions by solving the Schr odinger equation with boundary condition that the wavefunction should vanish at the horizon y →∞ [17, 15, 41]: \nω s,k = s 2 2 , f s,k ( x, y ) = { ( s sinh 2 πs 4 π 3 k ) 1 2 | Γ( is -b + 1 2 ) | e -ikx W b,is (2 ky ) , k > 0; ( s sinh 2 πs 4 π 3 | k | ) 1 2 | Γ( is + b + 1 2 ) | e -ikx W -b,is (2 | k | y ) , k < 0 . (4.25) \nwhere ω ks is giving the energy of the states labelled by s and k , and W is the Whittaker function. The additive constant in (4.18) was introduced to simplify this equation. We can think of s as the quantum number of the continuous series representation of SL (2) with spin j = 1 2 + is . \nAfter continuing b → iq we find that the gravitational system has a continuous spectrum \nE ( s ) = s 2 2 . (4.26)", '4.1 The Propagator': 'It is useful to compute the propagator for the non-relativistic particle in a magnetic field, K ( u, x 1 , x 2 ) = 〈 x 1 | e -uH | x 2 〉 . Here, x stands for x, y . The propagator for a real magnetic particle was obtained in [15]: \nG ( u, x 1 , x 2 ) = e iϕ ( x 1 , x 2 ) ∫ ∞ 0 dsse -u s 2 2 sinh 2 πs 2 π (cosh 2 πs +cos2 πb ) 1 d 1+2 is × × 2 F 1 ( 1 2 -b + is, 1 2 + b + is, 1 , 1 -1 d 2 ) . d = √ ( x 1 -x 2 ) 2 +( y 1 + y 2 ) 2 4 y 1 y 2 e iϕ ( x 1 , x 2 ) = e -2 ib arctan x 1 -x 2 y 1 + y 2 (4.27) \nIn the case that we have a real magnetic field the prefactor is a phase and it is gauge dependent. It is equal to the value of Wilson Line e i ∫ a stretched along the geodesic between x 2 and x 1 . Here we quoted the value in the gauge where the action is (4.18). The second equation defines the parameter d , which is a function of the geodesic distance between the two points. Note that d = 1 corresponds to coincident points. We can get the answer we want by making the analytic continuation b → iq of this formula. We can check that this is the right answer for our problem by noticing the following. First one can check that this expression is invariant under the SL(2) symmetry L a = L 1 a + L 2 a where L a are the generators (4.23) acting on x 1 and L 2 a are similar generators as in (4.23), but with q →-q . It is possible to commute the phase e iϕ ( x 1 , x 2 ) , in (4.27) past these generators which would remove the q dependent terms. This implies that the rest should be a function of the proper distance, which is the case with (4.27). Then we can check the equation \n0 = ( ∂ u + H 1 ) G ( u, x 1 , x 2 ) (4.28) \nwhich is also indeed obeyed by this expression. The s dependent prefactor is fixed by the requirement that the propagator composes properly, or more precisely, by saying that for u = 0 we should get a δ function.', '4.2 Partition Function': "The gravitational partition function is related with the particle partition function with inverse temperature β . The canonical partition function of the quantum mechanical system is \nZ Particle = Tre -βH = ∫ ∞ 0 ds ∫ ∞ -∞ dk ∫ M dxdy y 2 e -β s 2 2 f ∗ s,k ( x, y ) f s,k ( x, y ) = V AdS ∫ ∞ 0 dse -β s 2 2 s 2 π sinh(2 πs ) cosh(2 πq ) + cosh(2 πs ) . (4.29) \nFigure 1: Free Energy diagram with inverse temperature β . \n<!-- image --> \nThe volume factor V AdS arises because after momentum integration there is no position dependence. In a normal quantum mechanical system, the volume factor means that the particle can have independent configurations at different locations of our space, however for a gravitational system this should be thought as redundant and should be cancelled by the volume of SL (2 , R ) gauge group 2 πV AdS 4 . In gravitational system, there can also other contributions to the entropy from pure topological action. These give a contribution to the ground state entropy S 0 . Including the topological action in (3.11), we find a gravitational 'density of states' as \nρ ( s ) = e S 0 e 2 πq ︸ ︷︷ ︸ extra terms 1 2 π ︸︷︷︸ residue gauge s 2 π sinh(2 πs ) cosh(2 πq ) + cosh(2 πs ) ︸ ︷︷ ︸ particle in magnetic field = e S 0 e 2 πq s 2 π 2 ∞ ∑ k =1 ( -1) k -1 e -2 πqk sinh(2 πsk ) . (4.30) \nWe have not given an explicit description of these states in the Lorentzian theory. More details were discussed in [8, 18]. \nThis expression has some interesting features. Notice that the classical limit corresponds to large q and large s , where we reproduce (4.20). After approximating, the density of states are log ρ ( s ) ∼ S 0 +2 πs for s/q < 1 and S 0 +2 πq for s/q > 1. \nWe can also expand the partition function for very small and very large temperatures where we obtain \nZ JT ∼ e S 0 e 2 πq 1 4 π 2 β , β glyph[lessmuch] 1 q Z JT ∼ e S 0 1 √ 2 πβ 3 / 2 , β glyph[greatermuch] 1 (4.31) \nNotice that at leading order we get an almost constant entropy both at low and high \ntemperatures, with the high temperature one being higher. In both cases there are power law corrections in temperature. \nBefore we try to further elucidate the interpretation of this result, let us emphasize a couple of important defects of our discussion. First, when we replaced the partition function of the JT theory by the action of a non-relativistic particle in an electric field, we were summing over paths in H 2 . This includes paths that self intersect see figure 2(b). Such paths do not have an obvious interpretation in the JT theory and it is not even clear that we should include them. For example, the sum over k in (4.30) can be understood in terms of classical solutions which wind k times around the circle. These make sense for the problem of the particle in the electric field but apparently not in the JT theory. Maybe \nFigure 2: Density of States and the Two Instantons configuration \n<!-- image --> \n(a) Density of States \n(b) Two Instantons \nsuch paths could be given some interpretation in the gravity theory. Alternatively, we might want to sum over paths that do not self intersect. A second defect is that we would be eventually interested in adding some matter fields propagating in the bulk geometry. These matter fields have boundary conditions at the boundary of the region of hyperbolic space cut out by the boundary trajectory. The partition function of the fields with such an arbitrary boundary trajectory could also modify the results we described above. Of course, this issue does not arise if we have the pure JT theory. It is only important if we want to introduce bulk matter fields to define more complex observables. \nInstead of attempting to address the above issues, we will take an easy route, which is to consider the system only in the large q (or small glyph[epsilon1] ) limit. In this regime, we address the above issues, and we can still trust the description of the particle in the electric field. This large q or small glyph[epsilon1] limit is the same one that isolates the Schwarzian action from the JT theory [13, 19, 20]. It turns out that the limit can be taken already at the level of the mechanical system, a simple rescaled version of the above system. This provides an alternative method for quantizing the Schwarzian theory. It has the advantage of being a straightforward second order action of a particle moving in a region near the boundary of hyperbolic. Of course, the Schwarzian theory was already quantized using a variety of methods in [4, 5, 6, 7, 21]. We will simply provide yet another perspective, recover the old results, and write a few new expressions. \n<!-- image -->", '5 Quantum Gravity at Schwarzian Limit': 'Before getting into the details notice that the large q limit of (4.30) gives \nρ ( s ) = e S 0 s 2 π 2 sinh(2 πs ) , E = s 2 2 , Z JT = ∫ ∞ 0 dsρ ( s ) e -β s 2 2 = e S 0 1 √ 2 πβ 3 2 e 2 π 2 β . (5.32) \nThis reproduces what was found in [6, 7, 44] by other methods. We see that we get a finite answer and also that the contributions from the k > 1 terms in (4.30) have disappeared. Because the S 0 part decouples with JT gravity, from now on, we will drop it and discuss S 0 only when it is necessary.', '5.1 The Propagator': "To get a limit directly at the level of the mechanical system it is useful to define a rescaled coordinate, z , via \ny = z/q. (5.33) \nAfter taking the large q limit, the boundary particle propagator becomes 5 : \nG ( u, x 1 , x 2 ) = 1 q e -2 πqθ ( x 2 -x 1 ) ˜ K ( u, x 1 , x 2 ); q glyph[greatermuch] 1 . (5.34) ˜ K ( u, x 1 , x 2 ) = e -2 z 1 + z 2 x 1 -x 2 2 √ z 1 z 2 π 2 | x 1 -x 2 | ∫ ∞ 0 dss sinh(2 πs ) e -s 2 2 u K 2 is ( 4 √ z 1 z 2 | x 1 -x 2 | ); (5.35) = e -2 z 1 + z 2 x 1 -x 2 √ 2 π 3 / 2 u 3 / 2 √ z 1 z 2 | x 1 -x 2 | ∫ ∞ -∞ dξ ( π + iξ ) e -2 ( ξ -iπ ) 2 u -4 √ z 1 z 2 | x 1 -x 2 | cosh ξ . (5.36) \nThe original phase factor e iϕ ( x 1 , x 1 ) factorizes into a product of singular 'phase' e -2 πqθ ( x 2 -x 1 ) , with θ the step function, and a regular 'phase' e -2 z 1 + z 2 x 1 -x 2 . The singular 'phase' is the same order as the topological piece in (3.11). In order to have a finite result they should cancel between each other. This can only be satisfied if the x i s are in cyclic order. As shown in figure (3(a)), the product of singular 'phase' gives -2 πq for cyclic order x i s and this would cancel with the topological action 2 πq . While for other ordering of the x i s, this would have -2 πnq for n = 2 , 3 , ... and is highly suppressed in the limit q goes to infinity. This cyclic order is telling us where the interior of our space time is. The magnetic field produces a preferred orientation for the propagator. After fixing the order, all our formulas only depend on ˜ K ( u, x 1 , x 2 ) which has no q dependence. The residual q factor in 5.34 cancels out the additional q from the measure of coordinate integral, dxdy y 2 → q dxdz z 2 . In conclusion, after taking the limit we get a finite propagator equal to (5.35), which should be multiplied by a step function θ ( x 1 -x 2 ) that imposes the right order. \nThe final function ˜ K ( u, x 1 , x 2 ) has the structure of e -2 z 1 + z 2 x 1 -x 2 f ( u, z 1 z 2 ( x 1 -x 2 ) 2 ). This can be understood directly from the SL (2) symmetry. After taking the large q limit, the SL (2 , R ) charges become \nL 0 = i ( x∂ x + z∂ z ); L -1 = i∂ x ; L 1 = -ix 2 ∂ x -2 ixz∂ z -2 iz. (5.37) \nWe can check that they still satisfy the SL(2) algebra. If we drop the last term in L 1 , the SL (2 , R ) charges become the usual differential operators on EAdS 2 . And the propagator will have only dependence on the geodesic distance. When L 1 operator is deformed, the condition of SL (2 , R ) invariance fixes the structure of the propagator as follows. The L 0 and L -1 charges are not deformed and they imply that the only combinations that can appear are \nv ≡ z 1 + z 2 x 1 -x 2 and w ≡ z 1 z 2 ( x 1 -x 2 ) 2 . (5.38) \nWriting the propagator as ˜ K ( u, x 1 , x 1 ) = k ( v, w ) and requiring it to be invariant under L 1 gives the following equation for α : \n∂ v k +2 k = 0 -→ k = e -2 v h ( w ) (5.39) \n˜ K ( u, x 1 , x 2 ) = e -2 z 1 + z 2 x 1 -x 2 f ( u, z 1 z 2 ( x 1 -x 2 ) 2 ) , (5.40) \nThe full function can also be determined directly as follows. Again we impose the propagator equation (or heat equation) \n0 = [ ∂ u + 1 2 ( L 2 0 + 1 2 L -1 L 1 + 1 2 L 1 L -1 ) -1 8 ] ˜ K 0 = [ s 2 2 + w 2 2 ∂ 2 w +2 w + 1 8 ] K s ( w ) (5.41) \nwhere L a are given in (5.37) and are acting only on the first argument of ˜ K . The solution of the last equation which is regular at short distances ( w →∞ ) is √ w times the Bessel K function in (5.34). \nWe can also directly determine the measure of integration for s by demanding that the propagator at u = 0 is a δ function or by demanding the propagator compose properly. This indeed is the case with the s sinh 2 πs function in (5.35). To explicitly show the above statement, it will be useful to use spectral decomposition of the propagator: \n˜ K ( u, x 1 , x 2 ) = ∫ ds 2 s sinh(2 πs ) π 3 e -s 2 u 2 ∫ dk √ z 1 z 2 e ik ( x 1 -x 2 ) K 2 is (2 √ 2 ikz 1 ) K 2 is (2 √ 2 ikz 2 ) . \n(5.42) \nIt can be easily checked that the special functions f k,s ( x, z ) = √ ze ikx K 2 is (2 √ 2 ikz ) are delta function normalizable eigenmodes of the large q Hamiltonian: \n∫ dxdz z 2 f k 1 ,s 1 f k 2 ,s 2 = δ ( k 1 -k 2 ) δ ( s 1 -s 2 ) π 3 2 s sinh(2 πs ) (5.43) \nNotice that the inner product fixes the integral measure completely in (5.42), and the composition relation is manifestly true: \n∫ dxdz z 2 ˜ K ( u 1 , x 1 , x ) ˜ K ( u 2 , x , x 2 ) = ˜ K ( u 1 + u 2 , x 1 , x 2 ) (5.44) \nAt short time the propagator has the classical behavior: \n˜ K ( u, x 1 , x 2 ) ∼ δ ( x 1 -x 2 + uz 2 ) e -( z 1 -z 2 ) 2 2 uz 2 (5.45) \nThis form of singularity is expected since we are taking the large q limit first and thus the velocity in x direction is fixed to be z . In the original picture of finite q we are looking at the time scale which is large compare to AdS length but relatively small such that the quantum fluctuations are not gathered yet. \nThe integral structure in the propagator (5.35) has an obvious meaning: integrating over s represents summing over all energy states with Boltzmann distribution e -Eu , and the Bessel function stands for fixed energy propagator. We want to stress that the argument in the Bessel function is unusual, and at short distance it approaches a funny limit: \nK 2 is ( 4 glyph[lscript] ) glyph[similarequal] √ π 8 glyph[lscript] e -4 glyph[lscript] , glyph[lscript] = | x 1 -x 2 | √ z 1 z 2 → 0 . (5.46) \nOne should contrast this exponential suppression with the short distance divergence in QFT which is power law. In our later discussion of exact correlation function with gravity backreaction, we will see that this effect kills UV divergence from matter fields. \nTo obtain the expression (5.36), we use the integral representation for the Bessel function and the final result has some interesting physical properties: \nFirstly, we see that at large u the time dependence and coordinate dependence factorized. So, at large time we have a universal power law decay pointed out in [4]. \nSecondly, as we said before, the phase factor e -2 z 1 + z 2 x 1 -x 2 is equal to the (regularized) Wilson line e -q ∫ 2 1 a stretched along the geodesic connection between location 1 and 2 (Figure 3(b)). The field a depends on our choice of gauge, our convention corresponds to fix the minimum value of a at infinity and then the Wilson line is equal to e -qA , where A is the area of a hyperbolic triangle spanned by 1 , 2 and ∞ . \nThirdly, defining 2 π + 2 iξ as θ , then θ has the meaning of the spanned angle at the horizon (Figure 3(b)). Then the gaussian weight e -2 ( ξ -iπ ) 2 u = e θ 2 2 u can be understood from the classical action along the boundary with fixed span angle θ . The boundary drawn in the figure represents a curve with fixed (regularized) proper length u in H 2 . √ \nLastly, the factor e -1 2 | x 1 -x 2 | cosh ξ = e 4 cos θ/ 2 glyph[lscript] is equal to e -q ( α + β ) , which is a corner term that arise from JT gravity in geometry with jump angles. Here α and β are defined as the angle spanned by the geodesic with fixed length and the ray coming from horizon to the boundary. \n4 \nz \nz \nFigure 3: The singular 'phase' for different ordering and the geometric representation of the propagator \n<!-- image --> \n(a) singular 'phase' factor for different ordering \n(b) A geometric representation of the propagator. Here we fix the span angle θ , the propagator is a summation over such geometries. \n<!-- image --> \nIn summary the propagator can be understood as an integral of JT gravity partition functions over geometries 3(b) with different θ s. \nFinally, let us comment on the issues we raised in section 4.2. In the large q limit we are considering the propagator at relatively large distances and in a regime where locally in AdS the integration over paths that fluctuate wildly is suppressed. Alternatively we can say that in the integration over paths we put a UV cutoff which is large compared to 1 /q but small compared to the AdS radius. This is the non-relativistic regime for the boundary particle. The quantum effects are still important at much longer distances due to the large size of AdS . In addition, if we have quantum fields in AdS, then their partition functions for these fluctuating contours that have fluctuations over distances larger than the AdS radius are expected to depend on this shape in a local way. Due to the symmetries of AdS 2 , this is simply expected to renormalize the action we already have without introducing extra terms. This can be checked explicitly for conformal field theories by using the conformal anomaly to compute the effective action of the CFT 2 on a portion of H 2 (Appendix C).", '5.2 Wheeler-DeWitt Wavefunction': "In the pure JT theory we can think about quantizing the bulk theory and obtaining the Wheeler-DeWitt wavefunction. This was discussed in the classical limit by Harlow and Jafferis [10]. \nThe Wheeler-DeWitt wavefunction can be created by Euclidean evolution of the boundary and hence is closely related to the propagator we have discussed above. The wavefunction in Lorentzian signature could then be obtained by analytic continuation of the boundary time. The Euclidean evolution can be specified by either of the two parameters: \n૪ \nthe proper length u or energy E . Choosing a different parameter corresponds to imposing a different boundary condition in JT theory. In general there are four possible choices of boundary conditions in 2d dilaton gravity, there are two sets of conjugate variables: { φ b , K } , and { u, E } 6 . In preparing the wavefunction we fix the boundary value of dilaton and hence there are only two choices of the parameter ( u or E ). We denote the corresponding wavefunction as | u 〉 G and | E 〉 G respectively. In terms of holographic considerations, | u 〉 G represents a thermofield double state: \n| u 〉 G ∼ ∑ n e -E n u | E n 〉 L | ¯ E n 〉 R (5.47) \nand | E 〉 G is like an average of energy eigenstates in a window of energy E : \n| E 〉 G ∼ 1 δE ∑ | E -E n | <δE | E n 〉 L | ¯ E n 〉 R . (5.48) \nThe width of the energy window is some coarse graining factor such that the summation contains e S 0 states and does not show up clearly in gravity. 7 \nWith the definition of the states, one can evaluate them in terms of different basis. There are three natural bases turn out to be useful, we call them S , η and glyph[lscript] bases. Basis S corresponding to fix the horizon value of dilaton field φ h , or equivalently by BekensteinHawking formula, the entropy of the system. The canonical conjugate variable of S will be called η and that characterizes the boost angle at the horizon. glyph[lscript] stands for fixing geodesic distance between two boundary points. To see that the horizon value of the dilaton field is a gauge invariant quantity, one can do canonical analysis of JT gravity. With ADM decomposition of the spacetime metric, one can get the canonical momenta and Hamiltonian constraints of the system [23]: \nds 2 = -N 2 dt 2 + σ 2 ( dx + N x dt ) 2 ; (5.49) \nH = -Π φ Π σ + σ -1 φ '' -σ -2 σ ' φ ' -σφ ; H x = Π φ φ ' -σ Π ' σ ; (5.50) \nΠ φ = N -1 ( -˙ σ +( N x σ ) ' ) = Kσ ; Π σ = N -1 ( -˙ φ + N x φ ' ) = ∂ n φ. (5.51) \nThat is the dilaton field is canonically conjugate to the extrinsic curvature and boundary metric is canonical conjugate to the normal derivative of the dilaton field (both are pointing inwards). By a linear combination of the Hamiltonian constraints (5.50), one can construct the following gauge invariant quantity C : \n-1 σ ( φ ' H +Π σ H x ) = 1 2 (Π 2 σ + φ 2 -φ ' 2 σ 2 ) ' ≡ C [Π σ , φ, σ ] ' ∼ 0 (5.52) \nThe Dirac quantization scheme then tells us that the quantity C has a constant mode which is gauge invariant (commute with Hamiltonian constraint). Choosing the gauge that normal derivative of the dilaton is zero, we can solve the Hamiltonian constraint: \nφ 2 -( ∂ X φ ) 2 = 2 C ≡ S 2 → φ ( X ) = S cosh X, (5.53) \nwhere dX = σdx is the proper distance along the spatial slice. Because the normal derivative of dilaton field is zero, the minimum value of dilaton at this spatial slice is actually a local extremum in both directions. Therefore, the minimal value of dilaton field, namely S , is a global variable. The classical geometry in this gauge is a 'Pac-Man' shape (right figure in figure 4). Focusing on the intersection region of the spatial slice and the boundary, we have the spatial slice is orthogonal to the boundary. This is because we are gauge fixing ∂ n φ = 0 on the spatial slice, and φ = φ b on the boundary. The ADM mass of the system, after regularization, is then M = φ b ( φ b -∂ X φ ) [13]. Substituting the behavior of φ ( X ) we get: \nM = S 2 2 . (5.54) \nThis is the same relation in 4.26 and therefore we can interpret the s variable as entropy of our system S . \nFor the purpose of fixing geodesic distance, it is convenient to think of doing the path integral up to a slice L with zero extrinsic curvature. This picks out a particular slice (left figure in Figure 4) among the solutions obeying the Hamiltonian constraint. The WdW wavefunction can be evaluated as an Euclidean path integral with fixed (rescaled) geodesic distance d between the two boundary points: \nΨ( u ; d ) = ∫ D g D φe 1 2 ∫ φ ( R +2)+ ∫ L φK + φ b ∫ Bdy ( K -1)+ φ b ( α 1 + α 2 -π ) = ∫ D fe φ b ∫ Bdy ( K -1)+ φ b ( α 1 + α 2 -π ) = ∫ D x e -m ∫ Bdy √ g + q ∫ Bdy a + q ∫ L a + πq (5.55) \nHere we are fixing the total length of L to be d and the proper length of the boundary to be u . α 1 and α 2 in the last expression denotes the jump angle at the corner coming from the singular contribution of the extrinsic curvature and should be integrated over. Without the e q ∫ L a factor in (5.55), the path integral corresponds to the propagator (5.35). Remember that the phase factor is equal to e -q ∫ L a , so the wavefunction in glyph[lscript] basis is actually the propagator (5.35), with the phase factor stripped out \nΨ( u ; glyph[lscript] ) ≡ 〈 glyph[lscript] | u 〉 G = 2 π 2 glyph[lscript] ∫ ∞ 0 dss sinh(2 πs ) e -s 2 2 u K 2 is ( 4 glyph[lscript] ) , glyph[lscript] = | x 1 -x 2 | √ z 1 z 2 . (5.56) \nglyph[lscript] is a function of the regularized geodesic distance d between x 1 and x 2 : glyph[lscript] = e d 2 . The semiclassical of Ψ( u ; glyph[lscript] ) can be obtained using formula (5.36), in the exponent we get saddle \npoint result: \nΨ( u ; glyph[lscript] ) ∼ exp[ -2( ξ ∗ -iπ ) 2 u + 4 u ξ ∗ -iπ tanh ξ ∗ ]; ξ ∗ -iπ u = -sinh ξ ∗ glyph[lscript] . (5.57) \nThe same saddle point equation and classical action was obtained in [10] by a direct evaluation in JT gravity. \nThe wavefunction with fixed energy boundary condition can obtained by multiplying Ψ( u ; glyph[lscript] ) by e + Eu and integrating over u along the imaginary axis. This sets E = s 2 2 in the above integral over s . So this wavefunction has a very simple expression: \nΨ( E ; glyph[lscript] ) ≡ 〈 glyph[lscript] | E 〉 G = ρ ( E ) 4 glyph[lscript] K i √ 8 E ( 4 glyph[lscript] ) . (5.58) \nThe classical geometry for Ψ( E ; glyph[lscript] ) is the same as the left figure in Figure 4, with fixing energy on the boundary. We want to stress that it is important to have the ρ ( E ) factor in (5.58) for a classical geometry description since we are averaging over the states. We can roughly think of 4 glyph[lscript] K i √ 8 E ( 4 glyph[lscript] ) as a gravitational 'microstate' |E〉 with fixed energy E . Such a 'microstate' will not have a classical geometry representation and therefore is just a formal definition. The inner product between wavefunctions is defined as 〈 Ψ 1 | Ψ 2 〉 = ∫ ∞ 0 dglyph[lscript]glyph[lscript] Ψ ∗ 1 ( glyph[lscript] )Ψ 2 ( glyph[lscript] ). \nGoing to the entropy basis S , it is easy to start with Ψ( E ). Because of the identity E = S 2 2 , expanding Ψ( E ) in the S basis is diagonal: \nΨ( E ; S ) ≡ 〈 S | E 〉 G = √ ρ ( S ) δ ( E -S 2 2 ) (5.59) \nWe put this square root of ρ ( S ) factor in the definition of S basis such that inner product between different S state is a delta function 〈 S | S ' 〉 = δ ( S -S ' ). This factor is also required such that the classical limit matches with gravity calculation. Integrating over energy with Boltzman distribution, we can get the expression of thermofield double state in the S basis: \nΨ( u ; S ) ≡ 〈 S | u 〉 G = ∫ dEe -uE 〈 S | E 〉 G = √ ρ ( S ) e -uS 2 2 (5.60) \nIn the semiclassical limit, the wavefunction becomes gaussian and coincides with the on shell evaluation of the 'Pac-Man' geometry (Figure 4): \nΨ( u, S ) ∼ √ Se πS -uS 2 2 (5.61) \nThe on shell calculation is straightforward: JT action in this geometry contains two parts: the Schwarzian action ∫ ( K -1) on the boundary and a corner contribution at the center: S ( π -θ ), where θ is the span angle at the horizon (Figure 4). The Schwarzian action simply gives Eu = S 2 u 2 by direct evaluation. We can determine θ from u since they are related with redshift: θ = uS . Therefore the corner term gives: πS -S 2 u . Adding them \nFigure 4: Classical Geometry in glyph[lscript] and S basis \n<!-- image --> \nup then gives us the classical action. We can also expand S in terms of the glyph[lscript] basis, and relate Ψ( glyph[lscript] ) with Ψ( S ) by a change of basis: \n〈 glyph[lscript] | S 〉 = √ ρ ( S ) 4 glyph[lscript] K 2 iS ( 4 glyph[lscript] ); Ψ( E ; glyph[lscript] ) = ∫ ∞ 0 dS 〈 glyph[lscript] | S 〉〈 S | E 〉 G ; (5.62) \nBefore discussing our last basis, we want to stress the simplicity of the wavefunction in S basis (5.61) and the Gaussian factor resembles an ordinary particle wavefunction in momentum basis. We introduce our last basis η as canonical conjugate variable of S , with an analog of going to position space of the particle picture in mind: \n| η 〉 = ∫ ∞ 0 dS cos( ηS ) | S 〉 ; 〈 glyph[lscript] | η 〉 = ∫ ∞ 0 dS cos( ηS ) √ ρ ( S ) K 2 iS ( 4 glyph[lscript] ) 4 glyph[lscript] ;(5.63) Ψ( E,η ) = √ sinh(2 π √ 2 E ) 2 π 2 √ 2 E cos( η √ 2 E ); Ψ( u, η ) = ∫ ∞ 0 dS √ ρ ( S ) cos( ηS ) e -uS 2 2 . (5.64) \nTo understand the meaning of η better, we can look at the classical behavior of Ψ( u ; η ): \nΨ( u ; η ) ∼ 1 u exp[ π 2 2 u -η 2 2 u ] ( e iηπ u √ π + iη + e -iηπ u √ π -iη ) (5.65) \nWhen u is real, the wavefunction is concentrated at η = 0 and has classical action of a half disk in the exponent. When u = β 2 + it which corresponds to the case of analytically continuing into Lorentzian signature, the density of the wavefunction | Ψ( u, η ) | 2 is dominated \nFigure 5: Euclidean geometries with different cusps. \n<!-- image --> \nby: \n| Ψ( β 2 + it, η ) | 2 ∼ √ π 2 + η 2 β 2 +4 t 2 exp[ 2 π 2 β ] ( exp[ -2 β ( η -2 πt β ) 2 β 2 +4 t 2 ] + exp[ -2 β ( η + 2 πt β ) 2 β 2 +4 t 2 ] ) (5.66) \nshowing the fact that η is peaked at the Rindler time 2 πt β . We can therefore think of fixing η as fixing the IR time or the boost angle at the horizon. The classical intuition for the boost angle is most clear in Euclidean geometry, where for fixed boundary proper time there can be different cusps at the horizon (Figure 5). \nOne application of those wavefunctions is that we can take an inner product and get the partition function. However, there are also other ways to get the partition function. For example, we can concatenate three propagators and integrate over their locations. This also gives the partition function by the composition rule of propagator. By the relation between propagator and wavefunction, we can also view this as taking an inner product of three wavefunctions with an interior state as in figure 6, where the interior state can be understood as an entangled state for three universes. To be more precise, we can view the wavefunction as the result of integrating the bulk up to the geodesics with zero extrinsic curvature. Then the interior state is given by the area of the hyperbolic triangle in figure 6. The path integral for the hyperbolic triangle (denoted as I ( glyph[lscript] 12 , glyph[lscript] 23 , glyph[lscript] 31 ), where glyph[lscript] ij = | x i -x j | √ z i z j ), is a product of three phase factors, which satisfies a nontrivial equality (with ordering x 1 > x 2 > x 3 ): \nI ( glyph[lscript] 12 , glyph[lscript] 23 , glyph[lscript] 31 ) = e -2( z 1 + z 2 x 1 -x 2 + z 2 + z 3 x 2 -x 3 + z 3 + z 1 x 3 -x 1 ) = 16 π 2 ∫ ∞ 0 dττ sinh(2 πτ ) K 2 iτ ( 4 glyph[lscript] 12 ) K 2 iτ ( 4 glyph[lscript] 23 ) K 2 iτ ( 4 glyph[lscript] 31 ) . (5.67) \nRecalling that the Bessel function represents the fixed energy 'microstate' |E〉 (5.58) and τ 2 π 2 sinh(2 πτ ) is the density of state, this formula tells us that the interior state is a GHZ state for three universe: \nI 123 ∼ ∑ n |E n 〉 1 |E n 〉 2 |E n 〉 3 . (5.68) \nFigure 6: Partition function from inner product of three wavefunctions. \n<!-- image --> \nI can also been viewed as a scattering amplitude from two universes into one universe. It constrains the SL(2,R) representation of the three wavefunctions to be the same. 8 We can write down the partition function as: \nZ JT = ∫ ∞ 0 ∏ { ij }∈{ 12 , 23 , 31 } dglyph[lscript] ij Ψ( u 12 , glyph[lscript] 12 )Ψ( u 23 , glyph[lscript] 23 )Ψ( u 31 , glyph[lscript] 31 ) I glyph[lscript] 12 ,glyph[lscript] 23 ,glyph[lscript] 31 . (5.69) \nThis same result also holds if we repeat the process n times. It is interesting that we can view the full disk amplitude in these various ways. \nOne can also extend our analysis to include matter field. One type of such wavefunction can be created by inserting operator during Euclidean evolution, and is analysed in appendix B. Note that because of the SL(2,R) symmetry is a gauge symmetry, our final state has to be a gauge singlet including matter field.", '6.1 Gravitational Feynman Diagram': "The propagator enables us to 'dress' quantum field theory correlators to produce quantum gravity ones. Namely, we imagine that we have some quantum field theory in H 2 and we compute correlation functions of operators as we take the points close to the boundary where they take the form \n〈 O 1 ( x 1 ) ...O n ( x n ) 〉 QFT = q -∑ ∆ i z ∆ 1 1 ..z ∆ n n 〈 O 1 ( x 1 ) ...O n ( x n ) 〉 CFT (6.70) \nThe factor of q arises from (5.33), and the last factor is simply defined as the function that results after extracting the z dependence. For example, for a two point function we get \n〈 O 1 ( x 1 ) O 2 ( x 2 ) 〉 QFT = q -2∆ z ∆ 1 z ∆ 2 1 | x 1 -x 2 | 2∆ . (6.71) \nWe can now use the propagator (5.35) to couple the motion of the boundary and thus obtain the full quantum gravity expression for the correlator. The factors of q are absorbed as part of the renormalization procedure for defining the full quantum gravity correlators. In this way we obtain \nFigure 7: Summation of 1 N effects fluctuates the boundary of Witten Diagram \n<!-- image --> \n〈 O 1 ( u 1 ) ...O n ( u n ) 〉 QG = e 2 πq ∫ ∏ n i =1 dx i dy i y 2 i V(SL(2,R)) G ( u 12 , x 1 , x 2 ) G ( u 23 , x 2 , x 3 ) ...G ( u n 1 , x n , x 1 ) × ×〈 O 1 ( x 1 ) ...O n ( x n ) 〉 QFT q ∑ i ∆ i (6.72) \nwhere the left hand side is the full quantum gravity correlator by definition. The last factor is the usual renormalization necessary to get something finite. \nThe factor of e 2 πq cancels with the q dependent 'phase' factors in (5.34) to give one if we order the points cyclically ( x 1 > x 2 ... > x n ). This requires that we define more carefully the last propagator G ( u n 1 , x n , x 1 ) as: \ne -2 πq ˜ K ( u n 1 , x n , x 1 ) = e -2 πq e -2 zn + z 1 xn -x 1 2 √ z n z 1 π 2 | x n -x 1 | ∫ ∞ 0 dss sinh(2 πs ) e -s 2 2 u n 1 K 2 is ( 4 √ z n z 1 | x n -x 1 | ) . (6.73) \nThe factor 1 V(SL(2,R)) in (6.72) means that we should fix the SL (2 , R ) gauge symmetry (Appendix A). \nIn the end we can write down an expression where we have already taken the q →∞ limit \n〈 O 1 ( u 1 ) ...O n ( u n ) 〉 QG = ∫ x 1 >x 2 ..>x n ∏ n i =1 dx i dz i V(SL(2,R)) ˜ K ( u 12 , x 1 , x 2 ) ... ˜ K ( u n 1 , x n , x 1 ) z ∆ 1 -2 1 ..z ∆ n -2 n 〈 O 1 ( x 1 ) ...O n ( x n ) 〉 CFT . (6.74) \nThis is one of the main results of our paper and it gives a detailed expression for correlation function in 2 dimensional quantum gravity in terms of the correlation functions of the QFT in hyperbolic space, or AdS 2 . \nNotice that in usual AdS/CFT the correlators 〈 O 1 ( x 1 ) ...O n ( x n ) 〉 CFT are an approximation to the full answer. This is sometimes computed by Witten diagrams. We get a better approximation by integrating over the metric fluctuations. In this case, the non-trivial gravitational mode is captured by the boundary propagator. The formula (6.74) includes all the effects of quantum gravity in the JT theory (in the Schwarzian limit). The final diagrams consist of the Witten diagrams for the field theory in AdS plus the propagators for the boundary particle and we can call them 'Gravitational Feynman Diagrams', see figure 7.", '6.2 Two Point Function': 'Using formula (6.74), we can study gravitational effects on bulk fields such as its two point function: \n<!-- image --> \nThe explicit expression for 〈 O 1 ( u ) O 2 (0) 〉 QG with dimension ∆ at temperature 1 β is 9 : \n1 V(SL(2,R)) ∫ x 1 >x 2 dx 1 dx 2 dz 1 dz 2 z 2 1 z 2 2 ∫ ∞ 0 ds 1 ds 2 ρ ( s 1 ) ρ ( s 2 ) e -s 2 1 2 u -s 2 2 2 ( β -u ) K 2 is 1 ( 4 √ z 1 z 2 | x 1 -x 2 | ) K 2 is 2 ( 4 √ z 1 z 2 | x 1 -x 2 | )( √ z 1 z 2 | x 1 -x 2 | ) 2∆+2 . (6.76) \nTo fix the SL (2 , R ) gauge, we can choose z 1 = z 2 = 1 and x 2 = 0. Then the integral over H 2 space is reduced to a single integral over x 1 , with a Jacobian factor 2 x 1 (Appendix A): \n∫ ∞ 0 ds 1 ds 2 ρ ( s 1 ) ρ ( s 2 ) e -s 2 1 2 u -s 2 2 2 ( β -u ) ∫ ∞ 0 dx 1 ( 1 x 1 ) 2∆+1 K 2 is 1 ( 4 x 1 ) K 2 is 2 ( 4 x 1 ) . (6.77) \nthe last integral can be interpreted as a matrix element of two point operator O 1 O 2 between states | E 1 , ψ 〉 and | E 2 , ψ 〉 , where | ψ 〉 is the wavefunction of quantum field theory and | E 〉 G represents the fixed energy gravitational state. Integrating over x can be thought as integrating over a particular gravitational basis, and we can see that the gravity wavefunction suppress the UV contributions from quantum field theory ( K 2 is ( 4 x ) ∼ √ πx 8 e -4 /x \nfor x ∼ 0). The final expression for the two point function is: \n〈 O 1 ( u ) O 2 (0) 〉 QG = 1 N ∫ ds 1 ds 2 ρ ( s 1 ) ρ ( s 2 ) e -s 2 1 2 u -s 2 2 2 ( β -u ) | Γ(∆ -i ( s 1 + s 2 ))Γ(∆ + i ( s 1 -s 2 )) | 2 2 2∆+1 Γ(2∆) ;(6.78) = 1 N Γ(2∆) u 3 / 2 ( β -u ) 3 / 2 2 4∆+4 π 3 ∫ c + i ∞ c -i ∞ dθ 1 dθ 2 θ 1 θ 2 e θ 2 1 2 u + θ 2 2 2( β -u ) 1 (cos θ 1 2 +cos θ 2 2 ) 2∆ . (6.79) \nIn the second expression we write the integral in terms of variable θ using the second integral representation of the propagator (5.36). The normalization constant can be determined by taking the ∆ = 0 limit: N = Z JT . \nIf we contemplate the result (6.78) a little bit, then we find that the two integrals of s 1 and s 2 just represent the spectral decomposition of the two point function. Indeed, under spectral decomposition we have 〈 O ( u ) O (0) 〉 = ∑ n,m e -E n u -E m ( β -u ) |〈 E n | O | E m 〉| 2 . Compare \nwith (6.78), we can read out the square of matrix element of operator O : \nG 〈 E 1 | O L O R | E 2 〉 G = δE -2 ∑ | E n -E 1 | <δE | E m -E 2 | <δE |〈 E n | O | E m 〉| 2 = ρ ( E 1 ) ρ ( E 2 ) | Γ(∆ -i ( √ 2 E 1 + √ 2 E 2 ))Γ(∆ + i ( √ 2 E 1 -√ 2 E 2 )) | 2 2 2∆+1 Γ(2∆) . \n(6.80) \nRemember the notation is that | E 〉 G stands for a gravitational state with energy E and | E n 〉 stands for one side microstate (5.48). We have put the measure ρ ( E ) = 1 2 π 2 sinh(2 π √ 2 E ) in the definition of matrix element for the reason that in gravity it is more natural to consider an average of energy states as a bulk state. To understand this formula a little bit better, we can consider the classical limit, namely large E . In this limit the matrix element squared can be approximated as a nonanalytic function: \nG 〈 E 1 | O L O R | E 2 〉 G ∝ | E 1 -E 2 | 2∆ -1 e 2 π min( √ 2 E 1 , √ 2 E 2 ) . (6.81) \nIf we fix E 1 and varying E 2 from 0 to infinity, the matrix element changes from | E 1 -E 2 | 2∆ -1 ρ ( E 2 ) to | E 1 -E 2 | 2∆ -1 ρ ( E 1 ) after E 2 cross E 1 . We can understand this behavior qualitatively as a statistical effect: the mapping from energy subspace E 1 to E 2 by operator O is surjective when the Hilbert space dimension of E 2 is less that E 1 and is injective otherwise. Another understanding is the following: the two point function is finite in a fixed energy state | E n 〉 , which means the following summation of intermediate states | E m 〉 is order one: ∑ m |〈 E n | O | E m 〉| 2 . Looking at the case E m > E n , because of the density of states grows rapidly, the matrix element squared has to be proportional to 1 ρ ( E m ) to get a finite result. Multiplied by ρ ( E n ) ρ ( E m ), we have ρ ( E n ) ρ ( E m ) |〈 E n | O | E m 〉| 2 ∼ ρ (min( E n , E m )).', '6.3 ETH and the KMS condition': "The Eigenstate Thermalization Hypothesis (ETH) is a general expectation for chaotic system. It expresses that the operator expectation value in an energy eigenstate can be \napproximated by thermal expectation value with effective temperature determined from the energy. Such hypothesis can be tested with the knowledge of operator matrix elements. The two point function in microcanonical essemble is: \n1 δE ∑ | E n -E | <δE 〈 E n | O ( u ) O (0) | E n 〉 = ρ ( E ) e uE 〈 E | Oe -uH O | E 〉 〈 E | Oe -uH O | E 〉 = ∫ ∞ 0 dsρ ( s ) e -s 2 2 u | Γ(∆ + i ( s + √ 2 E )) | 2 | Γ(∆ + i ( s -√ 2 E )) | 2 2 2∆+1 Γ(2∆) . (6.82) \nNotice that | E 〉 is not | E 〉 G , the former represents a one side microstate, while the later is a gravitational state. Accordingly 〈 E | O ( u ) O (0) | E 〉 stands for a two point function in a microstate. To study ETH, we will consider the case of a heavy black hole E = S 2 2 = 2 π 2 β 2 glyph[greatermuch] 1. From the discussion in last section, we know that the matrix element tries to concentrate s around √ 2 E and thus we can approximate ρ ( E ) ρ ( s ) | Γ(∆ + i ( s + S )) | 2 as proportional to sS 2∆ -1 e π ( s + S ) . Using integral representation for | Γ(∆ + i ( s -S )) | 2 we derive the two point function in microcanonical essemble with energy E is proportional to: \nρ ( E ) S 2∆ -1 u 3 / 2 ∫ dξ ( π + iξ ) e -2 u ( ξ -i π 2 ) 2 -( π +2 iξ ) S + u S 2 2 1 (cosh ξ ) 2∆ . (6.83) \nThe ξ variable can be understood as the measure of time separation in units of effective temperature between two operators and its fluctuation represents the fluctuation of the effective temperature. And the final integral can be understood as a statistical average of correlation functions with different temperatures. If we put back the Newton Constant G N = 1 N , we have S ∼ N and u ∼ N -1 (2.5). As can be seen from the probability distribution, the fluctuation is of order 1 √ N , and hence for large N system we can use saddle point approximation: \nξ = i ( π 2 -S 2 u ) -u ∆tanh ξ 2 = i ( π 2 -πu β ) -u ∆tanh ξ 2 . (6.84) \nThe first piece gives the typical temperature of the external state, while the last piece comes from the backreaction of operator on the geometry. If we first take the limit of large N, one simply get that the two point function in microcanonical essemble is the same as canonical essemble. However, the euclidean correlator in canonical essemble is divergent as euclidean time approach to inverse temperature β because of KMS condition. Such singular behavior plays no role in the microcanonical essemble so is called a 'forbidden singularity' in ETH [26, 27]. In our situation we can see directly how the forbidden singularity disappears in the microcanonical essemble. When ξ approach -i π 2 at the forbidden singularity the backreaction on the geometry becomes large and hence the effective temperature becomes lower: \n2 π β ∗ → 2 π β -∆ π -πu β . (6.85) \nFigure 8: Bulk Diagram for Three Point Function \n<!-- image --> \nAt the time β -u β ∼ ∆ N , the backreaction is important and we expect to see deviation from thermal correlators. Therefore the correlation function in microcanonical essemble will never have singularity away from coincide point.", '6.4 Three Point Function': 'The bulk diagram of the three point function will be like Figure 6 with additional operator inserting at the intersection points (See Figure 8). The QFT three point correlation function in AdS 2 is fixed by conformal symmetry and we can write it down as: \n〈 O 1 O 2 O 3 〉 = C 123 z ∆ 1 1 z ∆ 2 2 z ∆ 3 3 | x 12 | ∆ 1 +∆ 2 -∆ 3 | x 23 | ∆ 2 +∆ 3 -∆ 1 | x 13 | ∆ 1 +∆ 3 -∆ 2 = C 123 glyph[lscript] ∆ 1 +∆ 2 -∆ 3 12 glyph[lscript] ∆ 2 +∆ 3 -∆ 1 23 glyph[lscript] ∆ 1 +∆ 3 -∆ 2 13 . (6.86) \n∆ i is the conformal dimension of O i . Putting them in formula 6.74 and rewrite the propagator in terms of the wavefunction (5.56), we have the quantum gravitational three point function: \n〈 O 1 O 2 O 3 〉 QG = ∫ x 1 >x 2 >x 3 ∏ 3 i =1 dx i dz i V(SL(2,R)) Ψ u 12 ,glyph[lscript] 12 Ψ u 23 ,glyph[lscript] 23 Ψ u 13 ,glyph[lscript] 13 I glyph[lscript] 12 ,glyph[lscript] 23 ,glyph[lscript] 13 C 123 glyph[lscript] ∆ 1 +∆ 2 -∆ 3 12 glyph[lscript] ∆ 2 +∆ 3 -∆ 1 23 glyph[lscript] ∆ 1 +∆ 3 -∆ 2 13 . (6.87) \nWe can view this expression as an inner product of three universe wavefunction with the interior, inserting three bilocal operators ˜ O ij ; k ˜ O ij ; k with dimension ˜ ∆ ij ; k = 1 2 (∆ i +∆ j -∆ k ) between them. One can fix the SL (2 , R ) symmetry and express the integral in terms of glyph[lscript] ij , it is the same exercise as in open string calculation to find the Jacobian factor (Appendix A). Here we can just argue that in order to get the partition function at ∆ = 0, the measure has to be flat. Therefore the three point function factorizes into form: \n〈 O 1 ( u 1 ) O 2 ( u 2 ) O 3 ( u 3 ) 〉 QG ∝ C 123 ∫ ∞ 0 dτρ ( τ ) I τ ( u 12 , ˜ ∆ 12;3 ) I τ ( u 23 , ˜ ∆ 23;1 ) I τ ( u 31 , ˜ ∆ 31;2 ) (6.88) \nwhile I τ ( u ij , ˜ ∆ ij ; k ) is an integral of glyph[lscript] ij which gives the two point function in microstate E τ (6.82) with the e uE τ factor stripped off: \nI τ ( u ij , ˜ ∆ ij ; k ) = 1 2 ∫ ∞ 0 dglyph[lscript] ij Ψ u ij ,glyph[lscript] ij 1 glyph[lscript] ∆ i +∆ j -∆ k ij K 2 iτ ( 4 glyph[lscript] ij ) = 〈 E τ | ˜ O ij ; k e -u ij H ˜ O ij ; k | E τ 〉 ; E τ = τ 2 2 . (6.89) \nAgain the normalization constant can be fixed by choosing O i to be identity.', '6.5 Einstein-Rosen Bridge': 'The Einstein-Rosen Bridge in a classical wormhole keeps growing linearly with time and this behavior was conjectured to related with the growth of computational complexity of the dual quantum state [28]. Based on the universal behavior of complexity growth, Susskind proposed a gravitational conjecture in a recent paper [11] about the limitation of classical general relativity description of black hole interior. The conjecture was stated as follows: \nClassical general relativity governs the behavior of an ERB for as long as possible. \nIn this section, we will test this conjecture using the exact quantum wavefunction of JT gravity (5.56). We will in particular focusing on the behavior of ERB at time bigger than 1. The size of Einstein-Rosen Bridge V in two dimensions is the geodesic distance d between two boundaries, and can be calculated in thermofield double state | u 〉 as: \nV = 〈 u | d | u 〉 (6.90) \nWe want to focus on the dependence of volume on Lorentzian time evolution. Therefore we do analytic continuation of u in Lorentzian time: u = β 2 + it . Using the WdW wavefunction in glyph[lscript] basis (5.56) and the relation between d and glyph[lscript] , we can calculate the expectation value exactly. This can be done by taking the derivative of the two point function (6.78) with respect to ∆ at ∆ = 0. Using the integral representation for | Γ(∆+ i ( s 1 -s 2 )) | 2 , the only time dependence of volume is given by: \nV ( t ) = 1 N ∫ ∞ -∞ dξ ∫ ∞ 0 ds 1 ds 2 ρ 1 ρ 2 e i ( s 1 -s 2 ) ξ -i ( s 1 -s 2 ) ( s 1 + s 2 ) 2 t -β 4 ( s 2 1 + s 2 2 ) log(2 cosh ξ 2 ) | Γ( is 1 + is 2 ) | 2 . (6.91) \nThe limit we are interested in is β glyph[lessmuch] 1 glyph[lessmuch] t 10 , in which case the integral has a saddle point at 11 : \ns 1 , 2 = 2 π β ; ξ = 2 πt β . (6.92) \nTherefore the volume has linear dependence in time: \nV ( t ) ∼ 2 πt β . (6.93) \nUsing the complexity equal to volume conjecture [28, 29], the complexity of thermofield double state is proportional to the maximum volume: \nC ( t ) = # V ( t ) = # 2 πt β . (6.94) \nThe proportionality constant is suggested in [30] to be S 0 based on classical calculation of near extremal black hole. This, however, is not very clear in our model since S 0 is the coupling constant of the pure Einstein-Hilbert action and decouples with JT theory (1.1). Since the saddle point (6.92) is actually valid from early time to late time, the proportionality constant can be fixed at classical level and once we fix it we can conclude that the length of Einstein-Rosen Bridge (or complexity of the state) keeps linearly growing even considering quantum gravity effects in JT theory. We want to comment that this is not an obvious result that one can expect from classical observables. For example, one might argue that we can extract the information of the ERB from two sided correlators for the reason that semiclassically we can approximate the correlator as e -md . Therefore one can conclude the ERB has linear growth from the quasinormal behavior of the correlator. However such observables can only give us information of ERB up to time order 1, which is the same time scale we can trust the classical general relativity calculation. After that the correlation function changes from exponential decay into universal power law decay 1 t 3 as one can directly derive from analytic continuation of result (6.78). If we still use such correlator to extract information about ERB we would get the wrong conclusion that it stops its linear growth after time order 1. The reason why it is incorrect is that at this time scale the operator disturbs the state and causes different energy states interfere each other strongly. It is simply that the correlator can no longer be described by the classical geometry, rather than the interior stops to behave classically. From our calculation, we see that if we probe of the state in a weaker and weaker way, we are still able to see the classical geometry. Lastly, we want to talk a little about when JT gravity needs to be modified. A naive estimate can be made from the partition function that when β approaches e 2 3 S 0 , the partition function becomes less than one and definitely at this time scale we need new physics. A recently study of gravitational physics at this time scale was discussed in [22].', '7 Conclusion': 'Our result gives an explicit formula (6.74) to calculate all order corrections to correlation functions from quantum gravity in two dimensions. The formula can be understood diagrammatically and we call it Gravitational Feynman Diagram. We also give the exact Wheeler-DeWitt wavefunction and discuss the growth of its complexity quantum mechanically. \nAlthough we are focusing on theoretical description of two dimensional black holes, the near-extremal black holes in nature should contain these features. Both Reissner-Nordstrom black holes and Kerr black holes have an AdS 2 throat near their extremality. For those black holes, the gravitational effects are enhanced by the their near extremal entropies (the coupling constant is φ h rather than φ 0 + φ h ) and therefore are better backgrounds to test gravitational effects. We should however point out that the observational black holes all have large near extremal entropies and thus are very classical [43]. In addition, the Thorne limit of Kerr black hole sets a lower bound on the near extremal entropy in nature. But for the Primordial black holes in early universe, our story might play a role and it will be interesting to study the physical consequence in that situation. \nAnother application of our result is to connect with SYK type models [31, 32, 33, 34] since those models all have an emergent Schwarzian action at low energy. On that account, the exact Schwarzian quantization can be used to test 1 N corrections to those models. For example, one can try to directly test the two point function with SYK numerics [35] or one can use SYK models to understand the microscopic description of WdW wavefunction and its complexity. It might also be interesting to consider the black hole information paradox [42] and late time traversable wormholes including the quantum fluctuations of the boundary [36, 37, 38, 39, 45].', 'Acknowledgments': 'The author wants to give special thanks to J.Maldacena and D.Stanford for patient guidance and help all through the project. We also want to thank A.Almheiri, L.Iliesiu, J.Jiang, J.Ripley, A.Kitaev, B.Kobrin, R.Mahajan, A.M.Polyakov, S.J.Suh, G.Turiaci and H.Verlinde for discussions. Z.Y is supported by Charlotte Elizabeth Procter Fellowship from Princeton University.', 'A Gauge Fix SL(2,R)': 'This section reviews the procedure to fix SL(2,R) gauge which is needed for calculating correlation functions in quantum gravity using formula (6.74). With the parametrization of group elements in SL(2,R) by g = e iglyph[epsilon1] α L α ( α = ± 1 , 0) near the identity, we have g acting on x as following (5.37): \ngx = x -glyph[epsilon1] -1 -glyph[epsilon1] 0 x + glyph[epsilon1] 1 x 2 ; gz = z -glyph[epsilon1] 0 z + glyph[epsilon1] 1 2 xz. (A.95) \nChoosing the gauge fixing condition as f α ( g x ) = 0, we can fix the SL(2,R) symmetry in (6.74) using Faddeev-Popov method. First we have the identity: \n1 = M ( x ) ∫ dgδ ( f α ( g x )) (A.96) \nBecause the measure is invariant under group multiplication, M ( x ) is equal to M ( g x ) and we can calculate it at the solution x 0 of the gauge constraints on its orbit: f α ( x 0 ) = 0, \nFigure 9: WdW wavefunction with matter \n<!-- image --> \nx 0 ∈ G ( x ). Since the Haar measure is flat near the identity we have 12 : \nM ( x ) = det ( ∂f α ( g x 0 ) ∂glyph[epsilon1] β )∣ ∣ ∣ ∣ glyph[epsilon1] β =0 . (A.97) \nInserting 1 in integrals of SL(2,R) invariant function F ( x ) like the one in (6.74), we have: \n∫ d x F ( x ) = ∫ d x M ( x ) ∫ dgδ ( f α ( g x )) F ( x ) = ∫ dg ∫ d x M ( x ) δ ( f α ( x )) F ( x ) . (A.98) \nWe see that the volume of SL(2,R) factorizes out and we have the gauge fixed expression: \n∫ d x det ( ∂f α ( g x 0 ) ∂glyph[epsilon1] β )∣ ∣ ∣ ∣ glyph[epsilon1] β =0 δ ( f α ( x )) F ( x ) . (A.99)', 'B Wavefunction with matter': "Including matter sector in JT gravity (action 1.1), we can discuss the exact wavefunction including matter backreaction. Schematically, since the geometry on which the matter field propagates is not changed, the WdW wavefunction Φ including matter sector will be : \n| Φ 〉 = ∑ n | Ψ n 〉 ⊗ | n 〉 , (B.100) \nwhere | n 〉 denotes the matter state in fixed AdS background, and | Ψ n 〉 means the gravitational wavefunction after backreaction from matter state | n 〉 . By specifying the boundary condition of the matter in Euclidean evolution, one can create different types of states such as vacuum state. The vacuum state in AdS is stable and will not backreact on the gravity sector and thus one will get the same story discussed in section 5.2. We will consider \nanother type of state that is created by inserting one boundary operator O during the euclidean evolution (Figure 9). The operator O creates a single SL(2,R) representation with conformal dimension ∆. For the reason that the boundary is fluctuating, O does not create only one asymptotic state, but a superposition of its descendants: | ∆ , n 〉 . The transition amplitude from | O ( x ) 〉 to | ∆ , n 〉 can be determined from the asymptotic behavior of two point function: \nlim x ' →∞ ∑ n 〈 O ( x ' ) | n 〉〈 n | O ( x ) 〉 = z ' ∆ z ∆ | x -x ' | 2∆ = z ' ∆ z ∆ x ' 2∆ (1 + 2∆ x x ' + ... + Γ(2∆ + n ) Γ(2∆)Γ( n ) x n x ' n + ... ) (B.101) \nTherefore we have: \n| O ( x ) 〉 = ∑ n √ Γ(2∆ + n ) Γ(2∆)Γ( n ) z ∆ x n | ∆ , n 〉 (B.102) \nNotice that because the matter carries SL(2,R) charge, the gravitational part is not a singlet and in particular will depend on the location of two boundary points x 1 and x 2 . By choosing our time slice to be the one with zero extrinsic curvature, we can get the backreacted gravitational wavefunction: \nΨ n ( x 1 , x 2 ) = e -2 z 2 + z 2 x 2 -x 1 ∫ d x ˜ K ( u 1 , x 1 , x ) ˜ K ( u 2 , x , x 2 ) √ Γ(2∆ + n ) Γ(2∆)Γ( n ) z ∆ x n . (B.103)", 'C Boundary effective action from CFT': "CFT partition function in two dimension has simple dependence on the shape of geometry by Liouville action. More precisely, the CFT partition function of central charge c on geometries related by g = e 2 ρ ˆ g is related: \nZ [ g ] = e c 6 S L Z [ˆ g ]; S L = 1 4 π [ ∫ ( ˆ ∇ ρ ) 2 + ρ ˆ R +2 ∫ ρ ˆ K ] . (C.104) \nOur strategy to get the effective action of the boundary shape is to first find a conformal map that maps the boundary into a circle, and then evaluate the Liouville action on that new metric. By Cauchy's Theorem, such a conformal map always exists and is uniquely determined up to SL(2,R) transformation. The SL(2,R) transformation does not change the weyl factor and therefore does not affect our final result. The original metric has constant negative curvature and we will parametrize it by a complex coordinate h as 4 (1 -| h | 2 ) 2 dhd ¯ h . If we denote the conformal map as h ( z ), where z is the coordinate in which the boundary is a circle | z | = 1, then the new metric in coordinate z is: \nds 2 = 4 ∂h ¯ ∂ ¯ h (1 -| h | 2 ) 2 dzd ¯ z. (C.105) \nThe holomorphic function h ( z ) determines the boundary location in h coordinate (parametrized by u ) at 13 : \nr ( u ) e i ˜ θ ( u ) = h ( e iθ ( u ) ) , (C.106) \nwhere r ( u ) and ˜ θ ( u ) are related by the metric boundary condition: \n4( r ' 2 + r 2 ˜ θ ' 2 ) (1 -r 2 ) 2 = q 2 ; r ( u ) = 1 -q -1 ˜ θ ' ( u ) + O ( q -2 ) . (C.107) \nCombine these two equations at large q we get a Riemann Hilbert type problem: \ne i ˜ θ ( u ) (1 -q -1 ˜ θ ' ( u )) = h ( e iθ ( u ) ) . (C.108) \nThis equation can be solved by the holomorphic property of h ( z ) and the solution is: \nh ( z ) = z ( 1 -1 2 πq ∫ dα e iα + z e iα -z θ ' ( α ) ) (C.109) \nChoosing our reference metric ˆ g to be flat, we have: \nρ = 1 2 (log ∂h +log ¯ ∂ ¯ h ) -log(1 -h ¯ h ) . (C.110) \nEvaluation of the Liouville action (C.104) is then straightforward and gives us a Schwarzian action: \nS L = -1 2 -1 4 πq ∫ duSch (tan( θ 2 )) . (C.111) \nWe want to remark that the sign in front of the Schwarzian action is negative so a naive attempt to get induced gravity from large number of quantum fields does not work. 14", 'D Connection with the relativistic particle and Pair Production': "We will start from a formal expression for the relativistic particle with mass m and charge q in an electric field and gradually implement these changes to get the partition function of the JT theory. The partition function for the relativistic particle has the form \nZ rel ( m 0 , q ) = ∑ Paths e -m 0 L e -q ∫ a = ∫ ∞ 0 dτ τ e -1 2 τµ 2 ∫ D x D y exp ( -∫ τ 0 dτ ' 1 2 ˙ x 2 + ˙ y 2 y 2 -q ∫ dx y ) (D.112) \nwhere L is the length of the path. In the right hand side τ is Schwinger's proper time, which is related by a renormalization factor to the actual proper time of the path [14] (Chapter 9). Also, we have that µ 2 = ( m 0 -m cr ) ˜ glyph[epsilon1] , where ˜ glyph[epsilon1] is a UV cutoff for the path integral (not to be confused with glyph[epsilon1] in (2.5)). If we are interested in the JT partition function at finite temperature, then we are interested in fixing the length of the paths. As we mentioned, this is the same as fixing the Schwinger time in (D.112). More explicitly, we can multiply Z rel ( m 0 , q ) by e m 0 β ˜ glyph[epsilon1] and then integrate over m 0 along the imaginary axis (with a suitable) real part to fix the length of the path. This then gives β = τ in the above expression. The precise value of µ 2 can be absorbed by shifting the ground state energy. It will be convenient for further purposes to set µ 2 = q 2 -1 / 4. The path integral in the right hand side of (D.112) has an infinite volume factor. We drop this factor when we divide by the volume of SL (2). In addition, the factor of 1 /τ should be dropped because we view configurations that differ by a shift in proper time as inequivalent. After all these modifications we find \nZ JT ( β ) = e S 0 e 2 πq 1 2 π G ( β, x, y ; x, y ) (D.113) \nwhere G denotes the propagator of the non-relativistic problem of a particle in an electric field \nG ( τ ; x, y ; x ' , y ' ) = 〈 x, y | e -τH | x ' , y ' 〉 = ∫ D x D y exp ( -∫ τ 0 dτ ' 1 2 ˙ x 2 + ˙ y 2 y 2 -q ∫ dx y ) (D.114) \nAt first sight, the statement that a gravitational system is equivalent to a particle makes no sense, since we know that the entropy of a particle is very small. Usually the partition function is of the form Z | p article ∼ ( glyph[planckover2pi1] ) # for a particle system, but black hole has entropy of order 1 glyph[planckover2pi1] , that is Z | BH ∼ e # glyph[planckover2pi1] . This is because in the particle case, the major contribution in functional integral is given by stationary solution, and the fluctuations near the stationary solution give the power of glyph[planckover2pi1] , while for the gravitational system, a stationary solution will corresponding to no geometry and we have the requirement of the boundary should have winding number one. A solution with winding number one in the particle system is an instanton contribution for particle pair production, which is usually very small and is in addition imaginary, so how can this matches with gravity system? The pair creation rate for a particle with charge q and mass m in AdS can be estimated from Euclidean solution which is a big circle with radius ρ b = arctanh( m q ): \nI = mL -qA ∼ 2 πm sinh ρ b -2 πq (cosh ρ b -1)+ π ( m sinh ρ b -q cosh ρ b ) δρ 2 ∼ 2 πq. (D.115) \nWe see that the damping factor is exactly cancelled out by our gravitational topological piece. The negative norm mode is related with the rescaling of the circle. That is not allowed in canonical essemble because of the temperature constraint. \n<!-- image --> \nFigure 10: Particle in real magnetic field \n<!-- image -->", 'E Details on the Schwarzian limit of Propagator': 'The main technical difficulty in finding the large q limit of propagator (4.27) is the hypergeometric function. To properly treat it, we can first use transformation of variables: \n1 d 1+2 is F ( 1 2 -iq + is, 1 2 + iq + is, 1 , 1 -1 d 2 ) = Γ( -2 is ) d 1+2 is Γ( 1 2 -is + iq )Γ( 1 2 -is -iq ) F ( 1 2 + is -iq, 1 2 + is + iq, 1 + 2 is, 1 d 2 ) + ( s →-s ) (E.116) \nIn the limit of large q ( d scales with q ), we have approximation of hypergeometric function: \nF ( 1 2 + is -iq, 1 2 + is + iq, 1 + 2 is, 1 d 2 ) ∼ Γ(1 + 2 is )( d q ) 2 is I 2 is ( 2 q d ) . (E.117) \nUsing reflection property of gamma function together with large q approximation of Γ( 1 2 -is + iq )Γ( 1 2 -is -iq ) ∼ 2 πe -πq q -2 is , we have: \n( E. 116) ∼ -ie πq 2 sinh(2 πs ) d ( I -2 is ( 2 q d ) -I 2 is ( 2 q d )) = e πq πd K 2 is ( 2 q d ) . (E.118) \nPutting everything together will give us (5.34).', 'F Trajectories in Real Magnetic Field': 'The equation of motions for a particle in real magnetic field b are: \n˙ x 2 + ˙ y 2 2 y 2 = E ; (F.119) \n˙ x y 2 + b y = k, (F.120) \nwhere E and k are the conserved energy and momentum respectively. Since we are only interested in the trajectories, we can introduce a time parametrization ξ such that dτ dξ = 1 2 ky . Then in coordinate ξ , we have: \n( ∂ ξ x ) 2 +( ∂ ξ y ) 2 = E 2 k 2 ; ∂ ξ x = ( y -b 2 k ) . (F.121) \nThis means that we have solutions: \nx 2 +( y -b k ) 2 = E 2 k 2 . (F.122) \nThose are circles with radius √ E 2 1 k and center at location (0 , b 2 k ). So classically we have two types of states as shown in Figure 10 : for E < b 2 2 , the particle is confined by magnetic field and becomes Landau level in the hyperbolic plane; for E > b 2 2 , the gravitational effect dominates and particle scatters out of the space.', 'References': '- [1] Andrew Strominger. Les Houches lectures on black holes. In NATO Advanced Study Institute: Les Houches Summer School, Session 62: Fluctuating Geometries in Statistical Mechanics and Field Theory Les Houches, France, August 2-September 9, 1994 , 1994.\n- [2] Roman Jackiw. Lower dimensional gravity. Nuclear Physics B , 252:343 - 356, 1985.\n- [3] Claudio Teitelboim. Gravitation and hamiltonian structure in two spacetime dimensions. Physics Letters B , 126(1):41 - 45, 1983.\n- [4] Dmitry Bagrets, Alexander Altland, and Alex Kamenev. Sachdev-Ye-Kitaev model as Liouville quantum mechanics. Nucl. Phys. , B911:191-205, 2016.\n- [5] Dmitry Bagrets, Alexander Altland, and Alex Kamenev. Power-law out of time order correlation functions in the SYK model. Nucl. Phys. , B921:727-752, 2017.\n- [6] Douglas Stanford and Edward Witten. Fermionic Localization of the Schwarzian Theory. JHEP , 10:008, 2017.\n- [7] Thomas G. Mertens, Gustavo J. Turiaci, and Herman L. Verlinde. Solving the Schwarzian via the Conformal Bootstrap. JHEP , 08:136, 2017.\n- [8] Alexei Kitaev and S. Josephine Suh. Statistical mechanics of a two-dimensional black hole. 2018.\n- [9] Alexei Kitaev. Workshop on the Chaos, the SYK Model and AdS 2 . \n| [10] Daniel Harlow and Daniel Jafferis. The Factorization Problem in Jackiw-Teitelboim Gravity. 2018. |\n|------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|\n| [11] Leonard Susskind. Black Holes and Complexity Classes. 2018. |\n| [12] Ahmed Almheiri and Joseph Polchinski. 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An SYK-Like Model Without Disorder. 2016. |\n| [34] Igor R. Klebanov and Grigory Tarnopolsky. Uncolored random tensors, melon dia- grams, and the Sachdev-Ye-Kitaev models. Phys. Rev. , D95(4):046004, 2017. |\n| [35] Bryce Kobrin and Norman Yao. Private communication. |\n| [36] Ping Gao, Daniel Louis Jafferis, and Aron Wall. Traversable Wormholes via a Double Trace Deformation. JHEP , 12:151, 2017. |\n| [37] Juan Maldacena, Douglas Stanford, and Zhenbin Yang. Diving into traversable worm- holes. Fortsch. Phys. , 65(5):1700034, 2017. |\n| [38] Leonard Susskind and Ying Zhao. Teleportation through the wormhole. Phys. Rev. , D98(4):046016, 2018. |\n| [39] Beni Yoshida and Alexei Kitaev. Efficient decoding for the Hayden-Preskill protocol. 2017. |\n| [40] A.H. Chamseddine. A study of non-critical strings in arbitrary dimensions. Nuclear Physics B , 368(1):98 - 120, 1992. |\n| [41] B. Pioline and J. Troost. Schwinger pair production in AdS(2). JHEP , 03:043, 2005. |\n| [42] Thomas M. Fiola, John Preskill, Andrew Strominger, and Sandip P. Trivedi. Black hole thermodynamics and information loss in two-dimensions. Phys. Rev. , D50:3987- 4014, 1994. | \n- [43] John Preskill, Patricia Schwarz, Alfred D. Shapere, Sandip Trivedi, and Frank Wilczek. Limitations on the statistical description of black holes. Mod. Phys. Lett. , A6:2353-2362, 1991.\n- [44] Thomas G. Mertens. The Schwarzian Theory - Origins. JHEP , 05:036, 2018.\n- [45] Juan Maldacena and Xiao-Liang Qi. Eternal traversable wormhole. 2018.\n- [46] Nele Callebaut and Herman Verlinde. Entanglement Dynamics in 2D CFT with Boundary: Entropic origin of JT gravity and Schwarzian QM. 2018.'}
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2016ASSL..418..263G
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Galaxy Bulges and Their Massive Black Holes: A Review
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2016-01-01
| 54 | 0.53 | 164 |
['-', '-', '-', '-', '-']
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[]
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With references to both key and often forgotten pioneering works, this article starts by presenting a review into how we came to believe in the existence of massive black holes at the centers of galaxies. It then presents the historical development of the near-linear (black hole)-(host spheroid) mass relation, before explaining why this has recently been dramatically revised. Past disagreement over the slope of the (black hole)-(velocity dispersion) relation is also explained, and the discovery of sub-structure within the (black hole)-(velocity dispersion) diagram is discussed. As the search for the fundamental connection between massive black holes and their host galaxies continues, the competing array of additional black hole mass scaling relations for samples of predominantly inactive galaxies are presented.
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[]
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https://arxiv.org/pdf/1501.02937.pdf
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{"TO APPEAR IN 'GALACTIC BULGES', E. LAURIKAINEN, R.F. PELETIER, D.A GADOTTI (EDS.), SPRINGER PUBLISHING": 'Alister W. Graham 1 \nCentre for Astrophysics and Supercomputing, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia. (Received 16 November, 2014; Revised 16 February, 2015 and 30 August, 2016.) Draft version September 7, 2016', 'ABSTRACT': 'With references to both key and oft-forgotten pioneering works, this article starts by presenting a review into how we came to believe in the existence of massive black holes at the centres of galaxies. It then presents the historical development of the near-linear (black hole)-(host spheroid) mass relation, before explaining why this has recently been dramatically revised. Past disagreement over the slope of the (black hole)-(velocity dispersion) relation is also explained, and the discovery of substructure within the (black hole)-(velocity dispersion) diagram is discussed. As the search for the fundamental connection between massive black holes and their host galaxies continues, the competing array of additional black hole mass scaling relations for samples of predominantly inactive galaxies are presented. \nSubject headings: black hole physics - galaxies: bulges - galaxies: nuclei - galaxies: fundamental parameters', '1. OVERVIEW': "Arguably one of the most exciting aspects of galaxy bulges are the monstrous black holes which reside in their cores, sometimes lurking quietly, other times beaming out their existence to the Universe. Not only are they the dominant species on the mass spectrum of individual objects, but they play host to such a range of extremely unusual phenomenon that they appeal to people of all ages and professions. \nFor extragalactic astronomers, one curious aspect is the apparent coupling between the mass of the black hole, M bh , and the host galaxy bulge or spheroid, M sph , within which it resides. The importance of this is because it suggests that the growth of the two is intimately intertwined, and unravelling this connection will provide insight into their co-evolution. While the M bh -M sph relation may arise from black hole feedback processes such that the black hole regulates the growth of the surrounding spheroid (a remarkable feat given the factor of a billion difference in physical size), correlations between both the central radial concentration of stars and the central stellar density of the spheroid with M sph might be telling us that it is instead the spheroid mass which (indirectly) dictates the black hole mass through these relations. \nThis article starts by providing a background briefing to the development of ideas (since Einstein introduced his theories of relativity) which have led to our current understanding of supermassive black holes in galactic nuclei (Section 2), and the eventual observational proof which ruled out alternative astrophysical suggestions for the dark mass concentrations identified there (Section 3). Some effort has been made to reference key papers and give credit to the original developers of ideas and solutions, of whom many have been poorly cited in the literature to date. \nNot surprisingly, many reviews have been written about supermassive black holes, and far more than the author was aware when approached to write this review. Enjoyable reports are provided by Kormendy & Richstone (1995), Longair (1996 and 2006) which includes a well-written historical perspective, and an impressively extensive overview of many sub-topics can be found in Ferrarese & Ford (2005) which remain highly relevant today. In it, they too provide an historical account of active galactic nuclei (AGN), detail the many methods used to measure the masses of black holes today, and compare the demographics of black holes in distant quasars with local galaxies. It is however their section 9, pertaining to the scaling relations between the masses of black holes and the properties of their host galaxy that is the main focus of this article. For references to other aspects of massive black holes, over the past decade or so the following astrophysical reviews have focussed on: Sagittarius A ∗ (Alexander 2005; Genzel et al. 2010); intermediate mass black holes (Miller & Colbert 2004; van der Marel 2004); massive black hole binaries (Merritt & Milosavljevi'c 2005); AGN activity and feedback (Brandt & Hasinger 2005; Ho 2008; McNamara & Nulsen 2007; Heckman & Best 2014), including hot accretion flows (Yuan & Narayan 2014) and cold accretion flows (Kato 2008; Abramowicz & Fragile 2013); connections with distant AGN (Shankar 2009a); redshifted fluorescent iron lines (Reynolds & Nowak 2003; Miller 2007); gravitational radiation (Berti et al. 2009, see also AmaroSeoane et al. 2012); black hole spin (Gammie et al. 2004; Reynolds 2013); black hole seeds (Volonteri 2010, see also Koushiappas et al. 2004); and a healthy mix of various topics (e.g. Kormendy & Ho 2013; Genzel 2014) as in Ferrarese & Ford (2005). \nAs noted by Ferrarese & Ford (2005), in 2004 direct black hole mass measurements were known for 30 galaxies, plus another 8 galaxies for which the dynamical mod- \nels might be in error. Recently, Savorgnan & Graham (2015), see also Kormendy & Ho (2013), tabulate 89 galaxies with reliably measured black hole masses. Not only has the sample size therefore tripled over the past decade, but new scaling relations have been uncovered and old relations have been revised - and dramatically so as we shall see in the case of the M bh -M sph and M bh -L sph relations (Section 4). The M bh -σ relation, involving the velocity dispersion of the galactic host, is reviewed in Section 5 and the controversial issue of its slope addressed. The apparent substructure in the M bh -σ diagram, reported in 2008 due to barred galaxies and/or pseudobulges, is additionally discussed. \nHaving dealt in some detail with the two most commonly cited black hole scaling relations in Sections 4 and 5, the assortment of related relations are presented. While not as popular in the literature, it may be one of these relations which provides the fundamental, or at least an important, link between the black hole mass and its host galaxy (an issue raised by Alexander & Hickox 2012). Therefore, Section 6 examines the connection between the black hole mass and the host spheroid's S'ersic index, i.e. how radially concentrated the spheroid's stellar distribution is; this dictates the radial gradient of the gravitational potential. Section 7 describes the expected association between the black hole mass and the central stellar density (prior to core depletion). Section 8 explores the link between the mass of the black hole and the missing stellar mass at the centres of giant spheroids. The connection between the black holes and the dense star clusters found in the nuclei of many galaxies - some of which may harbour intermediate mass black holes - is presented in Section 9. Section 10 discusses the black hole mass relation with the halo (baryons plus dark matter) mass, expected to exist for spheroid dominated galaxies, while Section 11 remarks on the existence of a correlation with the pitch angle of spiral arms in latetype galaxies. Finally, Section 12 considers the possibility that a third parameter may account for some of the scatter in the above bivariate distributions, leading to a more fundamental plane or hypersurface in 3-parameter space involving black hole mass and two galaxy/spheroid parameters.", '2. HISTORICAL DEVELOPMENT: FROM MATHEMATICAL SPECULATION TO WIDESPREAD SUSPICION': "Karl Schwarzschild (1916, 1999; see also Droste 1917 who independently derived the same solution in 1916) is widely recognised for having developed the 'Schwarzschild metric' for a spherical or point mass within Einstein's (1916) theory of general relativity 2 , but it was Finkelstein (1958, see also Kruskal 1960) who realised the true nature of what has come to be called the 'event horizon' bounding these gravitational prisons. Finkelstein eloquently describes this Schwarzschild surface as 'a perfect unidirectional membrane: causal influences can cross it but only in one direction'. Five years later, while working at the University of Texas, the New \nZealand mathematician Roy Kerr (1963) formulated the metric for the more realistic 3 rotating black hole. Interestingly, solutions to this space-time include closed time-like curves which, in theory, allow one to travel backwards in time (a concept popularised but also questioned by Thorne 1994). Kurt Godel (1949) was actually the first to derive such strange solutions to the equations of general relativity, although it is commonly suspected that all closed time-like curves are just a mathematical artifact, in the same way that the original singularity at the Schwarzschild radius was later explained away by a coordinate transformation (e.g. Eddington 1924; Georges Lemaˆıtre 1933), leaving just the singularity (i.e. black hole) at the centre. But even if we are to be denied our time machines 4 , black holes still offer the curious and unsuspecting property of evaporating over time - radiating like a black body - before possibly then exploding (Hawkings 1974, 1975). \nEvolving parallel to the above analytical developments, our acceptance of black holes as more than just a mathematical curiosity had additional connections with stellar evolution and dark stars 5 . As detailed by Yakovlev (1994), the Soviet physicist Yakov Frenkel (1928) was the first to derive equations for the energy density and pressure of super-dense stars comprised of a degenerate Fermi-gas of electrons of arbitrary relativistic extent. He is, however, not widely recognised for having done so. Also using results from Albert Einstein's (1905) theory of special relativity, Soviet physicist Wilhelm Anderson (1929) was the first to derive a maximum mass for the fermion degenerate stellar model of white dwarf stars, above which the Fermi pressure is insufficient to overcome gravity. It is however the British physicist Edmund Stoner (1929) who is somewhat better known for having presented the structure for the mass, radius and density of white dwarf stars composed of non-relativistic electrons. Using his uniformly distributed mass density model, Stoner (1930, see also Stoner 1932a,b) refined his work by formulating how the core becomes relativistic at sufficiently high densities (as had already been done by Frenkel 1928) and he too predicted a maximum stable mass (similar to Anderson 1929) for earth-sized, white dwarf stars. But it is Chandrasekhar (1931a, see also Chandrasekhar 1931b) who is well known for calculating, in a short two-page article using polytropic density models , that at masses above ≈ 0.91 M /circledot , electron-degenerate white dwarf stars are not stable. That is, there is a maximum mass (recognised today as 1.4 M /circledot ) that white dwarf stars can have. If more massive than this limit \n3 Collapsing stars, and (accretion disc)-fed black holes, are expected to have substantial angular momentum. \n4 Time-travel enthusiasts might appreciate a nod to the hypothetical Einstein-Rosen (1935) bridge (aka 'wormhole', a term introduced by John Wheeler in 1957, e.g. Misner & Wheeler 1957, and Klauder & Wheeler 1957) which are warped regions of spacetime within general relativity (Morris & Thorne 1988; Morris et al. 1988; Hawking 1988). There is additionally the cosmic string time machine of Gott (1991). \n5 John Michell (1783) was the first to calculate the existence of black holes, which he termed 'dark stars', whose gravity was so strong that light would not be able to escape from their surface (see McCormmach 1968 and Schaffer 1979). Interestingly, Herschel (1791) subsequently speculated that 'nebulæ' might be regions of space where gravitationally-retarded particles of light are endeavouring to fly off into space. \nthen they must undergo further gravitational compression. Soon after, Soviet physicist Lev Landau (1932) correctly identified that the next level of resistance to their gravitational collapse would be met in the form of the denser neutron star (see also Oppenheimer & Serber 1938; Oppenheimer & Volkoff 1939). Landau (1932) and Chandrasekhar (1932, 1935) 6 predicted that the ultimate fate of an evolved massive star would be to collapse to a singularity of infinite density 7 . Following further work on this idea (e.g. Baade & Zwicky 1934; Zwicky 1938; Datt 1938), Oppenheimer & Snyder (1939) carefully detailed how overly massive neutron stars are not stable and will collapse into stellar mass black holes. Quite simply, if a star is massive enough and the outward pressure from fusion is over, gravity will win over (e.g. Arnett 1967). \nWheeler (1966) wrote 'In all the physics of the postwar era it is difficult to name any situation more enveloped in paradox than the phenomenon of gravitational collapse' . Then, in the following year (1967), more than three decades after the initial prediction of neutron stars, pulsars were discovered, finally signalling the existence of neutron stars (Hewish et al. 1968; Pilkington et al. 1968; Hewish 1970). Not surprisingly, this bolstered belief in the existence of stellar mass black holes (e.g. Penrose 1965; Vishveshwara 1970), as did (i) mathematical proof that a singularity will form if an event horizon has formed (Penrose 1969; Hawking & Penrose 1970), (ii) the X-ray pulses from Cygnus X-1 (Oda et al. 1971; Thorne & Price 1975), and likely also (iii) the pioneering searches by Weber (1969, 1970) for gravitational radiation coming from even more massive objects at the centre of our Galaxy. \nAs detailed by Longair (1996, 2006, 2010), Ferrarese & Ford (2005) and Collin (2006), the notion that the centres of galaxies may contain massive black holes, millions to hundreds of millions times the mass of our Sun, stems from the discovery of the great distance to, and thus luminosity of, the quasi-stellar radio source 3C 273. The optical counterpart of this radio source was cleverly discovered by Hazard, Mackey & Shimmins (1963) using the Parkes radio telescope and lunar eclipsing. Its redshift was subsequently taken with the Palomar Observatory's Hale telescope and correctly interpreted by Schmidt (1963), see also Oke (1963) regarding 3C 273 and Greenstein & Matthews (1963a,b) in the case of 3C 48 (whose redshift had remained uninterrupted over the preceding couple of years). \nBaade & Minkowski (1954), Ambartsumian (1958), Woltjer (1959), Burbidge (1959), Burbidge et al. (1963, 1964), Lynds & Sandage (1963) and others had already recognised active galactic nuclei (AGN) to be incredibly energetic phenomena. 8 Radio galaxy 3C 273 and other active galactic nuclei emit vasts amount of energy from a small volume of space (as indicated by quasar \n- 6 Miller (2005) details the early work of Chandrasekhar on this topic.\n- 7 In passing, it is noted that quark stars (Ivanenko & Kurdgelaidze 1965) are also expected to have a stable configuration, en route between neutron stars and black holes.\n- 8 While relatively low-luminosity Seyfert (1943) galaxies - with broad emission lines as previously observed by Fath (1909) and Slipher (1917) - were of course already known in 1963, it was not yet fully appreciated that quasars are their high-energy kin, although similarities were noted by Burbidge et al. (1963) and Burbidge (1964). \nvariability on short time scales, Smith & Hoffleit 1963) 9 and were thus thought to be powered from the gravitational potential energy 10 released as matter falls onto a compact massive object (Salpeter 1964; Zel'dovich 1964; Zel'dovich & Novikov 1964; Ne'eman 1965; Shakura & Sunyaev 1973) 11 . Based upon Eddington-limiting arguments at the time, it was immediately realised that the central object has to be massive or else the radiation pressure of the quasar would literally blow the quasar apart. Hoyle et al. (1964) acknowledged the possibility of 'invisible mass' perhaps from imploded objects of very large mass 12 . \nJust two years after the high-redshifts were recorded for the star-like 13 radio sources, Sandage (1965) reported on the high abundance of radio-quiet quasars, referring to them as a 'major new constituent of the universe'. What he had revealed was that in addition to the radio-loud quasars, the Universe was teeming with many more quasars. Encapsulating the ideas of recent years, Lynden-Bell 14 (1969) and Lynden-Bell & Ress (1971) suggested that a massive black hole resides at the cores of many galaxies (see also Wolfe & Burbidge 1970), and that the infall of orbital matter builds an accretion disc (e.g. Thorne 1974) which heats up due to friction. For a rapidly spinning black hole, this process can liberate a substantial fraction (up to 0.42 for a maximally spinning black hole) of the infalling matter's rest mass energy 15 (Bardeen & Wagoner 1969; Bardeen 1970). Further support for the presence of massive black holes were the linear radio features emanating from the nuclei of galaxies -which were likely emitted from a stable gyroscope such as a spinning black hole - and the superluminal speed of these radio jets (e.g. Cohen et al. 1971; Whitney et al. 1971). \nThe moniker 'black hole' was used by Ann Ewing (1964) just a year after the redshift of 3C 273 was announced. She reportedly heard it at a meeting of the American Association for the Advancement of Science (AAAS), and it was later seen used in a scientific paper \n- 9 Reviews of AGN and their variability are given by Mushotzky et al. (1993), Ulrich et al. (1997), and Peterson (1997).\n- 10 As stated by Rees (1998), the black hole's gravitational well 'must be deep enough to allow several percent of the rest mass of infalling material to be converted into kinetic energy, and then radiated away from a region compact enough to vary on timescales as short as an hour.'\n- 11 Like many capable theorists, Zel'dovich and Novikov did not restrict themselves to one theory, and in Bisnovatyi-Kogan, Zel'dovich & Novikov (1967) they proposed that quasars may be billion solar mass stars burning brightly for tens of thousands of years, see also Hoyle & Fowler (1963), while Novikov (1965) additionally advocated what we now know as 'white holes'. \n12 Hoyle & Burbidge (1966) also speculated that quasars may be nearby objects and that their redshifts do not necessarily reflect the expansion of the universe, see also Hoyle et al. (2000) and Burbidge et al. (2006). \n13 Faint halos had been reported around some of these 'star-like' objects, which we now know is due to the host galaxy surrounding the bright AGN (e.g. Gehren et al. 1984; Hutchings et al. 1984, and references therein). \n14 Historical footnote: The daily commute along the A273 to Herstmonceux in Sussex prompted Donald Lynden-Bell to find a satisfactory explanation for the quasar 3C 273 (priv. comm. 2015). \n15 For comparison, nuclear fusion is known to release less than 1% of the rest mass energy (0.7% in the conversion of hydrogen to helium) and thus 'super-stars' are not as efficient sources of energy as rapidly spinning accretion discs around supermassive black holes. \nby John Wheeler (1968) 16 . By 1970 the label appears as a familiar term in the literature. It had, at times, previously been used to describe dusty dark patches in our own Galaxy (e.g. Barnard 1897; Campbell 1917). However it became the popular replacement for what the Soviet physicists (e.g. Zeldovich 1964) called a frozen star 17 , and Western physicists called a collapsed star or a 'collapsar' (e.g. Cameron 1971). The term 'singularity' had also been, and still is, regularly used by the mathematicians to indicate where any quantity in the field equations becomes infinite. \nDespite its strangely endearing name, the phrase 'black hole' is often noted to be somewhat unfortunate in that it implies a hole in space through which matter may fall through. The idea of an actual singularity - a point of infinite density which arises out of classical physics after division by 0 - is also not popular and considered rather old-school. While a Planck-sized mote may be a better description, what actually exists near the centre of a black hole's event horizon is hotly debated. Mathematically-inclined readers who are interested in what a black hole may be like, might enjoy reading about the 'fuzzball' picture from string theory ('t Hooft 1980; Mathur 2005, and associated references), or descriptions of black holes in quantum gravity theories such as spin foam networks (e.g. Penrose 1971a,b; Penrose & Rindler 1986; Rovelli 1998; Domagala & Lewandowski 2004; Perez 2004) or loop quantum gravity (e.g. Ashtekar & Bojowald 2005; Hayward 2006).", '3. ON FIRMER GROUND': "Acceptance of the idea that supermassive black holes reside at the centres of galaxies was not as straight forward as suggested above. During the 1960s and 1970s the AGN community battled it out amongst themselves before (largely) embracing the idea that black holes must be required to power the quasar engines of galaxies. Building on Sandage (1965), Soltan (1982) reasoned that there had to be a lot of mass locked up today in massive black holes because of all the past quasar activity, and Rees (1984) advocated further for the preponderance of massive black holes in the nuclei of galaxies. Then during the 1980s and early 1990s it was primarily the inactivegalaxy community, as opposed to the AGN-community, who remained skeptical until two key papers in 1995 (discussed shortly). \nAmong the pioneering observational papers for the presence of a massive black hole in individual, nearby, non-AGN galaxies, Sanders & Lowinger (1972) calculated that the Milky Way houses a 0 . 6 × 10 6 M /circledot black hole and Sargent et al. (1978) concluded that a 5 × 10 9 M /circledot black hole very probably exists in M87 (see also LyndenBell 1969 who predicted a 30 × 10 6 M /circledot black hole for the Milky Way 18 , and a 40 × 10 9 black hole in M87, i.e. an \n16 John Wheeler is first recorded to have used the term 'black hole' at his 27 December, 1967, AAAS invited lecture, a few years after Ann Ewing. However, given that he coined the term 'worm hole', it seems likely that he also introduced the expression 'black hole', although the author does not rule out that it may have been Fritz Zwicky. \n17 For an external observer, time appears to stop inside the Schwarzschild radius, giving rise to the term 'frozen star', because the collapse of a star will appear to freeze once the star is within the event horizon. \n18 With the benefit of hindsight, we now know that radio syn- \norder of magnitude higher). Although these works had revealed that very high masses in small volumes were required at the centres of these galaxies (see also Dressler 1984 and Tonry 1984 in the case of M31 and M32, respectively), it took some years before the observations / measurements improved and alternatives such as a dense cloud of stellar mass black holes or neutron stars could be ruled out. The three following observational works turned the tide of opinion among the remaining naysayers who demanded further proof before accepting the existence of what is indeed an extreme astrophysical object: the supermassive black hole. \n1) Before an object crosses within a black hole's event horizon, any radiation it emits away from the black hole will be gravitationally redshifted, the extent of which depending on how close the object is to the event horizon. Such a tell-tale signature of redshifting was reported on 22 June 1995 by Tanaka et al. (1995) who detected the highly broadened, ionised iron K α line (6.4 keV) from the galaxy MCG-6-30-15. This highly asymmetric, predominantly redshifted, X-ray emission line had a width corresponding to roughly one-third of the speed of light, and was thought to have been emitted at just 3 to 10 Schwarzschild radii from the black hole. Such relativistic broadening has since been shown to be commonplace (Nandra et al. 1997), thanks to the enhanced sensitivity and spectral resolution of the Japanese ASCA X-ray satellite (Tanaka et al. 1994). \n2) Additional convincing evidence for the reality of massive black holes had came from the very high mass density required to explain the central object in the Seyfert galaxy NGC 4258 (M106). Using the Very Long Baseline Array in New Mexico, Miyoshi et al. (1995) showed that the H 2 O maser emission from this galaxy originates from a thin, rotating nuclear gas disc/annulus displaying a clear Keplerian rotation curve and requiring a mass of 3 . 6 × 10 7 M /circledot within a size of just 0.13 parsec 19 (see also Haschick et al. 1994, Watson & Wallin 1994, and Greenhill et al. 1995a,b). In their January 12 paper, Miyoshi et al. (1995) note that the short collisional timescale ( < 10 8 years) for a swarm of solar mass dark stars with such density ( > 4 × 10 9 M /circledot pc -3 inside of the inner 4.1 milliarcseconds) implies that such a hypothetical star cluster could not survive (see also Maoz 1995, 1998); a single supermassive black hole is the only viable candidate. A second example of extreme mass density (3 . 2 ± 0 . 9 × 10 8 M /circledot pc -3 ) has since been shown in the Circinus galaxy by Greenhill et al. (2003). \n3) Several years later, high spatial resolution measurements of stellar orbits around the central object in our own Milky Way galaxy also eventually ruled out the possibility that it could be a swarm of neutron stars or stellar mass black holes, with the high density favouring the existence of a massive black hole (Schodel et al. 2002; Ghez et al. 2005, 2008; Gillessen et al. 2009). confirming earlier suspicions (Lacy et al. 1979, 1980; Eckart & Genzel 1996, 1997; Genzel et al. 1996, 1997; Ghez et al. 1998; see also Alexander 2005 and references therein). \nAs was appropriately emphasized by Merritt & Ferrarese (2001b), within the black hole's sphere-of-influence \nchrotron emission from Sagittarius A was first seen in the 5 GHz data from Ekers & Lynden-Bell (1971). \nFig. 1.Artist's impression of the horror at a galactic centre. Credit: Gabriel P'erez D'ıaz. [To appear in the published version.] \n<!-- image --> \n-whose radius is defined as r infl = GM bh /σ 2 sph (e.g. Peebles 1972; Frank & Rees 1976) where σ sph is roughly the host spheroid's velocity dispersion immediately beyond r infl -one expects to find Keplerian dynamics which are dominated by the black hole. The velocity dispersion of the stars (or the rotational velocity of a relatively lighter disc, as in the case of NGC 4258) inside r infl should thus decline with the inverse square root of the radius, i.e. σ ( R ) ∝ R -0 . 5 , just as rotational velocities of Keplerian discs or solar systems have v rot 1 / √ R . \nThe absence of this clear detection for many galaxies has led Merritt (2013) to question their reported black hole measurements, which may be better interpreted as upper limits until we are better able to resolve the sphereof-influence (see also Valluri, Merritt & Emsellem 2004). With this cautionary note, we proceed to the topic of black hole scaling relations, which at the very least would still be upper envelopes in the various diagrams of black hole mass versus host spheroid properties. It may however then be unusual that all of the clear-cut examples for a definitive black hole reside on this upper envelope (but see Ford et al. 1998; Ho 1999, his section 7; and Batcheldor 2010). \n∝ \nCommenting on the ratio of black hole mass to spheroid mass in M31 and M32, Dressler & Richstone (1988) suspected a relation, and used it to predict billion solar mass black holes in bright elliptical galaxies. While the prediction was not new, in the sense that the authors were aware that past theoretical papers had stated that quasars in big elliptical galaxies could have 10 9 solar mass black holes (e.g. Rees 1984; Begelman, Blandford & Rees 1984, and references therein) 20 the idea of a scaling relation with the spheroid does seem to be new 21 . Dressler (1989) further advocated this connection between the black hole and the host spheroid (not the disc), and from a sample of 5 galaxies he noted that there is a 'rough scaling of black hole mass with the mass of the spheroidal component'. \nThis differed slightly from Hutchings et al. (1984) who had reported that the 'black hole [] mass is related to that of the galaxy, increasing 60% faster than that of the galaxy'. The study by Hutchings et al. (1984) was of \n20 , \npoorly resolved, distant quasars which prevented them from performing a bulge/disc decomposition and as such they did not report on a black hole mass relation with the host spheroid. However, there is an upper limit to the brightness of quasars which has been observed to scale with the brightness of the host galaxy (which are typically spheroid-dominated for the brightest quasars). Using real data, Yee (1992) fit a linear relation to this limit, which he called the M QSO -M G relationship, and wrote that 'it may arise due to a correlation of the mass of the central engine and the galaxy mass', such that 'the brightest quasars for a given galaxy mass are the ones shining at or near the Eddington limit (which is set by the mass of the central engine), while others are at lower luminosities' 22 . As noted by McLeod (1997, see also McLeod et al. 1999 and result number 4 from Laor et al. 1997), Yee (1992) had effectively discovered the linear, high-mass end of the M bh -M sph distribution. \nWith three more galaxies than Dressler (1989), Kormendy & Richstone (1995, see also Kormendy 1993) wrote a review article in which they plotted this data and reiterated in mathematical form what Dressler had said, and Yee (1992) had shown for massive bulges, i.e. M bh ∝ M bulge . While they did not fit a relation to the data, they did report a mean M bh /M bulge ratio of 0.22% (including the Milky Way) and thereby effectively created a more quantitative basis for a linear M bh -M bulge relation. \nFollowing the prediction by Haehnelt & Rees (1993) that ≈ 30% of nearby galaxies likely house a central massive black hole, Kormendy & Richstone (1995) remarked that at least 20% of nearby galaxies possess such a black hole - while noting that alternatives such as massive concentrations of dark stars could not yet be ruled out. Magorrian et al. (1998) built on this and suggested that most nearby galaxies harbour a massive black hole (see also Sigurdsson & Rees 1997, and the reviews by Ford et al. 1998 and Richstone et al. 1998), supporting the strong suspicion held by many (e.g. Blandford 1986; Rees 1990). Moreover, this followed closely on the heels of the observation that many quiescent galaxies have weak central radio sources (e.g. Keel 1985; Sadler et al. 1989, 1995; Ho et al. 1997), likely signalling low-level accretion onto near-dead quasars. \nRather than the pure 'linear' scaling, a single powerlaw relation was introduced by Magorrian et al. (1998; see also Franceschini et al. 1998) to describe the distribution of 32 points in the M bh -M bulge diagram, such that the log-linear slope was 0.96 ± 0.12 (which is of course still consistent with a slope of 1 and thus a linear relation) 23 . In other works, using variously updated masses and samples, Ho (1999) reported a median M bh /M bulge ratio of 0.2%, and Merritt & Ferrarese (2001c) and Kormendy & \n22 In passing, and as noted by Alexander & Natarajan (2014), it is possible to exceed the Eddington limit to black hole growth (as noted by Begelman 1979 and Soffel 1982), due to an effective gas drag of the photons. Moreover, non-spherical accretion in the form of a disc can also result in black hole growth superseding the Eddington limit (Nayakshin et al. 2012). \n23 As suspected by Magorrian et al. (1998), and noted by van der Marel (1999) and Gebhardt et al. (2000), their use of a two-integral distribution function which ignores radial velocitydispersion anisotropy (see Binney & Mamon 1982) caused them to over-estimate the black hole masses by an average factor of 3-4.5 \nGebhardt (2001) reported a ratio of 0.13%, although with notable scatter. McLure & Dunlop (2002) noticed that the scatter was considerably reduced once the disc galaxies were excluded, suggestive of poor bulge/disc decompositions used to estimate the bulge masses. Marconi & Hunt (2003) subsequently performed careful bulge/disc decompositions on near-infrared K -band images, less effected by dust and star formation. They also showed that the dynamical/virial mass of the spheroid correlated linearly with the black hole mass, and Haring & Rix (2004) provided improved dynamical masses for the derivation of their near-linear relation. For the next decade, studies of the M bh -L bulge and M bh -M bulge diagram remained dominated by high-mass galaxies 24 having M bh > ∼ 0 . 5 × 10 8 M /circledot and, despite each paper's incremental improvements, continually recovered a single, near-linear M bh -M bulge relation (e.g. Ferrarese & Ford 2005; Lauer et al. 2007; Graham 2007b, 2008a, his section 6; Gultekin et al. 2009; Sani et al. 2011; Beifiori et al. 2012; Erwin & Gadotti 2012; Vika et al. 2012; van den Bosch et al. 2012; McConnell & Ma 2013; Rusli et al. 2013a). A recent notable exception has been Lasker et al. (2014b) who advocate, with a near-infrared sample of 35 galaxies, that the black hole mass correlates equally well with the total (bulge plus disc) luminosity as it does with the bulge luminosity at 2.2 µ m, and that one has M bh ∝ L 0 . 75 ± 0 . 10 bulge and M bh ∝ L 0 . 92 ± 0 . 14 galaxy . They attribute this to the smaller bulge fluxes obtained from their decomposition of the galaxies' light and the type of linear regression performed. Savorgnan et al. (2016) have, however, since included 17, rather than 4, spiral galaxies and found that it is indeed the bulge rather than galaxy mass which has the strongest correlation. \nThere were a few early deviations from the above (near) convergence of opinion on a linear relation that should be noted. First, while the Abstract of Laor (1998) largely supports the linear relation of Magorrian et al. (1998), the main text reports that M bh ∝ M 1 . 5 -1 . 8 bulge (although it suggests that this may be partly due to the fact that all their lower mass quasar hosts are disc galaxies for which they may have over-estimated the bulge mass) and Second, it also notes that the low-mass inactive galaxies from Magorrian et al. (1998) better match their steeper M bh -M bulge relation than the linear one. Third, Wandel (1999) reported a mean log( M bh /M bulge ) ratio of -3 . 5 for a sample of Seyfert galaxies with black hole masses predominantly less than 10 8 M /circledot . This is 0.6 dex, i.e. a factor of 4, smaller than reported by Merritt & Ferrarese (2001c) and Kormendy & Gebhardt (2001) who used a sample with ∼ 80% of the galaxies having M bh > 0 . 8 × 10 8 M /circledot . Wandel (1999) argued and wrote 'It is plausible, therefore, that the Seyfert galaxies in our sample represent a larger population of galaxies with low BBRs [black hole to bulge mass ratios], which is underrepresented in the Magorrian et al. sample' 25 . \n24 Studies were also biased by the inclusion of one or two rare 'compact elliptical' galaxies (e.g. M32 in Graham 2007b and Gultekin et al. 2009, their Fig.4) that do not represent the population at large. \n25 McLure & Dunlop (2001) correctly noted that a better bulge/disc decomposition reduces the observed flux attributed to the bulges by Wandel (1999), however the dust corrections which were not applied can largely cancel this reduction (compare fig- \nFourth, while Wandel reported M bh ∝ L 1 . 4 bulge (which equates to M bh ∝ M 1 . 2 bulge when using the same M/L ∝ L 0 . 18 relation as Laor 1998 and Magorrian et al. 1998), the data in Wandel (1999, their figure 1) reveal that a relation with a slope steeper than 1.4 would be likely from a symmetrical regression. Fifth, using upper limits for black hole masses, Salucci et al. (2000) reported on hints that the M bh -M bulge relation is significantly steeper in spiral galaxies than in [massive] elliptical galaxies. Finally, Laor (2001) reinforced his claim that a steeper, single power-law seems more applicable than a linear relation, finding M bh ∝ M 1 . 53 ± 0 . 14 bulge . Related to this, Ryan et al. (2007) further reveals that the linear M bh -M bulge relation over-estimates the masses of black holes in lowmass Seyfert galaxies.", '4.1. A bend in the road': "Before beginning this section, it is necessary to introduce some nomenclature which may be unfamiliar to some readers. The term 'S'ersic galaxy' or 'S'ersic spheroid' shall be used to denote galaxies or spheroids (elliptical galaxies and the bulges of disc galaxies) whose surface brightness profile is well described by the S'ersic (1963, 1968) model all the way into the centre of the galaxy. Two decades ago Caon et al. (1993) demonstrated that the S'ersic model fits the surface brightness profiles of early-type galaxies remarkably well over a large dynamic range. An historical and modern review of S'ersic's model can be found in Graham & Driver (2005). S'ersic galaxies may contain additional nuclear flux components above that of the host S'ersic spheroid. The term 'core-S'ersic galaxy' or 'core-S'ersic spheroid' refers to a galaxy whose main spheroidal component has a partiallydepleted core (i.e. a central stellar deficit of light that is not due to dust) such that the surface brightness profile is well described by the core-S'ersic model (Graham et al. 2003b). The history of galaxy surface brightness models and the impact that the above systematically (with luminosity) varying structures (i.e. non-homology and depleted cores) have on galaxy scaling laws and the unification of bright and faint early-type S'ersic galaxies is discussed at length in Graham (2013). \nRe-analysing the dynamical spheroid mass and (updated) black hole mass data for 30 galaxies studied by Haring & Rix (2004), but this time separating the galaxies depending on whether or not they have a partially depleted core, Graham (2012a) found that the two populations follow different relations in the M bh -M sph , dyn diagram. While the dozen core-S'ersic spheroids, which are the more massive spheroids, followed the near-linear relation M bh ∝ M 1 . 01 ± 0 . 52 sph , dyn , the S'ersic spheroids followed a much steeper power-law relation, such that M bh ∝ M 2 . 30 ± 0 . 47 sph , dyn . Excluding the barred galaxies, the S'ersic relation was M bh ∝ M 1 . 92 ± 0 . 38 sph , dyn . This nearquadratic relation for the low- and intermediate-mass spheroids had never been reported before and it signalled a bend in the M bh -M sph , dyn diagram. \nWith an increased sample size of 72 galaxies with directly measured black hole masses, Graham & Scott \nures 1 and 7 in Graham & Worley 2008). \n(2013) confirmed this behavior using near-infrared K s -band magnitudes. Their sample of two dozen coreS'ersic spheroids gave M bh ∝ L 1 . 10 ± 0 . 20 sph , while the four dozen S'ersic spheroids gave the relationship M bh ∝ L 2 . 73 ± 0 . 55 sph , which reduced to M bh ∝ M 2 . 34 ± 0 . 47 sph , dyn when using M dyn /L K ∝ L 1 / 6 K (e.g., Magoulas et al. 2012; La Barbera et al. 2010). Employing the archangel photometry pipeline (Schombert & Smith 2012) applied to Two Micron All-Sky Survey images (Skrutskie et al. 2006), which effectively corrects for missing light at large radii, Scott et al. (2013) converted the K s -band magnitudes of the spheroids into stellar masses. They found that M bh ∝ M 0 . 97 ± 0 . 14 sph , ∗ and M bh ∝ M 2 . 22 ± 0 . 58 sph , ∗ for the S'ersic spheroids and core-S'ersic, respectively. \nWe therefore now have a situation which is dramatically different to what was believed for the past two decades. It is not simply that we no longer have a single, near-linear M bh -M sph relation for all spheroids, but the main growth phase of black holes and bulges, involving gas rich processes, follows a near-quadratic relation, with gas-poor 'dry' mergers subsequently creating the core-S'ersic galaxies which depart from the high-mass end of this near-quadratic relation 26 . That is, the growth of massive black holes has been much more rapid than that of their host spheroids. \nNaturally, the simple addition of galaxies and their black holes, through dry merging, will establish the observed near-linear relation for the core-S'ersic galaxies. The average M bh /M sph ratio of these core-S'ersic galaxies then reflects the value obtained at the high-mass end of the near-quadratic S'ersic M bh -M sph relation from which they peeled off. In late 2012 Graham & Scott (2013) reported this mass ratio to be 0.49%, in agreement with that already noted by Laor (2001) for massive spheroids. This ratio is basically the calibration for the Yee (1992) relation between black hole mass and galaxy mass in massive galaxies, modulo the fact that some coreS'ersic galaxies contain large discs. Furthermore, our own galaxy, with an M bh /M sph ratio of 0.05%, is no longer a low outlying point requiring explanation in the M bh -M sph diagram. It has a mass ratio in accord with the near-quadratic scaling relation for S'ersic spheroids. \nAdding AGN data from half a dozen recent papers which had observed the AGN black hole masses to reside below the original M bh -M sph relation, Graham & Scott (2015) revealed that they depart from the nearlinear M bh -M sph relation in a systematic manner consistent with the near-quadratic M bh -M sph mass scaling relation for S'ersic galaxies. That is, they are not randomly offset. This is shown in Figure 2. This also provides the picture with which we can now interpret the observations by Laor (1998, 2001) and Wandel (1999), who were on the right track over a decade ago. \nIf one was to separate the galaxies in Figure 2 at M bh = 2 × 10 6 M /circledot , one would (understandably but inappropriately) conclude that the lower mass spheroids do not follow an M bh -M sph , ∗ relation (Jiang et al. 2011). \n26 Some S'ersic galaxies may follow the near-linear M bh -M sph relation, having experienced a major dry merger event in which the nuclear star clusters from the progenitor galaxies have been eroded away but an obvious partially depleted core is not yet formed (see Bekki & Graham 2010). These may well be the galaxies at -19 . 5 > M B > -20 mag in Cˆot'e et al. (2007. their figure 3e). \nFig. 2.Black hole mass versus host spheroid's stellar mass (in units of solar mass). Core-S'ersic spheroids are shown with open red circles, while S'ersic spheroids are shown by the large blue dots. A sample of 139 low mass AGN from Jiang et al. (2011) are denoted by the small dots, while an additional 35 higher mass AGN (which may have had their host spheroid masses over-estimated by overly-high ( M/L ) stellar ratios, see Busch et al. 2014) are denoted by the cross hairs. The optimal near-linear and near-quadratic scaling relations from Scott et al. (2013) are shown as the red (solid and dashed) and blue (solid) line for the core-S'ersic and S'ersic spheroids, respectively. Of note is that 68% of the 139 AGN (i.e. +/-34%) are contained within 0.83 dex in the horizontal direction, representing a level of scatter equal to that about the near-linear relation observed at the high-mass end. The non-AGN S'ersic galaxies have more scatter than the non-AGN core-S'ersic galaxies because of the crude way in which their bulge masses were estimated (see Graham & Scott 2015, from which this figure is taken). \n<!-- image --> \nThis had resulted in these lower mass spheroids being considered distinct by some, and sometimes labelled 'pseudobulges' as opposed to 'classical' bulges (Gadotti & Kauffmann 2009; Kormendy, Bender & Cornell 2011) with the separation said to occur at n = 2. This is also where the alleged divide between dwarf elliptical and ordinary elliptical galaxies was said to occur ( M B = -18 mag, M gal , ∗ ≈ 2 × 10 10 M /circledot n ≈ 2-2.5, σ ≈ 100-120 km s -1 ). However, without the fuller parameter baseline that we now have, or artificially subdividing the data at a S'ersic index of 2, or at M B = -18 mag, or where the curvature in relations using 'effective' radii and surface brightnesses are a maximum (see Graham 2013 for an explanation of this), the continuity between the low- and intermediate-luminosity S'ersic galaxies can be missed, even if the data itself is accurate. This issue is discussed further in section 5.2.1. \nThe distribution of points in Figure 2 reveals that black holes grow faster than the stellar population of their host spheroids, for which abundant evidence is now appearing (e.g. Diamond-Stanic & Rieke 2012; Seymour et al. 2012; Trakhtenbrot & Netzer 2012; Agarwal et al. 2013; Alonso-Herrero et al. 2013; LaMassa et al. 2013 Lehmer et al. 2013; Drouart et al. 2014). For example, DiamondStanic & Rieke (2012) report that the black hole growth rate is proportional to the 1.67 (=1/0.6) power of the star formation rate within the inner kpc (roughly the \nbulge half-light radii) of their Seyfert galaxies, while the analysis from LaMassa et al. (2013) gives an exponent of 2.78 (=1/0.36) for their sample of ∼ 28,000 obscured active galaxies, quite different from the linear value of 1. \nFigure 2 also reveals that classical bulges, pseudobulges, clump-bulges (Noguchi 1999), and mixed-bulges containing both a classical bulge and a pseudobulge, all follow the steeper scaling relation, until the onset of relatively dry mergers revealed by the scoured cores seen in the centres of (many of) the most massive spheroids. \nWith their supernova feedback producing a steeper relation than their AGN feedback prescription, the models of Cirasuolo et al. (2005, their Figure 5) show a bend in the M bh -M sph (and M bh -M σ ) relation at M bh ≈ 10 8 M /circledot . At these lower masses, a steeper than linear M bh -M sph relation can also be seen in the differing models of Dubois et al. (2012), Khandai et al. (2012, their Figure 7); Bonoli et al. (2014, their Figure 7) and Neistein & Netzer (2014, their Figure 8). \nWhat happens in the M bh -M sph diagram at black hole masses less than 10 5 M /circledot is not yet known, although LEDA 87300 suggests that the steep relation continues (Graham, Ciambur & Soria 2016). While the absence of a definitive black hole detection in M33 (Kormendy & McClure 1993; Gebhardt et al. 2001; Merritt et al. 2001) had reinforced the idea that black holes are associated with bulges (e.g. Dressler & Richstone 1988; Kormendy & Gebhardt 2001), bulgeless galaxies with massive black holes have since been detected (e.g. Reines et al. 2011; Secrest et al. 2012; Schramm et al. 2013; Simmons et al. 2013; Satyapal et al. 2014). Obviously these galaxies do not (yet?) participate in the observed M bh -M sph , ∗ scaling relation. As noted in Graham & Scott (2013), there are however tens of galaxies known to contain AGN in bulges whose spheroid magnitudes suggest, based on this near-quadratic M bh -M sph , ∗ scaling relation, that they harbour intermediate mass black holes (10 2 < M bh /M /circledot < 10 5 ). It will be interesting to see a) if this missing population of intermediate-mass black holes exists and b) where they reside in the M bh -M sph diagram.", '4.1.1. Implications': "Of course the above represents a dramatic revision to the bulge-(black hole) connection , i.e. a completely different relation connecting supermassive black holes with their host bulges, and as such has wide-spread implications. For one, the many-merger scenario proposed by Peng (2007), and explored further by Jahnke & Macci'o 2011 and Hirschmann et al. (2010), to produce a linear one-to-one scaling via the central limit theorem can be ruled out. Using a sample of galaxies with a range of initial M bh /M gal , ∗ mass ratios, Peng (2007) noted that after many mergers it would naturally create an M bh -M sph, ∗ relation with a slope of 1. Although this concept was independently ruled out by Angl'es-Alc'azar et al. (2013) who had emphasized that the number of actual major mergers are not frequent enough to have established such a linear relation, the quadratic slope of the M bh -M sph relation confirms this ruling. \nSome additional implications of the new relation include obvious things like (i) black hole mass predictions in other galaxies, (ii) estimates of the local black hole \nmass function (e.g. Shankar et al. 2004,2012; Comastri et al. 2015) and mass density based on local spheroid luminosity functions, and (iii) evolutionary studies of the M bh /M sph mass ratio over different cosmic epochs. In particular, the local M bh /M sph ratio was thought to be 0.14%-0.2% (e.g., Ho 1999; Kormendy 2001; Marconi & Hunt 2003; Haring & Rix 2004). However Graham (2012a) reported a larger value of 0.36% for the coreS'ersic galaxies, which was, as noted above, increased that same year to 0.49% by Graham & Scott (2013) 27 . Nearly a year later this higher ratio for massive spheroids was again noted in the review by Kormendy & Ho (2013) due to its significance. \nAddiionally impacted areas of research include (iv) galaxy/black hole formation theories, which extends to (v) AGN feedback models, (vi) predictions for spacebased gravitational wave detections, (vii) connections with nuclear star cluster scaling relations, (viii) derivations of past quasar accretion efficiency as a function of mass (e.g. Shankar et al. 2009b), (ix) searches for the fundamental, rather than secondary, black hole scaling relation, and (x) calibrations matching inactive galaxy samples with low-mass AGN data to determine the optimal virial factor for measuring black hole masses in AGN. Given that most of these topics could generate a review in their own right, only feedback is briefly commented on here. \nA large number of clever theoretical papers have tried to explain the nature of the M bh -M sph relation in terms of feedback from the AGN (e.g. Silk & Rees 1998; Haehnelt, Natarajan & Rees 1998; Fabian 1999; Kauffmann & Haehnelt 2000; Wilman, Fabian & Nulsen 2000; Benson et al. 2003; Wyithe & Loeb 2003; Granato et al. 2004; Di Matteo et al. 2005; Springel et al. 2005; Hopkins et al. 2005, 2006; Cattaneo et al. 2006; Sijacki et al. 2007; Somerville et al. 2008; Booth & Schaye 2009, to mention just a fraction). Some papers (but not all those listed here) which have claimed success because they obtained, through gaseous processes, a linear M bh -M sph relation over a wide range of mass, now appear in need of tweaking. Encouragingly, while not quite finding a quadratic relation with slope of 2, Hopkins & Quataert (2010) report that the black hole growth rate in their models is proportional to the 1.43 (=1/0.7) power of the star formation rate. \nThe so-called 'quasar' or 'cold' mode of black hole growth during gas-rich processes, as implemented in semi-analytical models, has typically assumed that the growth occurs via accretion which is linearly proportional to the inflowing mass of cold gas (which also produces the host spheroid), modulated by an efficiency which is lower for both unequal mass mergers (Croton et al. 2006) and less massive (more gas-rich) systems with lower virial velocities (e.g., Kauffmann & Haehnelt 2000, their eq 2; Croton et al. 2006, their eq. 8; Guo et al. 2011, their eq. 36) 28 . Graham & Scott (2013) therefore presented a new prescription for the increase in black hole mass, due to gas accretion during wet mergers, such that the black hole would grow quadratically relative to the host \nspheroid. The short duty (on) cycle of quasars ( ∼ 10 7 -10 8 years) may then imply that the bulk of a spheroid's stars are also formed rapidly. Once the gas is largely gone, and significant galaxy/(black hole) growth is attained via major dry merger events, the low-accretion model (e.g., Blandford & Begelman 1999) presumably results in the so-called 'mechanical' or 'radio mode' feedback maintaining the spheroid-(black hole) mass ratio, as is roughly observed for the core-S'ersic galaxies.", 'Updates': "Measurements of black hole masses that include the impact of a dark matter halo on the observed galaxy dynamics have led to the upward revision of some M bh estimates. While this increased the black hole mass in M87 by a factor of 2 (Gebhardt & Thomas 2009), the impact on other galaxies has not only been shown to be less than a factor of 2, but the 1-sigma uncertainties on the new masses encompass the old values. That is, no significant change of mass. For example, the change in mass was a mere 2% and -5% for NGC 3608 and NGC 4291, just 0.08 dex for NGC 3377 and NGC 5845, and 0.21 dex for NGC 821 (Schulze & Gebhardt et al. 2011). Using these slightly revised masses, plus 4 extra galaxies from the then newly-published Rusli et al. (2013a) paper, and excluding several other published black hole masses for a plethora of reasons, Kormendy &Ho(2013) subsequently reported the calibration midpoint of their M bh -M sph relation for large spheroids to be M bh /M sph = 0.49%. This agreement with the previously reported mass ratio is not particularly surprising given that the masses used by Kormendy & Ho (2013) for the large number of galaxies in common with Graham & Scott (2013) differed by more than 0.2 dex for just 8 galaxies, and by more than 0.3 dex for only 5 galaxies. \nThere are of course two quantities that define the M bh -M sph relation, and Savorgnan & Graham (2016a) have recently completed a thorough analysis of the spheroid masses for the galaxies listed in Graham & Scott (2013) and Rusli et al. (2013). Savorgnan & Graham (2016a) not only explain for every galaxy why many published spheroid masses have often disagreed - invariably due to inadequate bulge/disc/etc. decompositions - but they performed the most careful galaxy decompositions to date, effectively reclassifying many galaxies' morphological type, a process started in Graham & Scott (2013, 2015) - which also included the use of accurate distances to each galaxy. Galaxy reclassification typically occurred when a disk or a bar had been over-looked (e.g. Graham, Ciambur & Soria 2016 and Graham et al. 2016), or when the contribution from a disk had been over-estimated in a disky ES type galaxy (as also discussed in Savorgnan & Graham 2016b and Graham, Ciambur & Savorgnan 2016). The new 2016 spheroid masses, derived from 3.6 µ m images which are not affected by dust obscuration, supercede past efforts on many fronts (see Savorgnan & Graham 2016a for details). The revised M bh -M sph relation for large spheroids in early-type galaxies still has a slope consistent with unity, while the median M bh /M sph ratio has risen to 0 . 68 ± 0 . 04%. \nIntriguingly, while the median M bh /M sph ratio has increased, two points should be made. As warned by Merritt (2013), the importance of being able to better resolve the black hole's sphere-of-influence was illustrated with NGC 1277, whose black hole mass measurement dropped by an order of magnitude as the spatial resolution increased by an order of magnitude (van den Bosch et al. 2012; Emsellem 2013; Walsh et al. 2016; Graham et al. 2016). Second, Batcheldor (2010) and Shankar et al. (2016) have suggested that the sample of galaxies with directly measured black hole masses may reflect the upper envelope of points in the M bh -M sph diagram, because it is preferentially galaxies with a bigger black hole and thus a bigger sphere-of-influence that can have their black hole mass measured. \nAt the low-mass end of the distribution in the M bh -M sph diagram, defined by the bulges of 17 late-type galaxies in Savorgnan et al. (2016), the logarithmic slope of the relation varies from 2 to 3 depending on the type of linear regression used. This steeper slope (than observed at the high-mass end) is required for consistency with a wide body of literature, as we shall see in the coming sections. We are in the process of acquiring yet further reliable spheroid masses as there are now (as of mid-2016) 126 galaxies with directly measured black hole masses, including close to 50 spiral galaxies. As discussed at the 2012 IAU General Assembly Special Session 3, 'Galaxy Evolution Through Secular Processes', Graham (2015b) notes many reasons why pseudobulges cannot be reliably identified. Aside from the observation that galaxies can have both a classical bulge and a pseudobulge - thus voiding attempts to subsequently bin galaxies according to whether they have one type or the other - the fact that a continuity of morphological criteria exists from high to low bulge masses has led to considerable confusion regarding the picture of secular versus non-secular processes (Graham 2014). However, Figure 2 suggests that pseudobulges and classical bulges alike are (directly or indirectly) broadly aware of their central black hole mass.", '4.2. The L sph -σ relation': "Around the time that quasars were identified to be at large redshifts, Minkowski (1962) discovered a correlation between velocity dispersion and absolute magnitude for early-type galaxies. He refrained from fitting an equation to it, noting the need to extend the observations to low absolute magnitudes. While Morton & Chevalier (1973) achieved this, finding a continuous distribution of velocity dispersions, it was Faber & Jackson (1976) who were the first to fit an equation to Minkowski's relation. For their sample of 25 galaxies, they reported that L ∝ σ 4 , which has since become known as the FaberJackson relation. A few years later, exploring the bright end of Minkowski's relation, Schechter (1980) discovered that L ∝ σ 5 , a result confirmed by Malumuth and Kirshner (1981; see also von der Linden et al. 2007). Recent studies have suggested that the exponent may be 5.5 in brightest cluster galaxies (Liu et al. 2008) and as high as 6.5 ± 1.3 in core galaxies (Lauer et al. 2007). Shortly after this, Schechter co-authored Davies et al. (1983) in which they revealed that L ∝ σ 2 for low- and intermediate-luminosity early-type galaxies. Many stud- \nFig. 3.Dynamical galaxy mass ( M dyn ) - equal to twice the Jeans Anisotropic Multi-Gaussian-Expansion mass within the effective half-light radius R e - versus the velocity dispersion σ e within R e for the ATLAS 3D early-type galaxies (Cappellari et al. 2013, see their Fig.1). Core galaxies ( γ < 0 . 3 according to the Nuker model (Grillmair et al. 1994; Lauer et al. 1995) as used by Krajnovi'c 2013) are shown by the large red circles, while galaxies having steeper inner profiles ( γ > 0 . 5) are shown by the large blue dots. Galaxies with an unknown inner surface brightness profile slope, or those with 0 . 3 < γ < 0 . 5 are shown by the small dots. \n<!-- image --> \nince shown that this result holds from the lowest luminosity dwarf elliptical galaxies up to M B ≈ -20 to -21 mag (Held et al. 1992; de Rijcke et al. 2005; Matkovi'c & Guzm'an 2005; Balcells et al. 2007b; Lauer et al. 2007; Chilingarian et al. 2008; Forbes et al. 2008; Cody et al. 2009; Tortora et al. 2009; Kourkchi et al. 2012). This explained why past samples of intermediateto-bright early-type galaxies had a slope of around 4, or 3 (Tonry 1981), and confirmed the observation by Binney (1982) and Farouki et al. (1983) that a single power-law was not appropriate to describe the distribution of earlytype galaxies in the L -σ diagram. Most recently, Davies has again illustrated this bend, this time in the M gal -σ diagram for early-type galaxies, through co-authorship of Cappellari et al. (2013). Their bent M gal -σ diagram is reproduced in Figure 3. \nThe bend in Minkowski's relation has been explained by Matkovi'c & Guzm'an (2005) in terms of S'ersic galaxies (which have low- and intermediate-luminosity) following the L ∝ σ 2 relation of Davies et al. (1983) while coreS'ersic galaxies (which have high-luminosity) follow the L ∝ σ 5 relation of Schechter (1980). This continuity for the low- and intermediate-luminosity S'ersic galaxies, and the break-away of bright galaxies with partially depleted cores, is illustrated further in the L -µ 0 and L -n distributions seen in Graham & Guzm'an (2003, their figures 9c and 10; see also Cˆot'e et al. 2007, their figure 3e). As noted in footnote 26 of this article, some galaxies may have experienced a major dry merger event but not display a partially depleted core - such as the merger remnants NGC 1316 (Fornax) and NGC 3115 (Schauer et al. 2014; Menezes et al. 2014) - which could explain why some of the high-mass galaxies in Figure 3 do not \nhave depleted cores 29 . \nThe bend in the M gal -σ diagram, and the M bh -M sph diagram, is likely to have ties with the flattening that is also observed at the bright end of the colour magnitude diagram for early-type galaxies (Tremonti et al. 2004; Jim'enez et al. 2011). Dry merging will increase the luminosity while preserving the colour (modulo passive evolution) among the core-S'ersic elliptical galaxies. In contrast, the S'ersic early-type galaxies display a continuous mass-metallicity relation which unites the dwarf and ordinary early-type galaxies (e.g. Caldwell 1983; Caldwell & Bothun 1987). \nIf the M bh -σ relation (Section 5) is roughly described by a single power-law, and given that the L -σ (and M gal -σ ) relation is notably bent (Figure 3), then the M bh -L relation has to be bent, just as observed and discussed in Figure 2 and Section 4.1.", '5. THE M BH -σ RELATION': 'While the work on the M bh -L relation from Magorrian received considerable attention, it was the M bh -σ relation (Ferrarese & Merritt 2000; Gebhardt et al. 2000) which really sparked off wide-spread global interest in black hole scaling relations. The reason may likely have been because, after having identified and removed galaxies with less secure black hole mass estimates, the M bh -σ relation was reported by both teams to be consistent with having zero intrinsic scatter (see also Kormendy & Gebhardt 2001) 30 . That is, after accounting for the measurement errors, all the scatter was accounted for, suggesting that a new law of physics had been discovered. However, the slope of this potential new law was not agreed upon. Ferrarese & Merritt (2000) had reported M bh ∝ σ 4 . 8 ± 0 . 5 , while Gebhardt et al. (2000) reported an exponent of 3 . 75 ± 0 . 3. The former slope agreed with the energy-balancing prediction by Silk & Rees (1998, see also Haehnelt, Natarajan & Rees 1998) that M bh ∝ σ 5 , while the latter slope agreed with the momentum-balancing prediction by Fabian (1999) that M bh ∝ σ 4 . This discrepancy was to become a major source of controversy and uncertainty in what has become one of the most famous astronomical relations of recent years. As such, some space is dedicated to this issue here. In the following subsection, the main reason for the different slopes is presented, as this continues to be somewhat misunderstood today.', '5.1. Slippery slopes': "Ferrarese & Merritt (2000) performed a symmetrical linear regression, using the bces routine from Akritas & Bershady (1996) which allowed for intrinsic scatter and unique measurement errors on both variables, M bh and σ (which they took to be 13% for the velocity dispersion of external galaxies). Gebhardt et al. (2000), on the other hand, performed a non-symmetrical ordinary least squares regression by minimising the vertical offsets (i.e. in the log M bh direction) about their M bh -σ relation. \n29 It will be interesting in the future to careful apply the coreS'ersic model to see how all the points are distributed in terms of galaxies with and without partially-depleted cores. \nThis approach effectively assumed that the uncertainty on the velocity dispersion was zero and that the black hole masses all had the same uncertainty. \nMerritt & Ferrarese (2001a) addressed the issue of the differing slopes, using four different types of linear regression, two which treated the ( M bh , σ ) data symmetrically and two which did not. They revealed how the slope of the M bh -σ relation increased as one assigned an increasing uncertainty to the velocity dispersion and presented a best fit slope of 4.72 0.36 for their expanded sample. \nTremaine et al. (2002) also looked at this issue of different slopes and noted that under certain conditions 31 the minimisation routine from Akritas & Bershady, which was used by Ferrarese & Merritt (2000), can be biased. As noted above, Merritt & Ferrarese (2001a) had additionally used a second symmetrical regression routine, referred to as the 'Orthogonal distance regression' which had been implemented by Press et al. (1992, their Section 15.3) as FITEXY . It was such that the following quantity was minimised during the task of fitting the line y = a + bx \n± \nχ 2 = N ∑ i =1 [ y i -( a + bx i )] 2 δy i 2 + b 2 δx i 2 , (1) \nwhere N data pairs of y and x values are available in one's sample, and they have measurement errors δy and δx , respectively. Merritt & Ferrarese (2001a) pointed out that Feigelson & Babu (1992) had already noted that this routine is fine unless the distribution to be fit contains intrinsic scatter, i.e. real departures of the data from the optimal line which are not due to measurement errors. At that time, the M bh -σ relation was thought to contain no intrinsic scatter, or was at least consistent with having no intrinsic scatter. \nTremaine et al. (2002) subsequently developed their own modified version of FITEXY . I t was such that it minimised the quantity \nχ 2 = N ∑ i =1 [ y i -( a + bx i )] 2 δy i 2 + b 2 δx i 2 + /epsilon1 2 y , (2) \nwhere the intrinsic scatter /epsilon1 y is solved for by repeating the fit until χ 2 / ( N -2) equals 1. Although Tremaine et al. (2002) claimed this expression still gave a symmetrical treatment of the data, it did not. By trying to allow for intrinsic scatter, they had inadvertently converted a symmetrical expression into a non-symmetrical expression by minimising the offsets under the assumption that all of the intrinsic scatter lay in the y -direction. They reported a slope of 4 . 02 ± 0 . 32 for their M bh -σ relation using the smaller uncertainty of 5% (compare 13%) for the velocity dispersions of the external galaxies. \nHere we look at this a little more carefully, as it continues to cause confusion more than a decade later. If one was to minimise the offsets in the x -direction, about the line y = a + bx , or equivalently x = ( y -a ) /b , the \n31 The slope can be biased if (i) the uncertainty on the x values is large compared to the range of x values, or (ii) the sizes of all the x and y uncertainties are not roughly comparable to each other. \nexpression would be \nχ 2 = N ∑ i =1 [ x i -( y i -a ) b ] 2 δy i 2 /b 2 + δx i 2 + /epsilon1 2 x , N ∑ i =1 [ -y i +( a + bx i )] 2 δy i 2 + b 2 δx i 2 + b 2 /epsilon1 2 x , (3) \nwhere /epsilon1 x is the intrinsic scatter, but this time implicitly assumed to reside in the x -direction. The difference between equations 2 and 3 is the final term in the denominator, which has that /epsilon1 y = b/epsilon1 x . Given this (not surprising) dependence on the slope between /epsilon1 y and /epsilon1 x , the solution reached by solving for χ 2 / ( N -2) = 1 in equations 2 and 3 has a different value of b , i.e. a different slope. To obtain a symmetrical regression therefore requires an average of these two regressions as discussed in Novak et al. (2006) 32 , which are sometimes referred to as the forward and the inverse regression. \nPerforming a non-symmetrical linear regression analysis and minimising the offsets in just the log M bh direction is preferred if one wishes to obtain a relation useful for predicting black hole masses in other galaxies, simply because this relation has the smallest offsets in the log M bh direction (see Feigelson & Babu 1992; Andreon & Hurn 2012). If, on the other hand, one is interested in the underlying / fundamental relation connecting M bh and σ , then one should perform a symmetrical regression. This is discussed by Novak et al. (2006) in terms of the Observer's Question and the Theorist's Question. \nAnalysing the same data 33 from Tremaine et al. (2002), and assigning a 5% uncertainty to the velocity dispersion of each galaxy (including the Milky Way), Novak et al. (2006) reported a slope of 4.10 ± 0.30 using Eq. 2 and 4.59 ± 0.34 using Eq. 3. Had they used an uncertainty of 13%, they would have reported slopes of 4.39 and 4.59, giving an average value slope of 4.49 that was consistent with Merritt & Ferrarese (2001a) who reported an optimal slope of 4.72 0.36. \nTo make a point about the ongoing concerns regarding different minimisation routines, and in particular to show that the symmetrical bisector regression routine from Akritas & Bershady was not producing a biased fit in regard to the ( M bh , σ ) data, Graham & Li (2009) used three symmetrical regression routines, one from Akritas & Bershady (1996), the expression from Tremaine et al. (2002) operating in both forward and inverse mode, and an IDL routine from Kelly (2007) based on a Bayesian estimator. All were shown to give very similar results when the same uncertainty on the velocity dispersion was consistently used, a test that was recently confirmed in Park et al. (2012) who additionally used a fourth (maximum likelihood) estimator. \n±", '5.2. substructure and escalating slopes': "In 2007 Graham noticed that all of the barred galaxies in the M bh -σ diagram were offset, to either lower black hole masses and/or higher velocity dispersions, relative \nto the best-fitting line defined by the non-barred galaxies, and that excluding the barred galaxies resulted in a reduced scatter about the M bh -σ relation (Graham 2007a). At the same time, Hu (2008) had compiled a larger sample and shown the same apparent substructure within the M bh -σ diagram. Hu considered all of his offset galaxies to contain 'pseudobulges', built from the secular evolution of their surrounding disc and containing relatively under-developed black holes. They were also all barred galaxies. Graham (2008a) similarly considered the offset galaxies to have undermassive black holes, due to secular evolution over-developing the bulge, or to have elevated velocity dispersions due to the dynamics of the bar. The choice appears answered because Hartmann et al. (2014) have shown that bars are indeed capable of increasing the velocity dispersion in galaxies, and by exactly the average offset observed in the M bh -σ diagram (see also Debattista et al. 2013 and Monari et al. 2014). Furthermore, Figure 2 shows that pseudobulges and classical bulges (and clump bulges) follow the same broad distribution in the M bh -M sph diagram; at low spheroid masses they both reside systematically below the near-linear relation defined by the massive core-S'ersic spheroids. There is not yet evidence that pseudobulges contain smaller black hole masses than classical bulges of the same mass, although more data would be welcome. In particular, removing the contribution of the bar 34 , and the rotational contribution 35 , from the observed central velocity dispersions of the spheroids would be helpful. It may also make more sense to use the quantity √ 3 σ 2 sph + v 2 sph , rot (Busarello et al. 1992). Although, much of this may be moot in regard to pseudobulges due to the difficult task of actually identifying them, as discussed in the following subsection. \nOne thing that was clear from Hu (2008) and Graham (2008b) was that the growing sample size had generated an increased scatter about the M bh -σ relation 36 , and the intrinsic scatter no longer appeared consistent with zero, a result shown further by Gultekin et al. (2009). The M bh -σ diagram was therefore falling from grace, and it also now presented quite a contrast to early claims which had reported that classical bulges and pseudobulges follow the same black hole scaling relations (e.g. Kormendy 2001; Kormendy & Gebhardt 2001). In Kormendy et al. (2011) the offset nature of the pseudobulges was acknowledged, and it was now claimed that black hole masses do not correlate with the properties of pseudobulges. However, the range in absolute magnitude of the pseudobulges was restricted to just 2 mag, making it challenging to identify if there is a relation present. With a fuller data set, Figure 2 reveals that all bulge types appear to follow an M bh -M sph relation. \nWith a sample size of 72 galaxies, McConnell & Ma (2013) used the non-symmetrical, modified FITEXY routine, as coded by Williams et al. (2010) in MPFITEXY . They reported a slope of 5.64 ± 0.32 for their optimal M bh -σ relation (their figure 1, which included the \n≈ \nalleged over-massive black hole in NGC 1277 from van den Bosch et al. 2012 which has since been rescinded). If they had of additionally used the inverse of this regression, in which the unknown intrinsic scatter is assigned to the log σ direction, they would have obtained a slope of 6.64, and thus an average slope of 6.14. This is steeper than previously reported, and is in part due to their inclusion of the offset barred galaxies at low masses. While McConnell & Ma (2013) do report that their 19 late-type galaxies (with both classical bulges and pseudobulges) have an M bh -σ relation with a zero point (i.e. the term ' a ' in y = a + bx ) that is 0.29 dex lower than for their 53 early-type galaxies (8.36 vs 8.07), i.e. offset by a factor of 2, they did not perform a fit to the barred and non-barred galaxies. Given that the early-type galaxies dominate at the high-mass end of the diagram, and the late-type galaxies at the low-mass end, they combine to produce the steeper relation with a slope of 6. \nGraham et al. (2011) highlighted a potential sample selection bias such that the need to resolve (or nearly resolve) the sphere-of-influence of the black holes may be resulting in an artificial floor to the distribution of points in the M bh -σ diagram. As such, they additionally used a non-symmetrical regression, but one which minimised the offsets in the horizontal direction, i.e. they performed the 'inverse' regression as this should provide the least biased fit (see Lynden-Bell et al. 1988). Adding eight black hole masses to the compilation of 64 data pairs in Graham et al. (2011), Graham & Scott (2013) reported a slope of 6.08 ± 0.31 using their preferred inverse regression on their sample of 72 galaxies (see Figure 4). For the 51 non-barred galaxies, their optimal slope using the inverse regression was 5.53 ± 0.34. While this is at first glance in agreement with the preferred value of 5.64 ± 0.32 reported by McConnell & Ma 2013, it should be realised that it is a coincidence as different things have been measured: a forward regression for all galaxy types versus an inverse regression for non-barred galaxies. \nUsing updated and expanded data for 57 non-barred galaxies, taken from the sample of 89 galaxies in Savorgnan & Graham (2015), the forward, inverse and average regression give a slope of 5.10, 6.48 and 5.79. Folding in the offset barred galaxies results in steeper slopes still, as seen with the McConnell & Ma (2013) data. The increase to the slope over the past few years (see also Sabra et al. 2015 who report a slope of 4.60 using 89 galaxies and the 'forward' linear regression) has largely come from increased black hole masses, and new data, at the high mass end. McConnell & Ma (2013) additionally note that the flux-weighted velocity dispersion within one effective radius can be as much as 10-15% lower in their massive galaxies when excluding data within the black hole's sphere-of-influence. This follows Graham et al. (2011) who noted that the velocity dispersion for M32's spheroid should be reduced from ∼ 75 km s -1 to ∼ 55 km s -1 (Tonry 1987) for exactly this reason. Increases to black hole masses have also come from efforts to account for dark matter halos, resulting in an average increase of ∼ 20% (Schulze & Gebhardt 2011; Rusli et al. 2103a), but as high as a factor of 2 in the case of M87 (Gebhardt & Thomas 2009). Incorporating a dark matter halo is akin to relaxing the past assumption/simplification that \nFig. 4.M bh -σ diagram taken from Graham & Scott (2013). Red circles represent core-S'ersic galaxies; blue dots represent S'ersic galaxies. The crosses designate barred galaxies, which tend to be offset to higher velocity dispersions. The three lines are linear regressions, in which the barred S'ersic galaxies and the non-barred S'ersic galaxies have been fit separately from the core-S'ersic galaxies (which are not barred). \n<!-- image --> \nthe stellar mass-to-light ratio is constant with radius 37 . This new, slightly steeper, M bh -σ relation for the nonbarred galaxies suggests that if L sph ∝ σ 6 (Lauer et al. 2007) for the core-S'ersic galaxies, then one can expect to recover M bh ∝ L sph for the core-S'ersic galaxies. If L sph ∝ σ 5 (e.g. Schechter 1980) then one can expect to find M bh ∝ L 6 / 5 sph , suggestive of a second order effect on the picture of dry mergers maintaining a constant M bh /L sph and M bh /M sph ratio. Resolution to this minor query may simply require consistency with the regression analyses, or perhaps a careful bulge/disc separation of the galaxies involved (e.g. Laurikainen et al. 2005, 2011; Balcells et al. 2007a,b; Gadotti 2008; Lasker et al. 2014a), because core-S'ersic galaxies can contain a fast-rotating disc (e.g. Dullo & Graham 2013; Krajnovi'c et al. 2013).", '5.2.1. Pseudobulges': "Pseudobulges are particularly hard to identify, for the multitude of reasons presented in Graham (2013, 2014). Furthermore, many galaxies contain both a disc-like 'pseudobulge' and a classical bulge (e.g. Erwin et al. 2003, 2014; Athanassoula 2005; Gadotti 2009; MacArthur, Gonz'alez & Courteau 2009; dos Anjos & da Silva 2013; Seidel et al. 2014), including the Milky Way it seems (e.g. D'ek'any et al. 2013; Kunder et al. 2016; see also Saha 2015). In addition, some may have formed from the (secular) inward migration and (classical) merging of stellar clumps (e.g. Noguchi 1999; Bournaud et al. 2007; \n37 This raises another issue which is yet to be properly addressed in the literature: not only do many spheroids have radial stellar population gradients, but most S'ersic galaxies have nuclear star clusters in addition to massive black holes, and the assumption of a single stellar mass-to-light ratio when modelling the data to derive a black hole mass is therefore not appropriate. \nInoue & Saitoh 2012, and references therein). All of this makes the task of labelling galaxies as either containing a pseudobulge or a classical bulge highly problematic and untenable. In the M bh -σ analysis by Graham et al. (2011) and Graham & Scott (2013), they avoided the issue of pseudobulges and separated galaxies based on the presence (or not) of a bar and revealed that the masses of black holes in barred galaxies correlate with the velocity dispersion, despite their heightened dynamics. Given that the majority of S'ersic spheroids (i.e. those without partially depleted cores) also follow the near-quadratic M bh -L relation, it appears that the masses of black holes in pseudobulges correlate with at least one property of their host bulge, and unless pseudobulges are restricted to have a narrow range of velocity dispersion, then their black hole masses also correlate with velocity dispersion (or at least define an upper envelope in the M bh -σ diagram). \nA few of the (often not properly recognised) difficulties with identifying pseudobulges are noted here, in case it is helpful to some readers. From a kinematical perspective, just as with the formation of rotating elliptical galaxies via mergers, mergers can also create bulges which rotate (e.g. Bekki 2010; Keselman & Nusser 2012) and bars can spin-up classical bulges (e.g. Saha et al. 2012, 2016), and the smaller the bulges are the easier it is. Rotation is therefore not a definitive signature of a pseudobulge. In spiral galaxies, the observable presence of the disc's inner spiral arms, which cohabit the inner region of the galaxy where the bulge also resides, are of course easier to detect in fainter bulges (which are those that have smaller S'ersic indices) due to the greater bulge/arm contrast. However the detection and presence of these underlying features does not necessitate the presence of a pseudobulge (e.g. Eliche-Moral et al. 2011; dos Anjos & da Silva 2013). \nFrom a selection of hundreds of disc galaxies imaged in the K -band, Graham & Worley (2008) observe no bimodality in the bulge S'ersic indices, questioning the suitability of a divide at a S'ersic index of n = 2 which has frequently been used in the recent literature. This divide is roughly halfway between n = 1 (which describes the light-profiles of flattened rotating discs) and n = 4 (which was in the past thought to describe the majority of elliptical galaxies and large bulges). While pseudobulges are expected to have S'ersic indices n ≈ 1 having formed from their surrounding exponential disc (e.g. Bardeen 1975; Hohl 1975; Combes & Sanders 1981; Combes et al. 1990; Pfenniger & Friedli 1991) - the problem is that mergers do not only produce R 1 / 4 -like light profiles. Mergers can also create bulges with n < 2 (e.g. Eliche-Moral et al. 2011; Scannapieco et al. 2011; Querejeta et al. 2015), just as low-luminosity elliptical galaxies (not built from the secular evolution of a disc) are well known to have n < 2 and even < 1 (e.g. Davies et al. 1988; Young & Currie 1994) 38 . \nPrior to the realisation that the S'ersic index changes monotonically with spheroid luminosity and size (e.g. \n38 The occurrence of large-scale, rotating stellar discs and kinematical substructure in early-type galaxies on either side of the alleged divide at M B = -18 mag ( n ≈ 2) further reveals the continuity of dwarf and ordinary early-type galaxies (e.g., Emsellem et al. 2007; Krajnovi'c et al. 2008; Scott et al. 2014; Toloba et al. 2014). \nCaon et al. 1993; Andredakis et al. 1995) - referred to as structural nonhomology - the curved but continuous scaling relations involving the 'effective' half-light radii and 'effective' surface brightness (which have a maximum curvature around n = 2) had suggested that spheroids with n < 2 may be a distinct species rather than the low mass extension of spheroids with n > 2 (see Graham 2013). However we now know that this was a red-herring, and that all relations involving the 'effective' parameters are curved (e.g. Graham & Guzm'an 2003; Gavazzi et al. 2005; Ferrarese et al. 2006a; Cˆot'e et al. 2006, 2007). As such, the Kormendy (1977) relation cannot be used to separate dwarf early-type galaxies from ordinary earlytype galaxies, nor to separate pseudobulges from classical bulges, because at low-luminosities both types of bulge (classical and pseudo) depart from this relation, which is the tangent to the bright arm of the curved µ e -R e distribution.", '6. THE M BH -N RELATION': "As noted in Graham et al. (2001), it may not be the total amount of mass in a spheroid, but rather how that mass is distributed, when it comes to the connection with the central supermassive black hole. Similarly, the velocity dispersion is but a tracer of the underlying mass distribution, and as such it can not be the fundamental parameter driving the black hole mass scaling relations. \nIntriguingly, what Graham et al. (2001) revealed is that the central radial concentration of light, within the inner effective half light radii of spheroids, correlates strongly with the black hole mass. The concentration index which they used, taken from Trujillo et al. (2001), is monotonically related with the S'ersic index n , and thus an M bh -n relation also exists, as shown in Graham et al. (2003a). With an expanded data set, Graham & Driver (2007) revealed that this relation is no longer well described by a single log-linear power-law, and that a log-quadratic relation performs noticeably better (see Figure 5a). Given the log-linear L -n relation observed for both elliptical galaxies (e.g. Young & Currie 1994; Jerjen & Binggeli 1997; Graham & Guzm'an 2003; Ferrarese et al. 2006a) and the bulges of disc galaxies (e.g. Andredakis et al. 1995; Graham & Worley 2008, and references therein), and the bent M bh -L sph relation (Section 4), the M bh -n relation must be bent, such that galaxies which have experienced major, relatively dry, merger events are responsible for the flattening which is seen in Figure 5 at high masses. \nThe existence of the M bh -L sph relation, coupled with existence of the L sph -n relation, necessitates the existence of the M bh -n relation. Although, as illustrated by Savorgnan et al. (2013), there is a need for care when measuring S'ersic indices, and studies which fail to recover the M bh -n relation for the sample of galaxies with directly measured black hole masses may be dominated by poorly measured S'ersic indices, and in turn erroneous bulge magnitudes which depend on an accurate S'ersic index. Within the literature, measurements for individual galaxies have varied dramatically (e.g. Graham & Driver 2007; Laurikainen et al. 2010; Sani et al. 2011; Vika et al. 2012; Beifiori et al. 2012; Rusli et al. 2013a; L'asker et al. 2014a). Shown in Figure 5b are the average values, after the rejection of extreme outliers, plotted against black hole mass. Savorgnan et al. (2013) divided the sample \n<!-- image --> \nFig. 5.Left panel: M bh -n diagram taken from Graham & Driver (2007). The core-S'ersic spheroids are shown here by the red circles, while the S'ersic spheroids are shown by the blue dots. The lone S'ersic spheroid at the high-mass end is the S0 galaxy NGC 3115, identified to not have a core by Ravindranath et al. (2001). Right panel: M bh -n diagram from Savorgnan et al. (2013). Rather than a single log-quadratic relation, two log-linear relations are shown here, one for the S'ersic spheroids and one for the core-S'ersic spheroids. \n<!-- image --> \ninto S'ersic and core-S'ersic spheroids, and fit separate linear regressions for each sub-population. \nSavorgnan (2016, in prep.) will present an investigation based on a careful multi-component analyses (of the 72 galaxies used by Graham & Scott 2013) which reconciled the differences between past attempts to measure the S'ersic index. For example, sometimes these discrepancies arise because a lenticular disc galaxy may have been modelled with either a single S'ersic component or more correctly as the sum of a S'ersic-bulge plus an exponential disc by a different author. Other times the presence of an unaccounted for nuclear disc, or a partially depleted core, has biased the luminosity-weighted fits in some studies. Despite the need for care when measuring the S'ersic index, the advantage is that one only requires uncalibrated photometric images. \nReaders interested in the development of fitting bulge light profiles since de Vaucouleurs (1959) first noted departures from his R 1 / 4 model, may appreciate the references in section 4.1 of Graham (2013). Andredakis et al. (1995) were the first to model the bulges of disc galaxies with S'ersic's (1963) light profile model, following its application to elliptical galaxies by Davies et al. (1988) and Caon et al. (1993), and the earlier advocation of its use by Capaccioli (1985, 1987). Some of the difficulty with, and the impact of getting, the S'ersic index correct is illustrated by Gadotti & S'anchez-Janssen (2012) in the case of the Sombrero galaxy.", '7. THE M BH -µ 0 DIAGRAM': "It is not unreasonable to expect that the growth of massive black holes may be related to the growth, and subsequent space density, of stars in its immediate vicinity. Gas processes have contributed to the development of both, and the black hole mass may be more connected with the local stellar density than the total stellar mass of the host spheroid. While the de-projected stellar density, ρ 0 is ideally the quantity we would like to have (e.g. Mer- \nitt 2006b, his figure 5), and this can be derived under certain assumptions (e.g. Terzi'c & Graham 2005, their Eq. 4), it is of course the projected surface brightness that is observed. \nBinggeli, Sandage & Tarenghi (1984) and Sandage & Binggeli (1984) provide a nice historical account of the detection of dwarf galaxies, and wrote that it was established that 'the dwarf elliptical galaxies form a continuum in luminosity with the brighter E systems'. Caldwell (1983; his Figure 6) and Bothun et al. (1986, their figure 7) revealed this continuum was such that fainter than M B ≈ -20 . 5 mag, there is a log-linear relation between the luminosity and the central surface brightness, µ 0 . In addition to this, Binggeli et al. (1984, their figure 11) and Binggeli & Cameron (1991, their figures 9 and 18) found that, when using the inward extrapolation of King models, this L -µ 0 relation extends from -12 > M B > -23 mag. This was further highlighted by Jerjen & Binggeli (1997) and Graham & Guzm'an (2003) when using the inward extrapolation of the S'ersic model; extrapolated over partially depleted cores in the case of the brightest spheroids whose cores have been eroded away by coalescing supermassive black holes. \nGiven this log-linear L -µ 0 relation, and the bent M bh -L sph relation (Section 4), there must be a bent M bh -µ 0 relation. It should again be emphasized that this particular value of µ 0 refers to the extrapolated / expected value prior to core depletion. Given the difficulties in routinely obtaining robust S'ersic indices for the spheroids with black hole masses (Section 6), it is perhaps not surprising that this diagram is yet to be published. Although it may be the fundamental parameter linking black holes with their bulges, to date there is only a prediction by Graham & Driver (2007) for its form. This was derived by coupling the log-quadratic M bh -n relation from Graham & Driver with the log-linear n -µ 0 relation from Graham & Guzm'an (2003), and is reproduced here in Figure 6. \nGiven our current understanding, it makes more sense \nto construct the M bh -µ 0 relation using the log-linear M bh -L relations for the S'ersic and core-S'ersic spheroids given in Graham & Scott (2013, their table 3) together with the log-linear L -µ 0 relation given in Graham & Guzm'an (2003, their figure 9c). Because the latter was derived in the B -band, we use the B -band M bh -L relation from Graham & Scott (2013). For the S'ersic galaxies, this gives the relation \nlog( M bh /M /circledot ) = 17 . 24 -0 . 63 µ 0 , (4) \nand for the core-S'ersic galaxies one has the relation \nlog( M bh /M /circledot ) = 13 . 62 -0 . 36 µ 0 . (5) \nThese predictions are shown in Figure 6. From the multicomponent modelling by Savorgnan & Graham (2016a) of galaxies with directly measured black hole masses, it will be possible to populate this diagram and (under certain assumptions) its deprojected cousin.", '8. DEPLETED GALAXY CORES AND THE M BH -M DEF RELATION': "As noted previously, the merger of two galaxies without substantial gas, referred to as a dry merger, will result in the supermassive black holes from the progenitor galaxies sinking to the bottom of the newly wed galaxy by transferring much of their orbital angular momentum to the stars near the new galaxy's core (Begelman, Blandford & Rees 1980; Ebisuzaki et al. 1991). Such collisional construction of galaxies results in an evacuated 'loss cone' showing up as a partially depleted core 39 in the images of nearby galaxies (e.g. King & Minkowski 1966, 1972; Kormendy 1982; Lauer 1983). Typical core sizes, as quantified by the break radius R b of the core-S'ersic model, are \nFig. 6.Predictions for the M bh -µ 0 diagram. The dashed curve is from Graham & Driver (2007), while the thin blue and thick red lines show equations 4 and 5 for the S'ersic and core-S'ersic spheroids, respectively. Clearly the uncertainty on these lines is still quite large, given that the solid lines do not trace the dashed curve, but a bend is nonetheless expected. \n<!-- image --> \n39 See Dullo & Graham (2013, their Section 6.1) for a discussion of alternative concepts for core depletion. \nFig. 7.Cartoon showing a pair of supermassive black holes kicking stars away as they dance towards coalescence at the centre of a galaxy. Credit: Paolo Bonfini. \n<!-- image --> \ntens to a few hundred parsec (e.g. Trujillo et al. 2004; Ferrarese et al. 2006a; Cˆot'e et al. 2007; Hyde et al. 2008; Richings et al. 2011; Rusli et al. 2013b; Dullo & Graham 2013, 2014; Bonfini 2014), and roughly a factor of 2 smaller than Nuker model break radii (Lauer et al. 1995). Whether or not coalescence of the black holes has already occurred in these galaxies with partially depleted cores is not clear, although see Khan et al. (2011, 2013, and references therein) in regard to the 'final parsec problem'. \nUsing the core-S'ersic model to quantify the central flux deficit, and in turn the stellar mass deficit, Graham (2004) discovered M def ≈ 2 M bh . Previously it was thought that M def /M bh was, on average, an order of magnitude greater (e.g. Milosavljevi'c et al. 2002; Ravindranath, Ho & Filippenko 2002), which required a troublingly large number of merger events given that the ejected mass should roughly scale with N M bh , where N is the cumulative number of (equivalent major) dry merger events (Milosavljevi'c & Merritt 2001; Merritt 2006a). Using the core-S'ersic model, these new lower mass ratios were also found by Ferrarese et al. (2006a) and Hyde et al. (2008). Using the idea from Graham et al. (2003) that cores can be measured as a deficit of light relative to the inward extrapolation of the outer S'ersic profile, but fitting the S'ersic model rather than coreS'ersic model and identifying the sizes of depleted cores by eye, Kormendy & Bender (2009) reported notably larger mass ratios (typically close to 10 or higher). Hopkins & Hernquist (2010) subsequently resolved this issue in a model-independent manner and revealed that the coreS'ersic model measurements of the central mass deficits were correct. Most recently, Rusli et al. (2013b) found that ∼ 80% of their 23 galaxies have 1 < M def /M bh < 5, while Dullo & Graham (2014) reported typical values for their sample of 31 galaxies to be 0 . 5 < M def /M bh < 4. \nAlthough the central mass deficit and break radius are obviously not fundamental parameters in establishing the spheroid-(black hole) connection - simply because many galaxies have black holes but not partially depleted cores - there is nonetheless an M bh -R b re- \nlation (Lauer et al. 2007) 40 and an M bh -M def relation (e.g. Graham 2004; Rusli et al. 2013b; Dullo & Graham 2014). This relation simply exists over a restricted mass range. Dullo & Graham (2014, their Eq. 18) reported that M def ∝ M 3 . 70 ± 0 . 76 bh for the population ensemble (not to be confused with growth in individual galaxies). This is of interest for several reasons. One of which is that it may provide insight into the merging scenario, which currently has an unresolved problem. In general, galaxies with the greatest M def /M bh ratio should have experienced the highest number of major dry mergers, and due to the increase in black hole mass but stagnation in velocity dispersion associated with such mergers (e.g. Ostriker & Hausman 1977; Hausman & Ostriker 1978; Ciotti & van Albada 2001), they should be offset to high black hole masses in the M bh -σ diagram (see Volonteri & Ciotti 2013). However, they are not (Savorgnan & Graham 2015). \nWithin low-luminosity early-type galaxies, the nuclear star cluster can be slightly offset ( ∼ 100 parsec) from the galaxy's photometric centre (Binggeli et al. 2000; Barazza et al. 2003). This is thought to be due to the dense star cluster's harmonic oscillation within the weak gravitational gradient of the galaxy's core. The amplitude of the nuclear cluster's rocking back and forth motion is expected to be greater in spheroids with lower S'ersic index, because they have lower central stellar densities and shallower inner density profiles, and thus less well defined gravitational centres over a greater fraction of their half-light radii (see Terzi'c & Graham 2005, their figure 2). Similarly, high-luminosity core-S'ersic spheroids have somewhat weakened gravitational centres (Terzi'c & Graham 2005, their figure 3) due to the partial depletion of stars in their cores. One may then expect to find the supermassive black holes slightly offset from the photometric centres of core-S'ersic galaxies (Miller & Smith 1992; Taga & Iye 1998). However a mechanism capable of creating more extreme ( > 1 kpc) offsets is the recoil from the emission of anisotropic gravitational radiation that a newly merged black hole may receive (e.g. Bonnor & Rotenberg 1961; Peres 1962; Bekenstein 1973). The linear momentum carried away by the gravitational wave is balanced by a kick imparted to the black hole. This recoil process has the ability to evacuate a much greater loss cone, and has been proposed as an explanation for some cores having large M def /M bh ratios (e.g. Boylan-Kolchin et al. 2004; Campanelli et al. 2007; Gualandris & Merritt 2008, 2012), which have been observed in NGC 1399 and NGC 5061. While only small spatial offsets are known for black holes in galaxies with directly measured black hole masses (e.g. Batcheldor et al. 2010; Lena et al. 2014), if this process is operating one might expect to see greater displacements (e.g. Blecha et al. 2012) of black holes in galaxies with larger M def /M bh ratios. However, if the damping timescale of the recoilinduced oscillation is sufficiently short, one may not find this correlation. \nIn passing, it might be remiss if a few words were \n40 Lauer et al. (2007) found that using the radius where the negative, logarithmic slope of the surface brightness profile equals 0.5 (which matches well with the core-S'ersic break radius: Dullo & Graham 2012, their section 5.2) produces a stronger relation than obtained when using the Nuker model break radii. \nnot said about the gravitational wave signals expected from the final coalescence of massive black holes after they have scoured out the cores of massive spheroids, preferentially removing stars on plunging radial orbits (e.g. Quinlan & Hernquist 1997; Milosavljevi'c & Merritt 2001; Thomas et al. 2014). Binary AGN, and thus massive black holes, are now known in several galaxies (e.g. Komossa et al. 2003; Liu et al. 2014, and references therein). The rapidly changing gravitational field as the black holes spiral (and thus accelerate) around each other, generates a gravitational wave-like ripple which radiates out into space (e.g. Buonanno & Damour 2000; Barack & Cutler 2004; Baker et al. 2006; Blanchet 2006; Sesana 2010; Amaro-Seoane et al. 2012). Travelling at the speed of light, the amplitude of the wave decays linearly (rather than quadratically) with distance and, also unlike light, passes unimpeded through both space and matter. Due to the large orbital size of the binary black hole, space-based interferometers at great separations are required to sample the long wavelength of the waves generated by the black hole binary. Building on the hopes of the Laser Interferometer Space Antenna (LISA: Danzmann & Rudiger 2003), the European LISA Pathfinder mission 41 (LPF: Anza et al. 2005; McNamara 2013), formerly known as SMART-2, offers the very exciting promise of detecting these waves predicted by Einstein's theory of relativity but not yet observed (Will 2006).", '9. INTERMEDIATE MASS BLACK HOLES AND THE (BLACK HOLE)-(NUCLEAR CLUSTER) CONNECTION': "As was noted in section 4.1, the bent M bh -M sph relation offers hope for detecting the missing population of intermediate mass black holes. This is because the linear M bh -M sph relation predicts 10 2 < M bh /M /circledot < 10 5 black hole masses in smaller / fainter spheroids. Although we may not have the spatial resolution at optical/nearinfrared wavelengths to resolve the sphere-of-influence of these black holes, and thus directly measure their masses from Keplerian kinematics, there is an independent method which can be used to predict (strengthen / reject) the likely existence of such intermediate mass black holes. It is based on the observation that the black hole mass correlates with the AGN radio and X-ray flux in such a way that they define a 2-dimensional surface in 3-parameter space, which has been dubbed the 'fundamental plane of black hole activity' (Merloni et al. 2003). Therefore, obtaining radio and X-ray data is expected to prove fruitful in the hunt for the elusive intermediate mass black holes. Preferably, this data should be obtained simultaneously because the AGN are known to vary in their flux output over timescales of days. \nOne of the best candidates for an intermediate mass black hole is the ultraluminous X-ray source HLX-1 in the galaxy ESO 24349 (Farrell et al. 2009; Webb et al. 2014). Interestingly, this 9,000 solar mass black hole candidate does not reside near the centre of its host galaxy but in a compact star cluster (Soria et al. 2010; Wiersema et al. 2010; Farrell et al. 2012) located at a projected distance of ∼ 3 kpc from the galaxy's nucleus, perhaps shedding insight into the formation location of intermediate \nFig. 8.Predicted black hole masses. The solid histogram was obtained using the M bh -L K relation for S'ersic spheroids applied to the K -band bulge magnitudes in Graham & Scott (2013, their table 6). The open histogram was obtained using the M bh -M sph relation for S'ersic spheroids (shown in Figure 2) applied to the dwarf galaxy masses in Reines et al. (2013, their table 1). The shaded histogram was obtained in the same way but using the dwarf galaxy stellar masses in Moran et al. (2014, their table 1). The fainter bulges are expected to contain the least massive black holes. \n<!-- image --> \nmass black holes (see also Mezcua et al. 2013, 2015 in regard to an off-centered intermediate mass black hole candidate in NGC 2276). Despite early hopes for intermediate mass black holes in globular clusters (e.g. Gerssen et al. 2003; Gebhardt et al. 2005; Noyola et al. 2010; Lutzgendorf et al. 2013, and references therein), there are not yet any definite candidates (e.g. van den Bosch et al. 2006; Hurley 2007; Anderson & van der Marel 2010; Vesperini & Trenti 2010; Lanzoni et al. 2013; Lanzoni 2015). Observational research programs (e.g. Bellini et al. 2014; Lapenna et al. 2014) continue the hunt as the formation of intermediate mass black holes in dense star clusters seems probable (e.g. Miller & Hamilton 2002; Baumgardt et al. 2004; Gurkan et al. 2004; Portegies Zwart et al. 2004). \nAside from globular clusters, some of the dense star clusters found in the nuclei of many low- and intermediate-luminosity spheroids (e.g. Reaves 1983; Binggeli et al. 1985; Phillips et al. 1996; Carollo et al. 1997) are already known to house massive black holes. Ferrarese et al. (2006b) and Wehner & Harris (2006) originally suggested that these star clusters may be the low-mass extension of the supermassive black holes, in the sense that galaxies housed one type of nucleus or the other. However this idea was soon modified when it was realised that such clusters and massive black hole coexist in substantial numbers of galaxies (e.g. Gonz'alez Delgado et al. 2008; Seth et al. 2008; Graham & Spitler 2009). Ongoing efforts have revealed that nuclear star clusters do not follow the same mass scaling relations as supermassive black holes (Graham 2012b; Leigh et al. 2012; Neumayer & Walcher 2012; Scott & Graham 2013), and the search for intermediate mass black holes continues. Among the most promising targets are the low mass bulges of disc galaxies hosting an AGN (Graham & Scott 2013) and the low mass dwarf galaxies which also display AGN activity (e.g. Reines et al. 2013; Moran et al. 2014); see Figure 8. \nJust as there is a relation between spheroid luminos- \nity and the central surface brightness 42 of the spheroid - until the onset of partially depleted cores in massive spheroids - there is also a relationship between spheroid luminosity and the brightness of the nuclear star clusters that they host (Balcells et al. 2003; Graham & Guzm'an 2003). In a somewhat similar manner to the establishment of the M bh -µ 0 relation presented in Section 7, one can predict what the M bh -M nc relation should be like. Graham (2015a) combined the relation M bh ∝ M 2 sph for the S'ersic spheroids (Section 4.1) with the relation M nc ∝ M 0 . 6 -1 . 0 sph (references above) to obtain M bh ∝ M 2 -3 . 3 nc . A consistent result was obtained by coupling the relation M bh ∝ σ 5 . 5 (Section 5) with M nc ∝ σ 1 . 6 -2 . 7 (references above) to give M bh ∝ M 2 . 0 -3 . 4 nc . Massive black holes therefore grow rapidly within their host star cluster, until it is evaporated (e.g. Bekki & Graham 2010) or partially devoured (e.g. Hills 1975; Frank & Rees 1976; Murphy et al. 1991; Komossa 2013; Donato et al. 2014; Vasiliev 2014). However, disentangling which came first may be an interesting pursuit, and just as there are different types of bulges, there may be different types of nuclear star clusters (e.g. Turner et al. 2012). This M bh -M nc relation is somewhat complementary to the M bh -M def relation, with each applicable at opposing ends of the black hole mass range currently accessible. Such co-occupancy of black holes and nuclear star clusters is a likely source of stellar tidal disruption events (Komossa et al. 2009, 2013 and references therein) and gravitational wave emission from the inspiralling of compact stellar remnants (e.g. Hils & Bender 1995; Amaro-Seoane et al. 2007 and references therein), predictions for which are dramatically modified when using the new, near-quadratic M bh -M sph relation (Mapelli et al. 2012). Further quantifying the coexistence of massive black holes in dense, compact, nuclear star clusters should help us to predict the occurrence of, and better understand, these exciting phenomenon.", '10. THE M BH -M HALO RELATION': "Ferrarese et al. (2002) have revealed that there is a relationship between the black hole mass and the galaxy halo mass (baryons plus dark matter), as traced by the circular velocity at large radii (used as a proxy for the halo's virial radius). Due to the relation between this rotational velocity and the galaxy's velocity dispersion (see also Baes et al. 2003; Pizzella et al. 2005; Ferrarese & Ford 2005, their Eq. 21) 43 one can expect an M bh -M halo relation. The extent of this relationship may be applicable only to galaxies with large bulges (or v circ > ∼ 100 km s -1 or σ > ∼ 100 km s -1 ), because of the breakdown in the relationship between circular velocity and velocity dispersion for lower mass systems (e.g. Zasov et al. 2005; \n42 Technically it is the central surface brightness of the spheroid excluding blips from additional nuclear components such as star clusters. \n43 It should be noted that the dynamical study by Kronawitter et al. (2000) and Gerhard et al. (2001), which led to the relationship between the circular velocity and the velocity dispersion for elliptical galaxies, was based on a sample of elliptical galaxies that had very similar absolute magnitudes. Consequently, these galaxies will have similar structural and dynamical profiles, and thus their v circ -σ relationship may not be applicable to lower- or higher-luminosity elliptical galaxies with different S'ersic indices, i.e. concentration, and dynamical profiles (e.g. Ciotti 1991). \nHo 2007; Courteau et al. 2007). Nonetheless, this would make the relationship exist over a larger mass range than the M bh -R b and M bh -M def relations (Section 8). \nFor galaxies built from major dry merger events, in which the black hole mass and the galaxy stellar mass simply add together, the dark matter must also add in this linear fashion. This would then establish a linear M bh -M halo relation - just as there is a linear M bh -M sph relation preserving the M bh /M sph ratio - at high masses ( M bh > ∼ 10 8 M /circledot ). This appears to be consistent with the data in Ferrarese et al. (2002, their figure 5). However, their linear regression to the fuller sample gives M bh ∝ M 1 . 65 -1 . 82 halo , which is in remarkable agreement with the prediction M bh ∝ M 5 / 3 halo by Haehnelt, Natarajan & Rees (1998). Although, with a different sample, Baes et al. (2003) reported M bh ∝ M 1 . 27 halo . Curiously, for elliptical galaxies not built from dry mergers 44 , the prediction by Haehnelt et al. (1998) transforms into M bh ∝ L 20 / 9 gal (= L 2 . 22 gal ) if M halo /L gal ∝ L 1 / 3 gal (Jørgensen et al. 1996; Cappellari et al. 2006). This near-quadratic relation has been seen before in Section 4.", '10.1. Globular cluster systems': "Lending support to the M bh -M halo relation is the connection between black hole mass and the halo of globular clusters that swarm around galaxies, both in terms of their number (Burkert & Tremaine 2010; Harris & Harris 2011; Rhode 2012; Harris et al. 2014) and their velocity dispersion (Sadoun & Colin 2012; Pota et al. 2013). In Burkert & Tremaine (2010) they used a (selfadmittedly limited) sample of 13 galaxies for which the black hole mass and the number of globular clusters was known. They observed an rms scatter of just 0.21 dex about their optimal relation in the log( M bh ) mass direction. Not surprisingly this attracted some interest (e.g. Snyder et al. 2011) because it was half of the value observed in the M bh -σ diagram. However as more galaxies have been added, the scatter about the relation involving the globular clusters has increased. \nThe globular cluster system around individual galaxies are known to display a bimodality in their colour, with the red (metal rich) globular clusters thought to be associated with the galaxy's bulge while the blue (metalpoor) globular clusters are thought to be connected with the halo (Ashman & Zepf 1992; Forbes et al. 1997). Using both the observed velocity dispersion of the globular cluster system, and the velocity dispersion with the rotational component of the system subtracted, Pota et al. (2013) report that while a correlation with black hole mass is evident, it is not yet clear if the black hole mass is better correlated with the red (bulge) or the blue (halo) globular cluster sub-population.", '11. THE M BH -(SPIRAL ARM PITCH ANGLE) CONNECTION': "While the applicability of the M bh -M halo relation in lower mass spiral galaxies is unclear, there is a somewhat complementary relation which only operates in spiral galaxies. Seigar et al. (2008; see also Ringermacher \n- 44 Equal mass, (major) dry mergers preserve the M halo /L ratio and therefore galaxies built from major dry mergers follow the sequence M halo /L ∝ L 0 . \n& Mead 2009; Treuthardt et al. 2012 and Berrier et al. 2013) have presented the relation between black hole mass and spiral arm pitch angle. The spiral arm pitch angle (e.g. Puerari et al. 2014, and references therein) is of course known to vary along the Hubble-Jeans sequence, as does the bulge-to-total flux ratio, or more correctly the luminosity of the bulge (e.g. Yoshizawa & Wakamatsu 1975; Ostriker 1977; Meisels & Ostriker 1984; Trujillo et al. 2002), which may explain the black hole connection with the pitch angle. As with the radial concentration of the bulge light, the pitch angle has the advantage that it can be measured from photometrically uncalibrated images and therefore offers an easy means to predict black hole masses (perhaps even when there is no bulge 45 ), from which one can then do clever things like determine the black hole mass function in spiral galaxies (Davis et al. 2014). \nGiven that this is obviously a secondary relation, although the low level of scatter reported by Davis et al. (2014) is intriguing, less shall be said about this than the relations involving a spheroid's central concentration and density of stars (Sections 6 and 7).", '12. FUNDAMENTAL PLANES: ADDING A THIRD PARAMETER': "As noted in Section 9, stellar and supermassive black holes roughly define a plane within the 3-dimensional space of black hole mass, radio power and X-ray luminosity (Merloni et al. 2003; Heinz & Sunyaev 2003; Falcke et al. 2004; Kording et al. 2006; Li et al. 2008). While this is both interesting in its own right and highly useful, the relationship between the black hole mass, accretion disc and jet is of a different nature to the other relations presented in this article and as such is not detailed here as it is an AGN phenomenon. \nOne of the early attempts to introduce a third parameter into the (black hole)-(host galaxy) scaling relations was by Marconi & Hunt (2003). They used the effective half light radius ( R e ) of the spheroid, together with the velocity dispersion ( σ ), to derive a rough virial mass for the spheroid ( M virial ∝ σ 2 R e ). They found that the total vertical scatter about their M bh -M virial relation was slightly less than that about their M bh -σ relation (0.25 dex vs 0.30 dex). Using a sample of elliptical galaxies, Feoli & Mele (2005; see also Feoli & Mancini 2009, 2011) reported on a black hole mass relation with the kinetic energy of the host galaxy such that M bh ∝ ( M gal σ 2 ) α , where 0 . 87 < α < 1 . 00 and M gal was derived assuming R 1 / 4 light profiles 46 . Given that M gal roughly scales as σ 2 R e , their kinetic energy expression roughly scales with σ 4 R e . Additional variations of this theme, searching for a fundamental plane using combinations of σ and R e can be found in de Francesco et al. (2006), who effectively suggested independent exponents for σ and R e , in Aller & Richstone (2007) in terms of the gravitational binding energy, and in Hopkins et al. (2007) and Soker & Meiron (2011). Given the existence of the Fundamental Plane (Djorgovski & Davis 1985) linking the velocity dis- \n- 45 The M bh -(pitch angle) relation is yet to be established for a sample of bulgeless galaxies. \npersion with the mean effective surface brightness ( 〈 µ 〉 e ) and effective half light radius, the presence of the M bh -σ relation additionally suggests that there should be an M bh -( µ 〉 e , R e ) plane (Barway & Kembhavi 2007). \n〈 \n〉 With all of these attempts to define different planes, there are two issues that require attention: (i) barred galaxies, and (ii) the accuracy 47 and thus usefulness of R e . \nFirst, the increased scatter in the M bh -σ diagram due to the inclusion of barred galaxies was reported by Graham (2008a,b) and Hu (2008). Moreover, Graham (2008a) showed that once the barred galaxies were removed, there was no reduction in scatter when going from the M bh -σ diagram to the M bh -( σ, R e ) diagram. If there is a more fundamental relation with some combination of σ and R e , than compared with σ alone, this should not have been observed. The simulations of Younger et al. (2008, their Fig.9) show that (merger built) classical bulges follow a plane, without the need to include (secular-disk-evolution built) pseudobulges. Therefore, if the lower scatter about the hybrid relations is only achieved when including the barred galaxies, it suggests that something else is responsible for the reduction, such as barred galaxies having smaller R e values than the elliptical galaxies which dominate at the high mass end of one's sample. Younger et al. (2008) suggested that the relatively small dynamic range among the non-barred galaxies with direct black hole mass measurements may have been inadequate to provide a significant detection of this third parameter and thus a plane. It would be interesting to repeat the tests which searched for an optimal plane among the non-barred galaxies, but now using the larger galaxy samples which are available. However this brings us to the second issue. \nGiven that there have been errors in the measurement of the S'ersic indices n (as revealed by Savorgnan et al. 2013), there are thus errors in the measurements of the published, effective half light radii R e (see also Bernardi et al. 2014). Harris et al. (2014) show the large range of R e values (for the same spheroid) reported by different authors for spheroids with directly measured black hole masses. A similar plot is shown in Figure 9 but this time restricting the data to that obtained from S'ersic R 1 /n model fits by different authors. Consequently, attempts to use R e for measuring dynamical masses ( ∝ σ 2 R e ) or as a third parameter to mop up some of the scatter about the M bh -σ relation should at this time be treated with caution.", '13. CONCLUDING REMARKS': "The 'attraction' of black holes is vast, as evinced by a huge literature on the subject, of which but a small fraction is noted here. The fundamental physical connection \nbetween black-hole and bulge growth still awaits discovery. While it is expected that we may narrow in on the solution as we keep plugging away at more black hole mass measurements, coupled with improving the accuracy of all quantities involved, it is reasonable to expect that something unexpected may be discovered, such is the nature and joy of our collective pursuit. \nFig. 9.Major-axis effective half-light radii R e for the spheroidal component of 43 galaxies (having directly measured black hole masses) as determined by different authors. Figure taken from Savorgnan & Graham (2016a). Legend: red = Graham & Driver (2007); blue = Laurikainen et al. (2010); green = Sani et al. (2011); yellow = Vika et al. (2012); gray = Beifiori et al. (2012); orange = Lasker et al. (2014a). \n<!-- image --> \nGiven the role that pulsars played in convincing the community that black holes may exist in 1967-8, it is perhaps fitting that arrays of pulsar beacons are used today (e.g. Sesana et al. 2008; Hobbs et al. 2010 ; Kramer & Champion 2013) to try and detect the bob and sway of the space antennae as anticipated gravitational waves - from the inspiral of supermassive black holes at the centres of newly merged galaxies - wash by oblivious to our solar system. 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2015PhRvL.114o1102G
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No-Hair Theorem for Black Holes in Astrophysical Environments
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2015-01-01
| 32 | 0.45 | 164 |
['-', '-', '-', '-', '-', 'methods analytical', '-', '-', '-', '-']
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According to the no-hair theorem, static black holes are described by a Schwarzschild spacetime provided there are no other sources of the gravitational field. This requirement, however, is in astrophysical realistic scenarios often violated, e.g., if the black hole is part of a binary system or if it is surrounded by an accretion disk. In these cases, the black hole is distorted due to tidal forces. Nonetheless, the subsequent formulation of the no-hair theorem holds: The contribution of the distorted black hole to the multipole moments that describe the gravitational field close to infinity and, thus, all sources is that of a Schwarzschild black hole. It still has no hair. This implies that there is no multipole moment induced in the black hole and that its second Love numbers, which measure some aspects of the distortion, vanish as was already shown in approximations to general relativity. But here we prove this property for astrophysical relevant black holes in full general relativity.
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https://arxiv.org/pdf/1503.03240.pdf
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{'No-hair theorem for Black Holes in Astrophysical Environments': 'Norman Gurlebeck \nCenter of Applied Space Technology and Microgravity (ZARM), University of Bremen, Am Fallturm, 28359 Bremen, Germany, EU \nAccording to the no-hair theorem, static black holes are described by a Schwarzschild spacetime provided there are no other sources of the gravitational field. This requirement, however, is in astrophysical realistic scenarios often violated, e.g., if the black hole is part of a binary system or if it is surrounded by an accretion disk. In these cases, the black hole is distorted due to tidal forces. Nonetheless, the subsequent formulation of the no-hair theorem holds: The contribution of the distorted black hole to the multipole moments that describe the gravitational field close to infinity and, thus, all sources is that of a Schwarzschild black hole. It still has no hair. This implies that there is no multipole moment induced in the black hole and that its second Love numbers, which measure some aspects of the distortion, vanish as was already shown in approximations to general relativity. But here we prove this property for astrophysical relevant black holes in full general relativity. \nPACS numbers: 04.20.Cv, 04.20.Ha, 04.70.Bw, 04.20.Jb Keywords: black holes, no-hair theorem, tidal distortion', 'INTRODUCTION': "The no-hair theorem states that any isolated static black hole is necessarily a Schwarzschild black hole and that there is only one free parameter describing the spacetime - the mass M . Although the black hole has actually one hair, M , this property is still called the no-hair theorem and, thus, the black hole is called bald. This theorem is, of course, very appealing for astrophysics, since just one parameter has to be measured to determine the entire spacetime around a black hole. Even if the black hole rotates, a similar theorem holds and only the mass and the spin have to be measured. \nIn fact, these two parameters are already observable, see, e.g., Refs. [1-3]. Additionally, new observatories like GRAVITY (see Ref. [4]) will improve these measurements further. An independent and promising approach is the measurement of the shadow of a black hole, see, e.g., Refs. [5, 6] for recent results. The shadow will be resolvable in the millimeter/submillimeter range with the Event Horizon Telescope [7]. Moreover, the potential discovery of a binary system containing a black hole promises headway for characterizing black holes either via pulsar timing if the companion is a pulsar or via the detection of gravitational waves. \nSince the no-hair theorem dictates in the rotating case that the quadrupole moment of a black hole is determined by its mass and spin, an independent measurement of all three parameters allows for a test of alternative theories of gravity or of the assumptions of the no-hair theorem, for recent approaches see, e.g., Refs. [8-10]. For example, the source could be described by matter distributions like boson stars for which the no-hair theorem does not hold. \nA crucial assumption for the no-hair theorem is that the black hole is isolated, i.e., the spacetime is asymptotically flat and contains no other sources. However, in \nmany astrophysical situations this requirement is not fulfilled, e.g. for black holes in binary systems, if the black hole is surrounded by plasma, an accretion disk or if jets are formed in its vicinity, i.e., it might put on different types of wigs. These additional sources contribute also to the total multipole moments of the spacetime. Hence, a formulation like the standard no-hair theorem for isolated black holes cannot hold anymore, whereas a formulation solely for the part of the total multipole moment sourced by the black hole might still be correct. The latter is shown in this Letter. In the exterior field of additional sources, the black hole is distorted and the inner geometry of the horizon changes. This is measured by the Love numbers of the first kind or the multipole moments of isolated horizons, see Refs. [11-13]. Nonetheless, we show here that this does not imply that the black holes are not anymore bald. More precisely: Although the total multipole moments of the spacetime measured at infinity change, this is solely due to the external sources and not to a different contribution of the black holes themselves. In fact, distorted black holes have only a mass monopole . Thus, even though the black hole might put on a wig it still looks bald. Note that we assume here for simplicity static black holes. \nDistortions of static black holes and neutron stars are of particular interest for inspirals treated in an adiabatic regime, see, for details on the validity of this regime, Refs. [14, 15]. In such a quasistatic approximation, the black hole or the neutron star is distorted due to the external field of the companion instantaneously. Thus, the system is additionally axially symmetric with respect to the axis joining the two constituents of the binary system and the metric of distorted black holes of Ref. [16] is applicable. The imprints of such distortions in the gravitational waves emitted by inspiraling binaries give information on the equation of state of neutron stars [14, 17]. Moreover, \nthose imprints can be used to experimentally reveal if a constituent of a binary system is a black hole. This provides an avenue along which the existence of black holes can be directly inferred. On the other hand, if the existence of a black hole in a binary system is established independently by observing, say, gamma ray bursts at later stages of the inspiral, then the measurement of its distortions using gravitational waves allows us to test general relativity via the here presented no-hair theorem. \nThe distortions of the black holes and neutron stars are characterized by the Love numbers of first and second kind, h r and k r , cf. Refs. [15, 18, 19]. Roughly speaking, the h r measure the changes in the shape of the horizon or the neutron star and the k r measure the change in the asymptotic multipole moments caused by the distortion due to an external source, see Refs. [13, 15, 19-21] for their use in general relativity. In the latter four works, it was established using approximation methods that the k r vanish for four-dimensional black holes. However, it was debated if this result is still valid in the case higher orders in the approximations scheme are taken into account. We resolve this dispute here by proving the result analytically without any approximation. Although these distortions are not crucial to detect gravitational waves in prospective data from Advanced LIGO and Advanced Virgo [22], they will be important for a detailed analysis of the data and for future detectors with an increased sensitivity, see, e.g., Refs. [23, 24]. \nThe Love numbers of the second kind were also applied to establish universal relations, i.e., relations that are independent of the equation of state, between certain physical parameters describing neutron stars, see Refs. [25-27] but also Refs. [28-31]. The here considered black hole case is solved analytically in full general relativity and, thus, it serves as a test for the various approximation schemes employed for neutron stars also in this respect. \nSubsequently, we use geometric units, in which G = c = 1, where c is the velocity of light and G Newton's gravitational constant. The metric has the signature ( -1 , 1 , 1 , 1). Greek indices run from 0 to 3 and Latin indices run from 1 to 3.", 'DISTORTED BLACK HOLES': "The metric of arbitrary static and axially symmetric spacetimes can be written in the Weyl form under standard assumptions, cf. Ref. [32]: \nds 2 = e 2 k -2 U dρ 2 + dζ 2 ) + W 2 e -2 U dϕ 2 -e 2 U dt 2 , (1) \n( \n) where the functions U, k , and W depend on ρ and ζ . Note that the metric functions U and W can be expressed by the timelike Killing vector ξ α and the spacelike Killing vector η α : \ne 2 U = -ξ α ξ α , W 2 = -η α η α ξ β ξ β . (2) \nIn case the exterior sources are static and axially symmetric or allow for a quasistatic description, the general metric near the horizon H of a distorted black hole was found by Geroch and Hartle in Ref. [16] in the form of Eq. (1). In a neighborhood of H , we assume pure vacuum, which is physically reasonable if the matter can be treated quasistatically and satisfies the energy conditions, cf. Ref. [33]. Thus, there exists a surface S H , which encloses H and no other sources. If S H is sufficiently close to H , the metric functions in Eq. (1) read between S H and H \nU = U S + U D , k = k S + k SD , W = ρ. (3) \nThe functions U S and k S are given by the respective quantities of the Schwarzschild black hole: \nU S = 1 2 log [ r + + r --2 M r + + r -+2 M ] , k S = 1 2 log [ ( r + + r -) 2 -4 M 2 4 r + r -] , r 2 ± = ρ 2 +( ζ ± M ) 2 . (4) \nSubsequently, we find that the parameter M coincides with the Komar mass of the distorted black hole. The function U D is determined by the exterior matter and it solves a Laplace equation \n( ∂ 2 ∂ρ 2 + 1 ρ ∂ ∂ρ + ∂ 2 ∂ζ 2 ) U D = 0 . (5) \nIf U D vanishes, the spacetime describes a Schwarzschild black hole. The function k SD follows from a line integration once U D + U S is known, cf. Ref. [32]. However, we do not require its explicit form subsequently. \nThe horizon of the distorted black hole is located at the symmetry axis ( ρ = 0, ζ ∈ [ -M,M ]) like for the Schwarzschild black hole. In fact, in canonical Weyl coordinates the horizon can always be located at ρ = 0, see Ref. [34]. These coordinates allow a shift in the ζ coordinate. We employed this freedom to place the horizon symmetrically with respect to that coordinate, i.e., that the 'north or south pole' of the horizon are characterized by ζ N/S = ± M . At these points, U D has to take the same value to avoid struts, which we want to exclude for simplicity, see Ref. [16]. If the external matter is reflection symmetric, like for accretion disks or jets, this is trivially satisfied. Note that the metric functions take the form of Eq. (3)-(5) only in a neighborhood of H and they neither describe directly the asymptotic behavior nor the metric in the interior of the external source. Nonetheless, we will be able to conclude with the help of the source integrals the contributions of the distorted black hole to the asymptotic multipole moments without specifying the exterior sources in detail. These could also include other black holes. We only require that the spacetime is asymptotically flat and that all external sources \nare contained in a region, which does not contain H and which does not extend to infinity. We denote its boundary by S ext .", 'THE SOURCE INTEGRALS': 'To disentangle the contributions of the black hole and the external sources to the asymptotic multipole moments, the source integrals proved to be the essential tool. They were recently derived in Ref. [35] and they make it possible to define the asymptotics of the spacetime including the Geroch multipole moments by evaluating quasilocal surface or volume integrals. The respective surfaces and volumes need only to envelope or contain all regions with a nonvanishing stress-energy tensor. Here we need the surface integrals and introduce the required quantities, subsequently. \nThe Weyl multipole moments U ( r ) are defined as the expansion of U along the axis of symmetry close to infinity, i.e., \nU = ∞ ∑ r =0 U ( r ) | ζ | r +1 . (6) \nAs we will see later the coordinate ζ can be defined geometrically so that this definition is also covariant. Indeed, it was shown in Ref. [36] that from the U ( r ) the Geroch multipole moments m r can be determined uniquely by nonlinear algebraic relations. To calculate the m r , the U ( k ) need to be known for 0 ≤ k ≤ r . Thus, it is sufficient for us to consider here the U ( r ) . Note that the origin with respect to which the multipole moments are measured was chosen by requiring ζ N/S = ± M . \nFurthermore, we use the functions \nN ( r ) -( x, y ) = /floorleft r 2 /floorright ∑ k =0 2( -1) k +1 r ! x 2 k +1 y r -2 k 4 k ( k !) 2 ( r -2 k )! , N ( r ) + ( x, y ) = /floorleft r -1 2 /floorright ∑ k =0 2( -1) k +1 r ! x 2 k +2 y r -2 k -1 4 k ( k !) 2 ( r -2 k -1)!(2 k +2) . (7) \nIt can easily be checked that these functions obey the equations \nN ( r ) + ,x -N ( r ) -,y = 0 , N ( r ) + ,y + N ( r ) -,x -N ( r ) -x = 0 . (8) \nCommas denote partial derivatives. Additionally, let us introduce the 1-form \nZ α = /epsilon1 αβγδ W ,β W -1 η γ ξ δ , (9) \nwhere /epsilon1 αβγδ is the volume form of the spacetime. In vacuum, Z α is exact and it is hypersurface orthogonal in the entire spacetime. Since the surfaces of interest, \nS H and S ext , lie in the vacuum region or its boundaries, we can introduce a scalar Z via Z ,α = Z α , for technical details and a more general treatment see Ref. [35]. It turns out that Z = ζ in canonical Weyl coordinates if the constant of integration is suitably chosen. \nWith this notation at hand, we can express the Weyl multipole moments by \nU ( r ) = ∫ S H η ( r ) a ˆ n a d S H + ∫ S ext η ( r ) a ˆ n a d S ext , η ( r ) a = 1 8 π e U W ( N ( r ) -U ,a -N ( r ) + ,W Z ,a U + N ( r ) + ,Z W ,a U ) , (10) \nwhere ˆ n a denotes the outward pointing unit normal to the surfaces S H and S ext and the functions N ( r ) ± depend on ( x, y ) = ( W,Z ), see [35]. d S H and d S ext are the proper area elements of S H and S ext , respectively. In vacuum, we can always choose canonical Weyl coordinates such that W = ρ and Z = ζ .', 'THE INDUCED MULTIPOLE MOMENTS OF DISTORTED BLACK HOLES': "With Eq. (10), we can identify the contribution of the different sources to the asymptotic Weyl multipole moments covariantly. The first term in Eq. (10), which we denote U ( r ) H , gives the contribution of the distorted black hole and the second term, U ( r ) ext , the contribution of the external sources. The induced multipole moment of a distorted black hole is now simply defined as U ( r ) ind = U ( r ) H -U ( r ) S , where the U ( r ) S are the Weyl multipole moments of an undistorted Schwarzschild black hole. They coincide with the Newtonian multipole moments of a line mass of uniform density, see Ref. [32]. We parameterize S H for constant angles ϕ from the 'north pole' to the 'south pole' ( s ∈ [ s N , s S ] ↦→ ( ρ ( s ) , ζ ( s ) , ϕ = const . )), cf. Ref. [35]. Then we obtain with Eq. (3) the U ( r ) H : \nU ( r ) H = 1 4 s S ∫ s N [ N ( r ) -( U S + U D ) ,n -( N ( r ) + ,W Z ,n -N ( r ) + ,Z W ,n ) ( U S + U D ) ] d s, (11) \nwhere we denote by f ,n the normal derivative -f ,ρ d d s ζ ( s ) + f ,ζ d d s ρ ( s ). The multipole moments of a Schwarzschild black hole can be inferred from Eq. (11) by setting U D = 0. For the induced multipole moments, we have in turn only to subtract this Schwarzschild con- \nution from Eq. (11) and get \nU ( r ) ind = 1 4 s S ∫ s N [ N ( r ) -U D,n -N ( r ) + ,W Z ,n U D + N ( r ) + ,Z W ,n U D ] d s. (12) \nApplying the divergence theorem and Eq. (5), we can rewrite U ( r ) ind : \nU ( r ) ind = 1 8 π ∫ V H 1 ρ [ U D,ρ ( N ( r ) -,ρ + N ( r ) + ,ζ -N ( r ) -ρ ) + U D,ζ ( N ( r ) -,ζ -N ( r ) + ,ρ )] d V H , (13) \nwhich vanishes by virtue of Eq. (8). V H is the coordinate volume enclosed by S H and H in canonical Weyl coordinates. Thus, the induced multipole moments vanish and the contribution of the distorted black hole to the asymptotic Weyl multipole moments is the same as that of a Schwarzschild black hole . With the results in Refs. [35, 36], this can readily be translated to Geroch's multipole moments. This generalizes the no-hair theorem to black holes that are distorted by external matter. Note that this matter sources the gravitational field, too. Hence, the total asymptotic multipole moments differ in general from those of a Schwarzschild spacetime. This holds in particular for systems of two black holes. In the above derivation, the origin with respect to which the multipole moments are measured is chosen in the center of one of the black holes. Thus, the other black hole regarded as external matter contributes, for instance, a nonvanishing quadrupole moment to the total one, cf. the comment after Eq. (6). \nThe vanishing of the induced multipole moments implies that the second Love numbers k r vanish, too, because they are proportional to U ( r ) ind . This is corroborated by the results in Refs. [15, 19-21]. But here we did not use any approximation or linearization. The result holds in full general relativity . Thus, one can assume k r = 0 for black holes, which rotate sufficiently slowly, in binary systems when calculating the emitted gravitational radiation during the adiabatic regime. Note that k r = 0 is specific to black holes. It does not hold for neutron stars, cf. Ref. [19]. Nonetheless, the source integrals of the Weyl multipole moments are still tailored to calculate their k r , since the contributions from the individual sources to the Weyl multipole moments are separated covariantly and a definition of an induced multipole moment becomes possible in full general relativity. Moreover, it simplifies the evaluation of the source integrals in Ref. [35] in the presence of black holes considerably, since we have only to calculate the mass of the individual black holes to know all U ( r ) H . \nIf the black holes rotate sufficiently slowly, there are several implications for astrophysics: On the one hand, \nmeasuring the mass of the black hole determines its contribution to the multipole moments completely. In binary systems containing a black hole or for a black hole with an accretion disc, the mass of the black hole can be inferred from the mass of the entire system measured by the motion of distant stars and the mass of the companion star or disc. After that all multipole moments of the black hole are fixed. Thus, every measurement of the multipole moments of the entire system, say, the quadrupole moment determines the quadrupole moment of the companion or the disc. On the other hand, if the quadrupole moment of both, the entire system and the companion star or disc can be measured, then general relativity can be tested.", 'THE MULTIPOLE MOMENTS OF THE HORIZON': "Whereas the distorted black hole has the same asymptotic multipole moments as a Schwarzschild black hole, the horizon geometry clearly changes. This can be easily seen by evaluating the covariantly defined multipoles M n of isolated horizons following Ref. [11, 12]. The scheme outlined therein was independently carried out in Ref. [13] using Schwarzschild-like coordinates. In that paper, it was found that the multipole moments of the distorted horizon are different from those of the Schwarzschild black hole. In fact, these deviations were used to define a01 relativistic analogue of the first Love numbers for black holes, which do not vanish in contrast to the second Love numbers. \nThe change in the geometry of the horizon is, however, not reflected in the asymptotic multipole moments. This, at first glance, counterintuitive behavior can be understood with a trivial Newtonian example. Consider a point mass and its multipoles. All multipole moments but the mass vanish and the equipotential surfaces are spheres. If an additional gravitational field generated by, say, a second point mass separated from the first is introduced, the multipole moments of the original point mass, which can be evaluated with Newtonian source integrals, are unchanged. In fact, the point particle has no inner structure and, thus, cannot be distorted by an external gravitational field and the source stays the same. Nonetheless, the equipotential surfaces are no longer spheres analogously to the distorted horizon. The situation changes of course, if an internal structure is assumed, like one would have to do for the description of neutron stars. Then the external gravitational field can indeed deform the matter distribution and the sources, which can be measured in the asymptotic multipole moments. \nN.G. gratefully acknowledges support from the DFG within the Research Training Group 1620 'Models of Gravity'. Partial support comes also from NewComp- \nStar, COST Action MP1304. The author thanks A. Ashtekar and J. Steinhoff for helpful discussions. \n- [1] C. Done, C. Jin, M. Middleton, and M. Ward, Mon. Not. R. Astron. Soc. 434 , 1955 (2013).\n- [2] C. S. Reynolds, Classical Quant. Grav. 30 , 244004 (2013).\n- [3] G. Risaliti et al. , Nature 494 , 449 (2013).\n- [4] F. Eisenhauer et al. , GRAVITY: Microarcsecond Astrometry and Deep Interferometric Imaging with the VLT, in Science with the VLT in the ELT Era , edited by A. Moorwood, Springer, Netherlands, p. 361, 2009.\n- [5] Z. Li and C. Bambi, J. Cosmol. Astropart. Phys. 1 , 41 (2014).\n- [6] A. Grenzebach, V. Perlick, and C. Lammerzahl, Phys. Rev. D 89 , 124004 (2014).\n- [7] S. S. Doeleman et al. , Nature 455 , 78 (2008).\n- [8] A. E. Broderick, T. Johannsen, A. Loeb, and D. Psaltis, Astrophys. J. 784 , 7 (2014).\n- [9] T. Johannsen and D. Psaltis, Astrophys. J. 773 , 57 (2013).\n- [10] M. Wade, J. D. E. Creighton, E. Ochsner, and A. B. Nielsen, Phys. Rev. D 88 , 083002 (2013).\n- [11] A. Ashtekar, J. Engle, T. Pawlowski, and C. v. d. Broeck, Classical Quant. Grav. 21 , 2549 (2004).\n- [12] A. Ashtekar and B. Krishnan, Living Rev. Relat. 7 , (2004).\n- [13] T. Damour and O. M. Lecian, Phys. Rev. D 80 , 044017 (2009).\n- [14] E. E. Flanagan and T. Hinderer, Phys. Rev. D 77 , 021502(R) (2008).\n- [15] T. Damour and A. Nagar, Phys. Rev. D 80 , 084035 (2009).\n- [16] R. Geroch and J. B. Hartle, J. Math. Phys. 23 , 680 (1982).\n- [17] T. Hinderer, Astrophys. J. 677 , 1216 (2008), Astrophys. J. 697 , 964(E) (2009).\n- [18] A. Love, Some Problems of Geodynamics (Cornell University Library, Ithaca, 1911).\n- [19] T. Binnington and E. Poisson, Phys. Rev. D 80 , 084018 \n(2009). \n- [20] B. Kol and M. Smolkin, J. High Energ. Phys. 2 , 10 (2012).\n- [21] S. Chakrabarti, T. Delsate, and J. Steinhoff, arXiv:1304.2228 [gr-qc].\n- [22] C. P. L. Berry, I. Mandel, H. Middleton, L. P. Singer, A. L. Urban, A. Vecchio, S. Vitale, K. Cannon, B. Farr, W. M. Farr, P. B. Graff, C. Hanna, C.-J. Haster, S. Mohapatra, C. Pankow, L. R. Price, T. Sidery, and J. Veitch, arXiv:1411.6934 [astro-ph].\n- [23] L. Baiotti, T. Damour, B. Giacomazzo, A. Nagar, and L. Rezzolla, Phys. Rev. Lett. 105 , 261101 (2010).\n- [24] J. S. Read, L. Baiotti, J. D. E. Creighton, J. L. Friedman, B. Giacomazzo, K. Kyutoku, C. Markakis, L. Rezzolla, M. Shibata, and K. Taniguchi Phys. Rev. D 88 , 044042 (2013).\n- [25] K. Yagi and N. Yunes, Science 341 , 365 (2013).\n- [26] K. Yagi and N. Yunes, Phys. Rev. D 88 , 023009 (2013).\n- [27] A. Maselli, V. Cardoso, V. Ferrari, L. Gualtieri, and P. Pani, Phys. Rev. D 88 , 023007 (2013).\n- [28] D. D. Doneva, S. S. Yazadjiev, N. Stergioulas, and K. D. Kokkotas, Astrophys. J. Lett. 781 , L6 (2014).\n- [29] B. Haskell, R. Ciolfi, F. Pannarale, and L. Rezzolla, Mon. Not. R. Astron. Soc. 438 , L71 (2014).\n- [30] S. Chakrabarti, T. Delsate, N. Gurlebeck, and J. Steinhoff, Phys. Rev. Lett. 112 , 201102 (2014).\n- [31] G. Pappas, T.A. Apostolatos Phys. Rev. Lett. 112 , 121101 (2014).\n- [32] H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt, Exact Solutions of Einstein's Field Equations (Cambridge University Press, Cambridge, 2003).\n- [33] J. M. Bardeen, Rapidly Rotating Stars, Disks, and Black Holes., in Black Holes (Les Astres Occlus) , edited by C. Dewitt and B. S. Dewitt, pp. 241, Gordon and Breach Science Publishers, New York (1973).\n- [34] B. Carter, Black hole equilibrium states, in Black Holes (Les Astres Occlus) , edited by C. Dewitt and B. S. Dewitt, pp. 57, Gordon and Breach Science Publishers, New York (1973).\n- [35] N. Gurlebeck, Phys. Rev. D 90 , 024041 (2014).\n- [36] G. Fodor, C. Hoenselaers, and Z. Perj'es, J. Math. Phys. 30 , 2252 (1989)."}
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2016MNRAS.456.3929S
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Three-dimensional simulations of supercritical black hole accretion discs - luminosities, photon trapping and variability
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2016-01-01
| 30 | 0.52 | 164 |
['accretion', 'accretion disks', 'black hole physics', 'relativity', 'methods numerical', '-']
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We present a set of four three-dimensional, general relativistic, radiation magnetohydrodynamical simulations of black hole accretion at supercritical mass accretion rates, dot{M} > dot{M}_Edd. We use these simulations to study how disc properties are modified when we vary the black hole mass, the black hole spin, or the mass accretion rate. In the case of a non-rotating black hole, we find that the total efficiency is of the order of 3 per cent dot{M} c^2, approximately a factor of 2 less than the efficiency of a standard thin accretion disc. The radiation flux in the funnel along the axis is highly super-Eddington, but only a small fraction of the energy released by accretion escapes in this region. The bulk of the 3 per cent dot{M} c^2 of energy emerges farther out in the disc, either in the form of photospheric emission or as a wind. In the case of a black hole with a spin parameter of 0.7, we find a larger efficiency of about 8 per cent dot{M} c^2. By comparing the relative importance of advective and diffusive radiation transport, we show that photon trapping is effective near the equatorial plane. However, near the disc surface, vertical transport of radiation by diffusion dominates. We compare the properties of our fiducial three-dimensional run with those of an equivalent two-dimensional axisymmetric model with a mean-field dynamo. The latter simulation runs nearly 100 times faster than the three-dimensional simulation, and gives very similar results for time-averaged properties of the accretion flow, but does not reproduce the time-variability.
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https://arxiv.org/pdf/1509.03168.pdf
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{'Aleksander S˛adowski 1 ? and Ramesh Narayan 1 ?': '1 MIT Kavli Institute for Astrophysics and Space Research 77 Massachusetts Ave, Cambridge, MA 02139, USA \n- 2 Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02134, USA \n11 September 2015', 'ABSTRACT': 'We present a set of four three-dimensional, general relativistic, radiation MHD simulations of black hole accretion at super-critical mass accretion rates, ˙ M > ˙ M Edd. We use these simulations to study how disk properties are modified when we vary the black hole mass, the black hole spin, or the mass accretion rate. In the case of a non-rotating black hole, we find that the total e GLYPH<14> ciency is of order 3% ˙ Mc 2 , approximately a factor of two less than the e GLYPH<14> ciency of a standard thin accretion disk. The radiation flux in the funnel along the axis is highly super-Eddington, but only a small fraction of the energy released by accretion escapes in this region. The bulk of the 3% ˙ Mc 2 of energy emerges farther out in the disk, either in the form of photospheric emission or as a wind. In the case of a black hole with a spin parameter of 0.7, we find a larger e GLYPH<14> ciency of about 8% ˙ Mc 2 . By comparing the relative importance of advective and di GLYPH<11> usive radiation transport, we show that photon trapping is e GLYPH<11> ective near the equatorial plane. However, near the disk surface, vertical transport of radiation by di GLYPH<11> usion dominates. We compare the properties of our fiducial three-dimensional run with those of an equivalent two-dimensional axisymmetric model with a mean-field dynamo. The latter simulation runs nearly 100 times faster than the three-dimensional simulation, and gives very similar results for time-averaged properties of the accretion flow. \nKey words: accretion, accretion discs - black hole physics - relativistic processes - methods: numerical', '1 INTRODUCTION': 'Black hole (BH) accretion disks are common in the Universe. It appears that virtually every galaxy harbours a supermassive black hole (SMBH) in its nucleus and it is likely that every one of these SMBHs has some kind of an accretion flow. Moreover, just in our own Galaxy, there are probably thousands of stellar-mass BHs in binaries accreting gas from their companions, of which a few dozen have been detected in X-rays and are widely studied. \nBecause of the compactness of BHs, accreting gas can liberate significant amounts of gravitational energy and make the accreting systems extraordinary luminous. Moreover, magnetic fields near the BH encourage the extraction of rotational energy of spinning BHs, leading to formation of powerful relativistic jets. \nEarly theoretical work on accretion disks was limited to onedimensional analytical models. Later, 1 + 1 and two-dimensional models with GLYPH<11> -viscosity were developed. Because accretion flows are by nature turbulent, such simplified models were not adequate in most cases. Moreover, they often made strong assumptions, e.g., no mass loss, constant GLYPH<11> , zero-torque at the innermost stable circu- \nlar orbit (ISCO), which may not be satisfied in real systems. This motivated the development of techniques for numerically modeling multi-dimensional turbulent accretion flows. \nThe key breakthroughs were the identification of the magnetorotational instability (MRI, Balbus & Hawley 1991) and the development of magnetohydrodynamical (MHD) numerical codes, both Newtonian and relativistic. Initial e GLYPH<11> orts were focused on simulating optically thin disks, as are likely to be present at the lowest accretion rates. Such systems are relatively simple to simulate because the radiation is weak and is, moreover, decoupled from the gas. Only in recent years have more advanced radiation-MHD (RMHD) codes been developed which can be used to study radiatively luminous systems. The pioneering initial work was based on Newtonian codes with crude (flux-limited di GLYPH<11> usion) radiative transport (Ohsuga et al. 2009; Ohsuga & Mineshige 2011). This was later followed by codes using more advanced radiative transport schemes, either still in the Newtonian or special-relativistic approximation (Jiang et al. 2012, 2014a; Ohsuga & Takahashi 2015), or in general relativity (S˛adowski et al. 2013; McKinney et al. 2014; Fragile et al. 2014). \nSo far global simulations of optically thick disks that evolve the radiation field in parallel to gas have been performed only for \nsuper-critical (exceeding the Eddington value, see equation 6) accretion rates 1 . Such disks are geometrically thick and do not require excessive resolution near the equatorial plane which, so far, makes self-consistent simulations of thinner disks too expensive. \nIn this paper we continue the numerical study of super-critical BH accretion flows by performing a set of four three-dimensional simulations, the parameters of which probe di GLYPH<11> erent accretion rates, BH spins, and BH masses. Such a comparative study using a single code (and set of assumptions) has not yet been performed. In addition, we compare the properties of our fiducial three-dimensional model with an equivalent axisymmetrical twodimensional run which is simulated using the mean-field dynamo model described in S˛adowski et al. (2015a). \nWe begin with a description of the numerical methods (Section 2) and details of the five simulations (Section 3). We then discuss the results, focusing on the luminosities of the simulated disks (Section 4), the e GLYPH<14> ciency of photon trapping (Section 5), and the variability of the emitted radiation (Section 6). Finally, we assess the strengths and weaknesses of 2D simulations (Section 7), and list the conclusions in the Summary (Section 8).', '2 NUMERICAL METHODS': "The simulations described in this paper were performed with the general relativistic radiation magnetohydrodynamical (GRRMHD) code KORAL (S˛adowski et al. 2013) which solves the conservation equations in a fixed, arbitrary spacetime using finite-di GLYPH<11> erence methods. The equations we solve are, \n( GLYPH<26> u GLYPH<22> ); GLYPH<22> = 0 ; (1) \n( T GLYPH<22> GLYPH<23> ); GLYPH<22> = G GLYPH<23> ; (2) \n( R GLYPH<22> GLYPH<23> ); GLYPH<22> = GLYPH<0> G GLYPH<23> ; (3) \nwhere GLYPH<26> is the gas density in the comoving fluid frame, u GLYPH<22> are the components of the gas four-velocity as measured in the 'lab frame', T GLYPH<22> GLYPH<23> is the MHD stress-energy tensor in this frame, \nT GLYPH<22> GLYPH<23> = ( GLYPH<26> + u g + p g + b 2 ) u GLYPH<22> u GLYPH<23> + ( p g + 1 2 b 2 ) GLYPH<14> GLYPH<22> GLYPH<23> GLYPH<0> b GLYPH<22> b GLYPH<23> ; (4) \nR GLYPH<22> GLYPH<23> is the stress-energy tensor of radiation, and G GLYPH<23> is the radiative four-force describing the interaction between gas and radiation (see S˛adowski et al. 2014, for a more detailed description). Here, u g and p g = ( GLYPH<0> GLYPH<0> 1) u g represent the internal energy and pressure of the gas in the comoving frame and b GLYPH<22> is the magnetic field 4-vector (Gammie et al. 2003). The magnetic pressure is p mag = b 2 = 2 in geometrical units. \nThe magnetic field is evolved via the induction equation, \n@ t ( p GLYPH<0> gB i ) = GLYPH<0> @ j GLYPH<16> p GLYPH<0> g ( b j u i GLYPH<0> b i u j ) GLYPH<17> ; (5) \nwhere B i is the magnetic field three-vector (Komissarov 1999), and p GLYPH<0> g is the metric determinant. The divergence-free criterion is enforced using the flux-constrained scheme of Tóth (2000). \nThe radiation field is evolved through its energy density and flux, and the radiation stress-energy tensor is closed by means of the M1 closure scheme (Levermore 1984; S˛adowski et al. 2013). The energy exchange between gas and radiation is by free-free emission / absorption as well as Compton scattering. The latter is treated \n1 A number of groups (e.g., Shafee et al. 2008a; Schnittman et al. 2013; Avara et al. 2015) have performed simulations of thin disks in pure hydrodynamical setup implementing arbitrary cooling function. \nin the 'blackbody' Comptonization approximation as described in S˛adowski & Narayan (2015c). \nFour of the five simulations described here were performed in three dimensions (3D), while the fifth was carried out in 2D, assuming axisymmetry and using the mean-field dynamo model described in S˛adowski et al. (2015a) with model parameters identical to those used there. \nWeuse modified Kerr-Shild coordinates with the inner edge of the domain inside the BH horizon. The simulations were run with a moderately high resolution of 252 grid cells spaced logarithmically in radius, 234 grid cells in the polar angle, concentrated towards the equatorial plane, and 128 cells in azimuth. \nAll details of the numerical method are given in S˛adowski et al. (2014).", '3.1 Units': 'We adopt the following definition for the Eddington mass accretion rate, \n˙ M Edd = L Edd GLYPH<17> c 2 ; (6) \nwhere L Edd = 1 : 25 GLYPH<2> 10 38 M = M GLYPH<12> ergs = s is the Eddington luminosity, GLYPH<17> is the radiative e GLYPH<14> ciency of a thin disk around a black hole with a given spin a GLYPH<3> GLYPH<17> a = M , \nGLYPH<17> = 1 GLYPH<0> r 1 GLYPH<0> 2 3 R ISCO ; (7) \nand r ISCO is the radius of the Innermost Stable Circular Orbit (ISCO). According to this definition, a standard thin, radiatively e GLYPH<14> cient disk (Shakura & Sunyaev 1973; Novikov & Thorne 1973; Frank et al. 1985) accreting at ˙ M Edd would have a luminosity of L Edd. For zero BH spin, ˙ M Edd = 2 : 48 GLYPH<2> 10 18 M = M GLYPH<12> g = s. \nHereafter, we use the gravitational radius r g = GM = c 2 as the unit of length, and rg = c as the unit of time.', '3.2 Initial setup': 'Each of the five simulations was initialized as in S˛adowski et al. (2014), viz., by starting with the hydrodynamical equilibrium torus of Penna et al. (2013a) with the angular momentum parameters listed in Table 1, and then redistributing the pressure between gas and radiation such that local thermal equilibrium is established with the total pressure unchanged. The resulting, radiation-pressure supported torus is close to equilibrium. The initial density was set through the torus entropy parameter and was adjusted to provide an optically thick torus that would give super-critical accretion once the simulation has reached steady state. \nThe initial torii were threaded by weak poloidal magnetic field. As each simulation proceeded, the field grew in strength and led to the onset of the magnetorotational instability, which triggered and maintained MHD turbulence in the disk. For most models we started with multiple loops with alternating polarity. For one model ( r011 ) we used a single loop. Both field configurations were initialized as in S˛adowski et al. (2015a). \nWe performed two simulations with BH mass 10 M GLYPH<12> and zero BH spin. One of these (run r001 ) resulted in a mean accretion rate of GLYPH<24> 10 ˙ M Edd, while the other ( r003 ), which was initialized with a more optically thick torus, accreted at GLYPH<24> 175 ˙ M Edd. These two \nmodels allowed us to study the role of the mass accretion rate on disk properties. \nThe third model ( r011 ) was initialized with a torus similar to model r001 , but we assumed a rotating BH with spin parameter a GLYPH<3> = 0 : 7. This model was the only one of the five that was initialized with a single poloidal loop of magnetic field. The hope was that the single loop would lead to a strong magnetic field at the horizon and would give a magnetically arrested disk (Igumenshchev et al. 2003; Narayan et al. 2003; Tchekhovskoy et al. 2011; McKinney et al. 2012). However, although this simulation was run up to a time of nearly 15,000, this duration was still insu GLYPH<14> cient to reach the MAD limit. Therefore, the magnetic field at the BH still did not reach the saturated level appropriate to the MAD state. Nevertheless, by comparing this model with r001 , we were able to study the e GLYPH<11> ect of BH spin 2 . \nIn the fourth model ( r020 ), we increased the BH mass to 1000 M GLYPH<12> BH, but kept the mass accretion rate at GLYPH<24> 10 ˙ M Edd. This enabled us to investigate the role of BH mass. \nAll of the above models were run in 3D. The fifth model ( d300 ) was a 2D axisymmetric simulation which used the meanfield dynamo model of S˛adowski et al. (2015a). This model was initialized with exactly the same torus configuration as in model r001 . The only di GLYPH<11> erence was that it was evolved in 2D instead of 3D. The purpose of this model was to assess how well the 2D dynamo model captures the properties of the 3D model.', '3.3 Accretion flow properties': "Each of the four 3D simulations was run up to a final time t max GLYPH<25> 15 ; 000 GLYPH<0> 20 ; 000, by which time all had developed quasi-steady, turbulent accretion via optically and geometrically thick disks. The time histories of the mass accretion rate through the BH horizon are shown in Fig. 1. In all the runs, gas starts crossing the BH horizon at a substantial rate only after t GLYPH<25> 2000. This is the time needed for the magnetorotational instability to make the disk turbulent, and for gas from the inner edge of the initial torus to accrete on the BH. Once accretion begins, the mass accretion rate increases rapidly. In fact, ˙ M overshoots and remains somewhat enhanced until time t GLYPH<25> 9000, and only beyond this time does the accretion rate settle down to a quasi-steady value. In the following, we focus on disk properties averaged over the latter quasi-steady stage of accretion, from t = 9000 to t = t max. \nThe radial profile of the mass accretion rate is given by \n˙ M = GLYPH<0> Z GLYPH<25> 0 Z 2 GLYPH<25> 0 p GLYPH<0> g GLYPH<26> u r d GLYPH<30> d GLYPH<18>: (8) \nFig. 2 shows the time-averaged profiles of this quantity as a function of radius r corresponding to the four 3D runs. The flat sections of the profiles at relatively small radii denote the region where the flow has reached inflow equilibrium. Given the somewhat limited duration of the simulations, and the relatively low radial velocity of the accreting gas in the disk, inflow equilibrium is reached only up to a radius r eq GLYPH<24> 20 GLYPH<0> 30. The outflows in the jet and wind regions have larger velocities, and therefore the outflowing regions are causally connected with the equatorial disk out to much larger distances from the BH. In particular, the funnel region, which is filled in most cases with gas escaping at v > 0 : 1 c , achieves equilibrium all the way out to the domain boundary at r out = 1000. \nFigure 3 shows poloidal and azimuthal slices through model \nFigure 1. Time history of the mass accretion rate at the BH in Eddington units (see §3.1 for definition) for the four three-dimensional models considered in this paper. Model parameters are given in Table 1. \n<!-- image --> \nFigure 2. Time- and azimuth-averaged radial profiles of the mass accretion rates (eq. 8) in the four three-dimensional simulations. \n<!-- image --> \nr001 at time t = 16400. The colors in the left halves of the panels show the magnitude of the radiation field, while those in the right halves show the gas density. The arrows in the two halves show the direction of the radiative flux and gas velocity, respectively, with the arrow thicknesses indicating the corresponding magnitudes. For clarity, the vector fields were computed from time-averaged output. \nThe gas is concentrated near the equatorial plane and forms a geometrically thick (or slim) disk with density scale-height H = R GLYPH<25> 0 : 25. Non-axisymmetric modes are clearly visible in the right panel, showing the value of running the simulations in 3D. Because of the large density, the gas is optically thick and advects with it a significant amount of radiation. This explains why the radiation flux has its largest magnitude in the bulk of the disk. \nThe gas in the disk region around the mid-plane is turbulent. On average the gas moves there slowly towards the BH. Outside the bulk of the disk, within GLYPH<24> 40 deg from the pole, the gas flows outward, being driven mostly by the radiation pressure force exerted by the radial radiative flux 3 . \nGas in the disk region orbits around the BH but because the \nin the radiative super-critical regime. Those represent a di GLYPH<11> erent class of accretion flows than the ones considered here. \nTable 1. Model parameters \n| | r001 | r003 | r011 | r020 | d300 (2D) |\n|--------------------------------|-------------------------------|-------------------------------|-------------------------------|-------------------------------|-------------------------------|\n| M BH | 10 M GLYPH<12> | 10 M GLYPH<12> | 10 M GLYPH<12> | 1000 M GLYPH<12> | 10 M GLYPH<12> |\n| ˙ M = ˙ M Edd | 10.0 | 175.8 | 9.7 | 10.1 | 8.9 |\n| a GLYPH<3> | 0.0 | 0.0 | 0.7 | 0.0 | 0.0 |\n| GLYPH<26> 0 ; init | 4 : 27 GLYPH<2> 10 GLYPH<0> 3 | 6 : 61 GLYPH<2> 10 GLYPH<0> 2 | 4 : 68 GLYPH<2> 10 GLYPH<0> 3 | 3 : 99 GLYPH<2> 10 GLYPH<0> 5 | 4 : 27 GLYPH<2> 10 GLYPH<0> 3 |\n| GLYPH<12> max | 10.0 | 10.0 | 10.0 | 10.0 | 10.0 |\n| initial B loops | multi. | multi. | poloidal | multi. | multi. |\n| NR x N GLYPH<18> x N GLYPH<30> | 252 x 234 x 128 | 252 x 234 x 128 | 252 x 234 x 128 | 252 x 234 x 128 | 252 x 234 x 1 |\n| r min / r max / r 0 / H 0 | 1.85 / 1000 / 0 / 0.6 | 1.85 / 1000 / 0 / 0.6 | 1.85 / 1000 / 0 / 0.6 | 1.85 / 1000 / 0 / 0.6 | 1.85 / 1000 / 0 / 0.6 |\n| GLYPH<24> / r 1 / r 2 / r in | 0.705 / 40 / 1000 / 22.5 | 0.705 / 40 / 1000 / 22.5 | 0.705 / 40 / 1000 / 10 | 0.705 / 40 / 1000 / 22.5 | 0.705 / 40 / 1000 / 22.5 |\n| t max | 20,000 | 19,000 | 16,100 | 19,200 | 190,000 | \n˙ M - average accretion rate \nGLYPH<26> 0 ; init - maximal density of the initial torus in g = cm 3 \n; GLYPH<12> max - maximal value of initial total to magnetic pressure ratio \nNR x N GLYPH<18> x N GLYPH<30> - resolution \nGLYPH<18> GLYPH<30> r min / r max / r 0 / H 0 - grid parameters defined in S˛adowski et al. (2015a) \nGLYPH<24> / r 1 / r 2 / r in - parameters of the initial torus as defined in Penna et al. (2013a) \nt max - duration of simulation in units of GM = c 3 \nFigure 4. Average radial profiles of the density-weighted gas angular momentum, u GLYPH<30> . The black lines show the Keplerian profiles for spin a GLYPH<3> = 0 : 0 (solid) and a GLYPH<3> = 0 : 7 (dashed). \n<!-- image --> \ndisk is geometrically thick the rotation is mildly sub-Keplerian, as shown by Fig. 4. Outside the ISCO, the deviation from the Keplerian profile is no more than 13% in any of the models. The angular momentum profile flattens towards the BH horizon, reflecting the fact that the e GLYPH<11> ective visosity, which transports angular momentum, is less e GLYPH<11> ective in the plunging region. The profile is flattest for the models with the lowest accretion rate. Extrapolating to sub-Eddington accretion rates, we thus expect the viscous torques to become insignificant for thin disks, in agreement with previous work Paczy'nski (2000); Afshordi & Paczy'nski (2003); Shafee et al. (2008a,b); Penna et al. (2010). \nThe top set of panels in Fig. 6 shows the time- and azimuthaveraged distribution of density for all the five simulations. Streamlines reflect the velocity field of the gas, with the thickness of lines denoting the density-weighted average velocity. For all the simulations the accretion flow is geometrically thick with density peaking at the equatorial plane. The density values are similar in all the runs except the simulation with the highest accretion rate, r003 , which has a significantly higher density of gas (and increased optical depth). In all cases the gas flows on average towards the BH deep in the disk. The velocity of gas in the funnel is pointing out- \nward outside the stagnation radius separating the polar region of inflowing gas (which is the boundary condition imposed by the presence of BH horizon) and the outflowing gas, driven either by radiation pressure or pressure gradients. In the case of simulations with non-rotating BHs and moderate accretion rates (three-dimensional models r001 and r020 ), the stagnation radius at the axis is located near r GLYPH<24> 10. For the run with much higher accretion rate ( r003 ) it shifts significantly outward to r GLYPH<24> 50 due to the much larger opacity, which prevents radiation from ejecting gas from the innermost region. On the other hand, the stagnation radius is very close to the BH horizon for the simulation with a rotating BH. In this model, gas is accelerated outward by an additional energy source, viz., the spin energy of the BH. The transition between the region of inflow (inside the disk), and outflow (in the funnel) is quite rapid - when gas particles are blown out of the disk and enter the funnel they quickly gain large outward radial velocity and join the outflow. \nThe bottom set of panels in Fig. 6 shows the averaged radiative flux for the five simulations. The colors denote the magnitude and arrows show the direction of the radiative flux. The radiation emitted by hot gas in the disk is mostly advected with the optically thick gas, but in regions close to the disk surface it also di GLYPH<11> uses down the local density gradient (Jiang et al. 2014b). As a result, some fraction of the radiation is advected into the BH and the rest naturally fills up the funnel region. There, radiation pressure accelerates gas outward along the axis of the funnel, resulting in a jet that travels at a modest fraction (0.2-0.5) of the speed of light (S˛adowski & Narayan 2015b). The properties of the radiation field depend on the accretion rate and BH spin. For model r003 , which has the largest accretion rate, a significantly higher fraction of all the radiative flux in the inner region is advected into the BH, even in the inner part of the funnel region. In simulation r011 , the rotational energy of the BH is extracted (although not very e GLYPH<14> ciently due to the sub-MAD level of magnetic flux accumulated at the horizon) through the Blandford-Znajek process. This extra energy is converted into radiation and therefore this model has the highest radiative flux magnitude in the funnel region among all the runs. \nThe qualitative properties of the 3D simulations described here agree well with accretion flows simulated in the recent years in axisymmetry e.g., by S˛adowski et al. (2014), and in threedimensions by McKinney et al. (2014). We comment on the di GLYPH<11> erences between our models and the simulation presented by Jiang et al. (2014b) in Section 8.1. Below we discuss in detail three as- \nquatorial plane \nFigure 3. Shows the magnitude of the radiative flux (orange colors in the left half of each panel) and the gas density (grey colors in the right half of each panel) for a snapshot taken near the end of the r001 simulation. The left and right panels correspond to slices through the poloidal and equatorial planes, respectively. Streamlines of the radiative flux and gas velocity are azimuth- and time-averaged. The thickness of the streamlines increases with the magnitude of the respective quantity. \n<!-- image --> \npects of our models - luminosities, e GLYPH<14> ciency of photon trapping, and variability.", '4 LUMINOSITIES': "Accretion takes place if there is an e GLYPH<14> cient mechanism for transporting angular momentum outward. In BH accretion disks, we believe that this transport results from turbulence sustained by magnetorotational-unstable magnetic field. The angular momentum flux is accompanied by fluxes of energy in various forms. In the standard picture of a thin disk, the accreting gas brings in kinetic (orbital and turbulent), thermal, and binding energy. The latter has a negative sign, so e GLYPH<11> ectively the flux of binding energy transports energy out (and deposits it at infinity). The exchange of angular momentum would not be possible without a shear stress ('viscosity') which again causes an outward flux of energy. For turbulent magnetic disks, this energy flux comes from the work done by magnetic fields. In addition, there is radiation flux. In a thin disk, radiation carries energy out to infinity. However, in geometrically thick super-Eddington accretion flows, the radiation energy flux can be either outward or inward, depending on how e GLYPH<11> ectively the radiation is trapped in the optically thick flow. In addition, these models can also have mechanical energy flowing out in a wind or jet. \nWe postpone a comprehensive discussion of the energetics in multi-dimensional accretion disk to a forthcoming paper. Below, we limit ourselves to simple angle-integrated luminosities of the simulated disks.", '4.1 Total luminosities': 'The most fundamental definition of the luminosity is, \nL total = GLYPH<0> Z GLYPH<25> 0 Z 2 GLYPH<25> 0 GLYPH<0> T r t + R r t + GLYPH<26> u r GLYPH<1> p GLYPH<0> g d GLYPH<30> d GLYPH<18>; (9) \nwhich is the total rate of energy flowing through a sphere at some radius r at some instant of time. This quantity accounts for all forms of energy except the rest-mass energy (which has been subtracted \nFigure 5. Total luminosity (solid lines, eq. 9) and radiative luminosity (dashed lines, eq. 11) integrated over the whole sphere for the four threedimensional simulations. \n<!-- image --> \nout via the term GLYPH<26> u r ). In particular, L total includes binding (gravitational), radiative, kinetic, thermal and magnetic energies. In an equilibrium steady state accretion disk, the the time-averaged luminosity L total is conserved, i.e., it has the same value at all radii. This is because of energy conservation: any energy that appears in one form must ultimately come from one of the other forms discussed above, and therefore the sum of all forms of energy remains constant. \nThe radial profile of total luminosity is plotted with solid lines in Fig. 5. We see that it is indeed constant to good accuracy for all the runs. All the simulations with non-rotating BHs have total luminosity close to 3% ˙ Mc 2 . For an accretion rate of 10 ˙ M Edd this corresponds to GLYPH<24> 5 L Edd (for the adopted definition of ˙ M Edd see equation 6). The luminosity is significantly higher GLYPH<24> 8% ˙ Mc 2 for the simulation ( r011 ) with a rotating BH. There are two reasons for this: (i) the disk around a spinning BH extends deeper into the potential well since the ISCO is at a smaller radius (correspond- \ny, the thin disk e GLYPH<14> ciency for a GLYPH<3> = 0 : 7 is GLYPH<17> thin GLYPH<25> 10 : 3% ˙ Mc 2 ), and (ii) the accumulated magnetic flux at the BH horizon allows for the extraction of BH kinetic energy through the Blandford-Znajek process. The predicted rate of that process (for the measured average magnetic flux parameter GLYPH<8> GLYPH<25> 15 and a GLYPH<3> = 0 : 7) is GLYPH<17> jet GLYPH<25> 6%. The total energy available is therefore GLYPH<17> thin + GLYPH<17> jet GLYPH<25> 16%. In actuality, GLYPH<24> 8% is extracted, which is roughly 50% of the total available. A similar 50% energy extraction is seen also in the case of non-rotating BHs: a thin disk around a non-spinning BH has an e GLYPH<14> ciency of 5.7% whereas, as mentioned above, the luminosity of our models is only 3%. Total luminosities for all the simulations are given in Eddington units in the third row of Table 2. \nThe total luminosity as defined above may be, in principle, sensitive to the initial conditions. In the ideal world, one would initiate a simulation by dumping marginally bound (zero Bernoulli number) gas from infinity. Because the duration of real simulations is limited, we have to start our simulatons from a bound torus located close to the BH with its inner edge at r in = 22 : 5. The Bernoulli number, defined as \nBe = GLYPH<0> T t t + R t t + GLYPH<26> u t GLYPH<26> u t ; (10) \nof the initial gas at the very inner edge is Be GLYPH<25> GLYPH<0> 0 : 014 and approaches zero inversely proportional to radius. Thus, in principle, the luminosity estimates given above may be overestimated by up to GLYPH<24> 1% ˙ Mc 2 (it cannot be more, but it may be less if there is significant radial mixing of gas in the initial torus). To estimate how strongly the initial Be a GLYPH<11> ects results we compared the total luminosities averaged over t = 12 ; 000 GLYPH<0> 13 ; 000 and t = 18 ; 000 GLYPH<0> 19 ; 000. At t GLYPH<24> 12 ; 000, the BH had accreted an amount of gas equivalent to that contained in the initial torus inside within r = 35, while at t GLYPH<24> 18 ; 000, the mass accreted was equivalent to the torus mass out to r GLYPH<25> 40. If accretion occurs radius after radius, i.e., without any radial mixing, then gas located initially near these radii should fall on the BH during these periods of time. The initial Bernoulli numbers at r = 35 and r = 40 were Be = GLYPH<0> 0 : 012 and GLYPH<0> 0 : 010, respectively. The meeasured total e GLYPH<14> ciencies averaged over the corresponding periods were GLYPH<17> total = 0 : 029 and GLYPH<17> = 0 : 031. respectively. If the total e GLYPH<14> ciency of energy extraction was to reflect the binding energy of the initial gas then it should drop by 0 : 002 between the first and second periods (less bound gas accreted on the BH e GLYPH<11> ecively deposits less energy at infinity). However, the e GLYPH<14> ciency increased by a similar amount. We conclude that the gas mixes e GLYPH<11> ectively before reaching the BH and that the measured total luminosities are not sensitive to how much the initial torus was bound. \nThe total luminosity indicates how much energy will be ultimately deposited at infinity, where binding energy is zero and therefore the outflowing flux of binding energy must have converted into other forms of energy. For supermassive black holes (SMBHs), we may expect that all this energy will ultimately a GLYPH<11> ect the interstellar medium around the galactic nucleus and will contribute to AGN feedback. \nIn analogy with equation (9), we can straightforwardly define an equivalent total radiative luminosity, i.e., radiative flux integrated over the whole sphere, \nL rad ; total = GLYPH<0> Z GLYPH<25> 0 Z 2 GLYPH<25> 0 R r t p GLYPH<0> g d GLYPH<30> d GLYPH<18>; (11) \nwhich describes the radiative component of the total luminosity L total. However, this quantity is not as fundamental as L total since it is no longer conserved. In particular, radiation can be emitted \nor absorbed by the gas and can also gain / lose energy via momentum transfer to the gas. Nevertheless, for certain limited purposes, L rad ; total can be useful. Radial profiles are shown with dashed lines in Fig. 5. For all the simulations, L rad ; total is negative in the inner region, increases with radius, and ultimately becomes positive. Negative values correspond to regions where more photons are dragged (advected) with the flow inward than manage to escape. These are the regions where the photon-trapping e GLYPH<11> ect (discussed in detail in Section 5) dominates. The fact that the total radiative luminosity becomes ultimately positive reflects the fact that the flow is slower at larger radii and so it is easier for photons to escape there.', '4.2 Optically thin and outflow regions': 'Because of the limited range of inflow / outflow equilibrium in the disk mid-plane, extending at best only up to r eq GLYPH<24> 25, it is impossible to determine say how the outflowing energy is finally distributed between radiation and other forms when it reaches infinity. However, the funnel and wind regions are converged to larger radii because of their higher velocities which allow them to be causaly connected with the equilibrium innermost region near the equatorial plane. This fact allows us to measure luminosities in the funnel to larger distances from the BH. \nWe divide the outflow region into two zones: (i) an optically thin zone which is visible to an observer at infinity, and (ii) an outflow region where the gas is on average energetic enough to reach infinity. In all the simulations, zone (i) is a subset of zone (ii). The border of this zone is defined to be the photosphere, which satisfies the following condition 4 , \nGLYPH<28> ( R ) = GLYPH<0> Z R max R GLYPH<26> ( GLYPH<20> a + GLYPH<20> es)( ut + ur ) p grr d r = 2 3 ; (12) \ni.e., the total optical depth integrated along fixed polar angle from the domain boundary equals 2 = 3. The outflow region is defined as the region where the relativistic Bernoulli parameter (eq. 10) is positive. The borders of the two regions are denoted by dashed blue (optically thin) and green (outflow) lines, respectively, in Fig. 6. Only for model r001 (and its axisymmetric counterpart d300 ) does the optically thin region extend down to the BH. For all the other simulations the density of gas in the funnel region is large enough to move the lower edge of the photosphere away from the BH. For model r003 , which is characterized by a significantly larger accretion rate, the photosphere formally is at the outer edge of the domain. The photosphere is far from the BH also for model r011 for which the accretion rate (in Eddington units) is almost twice as high as in the fiducial one. The photosphere is relatively close ( r GLYPH<25> 40 at the axis) for model r020 , which has a similar accretion rate as r001 , but a higher BH mass. McKinney et al. (2015) has recently showed that if strong magnetic field is present near the axis, one may get optically thin funnel down to the BH even for highly super-critical accretion rates. \nRadiation flowing out in the optically thin region is guaranteed to reach observers at infinity - no significant interaction with gas is taking place here. The radiative luminosity integrated over this region may thus be viewed as a lower limit on the total radiative luminosity. The photons trapped in the optically thick regions \ncan in principle ultimately escape and could increase the radiative luminosity. However, this additional luminosity is not expected to exceed a couple of L Edd, because if locally the radiative flux exceeds the Eddington flux L Edd = c 2 and the gas is optically thick, this flux would accelerate the gas and convert radiative energy into kinetic energy of the outflow 5 . \nTo obtain the radiative luminosities of the optically thin and outflow zones, we calculate, \nL rad ; thin = GLYPH<0> Z GLYPH<18> thin 0 Z 2 GLYPH<25> 0 R r t p GLYPH<0> g d GLYPH<30> d GLYPH<18>; (13) \nL rad ; out = GLYPH<0> Z GLYPH<18> out 0 Z 2 GLYPH<25> 0 R r t p GLYPH<0> g d GLYPH<30> d GLYPH<18>; (14) \nwhere GLYPH<18> thin and GLYPH<18> out denote the limits of the regions of optically thin and outflowing gas, respectively. The kinetic luminosities are calculated in a similar way \nL kin ; thin = GLYPH<0> Z GLYPH<18> thin 0 Z 2 GLYPH<25> 0 ( ut + p GLYPH<0> gtt ) GLYPH<26> u r p GLYPH<0> g d GLYPH<30> d GLYPH<18>; (15) \nL kin ; out = GLYPH<0> Z GLYPH<18> out 0 Z 2 GLYPH<25> 0 ( ut + p GLYPH<0> gtt ) GLYPH<26> u r p GLYPH<0> g d GLYPH<30> d GLYPH<18>: (16) \nIn the Newtonian limit one has: GLYPH<0> ( ut + p GLYPH<0> gtt ) ! 1 = 2 v 2 . \nTable 2 lists the radiative and kinetic luminosities in the two regions as measured at radii r = 50 and r = 250. For the fiducial model r001 (accreting at 10 : 0 ˙ M Edd) only 0 : 3 L Edd escapes in the optically thin region at radius r = 50. This value increases to 0 : 95 L Edd at radius r = 250 reflecting the fact that the optically thin region extends further from the axis and more radiation is able to enter the optically thin funnel. The radiative luminosity in the whole region of outflowing ( Be > 0) gas is 1 : 13 L Edd and 1 : 57 L Edd at r = 50 and r = 250, respectively. At the same time kinetic luminosities are much smaller. Hardly any kinetic energy escapes in the optically thin region, mostly because of the negligible mass flux of outflowing gas there. In the whole outflow region, the kinetic luminosity grows from 0 : 17 L Edd at radius r = 50 up to 0 : 35 L Edd at r = 250. These numbers correspond to 13% and 18% of the total (radiative plus kinetic) luminosities, respectively. The increase of the fractional contribution of kinetic energy reflects the fact that radiation gradually accelerates gas, thereby losing momentum / energy. \nThis e GLYPH<11> ect is clear especially for simulation r003 (accreting at 175 ˙ M Edd). The fractional contribution of kinetic luminosity grows from 39% to 58% between r = 50 and r = 250. The transfer of energy from radiation to gas is more e GLYPH<11> ective because of higher optical depth in this run. For the same reason, there is no optically thin region in this simulation within the computational domain. The fact that both luminosities in simulation r003 grow significantly between the two radii arises from the fact that there is strong inflow of gas along the axis within r GLYPH<25> 30 which prevents photons from escaping. Only further out on the axis does the standard, radiatively driven outflow region form. \nThe kinetic and radiative luminosities measured in the thin and outflow regions for runs r001 and r003 are significantly lower than the total luminosities measured according to equation (9). At radius r = 250 and in the outflow region, these luminosities contribute only 37% and 10% of the total, respectively. Where does the rest of the luminosity go? It remains still in the optically thick gas in the \ndisk interior, where the total energy budget is dominated by the outflowing flux of magnetic (viscous stresses) and binding energy. The energy carried out in these channels (certainly, the binding energy) will be ultimately converted into radiative or kinetic energy by the time it reaches infinity. Unfortunately, the limited range of the equilibrium solution in our simulations prevents us from addressing this question directly. \nThe total e GLYPH<14> ciency in simulation r011 , which is the only one with a rotating BH, is significantly higher than in the other simulations. So are the radiative and kinetic luminosities integrated over the outflow region. The extra input of energy from the rotating BH makes the funnel region very energetic (see the bottom-middle panel of Fig. 6). At radius r = 250 the energy flux in the funnel and optically thick outflow is dominated by radiative luminosity equal to 8 : 4 L Edd. Most of these photons are carried with the gas and ultimately will escape when they reach the photosphere, which for most angles is outside the computational domain. However, the radiative luminosity may decrease if radiation keeps transferring its momentum to the gas. The kinetic component of the luminosity is significant already at this radius ( r = 250) - roughly 1 : 5 L Edd is carried in the form of outflow kinetic energy. \nThe remaining three-dimensional run ( r020 ), which di GLYPH<11> ers from the fiducial run ( r001 ) in the mass of the BH (1000 M GLYPH<12> instead of 10 M GLYPH<12> ) shows very similar properties in all respects.', '4.3 Angular distribution of energy fluxes': 'Figure 7 shows the angular distribution of the (time- and azimuthaveraged and symmetrized) radiative and kinetic fluxes of energy in the funnel / outflow region for three simulations (top to bottom): r001 , r011 , and r003 . Blue and red lines correspond to fluxes measured at r = 50 and r = 250, respectively. Solid and dashed lines denote the radiative and kinetic energies, respectively. \nFor the fiducial model (top panel, ˙ M = 10 ˙ M Edd) the energy flux (as discussed in the previous section) is dominated by radiation. The angular distribution of radiative flux follows roughly a Gaussian with half-maximum width at GLYPH<18> = 0 : 35rad. The maximal flux at radius r = 50 is F GLYPH<25> 12 F Edd; it increases to F GLYPH<25> 19 F Edd at r = 250. The radiative fluxes decline significantly with increasing polar angle GLYPH<18> . For an observer at GLYPH<18> = 0 : 5rad ( GLYPH<24> 30 GLYPH<14> ) the observed flux (and the inferred source luminosity) is only GLYPH<24> 4 F Edd. \nThe numbers given above are meaningful for the optically thin region but less so for other angles where the wind is optically thick. In these regions, there is still significant interaction between gas and radiation. The radiation is likely to accelerate the gas further and the radiative flux is expected to decrease towards the Eddington limit. To study this e GLYPH<11> ect quantitatively one would have to perform simulations in a much bigger box and for a much longer time. \nAt radius r = 50 there is hardly any kinetic luminosity in the polar region of the fiducial run ( r001 ). Only further from the BH is the gas is accelerated. In contrast to the radiative flux, the kinetic energy flux is not concentrated at the polar axis but rather in a shell around GLYPH<18> = 0 : 35rad, similar to the jet / wind boundary discussed in S˛adowski et al. (2013b). For the fiducial model accreting at 10 ˙ M Edd the maximal kinetic flux at that inclination equals GLYPH<24> 4 F Edd. \nThe angular profiles for the run with a rotating BH ( r011 ) look qualitatively similar to the profiles of the fiducial run. However, the magnitudes of the fluxes are much higher, reflecting the increased e GLYPH<14> ciency of accretion due to energy being extracted from the BH. The maximal radiative and kinetic fluxes at the axis at r = 250 exceed 100 F Edd and 20 F Edd, respectively. \nA larger mass accretion rate changes the picture significantly. \n<!-- image --> \nFigure 6. Top row of panels: Logarithm of average gas density (colors) and streamlines of average gas velocity (thickness indicates the magnitude of the velocity) for five models: r001 (leftmost), r003 , r011 , r020 , and d300 (rightmost panel). Model r020 corresponds to M BH = 1000 M GLYPH<12> ), and so its density was scaled up by a factor of 100 to enable direct comparison with the other simulations, which had M BH = 10 M GLYPH<12> . The blue dashed contour reflects the location of the scattering photosphere as seen from infinity along fixed polar angle. The green dashed contour separates the bound ( Be < 0) and unbound ( Be > 0) gas. Bottom row of panels: Logarithm of the magnitude of radiation flux and its streamlines (thickness indicates the magnitude of the flux). For model r020 , the flux was scaled up by 100. \n<!-- image --> \nThe third panel of Fig. 7 shows the angular profiles of fluxes for run r003 . The radiative flux is still beamed at the axis, and the maximal flux exceeds 40 F Edd at r = 250. The kinetic energy flux has a different shape than it used to. It is no longer concentrated in a shell but increases with angle, reflecting the fact that the wind carries a significant amount of energy over a wide solid angle. However, even at the axis, the kinetic energy flux is much higher than in the fiducial model - it now exceeds 10 F Edd at r = 250. The kinetic flux in the funnel is a result of radiative energy being converted into kinetic energy within the optically thick region (S˛adowski & Narayan 2015b). \nThe angular distribution of energy fluxes discussed above should be considered only approximate due to the limitations of \nthe M1 closure scheme adopted in this work. Most importantly, we evolve only the first moments of the radiation field instead of evolving specific intensities directly as in Jiang et al. (2014a) and Ohsuga & Takahashi (2015). The radiation observed by a distant observer should, in principle, be calculated as an integral of the specific intensity pointing towards the observer over the whole accretion disk. The local radial flux gives only an approximation of this quantity. Furthermore, the M1 closure is known to have difficulties in treating the region closest to the polar axis. We have substantially mitigated this problem by including an extra radiative viscosity (S˛adowski et al. 2015a), but the coe GLYPH<14> cients involved had to be chosen somewhat arbitrarily. \n<!-- image --> \n| | r001 | r001 | r003 | r003 | r011 | r011 | r020 | r020 | d300 (2D) | d300 (2D) |\n|-----------------------|--------|--------|---------|--------|--------|--------|--------|--------|-------------|-------------|\n| ˙ M = ˙ M Edd | 10 | | 175.8 | | 17.6 | | 10.1 | | 8.9 | |\n| L total = L Edd | 5.26 | | 83.27 | | 14.27 | | 5.14 | | 5.31 | |\n| GLYPH<17> total | 0.03 | 0.030 | 0.027 | 0.027 | 0.084 | 0.084 | 0.029 | 0.029 | 0.034 | 0.034 |\n| r lum | 50 | 250 | 50 | 250 | 50 | 250 | 50 | 250 | 50 | 250 |\n| L rad ; thin = L Edd | 0.3 | 0.95 | 0 | 0.0 | 0 | 2.13 | 0.2 | 0.85 | 0.64 | 2.14 |\n| L kin ; thin = L Edd | 0 | 0.15 | 0 | 0.0 | 0 | 0.16 | 0 | 0.15 | 0.01 | 0.23 |\n| L rad ; out : = L Edd | 1.13 | 1.57 | 0.78 | 3.47 | 6.7 | 8.39 | 1.11 | 1.60 | 1.86 | 2.58 |\n| L kin ; out = L Edd | 0.17 | 0.35 | 0.5 | 4.82 | 1.41 | 1.54 | 0.21 | 0.39 | 0.31 | 0.41 | \n˙ M - average accretion rate \nL total = L Edd - total (conserved) luminosity of the system including the flux of binding energy \nGLYPH<17> total = L total = ˙ Mc 2 - e GLYPH<14> ciency of accretion in total luminosity \nL rad ; thin, L kin ; thin - radiative and kinetic luminosities in the optically thin polar region integrated at radius r lum \nL rad ; out, L kin ; out - radiative and kinetic luminosities in the region of unbound gas ( Be > 0) integrated at radius r lum \nFigure 7. Average energy fluxes in simulations r001 (top panel), r003 (middle), and r011 (bottom) as a function of the polar angle GLYPH<18> . Blue and red lines correspond to fluxes measured at r = 50 and r = 250, respectively. Solid lines show the radiative flux and dashed lines show the flux of kinetic energy. Bright line segments are within the outflow region ( Be > 0), while the dull shaded segments are outside this region. \n<!-- image -->', '5 PHOTON TRAPPING': 'At the time the thin disk theory was established (Shakura & Sunyaev 1973; Novikov & Thorne 1973), it was commonly understood that the standard radiation pressure dominated disk can extend up to and above the critical Eddington accretion rate (equation 6). Once the Eddington limit is exceeded, it was predicted that the most energetic, inner region would attempt to produce a luminosity exceeding the Eddington value and this would produce a radiatively driven outflow inside the so called spherization radius. At the same time the wind would modify the accretion rate on the BH. \nThis picture did not take an important e GLYPH<11> ect into account. Whenthe accretion flow is very optically thick, photons do not have enough time to di GLYPH<11> use vertically to the disk photosphere before they are dragged radially inward by the accreting gas and advected into the BH. This e GLYPH<11> ect was described in a simple spherical context by Begelman (1978) and included for the first time in a full-fledged accretion model by Abramowicz et al. (1988) who constructed the so-called slim disk model. These authors predicted that the radiative luminosity would no longer scale with the accretion rate above the critical rate. Also, in priniciple, the slim disk state would prevent the spherization phenomenon. \nAs we have shown in the previous section, and as we will explain in greater detail here, the truth is in between - there is a region, but only near the axis, where locally flux is significantly super-Eddington and where radiation drives gas out of the disk, but at the same time photons are trapped and advected towards the BH in the inflow region near the equatorial plane. \nIn this section we try to answer the question: where is photon trapping e GLYPH<11> ective or, in other words, where is the trapping radius, the border between the radiatively ine GLYPH<14> cient (slim disk) and radiatively e GLYPH<14> cient (thin disk) regimes?', '5.1 One-dimensional, luminosity-based trapping radius': 'It is relatively straightforward to define the trapping radius in a spherically symmetric flow (Begelman 1978). Assuming a Bondilike flow, we can compare the local outward radiative di GLYPH<11> usion velocity to the local gas inflow velocity and thereby estmate \nR trap ; Bondi GLYPH<25> ˙ M = ˙ M Edd : (17) \nThis approach gives only an upper limit on the location of the trapping radius in a real accretion flow; accretion disks are not spherically symmetric but have low density polar regions where radiation can more easily escape, and also the inflow velocity near the equa- \ntorial plane is significantly lower than the free-fall velocity assumed in the Bondi model (S˛adowski et al. 2014). \nThe simplest way of estimating the location of the trapping radius numerically is to look at the net flux of radiation, e.g., the dashed lines in Fig. 5. Negative values mean that more photons were moving inward (mostly because they are dragged by optically thick gas) than outward. The radius at which the radiative luminosity goes to zero could then be defined as the trapping radius. This approach is simple and provides a single e GLYPH<11> ective trapping radius. It does not, however, account for the non-uniform structure of the disk - photons easily leave the system at the axis and are more e GLYPH<11> ectively trapped near the disk mid-plane. The present simple definition weights both e GLYPH<11> ects and provides some kind of an average. By this definition, for simulations r001 and r020 , the e GLYPH<11> ective trapping radius is located around r = 35. Model r003 , which has a significantly larger accretion rate, has almost the same trapping radius, r GLYPH<25> 40. Model r011 , with a rotating BH, produces much more powerful radiation flux along the axis, and correspondingly the e GLYPH<11> ective trapping radius is much closer to the BH ( r GLYPH<25> 10, although the trapping in the bulk of the disk is as e GLYPH<11> ective as in the other cases). A caveat: The values given above for the simulations with non-rotating BHs should be taken with caution, because the bulk of the disk is outside the inflow / equlibrium region at these radii.', '5.2 Two-dimensional trapping - importance of the di GLYPH<11> usive flux': 'Because of the limitations of the definition given in the previous section, we discuss here a di GLYPH<11> erent approach which, in particular, seeks to account for local properties of the flow. \nJiang et al. (2014b) calculated energy-weighted, vertically integrated, radial and vertical velocities of radiation transport. By comparing the two one can distinguish whether more energy flows up (away from the midplane) or inward (towards the BH). not adequate to say where the photon trapping is e GLYPH<11> ective because it does not gives no importance to di GLYPH<11> usion. The vertical flow of radiation could be the result of advection (in a wind) or turbulence Jiang et al. (2014b). Thus, while it is a good way of comparing vertical and radial energy transfer, it provides no information on the transport mechanism. \nBelow we attempt to quantify photon trapping by calculating the fraction of the total radiative flux that comes from di GLYPH<11> usive transfer. To estimate the di GLYPH<11> usive flux we use the moment equations (3), assuming @ t = 0 and a diagonal form of the radiative stress energy tensor in the fluid frame ( b R ii = 1 = 3 b R tt ), to obtain \nF i di GLYPH<11> = 1 3 h GLYPH<20>GLYPH<26> i d dx i h b E i ; (18) \nwhich is the standard di GLYPH<11> usive flux formula (Rybicki & Lightman 1979). This expression should be used only inside the optically thick parts of the disk where the Rosseland approximation is satisfied. \nThe top panel of Fig. 8 shows the magnitude (colors) and direction (red streamlines) of the total radiative flux, F i tot = R i t , for the fiducial model r001 which accretes at 10 ˙ M Edd. Deep in the disk the radiation flows towards the BH, while the polar region is filled with optically thin radiation escaping along the axis. There is a transition region at intermediate angles, GLYPH<18> GLYPH<25> 45 GLYPH<14> , where radiation relatively smoothly switches from flowing radially inward to flowing outward. \n<!-- image --> \n<!-- image --> \nFigure 8. Top panel: Magnitude and streamlines of the total radiative flux F i rad = GLYPH<0> R i t . Middle panel: Magnitude and direction of the radiative di GLYPH<11> usive flux F i di GLYPH<11> (equation 18). Bottom panel: Magnitude of the comoving frame radiative energy density (colors) and the ratio of the di GLYPH<11> usive radiative flux to the total radiative flux ( F i di GLYPH<11> = F i rad ). All panels correspond to simulation r001 (10 ˙ M Edd). The yellow stars denote the location where we studied in detail the vertical fluxes plotted in Fig. 10. \n<!-- image --> \nFigure 10. Estimated polar di GLYPH<11> usive flux of radiation (equation 19) versus the excess of the total flux over the local advective flux of radiation (equation 20). Colors denote the local optical depth over a distance of one gravitational radius. The dashed line shows where the two fluxes agree, i.e., when the excess can be explained entirely by di GLYPH<11> usive transport. The data are from simulation r001 and the points range from t = 7 ; 500 GLYPH<4> 20 ; 000 with a cadence of dt = 50. \n<!-- image --> \nFigure 9. Magnitude of the comoving frame radiative energy density (colors) and the ratio of the di GLYPH<11> usive to advective radiation flux ( F i di GLYPH<11> = F i rad ) in simulation r003 (175 ˙ M Edd). \n<!-- image --> \nBlue streamlines in the same panel show the average gas velocity. If all the radiative flux was coming from photons advected with the gas, the total radiation flux vectors would follow everywhere the direction of the gas velocity. The agreement between the two sets of streamlines is very deep inside the disk, where both gas and radiation flow inward. This is where gas is most optically thick and where one expects e GLYPH<14> cient photon trapping. The streamlines agree again in the polar region, but this is not because of photon trapping. The low optical depth allows radiation to flow independently of the gas. The funnel geometry, however, makes both gas and radiation to flow upward, and the locally super-Eddington flux accelerates gas in its direction. In the intermediate region between the funnel and the disk interior, the gas velocity and radiation flux vectors do not point in the same direction. As mentioned previously, gas flows on average radially inward in the disk and only close to the surface is it blown away, causing the velocity streamlines to turn rapidly outward. Radiation flux, on the other hand, changes direction rather smoothly. This di GLYPH<11> erence between gas and radiation streamlins shows that there must be an additional component of the total flux besides the advective photon transport. This is of course di GLYPH<11> usive transport. \nThe middle panel in Fig. 8 shows on the same axes and with the same color scale the di GLYPH<11> usive flux estimated according to equation (18). As expected, the di GLYPH<11> usive flux follows the gradient of radiative energy density, i.e., radiation di GLYPH<11> uses in the vertical direction away from the equatorial plane. Then, it turns smoothly towards the axis and enters the funnel region. IN the funnel itself (the red region), the estimate of the di GLYPH<11> usive flux must be disregarded since gas there is optically thin and radiation is free streaming instead of di GLYPH<11> using. However, the di GLYPH<11> usive flux estimates are trustworthy in the disk interior and the transition region between the disk and the funnel. \nWe now check whether the di GLYPH<11> usive flux is, in fact, responsible for the deviation between the total radiative flux streamlines and the advected radiation flux streamlines (which would be aligned with the gas velocity). For this purpose we choose a location in the region of largest deviation, ( r ; GLYPH<18> ) = (20 ; 51 GLYPH<14> ) (denoted by the yellow stars in Fig. 8), and study local fluxes of radiation over time. Fig. 10 presents the estimated orthonormal polar component of the \ndi GLYPH<11> usive flux, calculated as, \nF ˆ GLYPH<18> di GLYPH<11> = GLYPH<0> 1 3 GLYPH<20>GLYPH<26> d rd GLYPH<18> b E ; (19) \nas a function of the di GLYPH<11> erence between the polar components of the total flux and the estimated advective flux, \nF ˆ GLYPH<18> rad GLYPH<0> F ˆ GLYPH<18> adv = R GLYPH<18> t + 4 3 b Eu GLYPH<18> ! r : (20) \nThe signs have been chosen such that the positive values correspond to fluxes pointed out of the disk, i.e., towards the axis. The sizes of markers measure the local optical depth for scattering as estimated by GLYPH<28> es = GLYPH<20> es GLYPH<26> RG . The dashed line denotes F ˆ GLYPH<18> di GLYPH<11> = F ˆ GLYPH<18> rad GLYPH<0> F ˆ GLYPH<18> adv . The plotted points cover the time period t = 7 ; 500 GLYPH<4> 20 ; 000 with cadence of GLYPH<1> t = 50. \nThe majority of points in Fig. 10 cluster around the dashed line, showing that the excess of radiative flux over advective flux has exactly the analytically predicted magnitude, i.e., the excess of polar radiative flux is due to di GLYPH<11> usion. The agreement is no longer perfect at the lowest optical depths. This is natural since, when the optical depth is low, radiation decouples from the gas and the di GLYPH<11> usive approximation no longer valid. In fact, this is where we expect the actual flux to lie below the analytically estimated di GLYPH<11> usive flux, as is indeed seen in the plot. \nThe colors of the markers in Fig. 10 denote the ratio of magnitudes of instantenous di GLYPH<11> usive and advective fluxes. Despite the fact that the time-averaged advective flux has hardly any polar component, instantenous advective flux in GLYPH<18> direction can be more than 30 times stronger than the corresponding di GLYPH<11> usive flux. However, because of turbulent motion of gas, it averages to a value significantly smaller than the di GLYPH<11> usive flux. Therefore, it is the di GLYPH<11> usive flux which dominates the average net radiation flux towards the axis near the surface of the disk. \nThe bottom panel in Fig. 8 shows the ratio of the magnitude of the di GLYPH<11> usive flux over the total flux plotted with contours on top of the distribution of fluid frame radiative energy density, b E . Deep in the disk, advection dominates strongly over di GLYPH<11> usion. The magnitude of the former flux is & 30 times larger than the magnitude \nof the di GLYPH<11> usive flux. This ratio is lower closer to the disk surface, where the density drops and the gas is no longer able to drag photons e GLYPH<14> ciently. In the transition region discussed above, the advective and di GLYPH<11> usive fluxes have comparable magnitudes, with the di GLYPH<11> usive flux dominating the polar component. \nFigure 9 shows the same ratio of the di GLYPH<11> usive to advective fluxes, but for simulation r003 which has almost 20 times larger accretion rate. The larger accretion rate implies higher gas density and optical depth, and, as a result, more e GLYPH<11> ective photon trapping. This is indeed the case. The advective radiative flux is now & 300 times larger than di GLYPH<11> usive in the disk near the equatorial plane. \nThe properties described above show that for moderately super-critical accretion rates, GLYPH<24> 10 ˙ M Edd, photon transport deep inside the disk is dominated by radial advection. However, di GLYPH<11> usive transport of energy becomes important further from the equatorial plane, near the transition between the disk and the funnel, where the gradient of the energy density is large, and the optical depth of the gas is no longer huge. Radiation in the optically thin funnel follows the axis and is decoupled from gas. Because of significant photon trapping deep in the disk, only photons generated close to the surface can escape the disk and join the funnel region. This explains the relatively low radiative luminosities of our simulated disks and is in agreement with the slim disk model (e.g., Abramowicz et al. 1988; S˛adowski 2011). For higher accretion rates, the region of dominant photon trapping extends further out, not only near the equatorial plane, but also in the (now optically thick) funnel. Likewise, one expects that, for lower accretion rates, photon trapping will become less and less e GLYPH<11> ective and the radiative e GLYPH<14> ciency will approach the thin disk value.', '5.3 Turbulent transport': 'So far we have shown that the di GLYPH<11> usive transport of radiation is important near the surface of the disk, and that the radial photon advection dominates over di GLYPH<11> usion deep inside the disk. There is potentially one more way of radiation transfer - turbulent transport pointed out by Jiang et al. (2014b). It is e GLYPH<11> ective if radiative energy is transported without transporting mass, similar to what happens in convection. Such behavior may result, e.g., from magnetic buoyancy. To assess the importance of this e GLYPH<11> ect it is enough to compare the density- and radiative energy-weighted vertical velocitities of the gas. If the former is zero and the latter is not, then radiation is transported by gas without transporting mass, and hence, it is different in nature from the standard photon trapping which results from mean motion of gas. \nIf turbulent transport is e GLYPH<11> ective then lighter gas, but containing more radiation, moves preferably towards the disk surface. In Fig. 11 we plot the polar velocity of gas as a function of radiation to rest-mass energy density ratio for the same location as in Fig. 10 (red), and for another point closer to the equatorial plane, located at ( r ; GLYPH<18> ) = (20 ; 80 GLYPH<14> ) (blue markers). The size of the markers denotes the optical depth as it did in Fig. 10. We see that there is no correlation between the relative radiative energy density content and the vertical motion of the gas, what suggests that the turbulent e GLYPH<11> ect cannot be strong. \nFrom the same sets of points we now calculate the densityand energy-weighted velocities, defined as, \nh v GLYPH<18> i GLYPH<26> = GLYPH<6> GLYPH<26> v GLYPH<18> GLYPH<6> GLYPH<26> ; (21) \nh v GLYPH<18> i b E = GLYPH<6> b E v GLYPH<18> GLYPH<6> b E ; (22) \nFigure 11. Vertical velocity of gas as a function of radiative to rest-mass energy ratio for simulation r001 and two points, located at ( r ; GLYPH<18> ) = (20 ; 851 GLYPH<14> ) (red) and ( r ; GLYPH<18> ) = (20 ; 80 GLYPH<14> ) (blue markers). \n<!-- image --> \nwhere the sums go through all the points in the set, corresponding to di GLYPH<11> erent moments of time and fixed location. For the point closer to the disk surface we get h v GLYPH<18> i GLYPH<26> = 0 : 0022 and h v GLYPH<18> i b E = 0 : 0013. For the other point, located almost at the equatorial plane, we have, h v GLYPH<18> i GLYPH<26> = 0 : 0038 and h v GLYPH<18> i b E = 0 : 0002. All the values are positive what indicates that both the gas and radiation is advected with the gas towards the equatorial plane. However, the magnitudes of these velocities are at least order of magnitude lower than of the corresponding radial velocities, what reflects the fact that the radial motion and advection dominates. One may, however, notice that the energy-weighted velocities are lower, i.e., they tend to deviate from the density-weighted velocities towards the surface. Nevertheless, the magnitude of the turbulent transport of radiation is not significant.', '5.4 Advection coe GLYPH<14> cient': 'Defining the e GLYPH<14> ciency of photon trapping in a non-spherical accretion flow is a challenging task because of the multiple dimensions involved. Here we try to calculate the advection coe GLYPH<14> cient which estimates the fraction of photons generated at given radius that are able to escape, the rest being advected to the BH. For this purpuse, at each radius r box, we consider a disk annulus extending from r 1 = r box GLYPH<0> 0 : 5 r G to r 2 = r box + 0 : 5 r G, and limited in GLYPH<18> to the range GLYPH<18> GLYPH<6> = 90 GLYPH<14> GLYPH<6> 37 GLYPH<14> (Fig. 12). The polar angle range is chosen to fit the center of the transition region discussed in the previous section where the radiation flux in the fiducial model is dominated by its polar component. The annulus covers extends over 2 GLYPH<25> in azimuthal angle. From the time averaged disk properties we extract luminosities of the radiative flux that crosses each surface of the annulus. For the radial sides, we compute \nLr ; (1 ; 2) = Z 2 GLYPH<25> 0 Z GLYPH<18> + GLYPH<18> GLYPH<0> R r t p GLYPH<0> g d GLYPH<18> d GLYPH<30>; (23) \nwhere the integration takes place either at the inner edge r = r 1 or at the outer edge r = r 2. The fluxes crossing the top and bottom surfaces are similarly integrated to give, \nL GLYPH<18> = 2 GLYPH<2> Z 2 GLYPH<25> 0 Z r 2 r 1 R GLYPH<18> t p GLYPH<0> g drd GLYPH<30>; (24) \nc GLYPH<13> 0000 RAS, MNRAS 000 , 000-000 \nFigure 13 shows the above advection coe GLYPH<14> cient q adv as a function of radius for the fiducial model r001 and the large accretion rate model r003 (the chosen vertical size of the box corresponds to the region of purely polar radiative flux only for the former). For the fiducial simulation, the advection factor increases towards the BH, as expected because of increasing inflow velocity. At radius r = 10 we find q adv = 0 : 8 which means that only GLYPH<24> 20% of photons generated at that radius manage to escape and enter the funnel. The coe GLYPH<14> cient drops down to q adv = 0 : 55 at the edge of the inflow / outflow equilibrium region ( r = 25). The profile suggests that the e GLYPH<11> ective photon trapping radius, defined by q adv = 1 = 2, is probably around r GLYPH<25> 35 (the point plotted at r = 30 is outside the region of inflow equilibrium and is a little suspect). \n<!-- image --> \nFigure 13. The advection coe GLYPH<14> cient, q adv (equation 25) as a function of radius for simulations r001 (blue) and r003 (red circles). \n<!-- image --> \nFigure 12. Shows the box and the definitions of the luminosities used in equation (25) to estimate the advective factor q adv = 1 GLYPH<0> L GLYPH<18> = ( Lr ; 1 GLYPH<0> Lr ; 2 + L GLYPH<18> ). \nwhere the integration is done at fixed polar angles, GLYPH<18> GLYPH<6> . \nThe integrated flux Lr ; 1 incoming through the outer radial sufrace tells how much radiation is advected into the volume. The corresponding luminosity Lr ; 2 crossing the inner surface tells how much radiation is advected out of the same volume. Meanwhile, the total radiation generated within the annulus is Lr ; 1 + L GLYPH<18> GLYPH<0> Lr ; 2. Thus Lr ; 1 GLYPH<0> Lr ; 2 divided by the latter quantity measures the fraction of energy that is advected radially, whereas L GLYPH<18> over the same quantity measures what fraction of the radiative energy escapes through the top and bottom surfaces. This then motivates the following definition of the advection coe GLYPH<14> cient, q adv, \nq adv = Lr ; 1 GLYPH<0> Lr ; 2 Lr ; 1 GLYPH<0> Lr ; 2 + L GLYPH<18> : (25) \nIn the limit of a radiatively e GLYPH<14> cient disk we would have Lr ; 1 = Lr ; 2 GLYPH<25> 0 and q adv GLYPH<25> 0. In the opposite limit of an advectiondominated flow we expect L GLYPH<18> = 0 and q adv GLYPH<25> 1. Tehrefore, it is natural to define the e GLYPH<11> ective trapping radius as the location where q adv = 1 = 2, i.e., where half the radiation generated at that radius manages to escape and half is advected to the BH. \nThe e GLYPH<14> ciency of advection for the simulation with higher accretion rate ( r003 ) is significantly larger because of the larger optical depth. Even at radius r = 25 only GLYPH<24> 15% of photons escape the disk. In the innermost regions of this model, q adv exceeds 1, reflecting the fact that the top-bottom luminosities are negative, i.e., photons are brought into the box, and no radiation escapes. \nIn our analysis in this subsection we focused solely on the \nradiative energy flux. In general, one should include also the flux of mechanical energy because dissipation may result in kinetic energy leaving the box. Including this component hardly a GLYPH<11> ects the values of the advection coe GLYPH<14> cient calculated above.', '6 VARIABILITY': 'Radiation coming from accretion disks is known to be highly variable (e.g., Done et al. 2007). In case of galactic BH binaries, this variability takes place on short timescales (the horizon crossing time for a 10 M GLYPH<12> BH is GM = c 3 GLYPH<25> 5 GLYPH<2> 10 GLYPH<0> 5 s). The variability is strongest in the hard state, and weakest in the thermal state. However, even in the latter, the power spectrum is far from featureless. \nStudying variability is a powerful tool in understanding accretion flows. The characteristic frequencies tell us where the modulated radiation come from. Features in the power-spectrum, e.g., quasi-periodic oscillations or breaks, can manifest more subtle properties of the disks (e.g. Ingram et al. 2009; Wellons et al. 2014). Modeling the variability and its power spectrum has so far been limited mostly to analytical models. However, the dynamics of the gas and the properties of the radiation field are complicated and highly non-stationary, so the analytical approach is limited, and ultimately we should model variability using time-dependent threedimensional simulations. \nHowever, this approach is not straightforward. Numerical modeling of radiation in MHD codes is limited by a number of factors. First of all, radiation is solved for in the grey approximation and some arbitrary (usually blackbody) shape is often assumed for the spectrum. Secondly, general relativistic e GLYPH<11> ects are rarely included. Moreover, Comptonization is either neglected or treated in a crude way. Last but not least, various approximations for radiation closure are adopted (with the exception of direct radiation transfer solvers operating on a fixed grid of angles, as implemented recently by Jiang et al. (2014a) and Ohsuga & Takahashi (2015)). Simplistic closure schemes may limit the information available for calculating the visible spectra. \nThe most reasonable way to proceed is to take the global, timedependent output of a disk simulation and postprocess it with a sophisticated radiation (and only radiation) solver which will not be as limited as full radiation-MHD simulations. Such codes, which \nTable 3. Fractional variability of radiative flux in model r001 \n| GLYPH<18> = 2 | GLYPH<14> = 27 |\n|-----------------|------------------|\n| 0.2 | 1.96 |\n| 0.23 | 0.64 |\n| 0.23 | 0.58 | \nsolving the frequency-dependent radiative transfer equation and account for relativistic e GLYPH<11> ects and Comptonization have recently been developed (e.g. Zhu et al. 2015) and are expected to be soon available for spectral modeling. \nIn the meantime, we attempt to directly estimate the variability of light curves from the simulations with KORAL described in this paper. Because of the limitations mentioned above, in particular the fact that we evolve only the first moments of the radiation (M1 closure), our approach is expected to provide only a rough qualitative understanding of the temporal properties of radiation coming from super-Eddington accretion flows.', '6.1 Light curves': 'In principle, the observer is located at infinity, i.e, r GLYPH<29> r out. The correct way of measuring the radiation reaching a distant observer is to integrate the specific intensities pointing towards the observer at the outer edge of the computational box. However, because of the limited range of inflow / outflow equilibrium near the equatorial plane, we are limited only to studying light escaping in the polar region near the axis, where the disk solution has converged to large enough radii. Moreover, the size of our computational box is obviously limited and we cannot measure the radiative fluxes at radii larger than r & r out = 2 = 500. Even if the duration of simulation was infinite, the photosphere would be located beyond the domain boundary except in the polar region, and even there we resolve the photosphere only at the lowest accretion rates. \nBecause of the limitations mentioned above, we decided to estimate the light curve by looking at the local flux of radiation at some location inside the polar region. To be sure that the light curve measured in this way is close to the light curve seen from infinity, we insist that the radiation is already decoupled from the gas; otherwise, continued interaction with (and acceleration of) the gas could drastically change the properties of the radiation that finally escapes. Once this condition is satisfied, and provided there is little radiation coming from other regions of the disk towards the chosen line of sight, the shape of the light curve should be independent of the radius it is measured at, only shifted in time by the light crossing time. Below we test if this criterion is satisfied for the locally measured fluxes in simulation r001 . \nTo extract the light curves we choose an arbitrary slice through the poloidal plane, define the inclination angle, and choose a radius where we measure the energy fluxes. Then we take the radial component of the lab-frame radiative flux, R r t , and plot it as a function of time. Fig. 14 shows such time series extracted at two polar angles ( GLYPH<18> = 2 GLYPH<14> and 27 GLYPH<14> , top and bottom panels, respectively), and at three radii ( r = 100, 250, and 500). Only the second half of the simulation, when the accretion has settled down to a quasi-steady state (compare Fig. 1), is shown. The fluxes measured at r = 100 and r = 500 were scaled to account for geometrical expansion to match the magnitude of the flux at r = 250. \nThe first panel corresponds to radiation escaping along the axis. For model r001 , gas is optically thin on the axis at all radii so one may expect the measured fluxes not to depend on the radius where the measument is performed. This is indeed the case. \nFigure 14. Variability of the radiative flux in model r001 measured at polar angle GLYPH<18> = 2 GLYPH<14> (top panel) and 27 GLYPH<14> (bottom panel) and at three radii: r = 100, 250, and 500. \n<!-- image --> \nThe profiles resemble each other to a good accuracy, at least for t > 12000, with the shift in time reflecting the propagation of light in the vertical direction. This gives some confidence that the light curve we calculate is what a distant observer might see. It is interesting, however, that the propagation speed down the funnel to match the time delay between radii is less then the speed of light. It equals approximately 0 : 25 c which is the characteristic velocity of the radiation field (the velocity of the radiation rest frame) inside the funnel. It is determined by the geometry of the funnel and the e GLYPH<11> ect was discussed close to 40 years ago by Sikora (1981). Basically, not all the photons go straight up; a significant fraction go sideways and some even go backward. The characteristic velocity is in some sense the average radial photon velocity. \nThe short duration of the light curves ( GLYPH<24> 5 s real time for a 10 M GLYPH<12> BH)) do not allow us to calculate power spectra. Instead, we calculate the fractional variability through \nf = GLYPH<27> ( L ( t )) h L ( t ) i ; (26) \nwhere GLYPH<27> denotes the standard deviation and hi stands for the mean luminosity. The second column of Table 3 gives the fractional variability calculated for light curves in the top panel of Fig. 14. Note that the fractional variability does not change much with radius. In particular, it does not change between r = 250 and r = 500 which again reflects the fact that the radiation flowing along the axis has established its temporal profile and is not a GLYPH<11> ected any more by interaction with gas or radiation at larger radii. \nThe bottom panel of the same figure presents radiative light curves measured at the same radii but at a larger inclination angle, GLYPH<18> = 27 GLYPH<14> , which corresponds to the edge of the funnel region, but located in the optically thick portion of the gas. One may expect therefore, that the radiation interacts (is absorbed or emitted) \nwith the gas up to large radii, even out to the domain boundary, and that the radiation fluxes measured at di GLYPH<11> erent radii will di GLYPH<11> er, even after taking the geometrical factor into account. The radiation flux measured at r = 100 (blue line) varies significantly, reaching at moments negative values. This behavior reflects the motion of the gas to which the radiation is coupled to. The gas on average moves out, but temporarily can move inward, dragging photons with it. The light curves become more smooth and the luminosities become larger in magnitude with increasing radius. These result from the fact that the gas moves in a more laminar way further out, and that radiation originating from a larger volume contributes to the measured luminosity. As a result, the corresponding fractional variability (third column of Table 3) decreases significantly with increasing radius. \nBecause of the significant coupling with gas, and the fact that we do not resolve the photosphere (because the computation box is not large enough box and the duration of the simulation is too short), the light curves for the larger inclination angle are not reliable. Only the light curve for radiation leaving the system almost exactly along the axis is robust. However, even this will not be the case if the accretion rate is larger since then, even on the axis, the photosphere will be located at large radii. This is a severe limitation for studies of variability in super-critical disks. In contrast, for thin disks, the photosphere is close to the equatorial plane and one can hope to extract useful variability information from simulations.', '7 AXISYMMETRIC SIMULATIONS': "Angular momentum in accretion disks is transported by MHD turbulence, which is driven primarily by the magnetorotational instability (Balbus & Hawley 1991). Turbulent motion in the poloidal plane leads not only to the exchange of angular momentum, but also to dissipation of magnetic field. In axisymmetric simulations this process quickly causes the magnetic field to decay, and at some point, shortly after the beginning of a simulation, typically after t GLYPH<24> 5000, the accretion stops. In reality, the poloidal magnetic field is revived by a dynamo process which results from breaking axisymmetry - non-axisymmetric perturbation can 'rotate' toroidal field into the poloidal plane and thereby regenerate poloidal field. Three dimensional simulations of turbulent magneto-fluids in shearing boxes (and also in global models) have shown that the saturated state is characterized by an average toroidal-to-radial magnetic field ratio GLYPH<18> B GLYPH<25> 0 : 25, and an average magnetic-to-gas pressure ratio GLYPH<12> 0 = 0 : 1. There is no way of obtaining such a saturated state in pure axisymmetric simulations. \nOn the other hand, the advantages of assuming axisymmetry and simulating accretion disks in two dimensions are obvious such simulations are cheaper by more than an order of magnitude in terms of the required computational time. Until recently, however, axisymmetric simulations were limited to extremely short durations because of the rapid decay of the magnetic field mentioned above. The situation has now changed. S˛adowski et al. (2015a) introduced a mean-field model of the dynamo e GLYPH<11> ect which mimics the properties of three-dimensional MRI-driven turbulence but can be applied to axisymmetric simulations. In this approach, the properties of the magnetic field are driven towards the prescribed characteristics of the saturated state, described by parameters GLYPH<18> B and GLYPH<12> 0 . This is achieved by 'pumping' new vector potential into the MHD flow, leading to a correction to the existing poloidal field. The poloidal field is enhanced in regions where the magnetic field is too weak, and the toroidal component of the field is damped in regions were \nthe magnetic pressure exceeds the prescribed saturation value. It has been shown that such an approach successfully allows for arbitrarily long simulations and indeed leads to a saturated state similar to that seen in three-dimensional simulations. \nOur aim in this section is to compare the properties of threedimensional simulated super-critical, radiative disks with corresponding two-dimensional axisymmetric simulations. For this purpuse we simulated an additional model ( d300 ) which was run in axisymmetry, with 252 x 234 grid points in the poloidal plane, and initialized in an identical manner to the fiducial model r001 (see Table 1). We used the mean-field dynamo model with the fiducial saturated state parameters GLYPH<18> B = 0 : 25 and GLYPH<12> 0 = 0 : 1. Model d300 was run until t = 190 ; 000, an order of magnitude longer than the threedimensional model r001 , yet it required less computer resources by almost an order of magnitude. This enormous saving is the reason why we feel it is important to explore the two-dimensional option fully. \nThe left- and right-most panels in Fig. 6 compare the density distribution on the poloidal plane, gas velocities, radiative flux, location of the photosphere, and border of the outflow region in the three- ( r001 ) and two-dimensional ( d300 ) simulations. Qualitatively, the two solutions (after averaging over time and azimuth in the three-dimensional case) agree very well. The gas shows the same dynamical properties, with outflow taking place only in the funnel region, the magnitude of the radiation flux is similar, and the photosphere is located at the same place. The only noticeable di GLYPH<11> erence is near the disk surface at larger radii ( r & 20) where the two-dimensional run shows more significant vertical motion. This may be because of the approximate treatment of the dynamo e GLYPH<11> ect which is constructed to satisfy only the vertically averaged criteria, without paying too much attention to the vertical profile of magnetic field properties. It should also be kept in mind that the three-dimensional model has achieved inflow / outflow equilibrium only for r . 20. It could be that even the relatively minor di GLYPH<11> erences in structure between the three- and two-dimensional models would be reduced once the former is run for a much longer time and reaches equilibrium out to radii GLYPH<24> 100. Very expensive simulations would be required to verify if this explanation is correct. \nTo quantify the comparison between models r001 and d300 , we have calculated radial profiles of the accretion rate, surface density, and density-weighted angular momentum for the two runs. They are shown respectively in the top, middle, and bottom panels in Fig. 15. The net accretion rates are similar. The two-dimensional simulation has ˙ M = 8 : 9 ˙ M Edd, roughly 10% lower than the threedimensional model, but close enough to allow a meaningful comparison of the two models. The extent of the flat section of the net accretion rate profile is a benchmark for the extent of inflow / outflow equilibrium. As discussed earlier, for simulation r001 the equilibrium region extends only to r GLYPH<24> 20. Thanks to the long duration of the d300 simulation, the equilibrium region here extends much further out, to r GLYPH<24> 100. Dashed lines in the same panel show the rate of mass lost in the wind. It is comparable for the two runs, and suggests that the amount of outflowing gas equals the net accretion rate around radius r = 60. \nThe middle panel in Fig. 15 compares the surface density profiles. They are indistinguishable where the equilibrium regions of the two simulations overlap (the sections of the curves outside the equilibria are marked with shaded colors). Similarly, the profiles of angular momentum also match well (bottom panel) except in the plunging region inside the marginally stable orbit, where the twodimensional run results in a slightly lower angular momentum. This di GLYPH<11> erence probably arises from the fact that the dynamics of the \nFigure 15. Radial profiles of the net and outflowing accretion rate (top panel), surface density (middle), and specific angular momentum (bottom) for the three-dimensional run r001 and the equivalent two-dimensional run d300 . The light shaded line segments in the second and third panels correspond to regions where inflow / outflow equilibrum has not been reached. \n<!-- image --> \nflow and the properties of the magnetic field in this region are not governed by the combined e GLYPH<11> ects of shear and the dynamo process (Penna et al. 2013b). \nIn summary, the above comparisons indicate that the average properties of the two- and three-dimensional simulations are remarkably similar. The temporal properties, on the other hand, are significantly di GLYPH<11> erent. In Fig. 16 we show radiative light curves measured at angle GLYPH<18> = 2 GLYPH<14> and at radius r = 250 (the light curve of simulation r001 corresponds to the one shown in the top panel of Fig. 14 for this particular radius). The two-dimensional run re- \nFigure 16. Time variability of the radiative flux measured at angle GLYPH<18> = 2 GLYPH<14> from the axis at radius r = 250 for the three-dimensional simulation r001 and the two-dimensional model d300 . \n<!-- image --> \nsulted in noticeably larger radiative luminosity in the outflow region and larger beaming towards the axis (but the same total luminosity, see Table 2). As a result, the flux in the axisymmetrical model is twice as large, and at the same time, it is also more variable - its fractional variability ( f = 0 : 64) significantly exceeds that of the three-dimensional run ( f = 0 : 23). The di GLYPH<11> erence must come from the fact that the axisymmetrical simulation neglects the many non-symmetrical modes which develop in three-dimensional simulations. Variations in independent patches in azimuth would tend to wash each other out when averaged, but no such averaging would be present in an axisymmetric model.", '8 SUMMARY': "In this paper we presented a set of four three-dimensional simulations of black hole super-critical accretion disks. Two of the simulations accreted roughly at 10 ˙ M Edd (models r001 and r020 ), the third simulation ( r003 ) had a significantly larger accretion rate of 176 ˙ M Edd, and the fourth simulation ( r011 ) was characterized by a rotating BH with spin a GLYPH<3> = 0 : 7 and accreted roughly at 17 ˙ M Edd. The fifth simulation we presented was performed assuming axisymmetry (hence this was a two-dimensional simulation) and corresponded to roughly 10 ˙ M Edd accretion rate. All the simulations resulted in optically thick and geometrically thick turbulent accretion disks. In the course of this study we reached a number of conclusions: \n(i) Photosphere: Only for accretion rates ˙ M . 10 ˙ M Edd does the photosphere extend down to the horizon, i.e., only for such low accretion rates will an observer at infinity viewing down the axis be able to see the innermost region of the accretion disk. For larger accretion rates, an on-axis observer will only observe a photosphere which is located relatively far from the black hole. In fact, even for relatively low accretion rates ˙ M . 10 ˙ M Edd, the optically thin region is limited to polar angles GLYPH<18> . 15 GLYPH<14> . For larger inclinations, an observer would only see the photosphere of the accretion disk located at large distance from the BH, often outside the computational box of the simulations. The limited box size and duration of the simulations do not allow us to study the radiation coming from such distant photospheres in any detail. We expect, however, that the spectrum of any such radiation will be relatively soft, because of the large distance from the BH, and that the observed isotropicequivalent luminosity will not significantly exceed the Eddington \nvalue - a super-Eddington flux in an optically thick wind causes acceleration of the wind and reduces the radiative flux towards the Eddington limit. \n(ii) Stagnation radius: For the simulations with the lowest accretion rates ( ˙ M . 10 ˙ M Edd), the stagnation radius in the funnel, r 0, which separates an inner region where gas falls into the BH from an outer region where gas flows out, is located near the BH ( r 0 . 10). In the case of simulation r011 (with a spinning BH), where the extraction of the rotational energy of the BH increases the energy flux through the funnel, the stagnation radius is very close to the horizon. For the run with the largest accretion rate ( r003 ), the stagnation radius moves significantly out - all gas in the funnel within r 0 GLYPH<25> 50 falls on the BH. This is a result of the increased optical depth of the gas, which traps photons and e GLYPH<11> ectively suppresses the outflow of radiation up to this radius. \n(iii) Total luminosity: All the simulations with non-rotating BHs show the same total e GLYPH<14> ciency of roughly 3% ˙ Mc 2 , approximately a factor of two less than the e GLYPH<14> ciency of a standard thin accretion disk. We obtain similar ratio of e GLYPH<14> ciencies for the simulation with a rotating BH ( a GLYPH<3> = 0 : 7), for which the measured total e GLYPH<14> ciency was GLYPH<24> 8% ˙ Mc 2 . These e GLYPH<14> ciencies were calculated from the total luminosity in all forms of energy: radiative, kinetic, magnetic, thermal and binding (only rest mass energy was not included). The total luminosity is thus fundamental and represents the total energy deposited at 'infinity' (the interstellar medium). There is no unique way of decomposing the above total luminosity into its constituent parts because of the limited size of the inflow / outflow equilibrium region in the simulations. When measured at the limit of inflowoutflow equilibrium ( r GLYPH<25> 25 in the disk interior), energy balance is dominated by the binding energy flux. This energy must be transformed into other forms of energy before reaching infinity (where the binding energy is zero). \n(iv) Radiative luminosity: Radiative luminosity can be measured reliably only inside the optically thin funnel region near the axis; here, radiation is decoupled from gas, and whatever radiation is flowing outward is guaranteed to reach the observer. However, only the simulations with the lowest accretion rates show an optically thin region inside the computational domain at all. Even in these cases, the optically thin radiative luminosity increases with radius (Table 2) because of radiation flowing into the funnel from the disk at larger radii. To obtain a useful estimate of the net radiative luminosity from the funnel, we would have to simulate accretion flows in much bigger computational box and for a much longer time. This is presently impractical. The radiative luminosities measured at radius r = 250 in the optically thin and outflowing regions are quite low. For accretion rates near 10 ˙ M Edd and a non-rotating BH, only GLYPH<24> 20% of the total liberated energy of 3% ˙ Mc 2 (mentioned above) comes out as optically thin radiation escaping through the funnel. The kinetic luminosity exceeds the radiative luminosity for the largest accretion rate considered, where the coupling between radiation and gas in the funnel is strongest, resulting in radiative acceleration of gas (S˛adowski & Narayan 2015b). The luminosities are significantly larger for the simulation with a rotating BH because the BH spin provides an extra source of energy. \n(v) Beaming: We have confirmed that radiation is beamed along the polar axis and the radiative fluxes here can be significantly super-Eddington. For the fiducial accretion rate of 10 ˙ M Edd, the simulations with non-rotating and spinning BHs show radiative fluxes of 20 and 100 F Edd, respectively, on the axis. In contrast, the kinetic energy is not beamed on the axis but peaks (Fig. 7) either in a conical shell (for lower accretion rates) or in the wind region (for larger accretion rates). \n(vi) Photon trapping: We compared the total flux of radiation with the di GLYPH<11> usive flux and showed that, for the accretion rates considered, advection of radiation (photon trapping) dominates over di GLYPH<11> usive transport deep inside the disk. For ˙ M GLYPH<25> 10 ˙ M Edd, the advective flux is more than 30 times stronger than the di GLYPH<11> usive flux near the equatorial plane, and this ratio increases by at least an order of magnitude when the accretion rate grows to GLYPH<24> 176 ˙ M Edd, which is explained by the increased optical depth. Closer to the disk surface or funnel boundary, the di GLYPH<11> usive flux of radiation dominates, which ultimately contributes to the optically thin radiation escaping through the funnel. This flux also helps to drive gas away from the disk surface into the funnel. \n(vii) Radiation transport An analysis of our simulations shows that radiation transport is dominated by advective and di GLYPH<11> usive transport. We do not see a significant component of turbulent radiative transport. \n(viii) Advection factor: For the fiducial model ( ˙ M = 10 ˙ M Edd) we estimated the fraction of photons generated in the disk that manage to escape vertically and enter the optically thin funnel region. At radius r = 10, only GLYPH<24> 20% of photons leave the disk while GLYPH<24> 80% end up in the BH. Corresponding numbers at radius r = 25 are approximately 45% and 55%, respectively. This suggests that the e GLYPH<11> ective trapping radius, where half the photons escape from the disk and half fall into the BH, is located at r & 30 for 10 ˙ M Edd. \n(ix) Variability: We attempted to extract from the simulations (frequency integrated) lightcurves as seen by observers at infinity. Because of the limited size of the equilibrium region, and the fact that the photosphere is located close to the axis even for the lowest accretion rate considered, we were able to obtain robust light curves only for radiation escaping along the axis, and that too only for the fiducial model. Extracting variability information from simulations of optically thick and geometrically very thick disks is, and will remain to be, challenging. One may expect that thin disks, with photospheres much closer to the equatorial plane, will be more amenable to study. \n(x) Impact of the BH mass: We performed one simulation ( r020 ) with BH mass 1000 M GLYPH<12> , which had the same Eddingtonscaled mass accretion rate as the fiducial model ( r001 ). The properties of the two simulations, after scaling down the latter by appropriate factors, were quite similar; in fact, it was di GLYPH<14> cult to distinguish the two (compare Fig. 6). This is because electron scattering opacity dominates in both simulations, and under these conditions the accretion equations scale very simply with BH mass. For much larger BH masses (corresponding to the SMBH regime), the absorption opacity will no longer be negligible and this will break exact scaling. \n(xi) Axisymmetric simulations: We compared the properties of the fiducial three-dimensional simulation ( r001 ) with a twodimensional axisymmetric simulation ( d300 ) that made use of an artificial magnetic dynamo and accreted at roughly the same rate. We showed that the time-averaged properties of the two simulations were remarkably similar. Because of the significantly lower computational cost, we were able to run the axisymmetric simulation for a much longer time and obtained a significantly larger equilibrium region (extending up to r GLYPH<24> 100 instead of only r GLYPH<24> 25 in the case of the 3D model). Thus, two-dimensional axisymmetric simulations (with magnetic dynamo) are a cheap and promising method for running models for long times and thereby extending the range of radii over which one obtains useful information. However, this must be done with caution. While time-averged quantities appear to be reliable, we note that the temporal properties of the 2D and 3D runs were di GLYPH<11> erent. In particular, because of the \nlack of non-axisymmetric modes in the axisymmetric simulation, it showed much larger variability in the radiative flux flowing out along the axis.", '8.1 Comparison with other studies': "Our study is the first to explore the parameter space relevant to radiation-dominated BH accretion disks using three-dimensional, general relativistic, radiation-MHD simulations. The properties of the simulated super-critical disks described in this work are qualitatively in agreement with previously published global simulations. In particular, significant photon trapping was identified already by Ohsuga & Mineshige (2007), and confirmed more recently by S˛adowski et al. (2014) and McKinney et al. (2014). The total luminosities given in the present work are also close to those obtained in the latter studies. The same is true for radiation beaming and the radiative luminosity of the funnel. \nThere are a number of di GLYPH<11> erences between our results and those described in Jiang et al. (2014b). The model presented in the latter work, which had an accretion rate of GLYPH<24> 15 ˙ M Edd, can be directly compared to our fiducial model r001 . Jiang et al. (2014b) found a very di GLYPH<11> erent spatial distribution of gas compared to our run; their disk is strongly concentrated at the equatorial plane whereas our disk is not. Perhaps because of this, they obtained a powerful radiative flux from the innermost ( r < 10) region; in fact, nearly all of their radiative luminosity, which is of the order of 5% ˙ Mc 2 , comes from such small radii. Our simulations show a similar total e GLYPH<14> ciency (3% ˙ Mc 2 ), but only a small fraction of the energy escapes as radiation at small radii. \nJiang et al. (2014b) used a Newtonian code with cylindrical coordinates (and a 'cylindrical BH' with radius r = 4). Their simulation was intialized with a di GLYPH<11> erent and more strongly magnetized torus compared to ours. They implemented a sophisticated radiation transfer solver which evolved in real time a number of specific intensities. In comparison, our simulations were done with the M1 closure scheme, which means that we evolved only four independent radiation quantities in each cell. On the other hand, Jiang et al. (2014b), for the sake of performance, had to make some approximations when treating the interactions between gas and radiation 6 . Which of these factors is responsible for the large discrepancy between their study and ours is presently unclear.", '9 ACKNOWLEDGEMENTS': 'The authors thank Jean-Pierre Lasota for his comments. AS acknowledges support for this work by NASA through Einstein Postdoctotral Fellowship number PF4-150126 awarded by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for NASA under contract NAS8-03060. AS thanks Harvard-Smithsonian Center for Astrophysics for hospitality. RN was supported in part by NSF grant AST1312651 and NASA grant TCAN NNX14AB47G. The authors acknowledge computational support from NSF via XSEDE resources (grant TG-AST080026N), and from NASA via the High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center.', 'REFERENCES': "- Abramowicz, M. A., Czerny, B., Lasota, J. P., & Szuszkiewicz, E. 1988, Astrophysical Journal, 332, 646 \nAfshordi, N., & Paczy'nski, B. 2003, Astrophysical Journal, 592, 354 \n- Avara, M. J., McKinney, J. C., & Reynolds, C. S. 2015, arXiv:1508.05323 Balbus, S. A., & Hawley, J. F. 1991, Astrophysical Journal, 376, 214\n- Begelman, M. C. 1978, Monthly Notices of the Royal Astronomical Society, 184, 53\n- Davis, S. W., Blaes, O. M., Hubeny, I., & Turner, N. J. 2005, Astrophysical Journal, 621, 372\n- Done, C., Gierli'nski, M., & Kubota, A. 2007, Astronomy and Astrophysics Review, 15, 1\n- Fragile, P. C., Olejar, A., & Anninos, P. 2014, arXiv:1408.4460\n- Frank, J., King, A. R., & Raine, D. J. 1985, Cambridge and New York, Cambridge University Press, 1985, 283 p.,\n- Gammie, C. F., McKinney, J. C., & Tóth, G. 2003, Astrophysical Journal, 589, 444\n- Hubeny, I., & Hubeny, V. 1997, Astrophysical Journal Letters, 484, L37 Igumenshchev, I. V., Narayan, R., & Abramowicz, M. A. 2003, Astrophysical Journal, 592, 1042\n- Ingram, A., Done, C., & Fragile, P. C. 2009, Monthly Notices of the Royal Astronomical Society, 397, L101\n- Jiang, Y.-F., Stone, J. M., & Davis, S. W. 2012, Astrophysical Journal Suppl. Ser., 199, 14\n- Jiang, Y.-F., Stone, J. M., & Davis, S. W. 2014, Astrophysical Journal Suppl. Ser., 213, 7\n- Jiang, Y.-F., Stone, J. M., & Davis, S. W. 2014, Astrophysical Journal, 796, 106\n- Kawashima, T., Ohsuga, K., Mineshige, S., et al. 2009, Publications of the Astronomical Society of Japan, 61, 769\n- Komissarov, S. S. 1999, Monthly Notices of the Royal Astronomical Society, 303, 343\n- Levermore, C. D. 1984, Journal of Quantitative Spectroscopy and Radiative Transfer, 31, 149 2\n- McClintock, J. E., Narayan, R., Davis, S. W., et al. 2011, Classical and Quantum Gravity, 28, 114009\n- McKinney, J. C., Tchekhovskoy, A., & Blandford, R. D. 2012, Monthly Notices of the Royal Astronomical Society, 423, 3083\n- McKinney, J. C., Tchekhovskoy, A., Sadowski, A., & Narayan, R. 2014, Monthly Notices of the Royal Astronomical Society, 441, 3177 \nMcKinney, J. C., Dai, L., & Avara, M. 2015, arXiv:1508.02433 Möller, A., S˛adowski, A. 2015, in prep \n- Narayan, R., Igumenshchev, I. V., & Abramowicz, M. A. 2003, Publications of the Astronomical Society of Japan, 55, L69\n- Novikov, I. D., & Thorne, K. S. 1973, Black Holes (Les Astres Occlus), 343\n- Ohsuga, K., & Mineshige, S. 2007, Astrophysical Journal, 670, 1283 Ohsuga, K., Mineshige, S., Mori, M., & Yoshiaki, K. 2009, Publications of the Astronomical Society of Japan, 61, L7\n- Ohsuga, K., & Mineshige, S. 2011, Astrophysical Journal, 736, 2\n- Ohsuga, K., & Takahashi, H. R. 2015, Astrophysical Journal, submitted Paczy'nski, B. 2000, arXiv:astro-ph / 0004129\n- Penna, R. F., McKinney, J. C., Narayan, R., et al. 2010, Monthly Notices of the Royal Astronomical Society, 408, 752\n- Penna, R. F., Kulkarni, A., & Narayan, R. 2013a, Astronomy & Astrophysics, 559, A116\n- Penna, R. F., Sa¸dowski, A., Kulkarni, A. K., & Narayan, R. 2013a, Monthly Notices of the Royal Astronomical Society, 428, 2255\n- Pozdnyakov, L. A., Sobol, I. M., & Syunyaev, R. A. 1983, Astrophysics and Space Physics Reviews, 2, 189\n- Rybicki, G. B., & Lightman, A. P. 1979, New York, Wiley-Interscience, 1979. 393 p.,\n- S˛adowski, A. 2011, Ph.D. Thesis, Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, arXiv:1108.0396\n- S˛adowski, A., Narayan, R., Tchekhovskoy, A., & Zhu, Y. 2013a, Monthly Notices of the Royal Astronomical Society, 429, 3533\n- S˛adowski, A., Narayan, R., Penna, R., & Zhu, Y. 2013b, Monthly Notices \nof the Royal Astronomical Society, 436, 3856 \n- S˛adowski, A., Narayan, R., McKinney, J. C., & Tchekhovskoy, A. 2014, Monthly Notices of the Royal Astronomical Society, 439, 503 \nS˛adowski, A., Narayan, R., Tchekhovskoy, A., Abarca, D., Zhu, Y., & McKinney J. C. . 2015a, Monthly Notices of the Royal Astronomical Society, 447, 49 \n- S˛adowski, A., & Narayan, R. 2015b, arXiv:1503.00654, MNRAS, submitted \nS˛adowski, A., & Narayan, R. 2015c, MNRAS, in press Schnittman, J. D., Krolik, J. H., & Noble, S. C. 2013, Astrophysical Journal, 769, 156 \n- Shafee, R., Narayan, R., & McClintock, J. E. 2008, Astrophysical Journal, 676, 549 \nShafee, R., McKinney, J. C., Narayan, R., et al. 2008, Astrophysical Journal Letters, 687, L25 \nShakura, N. I., & Sunyaev, R. A. 1973, A&A, 24, 337 \nSikora, M. 1981, Monthly Notices of the Royal Astronomical Society, 196, 257 \nStraub, O., Bursa, M., S˛adowski, A., et al. 2011, Astronomy & Astrophysics, 533, A67 \n- Tchekhovskoy, A., Narayan, R., & McKinney, J. C. 2011, Monthly Notices of the Royal Astronomical Society, 418, L79 \nTóth, G. 2000, Journal of Computational Physics, 161, 605 \nWellons, S., Zhu, Y., Psaltis, D., Narayan, R., & McClintock, J. E. 2014, Astrophysical Journal, 785, 142 \n- Zhu, Y., Narayan, R., S˛adowski, A., & Psaltis, D. 2015, Monthly Notices of the Royal Astronomical Society, 451, 1661"}
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2002PhRvD..66d4009K
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Quasinormal modes of a small Schwarzschild-anti-de Sitter black hole
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2002-01-01
| 2 | 0.45 | 163 |
['-', '-', '-', '-', '-', '-', '-', '-']
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[]
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We compute the quasinormal modes associated with the decay of the massless scalar field around a small Schwarzschild-anti-de Sitter black hole. The computations show that when the horizon radius is much less than the anti-de Sitter radius, the imaginary part of the frequency goes to zero as r<SUP>d-2</SUP><SUB>+</SUB> while the real part of ω decreases to its minimum and then goes to d-1. Thus the quasinormal modes approach the usual AdS modes in the limit r<SUB>+</SUB>-->0. This agrees with suggestions of Horowitz and Hubeny [Phys. Rev. D 62, 024027 (2000)].
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| 1 |
https://arxiv.org/pdf/hep-ph/0205142.pdf
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{'GAUGE SYMMETRY BREAKING ON ORBIFOLDS': "CARLA BIGGIO \nDepartment of Physics, University of Padova and I.N.F.N., Sezione di Padova, via Marzolo 8 I-35131 Padova, Italy \n<!-- image --> \nWe discuss a new method for gauge symmetry breaking in theories with one extra dimension compactified on the orbifold S 1 /Z 2 . If we assume that fields and their derivatives can jump at the orbifold fixed points, we can implement a generalized Scherk-Schwarz mechanism that breaks the gauge symmetry. We show that our model with discontinuous fields is equivalent to another with continuous but non periodic fields; in our scheme localized lagrangian terms for bulk fields appear. \nSymmetries (and their breaking) play a fundamental role in theoretical high energy physics. In theories with extra dimensions new methods for symmetry breaking appear, such as the Scherk-Schwarz mechanism 1 , the Hosotani mechanism 2 and the orbifold projection 3 . \nIn this paper we will focus on theories with one extra dimension compactified on the orbifold S 1 /Z 2 where the radius of the circle R is of order of 1 /M GUT 4 , with M GUT ≈ 10 16 Gev , and the orbifold Z 2 is constructed by identifying points ( x, y ) with points ( x, -y ), where x and y are respectively the coordinates of the four and the fifth dimensions. The orbifold projection introduces two fixed points, at y = 0 and y = πR . While on the circle the only boundary condition we can assign to fields is the one corresponding to the translation of 2 πR (twist condition), on the orbifold we have to fix the parity and the behavior of fields at the fixed points, that means new possibilities are allowed 5 . \nIf we consider a set of n 5-dimensional bosonic fields φ ( x, y ) 6 , the self-adjointness of the kinetic operator requires that we assign conditions also on the y -derivative of φ ( x, y ). We look for boundary conditions in the following class: \n( φ ∂ y φ ) ( γ + ) = V γ ( φ ∂ y φ ) ( γ -) , (1) \nwhere γ = (0 , π, β ), γ ± = (0 ± , πR ± , y ± ), and V γ are constant 2 n by 2 n matrices. We have \ndefined 0 ± ≡ ± ξ , πR ± ≡ πR ± ξ , with ξ small positive parameter, y -≡ y 0 and y + ≡ y 0 +2 πR , with y 0 for convenience chosen between -πR + ξ and -ξ . \nWe now constrain the matrices V γ . We require, first, that the boundary conditions preserve the self-adjointness of the kinetic operator -∂ 2 y and, second, that physical quantities remain periodic and continuous even if fields are not. If the theory is invariant under the global transformations of a group G, these requirements imply the following form for the V γ matrices: \nV γ = ( U γ 0 0 U γ ) , (2) \nwhere U γ is an orthogonal n -dimensional representation of G. Finally we should take into account consistency conditions among the twist, the jumps and the action of orbifold on the fields, defined by the following equation: \nφ ( -y ) = Zφ ( y ) . (3) \nWe have: \nU γ Z U γ = Z γ ∈ (0 , π, β ) [ U 0 , U β ] = 0 (4) [ U π , U β ] = 0 . \nIf [ Z, U γ ] = 0 and [ U 0 , U π ] = 0 there is a basis in which U γ are diagonal with elements ± 1; in this case twist and jumps are defined by discrete parameters. However if [ Z, U γ ] /negationslash = 0 or [ U 0 , U π ] /negationslash = 0 continuous parameter can appear in U γ and then in eigenfunctions and eigenvalues. \nTo show how we can exploit these boundary conditions to break a symmetry, we focus on the simple case of one real scalar field. We start by writing the equation of motion for φ \n-∂ 2 y φ = m 2 φ , (5) \nin each region y q < y < y q +1 , where y q ≡ qπR and q ∈ Z . We have defined the mass m through the 4-dimensional equation ∂ 2 φ = m 2 φ . The solutions of these equations can be glued by exploiting the boundary conditions V 0 and V π , imposed at y = y 2 q and y = y 2 q +1 , respectively. Finally, the spectrum and the eigenfunctions are obtained by requiring that the solutions have the twist described by V β . In this theory the group of global symmetry is a Z 2 , so that V γ can be ± 1. \nWe focus on one example in which we compare the usual Scherk-Schwarz mechanism with our generalized one. We consider even fields and the following two sets of boundary conditions: \n( U 0 , U π , U β ) = { (+1 , +1 , -1) (A) (+1 , -1 , +1) (B) . (6) \nIn the first case fields are continuous but antiperiodic; the solution to the eq. of motion with these boundary conditions is: \nφ A ( x, y ) = α ( x ) cos ( my ) mR = n + 1 2 . (7) \nIn the case (B) fields are periodic but with a jump in y = πR ; the solution is: \nφ B ( x, y ) = β ( x ) /epsilon1 ( y 2 + πR 2 ) cos ( my ) mR = n + 1 2 , (8) \nwhere /epsilon1 ( y ) is the sign function on S 1 . The spectrum is the same in both cases and it is the \nFigure 1: Eigenfunctions of -∂ 2 y versus y/ ( πR ), in cases (A) and (B). \n<!-- image --> \nKaluza-Klein tower n/R shifted by 1 / 2 R . This is just what we expected in case (A), because this is precisely the Scherk-Schwarz mechanism: when fields are not periodic, their spectra are shifted from the Kaluza-Klein levels by an amount depending on the twist and zero modes are removed. If φ is a gauge field of a multiplet containing also untwisted fields, this is a mechanism for symmetry breaking. As we have already observed the (B)-spectrum is identical to the (A)spectrum; this means that we can break a symmetry not only through a twist, but also with discontinuities. While the spectra are identical, eigenfunctions (see fig. 1) are different, but they are related by a simple field redefinition: \nφ B ( x, y ) = /epsilon1 ( y 2 + πR 2 ) φ A ( x, y ) . (9) \nThe equation (9) is a map between a system in which fields are continuous and twisted (A) and another in which fields are discontinuous and periodic (B). In order to show the two schemes are completely equivalent, we should perform the field redefinition at the level of the action and, using the eq. of motion, recover the eigenfunction and the spectrum. After the field redefinition, the lagrangian for the massless real scalar field becomes: \nL ( φ, ∂φ ) = 1 2 /epsilon1 2 ∂ M φ ∂ M φ -2 /epsilon1 δ πR φ ∂ y φ +2 δ 2 πR φ 2 , (10) \nwhere /epsilon1 = /epsilon1 ( y/ 2 + πR/ 2) and δ y 0 = δ ( y -y 0 ). The new lagrangian contains two quadratic terms localized at y = πR . Now we want to derive the equation of motion for φ , but the naive application of the variational principle would lead to inconsistent results: indeed this works with continuous functions, but φ are discontinuous. In order to derive the correct equation of motion we regularize /epsilon1 with a smooth function /epsilon1 λ wich reproduces /epsilon1 in the limit λ → 0. If we now rewrite the lagrangian (10) in term of /epsilon1 λ and continuous fields, derive the equation of motion and then perform the limit λ → 0, we obtain: \n/epsilon1∂ 2 y φ -4 δ πR ∂ y φ -2 δ ' πR φ + /epsilon1m 2 φ = 0 (11) \nAway from the point y = πR this equation reduces to the one for continuous fields, while integrating it around the fixed point the singular terms give us the expected jumps. With this procedure we have shown that the continuous framework is really equivalent to the discontinuous one. \nIf we consider two or more real scalar fields such as a complex scalar field or a gauge field, continuous parameters are allowed in the boundary conditions and then they are also present in the eigenfunctions and in the spectra. It is interesting to note that, being the shift from the Kaluza-Klein spectrum continuous, we can go continuously from a phase in which the symmetry is broken to another in which it is unbroken. \nTable 1: Boundary conditions, eigenfunctions and spectra for fields in the Kawamura model in the traditional scheme (2 nd , 3 rd and 6 th columns) and in our framework (4 th , 5 th and 6 th columns). \n| Fields | ( Z 2 , U β ) | Kawamura eigenfunctions | ( Z 2 , U π ) | Our eigenfunctions | Spectrum m |\n|------------------------------------------------|-----------------|---------------------------|-----------------|-------------------------------------|--------------|\n| A a µ , λ 2 a L , H D u , H D d | (+ , +) | cos ( my ) | (+ , +) | cos ( my ) | 2 n R |\n| A ˆ a µ , λ 2ˆ a L , H T u , H T d | (+ , - ) | cos ( my ) | (+ , - ) | /epsilon1 ( y 2 + πR 2 ) cos ( my ) | 2 n +1 R |\n| A ˆ a 5 , Σ ˆ a , λ 1ˆ a L , ˆ H T u , ˆ H T d | ( - , - ) | sin ( my ) | ( - , - ) | /epsilon1 ( y 2 + πR 2 ) sin ( my ) | 2 n +1 R |\n| A a 5 , Σ a , λ 1 a L , ˆ H D u , ˆ H D d | ( - , +) | sin ( my ) | ( - , +) | sin ( my ) | 2 n +2 R | \nWe use now our mechanism to break the symmetries of a grand unified theory defined on the orbifold S 1 /Z 2 7 , 8 . The model we consider is the one constructed by Kawamura 8 in which the lagrangian is invariant under the transformations of an SU (5) gauge group and N = 2 (from the 4-dimensional point of view) supersymmetry. The fields content is the following: there is a vector multiplet V = ( A α M , λ 1 α L , λ 2 α L , Σ α ), with α = 1 , ..., 24 and M = µ, 5 which forms an adjoint representation of SU (5), and two hypermultiplets H 1 and H 2 , equivalent to four chiral multiplets H 5 , ˆ H 5 , ˆ H 5 and H 5 which form two fundamental representations of SU (5). These are multiplets of N = 1 supersymmetry in 5 dimensions, that is equivalent to N = 2 supersymmetry in 4 dimensions. The gauge vector bosons are the fields A α M : A a M with a = 1 , ..., 12 are the vector bosons of the standard model, while A ˆ a M with ˆ a = 13 , ..., 24 are those of the coset SU (5) / SU (3) × SU (2) × U (1). \nIn table 1 (second column) parity and twist assignments for all fields are shown, together with the resulting eigenfunctions and spectra (third and last columns). We see that these parity assigments alone break supersymmetry from N = 2 to N = 1, because zero modes survive only for even fields (without twist the spectrum would be n/R in all cases). When we assign a non trivial twist to fields, their spectra are shifted from the usual Kaluza-Klein tower. With the assignments of table 1 the only fields that mantain a zero mode are the vector bosons of the standard model, A a µ , their supersymmetric partners, λ 2 a L , and two Higgs doublets H D u and H D d : precisely the fields of the Minimal Supersymmetric Standard Model, that is SU (3) c × SU (2) L × U (1) Y with N = 1 supersymmetry. We observe that with these boundary conditions also the doublet-triplet splitting problem is naturally solved, because the lightest triplets modes are of order of 1 /R , while doublets have zero modes. \nUp to now only gauge and Higgs fields and their supersymmetric partners have been considered. If we introduce fermions and we assign the appropriate boundary conditions 9 we can see that fast proton decay is avoided. Then this model can be a realistic Grand Unified Theory and we adopt it to show in details how our mechanism for gauge symmetry breaking works. \nIn our framework parity assignments are identical to the original ones, but, instead of a twist, we require that some fields jump in y = πR . Among gauge fields only the vector bosons of the coset SU (5) / SU (3) × SU (2) × U (1) jump. In the last three columns of table 1 boundary \nconditions, eigenfunctions and eigenvalues are shown. We can observe that the spectra are the same of the Kawamura model for all fields, while eigenfunctions are identical for continuous fields but different for jumping fields. If we look at their explicit forms we observe we are just in the case studied before with one real scalar field. Performing the field redefinition of eq. (9), where 'A' fields are Kawamura's eigenfunctions and 'B' fields are ours, we see the two frameworks are completely equivalent. \nWhat about the lagrangians? Following the lines of what we did in the scalar case, we can perform the field redefinition at the level of the action in order to find the localized terms related to jumps in this theory. For simplicity we consider only the Yang-Mills term of the lagrangian, neglecting both the supersymmetric part and the Higgs terms: \nL = -1 4 F α MN F αMN . (12) \nApplying the redefinition (9) this becomes: \n-1 4 ˜ F a MN ˜ F aMN + 1 2 /epsilon1 2 f a ˆ b ˆ c A ˆ b M A ˆ c N ˜ F aMN + -1 4 /epsilon1 4 f a ˆ b ˆ c f a ˆ d ˆ e A ˆ b M A ˆ c N A ˆ dM A ˆ eN -1 4 /epsilon1 2 F ˆ a MN F ˆ aMN + (13) -2 δ 2 πR ( A ˆ a N A ˆ aN -A ˆ a 5 A ˆ a 5 ) + 2 /epsilon1 δ πR F ˆ a 5 N A ˆ aN , \nwhere ˜ F a MN = ∂ M A a N -∂ N A a M -f abc A b M A c N . We observe that the last two terms are localized at the fixed point y = πR . While the first is simply a bilinear term, as the ones found in the case of one scalar field, the last contains also a trilinear part: \n2 /epsilon1 δ πR F ˆ a 5 N A ˆ aN = 2 /epsilon1 δ πR [( ∂ 5 A ˆ a N -∂ N A ˆ a 5 ) A ˆ aN + f ˆ ab ˆ c A ˆ aN A b N A ˆ c 5 ] . (14) \nThis last term is proportional to the fields A ˆ c 5 that are just the 'would-be' Goldstone bosons that give mass to the Kaluza-Klein modes of the gauge vector bosons of the coset SU (5) /SU (3) × SU (2) × U (1). \nIn the previuos pages we always started from the continuous theory and then we derived the localized lagrangian terms, exploiting a relation found between smooth and jumping eigenfunctions. But our reasoning can be reversed: we can try to reabsorb localized terms for bulk fields through a field redefinition. This kind of terms are very common in theories with extra dimensions 10 , and showing they are only the effect of some discontinuities of the fields would improve our control of the theory. In our work we have shown that in some cases this is possible. \nIn summary in this paper we have shown how the Scherk-Schwarz mechanism works with discontinuous fields, in particular with bosonic fields. We have analyzed in detail the constraints on the generalized boundary conditions and we have shown the connection between the smooth framework and the discontinuous one in the case of one real scalar field. Then we have applied our mechanism to a Grand Unified Theory based on the SU (5) gauge group that breaks down to SU (3) × SU (2) × U (1) and we have shown how localized terms for bulk fields appear in the lagrangian. We have observed that these terms are strictly related to discontinuities of fields at the fixed points and that in some cases they can be reabsorbed through a field redefinition.", 'Acknowledgments': "I would like to thank Ferruccio Feruglio for the enjoyable and fruitful collaboration on which this talk is based and Fabio Zwirner for useful discussions. Many thanks go also to the organizers of the 'Rencontres de Moriond' for the pleasant and relaxed atmosphere of the conference. This work was partially supported by the European Program HPRN-CT-2000-00148 (network 'Across the Energy Frontier') and by the European Program HPRN-CT-2000-00149 (network 'Physics at Colliders').", 'References': '- 1. J. Scherk and J.H. Schwarz, Nucl. Phys. B 153 , 61 (1979); Phys. Lett. B 82 , 60 (1979).\n- 2. Y. Hosotani, Phys. Lett. B 126 , 309 (1983); Ann. Phys. 190 , 233 (1989).\n- 3. L.J. Dixon, J.A. Harvey, C. Vafa and E. Witten, Nucl. Phys. B 261 , 678 (1985); Nucl. Phys. B 274 , 285 (1986).\n- 4. P. Fayet, Phys. Lett. B 146 , 41 (1984).\n- 5. J.A. Bagger, F. Feruglio and F. Zwirner, Phys. Rev. Lett. 88 , 101601 (2002).\n- 6. C. Biggio and F. Feruglio, in preparation.\n- 7. A lot of models have been constructed on orbifolds; see for example:\n- R. Barbieri, L.J. Hall and Y. Nomura, Phys. Rev. D 63 , 105007 (2001); hep-ph/0106190 ; Nucl. Phys. B 624 , 63 (2002); hep-ph/0110102 ;\n- L.J. Hall, H. Murayama and Y. Nomura, hep-th/0107245 ;\n- L.J. Hall and Y. Nomura, Phys. Rev. D 64 , 055003 (2001); hep-ph/0111068 ; Phys. Lett. B 532 , 111 (2002); hep-ph/0205067 ;\n- A. Hebecker and J. March-Russel, Nucl. Phys. B 613 , 3 (2001); Nucl. Phys. B 625 , 128 (2002); hep-ph/0204037 ;\n- E.A. Mirabelli and M.E. Peskin, Phys. Rev. D 58 , 065002 (1998);\n- I. Antoniadis, Phys. Lett. B 246 , 377 (1990);\n- I. Antoniadis, C. Munoz and M. Quiros, Nucl. Phys. B 397 , 515 (1993);\n- A. Pomarol and M. Quiros, Phys. Lett. B 438 , 255 (1998);\n- I. Antoniadis, S. Dimopoulos, A. Pomarol and M. Quiros, Nucl. Phys. B 544 , 503 (1999);\n- A. Delgado, A. Pomarol and M. Quiros, Phys. Rev. D 60 , 095008 (1999); JHEP 0001 , 030 (2000);\n- A. Delgado and M. Quiros, Nucl. Phys. B 559 , 235 (1999); Phys. Lett. B 484 , 355 (2000); Nucl. Phys. B 607 , 99 (2001);\n- A. Delgado, G. von Gersdorff, P. John and M. Quiros, Phys. Lett. B 517 , 445 (2001);\n- A. Delgado, G. von Gersdorff and M. Quiros, Nucl. Phys. B 613 , 49 (2001);\n- V. Di Clemente, S.F. King and D.A.J. Rayner, Nucl. Phys. B 617 , 71 (2001); hep-ph/0205010 .\n- 8. Y. Kawamura, Prog. Theor. Phys. 105 , 999 (2001).\n- 9. G. Altarelli and F. Feruglio, Phys. Lett. B 511 , 257 (2001).\n- 10. H. Georgi, A.K. Grant and G. Hailu, Phys. Rev. D 63 , 064027 (2001); Phys. Lett. B 506 , 207 (2001);\n- N. Arkani-Hamed, L.J. Hall, Y. Nomura, D.R. Smith and N. Weiner, Nucl. Phys. B 605 , 81 (2001);\n- J.A. Bagger, F. Feruglio and F. Zwirner, JHEP 0202 , 010 (2002);\n- C.A. Scrucca, M. Serone, L. Silvestrini and F. Zwirner, Phys. Lett. B 525 , 169 (2002);\n- C.A. Scrucca, M. Serone and M. Trapletti, hep-th/0203190 ;\n- R. Barbieri, R. Contino, P. Creminelli, R. Rattazzi and C.A. Scrucca, hep-th/0203039 ; L. Pilo and A. Riotto, hep-th/0202144 ;\n- S. Groot Nibbelink, H.P. Nilles and M. Olechowski, hep-th/0203055 ; hep-th/0205012 .'}
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2001CQGra..18.2877G
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Logarithmic correction to the Bekenstein-Hawking entropy of the BTZ black hole
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2001-01-01
| 2 | 0.45 | 163 |
['-', '-']
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[]
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We derive an exact expression for the partition function of the Euclidean BTZ black hole. Using this, we show that for a black hole with large horizon area, the correction to the Bekenstein-Hawking entropy is -<SUP>3</SUP>/<SUB>2</SUB>log area), in agreement with that for the Schwarzschild black hole obtained in the four-dimensional canonical gravity formalism and also in a Lorentzian computation of BTZ black hole entropy. We find that the correct expression for the logarithmic correction in the context of the BTZ black hole comes from the modular invariance associated with the toroidal boundary of the black hole.
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[]
| 3 |
https://arxiv.org/pdf/gr-qc/0104010.pdf
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{'T. R. Govindarajan, R. K. Kaul and V. Suneeta 1': 'a The Institute of Mathematical Sciences, CIT Campus, Chennai 600113, India', 'Abstract': "We derive an exact expression for the partition function of the Euclidean BTZ black hole. Using this, we show that for a black hole with large horizon area, the correction to the Bekenstein-Hawking entropy is -3 / 2 log ( area ), in agreement with that for the Schwarzschild black hole obtained in the four dimensional canonical gravity formalism and also in a Lorentzian computation of BTZ black hole entropy. We find that the right expression for the logarithmic correction in the context of the BTZ black hole comes from the modular invariance associated with the toral boundary of the black hole. \nPACS No.: 04.60.-m, 04.60.Kz, 04.70.Dy \nWithin the quantum geometry formulation of gravity, it has been argued that the quantum degrees of freedom of a (3+1)-dimensional black hole can be described in terms of a ChernSimons theory on the horizon[1], [2]. For a (3+1)-dimensional Schwarzschild black hole, this allows an exact calculation of the entropy [2] which for a large horizon area yields, besides the usual Bekenstein-Hawking entropy proportional to the area, a next order log(area) correction with a definite numerical coefficient, -3 / 2 [3]. SU (2) Wess-Zumino conformal field theory on the boundary plays an important role in this calculation. There are other methods which can been employed to evaluate black hole entropy. Exploiting the nature of corrections to the Cardy formula for density of states in a two-dimensional conformal field theory, Carlip has evaluated this logarithmic correction in several black hole models including those from certain string theories[5]. Contrary to the expectations that these corrections may lead to distinguishing various formulations of quantum gravity, as emphasized by Carlip, the coefficient of logarithmic correction to the Bekenstein-Hawking entropy for large horizon area may have a universal character. The same value of the coefficient appears for a variety of black holes independent of the dimensions. In particular, the same correction was obtained for the entropy of the (2+1)-dimensional Lorentzian BTZ black hole [4] in [5] by studying the growth of states in the asymptotic conformal field theory at the boundary of the black hole spacetime. The semi-classical entropy of the BTZ black hole has been earlier obtained in different Lorentzian [6], [7] and Euclidean [8], [9] formulations of gravity. However, the correction term to semiclassical entropy seen in [5] has not been reproduced in the Euclidean path integral calculations for BTZ black hole, which is surprising. \nIn this paper, we derive an exact expression for the partition function of the BTZ black hole in the Euclidean path integral approach. In this framework, three-dimensional gravity with a negative cosmological constant is described in terms of two SU (2) Chern-Simons theories [10], [11]. Then, SU (2) Wess-Zumino conformal field theories are naturally induced on the boundary [12]. The quantum degrees of freedom corresponding to the entropy of the black hole are described by these conformal field theories. From the exact expression of the partition function, we show that there is indeed a correction to the semi-classical entropy that is proportional to the logarithm of the area (horizon length in this case) with a coefficient -3 / 2 again in agreement with the result for a four dimensional black hole obtained in ref. [3]. \nThe gravity action I grav written in a first-order formalism (using triads e and spin connection ω ) is the difference of two Chern-Simons actions. \nI grav = I CS [ A ] -I CS [ ¯ A ] , (1) \nwhere \nA = ( ω a + i l e a ) T a , ¯ A = ( ω a -i l e a ) T a (2) \nare SL(2 , C ) gauge fields (with T a = -iσ a / 2). Here, the negative cosmological constant Λ = -(1 /l 2 ). The Chern-Simons action I CS [ A ] is \nI CS = k 4 π ∫ M Tr ( A ∧ dA + 2 3 A ∧ A ∧ A ) (3) \nand the Chern-Simons coupling constant is k = l/ 4 G . We note that we can rewrite the Lorentzian gravity action also as a Chern-Simons theory, and in that case, the Chern-Simons coupling constant k = -l/ 4 G ; Lorentzian gravity is obtained from the Euclidean theory by a continuation G →-G . \nNow, for a manifold with boundary, the Chern-Simons field theory is described by a WessZumino conformal field theory on the boundary. Under the decomposition \nA = g -1 dg + g -1 ˜ Ag , (4) \nthe Chern-Simons action (3) becomes [14], [15] \nI CS [ A ] = I CS [ ˜ A ] + kI + WZW [ g, ˜ A z ] , (5) \nwhere I + WZW [ g, ˜ A z ] is the action of a chiral SU (2) Wess-Zumino model on the boundary ∂M , \nI + WZW [ g, ˜ A z ] = 1 4 π ∫ ∂M Tr ( g -1 ∂ z g g -1 ∂ ¯ z g -2 g -1 ∂ ¯ z g ˜ A z ) + 1 12 π ∫ M Tr ( g -1 dg ) 3 . (6) \nThe 'pure gauge' degrees of freedom g are now true dynamical degrees of freedom at the boundary. \nWe are interested in computing the entropy of the Euclidean BTZ black hole. The Euclidean continuation of the BTZ black hole has the topology of a solid torus [16]. The metric for the Euclidean BTZ black hole in the usual Schwarzschild-like coordinates is \nds 2 = N 2 dτ 2 + N -2 dr 2 + r 2 ( dφ + N φ dτ ) 2 (7) \nwhere τ here is the Euclidean time coordinate and \nN = ( -M + r 2 l 2 -J 2 4 r 2 ) 1 2 , N φ = -J 2 r 2 (8) \nThe inner and the outer horizons of the Lorentzian black hole solution get mapped in the Euclidean continuation to ir -and r + respectively, where \nr 2 ± = Ml 2 2 1 ± ( 1 + J 2 M 2 l 2 ) 1 / 2 (9) \nM and J are the mass and angular momentum of the black hole respectively. \nAs shown by Carlip and Teitelboim [16], after a coordinate transformation, \nx = ( r 2 -r 2 + r 2 -r 2 -) 1 / 2 cos ( r + l 2 τ + | r -| l φ ) exp { r + l φ -| r -| l 2 τ } \ny = ( r 2 -r 2 + r 2 -r 2 -) 1 / 2 sin ( r + l 2 τ + | r -| l φ ) exp { r + l φ -| r -| l 2 τ } (10) z = ( r 2 + -r 2 -r 2 -r 2 -) 1 / 2 exp { r + l φ -| r -| l 2 τ } \nThe black hole metric is just the metric for hyperbolic three-space H 3 \nds 2 = l 2 z 2 ( dx 2 + dy 2 + dz 2 ) , z > 0 , (11) \nChanging to spherical coordinates \nx = R cos θ cos χ, y = R sin θ cos χ, z = R sin χ , (12) \nthe metric is \nds 2 = l 2 R 2 sin 2 χ [ dR 2 + R 2 dχ 2 + R 2 cos 2 χdθ 2 ] , (13) \nHowever, one must make global identifications to account for the periodicity of the φ coordinate in (7). These are \n(ln R, θ, χ ) ∼ (ln R + 2 πr + l , θ + 2 π | r -| l , χ ) . (14) \nThe Euclidean BTZ black hole is obtained from hyperbolic space H 3 by these global identifications. \nUsing (2), the connection A a corresponding to the metric (13) may be written as: \nA 1 = -csc χ ( dθ -i dR R ) , A 2 = i csc χ dχ, A 3 = i cot χ ( dθ -i dR R ) . (15) \nThe Chern-Simons formulation of gravity was used to describe the BTZ black hole first in [13], where for the Lorentzian black hole, the corresponding gauge fields were evaluated. \nIn order to compute the black hole partition function, we first evaluate the Chern-Simons path integral on a solid torus. This path integral has been discussed in [14], [17], [18] and [19]. Through a suitable gauge transformation, the connection is set to a constant value on the toral boundary. In terms of coordinates on the toral boundary x and y with unit period, we can define z = ( x + τy ) such that \n∫ A dz = 1 , ∫ B dz = τ (16) \nwhere A is the contractible cycle and B the non-contractible cycle of the solid torus and τ = τ 1 + iτ 2 is the modular parameter of the boundary torus. Then, the connection can be written as [17]: \nA = ( -iπ ˜ u τ 2 d ¯ z + iπu τ 2 dz ) T 3 (17) \nwhere u and ˜ u are canonically conjugate fields and obey the canonical commutation relation: \n[˜ u, u ] = 2 τ 2 π ( k +2) (18) \nThey can be related to the black hole parameters by computing the holonomies of A around the contractible and non-contractible cycles of the solid torus. These holonomies have been computed in [16] for the general case of a rotating BTZ black hole solution with a conical singularity (Θ) at the horizon such that the identifications (14) characterizing the black hole now generalize to \n(ln R, θ, χ ) ∼ (ln R, θ + Θ , χ ) (ln R, θ, χ ) ∼ (ln R + 2 πr + l , θ + 2 π | r -| l , χ ) (19) \nThe former identification corresponds to the A cycle and the latter to the B cycle. Then the trace of the holonomies around the contractible cycle A and non-contractible cycle B are: \nTr ( H A ) = 2cosh( i Θ) , Tr ( H B ) = 2cosh ( 2 π l ( r + + i | r -| ) ) (20) \nFor the classical black hole solution, Θ = 2 π . From (17), \nA z = -iπ τ 2 ˜ u, A ¯ z = iπ τ 2 u (21) \nwhere \nu = -i 2 π ( -i Θ τ + 2 π ( r + + i | r -| ) l ) , ˜ u = -i 2 π ( -i Θ¯ τ + 2 π ( r + + i | r -| ) l ) (22) \nWe note here that ˜ u is the canonical conjugate, but not the complex conjugate of u . This is so because A is a complex SU (2) connection. \nNow, we write the Chern-Simons path integral on a solid torus with a boundary modular parameter τ . For a fixed boundary value of the connection, i.e. a fixed value of u , this path integral is formally equivalent to a state ψ 0 ( u, τ ) with no Wilson lines in the solid torus. The states corresponding to having closed Wilson lines (along the non-contractible cycle) carrying spin j/ 2 ( j ≤ k ) representations in the solid torus are given by [14], [17], [18], [19]: \nψ j ( u, τ ) = exp { πk 4 τ 2 u 2 } χ j ( u, τ ) , (23) \nwhere χ j are the Weyl-Kac characters for affine SU(2). The Weyl-Kac characters can be expressed in terms of the well-known Theta functions as \nχ j ( u, τ ) = Θ ( k +2) j +1 ( u, τ, 0) -Θ ( k +2) -j -1 ( u, τ, 0) Θ 2 1 ( u, τ, 0) -Θ 2 -1 ( u, τ, 0) (24) \nwhere Theta functions are given by: \nΘ k µ ( u, τ, z ) = exp( -2 πikz ) ∑ n ∈Z exp 2 πik [ ( n + µ 2 k ) 2 τ + ( n + µ 2 k ) u ] (25) \nThe black hole partition function is to be constructed from the boundary state ψ 0 ( u, τ ). To do that, we note the following: \n- a) We must first choose the appropriate ensemble. Here, we choose the microcanonical ensemble. This corresponds in our picture, to keeping the holonomy around the non-contractible cycle B fixed [20]. The holonomy around the contractible cycle A is Θ, which has a value 2 π for the classical solution. But it is not held fixed any more, and we must sum over contributions to the partition function from all values of Θ. This corresponds to starting with the value of u for the classical solution, i.e. with Θ = 2 π in (20), and then considering all other shifts of u of the form \nu → u + ατ (26) \nwhere α is an arbitrary number. This is implemented by a translation operator of the form \nT = exp ( ατ ∂ ∂u ) (27) \nHowever, this operator is not gauge invariant. The only gauge-invariant way of implementing these translations is through Verlinde operators of the form \nW j = ∑ n ∈ Λ j exp ( -nπ ¯ τu τ 2 + nτ k +2 ∂ ∂u ) (28) \nwhere Λ j = -j, -j +2 , ..., j -2 , j . This means that all possible shifts in u are not allowed, and from considerations of gauge invariance, the only possible shifts are \nu → u + nτ k +2 (29) \nwhere n is always an integer taking a maximum value of k . Thus, the only allowed values of Θ are 2 π (1 + n k +2 ). We know that acting on the state with no Wilson lines in the solid torus with the Verlinde operator W j corresponds to inserting a Wilson line of spin j/ 2 around the non-contractible cycle. Thus, taking into account all states with different shifted values of u as \nin (29) means that we have to take into account all the states in the boundary corresponding to the insertion of such Wilson lines. These are the states ψ j ( u, τ ) given in (23). \nb) In order to obtain the final partition function, we must integrate over all values of the modular parameter, i.e. over all inequivalent tori with the same holonomy around the non-contractible cycle. The integrand, which is a function of u and τ , must be the square of the partition function of a gauged SU (2) k Wess-Zumino model corresponding to the two SU (2) Chern-Simons theories. It must be modular invariant - modular invariance corresponds to large diffeomorphisms of the torus, and the partition function must not change under a modular transformation. \nThe partition function is then of the form \nZ = ∫ dµ ( τ, ¯ τ ) ∣ ∣ ∣ ∣ ∣ k ∑ j =0 a j ( τ ) ψ j ( u, τ ) ∣ ∣ ∣ ∣ ∣ ∣ 2 (30) \n∣ \n∣ where dµ ( τ, ¯ τ ) is the modular invariant measure, and the integration is over a fundamental domain in the τ plane. Coefficients a j ( τ ) must be chosen such that the integrand is modular invariant. Under an S modular transformation, τ →-1 /τ and u → u/τ , the SU (2) k characters transform as \nχ j ( u, τ ) → exp ( -2 πik u 2 4 τ ) χ j ( u τ , -1 τ ) = exp ( -2 πik u 2 4 τ ) ∑ l S jl χ l ( u, τ ) (31) \nwhere matrix S jl given by \nS jl = √ 2 k +2 sin [ π ( j +1)( l +1) k +2 ] , 0 ≤ j, l ≤ k (32) \nWe note here that this S matrix is orthogonal: ∑ j S lj S jp = δ lp . \nWe are interested in the transformation property of the state ψ j ( u, τ ) under an S modular transformation. The prefactor in (23) transforms into itself under such a transformation apart from an extra piece that exactly cancels the prefactor in (31). Thus, under an S transformation ( τ →-1 /τ ), \nψ j ( u, τ ) → ∑ l S jl ψ l ( u, τ ) (33) \nUnder a T modular transformation ( τ → τ +1), ψ j ( u, τ ) picks up a phase, \nψ j ( u, τ ) → exp(2 πim j ) ψ j ( u, τ ) (34) \nwhere m j = ( j +1) 2 2( k +2) -1 4 . For the integrand in (30) to be modular invariant, the coefficient a j ( τ ) must transform under the S transformation as a j ( τ ) → ∑ p a p ( τ ) S pj and under the T transformation as a j ( τ ) → exp( -2 πim j ) a j ( τ ). Further, since the integrand must correspond to the square of the partition function of a gauged SU (2) k Wess-Zumino model, the coefficients a j ( τ ) are just the complex conjugate of SU (2) k characters corresponding to u = 0, i.e ( ψ j (0 , τ )) ∗ . The black hole partition function therefore is \nZ bh = ∫ dµ ( τ, ¯ τ ) ∣ ∣ ∣ ∣ ∣ k ∑ j =0 ( ψ j (0 , τ )) ∗ ψ j ( u, τ ) ∣ ∣ ∣ ∣ ∣ ∣ 2 (35) \n∣ \nFinally the modular invariant measure is \ndµ ( τ, ¯ τ ) = dτd ¯ τ τ 2 2 (36) \nThe expression (35) is an exact expression for the partition function of Euclidean black hole. To make a comparison with the semi-classical entropy of black hole, we evaluate the expression (35) for large horizon radius r + by the saddle-point method. Substituting from (23), (24) and (25), the saddle point of the integrand occurs when τ 2 is proportional to r + and therefore large. But for τ 2 large, the character χ j is \nχ j ( τ, u ) ∼ exp πi ( ( j +1) 2 k +2 -1 2 ) 2 τ sin π ( j +1) u sin πu (37) \nWe now use in (35) the form of the character for large τ 2 from (37). In the expression for u in (20), we replace Θ by its classical value 2 π . The computation has been done with positive coupling constant k and at the end, we must perform an analytic continuation to the Lorentzian black hole, by taking G →-G . It can be checked that after the analytic continuation, it is the spin j = 0 in the sum over characters in (35) that dominates the partition function. \nWe obtain the leading behaviour of the partition function (35) for large r + (and when | r -| << r + ) by first performing the integration over τ 1 in this regime. The τ 2 integration is done by the method of steepest descent. The saddle-point is at τ 2 = r + /l . Expanding around the saddle-point, by writing τ 2 = r + /l + x and then integrating over x , we obtain \nZ bh = l 2 r 2 + exp ( -2 πkr + l ) ∫ dx exp [ -πkl 2 r + x 2 ] (38) \nThe integration produces a factor proportional to √ r + . The partition function for the Lorentzian black hole of large horizon area 2 πr + after the analytic continuation G →-G is then \nZ Lbh = l 2 r 2 + √ 8 r + G πl 2 exp ( 2 πr + 4 G ) (39) \nupto a multiplicative constant. The logarithm of this expression yields the black hole entropy for large horizon length r + : \nS = 2 πr + 4 G -3 2 log ( 2 πr + 4 G ) + . . . . (40) \nThe leading contribution to the black hole entropy is the familiar Bekenstein-Hawking term. The next-order correction to the semi-classical entropy is the logarithm of the black hole area 2 πr + . The coefficient -3 / 2 of this correction is in agreement with that of the logarithmic correction of semi-classical entropy of four dimensional Schwarzschild black hole first observed in ref. [3] in the quantum geometry formulation of gravity. The semi-classical Bekenstein-Hawking entropy for Euclidean BTZ black hole was previously studied in the path integral formulation in ref. [8], but the logarithmic correction was not seen there. As described above, the right logarithmic correction is obtained by considering the correct modular invariant measure while integrating over all inequivalent tori (as the holonomy around the non-contractible cycle is held fixed). \nThe calculation presented here should be contrasted with an earlier calculation of partition function of a BTZ black hole coupled to a scalar field [21]. This is a perturbative one-loop calculation which incorporates a specific type of fluctuation, namely a scalar field. For small r + , this leads to a different coefficient of the the logarithmic correction in the entropy. On the other hand, our calculation is exact; it includes all possible quantum gravity fluctuations. It is therefore not surprising that the results differ. \nFinally, we make a few remarks on the AdS gas partition function. As is well known [22], the action for the AdS gas can be obtained from that of the BTZ black hole by a transformation. For the case of a non-rotating black hole, this transformation has the form r + /l → l/r + . With this change, the AdS gas partition function is \nZ AdS [ r + ] = ∫ dµ ( τ, ¯ τ ) ∣ ∣ ∣ ∣ ∣ ∣ k ∑ j =0 ( ψ j (0 , τ )) ∗ ψ j ( u ' , τ ) ∣ ∣ ∣ ∣ ∣ ∣ 2 (41) \nwhere u ' = -i 2 π ( -i 2 πτ + 2 πl r + ) . \nThe AdS gas partition function can again be evaluated by saddle-point method. Small r + leads to a saddle-point with τ 2 large. In this limit of small r + (i.e small temperature), the partition function is \nZ AdS [ r + ] = ( r + l ) 3 2 exp ( 2 πl 2 4 r + G ) (42) \nThis, at the leading order, agrees with the corresponding expression obtained in ref. [22]. \nV.S. would like to thank S. Carlip for useful discussions. We would also like to thank N. D. Hari Dass for comments and discussions on the results of our paper.", 'References': '- [1] A. Ashtekar, J. Baez, A. Corichi and K. Krasnov, Phys.Rev.Lett. 80 , 904 (1998).\n- [2] R. K. Kaul and P. Majumdar, Phys. Lett. B439 , 267 (1998).\n- [3] R. K. Kaul and P. Majumdar, Phys. Rev. Lett. 84 , 5255 (2000); S. Das, R. K. Kaul and P. Majumdar, Phys. Rev. D63 , 044019 (2001).\n- [4] M. Banados, C. Teitelboim and J. Zanelli, Phys.Rev.Lett. 69 , 1849 (1992); M. Banados, M. Henneaux, C. Teitelboim and J. Zanelli, Phys. Rev. D48 , 1506 (1993).\n- [5] S. Carlip, Class. Quant. Grav. 17 , 4175 (2000).\n- [6] A. Strominger, JHEP 9802 , 009 (1998).\n- [7] S. Carlip, Nucl. Phys. Proc. Suppl. 88 , 10 (2000).\n- [8] S. Carlip, Phys. Rev. D55 , 878 (1997).\n- [9] V. Suneeta, R. K. Kaul and T. R. Govindarajan, Mod.Phys.Lett.A 14 , 349 (1999).\n- [10] A. Achucarro and P. Townsend, Phys.Lett. B 180 , 89 (1986).\n- [11] E. Witten, Nucl.Phys.B 311 , 46 (1988); S. Carlip, Class. Quant. Grav. 16 , 3327 (1999).\n- [12] E. Witten, Commun. Math. Phys. 121 , 351 (1989); R.K. Kaul, Commun. Math. Phys. 162 , 289 (1994); R.K. Kaul, Chern-Simons theory, knot invariants, vertex models and three manifold invariants, hep-th/9804122, published in Frontiers of Field Theory, Quantum Gravity and Strings (Horizons in World Physics, Vol. 227) , NOVA Science Publishers, New York, (1999).\n- [13] D. Cangemi, M. Leblanc and R. Mann, Phys. Rev. D48 , 3606 (1993).\n- [14] S. Elitzur, G. Moore, A. Schwimmer and N. Seiberg, Nucl. Phys. B 326 , 108 (1989).\n- [15] W. Ogura, Phys. Lett. B229 , 61 (1989); S. Carlip, Nucl. Phys. B 362 , 111 (1991).\n- [16] S. Carlip and C. Teitelboim, Phys. Rev. D51 , 622 (1995).\n- [17] J. M. F. Labastida and A. V. Ramallo, Phys. Lett. B 227 , 92 (1989). \n- [18] J. M. Isidro, J. M. F. Labastida and A. V. Ramallo, Nucl. Phys. B 398 , 187 (1993).\n- [19] N. Hayashi, Prog. Theor. Phys. Suppl. 114 , 125 (1993).\n- [20] J. D. Brown, G. L. Comer, E. A. Martinez, J. Melmed, B. F. Whiting and J. W. York, Class. Quant. Grav. 7 , 1433 (1990).\n- [21] R. Mann and S. Solodukhin, Phys. Rev. D55 , 3622 (1997).\n- [22] J. Maldacena and A. Strominger, JHEP 9812 005 (1998).'}
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2005ApJ...623...23S
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The Gravitational Wave Signal from Massive Black Hole Binaries and Its Contribution to the LISA Data Stream
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2005-01-01
| 8 | 0.48 | 163 |
['black hole physics', 'cosmology theory', 'cosmology early universe', 'gravitational waves', 'relativity', 'astrophysics']
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Massive black hole binaries, with masses in the range 10<SUP>3</SUP>-10<SUP>8</SUP> M<SUB>solar</SUB>, are expected to be the most powerful sources of gravitational radiation at mHz frequencies, and hence are among the primary targets for the planned Laser Interferometer Space Antenna (LISA). We extend and refine our previous analysis, detailing the gravitational wave signal expected from a cosmological population of massive black hole binaries. As done in our previous paper, we follow the merger history of dark matter halos, the dynamics of the massive black holes they host, and their growth via gas accretion and binary coalescences in a ΛCDM cosmology. Stellar dynamical processes dominates the orbital evolution of black hole binaries at large separations, while gravitational wave emission takes over at small radii, causing the final coalescence of the pairs. We show that the GW signal from this population, in a 3 yr LISA observation, will be resolved into ~=90 discrete events with S/N>=5, among which ~=35 will be observed above threshold until coalescence. These ``merging events'' involve relatively massive binaries, M~10<SUP>5</SUP> M<SUB>solar</SUB>, in the redshift range 2<~z<~6. The remaining ~=55 events come from higher redshift, less massive binaries (M~5×10<SUP>3</SUP> M<SUB>solar</SUB> at z>~6) and, although their S/N integrated over the duration of the observation can be substantial, the final coalescence phase is at too high a frequency to be directly observable by space-based interferometeres such as LISA. LISA will be able to detect a fraction >~90% of all the coalescences of massive black hole binaries occurring at z<~5. The residual confusion noise from unresolved massive black hole binaries is expected to be at least an order of magnitude below the estimated stochastic noise.
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https://arxiv.org/pdf/astro-ph/0409255.pdf
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{'THE GRAVITATIONAL WAVE SIGNAL FROM MASSIVE BLACK HOLE BINARIES AND ITS CONTRIBUTION TO THE LISA DATA STREAM': 'Alberto Sesana 1 , Francesco Haardt 1 , Piero Madau 2 , & Marta Volonteri 2 ApJ, in press', 'ABSTRACT': "Massive black hole binaries, with masses in the range 10 3 -10 8 M /circledot , are expected to be the most powerful sources of gravitational radiation at mHz frequencies, and hence are among the primary targets for the planned Laser Interferometer Space Antenna ( LISA ). We extend and refine our previous analysis (Sesana et al. 2004), detailing the gravitational wave signal expected from a cosmological population of massive black hole binaries. As done in our previous paper, we follow the merger history of dark matter halos, the dynamics of the massive black holes they host, and their growth via gas accretion and binary coalescences in a ΛCDM cosmology. Stellar dynamical processes dominates the orbital evolution of black hole binaries at large separations, while gravitational wave emission takes over at small radii, causing the final coalescence of the pairs. We show that the GW signal from this population, in a 3 year LISA observation, will be resolved into /similarequal 90 discrete events with S/N ≥ 5, among which /similarequal 35 will be observed above threshold until coalescence. These 'merging events' involve relatively massive binaries, M ∼ 10 5 M /circledot , in the redshift range 2 ∼ < z ∼ < 6. The remaining /similarequal 55 events come from higher redshift, less massive binaries ( M ∼ 5 × 10 3 M /circledot at z ∼ > 6) and, although their S/N integrated over the duration of the observation can be substantial, the final coalescence phase is at too high frequency to be directly observable by space-based interferometers such as LISA . LISA will be able to detect a fraction ∼ > 90% of all the coalescences of massive black hole binaries occurring at z ∼ < 5. The residual confusion noise from unresolved massive black hole binaries is expected to be at least an order of magnitude below the estimated stochastic noise. \nSubject headings: black hole physics - cosmology: theory - early universe - general relativity gravitational waves", '1. INTRODUCTION': "Gravitational radiation, described as a tensor perturbation to the metric travelling at the speed of light, is a natural consequence of Einstein's general relativity. It has been recognised (e.g., Thorne 1987) that black hole binaries are among the most important sources of gravitational waves (GW), both for ground based interferometers such as LIGO (Abramovici et al. 1992) and VIRGO (Bradaschia et al. 1990), and for the planned Laser Interferometer Space Antenna ( LISA , Bender et al. 1994). \nInterferometers operate as all-sky monitors, and the data streams collect the contributions from a large number of sources belonging to different cosmic populations. A precise determination of stochastic GW backgrounds from different classes of astrophysical objects is therefore crucial to interpret the data. While a GW background may provide information on the number density, redshift evolution, and mass function of the emitting population, confusion noises add to the instrumental noise limiting the possibility of detecting other class of objects. Moreover, to optimize the subtraction of resolved sources from the data stream, it is important to have a detailed description of the expected rate, duration, amplitude, and waveforms of events. \nLISA will operate in the frequency range 0.01 mHz - 1 Hz, where GW emission from a cosmological population of massive black hole binaries (MBHBs) is expected to be important (Haehnelt 1994). Today, massive black holes \n(MBHs) are ubiquitous in the nuclei of nearby galaxies (see, e.g., Magorrian et al. 1998). If MBHs were also common in the past, and if their host galaxies experience multiple mergers during their lifetime, as dictated by popular cold dark matter hierarchical cosmologies, then MBHBs will inevitably form in large numbers during cosmic history. The formation and evolution of MBHs has been investigated recently by several groups (e.g., Menou, Haiman & Narayanan 2001; Volonteri et al. 2003), and the expected GW signal from inspiraling MBH binaries has been first discussed by Rajagopal & Romani (1995), and recently by Jaffe & Backer (2003), Wyithe & Loeb (2003), Sesana et al. (2004, hereafter Paper I), and Enoki et al. (2004). \nIn Paper I we computed the GW background from MBHBs and the number of coalescences observable by LISA in a 3-year mission, adopting the scenario for the assembly and growth of MBHs proposed by Volonteri et al. (2003a,b). In such model, 'seed' holes are placed within rare high-density regions (minihalos) above the cosmological Jeans and cooling mass at redshift 20. Their evolution is followed through Monte Carlo realizations of the halo merger hierarchy combined with semi-analytical descriptions of the main dynamical processes, such as dynamical friction against the dark matter background, the shrinking of MBH binaries via three-body interactions, their coalescence driven by the emission of gravitational waves, and the recoil associated with the non-zero net linear momen- \ntum carried away by GWs in the coalescence of two unequal mass black holes (the 'gravitational rocket'). Major halo mergers lead to MBH fueling and trigger quasar activity. In this paper we use the same model to provide a more detailed characterization of the GW signal from inspiraling MBHBs. Their contribution to the LISA data stream is twofold: unresolved sources will give origin to confusion noise to be compared to instrumental noise and other astrophysical stochastic backgrounds (e.g. from white dwarf binaries, Farmer & Phinney 2003), while resolved inspiraling binaries will probe gravity in extreme conditions (e.g., Vecchio 2004). Confusion noise and resolved sources should provide different cosmological information. The former, produced by a large number of unresolved MBHBs, will trace light MBHBs at very high redshift, placing constraints on black hole formation scenarios prior to the reionization epoch; the latter will be a formidable tool to follow the cosmic evolution of MBHs and the formation and dynamics of MBH binaries following galaxy mergers. \nThe plan is as follows. In § 2 we review the basics of the detection of GW from MBHBs, defining observable quantities such as the characteristic strain amplitude, signalto-noise ratio, and source detection rate. In § 3 we briefly summarize our scenario for the cosmological evolution of galaxy halos and associated holes. In § 4 we present confusion noise levels and source number counts. Finally, in § 5 we discuss our results.", '2.1. Bursts and periodic events': "Following Thorne (1996), an interferometer can be characterized by two different sensitivity curves, depending on the type of signal one expects to detect, i.e. a 'burst' or a 'periodic' GW source. A burst, a short-lived signal whose waveform can be utterly complicated, can be described in terms of a characteristic strain amplitude h c at the observed frequency f c ∼ 1 / ∆ t s , where ∆ t s is the duration of the signal (Thorne 1987). The spread of the power spectrum around f c will be ∆ f ∼ f c . At the other extreme, a perfectly periodic source emits, for the entire duration of the observation, at a fixed frequency f . The power spectrum will be peaked at f , with a spread ∆ f /similarequal f/N , where N is the number of wave cycles clipped into the observation. In this respect, a burst can be thought as a single complete waveform with f = f c . In the case of a periodic signal, the interferometer sensitivity is increased by the fact that, across the observing interval τ , the signal is repeated fτ times. \nThe sensitivity to bursts ( h B ) and to periodic signals ( h P ) are related by: \nh P ( f ) = h B ( f ) √ fτ . (1) \nIn Figure 1 the two curves h B and h P are compared for an assumed 3-year LISA observation. The curves are obtained combining the LISA single-arm Michelson sensitivity curve (taken from the URL www.srl.caltech.edu/ ∼ shane/sensitivity) with the recent analysis of the LISA instrumental noise below 10 -4 Hz (Bender 2003, extended from 3 × 10 -6 Hz to 1 × 10 -6 Hz with a constant slope). \nConsider now a periodic signal of finite duration, with strain amplitude h . The total energy carried by the wave will be proportional to the number of wave cycles n spent at that particular frequency. The quantity to be compared with h B is then the 'characteristic' strain h c ≡ h √ n . \nFig. 1.LISA single-arm Michelson sensitivity curve to bursts ( thick solid line ) and periodic signals ( thick dashed line ) in a 3year mission. Data are obtained from www.srl.clatech.edu/ ∼ shane/sensitivity, and Bender (2003). The strain h (dashed lines) and characteristic strain h c (solid lines) for a MBHB with M 2 = 0 . 1 M 1 = 10 5 M /circledot at z = 1 (upper lines), and M 2 = M 1 = 10 3 M /circledot at z = 7 (lower lines) are also shown. \n<!-- image --> \nNote that for a periodic signal at frequency f lasting for a time interval longer than the observation time τ , we have simply n = fτ . Then, the signal-to-noise ratio S/N increases by the same factor one would obtain comparing h to h P in equation (1). The former approach, i.e. comparing h c to h B rather than h to h P , is more general, as it allows us to characterize the S/N not only for perfectly periodic signals ( n = fτ ), or for bursts ( n = 1), but also for events in which the emitted frequency shifts to increasingly larger values during the spiral-in phase of the binary system. In the latter case, n = n ( f ) represents the number of cycles spent in a frequency interval ∆ f /similarequal f around frequency f , and hence h c is the strain in a logarithmic frequency interval (Flanagan & Hughes 1998). Typically, the timescale for frequency shift is long compared to the wave period, and short compared to the duration of the observation. Only close to the innermost stable circular orbit (ISCO), the GW frequency changes at a rate comparable to the frequency itself ( n ∼ 1 and hence h c ∼ h ). In Figure 1 we also show h and h c for two representative binary systems. One should note that the true observable GW signal is, for f > n/τ (the 'knee' frequency in the h c curves), lower than h , as for these high frequencies the source is not monochromatic over the observation time. \nSuch simply consideration naturally leads to define h c and h B .", '2.2. Characteristic strain': "Consider now a binary system at comoving distance r ( z ). The strain amplitude (sky-and-polarisation averaged) at the rest-frame frequency f r is \nh = 8 π 2 / 3 10 1 / 2 G 5 / 3 M 5 / 3 c 4 r ( z ) f 2 / 3 r , (2) \nwhere M = ( M 1 M 2 ) 3 / 5 / ( M 1 + M 2 ) 1 / 5 is the 'chirp mass' of the binary, and all the other symbols have their standard meaning. The strain is averaged over a wave period. The rest-frame energy flux (energy per unit area per unit time) associated to the GW is \ndE dAdt = π 4 c 3 G f 2 r h 2 . (3) \nAs discussed above, the important quantity to consider is the number of cycles spent in a frequency interval ∆ f /similarequal f around a given frequency f . Assuming that the backreaction from GW emission dominates the orbital decay of a binary, during the spiral-in phase one can write \nn /similarequal f 2 r / ˙ f r = 5 96 π 8 / 3 c 5 G 5 / 3 M 5 / 3 f -5 / 3 r , (4) \nwhere we have used the rest-frame frequency shift rate \n˙ f r = df r dt r = 96 π 8 / 3 G 5 / 3 5 c 5 M 5 / 3 f 11 / 3 r . (5) \nNote that n can be computed either in the rest or in the observer frame. The characteristic strain in an observation of (observed) duration τ is then \nh c = h √ n /similarequal 1 3 1 / 2 π 2 / 3 G 5 / 6 M 5 / 6 c 3 / 2 r ( z ) f -1 / 6 r , n < fτ, (6) \nand \nh c = h fτ ∝ f 7 / 6 r , n > fτ, (7) \nwhere f = f r / (1 + z ) is the observed frequency. Using Parseval theorem, it is easy to see that h c is related to the Fourier transform of the strain ˜ h , as h 2 c = 2 f 2 r ˜ h 2 ( f r ), where ˜ h is defined over the positive frequency axis. The specific energy per unit area is then \n√ \ndE dAdf r = π 4 c 3 G h 2 c , (8) \nand, from equation (6), we obtain \ndE df r = π 2 / 3 3 G 2 / 3 M 5 / 3 f -1 / 3 r . (9) \nNote that dE/df r ∝ f -1 / 3 r , while (eq. 3) dE/dt ∝ f 10 / 3 r .", '2.3. Signal-to-noise ratio': "In an operating interferometer, any stochastic signal will add up (in quadrature) to h B to form the effective rms noise of the instrument, h rms . An inspiraling binary is then detected if the signal-to-noise ratio integrated over the observation is larger than the assumed threshold for detection, where the integrated S/N is given by \nS/N ∆ f = √ ∫ f +∆ f f d ln f ' [ h c ( f ' r ) h rms ( f ' ) ] 2 . (10) \nHere, f is the (observed) frequency emitted at the starting time t = 0 of the observation, and ∆ f is the (observed) frequency shift in a time τ starting from f . The latter is implicitly given by \nτ = ∫ f +∆ f f df ' ˙ f ' . (11) \nwhere df/ ˙ f = (1 + z ) df r / ˙ f r . The frequency at the ISCO is, strictly speaking, defined only in the test particle limit M 2 /lessmuch M 1 . In the general case, various estimate of the transition point from in-spiral to plunge exist, and differ by a factor of 3 at most (e.g., Kidder, Will & Wiseman 1993; Cook 1994). Such uncertainties do not affect our results in any manner, so we use, for the observed frequency at the ISCO, the conventional Keplerian defintion: \nf ISCO = c 3 6 3 / 2 πG 1 ( M 1 + M 2 ) (1 + z ) -1 . (12) \nReplacing f +∆ f with f ISCO in equation (11) gives τ ISCO , the time needed to span the frequency interval [ f, f ISCO ], to be compared a priori to τ . In the case τ > τ ISCO , we then set f +∆ f = f ISCO in equation (10). In Figure 2 we plot h c for different MBHBs at different redshifts, compared to the LISA h rms (see § 4.1) multiplied by a factor of 5, assuming a 3-year observation. If h c > 5 h rms , then the signal has, approximately, an integrated S/N > 5. This is for illustrative purposes only, as the actual S/N must be integrated over the observing period using equation (10). \nAt frequencies higher than the 'knee', the time spent around a given frequency is less than 3 years, and h c ∝ f -1 / 6 . The signal shifts toward higher frequency during the observation, and reaches the ISCO and the coalescence phase in most cases. The lowest curve represents a low mass, high redshift equal mass binary. As we shall see below, these sources are common in our hierarchical model for MBH assembly. In terms of their detectability by LISA , they represent a somewhat different class of events. Contrary to the case of more massive binaries present at lower z , the final coalescence phase of light binaries lies at too high frequecies, well below the LISA threshold. \nFor frequencies much below the knee, the characteristic strain is proportional to f 7 / 6 , as the timescale for frequency shift is longer than 3 years. The signal amplitude is then limited by the observation time, not by the intrinsic properties of the source. The source will be observed as a 'stationary source', a quasi-monochromatic wave for the whole duration of the observation. An increase in the observation time will result in a shift of the knee toward lower frequencies. The time needed for the sources to reach the ISCO starting from the knee frequency is, approximatively, \nthe observing time. Figure 2 shows that very few stationary sources above threshold should be expected anyway. \nFig. 2.Characteristic strain h c for MBHBs with different masses and redshifts. From top to bottom, the first three curves refer to systems with log( M 1 /M /circledot ) = 7 , 6 , 5, respectively, and M 2 = 0 . 1 M 1 . The solid, long-dashed, and short-dashed lines assumes the binary at z = 1 , 3 , 5, respectively. A 3-year observation is considered. The lowest solid curve assumes an equal mass binary M 1 = M 2 = 10 3 M /circledot at z = 7. The small diamonds on each curve mark, from left to right, the observed frequency at 1 year, 1 month and 1 day before coalescence. The thick curve is LISA 5 h rms (see § 4.1), approximatively the threshold for detection with S/N ≥ 5. \n<!-- image -->", '2.4. Coalescence rate and number counts': 'Given a coalescence rate R , using the frequency shift rate ˙ f (eq. 5), we can solve for the mean number of individual binaries resolved during an observing period τ . We begin considering that a MBHB spans, during its lifetime, a finite frequency range, f min < f < f ISCO , where the lower limit is set to the observed frequency at the hardening radius (Quinlan 1996). Then, from continuity, the number of individual observable MBHBs can be computed as \nN τ = R ∫ f ISCO f min df ˙ f + Rτ. (13) \nThe first term is simply the integrated density of sources in the frequency domain, and does not depend on τ . It is the number of sources caught in a snapshot of the entire sky. The second term is the number of new binaries born (at frequency f min ) during the observation time τ , and must be equal to the number of coalescences within the same period. \nThe general argument above does not consider that real detections must be above a specified minimum S/N , where the S/N is given by equation (10). Including a threshold criterium, the number of MBHBs with S/N > s in an \nobservation of duration τ is then \nN τ ( > s ) = R ∫ f ISCO f min df ˙ f H s (∆ f ) + R ∫ f max f min df ˙ f H s (∆ f min ) , (14) \nwhere \nH s (∆ f ) = { 1 , S/N ∆ f ≥ s 0 , S/N ∆ f < s . (15) \nIn the second term of equation (14), which again accounts for the new binaries formed at the hardening radius, f max is the frequency reached after 3 years starting from f min , and the function H s (∆ f min ) is evaluated by integration of the S/N from f min to f . Given the exceedingly low value of f min , this second term is totally negligible for an experiment such as LISA .', '3. HIERARCHICAL GROWTH OF MASSIVE BLACK HOLES': "The theory and method outlined in the previous sections allow us to fully characterize the expected contribution of MBHBs in the spiral-in phase to the LISA data stream, once the coalescence rate of MBHBs is specified. In this work a hierarchical structure formation scenario for the assembly and growth of MBHs in which seed holes form far up in the dark halo 'merger tree' is assumed. We use exactly the same model discussed in Volonteri et al. (2003a, 2003b) and in Paper I. Its main features are briefly summarized in this section. \nWe track backwards the merger history of 220 parent halos with present-day masses in the range 10 11 -10 15 M /circledot with a Monte Carlo algorithm based on the extended Press-Schechter formalism (see, e.g., Cole et al. 2000). Seed holes with m seed = 150M /circledot are placed within rare high-density regions (minihalos) above the cosmological Jeans and cooling mass at redshift 20. Their evolution and growth is followed through Monte Carlo realizations of the halo merger hierarchy combined with semi-analytical descriptions of the main dynamical processes, such as dynamical friction against the dark matter background, the shrinking of MBHBs via three-body interactions, their coalescence from the emission of gravitational waves, triple MBH interactions, and the effect of gravitational recoil. \nFig. 4.Estimated LISA rms confusion noise ( solid line ), as the quadratic sum of the LISA instrumental single-arm Michelson noise h B ( dotted line ), the confusion noise from unresolved galactic (Nelemans et al. 2001, long-dashed line ), and extragalactic (Farmer & Phinney 2003, short-dashed line ) WD-WD binaries, and our estimate of the confusion noise from unresolved MBHBs ( thick-solid line ). \n<!-- image --> \nFig. 3.Number of coalescences of MBHBs observed per year at z = 0 per unit redshift. Our fiducial rate ( thick solid line ) is compared to a case in which the hardening timescale is increased by a factor of 3 ( dotted line ) or reduced by the same factor ( dashed line ). \n<!-- image --> \nQuasar activity is triggered during major mergers. We assume that the more massive hole accretes, at the Eddington rate, a gas mass fraction that scales with the fifth power of the host halo circular velocity (Ferrarese 2002). \nIn a typical merger event, dynamical friction drives the satellite halo toward the centre of the new forming system, leading to the formation of a bound MBHB in the violently relaxed stellar core. As the binary separation decays, the effectiveness of dynamical friction slowly declines; the bound pair then hardens by capturing stars passing within a distance of the order of the binary semi-major axis and ejecting them at much higher velocities (gravitational slingshot). The heating of the surrounding stars by a decaying MBH pair creates a low-density core out of a preexisting stellar cusp, slowing down further binary hardening (see, e.g., Milosavljevic & Merritt 2001). If the hardening continues sufficiently far, GW emission takes over, driving the pair to coalescence. Figure 3 shows the number of MBHB coalescences per unit redshift per unit observed year predicted by our model: we expect ∼ 60 coalescences per year, the vast majority involving quite light binaries ( M 1 + M 2 ≤ 10 5 M /circledot ). The model was shown to reproduce rather well the observed luminosity function of optically-selected quasars in the redshift range 1 < z < 5 and the evolution of the nuclear MBH mass density with cosmic time (Volonteri et al. 2003a), and to provide a quantitative explanation to the stellar cores observed today in bright ellipticals as a result of the cumulative eroding action of shrinking MBHBs (Volonteri et al. 2003b).", '4.1. Stochastic noise from MBHBs': "The customary definition of GW confusion noise level is the amplitude at which there is, on average, at least one source per frequency resolution bin. The frequency bin width is ∆ f = 1 /τ , so the longer the observation, the smaller the noise. As pointed out by Cornish (2003), the crude 'one bin rule' is much too simple to properly describe a binary system. Using detailed information theory, Cornish (2003) shows that a GW background becomes unresolvable when there is, on average, at least one source per eight bins. \nIn the last decade a considerable effort has gone into quantifying the galactic and extragalactic confusion noise in the band 0.01 mHz - 1 Hz (e.g., Schneider et al. 2000; Freitag 2001; Nelemans et al. 2001; Farmer & Phinney 2003). We have then applied the 'eight bin rule' to asses the confusion noise associated with the evolving population of MBHBs, compared to the most recent estimates of the noises from galactic (Nelemeans et al. 2001, 'one bin' rule, 1 year observation) and extragalactic (Farmer & Phinney 2003, 'one-bin' rule, 3-year observation) white dwarf (WD) binaries. As shown in Figure 4, MBHBs produce confusion noise at f < 4 × 10 -4 Hz. \n∼ \n× Figure 4 also shows the global LISA h rms , along with separate contributions from different source populations. Though, as expected, MBHB stochastic noise dominates over WD-WD signals at low frequencies, it lies more than \nan order of magnitude below the instrumental LISA sensitivity curve, and hence its contribution to the LISA h rms can be ignored. On the other hand, this hampers the possibility that LISA could take advantage of the MBHB noise to probe the cosmological evolution of such particular parent population.", '4.2. Mass function and redshift distribution': "We have divided the resolved sources into 'merging' and 'in-spiral' binaries (MBs and IBs, respectively). The former are those binaries that reach the ISCO during the duration of the observation with a signal above threshold. \nFig. 5.Mass distribution of the more massive member of MBHBs resolved with S/N > 5 by LISA in a 3-year mission ( solid line ). The separate counts for MBs ( short-dashed line ) and IBs ( long-dashed line ) are also shown. \n<!-- image --> \nThese events are of particular importance, as they probe strong field effects and represent a unique chance of observing the coalescence and ring-down phases of MBHBs. Resolved IBs, instead, do not allow a direct observation of the coalescence phase. These events arise from light binaries whose final coalescence phase lies below threshold, and from binaries of all masses with τ ISCO > 3 years at the very start of the observation. We expect very few IBs in this last stage anyway, because binaries above threshold have typically τ ISCO < 3 years when the observation starts (see Fig. 2). An example of a resolved IB is represented by the lowest curve of Figure 2. MBHBs in this class have an integrated S/N above threshold, though the coalescence phase occur at too high frequency to be directly observed by LISA . \nFig. 6.Differential redshift distribution of MBHBs resolved with S/N > 5 by LISA in a 3-year mission. Line style as in Fig. 5. \n<!-- image --> \nAn obvious consequence of our classification is that MBs have larger mass and a lower redshift than IBs. The mass distribution of the most massive member of the binary M 1 is shown in Figure 5. The differential and cumulative redshift distributions are plotted in Figure 6 and Figure 7, respectively. Detectable IBs are ∼ 55 in total, and account for almost all of the MBHBs observed at z ∼ > 7. Conversely, the rarer MBs ( /similarequal 33 in total) are confined to the redshift interval 2 ∼ < z ∼ < 7. Figure 8 shows the average mass ratio M 2 /M 1 for MBs and IBs. As expected, this is larger for IBs, given their larger average redshift. The ratio decreases at low redshift, as a consequence of the complicated merger history of host dark matter halos. The reason why the average mass ratio of MBs peaks at M 2 /M 1 /similarequal 0 . 15 lies in the fact that both the probability of halo mergers (because of the steep P&S halo mass function) and the dynamical friction timescale increase with decreasing halo mass ratio. Hence, fast equal mass mergers are rare, while in more common unequal mass mergers it takes longer than an Hubble time to drag the satellite hole to the centre. \nFig. 7.Cumulative (integral from z to ∞ ) redshift distribution of MBHBs resolved with S/N > 5 by LISA in a 3-year mission. Line style as in Fig. 5. \n<!-- image --> \nFig. 8.Mass ratio distribution of MBHBs resolved with S/N > 5 by LISA in a 3-year mission. Line style as in Fig. 5. \n<!-- image --> \nWe can define the detection efficiency of a specific mission as the number of observable events divided by the expected number of coalescences in the same time interval. Figure 9 shows the global (MBs+IBs) detection efficiency for LISA and the efficiency considering as 'detections' \nonly MBs. The large GW-brightness of MBHBs is such that LISA will observe ∼ > 90% of all coalescences occurring at z ∼ < 5. The efficiency falls below 0 . 5 only for MBHBs at z ∼ > 8. The efficiency to MBs only is, obviously, lower. Figure 9 shows that a space-based interferometer such as LISA can directly observe the final stage of the spiral-in phase of about half of all MBHBs coalescing at z /similarequal 5. \nFig. 9.Detection efficiency ( solid line ), defined as the number of detected events (MBs+IBs) divided by the total number of coalescences in the same time interval, as a function of redshift. The efficiency considering only MBs as detections is also shown ( dashed line ). \n<!-- image -->", '5. DISCUSSION': "In this paper we have characterized the GW signal produced by cosmological MBHBs, and we have then folded the signal into the LISA performance capabilities. We find that LISA should resolve more than 90% of all cosmological coalescences of MBHBs occurring at z ∼ < 5. The detection efficiency is already ∼ > 0 . 5 for MBHBs at z /similarequal 8. We showed that the confusion noise from residual unresolved MBHBs is expected to be at least an order of magnitude below LISA instrumental noise. \nWe have divided the resolved events into merging and in-spiral binaries. MBs are associated with systems at relatively low redshift involving heavy pairs (10 4 -10 7 M /circledot ). Their strong GW signal can be used to study the orbital evolution of the pair until the ISCO, allowing to test GR in extreme conditions. On the contrary, IBs are less massive pairs at higher redshift. Such systems can be generally observed, with moderate integrated S/N , only for a relatively short amount of time, from few weeks to few months, before the ISCO is reached. IBs are nevertheless important as they push the limit of observable MBHBs out to z /similarequal 12, and allow studies of the formation and assembly of seed holes of intermediate masses. Binaries at even earlier \nepochs, while common in our model, are unfortunately too light to be observed by LISA with a relevant S/N . \nIn our refined study, the ∼ 20 'stationary sources' discussed in Paper I, i.e. sources whose shift in frequencies ∆ f during the observing period is ∼ < f , appear to be undetectable. This is because a too optimistic sensitivity threshold for LISA was assumed in Paper I for frequencies below 0.1 mHz, where most of the stationary sources are expected. Indeed, an order of magnitude improvement of the LISA sensitivity below 0.1 mHz would lead to the detection of ∼ 100 of such events. We stress here that our results may be sensitive to different model parameters and assumptions. As done in Paper I, we explore such possibility by running test models in which the hardening timescale t h was divided and multiplyied by a factor of 3. The resulting coalescence rates are plotted in Figure 3. We find 50 coalescences per observed year in the 'slow hardening case' and 78 in the 'fast hardening' case, compared to 64 in our reference model. In terms of LISA number counts, the effects are small. In the slow hardening case, the coalescence rate decreases at z ∼ > 9 as a large fraction of binaries has t h longer than the then Hubble time t H . At lower redshifts it is t h < t H anyway, the coalescence rate is basically unaffected with respect to the fiducial case, and so are the number counts (both for MBs and IBs). To obtain a significative reduction of observable sources, t h must increase by a larger factor, so that also MBHBs in the range 5 ∼ < z ∼ < 10 will have t h ∼ > t H . By increasing t h by an order of magnitude, we find the number of coalescences per observed year decreasing to 30: in a 3-year observation LISA would detect 20 (40) MBs (IBs). Note that, as increasing t h results in a lower average redshift of coalescences, this also increases the global detection efficiency. In other words, there are less sources, but a larger fraction of them is detectable. \nIn the fast hardening case, more binaries can coalesce at early times, and the number of surviving MBHBs at 6 ∼ < z ∼ < 12 decreases. This ultimately causes a slight reduction in the number of IBs observable by LISA (49 in 3 years, compared to 55). From Figure 3, it is also evident that, for z ∼ < 5, the coalescence rate is almost identical to the standard case, implying that the number of detectable MBs will remain approximately the same. \nIn Figure 8, we showed that IBs have an average mass ratio higher than MBs. To check whether this result depends on our assumption of equal-mass seed holes, we ran a test model with a flat initial mass function for the seed, 10 < m seed < 500M /circledot . The resulting binary mass distribution relevant for LISA was basically unaffected. \nThe vast majority of IBs are low mass systems at fairly high redshift. Their characteristic strain lies just above the LISA threshold at frequencies of the order of 10 -3 -10 -2 Hz (see Fig. 2) where confusion noises from unresolved galactic and extragalactic WD-WD binaries dominate the \nsensitivity curve (see Fig. 4). WD-WD confusion noise levels are difficult to compute because of the many uncertainties in stellar population synthesis models, and in estimating the fraction of binary stars in galaxies. In our fiducial sensitivity curve we have added to the LISA effective noise the galactic WD-WD confusion noise computed by Nelemans et al. (2001), and the extragalactic WD-WD confusion noise estimated by Farmer & Phinney (2003). Note that Nelemans et al. (2001) assumed 1, rather than 3, year integration, and both estimates employ the one bin rule. Using the eight bin rule we expect these noises to increase to some extent, but differences should be small. An alternative accurate estimation for galactic WD-WD confusion noise was performed by Hils & Bender (2000), who assumed a three bin rule. Using their noise level, which is somewhat higher than that computed by Nelemans et al. (2001), the number of IBs observed by LISA in 3 years slightly decreases, from 55 to 50. \nCompared to unresolved galactic WD-WD binaries, the uncertainties in the extragalactic WD-WD confusion noise are much larger, and may have a more relevant impact on number counts. The estimate of Schneider et al. (2000) lies nearly a factor of four above Farmer & Phinney's 'fiducial' model, and this leads to a significant reduction of observable IBs, from 55 to 44. The lowest estimate we could find in the literature was Farmer & Phinney's 'pessimistic' model, which is about a factor of four lower than their fiducial one; in this case we count 63 observable IBs. \nAnother potential source of noise in the frequency range 1-10 mHz is captures of compact objects (white dwarfs, neutron stars, and stellar-mass black holes) by MBHs in galaxy centers. Capture rates are quite uncertain, and estimates of relevant confusion noises span more than an order of magnitude in h c (see Barack & Cutler 2004 for a detailed discussion). We have estimated the impact of compact object captures on LISA number counts assuming the more optimistic rates calculated by Freitag (2001). The number of observable IBs decreases in this case to 43. More conservative rate estimates do not affect appreciably the counts. \nTo summarise, in the context of our model, we can assign an approximate error of /similarequal 50% to the number of highz MBHBs detectable by LISA . We conclude remarking that the bulk of detections involves binaries with masses in the interval 10 3 -10 5 M /circledot , a range where black holes have never been observed. Genuine supermassive BH binaries, whose existence is more secure on observational grounds, appear too 'heavy' for interferometers working in the band 0.01 mHz - 1 Hz. \nWe thank P. Bender and A. Vecchio for several enlightening discussions during the preparation of this paper. 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2000ApJ...531L..41C
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Correlation among Quasi-Periodic Oscillation Frequencies and Quiescent-State Duration in Black Hole Candidate GRS 1915+105
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2000-01-01
| 10 | 0.47 | 163 |
['accretion', 'accretion disks', 'black hole physics', 'hydrodynamics', 'shock waves', '-', 'astronomy x rays', 'astrophysics']
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[]
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We discover a definite correlation between the frequency of the quasi-periodic oscillations (QPOs) in quiescent states and the duration of the quiescent state of the transient X-ray source GRS 1915+105. We find that while the QPO frequency can be explained by the oscillation of shocks in accretion flows, the switching of burst to quiescent states (and vice versa) and their duration can be explained by assuming an outflow from the postshock region. The duration of the quiescent state is inversely related to the QPO frequency. We derive this relation. We also find the correlation between the observed low (~0.001-0.01 Hz) and the intermediate (1-10 Hz) QPO frequencies. Our analytical solutions are verified by analyzing several days of public domain data from the Rossi X-Ray Timing Explorer.
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https://arxiv.org/pdf/astro-ph/9910012.pdf
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{'GRS 1915+105': 'Sandip K. Chakrabarti and Sivakumar G. Manickam \nS. N. Bose National Centre for Basic Sciences, JD-Block, Salt Lake, Calcutta, 700091,', 'INDIA': 'e-mail: [email protected] & [email protected] \nReceived \n; \naccepted', 'ABSTRACT': 'We discover a definite correlation between the frequency of the quasi-periodic oscillations (QPO) in quiescence states and the duration of the quiescence state of the transient X-ray source GRS 1915+105. We find that while the QPO frequency can be explained with the oscillation of shocks in accretion flows, the switching of burst to quiescence states (and vice versa) and their duration can be explained by assuming an outflow from the post-shock region. The duration of the quiescence state is inversely related to the QPO-frequency. We derive this relation. We also find the correlation between the observed low ( ∼ 0 . 001 -0 . 01Hz) and the intermediate (1 -10Hz) QPO frequencies. Our analytical solutions are verified by analyzing several days of public-domain data from RXTE. \nSubject headings: accretion, accretion disks - black hole physics hydrodynamics - shock waves - stars: individual (GRS 1915+105) - X rays: stars', '1. Introduction': "X-ray transient source GRS 1915+105 in our galaxy exhibits various types of quasi-periodic oscillations with frequencies ranging from ∼ 0 . 001 -0 . 01 Hz to ∼ 67 Hz (Morgan et al, 1997; Paul et al. 1998; Yadav et al. 1999). The object is sometimes in a flaring state with regular and quasi-regular bursts and quiescences, while at some other time it is in usual low-hard and high-soft states. While the light curves look very chaotic with no apparent similarity between observations in two different days, some of the features are classifiable: (a) low-frequency QPO ( ν L ∼ 0 . 001 -0 . 01Hz) is due to the transition between burst and quiescence states (which we term as 'on'-state and 'off'-state respectively) and vice versa; (b) the intermediate frequency QPO ( ν I ∼ 1 -10Hz) could be due to oscillations of shocks located at tens to hundreds of Schwarzschild radii R g (= 2 GM/c 2 is the Schwarzschild radius. Here, M is the mass of the black hole, G , and c are the gravitational constant and velocity of light respectively) and (c) very high frequency QPO ( ν H ∼ 67Hz), if at all present, could be due to oscillations of the shocks located at several R g . ν I is generally observed during quiescence states. Typically, a shock located at R s (unless mentioned otherwise, measured hereafter in units of R g ), produces an oscillation of frequency, \nν I = 1 t ff ∼ 1 R R -α s cv 0 R g s (1) \nwhere, R is the compression ratio of the gas at the shock. Here we used a result of Molteni, Sponholz & Chakrabarti (1996, hereafter referred to as MSC96) which states that the time-period of QPO oscillation is comparable to the infall time ( t infall = t ff R ∼ R 3 / 2 s ) in the post-shock region. However, we assume now that the post-shock velocity is not necessarily R -1 / 2 s dependent as in a free fall but could be slowly varying, especially when angular momentum is high. In this case, t infall ∝ R α s . Clearly, α ∼ 3 / 2 for a low angular momentum freely falling matter and α ∼ 1 for a post-shock flow of constant velocity v 0 c/R . \nHere v 0 is a dimensionless quantity which is exactly unity for a free-fall gas. For a gas of γ = 4 / 3, R ∼ 7 and for γ = 5 / 3, R ∼ 4, when the shock is strong. Thus, for instance, for a ν I = 6Hz, R s ∼ 38 for M = 10 M /circledot and γ = 4 / 3. For ν H = 67Hz, R s ∼ 8 for the same parameters. MSC96 and Chakrabarti & Titarchuk (1995, hereafter CT95) postulated that since black hole QPOs show a large amount of photon flux variation, they cannot be explained simply by assuming some inhomogeinities, or perturbations in the flow. \nIf the QPOs are really due to shock oscillations, they should almost disappear at low energy soft X-rays, since these X-rays are produced in pre-shock flow which does not participate in large-scale oscillations. Second, a shock-compressed gas with compression ratio R > 1, must produce outflows or extended corona which pass through sonic points located at at R c = f 0 R s / 2, where f 0 = R 2 / ( R -1), if the flow is assumed to be isothermal till R c (Chakrabarti 1998, 1999, hereafter C98 and C99 respectively). In this solution the location of the sonic point R c and ratio between outflow and inflow rates are functions of the compression ratio R of the shock alone. Till the sonic point R c , matter is subsonic and this subsonic volume is filled in a time of (C99), \nt fill = 4 πR 3 c < ρ > 3 ˙ M out , (2) \nwhere, < ρ > is the average density of the sonic sphere, and ˙ M out is the outflow rate. The Compton cooling becomes catastrophic when < ρ > R c k es > ∼ 1, k es = 0 . 4 is the Thomson scattering opacity. Thus the duration of the off-state (i.e., duration between the end of a burst and the beginning of the next burst) is given by, \nt off = 4 πR 2 c 3 ˙ M out k es . (3) \nWe use now a simple relation between inflow and outflow rates given by (C98, C99), \n˙ M out ˙ M in = R ˙ m = Θ out Θ in R 4 [ R 2 R -1 ] 3 / 2 exp ( 3 2 -R 2 R -1 ) (4) \nwhere, Θ in and Θ out are the solid angles of the inflow and the outflow respectively. Because of the uncertainties in Θ in , Θ out and ˙ M in (subscript 'in' refers to the accretion rate) we define a dimensionless parameter, \nΘ ˙ M = Θ out Θ in ˙ M in ˙ M Edd . (5) \nwhere, ˙ M Edd is the Eddington rate. Using Eqs. (4-5), we get the following expression for t off as, \nt off = 10 . 47 ( R -1) 1 / 2 R 2 s R 2 g exp ( f 0 -3 2 ) ˙ M Edd Θ ˙ M s. (6) \nOr, eliminating shock location R s using eq. (1) and α = 3 / 2, v 0 = 1, we obtain, \nt off = 14 . 1 exp ( f 0 -3 2 ) R 4 / 3 ( R -1) 1 / 2 Θ ˙ M ( M 10 M /circledot ) -1 / 3 ν -4 / 3 I s (7) \nFor an average shock of strength 2 . 5 < ∼ R < ∼ 3 . 3, the result is insensitive to the compression ratio. Using average value of R = 2 . 9 and for Θ ˙ M ∼ 0 . 1 (which corresponds to 0 . 1 Eddington rate for Θ out ∼ Θ in ) we get, \nt off = 461 . 5( 0 . 1 Θ ˙ M )( M 10 M /circledot ) -1 / 3 ν -4 / 3 I s. (8) \nThus, the duration of the off-state must go down rapidly as the QPO frequency increases if the flow geometry and the net accretion rate remains fixed. When one considers a constant velocity post-shock flow, α = 1, and v 0 = 0 . 066 (chosen so as to keep the same numerical coefficient as in eq. 8) the above equation is changed to, \nt off = 461 . 5( 0 . 1 Θ ˙ M )( M 10 M /circledot ) -1 ( v 0 0 . 066 ) 2 ν -2 I s. (9) \nInterestingly, v 0 = 0 . 066 (i.e., a constant velocity of seven percent of the velocity of light) is very reasonable for a black hole accretion. \nIf the hot post-shock gas of height ∼ R s intercepts n soft photons per second, from the pre-shock Keplerian component (CT95), it should intercept about nf 2 0 / 4 soft photons per \nsecond when the sonic sphere of size R c is filled in. Thus, the photon flux in the burst state should be about f 2 0 / 4 > ∼ 4 times larger compared to the photon flux in off-state. Depending on the degree of flaring, and the fact that the wind is bent backward due to centrifugal force, the interception may be higher. \nSince ν L is basically due to recurrences of on and off-states, it is clear that ν L ∼ 1 / < t off + t on > . Here, t on is the duration of the burst state which may be very small for extremely regular (spiky) bursts reported by in Taam, Chen & Swank (1997) and Yadav et al. 1999. In this case, \nν L = 0 . 0022( Θ ˙ M 0 . 1 )( 10 M /circledot M ) ν 2 I Hz . (10) \nWhen on-state has a non-negligible duration ( t on /negationslash = 0), it is found to be directly related to the t off (Belloni et al., 1997; Yadav et al. 1999). Assuming t on ∼ t off , the ν L would be less by a factor of two when on states are broad. When the burst is very regular but 'spiky' (i.e., with momentary on-state), t on ≈ 0. The presence of ν L for these regular 'spiky' bursts are reported in Manickam and Chakrabarti (1999a) \nThus, if our shock oscillation solution for QPO is correct, the observations must pass all the following tests: (a) the QPO in the off-state must disappear at low energies, (b) the QPO must generally be absent in the on-state, when the sonic sphere is cooled down, (c) the intermediate QPO frequency must be correlated with t off as in Eqs. (8-9) and (d) the photon flux must jump at least a factor of 4 or more when going from quiescence to burst state. In addition, (e) lowest frequency ν L observed must be correlated to the intermediate QPO frequency ν I by eq. (10). There are uncertainties regarding the inflow velocity and actual volume-filling time, but we expect that above relations to be satisfied in general. \nIn the present Letter we show that observations do pass through these tests and therefore the shock oscillation model may be the correct picture. In the next Section, we present detailed analysis of some of the observational results on GRS1915+105 available in \npublic archive and show how they point to the shock oscillation model. Finally in § 3, we make concluding remarks.", '2. Observational Results': 'Figure 1 shows a light curve of the first phase of observation of June 18th, 1997, on the right panel. The average count rate (per second) vary from around ∼ 5000 in the off-state to about ∼ 24 , 000 in the on-state. The ratio of the fluxes is about ∼ 5. The duration of these states vary chaotically. At the mean location of a few off-states (arrows on right axis), observation time is marked in seconds. For each of these off-states, the power density spectrum (PDS) in arbitrary units is drawn in the left panel. The most prominent QPO frequencies ( ν I in our notation) are connected by a dashed curve just to indicate its variation with time. There are some weaker peaks which follow the short dashed curve, indicating that they may be higher harmonics. Observations of this kind for several other days show similar variations in QPO frequencies and details are presented elsewhere (Manickam & Chakrabarti, 1999ab). \nIn Figure 2, variation of ν I with the duration t off of the off-states (triangles) for the whole observation period on June 18th, 1997 in the log-log scale. Observational results from several other days (May 26th, 1997; June 9th, 1997; June 25, 1997; October 7th, 1996; October 25th, 1996) are also plotted on the same curve with circles, filled squares, squares, filled circles and stars respectively. We did not put error bars since in duration scale is it uniformly ± 2 seconds, and in frequency scale error bar is decided by the chosen bin-size while obtaining the PDS (In our analysis it remains around 0.15-0.25 Hz, increasing monotonically with QPO frequency). Times at which the photon flux is halved during the on-to-off and off-to-on transitions are taken respectively to be the beginning and the end of an off-state. Equation (9) is plotted in dashed lines with (from uppermost to the lowermost \nline) Θ ˙ M = 0 . 0034, 0 . 0123, 0 . 0163, 0 . 028, 0 . 0293 and 0 . 043 respectively indicating a slow variation of the accretion rate, provided the collimation property remains the same. Two dotted curves, on the other hand, represent eq. (8), with Θ ˙ M = 0 . 06 (top) and 0 . 093 (bottom) respectively. We find that the inverse-squared law (eq. 9) may be a better fit to the observations. Since on Oct. 7th, 1996 the points are lumped closed to the lower right corner, not much could be said about whether it follows our relation or not, but it is to be noted that its general behaviour (low frequency, high duration) follows our result for any reasonable Θ ˙ M . \nTable 1 shows the variation of the ν I (taken from Fig. 2) with days of observations. In 3rd column, the expected ν L has been put ( ν L / 2 for Oct. 7 and June 18 results as they show t on ≈ t off ). In 4th Column, the observed ν L is given. Generally, what we observe is that, when the drift of ν I is large, PDS around ν L is also broad In any case, observed ν L agrees with our expectations. \nFigure 3 shows the PDS of the off-state centered at 1576s (see Fig. 1) of the June 18th observation. The energy range is given in each panel Clearly, QPO disappears completely at low energies, exactly as is expected in the shock oscillation model (MSC96) though QPO frequencies, when present, seem to be energy independent (see, Manickam & Chakrabarti, 1999b for details). The pre-shock flow which emits soft radiation participates little in the oscillation as the fractional change in the Keplerian disk due to shock oscillation is negligible. On the contrary, the fractional change in the size of the post-shock flow during the oscillation is large. Thus, the flux of hard X-rays oscillates as the size of the post-shock region oscillates. This is the cause of QPO in our model. In on-states, when they exist, the QPO is found to be very weak.', '3. Discussion and Conclusions': "In this Letter , we have discovered a relation between the QPO frequency in 1-10Hz range with the duration of the quiescence state at which the QPO is observed. We also derived a relation between the low QPO frequency and the intermediate QPO frequency. We analyzed several days of RXTE observations and showed that our relations are satisfied, especially when the average bulk velocity in the post-shock region is constant. We showed that the QPO disappears in the low energy, but is very strong in high energies. The photon flux is found to fluctuate, typically by a factor of 4 or more, indicating that a vertically inflated post-shock region is responsible for interception of the soft photons from a Keplerian disk. This factor seems to be similar to f 2 0 / 4 (C98, C99) for any reasonable compression of the gas, which strengthens our belief that the quasi-periodic cooling of the sonic sphere of the outflow from the post-shock region may be responsible for the rapid transitions between on and off-states. Our computation of the duration of the off-states from this considerations are found to be quite reasonable. We find that for ν -4 / 3 law, t off is insensitive to the mass of the black hole while for ν -2 law, t off is inversely proportional to the mass. Trudolyubov et al. (1999) recently found the duration of 'hard' states varies as -7 / 3 power of the lowest centroid frequency for a group of data while we find an inverse-squared law when we choose the QPO frequency where the power is strongest. \nAlthough we chose a specific model for the outflow (C99) for concreteness, the physical processes invoked are generic and the explanation should be valid even when other models for outflows are used (except self-similar models). The shock location near the inner edge of the Keplerian disk can drift on viscous time scale (see appendix of CT95 where the transition from Keplerian to sub-Keplerian is plotted as a function of viscosity). The shocks can evacuate the disk, and form once again, very similar to what was seen in the numerical simulation of Ryu et al. (1997). This drift would cause a drift in frequency as Trudolyubov \net al. (1999) recently showed (see also, Belloni et al. 1997; Markwardt, Swank & Taam 1999, Muno et al. 1999). Our model invokes also outflows (which we believe form naturally in the post-shock region and from the transition region from Keplerian to a sub-Keplerian flow) which we find useful to explain variations in photon counts between 'off' and 'on' states. \nThis work is partly supported by a project (Quasi Periodic Oscillations in Black Hole Candidates) funded by Indian Space Research Organization (ISRO). The authors thank NASA for making RXTE data available and ISRO for creating a Data Bank at their Centre where these data are stored. \nTABLE 1 Correlation Between Low And Intermediate Frequencies of QPO (in Hz) \n| Day | ν I | ν L from Eq. 10 | ν L observed | Remarks |\n|---------------|-----------|-------------------|----------------|-------------|\n| Oct. 7, 1996 | 1.7-1.90 | 0.0013-0.0016 | 0.001 | broad on |\n| Oct. 25, 1996 | 2.9-3.5 | 0.0006-0.0009 | -- | sparse data |\n| May 26, 1997 | 5.32-6.54 | 0.0076-0.0115 | 0.01 | spiky |\n| June 9 1997 | 6.22-7.42 | 0.0138-0.0197 | 0.014 | spiky |\n| June 18 1997 | 4.0-7.51 | 0.003-0.011 | 0.01 | broad on |\n| June 25 1997 | 6.61-7.42 | 0.016-0.02 | 0.022 | spiky |", 'REFERENCES': "Belloni, T, M'endez, King, A.R., Van der Klis, M. & Van Paradijs, J. 1997, APJ 488, L109 Chakrabarti, S.K. & Titarchuk, L.G. 1995, ApJ 455, 623 (CT95) \nChakrabarti, S.K. 1998 in Observational Evidence for Black Holes in the Universe, ed. S K Chakrabarti 19 (Dordrecht: Kluwer Academic Publishers) (C98) \nChakrabarti, S.K., 1999, A&A, 351, 185, (C99) \nManickam, S.G. & Chakrabarti, S.K. 1999a in Proceedings of Young Astrophysicists of Today's India , Ind. J. Phys. 73(6), 967. \nManickam, S.G. & Chakrabarti, S.K. 1999b A&A (submitted) \nMarkwardt, C.B., Swank, J.H. & Taam, R.E. 1999, ApJ 513, L37 \nMolteni, D., Sponholz, H. & Chakrabarti, S.K. 1996, ApJ 457, 805 (MSC96) \nMorgan, E. H., Remillard, R. A. & Greiner, J. 1997, ApJ 482, 993 \nMuno, M.P., Morgan, E.H. & Remillard, R.A. 1999, ApJ (submitted) \nPaul, B., et al. 1998, ApJ 492, L63 \nRyu, D., Chakrabarti & Molteni, D. 1997, ApJ 474, 378 \nTaam, R.E., Chen, X.-M., Swank, J.H., ApJ, 485, L83 \nTrudolyubov, E. Churazov & M. Gilfanov, A & A Letters 351, L15 \nYadav, J.S., et al. 1999, ApJ 517, 935 \nThis manuscript was prepared with the AAS L A T E X macros v4.0. \nFig. 1.- Plot of the light curve (right panel) and evolution of power density spectrum (left panel) of the first phase of June 18th, 1997 observation. Off-states analyzed are marked by the time of observations on the right axis. The QPO frequencies (where the power is strongest) are connected by a dashed curve to highlight the evolution of ν I with time. \nFig. 2.- Variation of QPO frequency ν I with duration of quiescence states t off . Dotted curves are the t off ∝ ν -4 / 3 law (eq. 9) derived using simple free-fall velocity assumption. Dashed curves are the t off ∝ ν -2 (eq. 10) with constant post-shock velocity law. The general agreement strongly points to the shock oscillation model. \nFig. 3.- Power density spectrum of the off-state (duration 87s) centered at 1576s constructed from selected channel intervals of the binned PCA data. QPO is seen only in high energies, strongly pointing to the shock oscillation model. \n<!-- image --> \n<!-- image --> \n<!-- image -->"}
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2005GReGr..37.1255B
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(Anti-)de Sitter black hole thermodynamics and the generalized uncertainty principle
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2005-01-01
| 5 | 0.45 | 163 |
['-', '-', '-', '-', '-']
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[]
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We extend the derivation of the Hawking temperature of a Schwarzschild black hole via the Heisenberg uncertainty principle to the de Sitter and anti-de Sitter spacetimes. The thermodynamics of the Schwarzschild-(anti-)de Sitter black holes is obtained from the generalized uncertainty principle of string theory and non-commutative geometry. This may explain why the thermodynamics of (anti-)de Sitter-like black holes admits a holographic description in terms of a dual quantum conformal field theory, whereas the thermodynamics of Schwarzschild-like black holes does not.
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https://arxiv.org/pdf/gr-qc/0411086.pdf
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{'(Anti-)de Sitter Black Hole Thermodynamics and the Generalized Uncertainty Principle': "Brett Bolen ∗ , Marco Cavagli'a † \nDepartment of Physics and Astronomy, University of Mississippi University, MS 38677-1848, U.S.A.", 'Abstract': 'We extend the derivation of the Hawking temperature of a Schwarzschild black hole via the Heisenberg uncertainty principle to the de Sitter and anti-de Sitter spacetimes. The thermodynamics of the Schwarzschild-(anti-)de Sitter black holes is obtained from the generalized uncertainty principle of string theory and non-commutative geometry. This may explain why the thermodynamics of (anti-)de Sitter-like black holes admits a holographic description in terms of a dual quantum conformal field theory, whereas the thermodynamics of Schwarzschild-like black holes does not. \nPACS numbers: 04.70.-s, 04.70.Dy, 03.65.Ta', 'I. INTRODUCTION': "The Heisenberg uncertainty principle of quantum mechanics allows a heuristic derivation of the Hawking temperature [1] of a Schwarzschild black hole. The derivation proceeds as follows [2]. The uncertainty in the linear position x of an emitted quantum is approximately equal to the Schwarzschild radius r s . By modelling the black hole as an object with linear size r s , and assuming that the radiation satisfies the condition of minimum uncertainty, the uncertainty in the energy of the emitted quanta is \n∆ E ∼ c ∆ p ∼ ¯ hc ∆ x ∼ ¯ hc r s , → ∆ E = κ ¯ hc r s , (1) \nwhere κ is a proportionality constant. ∆ E is identified with the temperature T of the radiation. Setting κ = ( d -3) / 4 π , Eq. (1) gives the Hawking temperature for a d -dimensional Schwarzschild black hole \nT H = d -3 4 πr s ¯ hc . (2) \nThe above derivation deserves some comments. Black hole emission is usually regarded as being originated by quantum effects in the region around the black hole horizon, such as semiclassical wave scattering or particle tunnelling. (See, e.g. Ref. [3].) The uncertainty principle does not describe the origin of these effects, but only their consequence on the measurement process. Explaining the origin of black hole emission requires the knowledge of the quantum states that describe the black hole, from which the exact form of the uncertainty principle for the black hole can be derived. On the other hand, Eq. (1) seems to suggest that black hole thermodynamics is a generic low-energy effect of small scale physics. Since any quantum theory of gravity must include some kind of uncertainty principle that reduces to Heisenberg principle at low-energy scales, black hole thermodynamics should not depend too much on the details of the quantum gravity theory. This seems to agree in spirit with Visser's conclusion that the Hawking radiation only requires ordinary quantum mechanics plus a slowly evolving future horizon, and thus the knowledge of quantum gravity is unnecessary to explain the features of black hole thermodynamics [4]. \nThe above derivation, although appealing, is only known for the Schwarzschild black hole. The aim of paper is to extend the uncertainty principle derivation of the Hawking temperature to the de-Sitter (dS) and anti-de Sitter (adS) black holes.", 'II. ADS AND DS THERMODYNAMICS': "The line element of a d -dimensional Schwarzschild-(a)dS black hole ( d > 3) with mass M is (see, e.g., Refs. [5, 6]) \nds 2 = -( 1 ± λ 2 r 2 -ω d G d M c 2 r d -3 ) c 2 d t 2 + ( 1 ± λ 2 r 2 -ω d G d M c 2 r d -3 ) -1 d r 2 + r 2 dΩ 2 d -2 , (3) \nwhere G d is Newton's constant, λ = 1 /b is the inverse of the (a)dS radius, and the ± sign is for adS and dS, respectively. The constant ω d is equal to 16 π/ ( d -2)Ω d -2 , where Ω d -2 is the volume of the unit d -2 sphere. The Hawking temperature of the black hole horizon r h is \nT S ( a ) dS = d -3 4 π ( 1 r h ± γ 2 r h ) ¯ hc , (4) \nwhere γ is proportional to the inverse of the curvature radius of the (a)dS spacetime \nγ = b -1 √ ( d -1) / ( d -3) . (5) \nTwo limits of the temperature may be realized. In the Schwarzschild limit, the radius of the event horizon is negligible in comparison to the radius of curvature of the (a)dS spacetime. The Schwarzschild-(a)dS solution reduces to the asymptotically Schwarzschild solution with temperature Eq. (2). In the (a)dS limit, the radius of the black hole event horizon is large in comparison to the radius of curvature of the (a)dS spacetime. The temperature of the (cosmological) horizon is \nT ( a ) dS = ( d -3) γ 2 r h 4 π ¯ hc . (6) \nClearly, the Heisenberg uncertainty principle cannot reproduce Eq. (6). However, the (a)dS temperature may be obtained by substituting the standard Heisenberg relation with its generalized version.", 'III. GENERALIZED UNCERTAINTY PRINCIPLE': "The generalized version of the Heisenberg uncertainty principle is usually given by \n∆ x ∆ p > ∼ ¯ h [ 1 + α 2 /lscript 2 p ∆ p 2 ¯ h 2 ] , (7) \nwhere /lscript p = (¯ hG d /c 3 ) 1 / ( d -2) is the Planck length, and α is a numerical constant [2, 7]. Equation (7) is quite generic, and describes the quantum mechanical uncertainty when \nthe microscopic structure of spacetime is taken into account. Non-commutative quantum mechanics [8] and black hole gedanken-experiments [9] provide heuristic proofs of the generalized uncertainty principle. The two limits of Eq. (7) (see below) have been derived in the context of string theory in Refs. [10, 11]. \nThe generalized uncertainty principle (7) has both low-energy (quantum mechanical) and high-energy (quantum gravity) limits. The quantum mechanical limit is obtained when the second term in the r.h.s. of Eq. (7) is negligible: \nα 2 /lscript 2 p ∆ p 2 ¯ h 2 /lessmuch 1 → ∆ p M p c /lessmuch 1 α . (8) \nwhere M p = [¯ h d -3 / ( c d -5 G d )] 1 / ( d -2) is the Planck mass. From this limit, it follows that α = O (1). The quantum gravity limit is obtained when \nα 2 /lscript 2 p ∆ p 2 ¯ h 2 ∼ 1 → ∆ p M p c ∼ 1 α , (9) \nEquation (7) implies the existence of a minimum length l min of order of the Planck length. This can be seen by inverting Eq. (7): \n∆ x 2 α 2 /lscript 2 p 1 -√ 1 -4 α 2 /lscript 2 p ∆ x 2 < ∼ ∆ p ¯ h < ∼ ∆ x 2 α 2 /lscript 2 p 1 + √ 1 -4 α 2 /lscript 2 p ∆ x 2 . (10) \nThe lower limit on the uncertainty in position is \n∆ x > ∼ 2 α/lscript p ≡ l min . (11) \nThe standard Heisenberg uncertainty relation is obtained when l min is negligible compared to the scale of the process, i.e. when ∆ x /greatermuch /lscript p or α → 0. In the opposite limit, i.e. ∆ x ∼ l min , the uncertainty principle reads \n∆ p M p c ∼ ∆ x 2 α 2 /lscript p . (12) \nEquation (12) holds when strong quantum gravitational effects are present, and can be derived directly from the conformal invariance property of the fundamental string [10, 11]. In the stringy regime, the position uncertainty is proportional to the momentum uncertainty. Equation (7) is obtained by interpolating Eq. (12) with the standard uncertainty principle. \nEquation (7) is not the most general form of the generalized uncertainty principle [12]. The symmetry of the symplectic space suggests to write \n∆ x ∆ p ≥ ¯ h [ 1 + β 2 ∆ x 2 /lscript 2 p ] . (13) \nwhere β is a constant parameter. Combining Eq. (7) and Eq. (13) we find the general form \n∆ x ∆ p > ∼ ¯ h [ 1 + α 2 /lscript 2 p (∆ p ) 2 ¯ h 2 + β 2 (∆ x ) 2 /lscript 2 p ] . (14) \nEquation (14) possesses identical quantum mechanical limit and quantum gravity limit of Eq. (7). Thus Eq. (14) is consistent with the string theory derivation of the generalized uncertainty principle. Derivation of Eq. (14) in non-commutative quantum mechanics is discussed in Refs. [12]. \nIt is worthwile to discuss in detail the 'dual' form (13) of the generalized uncertainty principle (7). This will make clear why the general form of the uncertainty principle, Eq. (14), has been mostly overlooked in the literature in favor of Eq. (7). Equation (13) gives a different interpolation between the quantum mechanical limit and the quantum gravity limit than Eq. (7). The quantum mechanical limit is obtained when \nβ ∆ x /lscript p /lessmuch 1 → ∆ x /lessmuch /lscript p β . (15) \nTherefore, it follows that β /lessmuch 1. The quantum gravity limit is obtained when \nβ ∆ x /lscript p ≈ 1 → ∆ x ≈ /lscript p β . (16) \nSince β /lessmuch 1, one obtains the interesting result that quantum gravitational effects manifest themselves at very large distances. When the generalized uncertainty principle was first derived, the idea of modifications of gravity at great distances had not yet been seriously considered in the literature. Thus the interpolation (13) was overlooked. Inverting Eq. (13), \n∆ p 2 β 2 M p c 1 -√ √ √ √ 1 -4 β 2 M 2 p c 2 ∆ p 2 ≤ ∆ x /lscript p ≤ ∆ p 2 β 2 M p c 1 + √ √ √ √ 1 -4 β 2 M 2 p c 2 ∆ p 2 , (17) \none obtains a lower bound on the momentum uncertainty. This defines the minimum momentum P min = 2 βM p c .", 'IV. SCHWARZSCHILD-ADS THERMODYNAMICS WITH THE GENERALIZED UNCERTAINTY PRINCIPLE': 'The Hawking temperature of the adS black hole can be obtained by repeating the derivation of Sect. I with the generalized uncertainty principle. For semiclassical black holes, \n∆ x /greatermuch /lscript p and ∆ p /lessmuch M p c , and the form (13) of the generalized uncertainty principle applies. If we identify the parameter β with γ/lscript p , Eq. (13) reproduces the Schwarzschild-adS Hawking temperature \nT adS ∼ c ∆ p ∼ ( 1 ∆ x + β 2 /lscript 2 p ∆ x ) ¯ hc → T adS = d -3 4 π ( 1 r h + γ 2 r h ) ¯ hc . (18) \nThe two thermodynamical limits of the Schwarzschild-adS black hole follow from the two limiting relations between position and momentum (∆ p ∼ ¯ h/ ∆ x and ∆ p ∼ ¯ h ∆ x//lscript 2 p ) of the generalized uncertainty principle. \nThe above identification suggests that the Hawking temperature in adS and Schwarzschild spacetimes may have different origins. Since the adS temperature can be derived from the high-energy limit of the generalized uncertainty principle, the adS thermodynamics seems to have a quantum gravitational nature. It is interesting to note that the generalized uncertainty principle is a consequence of string theory, which can be consistently formulated in adS spacetime, whereas there is no consistent formulation of string theory in the Schwarzschild geometry, where the ordinary uncertainty principle suffices to derive the black hole thermodynamics. \nA word of explanation is required on the identification of the inverse adS radius with the generalized uncertainty principle parameter. In the context of known generalized uncertainty models, the coefficient of the correction term in the generalized uncertainty principle is proportional either to the fundamental gravitational length or the inverse string tension. Whereas the functional form of the generalized uncertainty principle seems to be rather generic and model-independent, the exact value of the correction depends on the quantum gravity states of the specific geometry. In the stringy derivation of Ref. [11], for instance, the parameter α of Eq. (7) is inversely proportional to the total momentum uncertainty of the string in a flat background. If the string propagates in a curved background, we expect its momentum uncertainty, and thus α , to be different. \nThe Schwarzschild-adS geometry is characterized by two length scales (the fundamental Planck length and the adS radius). The existence of the latter allows to set β ∝ /lscript p /b and α ∝ b//lscript p . The exact proportionality constants can be obtained by matching the quantum \ngravity limits of the generalized uncertainty principle to the black hole temperature in the adS regime, Eq. (18). Since β /lessmuch 1, the first identification applies to the b /greatermuch /lscript p regime, whereas the second identification applies to the b ∼ /lscript p regime. If the adS quantum states were known, the exact constant of proportionality between α , β and b could be formally derived. Unfortunately, in absence of a definite quantum gravity theory, the derivation of the exact geometry-dependent generalized uncertainty principle remains an open issue. A heuristic argument that illustrates the connection between the generalized uncertainty principle parameter in the adS spacetime and the adS radius is the following. Let us suppose to measure the momentum of particle by a scattering with a photon. The uncertainty in the measurement of the particle momentum is bounded from below by the value of the photon momentum, ∆ p > ∼ p γ ∼ h/λ . Since the photon wavelength cannot exceed the radius of the spacetime, the minimum uncertainty is ∆ p ∼ h/b . \nThe generalized uncertainty principle derivation applies also to the three-dimensional Ba˜nados-Teitelboim-Zanelli (BTZ) black hole [13] \nds 2 = -( -8 G 3 M c 2 + r 2 b 2 ) c 2 d t 2 + ( -8 G 3 M c 2 + r 2 b 2 ) -1 d r 2 + r 2 d φ 2 , (19) \nwhere the black hole radius is r BTZ = 2 b √ 2 G 3 M/c . From Eq. (12) we obtain the Hawking temperature of the BTZ black hole \nT BTZ ∼ r BTZ 2 α 2 /lscript p M p c 2 , → T BTZ = r BTZ 2 πb 2 ¯ hc , (20) \nwhere α = b √ π//lscript p .', 'V. SCHWARZSCHILD-DS THERMODYNAMICS WITH THE GENERALIZED UNCERTAINTY PRINCIPLE': 'The Hawking temperature of the Schwarzschild-dS black hole can be obtained from Eq. (13) by analytical continuation of the parameter β into the imaginary plane. This can be shown to be consistent with the topological structure of the dS spacetime as follows. For the dS spacetime, the analytic continuation of Eq. (13) reads: \n∆ x ∆ p > ∼ ¯ h [ 1 -β 2 ∆ x 2 /lscript 2 p ] . (21) \nThe inverse of Eq. (21) is \n∆ x /lscript p > ∼ ∆ p 2 β 2 M p c √ √ √ √ 1 + 4 β 2 M 2 p c 2 ∆ p 2 -1 (22) \nSince ∆ p is a positive-definite quantity the position uncertainty is limited from above by \n∆ x < ∼ b √ d -3 d -1 . (23) \nThis relation is a statement that the uncertainty in the measurement of position may not exceed the size of the de-Sitter spacetime.', 'VI. CONCLUSIONS': 'We have shown that the uncertainty principle derivation of the Hawking temperature can be extended to (a)dS-like black holes, provided that we consider the generalized uncertainty principle instead of the standard Heisenberg relation. The two thermodynamical limits of Schwarzschild-(a)dS follow from the quantum-regime limit and the standard limit of the generalized uncertainty principle. This result seems to indicate different origins for the thermodynamics of Schwarzschild- and (a)dS-like black holes. This could explain why only (a)dS-like black holes seem to admit a holographic description in terms of a dual quantum conformal field theory.', 'Acknowledgments': "We are grateful to M. Cadoni, S. Hossenfelder and L. Parker for interesting discussions. \n- [1] S. W. Hawking, Commun. Math. Phys. 43 , 199 (1975).\n- [2] R. J. Adler, P. Chen and D. I. Santiago, Gen. Rel. Grav. 33 , 2101 (2001) [arXiv:gr-qc/0106080].\n- [3] I. D. Novikov and V. P. Frolov, 'Physics Of Black Holes,' Fundamental theories of physics 27 (Kluwer Academic, Dordrecht, Netherlands, 1989).\n- [4] M. Visser, Int. J. Mod. Phys. D 12 , 649 (2003) [arXiv:hep-th/0106111].\n- [5] M. Cadoni, Phys. Rev. D 69 , 084021 (2004) [arXiv:gr-qc/0311056].\n- [6] R. G. Cai, Phys. Lett. B 525 , 331 (2002) [arXiv:hep-th/0111093].\n- [7] P. Chen and R. J. Adler, Nucl. Phys. Proc. Suppl. 124 , 103 (2003) [arXiv:gr-qc/0205106]; M. Cavagli'a, S. Das and R. Maartens, Class. Quant. Grav. 20 , L205 (2003) [arXiv:hep-ph/0305223];\n- M. Cavagli'a and S. Das, Class. Quant. Grav. 21 , 4511 (2004) [arXiv:hep-th/0404050];\n- S. Hossenfelder, M. Bleicher, S. Hofmann, J. Ruppert, S. Scherer and H. Stocker, Phys. Lett. B 575 , 85 (2003) [arXiv:hep-th/0305262].\n- [8] M. Maggiore, Phys. Rev. D 49 , 5182 (1994) [arXiv:hep-th/9305163];\n- M. Maggiore, Phys. Lett. B 319 , 83 (1993) [arXiv:hep-th/9309034].\n- [9] M. Maggiore, Phys. Lett. B 304 , 65 (1993) [arXiv:hep-th/9301067];\n- F. Scardigli, Phys. Lett. B 452 , 39 (1999) [arXiv:hep-th/9904025];\n- F. Scardigli and R. Casadio, Class. Quant. Grav. 20 , 3915 (2003) [arXiv:hep-th/0307174].\n- [10] D. Amati, M. Ciafaloni and G. Veneziano, Phys. Lett. B 216 , 41 (1989);\n- D. Amati, M. Ciafaloni and G. Veneziano, Nucl. Phys. B 347 , 550 (1990);\n- D. Amati, M. Ciafaloni and G. Veneziano, Nucl. Phys. B 403 , 707 (1993).\n- [11] K. Konishi, G. Paffuti and P. Provero, Phys. Lett. B 234 , 276 (1990).\n- [12] A. Kempf, J. Math. Phys. 35 , 4483 (1994) [arXiv:hep-th/9311147]; A. Kempf, arXiv:hep-th/9405067;\n- A. Kempf, J. Math. Phys. 38 , 1347 (1997) [arXiv:hep-th/9602085];\n- H. Hinrichsen and A. Kempf, J. Math. Phys. 37 , 2121 (1996) [arXiv:hep-th/9510144].\n- [13] M. Banados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 69 , 1849 (1992) [arXiv:hep-th/9204099]."}
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2007ApJ...671...85N
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Production of TeV Gamma Radiation in the Vicinity of the Supermassive Black Hole in the Giant Radio Galaxy M87
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2007-01-01
| 16 | 0.47 | 163 |
['black hole physics', 'galaxies active', 'galaxies', 'gamma rays', 'astrophysics']
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[]
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Although the giant radio galaxy M87 harbors many distinct regions of broadband nonthermal emission, the recently reported fast variability of TeV γ-rays from M87, on a timescale of days, strongly constrains the range of speculations concerning the possible sites and scenarios of particle acceleration responsible for the observed TeV emission. A natural production site of this radiation is the immediate vicinity of the central supermassive black hole (BH). Because of its low bolometric luminosity, the nucleus of M87 can be effectively transparent for γ-rays up to an energy of 10 TeV, which makes this source an ideal laboratory for the study of particle acceleration processes close to the BH event horizon. We critically analyze different possible radiation mechanisms in this region and argue that the observed very high energy γ-ray emission can be explained as the inverse Compton emission of ultrarelativistic electron-positron pairs produced through the development of an electromagnetic cascade in the BH magnetosphere. We demonstrate, through detailed numerical calculations of acceleration and radiation of electrons in the magnetospheric vacuum gap, that this ``pulsar magnetosphere-like'' scenario can satisfactorily explain the main properties of the TeV γ-ray emission from M87.
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https://arxiv.org/pdf/0704.3282.pdf
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{'PRODUCTION OF TEV GAMMA-RADIATION IN THE VICINITY OF THE SUPERMASSIVE BLACK HOLE IN THE GIANT RADIOGALAXY M87': "A.Neronov \nINTEGRAL Science Data Center, 16 ch. d'Ecogia, CH-1290, Versoix, Switzerland \nand \nGeneva Observatory, University of Geneva 51 ch. des Maillettes, CH-1290 Sauverny, Switzerland \nand", 'Felix A. Aharonian': 'Dublin Institute for Advanced Studies, 5 Merrion Square, Dublin 2, Ireland \nand \nMax Planck Institut fur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany Draft version October 27, 2018', 'ABSTRACT': "Although the giant radiogalaxy M 87 harbors many distinct regions of broad-band nonthermal emission, the recently reported fast variability of TeV γ -rays from M 87 on a timescale of days strongly constrains the range of speculations concerning the possible sites and scenarios of particle acceleration responsible for the observed TeV emission. A natural production site of this radiation is the immediate vicinity of the central supermassive mass black hole (BH). Because of the low bolometric luminosity, the nucleus of M 87 can be effectively transparent for γ -rays up to energy of 10 TeV, which makes this source an ideal laboratory for study of particle acceleration processes close to the BH event horizon. We critically analyse different possible radiation mechanisms in this region, and argue that the observed very high-energy γ -ray emission can be explained by the inverse Compton emission of ultrarelativistic electron-positron pairs produced through the development of an electromagnetic cascade in the BH magnetosphere. We demonstrate, through detailed numerical calculations of acceleration and radiation of electrons in the magnetospheric vacuum gap, that this 'pulsar magnetosphere like' scenario can satisfactorily explain the main properties of TeV gamma-ray emission of M 87. \nSubject headings: gamma rays: theory - black hole physics - galaxies: active - galaxies: individual (M 87)", '1. INTRODUCTION': "M 87, a nearby giant radio galaxy, located at a distance of d /similarequal 16 Mpc (Tonry 1991), hosts one of the most massive ( M /similarequal 3 × 10 9 M /circledot ) black holes (BH) in the nearby Universe (Marconi et al. 1997). M 87 contains a famous kpc-scale jet the high-resolution images of which detected at radio, optical and X-ray wavelengths show several prominent structures. Nonthermal processes play important, if not the dominant, role across the entire jet. The apparent synchrotron origin of the detected nonthermal emission, which extends from radio to X-ray bands, implies effective acceleration of electrons to multiTeV energies. One may expect that protons and nuclei, which do not suffer radiative losses, are accelerated to much higher energies. Both the inner (sub-parsec) and large (kpc) scale parts of the jet of M 87 are possible sites of acceleration of protons to extremely high energies ( E ∼ 10 20 eV). Therefore production of gamma-rays in different segments of the jet due to electromagnetic or hadronic processes is not only possible, but, in fact, unavoidable. The jet of M 87 is observed at large angle, ∼ 20 · (Biretta et al. 1999). Therefore, unlike blazars, we do not expect a strong Doppler boosting of the γ -ray flux. On the other hand, the nearby location of M 87 compensates this disadvantage (compared to blazars) and makes \nseveral prominent knots and hot spots of the jet as potentially detectable TeV γ -ray emitters. \nIn this regard, the discovery of a TeV γ -ray signal from M 87 by the HEGRA array of Cherenkov telescopes (Aharonian et al. 2003) and its confirmation by the HESS array of telescopes (Aharonian et al. 2006), was not a big surprise, especially given the rather modest apparent TeV γ -ray luminosity (few times 10 40 erg / cm 2 s). Several electronic and hadronic models have been suggested for explanation of TeV γ -ray emission of M87. The suggested sites of TeV γ -ray production range from large scale structures of the kpc jet (Stawarz et al. 2005) to a compact peculiar hot spot (the so-called HST -1 knot) at a distance 100 pc along the jet (Stawarz et al. 2006) and inner (sub-parsec) parts of the jet (Georganopoulos et al. 2005; Reimer et al. 2005). \nWhile gamma-ray observations cannot provide images with an adequate resolution which would allow localisation of sites of γ -ray production, the variability studies can discard or effectively constrain the suggested models. \nThe continuous monitoring of M 87 with the HESS telescope array during the period 2003-2006 not only revealed statistically significant fluctuations of the TeV flux on a yearly basis, but, more excitingly, an evidence of fast variability on timescales of ∆ t ∼ 2 days was found in the 2005 dataset (Aharonian et al. 2006). This requires a very compact region with a characteristic linear \nsize R ≤ ∆ tδ j /similarequal 5 × 10 15 δ j cm, where δ j is the Doppler factor of the relativistically moving source (throughout the paper we will use the system of units in which the speed of light c = 1). Note that since the mass of the BH in M 87 is well established (Marconi et al. 1997), the Schwarzschild radius is estimated quite accurately R Schw = 2 GM /similarequal 10 15 [ M/ 3 × 10 9 M /circledot ] cm ( G is the gravitational constant). The expected minimal variability time scale (light crossing time of the black hole) for a non-rotating BH is T ls = 2 R Schw / /similarequal 6 × 10 4 s /similarequal 1 day, while it is two times less for the maximally rotating Kerr BH. The observed variability time scale of TeV emission of ∼ 2 days indicates that the emission originates within the last stable orbit of rotation around the black hole, unless the radiation is produced in relativistically moving outflow with a Doppler factor ≥ 10. Although there are sound arguments against M 87 being a blazar (Fabian 2006), one cannot in principle exclude that at the base of its formation (close to the BH), where γ -rays are produced, the jet is pointed to us, and only later, it deviates from our line of sight. In this regard, both leptonic (Georganopoulos et al. 2005) and hadronic (Reimer et al. 2005) models suggested for TeV radiation of M 87 cannot be discarded. Note however, that both models predict rather steep energy spectra contrary to the observed hard γ -ray spectrum extending to E ≥ 10 TeV (Aharonian et al. 2006). Whether these models are flexible enough to reproduce the observed TeV γ -ray spectrum by tuning the relevant model parameters and introducing additional assumptions, should show further detailed theoretical studies. \nFinally, one should mention that formally one may assume that γ -rays are produced in a compact region far from the central engine. In this regard, the famous HST -1 knot has certain attractive features which make this compact structure a potential site of particle acceleration and γ -ray production (Stawarz et al. 2006). Although the favored size of this structure is in the range between 0.1 and 1 pc, and thus contradicts the observed TeV γ -ray variability, the lack of robust lower limits on the size of HST -1 leaves the γ -ray production in this peculiar knot as a possible option. Nevertheless one should note that the location of HST -1 at a distance of 100 pc from the central BH requires (almost) unrealistically tight collimation of the jet. \nIn this paper we assume that the TeV γ -ray production takes place close to the event horizon of the central supermassive BH, and show that the acceleration of electrons in a vacuum gap in BH magnetosphere can explain the general characteristics of the TeV γ -ray emission observed from M87. \nSuch mechanism of γ -ray emission from the vicinity of a black hole is a close analog of the mechanism of pulsed γ -ray emission from the vicinity of neutron stars in pulsars. The similarity between the electrodynamics of the pulsar and black hole magnetospheres was discussed in the seminal paper of Blandford & Znajek (1977) in the context of pair production and energy transfer of rotational energy of a black hole through the Poynting flux. The mechanisms of emission of high energy γ -rays from the direct vicinity of black holes by electrons and protons accelerated in the electric field of vacuum gaps were discussed by Blandford & Znajek (1977) (electron curvature \nFig. 1.Top panel: optical depth for γ -rays produced in the vicinity of the black hole in the cases of the infrared source of size 50 R Schw (black solid curve) and R Schw / 2 (grey dashed curve). The estimate of (Cheung et al. 2007) is shown by a cross. The spectrum of the infrared background in the source is the one shown by red dashed curve in Fig. 8). Bottom panel: attenuation of γ -rays in M 87 due to photon-photon pair production. The internal absorption (thin solid curve) is dominated by interactions with the infrared radiation of the compact source in the core of M 87. The external absorption due to the interaction with the diffuse radiation fields within the elliptical galaxy M 87, the 2.7 K CMBR and the diffuse extragalactic infrared background photons leads to a further suppression of the γ -ray flux, shown by thick solid curve. Attenuation in the case when the γ -ray emission is distributed throughout the infrared source is shown by thin (intrinsic absorption in the source) and thick (modification during the propagation through the galactic and extragalactic background light) dashed curves. Dashed grey curve shows the absorption of the γ -ray flux in the case of a 'maximally compact' infrared source of the size R IR = R Schw / 2 (tee text). \n<!-- image --> \nradiation), Beskin et al. (1992) (inverse Compton scattering by electrons) and Levinson (2000) (proton curvature radiation).", '2. INTERNAL ABSORPTION OF γ -RAYS': "The observed infrared luminosity of the nucleus of M 87, νL ν ∼ 10 40 ÷ 41 erg/s (Perlman et al. 2001; Whysong & Antonucci 2004) is 6.5 orders of magnitude lower than the Eddington luminosity of a 3 × 10 9 M /circledot BH. In this regard the BH of in M 87 is similar to the supermassive BH in the center of our galaxy. In both cases the low bolometric luminosity of the nucleus makes the 'central engine' of activity, i.e. the vicinity of the event horizon of the supermassive BH, transparent to the very high energy (VHE) γ -rays (Aharonian & Neronov 2005). \nIn the isotropic field of background photons, the crosssection of photon-photon pair production depends on the product of energies of colliding photons, s = E/epsilon1/m 2 e . Starting from the threshold at s = 1, the cross-section σ γγ rapidly increases achieving the maximum σ 0 ≈ σ T / 5 /similarequal 1 . 3 × 10 -25 cm 2 at s ≈ 4, and then decreases as s -1 ln s . Because of relatively narrow distribution of σ γγ ( s ), gamma-rays interact most effectively with the infrared background photons of energy \n/epsilon1 ≈ 1( E/ 1 TeV) -1 eV . (1) \nThus the optical depth for a gamma-ray of energy E produced in the center of the infrared source of the size R IR and the luminosity L IR at energy given by Eq.(1) can be written in the form \nτ ( E,R IR ) = L IR σ γγ 4 πR IR /epsilon1 /similarequal (2) 0 . 1 [ L IR (1[ E/ 1TeV] -1 eV) 3 × 10 40 erg/s ][ R IR 50 R Schw ] -1 [ E 1 TeV ] . \nThe dependence of the optical depth on gamma-ray energy is determined by the spectral form of background radiation n ( /epsilon1 ) = L IR ( /epsilon1 ) / (4 πR/epsilon1 ). In particular, in the case of power-law spectrum with photon index Γ ( n ( /epsilon1 ) ∝ /epsilon1 -Γ , or L IR ( /epsilon1 ) ∝ /epsilon1 -Γ+1 , one has τ ( E,R IR ) ∝ E Γ -1 . Accurate numerical calculations of the optical depth for the spectral energy distribution of the compact infrared source in the nucleus of M87 (see Fig.8, Section 5) and normalized to the source size R IR = 50 R Schw is shown by black solid curve in the upper panel of Fig.1. Since τ ( E,R IR ) ∝ R -1 IR , the optical depth does not exceed 1 even at the highest detected energies of gamma-rays of about 10 TeV, provided that infrared source is larger than 50 R Schw . This is demonstrated in the lower panel of Fig. 1. Note that the recent claim by Cheung et al. (2007) that the central region of M87 is excluded as a site of the TeV emission because of absorption of γ -rays, is misleading. The authors obtained very large optical depth relevant to the energy /similarequal 25 TeV and assuming an extremely compact infrared source with a linear size of R = R g = R Schw / 2 (the estimate of the optical depth by Cheung et al. (2007) is shown by a cross in the upper panel of Fig. 1, and the dependence of the optical depth on the γ -ray energy is shown by the grey dashed curve). Although formally one cannot rule out such a compact size of the IR source, a significantly larger size cannot be a priory excluded either. Moreover, there are not special reasons to assume that the infrared source is located very close to the event horizon. \nFor the nucleus of M 87, there are no direct measurements of the size of the infrared source. Observations in the microwave band at 43 GHz suggest that the size of the source at the mm wavelength is limited by 5 × 10 16 cm, or approximately 50 R Schw . Lower angular resolution of the infrared telescopes does not allow us to constrain (or marginally resolve, see Perlman et al. (2001)) the size of the nuclear source to ≤ 10 pc (Whysong & Antonucci 2004). However, even assuming that the size of the infrared source is comparable to the size of the microwave source, one finds from the above estimate that the nucleus can be transparent to γ -rays with energies up to 10 TeV. \nEven in the case of the 'maximally compact' infrared source of the size R IR ∼ R Schw / 2, the source is partially transparent for γ -rays. In spite of the fact that the optical depth for γ -rays produced in the center of the infrared source γ -rays becomes very large at energies E > 10 TeV (see the dashed grey curve on the upper panel of Fig. 1), there is no catastrophic absorption of multi-TeV γ -rays. The reason is that in this case the source(s) of γ -ray photons are distributed throughout the infrared source and the thickness H of the transparent surface layer of the source is determined from the condition τ ( E,H ) /similarequal 1. \n∼ \nAssuming a homogeneous γ -ray source, one can find that the luminosity of the last transparent layer ( τ ≤ 1) is only moderately, by a factor of H/R IR ∼ τ ( E,R IR ) (rather than by a factor of exp( -τ ( E,R IR )) lower than the total luminosity of the source. The attenuation of the γ -ray flux in the case of the γ -ray source distributed throughout the infrared source is shown by dashed curves in the lower panel of Fig. 1 for both cases of 50 R Schw (black) and R Schw / 2 (grey) size of the infrared source. We assume the suppression factor (1 + τ ( E,R IR )) -1 which is an interpolation between no supression in the τ = 0 limit and 1 /τ suppression in the τ /greatermuch 1 limit. \n/greatermuch The γ -rays after they escape the nucleus are further attenuated due the pair production in the radiation fields both inside and outside the elliptical galaxy M87. The spectrum of emission from the galactic bulge of M 87 sharply peaks at photon energies around /epsilon1 bulge /similarequal 1 eV. Interactions of nuclear γ -rays with the photon background in M 87 galaxy should therefore lead to maximum absorption at γ -ray energies around E γ /similarequal 1 TeV. The column density of infrared/optical photons in the bulge of the size R bulge and luminosity L bulge along the line of sight is estimated as \nN ph (1 eV) = ∫ R bulge 0 n ph ( r ) dr /similarequal (3) 5 × 10 23 [ L bulge 10 45 erg/s ][ 1 kpc R bulge ] cm -2 , \nwhich allows to estimate the optical depth of gamma-rays at 1 TeV: \nτ M 87 (1 TeV) = σ 0 N ph (1 eV) /similarequal (4) 0 . 08 [ L bulge 10 45 erg/s ][ R bulge 1 kpc ] -1 \nPropagation of γ -rays through the cosmic microwave and infrared backgrounds over the way from M 87 to the observer leads to further absorption of the highest energy quanta. Photons with energies above 10 15 eV are completely absorbed due to interactions with the 2.7 K microwave background (the minimal propagation distance is ∼ 8 kpc). Photons with energies above 100 TeV interact most efficiently with the far-infrared background photons whose density is some 3 orders of magnitude lower than the density of the microwave photons. However, the mean free path of E ≥ 10 TeV γ -rays interacting with far-infrared background, is still shorter than the distance to M 87 (16 Mpc).", '3. PHYSICAL PARAMETERS OF THE CENTRAL ENGINE': 'The high resolution observations of the nucleus of M 87 in X-rays with the Chandra observatory provide an important information about the accretion onto the supermassive BH (Di Matteo et al. 2003). In particular, they give an estimate of the electron density of plasma with a temperature kT ∼ 1 keV, \nn e /similarequal 0 . 1 cm -3 (5) \nat the distance of the order of Bondi accretion radius, R Bondi /similarequal 5 × 10 5 R Schw . The corresponding accretion rate inferred from this estimate is \n˙ M Bondi /similarequal 0 . 1 M /circledot yr -1 . (6) \nInterestingly, the observed bolometric luminosity of the nucleus of M 87 is 4 orders of magnitude below the expected nuclear luminosity corresponding to this accretion rate \nL Bondi /similarequal 10 45 [ η 0 . 1 ] [ ˙ M Bondi 0 . 1 M /circledot / yr ] erg/s . (7) \nwhere η ∼ 0 . 1 is the efficiency of conversion of the rest energy of accreting particles into radiation. This indicates that either the accretion proceeds in a radiatively inefficient way, or the actual accretion rate is still lower than the one inferred from X-ray observations. \nTo estimate the plasma density close to the event horizon of the black hole, one has to assume a certain radial density profile, n ( r ) ∼ r -γ . Depending on the model of accretion flow, the index γ can vary between 1 / 2 (this value is, in fact, a lower limit which can be realized for collisionless motions of individual particles in the central gravitational field) and 3 / 2. The lack of information about the accretion regime leads to a significant uncertainty of the plasma density near the event horizon, \n10 1 . 5 cm -3 < n < n max /similarequal 10 6 . 5 cm -3 . (8) \nRegardless of the uncertainty of this estimate, one may conclude that the strength of magnetic field in the vicinity of the BH can not be very high. Indeed, assuming that the magnetic field is generated by the accreting matter, one can find that the energy density of magnetic field can not exceed the density of the total kinetic energy stored in the particles of the accretion flow. In this case even if the accreting matter moves with relativistic speed, the estimate of maximal possible magnetic field is (assuming that the matter density is n ∼ n max ) \nB eq /similarequal (8 πn max m e ) 1 / 2 ∼ 10 G. (9) \nThus, particle acceleration close to the BH horizon proceeds in the relatively low-density and low-magnetic field environment which significantly limits the range of possible mechanisms of VHE γ -ray emission. Even for the maximally possible acceleration rate, dE/dt /similarequal eB eq , one can find that particles accelerated in a region of a linear size of about the Schwarzschild radius can not reach energies higher than \nE max ≤ eB eq R Schw /similarequal 10 18 eV, (10) \nunless the magnetic field is significantly larger than the equipartition estimate, given by Eq. (9).', '4. GAMMA-RAY EMISSION FROM ACCELERATED PROTONS.': 'Protons accelerated near the BH horizon can produce γ -ray emission in the VHE band through several radiation mechanisms. For example, TeV emission can be synchrotron or curvature γ -ray emission which accompanies proton acceleration (Levinson 2000; Aharonian et al. 2002; Neronov et al. 2005). The energy loss time for protons emitting synchrotron radiation at the energy /epsilon1 synch ,p is short enough to explain the observed day-scale variability of the signal, \nt synch ,p /similarequal 2 . 5 [ B 10 G ] -3 / 2 [ /epsilon1 synch ,p 1 TeV ] -1 / 2 d, (11) \nHowever, the energy of synchrotron and/or curvature photons produced by protons accelerated to the energy E max , given by Eq. (10) is too low to explain the emission at 1-10 TeV, \n/epsilon1 synch ,p /similarequal 0 . 1 [ B 10 G ][ E p 10 18 eV ] 2 GeV (12) \nand \n/epsilon1 curv ,p /similarequal 0 . 01 [ E p 10 18 eV ] 3 [ R Schw R curv ] GeV (13) \n(assuming that typical curvature radius of proton trajectories is R curv ∼ R Schw ). The γ -ray emission from the accelerated protons is thus expected in the 10 MeV 10 GeV energy region observable by GLAST , rather than in the TeV region visible by HESS. \nIt is, in principle, not excluded that during short episodes of enhanced accretion the magnetic field can rise up to 10 3 G, which would, in principle, allow proton acceleration up to the energies E max ∼ 10 20 eV. Thus, the energy of curvature emission given by Eq.(13) can extend up to 10 TeV. Note, however, that even in this case the observed emission can not be related to the proton synchrotron radiation which has an intrinsic self-regulated synchrotron cut-off at /epsilon1 synch ≤ 0 . 3 TeV (Aharonian 2000), if the region of proton acceleration is spatially coincident with the region of synchrotron emission. A potential problem of assumption about transient enhancement (by 4 orders of magnitude, to produce the necessary increase of equipartition magnetic field, see Eq.(9)) of accretion rate is that it should result, in general, in a broad-band flaring activity of the nucleus of M 87, which, however, is not observed. \nVHE γ -rays are produced also in proton-proton ( pp ) collisions. However, the interaction time of high-energy protons propagating through the low density medium ( n ≤ 10 7 cm -3 ) is quite large, \nt pp /similarequal 1 σ pp n max /similarequal 10 [ 10 7 cm -3 n max ] yr (14) \n( σ pp ∼ 10 -26 cm 2 is the proton-proton interaction crosssection). Even if pp interactions would significantly contribute to the overall VHE γ -ray emission, they cannot explain the fast day-scale γ -ray flux variability of M 87. Therefore the observed variability should be referred to fast changes in the concentration of multi-TeV protons in the source, i.e. due to adiabatic or escape losses of protons, on timescales comparable to t var . The fast non-radiative losses versus slow rates of γ -ray production at pp collisions implies very low efficiency of conversion of the energy of parent protons to the VHE γ -rays, κ = t var /t pp ≤ 3 × 10 -4 . Thus to explain the γ -ray luminosity L γ ∼ 3 × 10 40 erg/s, the proton acceleration power should exceed κ -1 L γ ∼ 10 44 erg/s which is just about the luminosity L Bondi given by Eq. (7), assuming a conventional, 10% or so, efficiency of conversion of the rest mass energy of accreting particles into radiation. (Here we ignore a possible formation of gamma-rays in a relativistic outflow moving towards the observer which in principle would reduce by an order of magnitude this requirement). \nThe energy requirements to the proton acceleration power can be somewhat relaxed if one invokes interactions of protons with the the surrounding radiation fields. Although TeV γ -rays can be produced in a two-step process which includes Bethe-Heitler pair productions ( pγ → pe + , e -) and synchrotron radiation of secondary electrons, pγ interactions become efficient when they proceed through the photomeson production channel. In order to interact with the photons of the infrared source with average energy /epsilon1 IR ∼ 10 -2 eV, protons should be accelerated to E p ∼ [200 MeV //epsilon1 IR ] m p ∼ 2 × 10 19 eV. The number density of IR photons in the compact infrared source is \nn IR = L IR 4 πR 2 IR /epsilon1 IR /similarequal (15) \n7 × 10 9 [ L IR ( /epsilon1 IR ) 10 41 erg/s ][ 0 . 01 eV /epsilon1 IR ][ 50 R Schw R IR ] 2 cm -3 \nFor the average cross-section of the photo-pion production cross-section, σ pγ ∼ 10 -28 cm 2 , the interaction time of protons with infrared photons is \nt pγ = 1 σ pγ n IR /similarequal 1 . 7 [ 10 41 erg/s L IR (0 . 01eV) ][ R IR 50 R Schw ] 2 yr (16) \nIf the infrared source is very compact, R IR ∼ R Schw / 2, and the the accretion rate is transiently increased by 2 orders of magnitude (to allow an order-of-magnitude increase in equipartition magnetic field and, as a consequence, proton acceleration to E > 10 19 eV), the pγ cooling time can be as short as the observed TeV variability time scale. \nNote that the hypothesis of TeV γ -ray emission of M87 based on the assumption of pγ interactions, requires a very compact IR source with a size R IR ∼ R Schw . This implies strong absorption of gamma-rays with fast multiplication of electron-positron pairs via Klein-Nishina cascades. Actually, photon-photon pair production is an important element of any pγ model; the observed spectrum of TeV γ -rays cannot be explained by first generation of ultra-high energy ( ≥ 10 15 ) photons from π 0 decays, and therefore requires production of secondary electrons which would provide broad-band emission in the TeV energy band. On the other hand, the copious pair production may lead to neutralization of the large scale ( ≥ R Schw ) electric field, and thus to significant reduction of the maximum achievable energy of protons given by Eq.(10). Since the rate of photomeson processes in the nucleus of M87 is very sensitive to the energy of protons, namely, it requires E p ≥ 10 18 , the generation of large amount of secondary electrons may result in a dramatic drop of the rate of photo-meson production. A non-negligible contribution to the secondary electrons may come also from the Bethe-Heitler pair production, especially when the efficiency of photomeson production is suppressed (the energy threshold of this process is two orders of magnitude smaller than the energy threshold of the photomeson production). Whether this mechanism can explain the observed spectral and temporal characteristics of TeV γ -ray emission from M87, is a question which needs detailed numerical calculations. In any case it is clear that pγ models can provide adequate efficiency only in the case of a very compact IR source with a size \nclose to the Schwarzschild radius.', '5.1. Order-of-magnitude estimates': 'The tough requirements of acceleration of protons to ultrahigh energies ( E ≥ 10 18 eV), as well as the relevant long cooling times challenge any interpretation of the day-scale variability of TeV γ -rays in terms of interactions of high-energy protons. The models based on acceleration of electrons do not face such problems, and are likely to be responsible for the observed TeV γ -ray emission. \nThe main emission mechanisms by electrons in the vicinity of the supermassive BH are synchrotron/curvature radiation and inverse Compton (IC) scattering. Electrons can be accelerated to multi-TeV energies only if the strength of the chaotic component of the magnetic field, B rand , in the acceleration region is not too high. Assuming that electrons are accelerated at a rate dE/dt ∼ κeB ord ( κ ≤ 1 and B ord is the ordered component of the magnetic field), from the balance of the acceleration and synchrotron energy loss rates one finds \nE e ≤ κ 1 / 2 m 2 e B 1 / 2 ord e 3 / 2 B rand /similarequal (17) 4 × 10 13 [ B ord 1 G ] 1 / 2 [ B rand 1 G ] -1 κ 1 / 2 eV . \nThus, even in the case of maximum possible acceleration rate ( κ = 1) electrons cannot emit in the 10-100 TeV band unless \nIn the ordered field the energy dissipation of electrons is reduced to curvature radiation loses. From the balance between the curvature loss rate and the acceleration rate, assuming that the typical curvature radius R curv of magnetic field is comparable to the gravitational radius, one finds \nB rand ∼ < 1 G . (18) \nE e = [ 3 m 4 e R 2 curv κB ord 2 e ] 1 / 4 /similarequal (19) 4 × 10 15 [ B ord 1 G ] 1 / 4 [ R curv R Schw ] 1 / 2 κ 1 / 4 eV . \nThus if the energy losses of electrons are dominated by curvature radiation, the maximum energy of accelerated electrons only weakly depends on the strength of the magnetic field. \nThe IC loss rate is determined by the energy density of infrared radiation in the nucleus, U ph = L IR / (4 πR 2 IR c ). The condition of the balance between IC loss rate and the electron acceleration rate gives \nE e /similarequal 3 3 / 4 m 2 e B 1 / 2 ord R IR 2 5 / 4 e 3 / 2 L 1 / 2 IR /similarequal (20) 1 × 10 15 [ B ord 1 G ] 1 / 2 [ R IR 50 R Schw ] κ 1 / 2 eV . \nNote that this estimate is obtained assuming that the IC scattering takes place in the Thompson regime. However, \nhighest energy electrons upscatter the infrared/optical radiation in the Klein-Nishina regime in which the efficiency of the IC scattering is reduced. A proper account of the decrease of the IC loss efficiency would result in higher electron energies exceeding the estimate of Eq. (20).', '5.2. Electron acceleration in the vacuum gaps of BH magnetosphere': "So far we did not specify the particular mechanism of particle acceleration. In principle, several mechanisms can be responsible for the electron acceleration, but an obvious requirement which follows from the above estimates is that the 'efficiency' parameter κ for the acceleration rate should be close to one, otherwise electrons would not reach the ∼ 100 TeV energies, as it follows from Eqs. (17),(19),(20). Also, the irregular component of the magnetic field should not exceed 1 G. \nLarge scale ordered electric fields, induced by rotation of black hole, are known to be responsible for particle acceleration and high-energy radiation in pulsars. A similar mechanism of generation of large electric fields can be realised in the vicinity of a rotating BH placed in an external magnetic field (Wald 1974; Bicak et al. 1976). In the case of pulsars, it is known that the force-free magnetosphere possesses so-called 'vacuum gaps' in which the rotation-induced electric field is not neutralized by redistribution of charges. The vacuum gaps work as powerful particle accelerators and sources of pulsed high-energy γ -ray emission. Vacuum gaps with strong rotation-induced electric field can be present also in the vicinity of a rotating black hole (Blandford & Znajek 1977; Beskin et al. 1992). Below we explore whether the observed VHE γ -ray emission from M 87 can be explained by the emission from the vacuum gaps formed in the magnetosphere of the supermassive black hole in M87.", '5.2.1. The magnetosphere of rotation-powered black hole': "Throughout the magnetosphere the component of electric field directed along the magnetic field lines is neutralized by the charge redistribution, so that a force-free condition /vector B ⊥ /vector E is satisfied. The characteristic charge density needed to neutralize the parallel component of electric field in the magnetosphere of a BH rotating with an angular velocity /vector Ω placed in an external magnetic field /vector B is the so-called 'Goldreich-Julian' density (Goldreich & Julian 1969) \nn q /similarequal /vector Ω · /vector B ord 2 πe /similarequal aB ord ( GM ) 2 (21) \n( a = Ω( GM ) 2 , the BH rotation moment per unit mass, 0 < a < GM , is a commonly used parameter of the Kerr metric describing the space-time of rotating black hole; see Appendix). \nIn general, the charge distribution in the magnetosphere is not static - additional free charges should be continuously supplied throughout the magnetosphere to compensate for the charge loss due to the magnetohydrodynamical outflow. The inefficiency of charge supply can lead to the formation of 'gaps' in the magnetosphere in which the parallel component of electric field is not zero and conditions for particle acceleration exist. \nIn the case of pulsars, there are several potential ways to supply charged particles to the magnetosphere. First of all, the charge can be extracted directly from the surface of the neutron star. Electrons and positrons can be generated also due to pair production in very strong magnetic field. Finally, electron-positron pairs can be created at interactions of γ -rays with low energy photons. \nApart from the extraction of free charges from the surface of the compact object, the same mechanisms can in principle, be responsible for the charge supply to the magnetosphere in the case of black holes. However, in the particular case of the black hole in M87, the pair production of γ -rays in the magnetic field, (a mechanism, assumed e.g. in the Blandford & Znajek (1977) scenario) is not efficient because (1) the magnetic field cannot significantly exceed 10 G and (2) the energy of γ -rays emitted by accelerated particles cannot exceed 100 TeV. On the other hand, the efficiency of the charge supply via the pair production by γ -rays on the soft infrared background depends on the compactness of the infrared source ( ∝ L IR /R ). This process can be efficient only if γ -rays with energies above 10 TeV are present in the compact source. \nSince the 10 TeV γ -rays have to be produced by particles accelerated to energies above 10 TeV, the gap(s), in which electric field component along the magnetic field lines is not neutralized by the charge redistribution, should be present in the magnetosphere. In a selfconsistent scenario the height of the gap(s) is limited by the condition that γ -rays emitted by the accelerated particles do not produce e + e -pairs within the gap. \nIn order to estimate whether particle acceleration and high-energy emission from the vacuum gaps can be responsible for the observed VHE luminosity of M 87 one has to estimate the total acceleration power output in the gap. In spite of the fact that the potential drop in the gap can be enough to accelerate charge particles to energies as high as 10 18 eV, strong radiative losses limit the maximum energy of electrons to 10 3 TeV. This means that the propagation of electrons through the gap proceeds in a 'loss-saturated' regime: all the work done by the gap's electric field is dissipated through the synchrotron/curvature and/or IC radiation. The rotation induced electric field near the BH horizon has a strength (see Appendix) E ∼ [ a/GM ] B ord . For each electrons propagating in the gap, the energy loss rate is estimated as dE/dt /similarequal e E ∼ e [ a/ ( GM )] B ord . 1 The density of electrons in the gap is limited by the Goldreich-Julian density, given by Eq. (21). If the size of the infrared source is large enough, so that the gap height is not limited by pair production, the size of the gap is estimated to be about H ∼ R Schw . Taking into account that the volume of the gap is roughly R 2 Schw H , the total number of electrons in the gap can be estimated as R 2 Schw Hn q . Then the total power output of the gap is \nP n q HR 2 Schw ( dE/dt ) (22) \n∼ 5 × 10 41 4 a 2 R 2 Schw [ M 3 × 10 9 M /circledot ] 2 [ H R Schw ][ B ord 10 G ] 2 erg/s \n/similarequal \n× 1 This implies that the acceleration efficiency κ is κ /similarequal [ a/ ( GM )]. For the extreme rotating black hole with a = GM , the acceleration reaches the maximum possible rate with κ /similarequal 1. \nThus, if the angular momentum of the black hole is large enough, the nonthermal power of the vacuum gap can be as large as the observed TeV gamma-ray luminosity.", '5.2.2. Numerical modelling of acceleration and radiation of electrons in the gap of the black hole magnetosphere': "Location of the gaps in the BH magnetosphere depends on the structure of both the accretion flow and the magnetic field near the event horizon. In this regard, it should be noted that even after four decades of intensive theoretical study of physics of pulsar magnetospheres, the details of the geometry of vacuum gaps remain uncertain. Nevertheless, the basic properties of particle accelerators operating in the vacuum gaps can be understood with a reasonable accuracy and confidence. \nWe have developed a numerical code which allows, for the given geometry of the gap and configuration of the magnetic field, a quantitative study of energy distributions of electrons accelerated in the vacuum gap and associated electromagnetic radiation. For demonstration of the importance of the VHE γ -ray emission from the vacuum gaps, in this paper we have chosen a simple geometry of the gap, namely we assumed that the gap occupies a spherical layer above the BH horizon and has the height of about the size of the event horizon. The geometry of electromagnetic field is assumed to be given by the solution of Maxwellian equations in Kerr metric, which corresponds to an asymptotically constant magnetic field inclined at an angle χ with respect to the rotation axis of the black hole (Bicak et al. 1976). The analytical solution of Maxwellian equations are given in Appendix. The initial locations of electrons are assumed to be homogeneously distributed either throughout the gap or over the boundary of the gap. The initial energies of electrons are assumed to be equal to the rest energy. Trajectories have been numerically integrated taking into account effects of General Relativity and energy losses of electrons due to synchrotron/curvature radiation and inverse Compton scattering (see Appendix for technical details). The spectra and angular distributions of the synchrotron and curvature radiation are calculated by tracing the photon trajectories through the Kerr space-time metric from the emission point to infinity. \nAs an example of numerical modelling, in Fig. 2 we show distributions of average energies of electrons propagating in the spherical layer occupied by the gap. The magnetic field is assumed to be inclined at an angle χ = 20 · with respect to the rotation axis. The two left panels show angular distributions of the average energy of individual electrons and the power of synchrotron/curvature emission, as measured in the Zero Angular Momentum Observer (ZAMO) frame (Bardeen et al. 1972), close to the black hole horizon (see Appendix for details). One can see that the maximum energies of photons are achieved in two oppositely situated hot-spots determined by the direction of magnetic field. At the same time, the total power of radiation does not strongly depend on the latitude and longitude coordinates. \nThe 'dark strips' (one along the equatorial plane and two snake-like dark strips above and below the equatorial plane) are clearly recognizable in the left panels of Fig. 2. The drop of energies is explained by the specific configuration of electromagnetic field in these regions. \nNamely, the dark strips surround the so-called 'forcefree' surfaces at which the rotation-induced electric field /vector E is orthogonal to the magnetic field /vector B ord . \nThe right panel of Fig. 2 shows the angular distribution of average photon energies and emissivity after the photon tracing to infinity through the Kerr space-time metric. One can see that polar 'hot spots' become more pronounced, mostly because the photons emitted from equatorial regions (from the ergosphere) have larger redshifts and a significant fraction of these photons is just absorbed by the black hole. \nFig. 3 shows evolution of the shape of the polar hot spots with an increase of the inclination angle of the magnetic field. One can see that the shape of the hot spots becomes wider and irregular. Also, with an increase of the inclination of the magnetic field the average photon energy decreases, which is explained by the fact that the acceleration of electrons is most efficient when the electric field is aligned with the magnetic field. With the increase of the magnetic field inclination angle χ , the regions of aligned electric and magnetic field (situated close to the rotation axis of the BH in the case χ = 0 · ) disappear. \nIn Fig. 4 a typical spectrum of electrons accelerated in the spherical vacuum gap close to the BH horizon is shown. It is assumed that an extreme rotating BH ( a = GM ) is placed in 1 G magnetic field inclined at an angle of 60 · with respect to the rotation axis. The size of the infrared emission region is assumed R IR = 10 R Schw . The three energy spectra shown in Fig. 4 correspond to different strengths of the random component of magnetic field. If the random magnetic field is smaller than 10 -3 fraction of the ordered field, electrons propagating in the gap reach energies up to ∼ 10 16 eV. With an increase of the random component of magnetic field, the synchrotron losses start to dominate which leads to reduction of the maximum electron energies energies, in a good agreement with the qualitative estimates of Section 5.1.", '5.3. Direct synchrotron/curvature and IC radiation from the acceleration process': "The radiative losses through both synchrotron/curvature and IC channels are released in the form of high energy γ -rays. The energy of Compton upscattered photons (in Thompson regime) is \n/epsilon1 IC = 0 . 4 [ /epsilon1 IR 10 -2 eV ] [ E e 1 TeV ] 2 TeV (23) \nThe IC scattering of E e ∼ > 10 TeV electrons on IR photons proceeds in the Klein-Nishina regime, thus /epsilon1 IC /similarequal E e . The curvature radiation peaks at significantly lower energies, \n/epsilon1 curv = 3 E 3 e 2 m 3 e R curv /similarequal 0 . 2 [ E e 10 15 eV ] 3 [ R Schw R curv ] GeV . (24) \nSince electron acceleration in the gap proceeds in the 'loss saturated' regime, the calculations of the spectral and angular distributions of radiation accompaning the acceleration process, requires 'self-consistent' approach in which the spectrum of radiation is calculated simultaneously with the spectrum of parent electrons. The algorithm of self-consistent calculations used in this work \nθ \nFig. 2.Angular distributions of synchrotron/curvature photon energies (top) and γ -rayproduction rate (bottom) for emission from a spherical vacuum gap close to an extreme rotating BH placed in an magnetic field B = 1 G inclined at an angle χ = 20 · with respect to the BH rotation axis. Two left panels show the angular distributions as seen in ZAMO frame close to the black hole horizon. Two right panels show the angular distributions after photon tracing to infinity. The scales are logarithmic and cover two decades from maximum (yellow on the top panels, red on the bottom panels) scale, max , down to the 0 . 01 max (black). Red and blue boxes on the bottom right panel show the regions used to extract spectra seen from 'on hot spot' and 'off hot spot' directions as shown in Fig. 5. \n<!-- image -->", 'is briefly described in the Appendix.': "Some results of self-consistent calculations of the γ -ray production spectra as functions of the viewing angle and the inclination of the magnetic field are shown in Figs. 5 and 6, respectively. Fig.5 demonstrates the difference of production spectra of γ -rays emitted along the direction of magnetic field (the region marked 'on' in Fig. 2) and away from this direction (the region marked 'off' in Fig. 2). Fig. 6 demonstrates the dependence of the γ -ray production spectra on the inclination angle of magnetic field. In both figures the low-energy (MeV-GeV) peak is due to the synchrotron/curvature radiation, while the high energy (TeV-PeV) peak is formed due to the IC scattering in the Klein-Nishina regime.", '5.4. Isotropic TeV emission from secondary pair-produced electrons': "The spectral energy distributions shown in Figs. 5 and 6 correspond to the production rates of the first generation γ -rays. They have essentially anisotropic distribution, thus the calculations of fluxes detected by an observer contain large uncertainties, mainly because of the poor knowledge of the source geometry. However, due to the internal and external absorption of ≥ 10 TeV γ -rays, the observer detects only a tiny fraction of the first generation γ -rays. While interactions with external photon fields lead to real attenuation of the γ -rayflux, the internal absorption is essentially recovered due to radiation of the pair produced electrons of second and further generations. Interestingly, the development of an electromagnetic cascade in radiation field of the infrared source may lead to 'isotropisation' of \nthe γ -ray source. Indeed, the absorption of first generation γ -rays leads to deposition of e + e -pairs throughout the infrared source volume, R IR . If the latter is significantly larger than the volume corresponding to the vacuum gap (i.e. R IR /greatermuch R Schw ), the magnetic field in the IR source can be dominated by the irregular component which would effectively isotropise the directions of secondary electrons. Correspondingly, the secondary radiation from e + e -pairs will be emitted isotropically. For any reasonable magnetic field, the synchrotron radiation of secondary electrons is produced at energies significantly below 1 TeV. Therefore for explanation of the observed TeV gamma-radiation one should assume that the energy losses of electrons are dominated by IC scattering, i.e. the magnetic field in the infrared source should be significantly less than B = ( L IR / 2 R 2 IR ) 1 / 2 ∼ 0 . 1 G. If so, the absorption of first generation gamma-rays will trigger an electromagnetic (Klein-Nishina) cascade. \nIn Fig. 7 we show the resulting spectrum of gammaradiation expected from the internal absorption of first generation gamma-rays. It consists of the isotropic component associated with the cascade in the infrared source (green dashed curve) and the primary anisotropic component whose intensity is uncertain since it strongly depends on the orientation of the observer with respect to the magnetic field direction. In Fig. 7 the thin solid red line corresponds to the sum of these two components, while the thick red solid line shows the result of absorption of the summary spectrum in the infrared source, in the elliptical galaxy M87 and in the intergalactic medium (see thick solid curve in Fig. 1). The curve \nFig. 3.Evolution of the shape of the polar hot spots with an increase of the inclination angle of magnetic field. The figures show the angular distribution of the energies of photons of synchrotron/curvature radiation traced to infinity. The color scale and parameters of numerical simulations are the same as in the top right panel of Fig. 2: maximum yellow corresponds to photon energies 10 GeV, minimum (black) to photon energies below 0.1 GeV. \n<!-- image --> \nis normalized to the observed flux of γ -rays at 0.5 TeV. The comparison of the calculated γ -ray spectrum shows quite a good agreement with the HESS measurements up to E ∼ 10 -20 TeV. One should note, however, that the agreement with the observations should not be overemphasized, since we consider a 'toy model' aimed to demonstrate the importance of TeV emission from the vacuum gaps in the magnetosphere. \nFinally in Fig. 8 we show the broad-band spectral energy distribution (SED) of the resulting radiation and compare the model curve with observed fluxes of the nucleus of M87 at infrared, X-ray and TeV gamma-rays. Two broad peaks in the SED correspond to synchrotron radiation and inverse Compton scattering of secondary (cascade) electrons in the infrared source. The condition that the synchrotron emission from the infrared source \nFig. 4.Spectrum of electrons accelerated in the vacuum gap above the horizon of a maximally rotating BH of a mass M = 3 × 10 9 M /circledot placed in an ordered magnetic field B = 1 G inclined at θ = 60 · with respect to the BH rotation axis. Black solid line: the random magnetic field is 0.1% of the ordered one; blue dotted line: the random magnetic field is 1%; red dashed line: random magnetic field is 10%. \n<!-- image --> \nFig. 5.The production rate of first generation gamma-rays emitted by the spherical vacuum gap. The physical parameters are the same as in Fig. 2. Thick solid line: the total spectrum integrated over all directions. Dashed line: the spectrum integrated over the direction around the 'hot spot' (the box marked 'on' in Fig. 2). Thin solid line: the spectrum collected from the box marked 'off' in Fig. 2. \n<!-- image --> \nFig. 6.The production rates of gamma-rays calculated for different values of the inclination angle of magnetic field. Solid line: χ = 20 · ; dashed line: χ = 60 · ; long-dashed line: χ = 90 · . Physical parameters are same as in Fig. 2. \n<!-- image --> \nFig. 7.Secondary emission from high-energy electrons injected into the compact infrared source in the nucleus of M 87 via the photon-photon pair production. Thick black solid line: omnidirectional spectrum of primary emission from accelerated particles; thin blue solid line: the primary spectrum only from the 'off' direction (same as in Fig. 5); black and blue thin dashed lines show the spectra attenuated by the pair production in the infrared source. Red dotted line shows the contribution of secondary cascade (isotropic) emission. \n<!-- image --> \nFig. 8.Spectral energy distribution of emission from the nucleus of M 87. Dashed blue line: isotropic synchrotron and IC emission from secondary electron positron pairs injected via the pair production. Short-dashed green line: direct synchrotron/curvature and IC emission from electrons accelerated in the vacuum gap (strongly anisotropic, depends on the geometry of the vacuum gap). Solid blue line shows the total emission spectrum, which is the sum of the direct and cascade contributions. Dashed red line shows the model spectrum of soft photon background used for the calculation of IC scattering. This radiation component can come from a larger region and is not necessarily related to the particle acceleration in the vacuum gap. \n<!-- image --> \nshould not exceed the observed flux in the X-ray band imposes an upper limit on the random magnetic field strength B < 0 . 1( R IR /R Schw ) -1 G. The existing upper limit on the M87 flux in the EGRET energy band (Sreekumar et al. 1994) imposes a restriction on the direct synchrotron/curvature emission from gap emitted in the direction of observer. \nBecause of uncertainties of model parameters as well as the strong variability of radiation, we do not attempt to make a detailed fit to the broad band spectrum of the source, especially taking into account that the measurements at different energy bands correspond different epochs. On the other hand, the future simultaneous studies of temporal and spectral properties of the broad-band emission can provide meaningful tests of the proposed model and significantly reduce the relevant parameter space.", '6. SUMMARY AND CONCLUSIONS.': 'We have studied the mechanisms of production of variable TeV γ -ray emission from vicinity of the supermas- \nive BH in the nucleus of M 87. Moderate accretion rate onto the black hole, inferred from the Chandra observations, limits the magnetic field strength close to the black hole horizon - it cannot significantly exceed B ≤ 10 G. This limits the maximum energy attainable by protons, E ≤ 10 18 eV. None of the known mechanisms of γ -ray emission by protons can satisfactorily explain the observed temporal and spectral characteristics of the observed TeV γ -ray emission. On the other hand, severe radiative losses of electrons in a compact region close to the black hole significantly constrains the range of possible acceleration scenarios. In particular the random component of the magnetic field cannot exceed 1 G. Thus, the acceleration takes place, most likely, in a region where the regular magnetic field significantly exceeds the random component of the field. Even so, the unavoidable energy losses due to the curvature radiation and inverse Compton scattering require an extremely effective mechanism of particle acceleration with a rate close to the maximum (theoretically possible) acceleration rate. \nIn this paper we show that the observed TeV gammaray emission from M87 can be explained by electrons accelerated in strong rotation induced electric fields in the vacuum gaps in black hole magnetosphere. Generally, this model has many similarities with models of particle acceleration in pulsar magnetospheres. Our detailed modelling shows that the gamma-radiation from the central engine of M 87 consists of both first generation photons emitted by particles accelerated in the gap (severely attenuated due to interactions with the internal and external radiation fields) and second and further generation (cascade) photons. If the first component dominates above 10 TeV, the cascade γ -rays contribute mainly to the ≤ 10 TeV energy domain. The electron acceleration and γ -ray production in a very compact region close to the event horizon of the black hole naturally explains the observed variability of TeV γ -ray emission from M87.', '7. ACKNOWLEDGEMENT.': 'We would like to thank V.Beskin for the clarifying comments on he manuscript.', 'DETAILS OF THE NUMERICAL MODELLING OF PARTICLE ACCELERATION IN THE VACUUM GAPS IN MAGNETOSPHERES OF ROTATING BLACK HOLE.': "A self-consistent modelling of acceleration of electrons in the direct vicinity of event horizon of a BH requires (a) a full account of the effects of General Relativity and (b) a full account of the radiation reaction on particle motion, since electrons propagate most of the time in the 'loss saturated' regime when the acceleration force is balanced by the radiation reaction force. Below we give some details of the modelling of trajectories electrons and photons in the vicinity of the black hole.", 'The Kerr space-time.': "A rotating BH is described by two parameters: its mass M and the angular momentum per unit mass a ≤ GM . The geometry of space-time in the vicinity of horizon is described by the Kerr metric \nds 2 = -α 2 dt 2 + g ik [ dx i + β i dt ] [ dx k + β k dt ] (A1) \nα = \nρ √ ∆ Σ ; \ng rr = ρ 2 ∆ ; \ng θθ = ρ 2 ; \ng φφ = Σ 2 sin 2 θ ρ 2 ; \nβ φ = -2 aGMr Σ 2 ; ∆ = r 2 + a 2 -2 GMr ; \nΣ 2 = ( r 2 + a 2 ) 2 -a 2 ∆sin 2 θ ; ρ 2 = r 2 + a 2 cos 2 θ (A2) \nThe horizon is situated at r H = GM + ( GM ) 2 -a 2 . \n√ \n-To understand the acceleration and energy losses of charged particles propagating close to the BH horizon, it is convenient to use an orthonormal (non-coordinate) frame \ne ˆ 0 = Σ ρ √ ∆ ∂ ∂t + 2 GMar Σ ρ √ ∆ ∂ ∂φ ; e ˆ r = √ ∆ ρ ∂ ∂r ; e ˆ θ = 1 ρ ∂ ∂θ ; e ˆ φ = ρ Σsin θ ∂ ∂φ . (A3) \ncarried by the so-called 'zero angular momentum' observers (ZAMO) (Bardeen et al. 1972). The corresponding covariant basis vectors e ˆ i are given by \ne ˆ 0 = ρ √ ∆ Σ dt ; e ˆ r = ρ √ ∆ dr ; e ˆ θ = ρdθ, e ˆ φ = Σsin θ ρ dφ -2 GMar sin θ ρ Σ dt. (A4) \nThe electromagnetic field. \nIn the reference frame (A3) the magnetic field inclined at angle χ with respect to the BH rotation axis is given by (Bicak et al. 1976) \nwhere \nB ˆ r = 1 Σ ρ 4 sin θ { B ‖ sin θ cos θ [ ∆ ρ 4 +2 GMr ( r 4 -a 4 ) ] + B ⊥ [ r cos ψ -a sin ψ ] [ ρ 4 ( r sin 2 θ + GM cos 2 θ ) -GM ( r 2 + a 2 )( r 2 cos 2 θ + a 2 cos 2 θ ) ]} B ˆ θ = -1 Σ ρ 4 √ ∆ { B ‖ ∆sin θ [ ρ 4 r + a 2 GM ( r 2 -a 2 cos 2 θ )(1 + cos 2 θ ) ] + B ⊥ cos θ [ ρ 4 ( (∆ r -GMa 2 ) cos ψ + a (∆ + GMr ) sin ψ ] -a 2 GMr sin 2 θ ( r 2 ( r -2 GM ) + 2 a 2 ( r sin 2 θ + GM cos 2 θ ) ) cos ψ -a (∆ -2 GMr -2 a 2 cos 2 θ ) sin ψ ]} (A5) \n√ \nψ = φ + a 2 ( GM ) 2 -a 2 ln [ r -GM + √ ( GM ) 2 -a 2 r -GM -√ ( GM ) 2 -a 2 ] ; B ‖ = B 0 cos χ, B ⊥ = B 0 sin χ. (A6) \n√ Rotation of the BH is responsible for the appearance of nonzero electric field whose components are \nE ˆ r = aGM Σ∆ ρ 6 { B ‖ ∆ [ 2 r 2 ρ 4 sin 2 θ -(Σ 2 -2 GMra 2 sin 2 θ )( r 2 -a 2 cos 2 θ )(1 + cos 2 θ ) ] -B ⊥ r sin θ cos θ [ 2 [ ( r ∆ -GMa 2 ) cos ψ -a (∆ + rGM ) sin ψ ] + (Σ 2 -2 GMra 2 sin 2 θ ) [ r 2 ( r -2 GM ) + 2 a 2 ( r sin 2 θ + GM cos 2 θ ) cos ψ - \na (∆ -2 GMr -2 a 2 cos 2 θ ) sin ψ ]]} E ˆ θ = aGM Σ √ ∆ ρ 6 { 2 B ‖ r sin θ cos θ [ ∆ ρ 4 -( r 2 -a 2 ) [ Σ 2 -2 GMr ( r 2 + a 2 ) ]] + B ⊥ [ 2 rρ 4 [ r ( r sin 2 θ + GM cos 2 θ ) cos ψ -a ( r sin 2 θ + GM cos 2 θ ) sin ψ ] -( r 2 cos 2 θ + a 2 cos 2 θ ) Σ 2 -2 GMr ( r 2 + a 2 ) ] ( a sin ψ -r cos ψ ) ]} (A7) \n[ \n] Equations of motion for a charged particle. \nThe components of the four-velocity of a particle v µ = dx µ /dt in the orthonormal frame e ˆ a (A4) are \nv µ = v ˆ a e µ ˆ a d ˆ t/dt (A8) \nwhere ˆ t is the time which would be locally by the ZAMO observers at a given point and t is the coordinate time which enters the metric (A1). For example, the φ components of particle velocity in coordinate and orthonormal reference frame are related through \nv ˆ φ = Σsin θ ρ [ dφ dt -2 GMar Σ 2 ] dt d ˆ t (A9) \nThe extra term Ω = 2 GMar/ Σ 2 is the angular velocity of the ZAMO frame at each point. From (A4) one can see that \nd ˆ t dt = ρ √ ∆ Σ (A10) \nIt is convenient to introduce a particle γ -factor in the orthonormal frame γ = 1 / √ 1 -( v ˆ a ) 2 . Equations of motion in the orthonormal basis (A3) have the same form as in the flat space (Thorne et al. 1986) \nwhere /vector p is the particle momentum \nd/vectorp d ˆ t = e ( /vector E + /vectorv × /vector B ) + mγ/vectorg + ˆ H/vectorp + /vector f rad (A11) \np ˆ a = mγv ˆ a (A12) \n/vectorg is the gravitational acceleration and ˆ H is the tensor of gravi-magnetic force. The force /vector f rad is the radiation reaction force. In the case of interest the time scales for acceleration in electromagnetic field and of the radiation reaction are orders of magnitude shorter than that of the motion of the particle in the gravitational field of the black hole. \nThe radiation reaction force /vector f rad for the ultra-relativistic particles moving in external electromagnetic field is (see, e.g. (Landau & Lifshitz 1975)) \n/vector f rad = 2 e 4 γ 2 3 m 2 [ ( /vector E + /vectorv × /vector B ) 2 -( /vectorv · ( /vector E + /vectorv × /vector B )) 2 ] /vectorv | v | (A13) \nNote that if particles move at large angle with respect to the magnetic field lines, this expression will describe mostly synchrotron energy loss. However, in the case when particles move almost along the magnetic field lines, the last equation will 'mimic' the effect of curvature energy loss (taking into account the fact that the typical curvature radius of the magnetic field lines in the considered case is about R Schw ).", 'REFERENCES': "Aharonian, F.A., 2000, New AR, 5, 377. \nAharonian F.A. and Neronov, A. 2006, Ap.J., 619, 306 Aharonian, F. A., Belyanin, A. A., Derishev, E. V., Kocharovsky, V. V., Kocharovsky, Vl. V. 2002, Phys ReV. D, 66, id. 023005 Aharonian, F. et al. (HEGRA collaboration) 2003, A&A, 403, L1 Aharonian, F. et al. (HESS collaboration) 2006, Science, 314, 1424 Bardeen, J.M., Press, W.H., Teukolsky, S.A., 1972, Ap.J. 178, 347. \nBeskin V.S., Istomin Ya.N., Par'ev V.I., 1992, Soviet Astronomy, 36, 642. \nBicak J., Dvorak L., 1976, Gen.Rel.Grav., 7, 959; see also Bicak J., Janis V., 1985, MNRAS, 212, 899. \nBiretta, J. A., Sparks, W. B., & Macchetto, F., 1999, ApJ, 520, 621. Blandford R.D., Znajek R.L., 1977, MNRAS, 179, 433. Cheng K.S., Ho C., Ruderman M., 1986, Ap.J. 300, 500. Cheung C.C., Harris D.E., Stawarz L., 2007, Ap.J.Lett. accepted, arXiv:0705.2448. \nDi Matteo T., Allen S.W., Fabian A.C., Wilson A.S. Young A.J. 2003, ApJ, 582, 133. \nFabian, A. 2006, Science, 314, 1398 Georganopoulos, M., Perlman, E. S., Kazanas, D. 2005, ApJ, 634, L33 Goldreich P., Julian W.H., 1969, Ap.J. 157, 869. Harris D.E., Biretta A.J., Junor W., Perlman E.S., Sparks W.B., Wilson A.S., 2003, Ap.J., 586, L41. Heinz S.; Begelman M.C., 1997, Ap.J., 490, 653 Landau L.D., Lifshitz, E.M., 1975 The Classical Theory of Fields , Oxford: Pergamon Press Levinson A., 2000, Phys. Rev. Lett., 85, 912. Marconi, A., Axon, D. J., Macchetto, F. D., Cappetti, A., Sparks, W. B., Crane, P. 1997, MNRAS, 289, L21 Michel F.C., 2004, Ad.Sp.R., 33, 542. Neronov A., Semikoz D., Aharonian F., Kalashev O., 2002, Phys.Rev. Lett., 89, 1101. Neronov A., Tinyakov P., Tkachev I., 2005, JETP, 100, 656. Perlman E.S., Sparks W.B., Radomski J., Packham C., Fisher R.S., Pina R., Biretta J.A. 2001, Ap.J., 561, L51. Reimer, A., Protheroe, R. J., Donea, A.-C. 2004, A&A, 419, 89 Sreekumar P. et al., 1994, Ap.J. 426, 105. \n- Stawarz, L., Siemiginowska, A., Ostrowski, M., Sikora, M. 2005 Ap.J., 626, 120\n- Stawarz, L., Aharonian, F., Kataoka, J., Ostrowski, M., \nSiemiginowska, A., Sikora, M. 2006, MNRAS, 370, 981 \n- Thorne K.S., Price R.H, Macdonald D.A., 1986, Black Holes: the Membrane Paradigm , Yale University Press.\n- Tonry, J. L. 1991, ApJ, 373, L1\n- Wald R.M., 1974, Phys.Rev., D10, 1680.\n- Whysong D., Antonucci R., 2004, Ap.J., 602, 16.\n- Young A.J., Wilson A.S., Mundell C.G., 2002, Ap.J., 579, 560."}
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2004ApJ...606..799J
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Constraining the Properties of Supermassive Black Hole Systems Using Pulsar Timing: Application to 3C 66B
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2004-01-01
| 14 | 0.49 | 163 |
['black hole physics', 'gravitational waves', '-', '-', 'astrophysics']
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General expressions for the expected timing residuals induced by gravitational wave (G-wave) emission from a slowly evolving, eccentric, binary black hole system are derived here for the first time. These expressions are used to search for the signature of G-waves emitted by the proposed supermassive binary black hole system in 3C 66B. We use data from long-term timing observations of the radio pulsar PSR B1855+09. For the case of a circular orbit, the emitted G-waves should generate clearly detectable fluctuations in the pulse-arrival times of PSR B1855+09. Since no G-waves are detected, the waveforms are used in a Monte Carlo analysis in order to place limits on the mass and eccentricity of the proposed black hole system. The analysis presented here rules out the adopted system with 95% confidence. The reported analysis also demonstrates several interesting features of a G-wave detector based on pulsar timing.
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https://arxiv.org/pdf/astro-ph/0310276.pdf
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{'Constraining the properties of the proposed supermassive black hole system in 3c66b: Limits from pulsar timing': 'Fredrick A. Jenet 1 , Andrea Lommen 2 , Shane L. Larson 3 , Linqing Wen 3', 'ABSTRACT': 'Data from long term timing observations of the radio pulsar PSR B1855+09 have been searched for the signature of gravitational waves (G-waves) emitted by the proposed supermassive binary black hole system in 3C66B. For the case of a circular orbit, the emitted G-waves would generate detectable fluctuations in the pulse arrival times of PSR B1855+09. General expressions for the expected timing residuals induced by G-wave emission from a slowly evolving, eccentric, binary black hole system are derived here for the first time. These waveforms are used in a Monte-Carlo analysis in order to place limits on the mass and eccentricity of the proposed black hole system. The reported analysis also demonstrates several interesting features of a gravitational wave detector based on pulsar timing. \nSubject headings: pulsar:general - pulsar:individual (B1855+09) - gravitational waves - black hole physics', '1. introduction': "This letter reports on the search for gravitational wave (G-wave) emission from the recently proposed Supermassive Binary Black Hole (SBBH) system in 3C66B (Sudou et al. 2003, S03 hereafter) using 7 years of timing data from the radio pulsar PSR B1855+09. Given the length of the available data set and this pulsar's low root-mean-square timing noise (1.5 µ s), these data are well suited for this analysis. The proposed binary system has a current period of 1.05 years, a total mass of 5 . 4 × 10 10 M /circledot , and a mass ratio of 0.1. \nGiven the close proximity of the radio galaxy 3C66B (z = 0.02), the G-waves emitted by this system could induce a detectable signature in the timing residuals of PSR B1855+09, with a maximum residual amplitude of order 10 µ s, assuming the eccentricity of the system is zero and the Hubble constant is 75 km s -1 Mpc -1 \nThe analysis of these data will demonstrate two interesting properties of a gravitational wave detector made up of radio pulsars. First, the amplitude of the observed signature increases with decreasing gravitational wave frequency. Second, the light travel time delay between the Earth and the pulsar can, depending on the geometry, allow one to observe the gravitational wave source at two distinct epochs of time simultaneously. For example, if the pulsar is 4000 light-years away and the Earth-pulsar line-of-sight is perpendicular to the G-wave propagation vector, then the observed timing residuals will contain information about the source both at the current epoch and 4000 years ago. If the G-wave emitter is a binary system, slowly inspiraling due to G-wave emission, then the observed residuals will contain both low and high frequency components. The difference in the frequencies of these components will depend on how quickly the system is evolving. Since pulsar timing is more sensitive to lower frequencies, the highest amplitude oscillations in the timing residuals will be due to the delayed (i.e. 4000 year old) component. This effect, referred to as the 'two-frequency response', is analogous to the three-pulse response occurring in spacecraft doppler tracking experiments (Estabrook & Wahlquist 1975) and the multi-pulse response from time-delay interferometry used in the proposed Laser Interferometer Space Antenna (LISA) mission (Armstrong, Estabrook, & Tinto 1999). \nThe next section describes the expected signature of G-wave emission from a general binary system and for the specific case of the proposed system in 3C66B. The observations of PSR B1855+09 used to search for G-waves are described in section 3. Section 4 discusses the search techniques employed as well as the Monte-Carlo simulation used to place limits on the mass and eccentricity of the system, and the results are discussed in section 5.", '2. The Signature of 3C66B': "The orbital motion of the proposed binary system in 3C66B will generate gravitational radiation. The emitted G-waves will induce periodic oscillations in the arrival times of individual pulses from radio pulsars. Given a model for the pulse arrival times in the absence of G-waves, one can generate a time series of 'residuals' which are the observed pulse arrival times minus the expected pulse arrival times. Ideally, the effects of known accelerations are removed from the timing residuals leaving only the variations due to the presence of G-waves. \nThe emitted G-waves are described by two functions of spacetime, h + and h × which correspond to the gravitational wave strain of the two polarization modes of the radiation field. As these waves pass between the Earth and a pulsar, the observed timing residuals, R ( t ), will vary as (Estabrook & Wahlquist 1975; Detweiler 1979) \nR ( t ) = 1 2 (1 + cos( µ ))( r + ( t ) cos(2 ψ ) + r × ( t ) sin(2 ψ )) , (1) \nwhere t is time, µ is the opening angle between the G-wave source and the pulsar relative to Earth, ψ is the G-wave polarization angle, and the '+' and ' × ' refer to the two G-wave polarization states. The functions r + and r × , referred to collectively as r + , × , are related to the gravitational wave strain by \nr + , × ( t ) = r e + , × ( t ) -r p + , × ( t ) (2) \nr e + , × ( t ) = ∫ t 0 h e + , × ( τ ) dτ (3) \nr p + , × ( t ) = ∫ t 0 h p + , × ( τ -d c (1 -cos( µ ))) dτ, (4) \nwhere h e + , × ( t ) is the gravitational wave strain at Earth, h p + , × ( t ) is the gravitational wave strain at the pulsar, τ is the time integration variable, d is the distance between Earth and the pulsar, and c is the speed of light. Note that the pulsar term, h p + , × , is evaluated at the current time minus a geometric delay. \nG-waves emitted by a system in a circular orbit (i.e. zero eccentricity) will vary sinusoidally as a function of time with a frequency given by twice the orbital frequency. For eccentric systems, the emitted waves will contain several harmonics of the orbital frequency. The 2nd harmonic will dominate at low eccentricities while the fundamental (i.e. the orbital) frequency will dominate at high eccentricities. In general, the period and eccentricity of a binary system will be decreasing with time due to the fact that the system is radiating away energy and angular momentum in G-waves. Hence, the frequencies present in h + , × ( t ) will vary with time. Since r e + , × and r p + , × may be generated by h + , × ( t ) at epochs separated by an extremely long time interval, the frequency content of these terms may differ significantly. \nThe G-wave strain, h ( t ), induced by a black hole binary may be calculated using the standard weak field approximation applied to two orbiting point masses (Wahlquist 1987). The expected residuals are found by integrating h ( t ) with respect to time (see Eqs. 2 - 4): \nr e + ( t ) = α ( t )( A ( t ) cos(2 φ ) -B ( t ) sin(2 φ )) (5) \nr e × ( t ) = α ( t )( A ( t ) sin(2 φ ) + B ( t ) cos(2 φ )) , (6) \nα ( t ) = M 5 3 c Dω 1 3 √ 1 -e ( t ) 2 1 + e ( t ) cos( θ ( t )) (7) \nwhere D is the distance to the source, φ is the orientation of the line of nodes on the sky, ω ( t ) is the orbital frequency, e ( t ) is the eccentricity, θ ( t ) is the orbital phase, and M c is the 'chirp mass' defined as \nM c = M t ( m 1 m 2 M 2 t ) 3 5 (8) \nwhere M t = m 1 + m 2 and m 1 and m 2 are the masses of the individual black holes. Note that all units from Equation 5 on are in 'geometrized' units where G = c = 1 4 . A ( t ) and B ( t ) are given by \nA ( t ) = 2 e ( t ) sin[ θ ( t )] { cos[ θ ( t ) -θ n ] 2 -cos[ i ] 2 sin[ θ ( t ) -θ n ] 2 } -1 2 sin[2( θ ( t ) -θ n )] { 1 + e ( t ) cos[ θ ( t )] }{ 3 + cos[2 i ] } (9) \nB ( t ) = 2cos[ i ] { cos[2( θ ( t ) -θ n )] + e ( t ) cos[ θ ( t ) -2 θ n ] } (10) \nwhere i and θ n are the orbital inclination angle and the value of θ at the line of nodes, respectively (Wahlquist 1987). θ ( t ) and e ( t ) are given by the following coupled differential equations (Wahlquist 1987; Peters 1964): \ndθ dt = ω ( t ) { 1 + e ( t ) cos[ θ ( t )] } 2 { 1 -e ( t ) 2 } 3 2 (11) \nde dt = -304 15 M 5 3 c ω 8 3 0 χ 0 e ( t ) -29 19 [1 -e ( t ) 2 ] 3 2 [1 + 121 304 e ( t ) 2 ] 1181 2299 , (12) \nwhere ω 0 is the initial value of ω ( t ) and χ 0 is a constant that depends on the initial eccentricity e 0 : \nω ( t ) is given by \nχ 0 = [1 -e 2 0 ] e -12 19 0 [1 + 121 304 e 2 0 ] -870 2299 . (13) \nω ( t ) = a 0 e ( t ) -18 19 [1 -e ( t ) 2 ] 3 2 [1 + 121 304 e ( t ) 2 ] -1305 2299 , (14) \nwhere a 0 is determined by the initial condition ω ( t = 0) = ω 0 . The above equations are accurate to first order in v/c , and valid only when both e ( t ) and ω ( t ) vary slowly with time. The expressions for r p + , × are identical to those for r e + , × . Note that r p + , × is evaluated at an earlier time than r e + , × (See Eqs. 3 and 4). \nFor the specific case of the S03 parameters for 3C66B, the high chirp mass (1 . 3 × 10 10 M /circledot ) together with the period of 1.05 years implies a lifetime of ≈ 5 years. The orbital period of such a system will be evolving rapidly. The angle between 3C66B and PSR B1855+09 \non the sky is 81 . 5 · and PSR B1855+09 lies 1 ± 0.3 kpc away (Kaspi, Taylor, & Ryba 1994, hereafter KTR94). The total time delay between the pulsar epoch and the Earth epoch is given by ( d/c )(1 -cos( µ )) which is equal to 3700 ± 1100 years (see Eq. 4) for these objects. Since the time delay between the Earth and the pulsar is much larger than the timescale for evolution of the system, the expected residual will contain a low frequency component due to the pulsar term ( r p + , × ) and a high frequency component due to the Earth term ( r e + , × ). The top panel in Figure 1 shows a theoretical set of timing residuals due to G-wave emission from the proposed binary system in 3C66B assuming that the distance to this galaxy is 80 Mpc and the distance to the pulsar is 1 kpc. This waveform was generated with i = θ n = φ = 0 and ψ = π/ 4. The chirp mass used was 1 . 3 × 10 10 M /circledot and the orbital period at the epoch of the S03 observations (i.e. MJD= 51981) was taken to be 1.05 years. The eccentricity at this epoch was taken to be .0001. Two distinct oscillation frequencies can be seen, one with a period of about 0.88 years and the other with a period around 6.24 years. The bottom panel in Figure 1 shows the Lomb periodogram of the simulated residuals. The Lomb periodogram is the analogue of the discrete Fourier transform for unevenly sampled data and is further discussed in § 4. The simulated residuals demonstrate two important features of an Earthpulsar gravitational wave detector. The first is that it is more sensitive to low frequency oscillations. The second is that a single set of timing residuals can contain information about the source from two widely separated epochs of time. The low frequency seen here was due to the orbital period 3700 years ago.", '3. Timing Observations of PSR B1855+09': 'We used observations of PSR B1855+09 made by Kaspi, Taylor, & Ryba (1994) (hereafter KTR94) at the Arecibo Observatory 300 m telescope 1 and made public therein. The KTR94 data set is made up of more than 7 years (1986-1993) of bi-weekly observations using the Princeton University MarkIV system. The data were corrected for small errors in the observatory UTC clock as compared to GPS time, for errors in GPS time as compared to UTC as maintained by the National Institute of Standards and Technology and finally for errors in UTC(NIST) as compared to terrestrial time (TT) as maintained by the Bureau International des Poids et Mesures (Guinot 1988). For more details on data acquisition, reduction, and clock correction, please see KTR94. \nFig. 1.- Top panel: Theoretical timing residuals induced by G-waves from 3C66B. The timing points are chosen to coincide with the actual timing residuals of B1855+09. Bottom panel: The corresponding normalized Lomb periodogram. \n<!-- image --> \nUsing the standard TEMPO software package 5 the TOAs were fit to the model published by KTR94. We used their best-fit values as our initial parameters. In the fitting we allowed the spin period ( P ), period derivative ( ˙ P ), right ascension, declination, proper motion, parallax, and the five Keplerian binary parameters to vary. Additionally, the Shapiro delay parameters were included in the model but were fixed at the optimum values published by KTR94. The best-fit values of all parameters were consistent with those published by KTR94. The resulting timing residuals used are shown in the top panel of Figure 2.', '4. Constraints from pulsar timing': "The timing residuals from PSR B1855+09 were searched for the signature of G-waves using the normalized Lomb periodogram (see Press, Teukolsky, Vetterling, & Flannery (1992), section 13.8) together with 'harmonic summing.' The Lomb periodogram (LP) is the analog of the discrete Fourier transform for unevenly sampled data. Harmonic summing is performed by adding together the periodogram power at harmonics of each frequency up to a chosen maximum harmonic (Lyne 1988). This process increases the sensitivity to periodic, non-sinusoidal waveforms like those expected from eccentric binaries. If a SBBH system existed in 3C66B with an eccentricity of zero and a chirp mass and period adopted by S03, then the LP should show the two-frequency response like that seen in Figure 1. Figure 2 plots the normalized LP for the residual data described above. The periodogram power was calculated for 542 frequencies ranging from 1/27.8 year -1 to 19.5 year -1 with a resolution of 1/27.8 year -1 . This corresponds to a frequency oversampling factor of 4. There are no significant peaks in this LP. For purposes of this paper, a significant peak has less than a . 1% chance of occurring in purely random data assuming Gaussian statistics. Harmonic summing was performed up to the 6th harmonic. Again, no significant features were found. \nSince the LP analysis was unable to detect the presence of G-waves in the timing residuals, limits can be placed on the possible chirp mass and eccentricity of the system. Since the general waveform given by Eqs. 5-7 depends on various unknown quantities that specify the orientation of the orbit and the viewing geometry, a Monte-Carlo simulation was performed in order to determine the probability of detecting a SBBH system in 3C66B with a given chirp mass and eccentricity. Aside from M c and e , the general wave form depends on 6 angles: two angles specify the plane of the orbit, two determine the orbital phase of binary at the beginning of each of the two relevant epochs, and two determine the initial location of the line of nodes at the start of each epoch. For a given M c and e , the initial eccentricities \nFig. 2.- Top Panel: Timing residuals for B1855+09. Bottom Panel: The corresponding normalized Lomb periodogram. \n<!-- image --> \nand periods were determined using Eqs. 11 and 12. An orbital period of 1.05 years at MJD = 51981 was chosen for the initial parameters in order to match the observations of S03. The distances to 3C66B and B1855+09 were taken to be 80 Mpc and 1 kpc, respectively. The 6 unknown angles were chosen at random from a uniform distribution that ranged from 0 to 2 π and a corresponding waveform was generated using Eqs. 5-14. The waveform was then added to the residual data. When processing the timing data, the program TEMPO will remove the effects of the Earth's orbit and parallax together with a linear trend. In order to simulate the effects of removing these components from the data, various functions were subtracted from the simulated data. A one year periodicity was removed by subtracting a function of the form y = a cos( ωt ) + b sin( ωt ) where t is time, ω = 2 π/ 1 year, and a and b were determined by a least-squares fit to the simulated data. A six month periodicity was removed in a similar fashion. A best fit first order polynomial was also removed. This combination of data plus simulated signal minus various fitted functions was then analyzed using the Lomb periodogram method described above. If a significant peak was found (see above), then the signal was considered to be detected. 1000 waveforms were tested for each M c and initial e . It was found that there is a 98% chance of detecting a system like that adopted by S03 if it has an eccentricity less than 0 . 03. As the eccentricity increases above 0 . 03, the system is evolving rapidly enough to make the period at the earlier epoch (i.e. the period in the pulsar term), much longer than the observation length. Hence, for eccentricities between 0 . 03 and 0 . 49, the probability drops to about 95%. The detection probability starts falling off again above eccentricities of 0 . 49. At this point, the period of binary system at the start of the observations is longer than the observing time. The results for this and other chirp masses are summarized in table 1. The first column lists the chirp mass in 10 10 M /circledot , the next four columns list the limiting eccentricities at the 98%, 95%, and 90% probability levels. For example, with M c = 1 . 0, if the eccentricity at the epoch of the S03 observations was less than 0 . 03, then there was at least a 95% chance of detecting the system in these data using the techniques described above.", '5. Conclusions': "The signature of G-waves emitted by the proposed system in 3C66B was not found in the analysis of the pulsar timing residuals of PSR B1855+09. The system adopted by S03 has a total mass of 5 . 4 × 10 10 M /circledot and a chirp mass of 1 . 3 × 10 10 M /circledot . The confidence with which such a system can be ruled out depends on its eccentricity, which is not constrained by the S03 observations. It is generally accepted that the eccentricity of a system near coalescence will be small, but exactly how small depends on many unknown aspects of the system's formation and evolution. If the eccentricity is less than 0 . 03, then the adopted \nsystem may be ruled out at the 98% confidence level. As the assumed eccentricity of the system increases, its expected lifetime will decrease. Given that the system had to exist for longer than one year and assuming that it will merge when it reaches the last stable orbit, it can be shown that the eccentricity must be less than 0.3 for a black hole binary system with negligible spins. In this case, the system can be ruled out at the 95% confidence level. \nEven though the adopted system is highly unlikely, it is possible that the system has a lower chirp mass. A system with a chirp mass less than 0 . 7 × 10 10 M /circledot cannot be ruled out from the timing data regardless of the eccentricity. Systems with chirp masses of 1 . 0 × 10 10 M /circledot and 0 . 8 × 10 10 M /circledot become more and more allowable when the eccentricities are larger than 0 . 18 and 0 . 03, respectively. \nThe above discussion assumed a value of 75 km s -1 Mpc -1 for the Hubble constant. For other values, the chirp masses listed in Table 1 need to be multiplied by a factor of ( H/ 75) -3 / 5 where H is the desired Hubble contant in units of km s -1 Mpc -1 . For Hubble constants within the range of 65 to 85 km s -1 Mpc -1 , the chirp masses listed in Table 1 are valid to within 10%. \nAside from a lower mass binary black hole system, there are other possible explanations for the S03 observations. The observed periodicity of 1.05 ± 0 . 03 years could be an artifact arising from the Earth's orbit. On the other hand, if the periodicity is real, then the observed position angles of the two ellipses may be explained by wandering of the emission region along the jet as various shocks propagate within the jet (see for example Marscher et al 1991). \nThis analysis demonstrates how pulsar timing measurements may be used to search for G-waves from SBBH systems. In the future, pulsar timing will become more sensitive to SBBH systems as radio astronomers learn how to reduce the observed noise in pulsar timing data and/or more stable radio pulsars are discovered. The residual waveforms presented here will be useful in searching such high quality data for the signatures of SBBH systems. The two-frequency response may also provided an interesting tool for studying the physics of such systems since it will provide information about the SBBH system at two distinct epochs of time. \nPart of this research was performed at the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. AL acknowledges support of NSF grant 0107342. LW acknowledges support of NSF grants PHY-0071050 and PHY-0107417. The authors wish to thank John Armstrong for useful discussions.", 'REFERENCES': "Armstrong, J. A., Estabrook, F. B. & Tinto, M. 1999, ApJ, 527, 814 \nDetweiler, S. 1979, ApJ, 234, 1100 \nEstabrook, F. B. & Wahlquist, H. D. 1975, General Relativity and Gravitation, 6, 439 \nGuinot, B. 1988, A&A, 192, 370 \nKaspi, V. M., Taylor, J. H., & Ryba, M. F. 1994, ApJ, 428, 713 \nLyne, A. G. in Gravitational Wave Data Analysis (NATO ASI Series), ed. Schutz D. (Reidel, Dordrecht), p. 95 \nMarscher, A. P., Zhang, Y. F., Shaffer, D. B., Aller, H. D., & Aller, M. F. 1991, ApJ, 371, 491 \nPeters, P. C., Phys. Rev., 136, 1224 \nPress, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1992, \nSudou, H., Iguchi, S., Murata, Y., & Taniguchi, Y. 2003, Science, 300, 1263 \nWahlquist, H. 1987, General Relativity and Gravitation, 19, 1101 \nTable 1. Detection Limits \n| M c | 98% | 95% | 90% | M c | 98% | 95% | 90% |\n|-----------------------|-----------------------|-----------------------|-----------------------|-----------------------|-----------------------|-----------------------|-----------------------|\n| (10 10 M /circledot ) | (10 10 M /circledot ) | (10 10 M /circledot ) | (10 10 M /circledot ) | (10 10 M /circledot ) | (10 10 M /circledot ) | (10 10 M /circledot ) | (10 10 M /circledot ) |\n| 1.3 | 0.03 | 0.49 | 0.51 | 1.0 | - | 0.03 | 0.18 |\n| 1.2 | 0.02 | 0.49 | 0.51 | 0.9 | - | 0.02 | 0.04 |\n| 1.1 | 0.02 | 0.16 | 0.23 | 0.8 | - | 0.01 | 0.03 |\n| 1.1 | | | | 0.7 | - | - | - | \nNote. Given a chirp mass, M c , and a minimum detection probability, this table lists the maximum eccentricity the proposed system can have at the epoch of the S03 observations. A '-' means that the probability of detecting the system never reached the specified value."}
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2004PhRvD..70l4009M
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Disappearance of the black hole singularity in loop quantum gravity
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2004-01-01
| 15 | 0.45 | 163 |
['-', '-', '-', '-', '-']
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[]
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We apply techniques recently introduced in quantum cosmology to the Schwarzschild metric inside the horizon and near the black hole singularity at r=0. In particular, we use the quantization introduced by Husain and Winkler, which is suggested by Loop Quantum Gravity and is based on an alternative to the Schrödinger representation introduced by Halvorson. Using this quantization procedure, we show that the black hole singularity disappears and spacetime can be dynamically extended beyond the classical singularity.
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[]
| 1 |
https://arxiv.org/pdf/gr-qc/0407097.pdf
|
{'Leonardo Modesto': "Centre de Physique Th'eorique de Luminy, Universit'e de la M'editerran'ee, Case 907, F-13288 Marseille, EU. Dipartimento di Fisica dell'Universit'a di Torino, INFN - Sez. di Torino, via P. Giuria 1, I-10125 Torino, EU", 'Abstract': 'We apply techniques recently introduced in quantum cosmology to the Schwarzschild metric inside the horizon and near the black hole singularity at r = 0. In particular, we use the quantization introduced by Husain and Winkler, which is suggested by Loop Quantum Gravity and is based on an alternative to the Schrodinger representation introduced by Halvorson. Using this quantization procedure, we show that the black hole singularity disappears and spacetime can be dynamically extended beyond the classical singularity.', 'Introduction': 'A remarkable result of loop quantum cosmology [2] is the disappearance of the initial cosmological singularity present in the classical theory. The main results of loop quantum gravity [3], indeed, are the quantization of area and volume partial observables [4], which suggest that in the complete theory there cannot be spacetime points with infinity matter density. If this is correct, the quantum theory should control all classical singularities of general relativity. In this work, we apply techniques analogous to the ones used in loop quantum cosmology to study the r = 0 singularity in the interior of a Schwarzschild black hole. \nIn particular, we use the non-Schrodinger procedure of quantization introduced by Halvorson [5] and utilized in quantum cosmology by Husain and Winkler [6]. We focus on the Schwarzschild solution inside the horizon and near the singularity. We use the method introduced in [7] to express 1 /r , and therefore the curvature invariant R µνρσ R µνρσ = 48 M 2 G 2 N /r 6 , in terms of the volume operator. Following [7], we write the Hamiltonian constraint as well in terms of the volume. This allows us to express the quantum evolution equation as a difference equation for the coefficients for the physical states, and to completely control the singularity. \nThe paper is organized as follow. In the first section we briefly recall the properties of the Schwarzschild solution for r < 2 MG N , namely inside the horizon. As well known, here the temporal and spatial (radial) coordinate exchange their role. In the second section we study the classical dynamics of a very simple model giving this solution. The Hamiltonian constraint depends on a single variable, and its classical solution yields the Schwarzschild metric inside the horizon, in the new temporal variable. In the third section we quantize the system using the non-Schrodinger procedure of quantization of references [5, 6]. In particular, we show that the singularity in r = 0 is resolved in quantum gravity and that the Hamiltonian constraint acts like a difference operator, as in loop quantum cosmology.', '1 The Schwarzschild Solution Inside the Horizon': 'Consider the Schwarzschild solution \nds 2 = -( 1 -2 MG N r ) dt 2 + dr 2 ( 1 -2 MG N r ) + r 2 (sin 2 θdφ 2 + dθ 2 ) (1) \nds 2 = -dT 2 ( 2 MG N T -1 ) + ( 2 MG N T -1 ) dr 2 + T 2 (sin 2 θdφ 2 + dθ 2 ) . (2) \nfor r < 2 MG N . This metric describe spacetime inside the horizon of a Schwarzschild black hole. The coordinate r is timelike and the coordinate t is spatial; for convenience we rename them as r ≡ T and t ≡ r with T ∈ ]0 , 2 MG N [ and r ∈ ] -∞ , + ∞ [. The metric reads then \nWe eliminate the coefficient of dT 2 by defining a new temporal variable τ via \ndτ = dT √ 2 MG N T -1 . (3) \nThe integration gives \nτ = -√ T (2 MG N -T ) + 2 MG N arctan ( √ T 2 MG N -T ) + const. (4) \nWe take const = 0 because lim T → 0 τ ( T ) = const . The function T = T ( τ ) is monotonic and convex, thus τ ∈ ]0 , 2 MG N π/ 2[. In this new temporal variable the metric becomes \nds 2 = -dτ 2 + ( 2 MG N T ( τ ) -1 ) dr 2 + T ( τ ) 2 (sin 2 θdφ 2 + dθ 2 ) . (5) \nWe introduce two function a 2 ( τ ) ≡ 2 MG N T ( τ ) -1 and b 2 ( τ ) ≡ T 2 ( τ ) and redefine τ ≡ t . The metric reads \nds 2 = -dt 2 + ( 2 MG N b ( t ) -1 ) dr 2 + b ( t ) 2 (sin 2 θdφ 2 + dθ 2 ) . (6) \nNotice that a metric written in terms of two functions a ( t ) and b ( t ) with the form \nds 2 = -dt 2 + a 2 ( t ) dr 2 + b 2 ( t )(sin 2 θdφ 2 + dθ 2 ) (7) \nis the metric of an homogeneous, anisotropic space with spatial section of topology R × S 2 . In our case, a ( t ) is a function of b ( t ), a = a ( b ( t )).', '2 Classical Theory': "The complete action for gravity can be written in the form \nS = M 2 p 16 π ∫ d 3 xdtNh 1 / 2 [ K ij K ij -K 2 + (3) R ] (8) \nIf we specialize this for metrics of the form (7), the action becomes [1] \nS = -M 2 p 16 π ∫ dt ∫ R 0 dr ∫ 2 π 0 dφ ∫ 2 π 0 dθ sin θ b 2 a [ 2 ˙ b 2 b 2 +4 ˙ a ˙ b ab -2 b 2 ] = = -M 2 p R 2 ∫ dt [ a ˙ b 2 +2 ˙ a ˙ b b -a ] . (9) \nRecalling from (6) that the two functions a ( t ) and b ( t ) are not independent, and satisfy \na 2 ( t ) = 2 MG N b ( t ) -1 , (10) \nwe can write the action in terms of a single function \nS = M 2 p R 2 ∫ dt [ √ b √ 2 MG N ( 1 -b 2 MG N ) -1 2 ˙ b 2 +2 √ b √ 2 MG N ( 1 -b 2 MG N ) 1 2 ] . (11) \nNow we calculate the Hamiltonian, which is also the Hamiltonian constraint (see Appendix B). The momentum is \np = M 2 p R √ b √ 2 MG N ( 1 -b 2 MG N ) -1 2 ˙ b, (12) \nH = p ˙ b -L = ( p 2 2 M 2 P R -M 2 P R 2 )[ √ 2 MG N √ b ( 1 -b 2 MG N ) 1 2 ] . (13) \nWe can now show that the Hamiltonian constraint produce the correct classical dynamics. We express the Hamiltonian constraint in terms of ˙ b \nThe solution is \nH = M 2 P R ˙ b 2 √ 2 MG N b -1 -√ 2 MG N b -1 = 0 (14) \n˙ b 2 = ( 2 MG N b -1 ) (15) \nand this is exactly the equation (3) with solution (4) that reproduces the Schwarzschild metric. \nWe now introduce an approximation. In the quantum theory, we will be interest in the region of the scale the Planck length l p around the singularity. We assume that the Schwarzschild radius r s ≡ 2 MG N is much larger than this scale, and that b ( t ) = T ( t ). In this approximation we can write \n1 -b 2 MG N ∼ 1 (16) \nand H becomes \nThe volume is \nV = ∫ dr dφ dθ N h 1 / 2 = 4 πRab 2 = 4 πR √ 2 MG N b 3 / 2 √ 1 -b 2 MG N ; (18) \nand so \nH = ( p 2 2 M 2 P R -M 2 P 2 ) √ 2 MG N √ b . (17) \nV = 4 πR √ 2 MG N b 3 / 2 ≡ l o b 3 / 2 . (19) \nThe canonical pair is given by b x and p , with Poisson brackets { x, p = 1. \n{ We now assume that x ∈ R (and introduce the absolute value where appropriate). This choice it not correct classically, because for b ≡ x = 0 we have the singularity. But it allows us to open the possibility that the situation be different in the quantum theory. We introduce an algebra of classical observables and we write the quantities of physical interest in terms of those variables. We are motivated by loop quantum gravity to use the fundamental variables x and \n} \nU γ ( p ) ≡ exp ( 8 πG N γ L i p ) (20) \n≡ \nwhere γ is a real parameter and L fixes the unit of length. The parameter γ is necessary to separate the momentum point in the phase space. (Choosing γ/L = 1 we obtain the some value of U for p and p +2 πn ). This variable can be seen as the analog of the holonomy variable of loop quantum gravity. \nA straightforward calculation gives \n{ x, U γ ( p ) } = 8 πG N i γ L U γ ( p ) U -1 γ { V n , U γ } = l n 0 U -1 γ {| x | 3 n 2 , U γ } = i 8 πG N l n 0 γ L 3 n 2 sgn( x ) | x | 3 n 2 -1 (21) \nThese formulas allow us to express inverse powers of x in terms of a Poisson bracket, following Thiemann's trick [7]. As we will see below, the volume operator has zero as an eigenvalue, therefore so we must take n /greaterorequalslant 0 for the second equation to be well define din the quantum theory. On the other hand, if we want that the power of x on the right hand side be negative we need n /lessorequalslant 2 / 3. The choice n = 1 / 3 gives \nsgn( x ) √ | x | = -2 Li (8 πG N ) l 1 3 0 γ U -1 γ { V 1 3 , U γ } . (22) \nWe use this relation in the next section to write physical operators. We are interested to the quantity 1 | x | because classically this quantity diverge for | x | → 0 and produce the singularity. We are also interested to the Hamiltonian constraint and the dynamics and we will use (22) for writing the Hamiltonian.", '3 Quantum Theory': 'We construct the quantum theory proceeding in analogy with the procedure used in loop quantum gravity. The first step is the choice of an algebra of classical functions to be represented as quantum configuration operators. We choose the algebra generated by the functions \nW ( λ ) = e iλx/L (23) \nwhere λ ∈ R . The algebra consists of all function of the form \nf ( x ) = n ∑ j =1 c j e iλ j x/L (24) \nwhere c j ∈ C , and their limits with respect to the sup norm. This is the algebra AP ( R ) of the almost periodic functions over R . The algebra AP ( R ) is isomorphic to C ( ¯ R Bohr ), the algebra of continuous \nfunctions on the Bohr-compactification of R . This suggests to take the Hilbert space L 2 ( ¯ R Bohr , dµ 0 ), where dµ 0 is the Haar measure on ¯ R Bohr . With this choice the basis states in the Hilbert space are \n| λ 〉 ≡ | e iλx/L 〉 〈 µ | λ 〉 = δ µ,λ (25) \nThe action of the configuration operators ˆ W ( λ ) on the basis is defined by \nˆ W ( λ ) | µ 〉 = e iλ ˆ x/L | µ 〉 = e iλµ | µ 〉 (26) \nThese operators are weakly continuous in λ . This implies the existence of a self-adjoint operator ˆ x , acting on the basis states according to \nˆ x | µ 〉 = Lµ | µ 〉 (27) \nNext, we introduce the operator corresponding to the classical momentum function U γ = e iγp/L . We define the action of ˆ U γ on the basis states using the definition (27) and using a quantum analog of the Poisson bracket between x and U γ \nˆ U γ | µ 〉 = | µ -γ 〉 [ ˆ x, ˆ U γ ] = -γL ˆ U γ (28) \nUsing the standard quantization procedure [ , ] → i /planckover2pi1 { , } , and using (21) we obtain \n-γ = /planckover2pi1 (8 πG N ) iγ L L = √ 8 π l p (29)', '3.1 Volume operator and disappearance of the singularity': 'Near the singularity we can use the approximation (19). The action of the volume operator on the basis states is \nˆ V | µ 〉 = l 0 | x | 3 2 | µ 〉 = l 0 | Lµ | 3 2 | µ 〉 . (30) \nRecall that the dynamics is all in the function b ( t ), which is equal to the the radial Schwarzschild coordinate inside the horizon b ( t ) = T ( t ). The function b ( t ) generated by the dynamics is monotonic and convex. The important point is that b ( t = 0) = 0 and this is the Schwarzschild singularity. We now show that the term 2 MG N b ( t ) does not diverge in the quantum theory and therefore there is no singularity in the quantum theory. \nWe use the relation (22) and we promote the Poisson brackets to commutators. In this way we obtain (for γ = 1) the operator \n̂ 1 | x | = 1 2 πl 2 p l 1 3 0 ( ˆ U -1 [ ˆ V 1 3 , ˆ U ]) 2 . (31) \nThe action of this operator on the basis states is \n̂ 1 | x | | µ 〉 = √ 2 πl 2 p ( | µ | 1 2 -| µ -1 | 1 2 ) 2 | µ 〉 . (32) \nWe can now see that the spectrum is bounded from below and so we have not singularity in the quantum theory. In fact, for example, the curvature invariant \nR µνρσ R µνρσ = 48 M 2 G 2 N r 6 ≡ 48 M 2 G 2 N T 6 = 48 M 2 G 2 N T ( t ) 6 ≡ 48 M 2 G 2 N b ( t ) 6 (33) \nis finite in quantum mechanics in fact the eigenvalue of 1 / | x | for the state | 0 〉 corresponds to the classical singularity and in the quantum case it is √ 2 /πl 2 p , which is the largest possible eigenvalue. For this particular value the curvature invariant it is not infinity \n̂ R µνρσ R µνρσ | 0 〉 = ̂ 48 M 2 G 2 N | x | 6 | 0 〉 = 384 M 2 G 2 N π 3 l 6 P | 0 〉 . (34) \nOn the other hand, for | µ | → ∞ the eigenvalues go to zero, which is the expected behavior of 1 / | x | for large | x | .', '3.2 Hamiltonian Constraint': "We now study the quantization of the Hamiltonian constraint near the singularity, in the approximation (17). There is no operator p in quantum representation that we have chosen, hence we choose the following alternative representation for p 2 . Consider the classical expression \np 2 = L 2 (8 πG N ) 2 lim γ → 0 ( 2 -U γ -U -1 γ γ 2 ) . (35) \nWe have can give a physical interpretation to γ as γ = l p /L phys , where L Phys is the characteristic size of the system. Using this, we write the Hamiltonian constraint as \nˆ H = A 1 l 1 / 3 0 [ ˆ U γ + ˆ U -1 γ -(2 -A 2 ) 1 ] sgn( x ) ( ˆ U -1 [ ˆ V 1 3 , ˆ U ]) (36) \nwhere A 1 = L 3 (8 πG N ) 5 / 2 γ 3 M 2 P Rl 1 / 3 0 /planckover2pi1 and A 2 = R 2 γ 2 8 πl 2 P . The action of ˆ H on the basis states is \nˆ H | µ 〉 = C V 1 2 ( µ ) [ | µ -γ 〉 + | µ + γ 〉 -(2 -C ' ) | µ 〉 ] , (37) \nwhere C = A 1 L 1 / 2 and C ' ≡ A 2 , and \nIf we calculate the action of ˆ H and 1 / | x | on the state of zero volume eigenvalue we obtain \nV 1 2 ( µ ) = { -∣ ∣ | µ -γ | 1 / 2 -| µ | 1 / 2 ∣ ∣ for µ /negationslash = 0 | γ | 1 / 2 for µ = 0 (38) \nˆ H | 0 〉 = C| γ | 1 2 [ | -γ 〉 + | γ 〉 -(2 -C ' ) | 0 〉 ] ̂ 1 | x | | 0 〉 = √ 2 πl 2 P | 0 〉 . (39) \nThis finite value of 1 | x | can be interpreted as the effect of the quantization on the classical singularity. \nWe now study the solution of the the Hamiltonian constraint. The solutions are in the C /star space that is the dual of the dense subspace C of the kinematical space H . A generic element of this space is \n〈 ψ | = ∑ µ ψ ( µ ) 〈 µ | . (40) \nThe constraint equation ˆ H | ψ 〉 = 0 is now interpreted as an equation in the dual space 〈 ψ | ˆ H † ; from this equation we can derive a relation for the coefficients ψ ( µ ) \nV 1 2 ( µ + γ ) ψ ( µ + γ ) + V 1 2 ( µ -γ ) ψ ( µ -γ ) -(2 -C ' ) V 1 2 ( µ ) ψ ( µ ) = 0 . (41) \nThis relation determines the coefficients for the physical dual state. We can interpret this states as describing the quantum spacetime near the singularity. From the difference equation (41) we obtain physical states as combinations of a countable number of components of the form ψ ( µ + nγ ) | µ + nγ 〉 ( γ ∼ l P /L Phys ∼ 1); any component corresponds to a particular value of volume, so we can interpret ψ ( µ + γ ) as the wave function describing the black hole near the singularity at the time µ + γ . A solution of the Hamiltonian constraint corresponds to a linear combination of black hole states for particular values of the volume or equivalently particular values of the time.", 'Conclusions': "We have applied the quantization procedure of [6] to the case of the Schwarzschild singularity. This procedure is alternative to the Schrodinger quantization and it is suggested by loop quantum cosmology. The main results are: \n- 1. The classical black hole singularity near r ∼ 0, which in our coordinate is b ( t ) ≡ T ( t ) ∼ 0, disappears from the quantum theory. Classical divergent quantities are bounded in the quantum theory. For instance: \nR µνρσ R µνρσ = 48 M 2 G 2 N b ( t ) 6 → ̂ R µνρσ R µνρσ | 0 〉 = ̂ 48 M 2 G 2 N | x | 6 | 0 〉 = 384 M 2 G 2 N π 3 l 6 P | 0 〉 . \n- 2. The quantum hamiltonian constraint gives a discrete difference equation for the coefficients of the physical states. \nIt is interesting to observe that beyond the classical singularity the function b ≡ x is negative. One can speculate, extrapolating the form of the metric that 'on the other side' of the singularity there is no horizon: a black hole and a naked singularity are connected [9].", 'Acknowledgements': 'I am grateful to Carlo Rovelli, and Eugenio Bianchi for many important and clarifying discussions. This work was partially supported by the Fondazione Angelo Della Riccia', 'Appendix A': 'We give here the explicit form of some tensors used in the paper. The spatial diagonal metric tensor is \nThe inverse spatial metric tensor is \nh ij = a 2 ( t ) 0 0 0 b 2 ( t ) sin 2 θ 0 0 0 b 2 ( t ) . (42) \nh ij = a -2 ( t ) 0 0 0 b -2 ( t ) sin -2 θ 0 0 0 b -2 ( t ) . (43) \nK ij = -a ˙ a 0 0 0 -b ˙ b sin 2 θ 0 0 0 -b ˙ b . (44) \nThe extrinsic curvature is K ij = -1 2 ∂h ij ∂t , and so \nK ≡ K ij h ij = -( ˙ a a +2 ˙ b b ) K ij K ij = ˙ a 2 a 2 +2 ˙ b 2 b 2 K ij K ij -K 2 = -( 2 ˙ b 2 b 2 +4 ˙ a ˙ b ab ) (45) \nThe Ricci curvature for the space section is \n(3) R = 2 b 2 (46)', 'Appendix B': 'In this appendix we report the Hamiltonian for our system and we show that reproduces the correct equation of motion. We can start from the Hamiltonian \nH = ( p 2 2 M 2 P R -M 2 P R 2 )[ √ 2 MG N √ b ( 1 -b 2 MG N ) 1 2 ] , (47) \nand calculate the Hamilton equation for b ( ˙ b = ∂H ∂p ) \n˙ b = p M 2 P R √ 2 MG N b -1 . (48) \nAt this point using the constraint H = 0 with (48), we obtain \n˙ b 2 = ( 2 MG N b -1 ) , (49) \nthat is the equation of motion for b ( t ) that reproduce the Schwarzschild solution.', 'References': "- [1] M. Cavaglia, V. de Alfaro, and A.T. Filippov, Hamiltonian formalism for black holes and quantization II , Int. Jou. Mod. Phys. D5 , 227 (1996); gr-qc/9508062. M. Cavaglia, V. de Alfaro, and A.T. Filippov, Hamiltonian formalism for black holes and quantization Int. Jou. Mod. Phys. D4 , 661 (1995); gr-qc/9411070.\n- [2] M. Bojowald, Inverse scale factor in isotropic quantum geometry , Phys. Rev. D64 084018 (2001). M. Bojowald, Loop Quantum Cosmology IV: discrete time evolution , Class. Quant. Grav. 18 , 1071 (2001). A. Ashtekar, M. Bojowald and J. Lewandowski, Mathematica structure of loop quantum cosmology , Class. Quant. Grav. (2003).\n- [3] C Rovelli, Quantum Gravity , (Cambridge University Press, Cambridge, 2004) to appear; a draft of the book can be found in http://www.cpt.univ-mrs.fr/ ∼ rovelli .\n- [4] C Rovelli, 'Partial observables', Phys Rev D65 (2002) 124013; gr-qc/0110035.\n- [5] H. Halvorson, Complementary of representations in quantum mechanics ; quant - ph/0110102.\n- [6] Viqar Husain and Oliver Winkler, On singularity resolution in quantum gravity ; gr-qc/0312094. \n- [7] T. Thiemann, Quantum Spin Dynamics , Class. Quant. Grav. 15 , 839 (1998).\n- [8] T. Thiemann, Introduction to Modern Canonical Quantum General Relativity ; gr -qc/0110034. Lectures on Loop Quantum Gravity , gr-qc/0210094.\n- [9] D. A. Easson and R. H. Brandenberger, Universe generation from black hole interiors , JHEP 0106 , 024 (2001); hep-th/0103019. D. A. Easson, Hawking radiation of nonsingular black holes in two dimensions , JHEP 0302 , 037 (2003); hep-th/0210016."}
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2019JCAP...11..012Y
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Primordial black hole formation and abundance: contribution from the non-linear relation between the density and curvature perturbation
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2019-01-01
| 33 | 0.45 | 163 |
['-']
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[]
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The formation and abundance of primordial black holes (PBHs) arising from the curvature perturbation ζ is studied. The non-linear relation between ζ and the density contrast δ means that, even when ζ has an exactly Gaussian distribution, significant non-Gaussianities affecting PBH formation must be considered. Numerical simulations are used to investigate the critical value and the mass of PBHs which form, and peaks theory is used to calculate the mass fraction of the universe collapsing to form PBHs at the time of formation. A formalism to calculate the total present day PBH abundance and mass function is also derived. It is found that the abundance of PBHs is very sensitive to the non-linear effects, and that the power spectrum Script P<SUB>ζ</SUB> must be a factor of Script O (2) larger to produce the same number of PBHs as if using the linear relation between ζ and δ (where the exact value depends on the critical value for a region to collapse and form a PBH). This also means that the derived constraints on the small-scale power spectrum from constraints on the abundance of PBHs are weaker by the same factor.
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[]
| 3 |
https://arxiv.org/pdf/1904.00984.pdf
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{'Primordial black hole formation and abundance: contribution from the non-linear relation between the density and curvature perturbation': "Sam Young 1 , ∗ Ilia Musco 2 , 3 , † and Christian T. Byrnes 4 ‡ \n1) Max Planck Institute for Astrophysics, \nKarl-Schwarzschild-Strasse 1, 85748 Garching bei Muenchen, Germany \n2) Institut de Ci'encies del Cosmos, Universitat de Barcelona, \nMart'ı i Franqu'es 1, 08028 Barcelona, Spain \n3) Laboratoire Univers et Th'eories, \nUMR 8102 CNRS, Observatoire de Paris, \nUniversit'e Paris Diderot, \n5 Place Jules Janssen, \nF-92190 Meudon, France \nand \n4) Department of Physics and Astronomy, \nUniversity of Sussex, Brighton BN1 9QH, United Kingdom \n(Dated: November 15, 2019) \nThe formation and abundance of primordial black holes (PBHs) arising from the curvature perturbation ζ is studied. The non-linear relation between ζ and the density contrast δ means that, even when ζ has an exactly Gaussian distribution, significant non-Gaussianities affecting PBH formation must be considered. Numerical simulations are used to investigate the critical value and the mass of PBHs which form, and peaks theory is used to calculate the mass fraction of the universe collapsing to form PBHs at the time of formation. A formalism to calculate the total present day PBH abundance and mass function is also derived. It is found that the abundance of PBHs is very sensitive to the non-linear effects, and that the power spectrum P ζ must be a factor of O (2) larger to produce the same number of PBHs as if using the linear relation between ζ and δ (where the exact value depends on the critical value for a region to collapse and form a PBH). This also means that the derived constraints on the small-scale power spectrum from constraints on the abundance of PBHs are weaker by the same factor.", 'CONTENTS': "| I. Introduction | 2 |\n|------------------------------------------------------------|-----|\n| II. Cosmological perturbations in the super horizon regime | 5 |\n| A. Gradient expansion approach | 6 |\n| B. Initial conditions | 8 |\n| III. Criterion for collapse | 10 |\n| IV. Numerical results of PBH formation | 14 |\n| A. Numerical scheme | 14 |\n| V. Calculation of PBH abundance | 15 |\n| VI. Summary | 22 |\n| Acknowledgements 4 | 24 |\n| A. Gaussianity and variance of - 3 r m ζ ' ( r m ) | 25 |\n| B. Correspondence of large peaks | 26 |", 'I. INTRODUCTION': "Primordial black holes (PBHs) could be formed from the gravitational collapse of large curvature perturbations created during cosmological inflation shortly after re-entering the cosmological horizon at early times [1-3]. If a density perturbation is above a threshold δ c , then an apparent horizon will form during the collapse, otherwise it will quickly disperse into the surrounding local environment. The mass of the resulting PBH is strongly related to the scale and amplitude of the perturbation from which it formed, with more massive black holes forming from larger-scale perturbations which enter the horizon at a later time. PBHs can theoretically form with any mass, and can provide a natural explanation for any observed black holes with masses which are not easily explained by the standard picture of black hole formation from collapsing stars. PBHs which \nformed with a mass below 10 15 g would have evaporated by today (ignoring the possible accretion after formation), but more massive PBHs would persist into the present epoch. \nPBHs still represent a viable dark matter candidate, although there are numerous constraints on the abundance of PBHs of varying masses (see [4-6] for recent discussions of the constraints for a broad mass spectrum), and clusters of PBHs could explain the early formation of super-massive black holes found in the centres of galaxies. In recent years, there have been several interesting observations which may hint towards the existence of PBHs [7]. For a review of PBH formation and constraints, see [8, 9]. \nThere have been many attempts to detect them by their indirect effects on the universe. Ignoring the possible observations described above, no positive detection has been made. However, the non-detection of PBHs constrains their abundance. The abundance of PBHs is typically stated in terms of β , the energy fraction of the universe which goes into PBHs at the time of formation. The abundance of small PBHs ( < 10 15 g ) in the early universe which would have evaporated by today can be constrained by looking for the effects of the radiation from their evaporation, whilst the abundance of more massive PBHs ( > 10 15 g ) can be constrained by their gravitational effects. Because PBHs of different masses form from different scale perturbations, the constraints on different mass PBHs can be used to place a constraint on different scales of the primordial power spectrum in the early universe - though these constraints are sensitive to primordial non-Gaussianity in the early universe. Because PBHs form on scales much smaller than those observable in the cosmic microwave background (CMB) or large-scale structure of the universe (LSS), they therefore place the only available constraints on the small-scale power spectrum - and can be used to constrain models of inflation. In order for an interesting number of PBHs to form however, the power spectrum must become orders of magnitude larger on small scales than is observed in the CMB or LSS ( O (10 -2 ) compared to O (10 -9 )). Therefore the derived constraints on the power spectrum are much weaker, but they span a much larger range of scales, including scales which cannot be probed by any other method [10]. \nThere are many different models for cosmological inflation which would predict a large number of PBHs to form in the early universe. For example, the running mass model [11], axion inflation [12, 13], or a waterfall transition during hybrid inflation [14-16], a quartic action during inflation or a variable sound speed [17], amongst many others. A metric perturbation in the form of the curvature perturbation ζ is typically used to study cosmological perturbations generated with the \ndifferent models and to predict their observable consequences 1 . The curvature perturbation ζ appears in the metric in the comoving uniform-density gauge as \nd s 2 = -d η 2 + a 2 ( t ) e 2 ζ d X 2 , (1) \nwhere η is the conformal time, a ( t ) is the scale factor and X represents the three comoving spatial coordinates. In order to translate the constraints on PBH abundance into constraints on models of inflation (or alternatively to predict PBH abundances from a given model) it is desirable to relate the primordial curvature perturbation power spectrum P ζ to the abundance of PBHs of different masses. \nThe formation of PBHs from a non linear metric perturbation was initially studied by Shibata and Sasaki [18], which was then used to derive a relation between the abundance of PBHs and the power spectrum P ζ [19]. Around the same time Niemeyer and Jedamzik performed a numerical study of PBH formation using instead an initial perturbation of the energy density [20, 21]. For a long time the abundance of PBHs was calculated assuming that regions where the curvature perturbation ζ was above a critical value ζ c of order unity. However, it has since been shown that the curvature perturbation ζ is not a suitable parameter to use to determine whether a region will form a PBH or not, due to the effect of super-horizon modes on the calculation, and that the density contrast should be used instead [22]. The effect of super-horizon modes on PBH formation is discussed in detail in [23]. Nonetheless, in the following years it has been typical to use the curvature perturbation directly to calculate the abundance - which is valid in the case that an approximation is being used (as described in [22]) or in the case that a narrow peak in the power spectrum is being considered (so that large perturbations only exist at one scale). Papers which have used the density contrast δρ/ρ b rather than the curvature perturbation for the calculation of the abundance have since used a linear relation between the two parameters (as in the recent paper [24] for example), \nδρ ρ b = -2(1 + ω ) 5 + 3 ω ( 1 aH ) 2 ∇ 2 ζ, (2) \nwhere ω = 1 / 3 is the equation of state parameter during the radiation dominated epoch of the early universe, ρ b is the background density and ( aH ) -1 is the comoving cosmological horizon scale. However, this expression ignores the non-linear relation between the curvature and the \nenergy density profile. Starting for the first time from simulations of PBH formation arising from perturbations in the curvature perturbation ζ , the aims of this paper are to investigate how the fully non-linear relation between ζ and δρ/ρ b affects the calculation of the abundance of primordial black holes, and to derive the most accurate relation to date between the primordial curvature perturbation power spectrum P ζ and the abundance of PBHs at formation β . \nWe note that reference [25] introduced a new calculation which included the non-linear effect. The abundance of PBHs was calculated using a method based on estimating the critical heights of peaks in the curvature perturbations ζ , and then calculating the abundance of PBHs by utilising the Gaussianity of ζ . Using the critical height of peaks in ζ is only valid for a narrow power spectrum, such that perturbations exist only on one scale (as they note in section 4), whereas the method presented here can be applied to power spectra of any width. Reference [25] stated that the abundance of PBHs they calculated was greatly increased compared to some previous calculations, whereas it is shown here that considering the non-linear relation decreases the abundance compared to the linear calculation. The apparent contradiction in conclusions can be attributed to differences in what is considered as the 'standard calculation' between that paper and ours. We compare our results using the same method from using the linear or non-linear relations between δ and ζ , whereas [25] compares very different methods. \nThe paper is organised as follows: in section II we will discuss the set up of the initial conditions in the density contrast δρ/ρ b arising from an initial curvature perturbation ζ . In section III we will discuss the criteria which should be used to determine whether an initial perturbation will eventually collapse to form a PBH. In section IV we will discuss the simulation procedure used and the numerical results obtained from the simulations. Finally, in section V we will show how the abundance of PBHs β can be obtained from the curvature perturbation power spectrum. Our findings are summarised in section VI, leaving some details of our calculations to an appendix.", 'II. COSMOLOGICAL PERTURBATIONS IN THE SUPER HORIZON REGIME': 'In this section we will first describe the general relation between the curvature perturbation ζ and the density contrast δρ/ρ b before analysing a specific parametrization of ζ that allows us to vary the profile of δρ/ρ b . This allows us, with the help of numerical simulations (see Section IV), to span almost all the possible range of values of δ c . Throughout this paper, we assume that perturbations large enough to form PBHs are spherically symmetric. This is justified because such \npeaks must be extremely rare [26], and the perturbation profile is therefore defined using only a radial coordinate r . \nNote that the curvature perturbation ζ in the literature is typically defined on a uniform density slicing, whilst the density contrast δρ/ρ b is defined on a constant cosmic time slicing. However, in the super-horizon regime described in the following section the difference between these two gauges is a higher-order correction which can be neglected (see [27] and the references therein).', 'A. Gradient expansion approach': "In the super-horizon regime perturbations have a length scale much larger than the cosmological Hubble horizon. In this regime it is possible to have an analytic treatment, usually called the gradient expansion or long-wavelength approximation [18, 28], based on expanding the exact solution in a power series of a small parameter ( glyph[epsilon1] glyph[lessmuch] 1) that is conveniently identified with the ratio between the Hubble radius 1 /H ( t ) and the length scale L characterising the perturbation \nglyph[epsilon1] ( t ) ≡ 1 H ( t ) L , (3) \nwhere a particular value of glyph[epsilon1] corresponds to a particular value of the time t . \nWhen glyph[epsilon1] glyph[lessmuch] 1 the curvature profile ζ ( r ) is conserved (time independent) and each spatial gradient is multiplied by glyph[epsilon1] , expanding the equations in a power series in glyph[epsilon1] up to the first non-zero order. Putting glyph[epsilon1] = 1, which defines the horizon crossing time, one obtains the spatial dependence of the different matter/geometrical variables, related to the right/left hand side of the Einstein equations, in terms of the conserved curvature profile ζ ( r ). The curvature profile represents the fundamental source of the perturbation, embedded into the metric (1) from the asymptotic limit, t → 0. Although this approach reproduces the time evolution of linear perturbation theory when glyph[epsilon1] glyph[lessmuch] 1, it also allows one to consider non-linear curvature perturbations if the spacetime is sufficiently smooth on scales greater than L [27]. This is equivalent to pressure gradients being a higher-order correction in glyph[epsilon1] , which corresponds to a self-similar growth of the perturbation, conserving the spatial profile. \nIn the gradient expansion the non-linear relation between the density contrast δρ/ρ b defined on a comoving uniform-cosmic time slicing and the curvature perturbation ζ is given by [23, 25, 29], which is exact up to the first non-zero terms in glyph[epsilon1] \nδρ ρ b ( r, t ) = -4(1 + ω ) 5 + 3 ω ( 1 aH ) 2 e -5 ζ ( r ) / 2 ∇ 2 e ζ ( r ) / 2 , (4) \nwhere ω ≡ p/ρ is the equation of state parameter, and \n∇ 2 = d 2 dr 2 + 2 r d dr \nis the Laplacian written assuming spherical symmetry. There are two non-linear effects contained within equation (4): the exponential term exp( -2 ζ ( r )) and the quadratic term of the first derivative ( ζ ' ( r )) 2 , where the prime denotes a derivative with respect to the radial coordinate r . Because PBHs form from large perturbations, the effect of the non-linear components is comparable with the linear term and should not be neglected. \nAs explained in [29], whether a cosmological perturbation is able to form a PBH depends on the amplitude measured at the peak of the compaction function defined as \nC ( r, t ) ≡ 2 M ( r, t ) -M b ( r, t ) R ( r, t ) , (5) \nwhere R ( r, t ) is the areal radius, M ( r, t ) is the Misner-Sharp mass within a sphere of the radius R and M b ( r, t ) = 4 πρ b ( r, t ) R 3 ( r, t ) / 3 is the background mass within the same areal radius calculated with respect to a Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) universe. In the superhorizon regime, applying the gradient expansion approximation, the compaction function is conserved and is related to ζ ( r ) as [23, 25, 29] \nC ( r ) = -f ( w ) rζ ' ( r ) [ 2 + rζ ' ( r ) ] , f ( ω ) = 3(1 + ω ) 5 + 3 ω . (6) \nWe can then compute the length-scale of the perturbation, identified as the location r m where the compaction function is maximized ( C ' ( r m ) = 0), which gives \nζ ' ( r m ) + r m ζ '' ( r m ) = 0 . (7) \nMeasuring the curvature perturbation with ζ ( r ) introduces an intrinsic rescaling of the comoving coordinate with respect to the background FLRW solution, because the exponential term appearing in the metric (1) introduces a local perturbation of the scale factor which depends on the local value of the curvature. This implies that the horizon crossing time t H is defined in real space when \na ( t H ) H ( t H ) r m e ζ ( r m ) = 1 , (8) \nand therefore according to the definition of glyph[epsilon1] given above, the physical length scale of the perturbation, to be called R m from here onwards, is given by \nR m ( r, t ) = a ( t ) r m e ζ ( r m ) . (9) \nThe perturbation amplitude can be measured as the mass excess of the energy density within the scale R m , measured at the horizon crossing time t H . Although in this regime the gradient expansion approximation is not very accurate, this represent a well defined criterion that allows a consistent comparison between the amplitude of different perturbations (see [29] for more details). Computing the mass excess as the integral of the density contrast averaged over the background volume V R m = 4 π 3 R 3 m , the amplitude of the perturbation is given by \nδ ( r m , t H ) ≡ 3 R 3 m R m ∫ 0 δρ ρ b R 2 d R = 3 ( r m e ζ ( r m ) ) 3 r m ∫ 0 δρ ρ b ( r, t H )( re ζ ( r ) ) 2 ( re ζ ( r ) ) ' d r , (10) \nand using the explicit expression of δρ ( r, t ) /ρ b in terms of the curvature profile seen in (4), we get δ m ≡ δ ( r m , t H ) = C ( r m ), which satisfies the fundamental relation [29] \nδ m = 3 δρ ρ b ( r m , t H ) (11) \nfor any curvature profile, ζ ( r ).", 'B. Initial conditions': "We will now study the main feature of the density profile when an explicit parameterization of the curvature profile ζ ( r ) is specified as \nζ α ( r ) = A exp [ -( r r m ) 2 α ] , (12) \nwhere A and r m respectively denote the amplitude and the scale of the perturbation. Inserting this into (4), the corresponding density profile is given by \nδρ α ρ b = ( 1 aH ) 2 4 3 f ( ω ) α ( r r m ) 2 α [ (2 α +1) -α ( r r m ) 2 α (2 + ζ α ( r )) ] ζ α ( r ) r 2 e 2 ζ α ( r ) , (13) \nand then inserting (12) into (6) and (7), we can calculate the corresponding perturbation amplitude \n-r m ζ ' α ( r m ) = 2 A α e ⇒ δ m = 4 f ( ω ) A α e ( 1 -A α e ) , (14) \nwhich gives the value of A in terms of the averaged amplitude δ m \nA = e 2 α ( 1 -√ 1 -δ m f ( ω ) ) . (15) \n<!-- image --> \nFIG. 1 . The left plots shows the critical curvature profiles given by (12) for α = 1 ( A c glyph[similarequal] 0 . 8 and δ c glyph[similarequal] 0 . 55), while the right plot shows the corresponding C ( r ) profile. The upper limit δ c,max = 2 / 3 represents the maximum theoretical upper limit of δ c . \n<!-- image --> \nThis behaviour of δ m in terms of A also shows that there is a maximum value of δ m = 2 / 3 corresponding to A = eα/ 2, that represents the transition between PBHs of type I and type II, after which formation of PBHs cannot be studied in terms of δ m but only in terms of ζ [30]. \nUsing numerical simulations we have calculated the critical values for PBHs using initial condition in terms of ζ ( r ) given by (12), finding that the value of δ c is varying in terms of α (see section IV for more details). In the particular case of α = 1 we get δ c glyph[similarequal] 0 . 55 which corresponds to a value of A glyph[similarequal] 0 . 80. In the left frame of figure 1 we have plotted the critical profile of ζ ( r ) as function of r/r m identifying the critical peak A c of the profile. In this plot we are also showing the value of A corresponding to the maximum limit of δ c,max = 2 / 3. In the right hand plot of figure 1 we show the corresponding compaction function C ( r ) with the peak amplitude of C ( r m ) being equal to δ c . \nIn figure 2 we plot the ζ -profiles (plotted in the left panel) and the corresponding density contrast (right panel) measured at horizon crossing, defined by (8), for the threshold values δ c associated to each shape. Although the ζ -profiles given by (12) are always centrally peaked, the energy density profile is centrally peaked only for α ≤ 1. In particular, only the case of α = 1, plotted with a dashed line, is smooth at r = 0, giving a finite value of the peak of the density contrast, while for smaller values of α the peak is diverging. Nevertheless the amplitude of the perturbation, measured \nby the averaged value δ m always remains finite. \nThe right panel of figure 2 also shows that for α > 1 the density contrast is off-centred, with an increasing value of the peak for larger values of α . One can see therefore that a ζ -profile with a centrally finite peak does not always corresponds to the same type of peaks in the density contrast, because of the non-linear expression given by (4) and the correspondence among the peaks is guaranteed only at the linear level. In general, the correspondence between the peaks in ζ and the peaks in δρ/ρ b requires the assumption that ζ ' ( r ) = 0 at r = 0, such that in the centre the only non-linear effect is given by the exponential term exp ( -2 ζ ( r )) which reduces the amplitude. \nA second issue already mentioned is that finite peaks of ζ do not always correspond to finite values of the peak of the density contrast, which happens here for α < 1. This is because the ζ -profiles for α < 1 are not smooth in the centre (they are not infinitely differentiable), and the term ζ ' ( r ) /r diverges in the limit r → 0. Such peaks are of course unphysical and this divergence can be removed with a transfer function or a smoothing function which removes the small scale power, but there is lack of knowledge in the literature about which is the correct form of the non-linear transfer function to be used for a radiation fluid. Note that the transfer function should always be taken into account, because strictly speaking the curvature is exactly conserved only for glyph[epsilon1] → 0. Because in practice a finite value of glyph[epsilon1] , corresponding to a finite initial time t i , needs to be chosen, the effects of the pressure gradients within a region of the size of the initial sound horizon, which in radiation is R s = ( √ 3 H ) -1 , are not completely negligible even at the initial time. For spiky shapes (which have ζ ' ( r ) /r glyph[negationslash] = 0 in the centre), like those obtained from (12) with α < 1 and the effect of the transfer function might significantly change the amplitude of the peak, while the value of the averaged δ m hardly changes. \nBecause for every ζ -peak with a finite amplitude there is always a peak of the compaction function C evaluated at r m , with finite amplitude equal to δ m , we have used the averaged amplitude δ m to calculate the abundance of PBHs (see section V), using the linear transfer function to correct the value of the peak of the density contrast at r = 0, leaving a study of the effects of the non-linear transfer function to future work.", 'III. CRITERION FOR COLLAPSE': "In order for a perturbation to collapse into a PBH, the density must exceed some critical threshold. The original work by Carr [3], using a Jeans length argument, provided an order of magnitude \n<!-- image --> \nFIG. 2 . The left plot shows the critical profiles ζ α ( r ) given by (13) plotted against r/r m for α = 0 . 5 , 0 . 75 , 1 . 0 , 1 . 25 , 1 . 50 , 2 . 0 where for increasing values of α the central value A c is decreasing. The right plot shows the corresponding critical profiles of δρ α ( r ) /ρ b given by equation (13) plotted against r/r m : for α < 1 the profile has a spiky shape, with increasing steepness for decreasing value of α ; for α = 1 the profiles is centrally peaked with ( δρ ) ' /ρ b → 0 when r → 0 while for α > 1 the profile has an increasing off-peak. In both plots the dashed line corresponds to the smooth centrally peaked case with α = 1. \n<!-- image --> \nestimate for the threshold δ c ∼ ω = 1 / 3 at horizon crossing for a radiation fluid. Since then, there has been extensive work to determine the collapse threshold [18, 21, 23, 29, 31-34], as well as discussions about the appropriate parameter to use to determine whether a perturbation will collapse [19, 22, 24, 25]. The collapse threshold is typically obtained from simulating the evolution of a perturbation as it reenters the cosmological horizon, although analytic attempts have been made, neglecting the effects of pressure gradients [35]. \nAs discussed previously, in order to determine a clear criterion to distinguish which perturbations are able to form a PBH, the density contrast δρ/ρ b should be used rather than a metric perturbation such as the curvature perturbation ζ . There has been much ambiguity in the literature about how this critical amplitude is calculated and used (especially between the different communities of relativists modelling PBH formation and cosmologists calculating the abundance of PBHs). It is the aim of this section of the paper to discuss how this should be defined. Spherical symmetry is typically assumed when modelling PBH formation, again justified by the fact that such peaks are large and rare [26] - although non-spherical symmetry have been considered [36]. In this section, \nwe will discuss several ambiguities within the literature over how the criteria for collapse should be defined 2 : \n- · The time at which PBH abundance should be calculated. The threshold for collapse is normally stated in terms of the time-independent component of the density contrast, during the linear regime whilst a perturbation is super-horizon [29]. In the linear regime, the density contrast grows proportionally to the parameter glyph[epsilon1] (equation (3)), which is the ratio of the perturbation scale to the horizon scale at a given scale. Taking the time-independent component is therefore equivalent to setting this parameter to unity, which has resulted in many papers treating this as the value of the density contrast at horizon crossing. Ideally, the abundance of PBHs should be calculated by considering the perturbations on super-horizon scales, long before they reenter the horizon.\n- · Should the peak value of the density contrast be used, or the smoothed density contrast? The peak value at the centre of a density perturbation was used in a recent paper [24] to determine the abundance of PBHs. This is valid when the distribution is already smooth on scales smaller than the scale being considered (as was considered in that paper), which is generally not the case unless a power spectrum with a very narrow peak is being considered. In order to investigate PBH formation over a wider range of scales, it is necessary to use a smoothing function. We also consider the fact that (for perturbations of arbitrary profile shapes), the threshold value for collapse of the central peak varies from 2 / 3 to infinity, whilst the critical value for collapse of the top-hat smoothed density contrast varies from 0 . 41 to 2/3 - a much smaller range of values. It was also discussed in section II that, for certain ζ -profiles, the peak in the density contrast may be off-centred 3 (when α > 1, meaning the central value is smaller than the peak) or infinite - a problem which is avoided by using the smoothed value (see appendix B).\n- · The choice of window function has a significant effect on the calculated abundance of PBHs (as was discussed in [38]). The threshold value for collapse is typically stated in terms of the volume-averaged density contrast (as for example in [29, 31]), which corresponds to a top-hat window function - suggesting a top-hat function should be used. However, in \nthe super-horizon regime, the smoothing function decreases as k 2 but perturbations grow as k 2 - meaning the top hat window function is typically not efficient enough at removing small-scale perturbations. For this reason a Gaussian window function is often used in the literature. This however has the drawback of changing the perturbation shape, introducing an unphysical change in the value of the threshold δ c . For this reason, we will follow the standard approach of using a top-hat smoothing function in this paper, whilst treating perturbations as if they are still linear at horizon entry, and employ the linear transfer function for subhorizon perturbations to reduce the effects of small-scale perturbations (although note that the linear transfer function might not very accurate for the large amplitude perturbations required to generate PBHs). \n- · The scale of a perturbation r m is best stated in terms of the radius at which the compactness function C ( r ) is maximised (see Section II). This is different to the previously used definition r 0 [31], where the scale of the perturbation was defined as the radius at which the density contrast becomes negative. As was shown in [29] computing the density contrast at r 0 does not give a sensible parameter to compare different shapes. The averaged value of the density contrast evaluated at r m is characterised by the general relation given by (11): thus it relates the local value of the density contrast with the smoothed averaged value, independently of any possible choice of the curvature profile. For this reason, measuring the amplitude of the perturbation at r m is a consistent way to quantify the effect of the curvature profile on the threshold. \nIn this paper, the criteria for a perturbation to collapse to form a PBH will be stated in terms of the volume-averaged density perturbation (averaged over a sphere of radius r m , corresponding to a top-hat window function with radius r m centred on the peak of the perturbation) at the time of horizon reentry where the perturbation is taken to behave linearly up to this point (although this is not assumed in the simulations). The formation criterion, and its effect on the calculated abundance of PBHs obtained is discussed in more detail in [37].", 'A. Numerical scheme': 'The calculations made in this paper to calculate the threshold of PBH formation for the different initial ζ -profiles described in Section II have been made with the same code as used in [29, 3133, 39]. This has been fully described previously and therefore we give only a very brief outline of it here. It is an explicit Lagrangian hydrodynamics code with the grid designed for calculations in an expanding cosmological background. The basic grid uses logarithmic spacing in a mass-type comoving coordinate, allowing it to reach out to very large radii while giving finer resolution at small radii. The initial data follow from the initial condition seen in Section II, specified on a space-like slice at constant initial cosmic time t i defined as a ( t i ) r m e ζ ( r m ) = 10 /H , ( glyph[epsilon1] = 10 -1 ), while the outer edge of the grid has been placed at 90 R m , to ensure that there is no causal contact between it and the perturbed region during the time of the calculations. The initial data is then evolved using the Misner-Sharp-Hernandez equations so as to generate a second set of initial data on an initial null slice which is then evolved using the Hernandez-Misner equations. During the evolution the grid is modified with an adaptive mesh refinement scheme (AMR), built on top of the initial logarithmic grid, to provide sufficient resolution to follow black hole formation down to extremely small values of ( δ -δ c ).', 'B. Threshold, scaling law and mass spectrum': "In the left panel of Figure 3 we plot the threshold values δ c against the parameter α used in (12) to vary the initial curvature profile. The lowest limit δ c glyph[similarequal] 0 . 41 corresponds to the analytic solution derived in [35] where the gravitational effect of pressure was taken into account while pressure gradients were instead neglected. It was shown in [29] that this corresponds to shapes of the density contrast with a very large peak ( α glyph[lessmuch] 1) and a smooth tail ( r 0 glyph[greatermuch] r m ): in this configuration most of the matter is already within in the initial cosmological horizon, and only a negligible amount of matter is spread away by the pressure gradients before the black hole is formed, without modifying the shape significantly during the collapse. The maximum value of δ c = 2 / 3, ( -r m ζ ' ( r m ) = 1), corresponds instead to shapes with r m = r 0 , ( α →∞ ), and the density contrast is very steep. In this case the pressure gradients are very large and a substantial modification of the matter configuration is produced during the collapse. \nThe right panel show the same results of δ c as a function of the corresponding behaviour of r 0 /r m : the range of values we have been able to compute here are 0 . 442 glyph[lessorsimilar] δ c glyph[lessorsimilar] 0 . 656, (0 . 34 ≤ α ≤ 2). Beyond this range the initial profile of the density contrast becomes too extreme, making the numerical simulations unstable. A more detailed analysis of the relationship between the density contrast and the morphology of the initial curvature profile can be found in [29] where different parameterizations of the curvature profiles has been considered, using more than one parameter, getting much closer to the lower limit of δ c glyph[similarequal] 0 . 41. \nAs has been shown in previous works [20, 21, 31, 33, 39] the mass spectrum of PBHs follows the critical collapse, characterized by a scaling law relation \nM PBH M H = K ( δ -δ c ) γ (16) \nwhere M H is the mass of the cosmological horizon at horizon crossing, γ glyph[similarequal] 0 . 36 is the critical exponent depending only on the value of ω of the equation of state and K is a parameter depending on the particular shape of the density contrast. Because this parameter will play some role in the next sectionto determine the abundance of PBHs, we have performed numerical simulations to quantify its variation, finding that it varies between 3 and 11 for α varying from 0 . 4 and 1 . 9 (corresponding to δ c varying between 0 . 45 and 0 . 65) for the profiles considered here, with a representative value of K glyph[similarequal] 4 when δ c glyph[similarequal] 0 . 55, ( α = 1 in (12)).", 'V. CALCULATION OF PBH ABUNDANCE': "In this section we will discuss how the abundance of PBHs can be calculated from the primordial curvature perturbation power spectrum P ζ . The abundance of PBHs will be stated in terms of the energy fraction of the universe (which will be) contained in PBHs at the time of formation, taken for simplicity to be the time of horizon reentry. In principle the time taken for a PBH to form depends slightly on the amplitude of the perturbation collapsing. A formalism for deriving the mass function from a given power spectrum P ζ taking into account the non-linear relation between ζ and δ m will also be derived. \nWe will assume throughout that the curvature perturbation has a Gaussian distribution, partly for simplicity and also motivated by the fact that any local-type non-Gaussianity with f NL glyph[greaterorsimilar] O (10 -3 ) will generate an unacceptably large dark-matter isocurvature perturbation in the CMB [40, 41] - although such bounds can be evaded if non-Gaussianity only couples scales smaller than those observable in the CMB or LSS. In addition, we note that the non-Gaussianity present in \n<!-- image --> \nFIG. 3 . The left plot show the value of δ c as function of α obtained with numerical simulations using an initial curvature profile given by (12), while the right panel show the behaviour of δ c as function of the corresponding behaviour if r 0 /r m . The bottom dashed horizontal line indicates the lowest limit of the threshold, obtained analytically assuming that the pressure gradients during the collapse are negligible. The upper dashed horizontal shows to the largest possible value of δ c , with shapes characterized by very large pressure gradients at the scale r m . \n<!-- image --> \nsingle-field inflation (e.g. the Maldacena consistency relation [42]) does not generate isocurvature perturbations [43, 44]. Even when taking ultra slow roll inflation into account, it remains uncertain whether the non-Gaussianity can have a relevant effect [45-48] unless the inflaton field has a non-canonical kinetic term [49-52]. \nThe density contrast δρ/ρ b is related to the curvature perturbation ζ as in equation (4). However, the key parameter to use for PBH formation is instead the smoothed density contrast δ m . Using a top-hat window function with areal radius R = a ( t ) exp( ζ ( r m )) r m , the amplitude of (spherically symmetric) peaks in the smoothed density contrast is related to the curvature perturbation in radiation domination as [29] \nδ m = -2 3 r m ζ ' ( r m ) [ 2 + r m ζ ' ( r m ) ] . (17) \nBecause ζ has a Gaussian distribution, its derivative will also have a Gaussian distribution. Therefore, equation (17) can be expressed in terms of a linear Gaussian component δ l = -4 3 r m ζ ' ( r m ) \nas 4 \nδ m = δ l -3 8 δ 2 l . (18) \nThe probability density function (PDF) of δ l then follows a Gaussian distribution \nP ( δ l ) = 1 √ 2 πσ 2 exp ( -δ 2 l 2 σ 2 ) . (19) \nδ l represents the linear component of the smoothed density contrast and its variance σ 2 can be calculated by integrating the linear component of the density power spectrum as follows: \nσ 2 = 〈 δ 2 l 〉 = ∞ ∫ 0 d k k P δ l ( k, r m ) = 16 81 ∞ ∫ 0 d k k ( kr m ) 4 ˜ W 2 ( k, r m ) T 2 ( k, r m ) P ζ ( k ) , (20) \nwhere ˜ W ( k, r m ) is the Fourier transform of the top-hat smoothing function, T ( k, r m ) is the linear transfer function, and the smoothing scale r m is equal to the horizon scale. The horizon scale r m is used to define the time at which P δl (and T ( k, r m )) should be evaluated 5 . For simplicity here, we assume that the relevant scale for PBH formation is given by kr m glyph[similarequal] 1, although this is not a very accurate approximation, and the exact relation between k and r m depends on the profile of the density contrast, which depends on the shape of the power spectrum [24]. \nThe Fourier transform of the top-hat smoothing function is given by \n˜ W ( k, r m ) = 3 sin( kr m ) -kr m cos( kr m ) ( kr m ) 3 , (21) \nand the linear transfer function, where we consider r m as a time dependent measure of the horizon, is given by [55] \nT ( k, r m ) = 3 sin( kr m √ 3 ) -kr m √ 3 cos( kr m √ 3 ) ( kr m √ 3 ) 3 . (22) \nThe most straight-forward method to calculate the abundance of PBHs from a non-Gaussian distribution is to work instead with the Gaussian component of the perturbations [56, 57]. To this \nend, equation (18) is solved with δ R = δ c to find the critical amplitude of the linear component δ c,l ± , \nδ c,l ± = 4 3 ( 1 ± √ 2 -3 δ c 2 ) , (23) \nwhere there are 2 solutions because equation (18) is quadratic. When necessary, we will take δ c = 0 . 55 in this paper, the critical value of the volume averaged density contrast when ζ is taken to have a Gaussian profile shape. However, we note that only the first solution δ c,l -corresponds to a physical solution - the second solution corresponds to type 2 perturbations. We will therefore take that a PBH will form in regions where δ c < δ < 2 / 3, where 2 / 3 is the maximum value for the density contrast given by equation (18). This corresponds to δ c,l -< δ l < 4 / 3. \nThe final mass of a PBH, M PBH , which forms during radiation domination depends on the shape and amplitude δ R of the initial perturbation, and the horizon mass at horizon reentry M H , \nM PBH = K M H ( δ m -δ c ) γ = K M H ([ δ l -3 8 δ 2 l ] -δ c ) γ , (24) \nwhere K depends on the profile shape and typically takes a value between 3 and 5. During radiation domination γ glyph[similarequal] 0 . 36, which is the value we will take [58]. The horizon mass M H is proportional to the horizon scale squared r 2 m , \nM H M glyph[circledot] ≈ ( r m r glyph[circledot] ) 2 = ( k k glyph[circledot] ) -2 . (25) \nFor a random Gaussian field, the number density of sufficiently rare peaks in a comoving volume is [26] \nN = µ 3 4 π 2 σ 3 ν 3 exp ( -ν 2 2 ) , (26) \nwhere σ is given by equation (20), \nν ≡ δ l /σ (27) \nand µ is given by \nµ 2 = ∞ ∫ 0 d k k P δ l ( k, r m ) k 2 = 16 81 ∞ ∫ 0 d k k ( kr m ) 4 ˜ W 2 ( k, r m ) T 2 ( k, r m ) P ζ ( k ) k 2 . (28) \nHere, the application of equation (26) relies on the assumption that peaks in the smoothed linear density field correspond to peaks in the smoothed non-linear density field, such that equation (17) \ncan be applied to calculate the height of peaks in the non-linear field. In general, this will not be true, but will be valid in the case that only sufficiently rare and large peaks are considered (which is the same assumption required for the validity of (26) and spherical symmetry) - discussed further in appendix B. \nThe fraction of the energy of the universe at a peak of given height ν which collapses to form PBHs, β ν , then depends on the mass of the PBHs relative to the horizon scale and the number density of the peaks: \nβ ν = M PBH ( ν ) M H ( r m ) N ( ν ) θ ( ν -ν c ) , (29) \nwhere the time dependance of the horizon mass M H ( r m ) is parameterised by the horizon scale r m . θ ( ν -ν c ) is the Heaviside step function which indicates that no PBH will form if ν is below the critical threshold. The total energy fraction of the universe contained within PBHs at a single time of formation is given by integrating over the range of ν which forms PBHs, \nβ = 4 3 σ ∫ ν c -d ν K 3 π ( νσ -3 8 ( νσ ) 2 -δ c ) γ ( µ aHσ ) 3 ν 3 exp ( -ν 2 2 ) , (30) \nwhere ν c -≡ δ c,l -/σ , see equation (23), and equations (24), (26), and (27) have been explicitly substituted into (29). The total energy fraction of the universe contained within PBHs today can be approximated by integrating over all times at which PBHs form, parameterised here by the horizon mass (more details of this integral can be found in [59]) \nΩ PBH = M max ∫ M min d(ln M H ) ( M eq M H ) 1 / 2 β ( M H ) , (31) \nwhere M H is the horizon mass at the time of formation and M eq is the horizon mass at the time of matter-radiation equality. We will use the value M eq = 2 . 8 × 10 17 M glyph[circledot] (using the same parameters as [60]). M min and M max are respectively the smallest and largest horizon masses at which PBHs form 6 . The derivation of this formula assumes that the matter density Ω m during radiation domination grows proportionally to the scale factor until the time of matter-radiation equality, whereupon the universe immediately becomes matter dominated. \nThe mass function f ( M PBH ) can then be obtained by differentiating Ω PBH with respect to the PBH mass: \nf ( M PBH ) = 1 Ω CDM dΩ PBH d(ln M PBH ) , (32) \n<!-- image --> \nFIG. 4 . Top: The energy fraction of the universe collapsing to form PBHs at the time of formation, β , is plotted against the amplitude of a scale invariant power spectrum, P ζ = A s . Bottom: The mass function of PBHs f ( M glyph[circledot] ) of PBHs (defined in equation (32)) with a mass of 1 M glyph[circledot] as a function of the amplitude of A s . Accounting for the non-linear relation means that a significantly smaller abundance of PBHs is calculated, by many orders of magnitude. For δ c = 0 . 5 then K = 5 . 5 and for δ c = 0 . 6 then K = 3 . 5. The abundance of PBHs is dominated by the uncertainty in δ c (which has an exponential effect), rather than K (which has a linear effect). \n<!-- image --> \nwhere the equations (20), (28) and (30) should be recast in terms of the horizon mass and PBH mass using the substitutions in equations (24) and (25). Ω CDM will be taken as 0.27 where necessary in this paper. \nBroad and narrow peaks in the power spectrum are often considered when calculating the PBH abundance. To give a concrete example, we will consider the two extreme cases - a scale invariant power spectrum, and an extremely narrow peak. For the scale invariant case, we will take P ζ = A s = constant . For the narrow peak, we will take the peak to be narrow enough such that it can be treated as a Dirac-delta function, P ζ = A s δ D (ln( k/k ∗ )), although note that such a power spectrum is unphysical (for example, [10] describes that the power spectrum cannot be steeper than k 4 , at least in the context of single-field inflation). \nFor the scale invariant case, equations (20) and (28) give scale invariant values of σ 2 ∼ 1 . 06 A ∫ and ( µ/ ( aH )) 2 ∼ 6 . 86 A s respectively. Figure 4 shows the abundance of PBHs, β (equation (30)), as a function of A s , whilst figure 4 shows the mass function f ( M ) (equation (32)) evaluated at M = M PBH as a function of the power spectrum amplitude. \nIn the case of the narrowly peaked spectrum, we assume without loss of generality that the power spectrum peaks at a scale corresponding to a horizon mass of 1 M glyph[circledot] , with corresponding horizon scale k ∗ = k glyph[circledot] , in order to allow a direct comparison to the broad power spectrum. We will \n<!-- image --> \nFIG. 5 . Left: The energy fraction of the universe collapsing to form PBHs at the time of formation, β , is plotted against the amplitude of the power spectrum. Right: the mass function of PBHs f ( M glyph[circledot] ) of PBHs with a mass of 1 M glyph[circledot] as a function of the amplitude of the power spectrum. In both plots, a Dirac delta power spectrum has been assumed, P ζ = A s δ D (ln( k/k ∗ )). Using the profile shape predicted from a Dirac-delta power spectrum, K = 4, and δ c = 0 . 51. \n<!-- image --> \nalso consider that PBH formation occurs only at the horizon scale corresponding exactly to the peak in the power spectrum, and that β corresponds to the total energy fraction collapsing into PBHs at all epochs, rather than integrating over ln M H as in equation (31). Whilst equation (20) will give a significant variance σ 2 for values of r m close to r glyph[circledot] (suggesting perturbations of varying scales exist), this is because δ m does not immediately become zero when the smoothing scale is not set exactly equal to the scale of the perturbation. However, the scale of all perturbations here is fixed, and so will all enter the horizon at the same scale. In this case, evaluated as k glyph[circledot] enters the horizon, equations (20) and (28) give σ 2 = ( µ/ ( aH )) 2 ∼ 0 . 151 A s . \nWhat can be learned from these figures is that accounting for the non-linear relationship between the density contrast and the curvature perturbation will always serve to reduce the calculated abundance of PBHs, compared to using the linear model. It can be seen from comparing the figures from the scale invariant power spectrum to the narrowly peaked power spectrum, that the abundance of PBHs is strongly dependent on the shape of the power spectrum rather than simply the amplitude - and so constraints on the power spectrum from constraints on PBH abundance should be calculated on a case-by-case basis for different inflationary models which predict different shapes for the power spectrum. This fact has been well known for some time and was investigated in more detail recently by [24]. \nHowever, we will here make a simple calculation to describe by what fraction the amplitude of the curvature perturbation power spectrum A s needs to increase in order to obtain the same \nvalue for β as when the linear relation is used. The dominant term in the calculation for the abundance of black holes, equation (30), is the exponential term exp( -ν 2 ). After integrating, this will give a dependance roughly proportional to exp( -ν 2 c -). In order to produce (approximately) the same number of PBHs from the linear calculation as from the non-linear calculation, we therefore require ν c,L = ν c,NL - where the subscripts L and NL denote the linear and non-linear calculations respectively, \nδ c,L σ L = δ c,l -σ NL , (33) \nwhere δ c,l -is given by equation (23). In both the linear and non-linear case, σ 2 for a given power spectrum shape is proportional to A s by the same constant of proportionality, σ 2 = C A s . Finally, we can write down the ratio between A L and A NL as \nA NL A L = ( δ c,l -δ c ) 2 = 4 ( 1 -√ 2 -3 δ c 2 ) 3 δ c 2 . (34) \nFor values 0 . 41 < δ c < 2 / 3, this factor varies approximately from 1.5 to 4. For more typical values 0 . 5 < δ c < 0 . 6, in order for the same number of PBHs to form, the amplitude of the power spectrum must be 1.78-2.31 times greater than previously calculated with the linear model. In particular, for the commonly considered case of a Gaussian profile in ζ , with δ c = 0 . 55, the power spectrum needs to be enhanced by a factor of 2.0 in order to get the same number of PBHs forming if one neglected the non-linear relationship between ζ and δ .", 'VI. SUMMARY': "In the radiation dominated epoch following the end of inflation, perturbations can collapse to form PBHs if the perturbation amplitude is large enough. Whether or not a PBH will form depends on the amplitude of the density contrast δ m , rather than the amplitude of the curvature perturbation ζ . The non-linear relation between the curvature perturbation ζ and the density contrast δρ/ρ b , given by equation (4), means that δ m will have a non-Gaussian distribution even when ζ is perfectly Gaussian, which has a significant effect on the number of PBHs which form in the early universe. \nWe have discussed briefly the formation criterion which should be used to determine whether a perturbation will collapse or not, and in this paper, we argue that the volume-averaged (smoothed) \ndensity contrast of a peak δ m should be used rather than the value δρ/ρ b evaluated at r = 0. The scale over which the perturbation should be averaged, r m , is the scale at which the compaction function C ( r ) (defined in equation (5)) is maximised, and note that at this scale C ( r ) is equal to δ m . In this paper, we use the amplitude of δ m measured at the cosmological horizon entry to determine whether a PBH will form - although point out that this is somewhat inconsistent as the expression for δρ/ρ b is valid in the super-horizon regime and the perturbation will not evolve linearly until horizon entry when they have a large amplitude. Further consideration of these factors is left for future study. \nMaking use of numerical simulations described in section IV, we have considered different profiles of ζ which form density perturbations to determine a threshold value for PBH formation and how this can depend on the shape of the profile in ζ . We noted in section II that the relation between peaks in ζ and δρ/ρ b is not straightforward and may not coincide. For a given profile shape of ζ , the profile of δρ/ρ b depends also on the amplitude of the perturbation, and we have calculated a relation between the amplitude of the perturbation δ m and the mass of the PBH formed accounting for this (often referred to as the extended mass function, rather than assuming monochromatic formation of PBHs at a given epoch). \nThere is a simple relation between δ m and ζ , given by equation (17), which can be used to fully describe the non-Gaussianity of δ m when ζ is taken as Gaussian. Making use of this relation we have derived a formalism to derive the abundance of PBHs. The abundance of PBHs may be expressed either in terms of the energy fraction collapsing into black holes at the time of formation β , the present-time density parameter for PBHs, Ω PBH , or the mass fraction of dark matter contained within PBHs of a given mass, f ( M PBH ). These expressions are calculated utilising the theory of peaks and accounting for the extended mass function of PBHs. \nWhen the non-Gaussianity of the density δ m is taken into account, we find that this always reduces the number of PBHs which form by many orders of magnitude, see figures 4-5. We reproduce the known result that the abundance of PBHs depends upon both the amplitude and shape of the curvature perturbation power spectrum. In order for a comparable number of PBHs to form compared to using the linear relation between ζ and δ , we find that the amplitude of the power spectrum must therefore be ∼ 2 -3 times larger, with the amount depending on the value taken for the collapse threshold, see (34), but otherwise being almost independent of the shape of the power spectrum. \nFinally we note that the non-linear relation for the smoothed density contrast in terms of the \nspatial derivative of the curvature perturbation makes a clear analogy to local non-Gaussianity, with a negative value of (local) f NL that suppresses PBH formation (see also the 'Note added' below). However, the analogy is potentially misleading since this non-Gaussian term only affects the one-point function of δ m and it does not generate a bispectrum because the derivatives of ζ are uncorrelated on scales much larger than the PBH scale, r m . Therefore it does not correspond to a coupling between long and short-wavelength modes and hence does not generate a large-scale dark matter isocurvature perturbation (which would be observationally ruled out [40, 41]). This also implies that the 'apparent' negative local f NL cannot lead to an enhanced formation of PBHs in the case of a primordial power spectrum with a broad peak, in contrast to a physical value of f NL < 0 of ζ , see [61] for details. \nThe relation used between ζ and δ , equation (4), is exact in ζ , only neglecting higher-order terms in glyph[epsilon1] , valid because glyph[epsilon1] glyph[lessmuch] 1 in the super-horizon limit. Therefore, we have quantified the effect of the complete non-linear relationship upon the abundance of PBHs. \nNote added: While this paper was approaching completion a related work by Kawasaki and Nakatsuka [53] appeared on the arXiv. Our broad conclusions are in agreement with their paper, principally that the non-linear relationship between ζ and δ makes the formation of PBHs less likely. For typical values of δ c ∼ 0 . 5, we confirm their finding that the suppression of PBH formation due to the non-linear terms requires the power spectrum to have an amplitude ∼ 1 . 4 2 times greater in order to form the same number of PBHs. \nProduced and appearing in parallel with our paper, the paper by De Luca et al. [62] studies the same topic. Although there are some significant differences in our methodology, our results are in broad agreement, both showing that the non-linear relation between δρ/ρ b and ζ leads to a suppression in the PBH formation rate. Unlike these two papers, we take critical collapse into account.", 'ACKNOWLEDGEMENTS': "We thank V. De Luca, G. Franciolini, A. Kehagias, M. Peloso, A. Riotto and C. Unal, the authors of [62], for sharing their draft of a paper which was written in parallel to this one and helpful discussions. We thank Nicola Bellomo and Eiichiro Komatsu for helpful comments on a draft of this paper and we thank Jaume Garriga, Cristiano Germani, Marcello Musso and Licia Verde for helpful discussions. \nSY is funded by a Humboldt Research Fellowship for Postdoctoral Researchers. IM is supported by the Unidad de Excelencia Mar'ıa de Maeztu Grant No. MDM-2014-0369. CB is funded by a Royal Society University Research Fellowship.", "Appendix A: Gaussianity and variance of -4 3 r m ζ ' ( r m )": "Here we will derive the variance and discuss the Gaussianity of the linear term in equation (17), δ l = -4 3 r m ζ ' ( r m ). The linear relation between the density contrast δρ/ρ b and the curvature perturbation ζ in radial coordinates r is given by \nδρ ρ b ( r, t ) = -4 9 glyph[epsilon1] 2 ( t ) r 2 m ( ζ '' ( r ) + 2 r ζ ' ( r ) ) , (A1) \nwhere the prime denotes a derivative with respect to the radial co-ordinate r , and the parameter glyph[epsilon1] describes the super-horizon time-evolution of the perturbation and is given in the linear approximation by glyph[epsilon1] = ( r m aH ) -1 , where r m is the perturbation scale and aH is the horizon scale. In this paper we consider the perturbations at the time of horizon reentry, t H , such that r m = ( aH ) -1 ⇒ glyph[epsilon1] ( t H ) = 1, and we will therefore drop the glyph[epsilon1] ( t ). The volume averaged density contrast used to determine PBH formation is given by \nδ m = 1 V r m ∫ 0 4 πr 2 δρ ρ b ( r, t H )d r , (A2) \nwhere in the linear approximation the volume is given by V = 4 3 πr 3 m . Although this integral is normally integrated over the comoving distance r com ≡ re ζ ( r ) , this difference represents higher-order effects neglected in the linear approximation. Combining the above equations gives \nδ m = -4 3 r m r m ∫ 0 d rr 2 ( ζ '' ( r ) + 2 r ζ ' ( r ) ) = -4 3 r m ζ ' ( r m ) = δ l , (A3) \nwhich is the linear term seen in equation (17). \nWe now consider the variance of δ m , noticing that equation (A2) can be considered as a top-hat window function centred on the peak convolved with the density contrast \nδ m ( X ) = 1 V ∞ ∫ 0 d r 4 πr 2 δρ ρ b ( r, t H ) θ ( r m -r ) = ∫ d x 3 δρ ρ b ( x, t H ) W ( X -x, r m ) , (A4) \nwhere X is the location of a peak in cartesian coordinates (corresponding to r = 0) and θ ( r ) is the Heaviside step function. The second equality is the same integral expressed in cartesian coordinates, \nwith the window function W ( x, r m ) given by: \nW ( x, r m ) = θ ( r m -| x | ) 4 3 πr 3 m . (A5) \nFor our purposes, it is more convenient to express equation (A4), a convolution in real space, as a multiplication in Fourier space, \nδ m ( k ) = ˜ W ( k, r m ) δ ( k ) , (A6) \nwhere the Fourier transform of the window function ˜ W ( k, r m ) is given by equation (21) and δ ( k ) is given by \nδ ( k ) = -4 9 T ( k, r m )( kr m ) 2 ζ ( k ) . (A7) \nwhere the linear transfer function T ( k, r m ) is given by equation (22) and δ m ( k ) can therefore be written as \nδ m ( k ) = -4 9 ˜ W ( k, r m ) T ( k, r m )( kr m ) 2 ζ ( k ) . (A8) \nSince we have assumed ζ ( x ) has a Gaussian distribution, it also has a Gaussian distribution in Fourier space: ζ ( k ) has a Gaussian distribution (being a linear combination of Gaussian variables ζ ( x )). δ m ( k ) is then related to ζ ( k ) by a linear factor, meaning that δ m ( k ) also has a Gaussian distribution. Finally, δ m ( x ) is a linear combination of the Gaussian Fourier modes; hence it also has a Gaussian distribution. \nFinally, we can calculate the variance σ 2 by integrating the power spectrum, \nσ 2 = 〈 δ 2 l 〉 = ∞ ∫ 0 d k k P δl ( k, r m ) = 16 81 ∞ ∫ 0 d k k ( kr m ) 4 ˜ W 2 ( k, r m ) T 2 ( k, r m ) P ζ ( k ) . (A9)", 'Appendix B: Correspondence of large peaks': "We will here consider the correspondence between peaks in the various fields considered within the context of this paper - the curvature perturbation field ζ , the non-linear density contrast field δρ/ρ b , the non-linear smoothed density field δ m and the linear smoothed density field δ l . For the purposes of this discussion, type 2 perturbations will not be considered (corresponding to δ m > 2 / 3, as the abundance of such perturbations is exponentially suppressed (even compared to the exponentially small number of perturbations which form PBHs). \nFIG. 6 . Profile shapes in the density contrast after a top-hat smoothing function has been applied. The profiles are calculated from a curvature perturbation profile given by equation (12), and each profile is smoothed on a scale r m . The values of α are α = 0 . 5 , 0 . 75 , 1 . 0 , 1 . 25 , 1 . 50 , 2 . 0, where increasing α corresponds to more negative troughs seen in the figure. The amplitude in each case is A = 0 . 5. \n<!-- image --> \nFigure 2 shows the critical profiles of peaks in ζ and δρ/ρ b (in spherical symmetry). All of the profiles in ζ have a central peak at r = 0, whilst the density field can have an off-centred peak or a divergence at the centre - and it cannot necessarily therefore be stated that a PBH forms at peaks in δρ/ρ b . As mentioned in section III this problem is overcome by using the smoothed density contrast - figure 6 shows the same profiles smoothed on a scale r m . The off-centred peaks are no longer seen because the smoothing scale r m will by definition larger than the radius at which the density peaks, and, being a feature smaller than the smoothing radius, is therefore removed in the process of smoothing. The divergences at the centre (while unphysical) are nonetheless removed because the divergence, whilst they represent infinite density as r → 0, they do not represent infinite mass - and the integral during the smoothing therefore converges. It can therefore be stated that, for isolated spherically-symmetric type 1 perturbations, the peaks in ζ correspond to peaks in the smoothed density field δ m . \nIn order to calculate the abundance, it has been assumed that peaks in the (Gaussian) linear smoothed density field correspond to peaks in the (non-Gaussian) non-linear field. In general, this is not expected to be true - however, for the very large and rare peaks from which PBHs form, it is expected that this should be a valid approximation. This is primarily due to the fact that the \nlarge peaks in the Gaussian fields ( ζ or δ l ) are very unlikely to be close to other large peaks, that is, that the local region surrouding large peaks contains only significantly smaller perturbations. This is expected to result in the large peak being (approximately) spherically symmetric peak [26]. When the non-linear smoothed density field is calculated, this spherical symmetry is preserved implying that the perturbation in the non-linear field must also be a peak (when smoothed on an appropriate scale, see above). \nThe correlation of peaks in ζ and the density δρ/ρ b was investigated in [62], concluding that 'one can associate the number of rare peaks in the overdensity with the number of peaks in the curvature perturbation which are spiky enough' , validating the assumptions applied here. \nFurthermore, only modes which are similar to the smoothing scale (which is considered equal to the horizon scale) have a non-negligible effect on the smoothed density field - larger-scale and smaller-scale modes are suppressed by the k 4 term and the smoothing and transfer functions respectively in equation (20). The smoothed density fields (linear and non-linear) therefore only inherit peaks from the ζ -field on a small range of scales. 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2020PhRvL.125w1101D
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Spin-Induced Black Hole Spontaneous Scalarization
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2020-01-01
| 37 | 0.45 | 163 |
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We study scalar fields in a black hole background and show that, when the scalar is suitably coupled to curvature, rapid rotation can induce a tachyonic instability. This instability, which is the hallmark of spontaneous scalarization in the linearized regime, is expected to be quenched by nonlinearities and endow the black hole with scalar hair. Hence, our results demonstrate the existence of a broad class of theories that share the same stationary black hole solutions with general relativity at low spins, but which exhibit black hole hair at sufficiently high spins (a /M ≳0.5 ). This result has clear implications for tests of general relativity and the nature of black holes with gravitational and electromagnetic observations.
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https://arxiv.org/pdf/2006.03095.pdf
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{'Spin-induced black hole spontaneous scalarization': "Alexandru Dima, 1, 2 Enrico Barausse, 1, 2, 3 Nicola Franchini, 1, 2 and Thomas P. Sotiriou 4 \n1 SISSA, Via Bonomea 265, 34136 Trieste, Italy and INFN Sezione di Trieste 2 IFPU - Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014 Trieste, Italy 3 Institut d'Astrophysique de Paris, CNRS & Sorbonne Universit'es, UMR 7095, 98 bis bd Arago, 75014 Paris, France 4 School of Mathematical Sciences & School of Physics and Astronomy, University of Nottingham, University Park, Nottingham, NG7 2RD, UK \n(Dated: November 18, 2020) \nWe study scalar fields in a black hole background and show that, when the scalar is suitably coupled to curvature, rapid rotation can induce a tachyonic instability. This instability, which is the hallmark of spontaneous scalarization in the linearized regime, is expected to be quenched by nonlinearities and endow the black hole with scalar hair. Hence, our results demonstrate the existence of a broad class of theories that share the same stationary black hole solutions with general relativity at low spins, but which exhibit black hole hair at sufficiently high spins ( a/M glyph[greaterorsimilar] 0 . 5). This result has clear implications for tests of general relativity and the nature of black holes with gravitational and electromagnetic observations. \nIntroduction: Direct and indirect detections leave little doubt that black holes (BH) exist in nature [1-8]. In general relativity (GR) the mass and the spin of an astrophysical BH fully determine its properties. An electric charge is also technically allowed, but is expected to be paltry for astrophysical BHs, see e.g. [9]. Any other quantity, hair in jargon, is not necessary according to nohair theorems [10-12]. Future gravitational wave detectors will finally allow us to confront theorems and observations with unprecedented precision [13-15], improving upon current observations, which are perfectly compatible with hairless BHs [16, 17]. \nIt is tempting to interpret an absence of BH hair as a vindication of GR minimally coupled to the Standard Model. However, new fundamental fields can be more elusive. It is illustrative to consider scalar fields: no-hair theorems exist for stationary BHs in scalar-tensor theories [18, 19], and static, spherically symmetric and slowly rotating BHs in shift-symmetric generalized (Horndeski) scalar-tensor theories [20, 21]. 1 In fact, it turns out that there is a single coupling term in the Horndeski class that gives rise to hair: a linear coupling between the scalar and the Gauss-Bonnet (GB) invariant [21, 26], given by \nG = R µνρσ R µνρσ -4 R µν R µν + R 2 . (1) \nConsidering that the Horndeski class contains all actions for a massless scalar nonminimally coupled to gravity that yield second order equations upon variation, absence of hair actually seems to be the norm rather than the exception for scalar fields. Indeed, known hairy BH solutions circumvent theorems by evading one or more of their assumptions, see e.g. [21, 27-32]. \nA further complication in attempting to detect new fields through BH hair is the possibility that, even within the context of the same theory, only certain BHs might actually exhibit it. This was realized only recently, as the first models of BH scalarization appeared in the literature [33, 34]. For concreteness, consider the action \nS = 1 2 ∫ d 4 x √ -g ( R -1 2 ∇ µ φ ∇ µ φ + f ( φ ) G ) , (2) \nwhere f is some function of φ , and where we have also set (as in the rest of this paper) 8 πG = c = 1. Varying the action with respect to φ yields \nglyph[square] φ = -f ' ( φ ) G , (3) \nwhere f ' ( φ ) ≡ df/dφ . Assume that f ' ( φ 0 ) = 0, for some constant φ 0 . Then solutions with φ = φ 0 are admissible and they are also solutions of GR. A no-hair theorem [33] ensures that they are unique if they are stationary, provided that f '' ( φ ) G < 0. \nThe fact that GR BHs are stationary solutions to this theory is not sufficient to conclude that there are no observable deviations from GR, as the perturbations over these solutions do not generally obey the GR field equations [35]. These perturbations may even grow unstable, thus rendering the GR solutions irrelevant. Indeed, one can think of -f '' G as the (square of the) mass of the scalar perturbation on a fixed background. Hence, the condition above ensures that this effective (squared) mass is positive. If the condition is violated and the effective (squared) mass becomes sufficiently negative, the GR solutions suffer a tachyonic instability and the scalar develops a nontrivial profile. \nA similar scalarization effect was shown to occur for neutron stars in a different class of scalar-tensor theories more that 25 years ago [36], and is triggered when the star compactness reaches a critical threshold. Related 'dynamical' scalarization effects [37-40] are present in the \nsame theories for neutron star binaries, whenever their separation is sufficiently small (or the binary's 'compactness' sufficiently large). However, in the class of theories considered in [36-40], scalarization is not present without matter, and BHs are vacuum solutions. 2 \nBlack hole scalarization is fairly well understood. It starts as a linear tachyonic instability and, as such, its onset is controlled only by terms that contribute to linear perturbations around GR solutions. In this sense, action (2) with f ( φ ) = ηφ 2 / 2 is sufficient to study the onset of scalarization [33, 41]. As the instability develops and the scalar grows, nonlinear terms become increasingly important and eventually quench the instability. Hence, the endpoint and properties of the scalarized solutions are actually controlled by the nonlinear interactions of the scalar [42, 43]. A characteristic example is that in models with different nonlinear interactions, scalarized solutions have different stability properties [42, 44]. \nHere we will focus exclusively on the onset of scalarization, so we will restrict our attention to quadratic scalar GB (qsGB) gravity, i.e. f ( φ ) = ηφ 2 / 2 (without loss of generality [41]). The effective (squared) mass of the scalar on a fixed background is then \nµ 2 eff = -η G . (4) \nFor the Schwarzschild solution, one has G = 48 M 2 /r 6 , which is always positive and decreasing with r , and which yields the horizon value G ( r = 2 M ) = 3 / (4 M 4 ). Hence, a tachyonic instability only occurs for η > 0, and the instability is expected to be more violent for smaller masses. 3 This is why the focus in the literature so far has been on η > 0 (or the equivalent condition in more complicated models). However, for a Kerr BH of mass M and spin parameter a in Boyer-Lindquist ( t, r, θ, ϕ ) one has \nG Kerr = 48 M 2 ( r 2 + χ 2 ) 6 ( r 6 -15 r 4 χ 2 +15 r 2 χ 4 -χ 6 ) (5) \nwhere, for brevity, χ ≡ a cos θ . Clearly, G Kerr is not monotonic, and can even become negative close to the horizon. This explains the results of [45, 46], where it was shown that rotation suppresses scalarization for η > 0. \nIn this Letter we focus on η < 0, which yields a real effective mass µ eff for low BH spins, but which can yield an imaginary µ eff for high spins. We investigate the behavior of linear scalar perturbations to the GR solution by evolving Eq. (3) on a Kerr background, with the goal of assessing for what BH spins and couplings η the perturbations become unstable. Indeed, at least two possible instability mechanisms may be at play in Eq. (3). \nThe first is the tachyonic instability associated to spontaneous scalarization, mentioned above. The second could be a superradiant instability, which is known to exist at high spins for constant real masses [47-50], and potentially also for non-constant effective masses [51] such as the one of Eq. (4). Superradiance occurs when bosonic waves with non-vanishing angular momentum are amplified when scattered by a spinning BH, at the expense of the rotational energy of the BH, which as a result spins down. For massive bosons, superradiant scattering can develop into an instability because the field is confined near the BH by its own mass. \nIt should be stressed that, in principle, both instabilites could be present. However, they have distinct features (timescales, the angular momenta involved, dependence on the BH spin). We show below that the tachyonic instability is by far the dominant effect for η < 0. More broadly, our results strongly suggest that there exist theories in which scalarization occurs only for rapidly rotating BHs. \nMethodology: For f ( φ ) = 0 and over a Kerr background, Eq. (3) separates into ordinary differential equations when φ is decomposed onto a basis of spheroidal harmonics. However, the choice f ( φ ) = ηφ 2 / 2 yields an intrinsically non-separable equation. We therefore resort to a time-domain numerical integration of this equation, by using techniques akin to those presented in [51, 52], to which we refer for more details. \nIn brief, the idea is to project Eq. (3) onto a basis of spherical 4 harmonics Y lm , which yields 1+1 evolutions equations (in t and r ) for the components of the scalar field, \nψ lm ( t, r ) ≡ ∫ Y ∗ lm ( rφ ) d Ω (6) \nThese equations are coupled and given explicitly by \n[ ( r 2 + a 2 ) 2 -a 2 ∆(1 -c m ll ) ] ¨ ψ l + a 2 ∆( c m l,l +2 ¨ ψ l +2 + + c m l,l -2 ¨ ψ l -2 ) + 4 iamMr ˙ ψ l \n-( r 2 + a 2 ) 2 ψ '' l -( 2 iam ( r 2 + a 2 ) -2 a 2 ∆ r ) ψ ' l +∆ [ l ( l +1) + 2 M r -2 a 2 r 2 + 2 iam r ] ψ l +∆ ∑ j 〈 lm | µ 2 eff ( r 2 + χ 2 ) | jm 〉 ψ j = 0 , (7) \n∆ ≡ r 2 -2 Mr + a 2 , (8) \nc m jl ≡ 〈 lm | cos 2 θ | jm 〉 \n= δ lj 3 + 2 3 √ 2 j +1 2 l +1 〈 j, 2 , m, 0 | l, m 〉 · 〈 j, 2 , 0 , 0 | l, 0 〉 , (9) \nwhere 〈 j 1 , j 2 , m 1 , m 2 | j 3 , m 3 〉 are the Clebsch-Gordan coefficients [53]. Note that the evolution of modes of different m decouples because of the axisymmetry of the problem. Moreover, because of reflection symmetry with respect to the origin, evenl and oddl modes also decouple: the evolution of a mode ( l, m ) is coupled to that of all the modes ( l +2 k, m ), with k = 1 , 2 , 3 , . . . . \nTo numerically evolve the system (7), we discretize the spatial grid and use a method of lines. By integrating in time using a fourth order explicit Runge-Kutta time-step inside the computational grid (as done e.g in [51]), it becomes apparent that the equations are stiff for large η , and that the numerical integration becomes unstable. To overcome this problem, we have used an Implicit-Explicit (IMEX) Runge-Kutta solver with adaptive time step, namely the IMEX-SSP3(3,3,2) and IMEXSSP(4,3,3) schemes of [54]. Note that implicit methods [55], while effective at dealing with stiff problems, are typically less accurate and more computationally expensive. However, implicit-explicit algorithms, by employing explicit steps for the non-stiff terms and implicit steps only for the stiff ones, can tackle stiff problems with limited computational overhead. We successfully compared our code to results from both frequency-domain techniques [56] and similar time-domain codes [52]. Our implementation was also tested by analysing the convergence of the results (and their overall robustness) vs timestep and spatial-grid resolution. \nResults: To investigate the possible presence of an instability, we evolve the scalar field by integrating the system given by Eq. (7), with l ranging from 0 to l max = 30 and | m | ≤ l , and with Gaussian initial conditions for each mode ψ lm . The results are robust against the choice of the cutoff l max - as long as that is sufficiently large and initial conditions, which only affect the early transient evolution of the scalar and not the unstable growth phase, if present. We consider BH spins a/M ∼ 0 . 5 0 . 999 and qsGB coupling | η | /M 2 ∼ 0 . 1 - 10 5 . \nFrom the simulations showing an exponential scalar growth, we extract the instability timescale τ of the reconstructed field | φ | = (∑ lm | ψ lm | 2 ) 1 / 2 ∝ exp( t/τ ) by fitting the time evolution of the scalar's amplitude after the initial transient. The contours in Fig. 1 show τ -1 as a function of a/M and | η | /M 2 . The instability becomes stronger as either the spin or the coupling increases. Moreover, there is a minimum spin a min below which the instability disappears. For | η | → ∞ , it appears that a min /M → 0 . 5 (up to percent level numerical errors). The solid green line denotes the combinations of parameters for which the instability disappears (i.e. τ → ∞ ). With the blue dotted line we show the same marginal instability curve for the reconstructed field, but excluding the m = 0 modes. As can be seen, when the latter are excluded the parameter space region yielding an instability shrinks, i.e. the main contribution to the \nFig. 1 - Instability timescale τ (color code) for the reconstructed field as a function of spin and GB coupling. The instability threshold for the total reconstructed field is shown by the solid green line, while the threshold when the m = 0 modes are excluded is shown by a blue dotted line. The red dashed line corresponds the instability threshold for the m = 0 odd modes, while the dot-dashed cyan line marks the instability threshold for the spherical mode l = m = 0 (see text for details). Note that all shown values of η are unconstrained by different observables (c.f. discussion in the conclusions). \n<!-- image --> \ninstability comes from the m = 0 modes. As a further test of this conclusion, we also computed the marginal instability curve for the m = 0 modes alone, and that does indeed match the solid green line in Fig. 1. \nEven and odd parity modes (i.e. modes with even and odd l ) automatically decouple in Eq. (7). In the m = 0 sector, which dominates the instability shown in Fig. 1, the odd and even modes give roughly comparable contributions. We have verified this by considering the marginal instability curves for the odd and even m = 0 modes separately, which are both very close to the solid green line of Fig. 1. As an example, the red dashed line in Fig. 1 represents the marginal instability curve for the m = 0 odd modes. \nIndeed, odd modes seem to have only marginally shorter instability times (by ∼ 1 -2%) than even ones for high spins and large couplings. Conversely, in the region | η | < 1, a/M > 0 . 9 the even modes are slightly more unstable, as can be seen from the somewhat increased distance between the red dashed and solid green line curves. \nNext we consider if some individual angular mode l, m gives the dominant contribution to the instability. To answer this question, we have to override the nonseparability of the problem. To this end, we have force- \nfully decoupled each l -mode in Eq. (7) , suppressing 'by hand' all the couplings between angular modes (i.e. 〈 lm | µ 2 eff ( r 2 + χ 2 ) | jm 〉 with l glyph[negationslash] = j ) generated by the GB invariant; we have only kept active the contributions to the effective mass of the single l -mode. We have then let the system evolve, selecting Gaussian initial data for the chosen mode only. By this technique, we have isolated, for instance, the instability parameter space for the spherical mode l = m = 0, whose marginal instability curve is shown in Fig. 1 by a cyan dot-dashed line. However, we could not find any single l, m mode for which the marginal instability curve obtained in this way matched, even roughly, the solid green line for the whole reconstructed field. We therefore conclude that the gravitational coupling between angular modes plays a fundamental role in the onset of the observed instability. \nWe now proceed to examine whether the instability is dominantly tachyonic or powered by superradiance. The growth times, as shown in Fig. 1, can be as small as ∼ 0 . 01 M . This seems to favor a tachyonic origin, as superradiance acts on longer timescales (see e.g. [50, 51]). Moreover, the fact that the instability is mostly due to the m = 0 modes, and that even the spherical mode l = m = 0 can be unstable (see cyan long-dashed critical line in Fig. 1) bodes ill for superradiance, as these modes can never satisfy the superradiance condition ω < m Ω (with ω and Ω respectively the wave and horizon angular frequencies). \nOne may naively expect the spherical mode l = m = 0 not to suffer from a tachyonic instability either, since µ 2 eff = -η G is positive everywhere in a Schwarzschild spacetime when η < 0 (as considered here). However, the (squared) effective mass for the l = m = 0 mode is actually -η 〈 00 |G Kerr | 00 〉 , which only matches the naive estimate -η G Schwarzschild at leading order in spin, correcting it by terms O ( a 2 ). This explains, in particular, why the spherical mode is stable at low spins. \nTo further confirm the tachyonic nature of the instabilities, we have conducted the following test. We re-ran our simulations with the (squared) effective mass replaced by its absolute value, µ 2 eff → | µ 2 eff | . This is enough to suppress the instabilities, and further shows that the latter were due to the change of sign of the GB invariant close to the horizon. One can also look at the scalar fluxes through the event horizon after the initial transient. In Fig. 2, we compare the scalar field's energy flux through the horizon for η = -10 M 2 (blue) vs the same fluxes for minimally coupled scalar fields with imaginary (orange) and real (magenta) constant masses. Clearly, the flux for a scalar coupled to the GB invariant resembles more closely the tachyonic (i.e. imaginary mass) scalar field evolution, both in timescale and sign. Note that the constant, real mass case, whose evolution is unstable due to superradiance, shows a slower growth and negative energy fluxes. The latter are indeed the hallmark \nFig. 2 - Energy flux F E through the BH horizon vs time, for a = 0 . 99 M . The blue, orange and magenta lines correspond respectively to η = -10 M 2 , to a tachyonic mass µM = i , and to a constant, real mass µM = 0 . 42. The inset zooms on the constant, real mass flux (of which we show a moving average to decrease the oscillations caused by the dynamics). That flux is negative, signaling energy extraction from the BH, as expected for superradiant instabilities. \n<!-- image --> \nof a superradiant instability, which removes rotational energy and angular momentum from the BH. \nThe most plausible explanation for why Kerr BHs in qsGB do not suffer from superradiant instabilities seems to be the rapid falloff of the GB invariant (thus of the effective mass) at large distances, G ( r → ∞ ) ∼ 1 /r 6 . Scalar perturbations with a position-dependent mass were studied in [51], which showed that a steep decay of the mass with distance quenches the superradiant instability. This happens because the effective potential for scalar perturbations does not develop wells, and thus quasi-bound states, unless the mass remains relatively constant till at least r ∼ 2 -3 M [51]. \nConclusions: We have shown that a coupling, with a suitable sign, between a scalar and the GB invariant can lead to an instability triggered by rapid rotation. We have also demonstrated that this instability is not related to superradiance, but is instead tachyonic in nature. Nonlinear effects, which our approach does not capture, are expected to quench that instability and lead to a BH with scalar hair. The process is analogous to the more conventional spontaneous scalarization, but the threshold is controlled by the black hole rotation instead of its curvature. \nThe action that we use is sufficient for studying the onset of the instability that we have found for BHs. However, the endpoint of this instability, and hence the amount of hair a BH would carry, will strongly depend on nonlinear (self)interactions. 5 There is no obvious reason to believe that this instability is restricted to BHs, \nand it could well affect rapidly rotating stars as well. Hence, our results demonstrate that there is a broad class of theories where rotation might control deviations from GR. Our findings also have clear implications for searches of new physics in the strong-field regime. Black hole scalar hair induces vacuum dipole gravitational emission, which is potentially observable in the low frequency inspiral of binary system by gravitational wave interferometers [14, 15], deviations from GR in the spectrum of the gravitational wave ringdown [13] or in the electromagnetic spectrum of accretion disks [59], and it may also impact the black hole shadow observed by the Event Horizon Telescope [8]. \nWe stress that we are not aware of any observational upper bounds on η , which we therefore allow here to reach very high values, for illustrative purposes and in order to excite higher modes. Note that slowly rotating black holes in qsGB would be identical to their GR counterpart. Compact stars can scalarize for η < 0 [33] and hence yield constraints. However, this effect could easily be quenched by adding a coupling between the scalar field and the Ricci scalar [41, 60]. 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2020JHEP...05..004R
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Information radiation in BCFT models of black holes
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2020-01-01
| 31 | 0.45 | 163 |
['-', 'black hole physics', '-', '-', '-', '-']
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[]
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In this note, following [1-3], we introduce and study various holographic systems which can describe evaporating black holes. The systems we consider are boundary conformal field theories for which the number of local degrees of freedom on the boundary (c<SUB>bdy</SUB>) is large compared to the number of local degrees of freedom in the bulk CFT (c<SUB>bulk</SUB>). We consider states where the boundary degrees of freedom on their own would describe an equilibrium black hole, but the coupling to the bulk CFT degrees of freedom allows this black hole to evaporate. The Page time for the black hole is controlled by the ratio c<SUB>bdy</SUB>/c<SUB>bulk</SUB>. Using both holographic calculations and direct CFT calculations, we study the evolution of the entanglement entropy for the subset of the radiation system (i.e. the bulk CFT) at a distance d > a from the boundary. We find that the entanglement entropy for this subsystem increases until time a + t<SUB>Page</SUB> and then undergoes a phase transition after which the entanglement wedge of the radiation system includes the black hole interior. Remarkably, this occurs even if the radiation system is initially at the same temperature as the black hole so that the two are in thermal equilibrium. In this case, even though the black hole does not lose energy, it "radiates" information through interaction with the radiation system until the radiation system contains enough information to reconstruct the black hole interior.
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https://arxiv.org/pdf/1910.12836.pdf
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{'Information radiation in BCFT models of black holes': "Moshe Rozali a \nJames Sully a \nMark Van Raamsdonk a \nChristopher Waddell a \nDavid Wakeham a \n- a Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Road, Vancouver, B.C. V6T 1Z1, Canada. \nE-mail: \[email protected], [email protected], [email protected], \[email protected], [email protected] \nAbstract: In this note, following [1-3], we introduce and study various holographic systems which can describe evaporating black holes. The systems we consider are boundary conformal field theories for which the number of local degrees of freedom on the boundary ( c bdy ) is large compared to the number of local degrees of freedom in the bulk CFT ( c bulk ). We consider states where the boundary degrees of freedom on their own would describe an equilibrium black hole, but the coupling to the bulk CFT degrees of freedom allows this black hole to evaporate. The Page time for the black hole is controlled by the ratio c bdy /c bulk . Using both holographic calculations and direct CFT calculations, we study the evolution of the entanglement entropy for the subset of the radiation system (i.e. the bulk CFT) at a distance d > a from the boundary. We find that the entanglement entropy for this subsystem increases until time a + t Page and then undergoes a phase transition after which the entanglement wedge of the radiation system includes the black hole interior. Remarkably, this occurs even if the radiation system is initially at the same temperature as the black hole so that the two are in thermal equilibrium. In this case, even though the black hole does not lose energy, it 'radiates' information through interaction with the radiation system until the radiation system contains enough information to reconstruct the black hole interior. \nContents \n| 1 | Introduction | 1 |\n|---------|----------------------------------------------------------------------|-----|\n| 2 | Basic setup | 6 |\n| | 2.1 Holographic Duals of BCFTs | 8 |\n| 3 | Two-dimensional models: static case | 11 |\n| | 3.1 Entanglement wedge after the transition | 17 |\n| 3.2 | CFT calculation | 20 |\n| 3.3 | Holographic replica calculation | 22 |\n| 4 | 2D evaporating and single sided examples | 23 |\n| 4.1 | Single-sided case | 24 |\n| 4.2 | Dynamical case | 25 |\n| 5 | Discussion | 31 |\n| 5.1 5.2 | A connection to behind-the-horizon physics of black hole microstates | 31 |", 'Background': "Within the context of holographic models of quantum gravity, the formation and evaporation of black holes is a manifestly unitary process in the sense that the underlying quantum system evolves through conventional Schrodinger evolution with a Hermitian Hamiltonian. However, in the gravity picture, the physics of the black hole interior and the mechanism through which information about the microstate of the black hole emerges in the Hawking radiation are still not fully understood. \nAcrucial piece of physics to understand is the evolution of the density matrix for the black hole radiation. Hawking's original calculation [4] suggests that the entropy of this density matrix continues to increase throughout the black hole's evaporation. But unitary evolution predicts that this entropy should begin decreasing at the 'Page time' when the black hole's \n(macroscopic) entropy has been reduced to half of its original value [5, 6] and the remaining black hole becomes maximally entangled with the radiation system. The specific increasing and then decreasing behavior of the entropy of the radiation system as a function of time is known as the Page curve. Understanding how this curve comes about from the gravity picture is a key challenge. \nA further mystery appeared in the work [7-11], in which the authors argued that assuming a unitary picture of black hole evaporation leads to the conclusion that there cannot be a smooth region of spacetime behind the horizon of an evaporating black hole past the Page time. The argument was based on an apparent inconsistency between having maximal entanglement between the black hole and its early Hawking radiation after the Page time and having entanglement between field theory degrees of freedom on either side of the black hole horizon, as required by smoothness. The proposed alternative is that the old black hole develops a 'firewall' at its horizon. \nA fascinating suggestion [12] to avoid this firewall conclusion, making use of the general idea that the connectivity of spacetime is related to quantum entanglement between underlying degrees of freedom [13, 14], is that the entanglement between the black hole and its early radiation past the Page time is actually responsible for the existence of a smooth geometry behind the black hole horizon, in the same way that the entanglement between two CFTs in the thermofield double state gives rise to a smooth wormhole geometry connecting the two black hole exteriors. 1 In this picture, the behind-the-horizon degrees of freedom are the radiation degrees of freedom, so there is no contradiction that both are entangled with outside-the-horizon modes of the black hole. \nVery recently, a series of papers [1-3] have provided more detailed insight into how the black hole radiation can be seen to have an entropy described by a Page curve yet avoid the firewall paradox by the mechanism of [12] (see also [16]). The examples in these papers make use of an auxiliary radiation system coupled to a system that would otherwise describe an equilibrium black hole 2 . The new insights come by making use of the quantum version [18, 19] of the Ryu-Takayanagi formula [20, 21], which gives the gravity interpretation of entanglement entropies for subsystems of a holographic quantum system. 3 Importantly, the \n3 For a subsystem A of a holographic system, the quantum RT surface ˜ A in the dual gravitational picture is a bulk surface which encloses a region corresponding to A at the boundary of the dual spacetime and has \nFigure 1 . Basic setup. A) Our thermal system, dual to a bulk black hole, is the red boundary. It interacts with a bulk CFT which can serve as an auxiliary system to which the black hole can radiate. B) Higher-dimensional bulk picture: the red surface is a dynamical ETW brane whose tension is monotonically related to the number of local degrees of freedom in the boundary system. For large tension, this ETW brane moves close to the boundary and behaves like a Randall-Sundrum Planck brane. C) The Planck-brane picture suggests an effective lower-dimensional description where a part of the CFT in the central region is replaced with a cutoff CFT coupled to gravity, similar to the setup in [3]. \n<!-- image --> \nprescription for calculating these entropies in the gravity picture requires the identification of a 'quantum extremal surface' on which the functional (1.1) is evaluated to calculate the entanglement entropy. A central observation of [1-3] is that during the evaporation of a black hole, the quantum extremal surface that computes the entanglement entropy of the radiation system can jump, leading to a first-order transition in the entanglement entropy that provides the necessary switch from increasing to decreasing behavior. \nFurther insights in [1-3] make use of the notion of the 'entanglement wedge' of a subsystem of a holographic system, which is the portion of the full spacetime that is dual to or reconstructable from the density matrix for the subsystem, and is understood to be the bulk region enclosed by the quantum extremal surface [22-28]. In the examples of [1-3], it is seen that after the transition in the quantum extremal surface, the entanglement wedge of the radiation system actually includes a portion of the black hole interior. Thus, the underlying degrees of freedom for this interior region after the transition are understood to be the degrees of freedom of the radiation system, in accord with the proposal of [12].", 'Summary and outline': 'In this paper, our first motivation is to further elucidate the observations of [1-3] by studying the evolution of black holes in a new class of models where the evolution of entanglement \nthe minimum value of the functional \nS grav ( A ) = Area( ˜ A ) 4 G + S bulk (Σ A ) (1.1) \namong extrema of this functional. Here S bulk (Σ A ) is the entanglement entropy of bulk fields in the bulk region Σ A enclosed by ˜ A . \nentropy and the entanglement wedge can be studied very explicitly through direct holographic calculations. Our models are similar to and motivated by the one in [3] in that they have a holographic description in one higher dimension than the original black hole of interest, and the full dynamics of entanglement entropy for the basic degrees of freedom is captured geometrically through the behavior of classical HRT surfaces. However, our systems are described somewhat more explicitly than the one in [3] and have an additional parameter that controls the Page time for the black hole. \nOur specific construction, described in section 2, starts with a d -dimensional holographic system on S d -1 in a high-energy state, or a thermofield double state with a second copy of the holographic system. These holographically describe one-sided or two-sided black holes in spacetimes that are asymptotically AdS if the theory that we start with is a CFT. The black holes are in equilibrium with their Hawking radiation, which reflects off the boundary of the spacetime. In order to have the black holes evaporate, we couple our holographic system to an auxiliary system as in [1-3, 15, 17]. Our auxiliary system is a CFT in one higher dimension living on a space whose boundary is S d -1 (or two copies of this), such that our original degrees of freedom provide boundary degrees of freedom for this higher-dimensional CFT. We can take the higher-dimensional CFT to be holographic, such that the full system is a holographic BCFT (or flows to one in the IR). We show in section 2 that the Page time for the black hole is proportional to the ratio c bnd /c bulk of the local number of boundary degrees of freedom to the local number of degrees of freedom in the bulk CFT. In the limit where c bnd is large and c bulk is fixed, the Page time that we calculate from CFT considerations matches the Page time obtained in the gravity picture in AdS with absorbing boundary conditions [29]. \nFor our explicit calculations, we consider various states of the BCFT constructed via Euclidean path integrals, so that the dual gravity geometries can be understood explicitly. For these states, we will consider the computation of entanglement entropy for the auxiliary system, considering a spatial region defined by the points at distance greater than a from the boundary system. We calculate the entanglement entropy for this system as a function of time and of the distance a . We perform the calculation holographically by finding the HRT surface in a dual d +1-dimensional gravitational system. We make use of a bottom-up holographic prescription for studying the dual BCFTs in which the CFT boundary extends into the bulk as a dynamical end-of-the-world brane whose tension is directly related to c bnd . We also reproduce the results of these holographic calculations through direct calculations in our BCFT system, making use of standard assumptions about holographic CFTs. \nAs hoped, our calculations show a first order phase transition of the entanglement entropy at the Page time after which the entropy of the radiation stops increasing; a sample result for \nFigure 2 . Time at which the subsystem of the radiation system greater than some distance from the BCFT boundary exhibits a transition in its entanglement entropy, for the case c bnd /c bulk ∼ 50. After the transition, the entanglement wedge of this subset of the radiation system includes a portion of the black hole interior. After a time equal to the Page time plus the light travel time from the boundary to our subsystem, there is enough information in the subsystem to reconstruct part of the black hole. \n<!-- image --> \nthe transition time is shown in figure (2). In the higher-dimensional gravity picture, we find that after the transition, the entanglement wedge of the radiation system includes a portion of the black hole interior. \nA new qualitative result of the present paper is that the phase transition described in the previous paragraph can occur even when the black hole is not evaporating, but simply coupled to an open radiation system which is in thermal equilibrium with the black hole. In this case, we find that while the energy density is static everywhere, the entanglement entropy for subsets of the radiation system still shows interesting dynamics, increasing with time until a phase transition after which it is constant. Again, the entanglement wedge of the radiation system includes a portion of the black hole interior after the transition. This static case is the focus of section 3. \nIn section 4, we consider more general states for which the initial radiation system is not in equilibrium with the black hole and the energy density is time-dependent. These more closely model evaporating black holes. Our detailed results are again in line with the expectations of [1-3] and confirm some of the qualitative predictions of [3]. \nWe end in section 5 with a discussion. There, we describe some directions for future work and describe further holographic constructions of evaporating black hole systems. We also \npoint out that the transition in extremal surfaces described in this paper and in [1-3] is closely related to a similar transition [30] that can occur when looking at the entanglement entropy for subsystems of a CFT on S d in a high-energy state dual to a single-sided black hole. For the CFT states described in [30], we can have a transition as the subsystem size is increased, after which the entanglement wedge of the subsystem includes part of the geometry behind the black hole horizon. Remarkably, in the case of 3D gravity, the CFT calculations that exhibit this transition are precisely the same CFT calculations that show the entanglement wedge transition in the present paper. \nNote added: While this manuscript was in preparation, the paper [31] appeared, which has some overlap with section 3 of this paper.', '2 Basic setup': 'A schematic of our basic setup is shown in figure 1A. We imagine starting with a holographic system on S d -1 whose high-energy states or high-temperature thermal states describe black holes in a dual gravitational picture. In these systems, the black hole is in equilibrium with its Hawking radiation, which reflects off the boundary of the spacetime. \nNext, following [1-3, 15] we augment our holographic model with additional degrees of freedom which will serve as an auxiliary radiation system, allowing the black hole to evaporate. As in [2, 3], our auxiliary degrees of freedom will take the form of a higher-dimensional CFT living on a space with boundary S d -1 , such that the original system now serves as a set of boundary degrees of freedom for the higher-dimensional CFT. We will denote by c bulk the local number of bulk CFT degrees of freedom and by c bdy the local number of boundary degrees of freedom. We have in mind that c bdy glyph[greatermuch] c bulk glyph[greatermuch] 1. This will allow the full system to be holographic, but as we show below, will give a parametrically large evaporation time. \nHolographic models of this type can arise in string theory by considering branes ending on other branes. For example, we can have a stack of n D3-branes in directions 0123 ending on various D5 and NS5 branes at some locations in the 3 direction [32, 33]. The low energy physics is N = 4 SYM theory on a half-space with some boundary conditions. We can have an additional N D3-branes of finite extent in the 3 direction which are stretched between some of the fivebranes. Without the original n D3-branes, these can give rise to a 3D CFT in the infrared. In the full setup, this 3D CFT is coupled to the N = 4 theory at its boundary. Here, in this setup, we have c bdy /c bulk = N 2 /n 2 .', 'Evaporation time in the CFT picture': 'Now, suppose we have some initial energy M in the boundary degrees of freedom such that the energy corresponds to a temperature above the Hawking-Page transition for that system. The relation between temperature, energy, and entropy is \nE ∼ c bdy R d -1 T d S ∼ c bdy R d -1 T d -1 . (2.1) \nIf this system is coupled to a higher-dimensional CFT with c bulk local degrees of freedom, we expect that the energy will be radiated away at a rate \ndE dt ∼ -ec bulk R d -1 T d +1 (2.2) \nwhere we are using a Boltzmann law, with emissivity e that presumably depends on the nature of the coupling. The factor of c bulk can be understood from a weak-coupling picture where we have c bulk light fields that can carry away the energy. \nUsing these results, we have that \ndT dt = -ˆ e c bulk c bdy T 2 , (2.3) \nwhere ˆ e is defined to absorb any numerical coefficients we are ignoring. Solving, we have \nT = 1 1 T 0 + ˆ e c bulk c bdy t . (2.4) \nThe Page time is when half the (macroscopic) entropy of the black hole has been radiated. This corresponds to a temperature \nT p = 1 2 1 d -1 T 0 . (2.5) \nIgnoring factors of order 1, we find that \nt Page ∼ c bdy c bulk 1 ˆ eT 0 (2.6) \nor \nt Page /R ∼ 1 c bulk ˆ e c 1+ 1 d bdy ( MR ) 1 d . (2.7) \nSince the initial energy is of order c bdy , it is also illustrative to write MR = xc bdy , so that \nt Page /R ∼ c bdy c bulk ˆ e 1 x 1 d . (2.8) \nWe see that the Page time is proportional to c bdy c bulk ; we can make the black hole evaporation take a long time by choosing c bdy glyph[greatermuch] c bulk .', 'Evaporation time for a black hole with absorbing boundary conditions': 'We can compare this to the calculation in [29] of Page (see also [34]), who considers perfectly absorbing boundary conditions for a large black hole in AdS. Using those results, one finds a Page time \nt Page ∼ L d +1 -2 d AdS G 1+ 1 d 1 M 1 d (2.9) \nwhere we have omitted some numerical factors. An energy of 1 /R in the field theory corresponds to energy 1 /L AdS on the gravity side, while field theory entropy c bdy R d -1 T d -1 corresponds on the gravity side to r d -1 H /G = T d -1 L 2 d -2 G so we can relate \nc bdy R d -1 = L 2 d -2 G . (2.10) \nRewriting (2.9) in terms of field theory parameters, we get \nt Page /R ∼ c 1+ 1 d bdy ( MR ) 1 d (2.11) \nComparing with the expression (2.7) above, we see that the expressions have the same dependence on c bdy and M ; to match the gravity calculation, should take c bulk e to be of order 1, at least in terms of scaling with c bdy . In order that the full system is holographic, we want to take c bdy glyph[greatermuch] c bulk glyph[greatermuch] 1.', '2.1 Holographic Duals of BCFTs': 'In this section, we briefly review the gravitational dual description of holographic BCFTs and explain how the dual of a BCFT with large c bdy glyph[greatermuch] c bulk can give rise to the physics of a Planck brane whose geometry is the geometry of the black hole we are studying. \nIn their vacuum state, BCFTs preserve the conformal invariance of a CFT in one lower dimension. Thus, the gravity dual of a d -dimensional CFT with boundary in its vacuum state will generally correspond to a spacetime that is a warped product of AdS d with some internal space, but which has an asyptotically AdS d +1 region with boundary geometry equal to the half space. For various supersymmetric examples, gravitational dual solutions corresponding to the vacuum state are known explicitly [35, 36]. For example, there is a family of halfsupersymmetric solutions to type IIB supergravity that correspond to the vacua of N = 4 SYM theory living on half-space with the various boundary conditions preserving half supersymmetry (e.g. [37-40]). \nIn general it is difficult to work with the fully microscopic examples and to find full solutions of the ten or eleven-dimensional supergravity equations that would correspond to various BCFT states. Thus, rather than employing this top-down approach, we will consider bottom-up models of BCFT duals, introduced in [41-43] 4 . Here, the bulk dual of a d -dimensional CFT with boundary is taken to be a d +1-dimensional gravitational theory on a space which has a dynamical boundary extending from the CFT boundary into the bulk. Just as we can consider various possibilities for the bulk gravitational effective action, we can choose various terms for the boundary effective action. We expect that for appropriate choices of the bulk and boundary effective actions, we can accurately capture the physics of various holographic CFTs. 5 In this paper, we consider the simple situation where the ETW brane couples only to the bulk metric field; its action is taken to include a boundary cosmological constant (interpreted as the brane tension) and a Gibbons-Hawking term involving the trace of the extrinsic curvature. The details of the action and equation of motion, and all the solutions that we will require in this paper may be found in [30]. \nThe work of [42] established a connection between the tension of the ETW brane and the boundary entropy (or higher-dimensional generalizations), which can be understood as a measure of the number of degrees of freedom associated with the boundary. One simple calculation that indicates this relation is the holographic calculation of entanglement entropy for a region of the BCFT that is the interior of a half-sphere centred on the boundary. Holographically, this is computed via the area of an extremal surface anchored to the halfsphere which extends into the bulk and ends on the ETW brane. For larger tension of the ETW brane, this brane enters the bulk at a larger coordinate angle from the vertical in Fefferman-Graham coordinates for the asymptotic region, as shown in figure 3. As a result and the area of the extremal surface becomes larger, indicating a larger boundary entropy. \nIn our application, we would like to consider the case where the number of local boundary degrees of freedom is large compared with the number of local bulk degrees of freedom. In this case, there is an independent way to motivate the ETW brane picture. Since we are considering the bulk CFT degrees of freedom to be much fewer than the boundary degrees of freedom, we expect that in some sense, they act as a small perturbation. Over short time scales (much shorter than the Page time), the physics of the boundary degrees of freedom is not significantly affected by the bulk CFT degrees of freedom. We can think of the d -dimensional geometry of the ETW brane as the usual holographic dual of the d -1-dimensional \nFigure 3 . An ETW brane with tension parameter T enters the bulk at coordinate angle Θ in Fefferman-Graham coordinates. Larger T gives a larger angle Θ. Shown in blue is the RT surface computing the entanglement entropy of the subsystem A which includes the boundary. The area to the right of the dashed line proportional to the boundary entropy. \n<!-- image --> \nboundary system in its state at a particular time. The d +1-dimensional system dual to the bulk CFT-degrees of freedom couples to this system, and this corresponds to adding in the bulk d +1-dimensional geometry coupled to the d -dimensional brane. Over long time scales, the bulk CFT degrees of freedom can have a significant impact (e.g. when the black hole evaporates). Thus, over long time scales, the full geometry of the ETW brane can be affected significantly by its coupling to the bulk gravity modes, so it is important to consider the full d +1-dimensional system when understanding the long-time dynamics of the system.', 'The Randall-Sundrum Planck brane and the effective gravity picture': "As we have reviewed above, a large number of boundary degrees of freedom corresponds to a large tension for the ETW brane and in this case, the ETW brane enters the bulk at a very large angle to the AdS boundary. For the case of a single sphere-topology boundary, the resulting dual gravity solutions have ETW branes that stay close to the boundary in some sense (e.g. they correspond to a cutoff surface in a complete AdS spacetime for which light signals can propagate out to the AdS boundary and back in small proper time). In this and similar cases, the ETW brane behaves as a 'Planck brane' in the Randall-Sundrum sense [45], cutting off a portion of the asymptotic region of the geometry so that this part of the spacetime now terminates with a dynamical brane. 6 This point of view suggests a third description of the physics of our situation: from the CFT point of view, the addition of a Planck brane to a region of the bulk corresponds to cutting off the CFT in some spatial region and coupling to gravity in this region. The cutoff goes to infinity at the boundary of the region. This picture corresponds to the '2D gravity with holographic matter' picture of [3]. This latter picture most closely aligns with the model in [2]. The three pictures are summarized in figure 1. Note that it is this last picture (figure 1C) where the coupling \nFigure 4 . a) BCFT path integral defining the thermofield double state of two 1+1 dimensional BCFTs. b) Euclidean geometry dual to the BCFT thermofield double. The red surface is an ETW brane. c) The same geometry represented as part of Euclidean Poincar'e-AdS. d) Lorentzian geometry of the original state, looking perpendicular to the boundary. Dashed lines represent horizons on the ETW brane, corresponding to the horizons of the two-sided black hole represented by the boundary system. \n<!-- image --> \nbetween the black hole system and the radiation system is strictly at the boundary of the gravitational system.", '3 Two-dimensional models: static case': "In this section, we will consider a very simple system that already exhibits all of the key features of the entanglement dynamics described in [1-3]. The system we consider is not an evaporating black hole, but one where the auxiliary radiation system has the same initial temperature as the black hole, so that the two systems are in equilibrium. The system we look at has a static energy density (in a particular conformal frame), but the entanglement entropy for various subsystems still evolves with time and the entanglement wedge exhibits a phase transition similar to the ones discussed in [1-3]. \nSpecifically, we consider a 1+1 dimensional BCFT which is in the thermofield double state with a second copy of this system. This can be constructed via a path integral on a quarter-cylinder y ≤ 0, 0 ≤ θ ≤ π , where θ is the Euclidean time direction, and the boundary of each CFT is at y = 0. This is shown in figure 4a. \nTo understand the gravity dual, we use the bottom-up prescription where the boundary \nsystem leads to a bulk ETW brane. For 1+1 dimensional CFTs, it is convenient to define \nc bdy = 6log g (3.1) \nwhere log g is the usual boundary entropy. Then, defining \nF = c bdy c bulk , (3.2) \nthe tension parameter T (defined explicitly in [30]) for the ETW brane is related to F and to the angle Θ in figure 3 by \nT = tanh F = sin Θ . (3.3) \nThe dual Euclidean solution corresponding to our state is a portion of Euclidean AdS, which we may describe using metric (setting L AdS = 1) \nds 2 = ( ρ 2 +1) dy 2 + dρ 2 ρ 2 +1 + ρ 2 dφ 2 . (3.4) \nThe specific solution we need was already constructed in [30, 43]. The bulk Euclidean solution terminates on an end-of-the-world (ETW) brane with locus \ny ( ρ ) = -arcsinh ( tan Θ √ ρ 2 +1 ) , (3.5) \nwhere Θ is related to the brane tension and the number of boundary degrees of freedom by (3.3). The Euclidean geometry is depicted in figure 4b. The Lorentzian geometry dual to our state is obtained by taking the geometry of the φ = 0 , π slice of the Euclidean solution as our initial data. \nTo analyze the extremal surfaces in the Lorentzian version of this geometry, it will be convenient to change coordinates to Poincar'e coordinates, via the transformations \ny = ln( r ) ρ = tan( θ ) (3.6) \nwhich bring us to spherical Poincar'e coordinates and \nz = r cos θ x = r sin θ cos φ τ = r sin θ sin φ . (3.7) \nwhich bring us to the usual Cartesian Poincar'e coordinates in which the metric is \nds 2 = 1 z 2 ( dz 2 + dx 2 + dτ 2 ) . (3.8) \nIn these coordinates, the CFT boundary is at x 2 + τ 2 = 1, while the ETW brane is the surface \nx 2 + τ 2 +( z +tanΘ) 2 = sec 2 Θ , (3.9) \nas shown in figure 4c. We obtain the Lorentzian solution by analytic continuation τ → it . This gives \nds 2 = 1 z 2 ( dz 2 + dx 2 -dt 2 ) , (3.10) \nCFT boundary at x 2 -t 2 = 1, and ETW brane at \nx 2 -t 2 +( z +tanΘ) 2 = sec 2 Θ . (3.11) \nThis is shown in figure 4d.", 'Horizons on the ETW brane': "Let's now understand the causal structure of the ETW brane geometry to map out the horizons of the black hole that it contains. Consider the ETW brane in the Lorentzian picture, where it is described as the surface 3.11 in the metric 3.10. We would like to find the future horizon for this surface, i.e. the boundary of the set of points from which it is possible to reach the right ETW brane boundary on a lightlike curve. The lightlike curves on the ETW brane satisfy \nx ( t ) 2 -t 2 +( z ( t ) + tan Θ) 2 = sec 2 Θ (3.12) \nand \n( dx dt ) 2 + ( dz dt ) 2 = 1 . (3.13) \nWe find that they are given by \nx ( t ) = vt ± √ 1 -v 2 cos Θ z ( t ) = | √ 1 -v 2 t ± v sec Θ | -tan Θ (3.14) \nfor | v | < 1. The right and left boundaries of the ETW brane are described by x = ± √ t 2 +1. The future horizons are the lightlike curves that asymptotes to this for t → ∞ . These are the trajectories \nx = ± t z = 1 -sin Θ cos Θ . (3.15) \nThus, independent of Θ, we have horizons on the ETW brane located at x = ± t and these lie at constant z . The black hole interior can be identified with the region | x | < t or alternatively z > 1 -sin Θ cos Θ", 'Extremal surfaces': "We would now like to investigate the HRT surfaces which calculate the entanglement entropy associated with the spacetime region spacelike separated from the interval [ -x 0 , x 0 ] at t = 0 (equivalently, the union of intervals [ ± x 0 , ±∞ ) at t = t 0 . \nIn general, there are two possibilities for this HRT surface. First, we have the connected surfaces described by the semicircle \nt = t 0 z 2 + x 2 = x 2 0 . (3.16) \nWe can also have disconnected surfaces that end on the ETW brane. We need to compare the areas to find out which one is the minimal area extremal surface that computes the entanglement entropy. \nIt will be somewhat simpler to perform our calculations in the Euclidean picture and then analytically continue the results to the Lorentzian case. That is, we will look at geodesics in the Euclidean geometry, evaluate their length and the length difference between the two cases, and find the phase boundary for transitions between the two surfaces. The Lorentzian version of all of these things can be obtained by analytic continuation. 7 \nTo find the areas, we note that the area of a geodesic semicircle of coordinate radius R from the point z = R of maximum z to some z min is \nA ( R,z min ) = arccoth 1 √ 1 -z 2 min R 2 = 1 2 ln 1 + √ 1 -z 2 min /R 2 1 -√ 1 -z 2 min /R 2 (3.17) \nFor z min = glyph[epsilon1] with infinitesimal glyph[epsilon1] , this reduces to ln(2 R/glyph[epsilon1] ). \nFrom this, the area of the connected extremal surface is \nA c = ln ( 2 x 0 glyph[epsilon1] ) (3.18) \nFor the disconnected surface, each part is the arc of a circle which lies at constant θ , intersecting the ETW brane orthogonally and intersecting one of the the points ( ± x 0 , τ 0 ). 8 This is shown in figure 5. \nUsing basic geometry (see figure 5), we find that the extremal surface has coordinate radius \nr H = r 2 -1 2 r (3.19) \nFigure 5 . Geometry of the ETW brane and half of the disconnected RT surface in the plane of the RT surface. We have OQ = 1 and OA = tanΘ. Thus, AQ = AH = sec Θ. Also HB ⊥ AH so AH 2 + HB 2 = OA 2 + OB 2 . This gives r H = ( r 2 -1) / (2 r ) r H = ( r 2 -1) / (2 r ) r H = ( r 2 -1) / (2 r ). Now OM = OA tan α = tan Θ tan α and AM = OA sec α = tan Θ sec α . So HM = HA -MA = sec Θ -tan Θ sec α . Finally, HM/HB = tan α gives r H = sec Θcot α -tan Θ csc α r H = sec Θcot α -tan Θ csc α r H = sec Θcot α -tan Θ csc α , while HP = HB sin α gives z = r H sin α z = r H sin α z = r H sin α . The boldface equations allow us to express z and r H in terms of r . \n<!-- image --> \nand intersects the ETW brane at z coordinate \nz c = cos Θ r 2 +1 r 2 -1 +sinΘ (3.20) \nwhere r 2 = x 2 0 + τ 2 0 . \nFrom (3.17), we find that the area of the disconnected surface (including both parts) is \nA d = ln ( r 2 -1 glyph[epsilon1] 1 + sin Θ cos Θ ) (3.21) \nThe difference in areas between the two possible extremal surfaces is \nA d -A c = ln ( x 2 0 + τ 2 0 -1 2 x 0 1 + sin Θ cos Θ ) . (3.22) \nFrom this, we see that there will be a transition when \nτ 2 0 + ( x 0 -1 -sin Θ cos Θ ) 2 = 2 1 + sin Θ . (3.23) \nIn the Lorentzian picture, this gives the trajectory of the phase boundary as \n( x 0 -1 -sin Θ cos Θ ) 2 = t 2 + 2 1 + sin Θ . (3.24) \nWe can now map back to the original conformal frame (corresponding to figure 4a) where the energy density is time-independent. \nUsing the coordinate transformations \nx = e y cos φ τ = e y sin φ (3.25) \nwe have that the phase boundary in Euclidean coordinates is \ne F sinh y = cos φ . (3.26) \nHere, φ is the Euclidean time, so in Lorentzian coordinates (where η is the time coordinate), this phase boundary becomes \ne F sinh y = cosh η . (3.27) \nFinally, if we consider an interval [ y 0 , ∞ ) (together with the equivalent interval in the other BCFT), we find that the entanglement wedge for this subsystem makes a transition to include geometry behind the black hole horizon when \nη = arccosh( e F sinh y 0 ) ∼ F + y 0 (3.28) \nwhere the last relation holds for large y 0 and F . Thus, for intervals that include most of the radiation system (when y 0 is some small order 1 number), we see a transition at the Page time after which the black hole interior can be reconstructed from the radiation system. For large y 0 the time is increased by an amount which is the time taken for the radiation to reach y 0 . The behavior of the transition time is shown in figure 2. In this frame, the entanglement entropy is constant after the transition, since each part of the disconnected extremal surface in this case is just a boosted version of the extremal surface for earlier times. Thus, the entanglement entropy increases from the initial time and then remains constant after the transition. Using the results above, the precise expression for the entropy as a function of time is 9 \nS = { c bulk 6 ln ( 2 glyph[epsilon1] cosh η ) η < arccosh( e F sinh y ) log g + c bulk 6 ln ( 2 glyph[epsilon1] sinh y 0 ) η < arccosh( e F sinh y ) , (3.29) \nso we have an approximately linear increase before the transition and a constant entropy afterwards. \nLet's understand the physics of this phase transition in the behavior of the entanglement. We have that the energy density in both BCFTs is completely time-independent. However, the entanglement entropy for the union of regions x > x 0 in the two CFTs increases with time, \nthen undergoes a first order phase transition after which it is constant. The entanglement wedge initially does not include the black hole system, but after the transition includes a portion of the interior of the black hole. \nThus, while everything is static from an energy point of view, the state is evolving in such a way that information about the black hole interior eventually becomes accessible in the auxiliary radiation system. \nTo understand this better, it is helpful to recall that for a free field theory in the thermofield double state, each mode in one copy of the system is purified by the corresponding mode in the other copy of the system. In our present case, we expect similarly that the boundary system is initially purified to a large extent by the other copy of the boundary system, while the bulk system is purified by the other copy of the bulk system. 10 However, as we evolve forward in time, the entanglement structure evolves, and the information initially contained within the boundary system (describing our black hole initial state) leaks out into the bulk degrees of freedom, eventually leading to the transition we observe.", '3.1 Entanglement wedge after the transition': "We would now like to understand where the boundary of the entanglement wedge lies on the ETW brane after the transition. \nConsider a point ( x 0 , τ 0 ) on the Euclidean transition surface (3.23). Just after the transition to a disconnected minimal area extremal surface, the part of the surface originating at ( x 0 , τ 0 ) will end on the ETW brane at a point ( x c , τ c ) = λ ( x 0 , τ 0 ). From figure 5 we see that the distance r c = √ x 2 c + τ 2 c from the origin for this point will satisfy \nr = r c + r H + √ r 2 H -z 2 c . (3.30) \nThis gives \nso we have \nλ = r c r = 2 ( x 2 0 + τ 2 0 )(1 + sin Θ) + (1 -sin Θ) = 1 x 0 cos Θ + 1 \nr c = 2 r r 2 (1 + sin Θ) + (1 -sin Θ) , (3.31) \nwhere we have used (3.23) in the last line. Thus, we have \nx c = x 0 x 0 cos Θ + 1 τ c = τ 0 x 0 cos Θ + 1 . (3.32) \nInverting these relations and plugging the resulting expressions for x 0 and τ 0 in (3.23), we find that the points ( x c , τ c ) lie on a curve \n(1 + (1 -sin Θ) 2 ) x 2 c +2tanΘ(1 -sin Θ) x c + τ 2 c = 1 . (3.33) \nFor the Lorentzian version of the problem, this becomes \n(1 + (1 -sin Θ) 2 ) x 2 c +2tanΘ(1 -sin Θ) x c = t 2 c +1 . (3.34) \nNote that x 0 > √ t 2 0 +1 > t 0 , so from 3.32, we see that we will also have x c > t c . Thus, while the curve (3.34) crosses the horizon, the part beyond the horizon isn't relevant to us. The extremal surface always ends at a point on the brane that is outside the horizon. \nLet's now calculate the proper distance to the horizon from the intersection point ( x c , t c , z c ) on the ETW brane. The ETW brane lies in the plane containing the origin and the point ( x 0 , t 0 ) and extending directly inward in the z direction. In this plane, the geometry is as in figure 5, where the outermost point is at distance r = √ x 2 0 -t 2 0 . \nThis is the proper distance along the blue curve in figure 5 from H to the top of the blue arc, which lies at \nz max = sec Θ -tan Θ . (3.35) \nThe distance is \nUsing \nwe find that the result is \nd = ∫ z max z c dz z √ dz 2 + dr 2 (3.36) \nr 2 +( z +tan θ ) 2 = sec 2 θ , (3.37) \nd = 1 cos Θ ln ( r +1 r -1 ) . (3.38) \nIn the y 0 coordinates and in terms of F, this is \nd = cosh( F ) ln ( 1 + e -y 0 1 -e -y 0 ) (3.39) \nWe see that for large y 0 the location of the HRT surface intersection with the ETW brane after the transition is very close to the horizon. \nFigure 6 . The blue shaded region is the portion of the black hole interior that is included in the latetime entanglement wedge of any subsystem | x | > a of the radiation system (for Poincar'e coordinates). \n<!-- image --> \nFinally, we can look at the trajectory of the intersection point as a function of time after the transition. For the interval with left boundary y 0 in the y -coordinates, the initial intersection point is at \nx c = sec Θ 1 + 2 (1+sin Θ)( e 2 y 0 -1) (3.40) \non the curve (3.34) and the later trajectory follows the curve \nx 2 c -t 2 c = e 2 y 0 (1 -x c cos Θ) 2 . (3.41) \nAt late times, independent of y 0 , this approaches the point \nx = t = sec Θ = cosh( F ) (3.42) \non the horizon. \nThe outgoing lightlike curve along the ETW brane from this point is x = t , while the ingoing lightlike curve along the ETW brane from this point is simply x = sec Θ for all t (using the result 3.14). We note that the corresponding lightlike curve x = -sec Θ on the other side of the black hole does not intersect this curve, but the ingoing lightlike curve from any closer point does intersect this curve. Thus, the points t = ± x = sec Θ are a distinguished pair of points on the horizon for which the ingoing lightlike curves barely meet at the future singularity. The late-time intersection between the entanglement wedge for the radiation system and the black hole geometry is shown in figure 6.", '3.2 CFT calculation': "The calculations of the previous section relied on holographic calculations of the entanglement entropy in a bottom-up holographic model where the number of boundary degrees of freedom on our BCFT is related to the tension of an ETW brane. While bottom-up models in AdS/CFT are widely studied and known to produce qualitative results that agree with those in systems that can be studied using a top-down approach, the bottom-up approach for BCFTs is less well studied, and one might thus worry whether our holographic results correctly capture the physics of genuine holographic CFTs. \nIn this section, we will attempt to alleviate these concerns by reproducing our results for the entanglement entropies using direct CFT calculations, invoking standard assumptions about the properties of holographic CFTs. \nRecall that entanglement entropy can be calculated from R'enyi entropies using the replica trick: \nS A = lim n → 1 S ( n ) A = lim n → 1 1 1 -n log Tr[ ρ n A ] . \nThe operator ρ n A can be related to the partition function of the n -fold branched cover, or replica manifold , of the original geometry. This, in turn, can be calculated for 2D CFTs by introducing certain twist operators Φ n at the entangling points of A [47]. The partition function is given by a correlator of these twists. For A = [ z 1 , z 2 ] for instance, we have \nTr[ ρ n A ] = 〈 Φ n ( z 1 )Φ -n ( z 2 ) 〉 . \nIn holographic theories, these correlation functions are dominated by the identity block in some channel. A change in dominance will lead to a phase transition in entanglement entropy. In an ordinary two-dimensional holographic CFT, this exchange causes a sudden shift from the disconnected to the connected entanglement wedge for two disjoint intervals. In a holographic BCFT, this exchange can occur for a two-point correlator of twists, corresponding to the entanglement entropy of a single interval. This is analogous to the four-point result in a CFT since the two-point function in a BCFT has the same symmetries as the four-point function, and can be evaluated using the method of images. \nConsider a BCFT with boundary condition b on the upper half-plane (UHP), {glyph[Ifractur] ( z ) ≥ 0 } . We can perform a global transformation to the complement of the disk of radius R via \nw = R ( 1 z -i/ 2 -i ) . (3.43) \nFor simplicity, we also define ϑ := w + iR . We then have \nz = R ϑ + i 2 , glyph[Ifractur] [ z ( w )] = | w | 2 -R 2 2 | ϑ | 2 , w ' ( z ) = -1 R ϑ 2 . (3.44) \nSince we have performed a global transformation, the energy density vanishes: \n〈 T ( w ) 〉 = c 12 { z ; w } = c 12 z ''' z ' -(3 / 2)( z '' ) 2 ( z ' ) 2 = 0 . (3.45) \nConsider a two-point function of twist operators, Φ n ( w 1 ) , Φ -n ( w 2 ), introducing an n -fold branched cover with branch cut from w 1 to w 2 . The twists are primary by definition, so the correlation function transforms as \n〈 Φ n ( w 1 )Φ -n ( w 2 ) 〉 ¯ D G = | w ' ( z 1 ) w ' ( z 1 ) | -d n 〈 Φ n ( z 1 )Φ -n ( z 2 ) 〉 UHP = ∣ ∣ ∣ ∣ ( ϑ 1 ϑ 2 ) 2 R 2 ∣ ∣ ∣ ∣ -d n 〈 Φ n ( z ( w 1 ))Φ -n ( z ( w 2 )) 〉 UHP . (3.46) \nFor holographic BCFTs, the correlator of twists on the UHP can be evaluated [48], using vacuum block dominance and an appropriate sparsity condition on the density of states, in a similar vein to [49]. Using this correlator and the replica trick, the entanglement entropy of the interval A = ( -∞ , w 1 ] ∪ [ w 2 , ∞ ) is calculated by \nS A = lim n → 1 1 1 -n log 〈 Φ n ( w 1 )Φ -n ( w 2 ) 〉 disk = c 6 [ 2 log ∣ ∣ ∣ ∣ ϑ 1 ϑ 2 R ∣ ∣ ∣ ∣ +min { 12 c g b +log ∣ ∣ ∣ ∣ ( | w 1 | 2 -R 2 )( | w 2 | 2 -R 2 ) ( ϑ 1 ϑ 2 ˜ glyph[epsilon1] ) 2 ∣ ∣ ∣ ∣ , log ∣ ∣ ∣ ∣ Rw 12 ϑ 1 ϑ 2 ˜ glyph[epsilon1] ∣ ∣ ∣ ∣ 2 }] (3.47) \nwhere g b := -log 〈 0 | b 〉 is the boundary entropy, and F is given by (3.2). We note the relations \ne F = 1 + T √ 1 -T 2 = 1 + sin Θ cos Θ , 1 -e -2 F = 2 sin Θ 1 + sin Θ , (3.48) \nwhich we will use momentarily. Note that a UV regulator glyph[epsilon1] in Poincar'e coordinates becomes ˜ glyph[epsilon1] = | w ' 1 , 2 | glyph[epsilon1] after the global transformation z ↦→ w [50]; the phase boundaries are unaffected. \nWe now specialize to the symmetric interval A at some fixed time glyph[Ifractur] ( w ) = τ 0 , with w 1 , 2 = ± x 0 + iτ 0 . Exponentiating (3.47), a phase transition occurs at \n( x 2 0 -e -F R ) 2 + τ 2 0 = R 2 (1 -e -2 F ) (3.49) \n= ⇒ ( x 2 0 -cos Θ 1 + sin Θ R ) 2 + τ 2 0 = 2 R sin Θ 1 + sin Θ , (3.50) \nusing (3.48). In Lorentzian signature, τ 2 0 →-t 2 0 , and we obtain \n( x 2 0 -cos Θ 1 + sin Θ R ) 2 = t 2 0 + 2 R sin Θ 1 + sin Θ . (3.51) \nThese phase boundaries precisely match (3.23) and (3.24) for R = 1. \n<!-- image --> \n<!-- image --> \nFigure 7 . Replica calculation of entanglement entropy. \n<!-- image -->", '3.3 Holographic replica calculation': "It is interesting to consider a replica version of the same calculation. 11 In calculating the entanglement entropy, we want to evaluate the Renyi entropies by calculating the BCFT partition function on a replica manifold obtained by gluing n copies of the Euclidean space shown in figure 7 across the cut. The topology of the replica manifold is a sphere with n boundaries, as shown in the second figure. Considering a larger and smaller portion of the radiation system corresponds to enlarging or shrinking the size of the boundaries relative to the size of the sphere. \nNow we can consider performing this path-integral calculation holographically, using the bottom-up approach where the boundaries extend into the bulk as ETW branes. In the case of a smaller portion of the radiation system, the holes in the second picture will be small, and we will have a set of disconnected ETW branes of disk topology that 'cap off' the boundary holes. On the other hand, as we consider a larger portion of the radiation system, the circles become large in the second picture, and we expect that the dominant saddle in the gravitational calculation will correspond to the topology shown in the picture on the right where we have a single connected ETW brane with multiple boundary components. \nIt seems immediately plausible that the transition to this new bulk topology is directly related to the transition of HRT surfaces in our original calculation, since the two calculations must agree. However, it also appears at first slightly confusing: the CFT calculation correctly reproduces the disconnected bulk HRT surface from the disconnected contribution to the twist correlation function alone, while this bulk saddle is a complicated connected geometry involving both twist operators. To align the CFT and bulk pictures, note that the same \nFigure 8 . BCFT models for single-sided black holes. \n<!-- image --> \nissue appears when calculating the entanglement entropy of two (or multiple) intervals in the vacuum of a 2D CFT [49]. There, the higher Renyi entropies are also computed by a connected bulk geometry [53], but the entanglement entropy is a sum of disconnected contributions. This is consistent because the semi-classical Virasoro block describing the connected geometry reduces to the identity exchange in the limit n → 1. Despite the slightly different setting, the same ideas and kinematics describe the BCFT Renyi calculation [48]. \nThus, taking into account the second HRT surface that correctly sees the decreasing branch of entanglement entropy corresponds in the gravity version of the replica calculation to including non-trivial topologies. Had we stuck with the original topology (as we would do if treating gravity perturbatively) it seems that we would get an answer which misses the transition, and is perhaps more akin to Hawking's original calculation.", '4 2D evaporating and single sided examples': 'In this section, we continue focusing on two-dimensional models, but generalize the simple example of the previous section to a case where we have a pure state of a single-sided black hole, and to cases with a dynamical energy density (as in the example of [3]) that more closely models the physics of a genuine evaporating black hole. 12', '4.1 Single-sided case': 'It is straightforward to come up with BCFT examples of single-sided black holes. For example, the first picture in figure 8 shows a path-integral defining the state of a BCFT with some boundary system (fat red line) with many degrees of freedom. Here, instead of evolving the full BCFT from τ = -∞ to define the vacuum state of this system, we only evolve the boundary system from some finite past Euclidean time, as for the SYK states in [54]. For prior Euclidean times, we have a different boundary condition (thin red line) that we take to be associated with a small number of boundary degrees of freedom. At the transition between these two boundaries we have an appropriate boundary condition changing operator. \nThis construction should place the boundary system in a high-energy state, while the bulk CFT degrees of freedom should be in a lower-energy state (through they are also affected by the change of boundary conditions in the Euclidean past). In this case, the dual gravity solution will involve ETW branes with different tensions, and a codimension-two brane associated with the boundary-condition changing operator. \nIt would be interesting to analyze this example in detail. For now, we point out that we can understand the physics of a very similar example using the results of the previous section. The second picture in figure (8) shows almost the same setup, but with a different geometry for the path-integral. This picture is similar to a Z 2 identification of our setup from the previous section. If we choose the lower boundary condition to correspond to a T = 0 ETW brane in the bulk and we choose the boundary-condition changing operator appropriately (so that the equation of motion at the codimension-two brane gives a constraint that the two-types of ETW branes should meet orthogonally), then the dual geometry for this setup will be precisely a Z 2 identification of the bulk geometries from the previous section, with a zero-tension ETW brane at the Z 2 fixed point. In this case, all of our calculations and qualitative conclusions go through almost unchanged. The only significant difference is that the connected RT surface from the previous section is now replaced by its Z 2 identification, which ends on the T = 0 brane. \nFigure 9 . 2D model for an evaporating black hole. \n<!-- image -->', '4.2 Dynamical case': 'We can also modify our two-sided example in order to introduce time evolution of the energy density more characteristic of an evaporating black hole. We would like to have a situation where our auxiliary system starts out in a state that is closer to the vacuum state, so that the energy in the initial black hole state will radiate into this system. \nA simple construction (similar to that discussed in [3]) is shown in figure 9. The left picture shows a state of four quantum systems. The outer systems are BCFTs with some boundary condition (denoted by a dark red boundary) that we imagine has a small boundary entropy. The path integrals shown place these systems into their vacuum state. The remaining part of the path integral constructs a thermofield double state of two systems, each of which is a BCFT living on a small interval with different boundary conditions on the two ends. The dark red boundary condition is the same as before, but the semicircular boundary (shown bright red) corresponds to a boundary system with many degrees of freedom as in the example of the previous section. \nIn order to make the two-sided black hole evaporate, we consider a modified system where we glue the systems together as shown on the right side of figure (9). In the final path integral, shown on the right, we are describing a state of the same system that we considered in the earlier part of this section. However, since our Euclidean path integral is in some sense a small modification of the picture on the left, we expect that far away from the black hole, the local physics of the reservoir system will be similar to the vacuum. In this case, the energy in the (bright red) boundary degrees of freedom will gradually leak out into the reservoir system. The dual gravitational picture will be that of an evaporating black hole. \nIn studying the dual system explicitly using the bottom-up approach, we will now have \ntwo types of branes, one with a larger tension corresponding to the blue boundary condition, and one with a smaller tension corresponding to the drak red boundary condition. The latter is what [3] refer to as the Cardy brane. We expect that the behavior of this system should match the qualitative picture described in [3], but now it should be possible to study everything quantitatively. Since the branes only couple to the metric and we are in three dimensions, the local geometry of the holographic dual will be that of AdS, and the dynamics of the system will be reflected in the trajectories of the ETW branes.', 'Phase Boundaries on the Annulus': "In order to study situations like the previous section, we can apply the methods of [50, 55] who were making use of a similar Euclidean setup (without the middle boundary) to study local quenches in a holographic CFT. For any specific shape of the boundaries in (9), it is possible to map the doubled picture describing the full CFT path integral conformally to an annulus, where the circular boundary maps to the inner edge of the annulus and the other boundaries (shown in dark red) together map to the outer boundary of the annulus. We can also map the annulus to a finite cylinder, so we see that the physics will be related to the physics of the thermofield double state of a pair of CFTs on a finite interval with different boundary conditions on the two ends. \nWe can again start with the global AdS metric (3.4) in which we know the ETW trajectories explicitly. Here, though, we consider a finite segment of the boundary cylinder, with a boundary condition corresponding to tension T at y = -L and a boundary condition corresponding to tension T = 0 (or some other tension) at y = 0. Changing to Poincar'e coordinates as in Section 3, the CFT region becomes an annulus with inner radius R = e -L and outer radius 1, centred at the origin. Also as in that section, the location of the ETW brane corresponding to the inner boundary is \nx 2 + τ 2 +( z + R tan Θ) 2 = R 2 sec 2 Θ , Θ = arcsin( T ) , (4.1) \nwhile that corresponding to the outer boundary is \nx 2 + τ 2 + z 2 = 1 . (4.2) \nFor sufficiently large L , the two BCFT boundaries are far apart and the phase boundaries for the transition between connected and disconnected HRT surfaces are those found previously for the case of a single boundary; the phase boundary for the transition between a connected surface and a disconnected surface ending on the inner ETW brane has locus \n( x -R (1 -sin Θ) cos Θ ) 2 + τ 2 = 2 R 2 1 + sin Θ , (4.3) \nwhile that for the outer ETW brane is \n( x +1) 2 + τ 2 = 2 . (4.4) \n(These are the phase boundaries in the region x > 0; the x < 0 phase boundaries are given by symmetry about τ = 0.) As L is decreased to some critical value \nL c ≡ -ln ( ( -1 + √ 2) cos Θ (1 -sin Θ) + √ 2(1 -sin Θ) ) , (4.5) \nthe phase boundaries will osculate within the annulus at τ = 0; for smaller L , a direct transition between disconnected HRT surfaces ending on the higher tension brane and surfaces ending on the lower tension brane can occur (see Figure 10). The phase boundary between these disconnected phases is given by \nx 2 + τ 2 = R ( (1 -sin Θ) + R cos Θ R (1 -sin Θ) + cos Θ ) ≡ glyph[lscript] 2 . (4.6) \nWe can now map to a new conformal frame with the desired dynamical Cardy brane; the phase boundaries should simply be pushed forward using the appropriate conformal transformation, then analytically continued to Lorentzian signature. Note [55] that, starting from Poincar'e coordinates \nds 2 = dη 2 + dζd ¯ ζ η 2 (4.7) \na map ζ = f ( w ) corresponds to a coordinate transformation \nζ = f ( w ) -2 z 2 ( f ' ) 2 ( ¯ f '' ) 4 | f ' | 2 + z 2 | f '' | 2 η = 4 z | f ' | 3 4 | f ' | 2 + z 2 | f '' | 2 \nin the dual asymptotically AdS geometry, which gives a metric \nds 2 = 1 z 2 ( dz 2 + dwd ¯ w + z 2 ( T ( w ) dw 2 + ¯ T ( ¯ w ) d ¯ w 2 ) + z 4 T ( w ) ¯ T ( ¯ w ) dwd ¯ w ) (4.8) \nwhere the holographic stress tensors (corresponding to the stress tensors in the CFT state) are given by \nT ( w ) = 3( f '' ) 2 -2 f ' f ''' 4( f ' ) 2 ¯ T ( ¯ w ) = 3( ¯ f '' ) 2 -2 ¯ f ' ¯ f ''' 4( ¯ f ' ) 2 . (4.9) \n<!-- image --> \nFigure 10 . Phase diagram for annulus with supercritical and subcritical L respectively. \n<!-- image --> \nFigure 11 . Example path-integral geometry generating a BCFT state corresponding to a two-sided black hole system with dynamical energy density. \n<!-- image -->", 'Conformal mapping': "As a specific example, we can take the 'single joining quench' geometry of [55] and add to it another boundary centered at the origin; this second boundary is taken to be the image of the inner boundary of the annulus under the conformal transformation \nw ( ζ ) = 2 ζ 1 -ζ 2 , (4.10) \nwhich takes us from the unit disk (with complex coordinate ζ = x + iτ ) to the single joining quench geometry (with coordinate w = ˆ x + i ˆ τ ). An example of the resulting path-integral \ngeometry is shown in figure 11. \nWe note a few important features of such a map. Firstly, the symmetry x →-x translates to a symmetry ˆ x → -ˆ x , and likewise symmetry τ → -τ translates to symmetry ˆ τ → -ˆ τ . Secondly, the outer annular boundary | ζ | = 1 maps to the intersection of the slits i [1 , ∞ ) and -i [1 , ∞ ), while the inner boundary maps to \nˆ x 2 + ˆ τ 2 = 1 2 cosh 2 ( L ) ( 1 + √ 1 + 4ˆ x 2 tanh 2 ( L ) ) . (4.11) \nFinally, we note that the energy density with respect to Euclidean time ˆ τ is defined by \nT ( w ) + ¯ T ( ¯ w ) = 3 4(1 + w 2 ) 2 + 3 4(1 + ¯ w 2 ) 2 = 3 2 ( ˆ τ 4 -2(3ˆ x 2 +1)ˆ τ 2 +(ˆ x 2 +1) 2 ((1 + ˆ x 2 -ˆ τ 2 ) 2 +4ˆ x 2 ˆ τ 2 ) 2 ) ; (4.12) \nthe Lorentzian analogue decays as we move away from the boundary which represents the black hole. \nIn the new coordinates, the phase boundary between connected HRT surfaces and disconnected surfaces ending on the outer ETW brane is ˆ x 2 +ˆ τ 2 = 1, while the phase boundary between connected surfaces and disconnected surfaces ending on the inner ETW brane is \n( α (ˆ x 2 + ˆ τ 2 ) -β ˆ x -sin Θ ) 2 = (ˆ x 2 + ˆ τ 2 +1) 2 -4ˆ τ 2 , (4.13) \nwith \nα = (1 + R 2 ) 2 (1 + sin Θ) -4 R 2 4 R 2 = cosh 2 ( L )(1 + sin Θ) -1 β = (1 + R 2 ) R cos Θ = 2 cosh( L ) cos Θ . (4.14) \nIf a transition between the two disconnected phases is present, the phase boundary has locus \nˆ x 2 + ˆ τ 2 = 2 glyph[lscript] 2 (1 + glyph[lscript] 2 ) 2 ( 1 + √ 1 + 4ˆ x 2 (1 + glyph[lscript] 2 ) 2 (1 -glyph[lscript] 2 ) 2 ) (4.15) \nSee Figure 12. We can analytically continue ˆ t = -i ˆ τ to determine the BCFT boundaries and phase boundaries in Lorentzian signature. For L > L c , the phase boundaries now meet at the point \nˆ x 0 = α -sin Θ 2 + β , ˆ t 0 = √ ˆ x 2 0 -1 . (4.16) \nFor | ˆ t | < ˆ t 0 we have three distinct phases, while for | ˆ t | > ˆ t 0 we just have the two disconnected phases. For L < L c , we just have the two disconnected phases (see Figure 13). \n<!-- image --> \nFigure 12 . Phase diagram for Euclidean modified (two boundary) single joining quench geometry with supercritical and subcritical L respectively. \n<!-- image --> \n<!-- image --> \nFigure 13 . Phase diagram for Lorentzian modified (two boundary) single joining quench geometry with supercritical and subcritical L respectively. \n<!-- image --> \nOne can now determine the time-dependence of the entanglement entropy along any desired trajectory. Recall from previous sections that, on the annulus, the HRT surfaces for symmetrically situated intervals (with inner endpoints ( ± x, τ )) are circular arcs, and the corresponding entanglement entropy is given by \nS ( x, τ ) = ln ( 2 x ˜ glyph[epsilon1] ( x,τ ) ) , connected ln ( ( x 2 + τ 2 -R 2 )(1+sin Θ) ˜ glyph[epsilon1] ( x,τ ) R cos Θ ) , disconnected T > 0 ln ( 1 -x 2 -τ 2 ˜ glyph[epsilon1] ( x,τ ) ) , disconnected T = 0 , (4.17) \nwhere we have recalled [50] that the UV regulator glyph[epsilon1] in the physical setup requires a position dependent regulator ˜ glyph[epsilon1] ( x, τ ) = | ζ ' ( w ) | glyph[epsilon1] in the annular setup. It is a simple matter to apply the appropriate conformal transformation and Wick rotate to Lorentzian signature, whence we recover the expression for the entanglement entropy of symmetrically situated intervals in the Lorentzian modified local quench geometry.", '5 Discussion': 'In this section we present a few additional observations and some directions for future work.', '5.1 A connection to behind-the-horizon physics of black hole microstates': 'There is an interesting connection between the transitions in entanglement entropy that we have observed in this paper and another type of transition for entanglement entropy pointed out in [30]. In that paper, the authors (including some of the present authors) considered black hole microstates for a holographic CFT on S d defined via a Euclidean path-integral on a finite cylinder, with a boundary at time τ 0 in the Euclidean past. This corresponds to the evolution of a boundary state | B 〉 by Euclidean time τ 0 . In the 2D CFT case for small enough τ 0 , this state corresponds to a single-sided black hole at temperature 4 /τ 0 , with a time-dependent ETW brane behind the horizon providing an inner boundary for the black hole. \nFor these states, the entanglement entropy for an interval can exhibit a phase transition as the interval size is increased, such that after the transition, the entanglement wedge of the interval includes a region behind the black hole horizon (terminating on the ETW brane). This is somewhat reminiscent of the entanglement wedge transition discussed in this paper, but it turns out that there is a precise connection between the two. \nIf we unwrap the circle on which the CFT lives, we obtain a planar black hole dual (above the Hawking-Page transition [56]) to the global quench geometry [57]. The holographic results for entanglement entropy in this situation are the same as in the compact case, since the gravity dual for the compact case is just a periodic identification of the gravity dual for the non-compact case. \nThe CFT calculation of entanglement entropy in the non-compact case is carried out via a correlation function of twist operators on an infinite strip. But a local conformal transformation maps this calculation to exactly the CFT calculation in section 3.2 used to deduce the phase transition in this paper. \nFigure 14 . BTZ black hole microstates have the same brane profile and hence entanglement entropy as the planar black hole dual to a global quench. The quench geometry is obtained from a local conformal transformation of the excised disk, so the transition in entanglement entropy for the static case described above, and the BTZ microstates in [30], are controlled by the same CFT correlator. \n<!-- image --> \nWe visual this connection in figure 14. In the single-sided microstates, there is a transition in the extremal surfaces as the boundary region is increased (blue and green regions in figure 14). In the CFT, this can be calculated by a correlator of twists in the largec limit and simple spectral constraints [48]. Remarkably, this is essentially the same correlator governing the transition in entanglement wedge, as a function of subsystem size, as the static 2D case described in section 3.', '5.2 CFT constructions for duals of higher-dimensional evaporating black holes': 'In future work, it would be interesting to study explicitly some higher-dimensional analogues of the constructions considered in this paper. We describe a few specific constructions in this final section. For these higher-dimensional examples, a detailed study will likely require some numerics as the bulk geometry will no longer be locally AdS. However, as the geometries depend on only two variables, such a study should be quite feasible.', 'BCFT microstate construction': 'Figure 15 shows on the left a Euclidean path integral for a high-energy CFT state obtained by placing some boundary conditions in the Euclidean past (at the red sphere). This corresponds to a black hole with some time-dependent behind-the-horizon physics, as described in [30]. We have in mind that the red boundary corresponds to a boundary condition with a large boundary entropy, so that the holographic description involves a brane with large tension. \nNow we couple this system to a bulk CFT as shown on the right. Here, we need to intro- \nFigure 15 . Higher dimensional construction based on BCFT microstates \n<!-- image --> \nduce an additional boundary component (shown in green) into the Euclidean path integral. Two possible choices for the topology of this boundary component are shown. We have in mind that this boundary has a small boundary entropy, perhaps corresponding to a T = 0 brane. This setup is the precise higher-dimensional analog of the single-sided setup of section 4.1. \nIn the dual holographic theory, using the bottom-up approach, we will have a bulk d +1dimensional gravity action, but also two different types of d -dimensional ETW branes corresponding to the two different boundary conditions. Finally, there will be another d -1 dimensional brane that serves as the interface between the two types of d -dimensional branes. This can have its own tension parameter independent of the others.', 'Vaidya-type construction': 'Another interesting case makes use of the setup of [58]. Figure 16 shows on the left a Euclidean path integral for a CFT state dual to a shell of matter that collapses to form a black hole. We \nFigure 16 . Higher-dimensional construction based on CFT-Vaidya states. \n<!-- image --> \nhave insertions of many operators at some small time in the Euclidean past. Alternatively, we could consider a smooth source for some operator, again localized around some particular time τ = -glyph[epsilon1] . We can take a limit where τ → 0 but the sources/insertions are chosen such that we end up with a finite energy state. \nNow we couple this system to a bulk CFT as shown on the right. Without the sources, this path-integral would give the vacuum state of the BCFT. We expect that the sources mainly excite boundary degrees of freedom, so the bulk part of the CFT is still nearly in the vacuum state. In this case, we expect that the state is dual to a shell that collapses to form a black hole but then evaporates.', 'Acknowledgments': 'We would like to thank Tarek Anous for pointing out that the Acknowledgments section was missing from the original version of the paper. 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2012PhRvD..86d3512B
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Primordial black holes as a tool for constraining non-Gaussianity
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2012-01-01
| 32 | 0.45 | 163 |
['-', 'cosmology dark matter', '-', '-']
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[]
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Primordial black holes (PBHs) can form in the early Universe from the collapse of large density fluctuations. Tight observational limits on their abundance constrain the amplitude of the primordial fluctuations on very small scales which cannot otherwise be constrained, with PBHs only forming from the extremely rare large fluctuations. The number of PBHs formed is therefore sensitive to small changes in the shape of the tail of the fluctuation distribution, which itself depends on the amount of non-Gaussianity present. We study, for the first time, how quadratic and cubic local non-Gaussianity of arbitrary size (parametrized by f<SUB>NL</SUB> and g<SUB>NL</SUB> respectively) affects the PBH abundance and the resulting constraints on the amplitude of the fluctuations on very small scales. Intriguingly we find that even nonlinearity parameters of order unity have a significant impact on the PBH abundance. The sign of the non-Gaussianity is particularly important, with the constraint on the allowed fluctuation amplitude tightening by an order of magnitude as f<SUB>NL</SUB> changes from just -0.5 to 0.5. We find that if PBHs are observed in the future, then regardless of the amplitude of the fluctuations, non-negligible negative f<SUB>NL</SUB> would be ruled out. Finally we show that g<SUB>NL</SUB> can have an even larger effect on the number of PBHs formed than f<SUB>NL</SUB>.
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[]
| 3 |
https://arxiv.org/pdf/1206.4188.pdf
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{'Primordial black holes as a tool for constraining non-Gaussianity': "Christian T. Byrnes, 1, ∗ Edmund J. Copeland, 2, † and Anne M. Green 2, ‡ \n1 CERN, PH-TH Division, CH-1211, Geneva 23, Switzerland. \n2 School of Physics and Astronomy, University of Nottingham, University Park, Nottingham, NG7 2RD, UK \nPrimordial Black Holes (PBH's) can form in the early Universe from the collapse of large density fluctuations. Tight observational limits on their abundance constrain the amplitude of the primordial fluctuations on very small scales which can not otherwise be constrained, with PBH's only forming from the extremely rare large fluctuations. The number of PBH's formed is therefore sensitive to small changes in the shape of the tail of the fluctuation distribution, which itself depends on the amount of non-Gaussianity present. We study, for the first time, how quadratic and cubic local non-Gaussianity of arbitrary size (parameterised by f NL and g NL respectively) affects the PBH abundance and the resulting constraints on the amplitude of the fluctuations on very small scales. Intriguingly we find that even non-linearity parameters of order unity have a significant impact on the PBH abundance. The sign of the non-Gaussianity is particularly important, with the constraint on the allowed fluctuation amplitude tightening by an order of magnitude as f NL changes from just -0 . 5 to 0 . 5. We find that if PBH's are observed in the future, then regardless of the amplitude of the fluctuations, non-negligible negative f NL would be ruled out. Finally we show that g NL can have an even larger effect on the number of PBH's formed than f NL . \nPACS numbers: 95.35.+d \nCERN-PH-TH/2012-167", 'I. INTRODUCTION': "Primordial Black Holes (PBH's) play a very special role in cosmology. They have never been detected but this very fact is enough to rule out or at least tightly constrain many cosmological paradigms. Convincing theoretical arguments suggest that during radiation domination they can form from the collapse of large density fluctuations [1]. If the density perturbation at horizon entry in a given region exceeds a threshold value, of order unity, then gravity overcomes pressure forces and the region collapses to form a PBH with mass of order the horizon mass. \nThere are tight constraints on the abundance of PBH's formed due to their gravitational effects and the consequences of their evaporation (for recent updates and compilations of the constraints see Refs. [2, 3]). These abundance constraints can be used to constrain the primordial power spectrum, and hence models of inflation, on scales far smaller than those probed by cosmological observations (e.g. Refs. [4-7]). These calculations usually assume that the primordial fluctuations are Gaussian. However since PBH's form from the extremely rare, large fluctuations in the tail of the fluctuation distribution non-Gaussianity can potentially significantly affect the number of PBH's formed. Therefore PBH formation probes both the amplitude and the non-Gaussianity of the primordial fluctuations on small scales. \nBullock and Primack [8] and Ivanov [9] were the first to study the effects of non-Gaussianity on PBH for- \nmation, reaching opposite conclusions on whether nonGaussianity enhances or suppresses the number of PBH's formed (see also Ref. [10]). Refs. [11, 12] used a nonGaussian probability distribution function (pdf) derived from an expansion about the Gaussian pdf [13, 14] to study PBH formation. However, since PBH's form from rare fluctuations in the extreme tails of the probability distribution, expansions which are only valid for typical size fluctuations can not reliably be used to study PBH formation. \nRef. [15] studied the constraints from PBH formation on the primordial curvature perturbation for the special cases where it has the form ζ = ± ( x 2 - 〈 x 2 〉 ), where x has a Gaussian distribution, see also Ref. [16]. The minus sign is expected from the linear era of the hybrid inflation waterfall (see also Ref. [17]), while the positive sign might arise if ζ is generated after inflation by a curvaton-type mechanism. \nIn this paper we go beyond this earlier work and calculate the constraints on the amplitude of the primordial curvature fluctuations, ζ , from black hole formation for both the quadratic and cubic local non-Gaussianity models (parameterised by f NL and g NL respectively). In the process we calculate the probability distribution function of the curvature perturbation for these models. Our results are valid for arbitrary values of these non-linearity parameters, and we recover the known limiting results for very small or large non-Gaussianity. In Sec. II we review the calculation of the PBH abundance constraints in the standard case of Gaussian fluctuations, before calculating the constraints for quadratic and cubic local nonGaussianity in Sec. III and IV respectively. We conclude with discussion in Sec. V.", 'II. PRIMORDIAL BLACK HOLE FORMATION CONSTRAINTS': "The condition for collapse to form a PBH is traditionally stated in terms of the smoothed density contrast at horizon crossing, δ hor ( R ). A fluctuation on a scale R will collapse to form a PBH, with mass M PBH roughly equal to the horizon mass, if δ hor ( R ) > δ c ∼ O (1) [1] 1 . If the initial perturbations have a Gaussian distribution then the probability distribution of the smoothed density contrast is given by (e.g. Ref. [19]): \nP ( δ hor ( R )) = 1 √ 2 πσ hor ( R ) exp ( -δ 2 hor ( R ) 2 σ 2 hor ( R ) ) , (1) \nwhere σ ( R ) is the mass variance \nσ 2 ( R ) = ∫ ∞ 0 ˜ W 2 ( kR ) P δ ( k, t ) d k k , (2) \nwhile P δ ( k, t ) is the power spectrum of the (unsmoothed) density contrast \nP δ ( k, t ) ≡ k 3 2 π 2 〈| δ k | 2 〉 , (3) \nand ˜ W ( kR ) is the Fourier transform of the window function used to smooth the density contrast. \nThe initial PBH mass fraction \nβ ( M PBH ) ≡ ρ PBH ( M PBH ) ρ tot , (4) \nis equal to the fraction of the energy density of the Universe contained in regions dense enough to form PBH's which is given by 2 \nβ ( M PBH ) = ∫ ∞ δ c P ( δ hor ( R )) d δ hor ( R ) . (5) \nThe PBH initial mass fraction is then related to the mass variance by \nβ = 1 2 erfc ( ζ c √ 2 σ ) , (9) \n, (6) where ζ c is the threshold for PBH formation. The variance of the probability distribution is related to the power spectrum of the curvature perturbation, \nP ζ ≡ k 3 2 π 2 〈| ζ k | 2 〉 , (10) \nby σ 2 ≈ P ζ . Here and subsequently for compactness we drop the explicit scale dependence of β and σ and the subscript 'hor' indicating that σ is to be evaluated at horizon crossing of the mass variance, i.e. σ σ hor ( R ). \nThe constraints on σ obtained via this method differ from those obtained from the full calculation involving the smoothed density contrast by O (10%): for β = 10 -20 the full calculation gives P 1 / 2 ζ = 0 . 12 [2], while using Eq. (9) gives σ = P 1 / 2 ζ = 0 . 11. We take ζ c = 1 for \n≡ \nβ ( M PBH ) = 1 √ 2 πσ hor ( R ) ∫ ∞ δ c exp ( -δ 2 hor ( R ) 2 σ 2 hor ( R ) ) d δ hor ( R ) , = 1 2 erfc ( δ c √ 2 σ hor ( R ) ) . (6) \nThe constraints on the PBH initial mass fraction, β ( M PBH ), can therefore be translated into constraints on the mass variance by inverting this expression. There are a wide range of constraints on the PBH abundance, from their various gravitational effects and the consequences of their evaporation, which apply over different mass ranges. These constraints are mass dependent and lie in the range β ( M PBH ) < 10 -20 -10 -5 [2, 3]. The power of these PBH abundance constraints is apparent when we consider the resulting constraints on σ hor ( R ) which are in the range σ hor ( R ) /δ c < 0 . 1 -0 . 2. In other words a small change in σ hor ( R ) /δ c leads to a huge change in β . Finally to impose constraints on the power spectrum of the curvature perturbation the transfer function which relates δ ( k, t ) to the primordial curvature perturbation is calculated (e.g. Refs. [2, 21]). \nFor an interesting number of PBH's to form the power spectrum on small scales must be several orders of magnitude larger than on a cosmological scales. This is possible in models, such as the running mass model [22-27], where the power spectrum increases monotonically with increasing wave-number [6, 7]. Another possibility is a peak in the primordial power spectrum, due to a phase transition during inflation or features in the inflation potential (see Ref. [28] and references therein). \nTo assess the affects of non-Gaussianity on the PBH constraints on the curvature perturbations it is sufficient to follow Ref. [15] and work directly with the curvature perturbation, with the PBH abundance being given by \nβ = ∫ ∞ ζ c P ( ζ ) d ζ . (7) \nFor Gaussian fluctuations \nP ( ζ ) = 1 √ 2 πσ exp ( -ζ 2 2 σ 2 ) , (8) \nand hence \ndefiniteness, however any variation in the threshold for collapse affects the constraints on σ in the Gaussian and non-Gaussian cases in the same way (since it is in fact the combination σ/ζ c that is constrained). We consider the upper and lower values of the constraints on β , 10 -5 and 10 -20 , and henceforth drop the explicit dependence on the PBH mass. \nBecause PBH's form from the rare large fluctuations in the tail of the distribution, this corresponds to ζ c / ( √ 2 σ ) /greatermuch 1 which allows a useful analytic approximation for σ ( β ) to be found. Using the large x limit of erfc( x ), Eq. (9) can be rewritten as \nβ ≈ σ √ 2 πζ c exp ( -ζ 2 c 2 σ 2 ) , (11) \nand hence \nσ ζ c ≈ √ 1 ln (1 /β ) . (12) \nUp to now we have been concentrating on the large tail limit of a Gaussian distribution. We will now see that the consequence of allowing even a small amount of nonGaussianity in the distribution can be dramatic in terms of the constraints from PBH formation - enough to potentially rule out certain values of the non-linearity parameters. We begin in Sec. III by considering the impact of adding quadratic local non-Gaussianity.", 'III. QUADRATIC LOCAL NON-GAUSSIANITY ( f NL )': "We take the model of local non-Gaussianity to be \nζ = ζ G + 3 5 f NL ( ζ 2 G -σ 2 ) ≡ h ( ζ G ) , (13) \nwhere ζ G is Gaussian with variance σ 2 , i.e. 〈 ζ 2 G 〉 = σ 2 . The constant term is subtracted from ζ such that 〈 ζ 〉 = 0, as required by the definition of the curvature perturbation. Note that we are not assuming single-field inflation, but the less restrictive assumption that only a single field direction generates ζ , known as single-source inflation [29], and furthermore this degree of freedom might be different from the one which generates the primordial temperature perturbation on the much larger CMB scales, this is for example the case in [15]. The variance of ζ is given by [30] \nP ζ = σ 2 +4 ( 3 f NL 5 ) 2 σ 4 ln( kL ) , (14) \nwhere the cut-off scale L /similarequal 1 /H is of order the horizon scale, the second term is a one-loop correction which dominates in the large f NL limit, k is the scale of interest and ln( kL ) is typically of order unity (see e.g. Ref. [31]). We have implicitly assumed here, for simplicity, that the \nprimordial power spectrum is close to scale-invariant on the scales of interest, as is the case for models where the power spectrum varies monotonically with wavenumber. \nThe non-Gaussian probability distribution function (pdf) for ζ , P NG ( ζ ), can be found by making a formal change of variables \n∣ \nP NG ( ζ ) d ζ = n ∑ i =1 ∣ ∣ ∣ d h -1 i ( ζ ) d ζ ∣ ∣ ∣ ∣ P G ( h -1 i ) d ζ , (15) \n∣ where P G has a Gaussian distribution and the sum is over the n solutions, h -1 i ( ζ ), of the equation h ( ζ G ) = ζ . For quartic local non-Gaussianity solving Eq. (13) for ζ G gives two solutions (which we denote with ' ± '): \nh -1 ± ( ζ ) = 5 6 f NL [ -1 ± √ 1 + 12 f NL 5 ( 3 f NL σ 2 5 + ζ ) ] , (16) \nand hence [14, 32]: \nP NG ( ζ ) d ζ = d ζ √ 2 πσ √ 1 + 12 f NL 5 ( 3 f NL σ 2 5 + ζ ) ( ε -+ ε -) , (17) \nwhere \nε ± = exp -1 2 ( h -1 ± ( ζ ) σ ) 2 . (18) \nThe log of the non-Gaussian pdf is shown in the upper panels of Fig. 1 for σ = 0 . 1 and f NL = 0 , ± 2 , ± 3 . 5 and ± 5. For positive f NL , as f NL is increased the amplitude of the large ζ tail of the pdf increases in amplitude. There is a minimum value of ζ , ζ min = ζ lim , \nζ lim = -5 12 f NL ( 1 + 36 f 2 NL σ 2 25 ) , (19) \nat which the pdf diverges, since d h -1 ( ζ ) / d ζ tends to infinity as ζ → ζ lim . Initially the peak in the pdf increases in amplitude and moves to negative ζ . As f NL is increased further the pdf becomes monotonic, increasing continuously with decreasing ζ down to ζ min . For negative f NL the pdf is the mirror image of that for positive f NL , and in this case there is a maximum possible value of ζ , ζ max = ζ lim . \nThe initial PBH mass fraction is given by \nβ = I + + I -, (20) \nwhere \nI ± ≡ ∫ ζ max ζ c 1 √ 2 πσ √ 1 + 12 f NL 5 ( 3 f NL σ 2 5 + ζ ) ε ± d ζ . (21) \nThe lower limit on the integral is ζ c /similarequal 1, the threshold value above which a black hole is formed. The upper \nFIG. 1: The log of the non-Gaussian probability distribution, top row f NL and bottom row g NL . We have fixed σ = 0 . 1 throughout and the solid line shows the Gaussian pdf. In the top left panel the dotted, short-dashed and long-dashed lines correspond to f NL = 2 , 3 . 5 and 5 respectively, while in the top right panel they correspond to f NL = -2 , -3 . 5 and -5. In the bottom left panel the dotted, short-dashed and long-dashed lines correspond to g NL = 10 , 20 and 30 respectively, while in the bottom right panel they correspond to g NL = -10 , -20 and -30. \n<!-- image --> \nFIG. 2: The constraints on the square root of the power spectrum of the curvature perturbation, P 1 / 2 ζ , for initial PBH abundances β = 10 -5 and 10 -20 (the upper and lower lines respectively) for the quadratic local non-Gaussianity model as a function of f NL . \n<!-- image --> \nlimit, ζ max , is the maximum possible value of ζ . For f NL > 0, ζ max = ∞ , while for f NL < 0, ζ max = ζ lim as discussed above, with ζ lim given by Eq. (19). \nThe initial PBH mass fraction is most easily calculated by making a transformation to a new variable y , \ny = h -1 ± ( ζ ) σ , (22) \nFIG. 3: The constraints on the square root of the power spectrum of the curvature perturbation, P 1 / 2 ζ , as in Fig. 2, for the quadratic local non-Gaussianity model as a function of the fraction of the power spectrum which is non-Gaussian, 4 sgn( f ) f 2 σ 2 / (1 + 4 f 2 σ 2 ), where f ≡ 3 f NL / 5. \n<!-- image --> \nwhich has unit variance so that \nf NL > 0 : β = 1 √ 2 π ( ∫ ∞ y c + e -y 2 / 2 d y + ∫ y c --∞ e -y 2 / 2 d y ) , (23) = 1 2 erfc ( y c + √ 2 ) + 1 2 erfc ( | y c -| √ 2 ) , (24) f < 0 : \nNL \nβ = 1 √ 2 π ∫ y c -y c + e -y 2 / 2 d y , (25) \nwhere y c ± are the values of y corresponding to the threshold for PBH formation, ζ c : \ny c ± = h -1 ± ( ζ c ) σ . (26) \nFor f NL > 0, y c + > 0, y c -< 0, and | y c + |-| y c -| = y c + + y c -= -3 / (5 f NL ). Consequently the first integral in the expression for β , Eq. (23), which corresponds to the positive branch, gives the dominant contribution to β . However in the limit of very large f NL , | y c + |-| y c -| tends to zero and the positive and negative branches contribute equally to β . \nThe constraints on the square root of the power spectrum of the curvature perturbation, P 1 / 2 ζ , which arise from the tightest and weakest constraints on the initial PBH mass fraction, β < 10 -20 and 10 -5 respectively, are shown in Figs. 2 and 3. In Fig. 2 we plot the constraints as a function of f NL for small f NL , while in Fig. 3 we plot the constraints as a function of the fraction of the power spectrum which is non-Gaussian i.e the ratio of the second non-Gaussian term in the expression for 〈 ζ 2 〉 in Eq. (14) to the full expression. The limit that this ratio \nis unity corresponds to a purely non-Gaussian ζ . We now discuss how the change in the PBH constraints depends on the amount of non-Gaussianity.", 'A. Very small | f NL |': "Expanding the pdf for ζ , Eq. (17), to second order in f NL we find \nP NG ( ζ ) = P G ( ζ ) [ 1 + ( ζ 2 σ 2 -3 ) 3 f NL ζ 5 + (3 f NL ζ ) 2 50 ( ζ 4 σ 4 -11 ζ 2 σ 2 +23 -5 σ 2 ζ 2 )] , (27) \nwhich agrees with the non-Gaussian pdf in Ref. [13] to linear order in f NL . The second order term has an extra 1 /σ 2 term, and hence the above expansion can only be expected to be accurate for f NL /lessmuch σ 2 /ζ 3 . Since ζ c /similarequal 1 and σ ∼ 0 . 1 in the Gaussian case, this means that this expansion is only valid when applied to PBH formation if f NL < 0 . 01. \nAs can be seen in Fig. 2 even a small level of nonGaussianity has a significant impact on the constraints on σ . This is because PBH's form from the fluctuations in the extreme tail of the distribution (e.g. for σ ∼ 0 . 2 and ζ c = 1 they are a 5 σ fluctuation) and it is in this regime that even a small skewness is important. The strong asymmetry between positive and negative f NL is because for f NL > 0 overdensities are enhanced, while for f NL < 0 the overdensity from a positive ζ is partially canceled by the Gaussian squared term, thereby σ has to become large in order for PBH formation to be possible at all. \n∼", 'B. Intermediate | f NL |': "In the regime where 0 < f NL σ /lessmuch 1 (which in practice corresponds to 0 < f NL /lessmuch 10 since σ ∼ 0 . 1), we have σ 2 /similarequal 〈 ζ 2 〉 . However f NL ζ c > ∼ 1 and the expression for h -1 ( ζ ), Eq. (16), simplifies substantially leading to \nσ ζ c /similarequal ( 5 3 f NL ζ c ) 1 / 2 √ 1 2 ln(1 /β ) . (28) \nThe square root term is the result in the Gaussian case, Eq. (12). Hence the constraint on P ζ is tightened by a factor of f NL compared to the Gaussian result. \nAs can be clearly seen from Fig. 2, the constraints are very asymmetric under a change of sign of f NL , with the constraints becoming very rapidly much weaker for f NL < 0, we discuss the case of negative f NL in the next section. \nand hence [15] \nP 1 / 2 ζ /similarequal 2 σ 2 /similarequal ζ c ln(1 /β ) , (31) \ni.e. the constraint on σ is approximately the square of the constraint in the Gaussian case, and is hence a lot tighter. \nThe case of a pure, negative chi-squared distribution is very different from the positive case. Using the transform h -1 this leads to \nβ = ∫ σ 2 ζ c 1 √ 2 πσ √ σ 2 -ζ e -( σ 2 -ζ ) / (2 σ 2 ) d ζ , (32) \n√ which using a further change of variables we transform to \nβ = ∫ y c 0 1 √ 2 πy e -y/ 2 d y , y c = σ 2 -ζ c σ 2 , (33) \nwhich leads to the relationship between σ 2 /similarequal P 1 / 2 ζ and β \nσ 2 = ζ c 1 + 8 πσ 2 β 2 ζ c + O ( β 4 ) ) /similarequal ζ c . (34) \n( \n) The constraint on σ is very weakly dependent on the limit on β , confirming the behaviour seen in Fig. 2 where the constraints for β = 10 -5 and 10 -20 merge for f NL < ∼ -0 . 5. For f NL < 0, once | f NL | σ > ∼ 0 . 4 the pdf for ζ increases monotonically with increasing ζ , before diverging at ζ max = ζ lim . If ζ max < ζ c then the number of PBH's formed is identically zero, while if ζ max > ζ c it is extremely sensitive to the precise value of σ . Therefore in this regime, unless there is extreme fine-tuning of σ , the number of PBH's formed will either be completely negligible or so large that the PBH abundance constraints are violated by many orders of magnitude. Although one can formally produce any given value of β with sufficient fine tuning of σ , in a realistic model the non-Gaussianity will lead to small spatial variations of σ in different patches (e.g. due to a small cubic term ) [33], which would probably rule this model out if one performed a more detailed", 'C. Large | f NL |': "In the case of a pure, positive non-Gaussianity the constraints on P ζ become a lot tighter. Since there is a degeneracy between f NL and σ in this case, ζ can be taken to be given by \nζ = ± ( ζ 2 G -σ 2 = h ( ζ G ) ≤ σ 2 , (29) \n( (c.f. Ref. [15]) and hence 〈 ζ 2 4 σ 4 . \n) \n〈 We first study the + case. Performing a similar calculation to the more general case with a linear term, the PBH initial mass fraction is given by \n〉 /similarequal \nβ = 2 √ 2 π ∫ ∞ y c e -y 2 / 2 d y , (30) \ncalculation. Hence we conclude that a future detection of PBH's would effectively rule out a negative f NL unless it has a tiny value, i.e. from Fig. 2, f NL < ∼ -0 . 5 would be ruled out regardless of the value of σ . \nHaving looked at the case of adding a quadratic type of local non-Gaussianity, we now consider the case of adding a cubic type to see what new constraints this may impose.", 'IV. CUBIC NON-GAUSSIANITY ( g NL )': 'The model of local non-Gaussianity with a cubic term (but assuming that f NL = 0) is defined by \nζ = ζ G + gζ 3 G ≡ h ( ζ G ) , g ≡ 9 25 g NL , (35) \nwhere we have introduced the definition of g in order to reduce the numerical factors which will appear in many expressions in this section. The variance of ζ for this model is given by \nP ζ = σ 2 1 + 6 gσ 2 ln( kL ) + 27 g 2 σ 4 ln( kL ) 2 ) , (36) \n( \n) where the second term is a one-loop contribution and the second term a two-loop contribution [30], which nonetheless dominates in the limit of large g , and ln( kL ) is again of order unity. \nFor g NL > 0 the cubic equation ζ = ζ G + gζ 3 G ≡ h ( ζ G ) has one real solution for all ζ : \nh -1 ( ζ ) = -( 2 1 / 3 3 )[ g 2 ( ζ + √ ζ 2 + 4 27 g )] -1 / 3 + 1 2 1 / 3 g [ g 2 ( ζ + √ ζ 2 + 4 27 g )] 1 / 3 . (37) \nThe PBH initial mass fraction is then given by \ng > 0 : β = 1 √ 2 π ∫ ∞ y c e -y 2 / 2 d y = 1 2 erfc ( y c √ 2 ) , (38) \nwhere y c = h -1 ( ζ c ) /σ , with h -1 ( ζ c ) given by Eq. (37). In the case of negative g NL , the cubic function has a local maximum for positive ζ G , with a peak value at ζ = ζ t where \nζ t ≡ 2 3 √ 3 √ -g . (39) \nHence the cubic polynomial has three real roots if | ζ | < ζ t , and otherwise only one real root. The transition between one and three real roots for ζ = ζ c occurs when g = g t where \ng t = -4 27 ζ 2 c ≈ -0 . 15 . (40) \nThe expression for the PBH initial mass fraction therefore has different forms depending on whether g is greater or less than g t . \nIf ζ t < ζ c , equivalently g < g t , then \ng < g t : β = 1 √ 2 π ∫ y c -∞ exp( -y 2 / 2) d y = 1 2 erfc ( | y c | √ 2 ) , (41) \nwhere y c = h -1 ( ζ c ) /σ and \nh -1 ( ζ c ) = -2 1 / 3 3( -g ) 2 / 3 [ ζ c + √ ζ 2 c -ζ 2 t ] -1 / 3 -1 2 1 / 3 ( -g ) 1 / 3 [ ζ c + √ ζ 2 c -ζ 2 t ] 1 / 3 . (42) \nIf ζ t > ζ c , equivalently g t < g < 0, there are three roots h -1 1 ( ζ c ) < 0 < h -1 2 ( ζ c ) < h -1 3 ( ζ c ), given by \nh -1 1 ( ζ c ) = -2 √ 3( g ) 1 / 2 cos ( θ/ 3) , (43) \nh -1 2 ( ζ c ) = 1 √ 3( -g ) 1 / 2 [ cos ( θ/ 3) -√ 3sin ( θ/ 3) ] , (44) \n- \n-h -1 3 ( ζ c ) = 1 √ 3( -g ) 1 / 2 [ cos ( θ/ 3) + √ 3sin ( θ/ 3) ] , (45) \nwhere \nθ = atan [ ( ζ 2 t -ζ 2 c ) 1 / 2 ζ c ] . (46) \nIt follows that \ng t < g < 0 : β = 1 √ 2 π (∫ y 1 -∞ exp( -y 2 / 2) d y + ∫ y 3 y 2 exp( -y 2 / 2) d y ) , (47) = 1 2 erfc ( | y 1 | √ 2 ) + 1 √ 2 π ∫ y 3 y 2 exp( -y 2 / 2) d y , (48) \nwhere y i = h -1 i ( ζ c ) /σ , with h -1 i ( ζ c ) given by Eqs. (43)(45). \nWe calculate the non-Gaussian pdf using the procedure described in Sec. III. The log of the pdf is shown in the lower panels of Fig. 1 for σ = 0 . 1 and g NL = 0 , ± 10 , ± 20 and ± 30. For positive g NL , as g NL is increased the large ζ tail of the pdf increases in amplitude, however (unlike the f NL case) the body of the pdf does not deviate significantly from the Gaussian pdf. This suggests that PBH formation is potentially a more sensitive probe of positive cubic local non-Gaussianity than structure formation. Negative g NL is very different from positive g NL . In particular there is a divergence at ζ = ζ t which arises from the d h -1 ( ζ ) / d ζ factor in the pdf. From Eq. (35) we see that \nd h -1 ( ζ ) d ζ = 1 1 + 3 gζ 2 G . (49) \nFIG. 4: The constraints on the square root of the power spectrum of the curvature perturbation, P 1 / 2 ζ , as in Fig. 2, for the cubic local non-Gaussianity model for -10 < g NL < 10. \n<!-- image --> \nFIG. 5: The constraints on the square root of the power spectrum of the curvature perturbation, P 1 / 2 ζ , as in Fig. 3, for the cubic local non-Gaussianity model as a function of the fraction of the power spectrum that is non-Gaussian, sgn( g )(6 gσ 2 +27 g 2 σ 4 ) / (1 + 6 gσ 2 +27 g 2 σ 4 ). \n<!-- image --> \nThis diverges when 1 /ζ 2 G = 3( -g ) which from Eq. (35) corresponds to ζ = ζ t . Expanding around this point with ζ = ζ t -δ, δ /lessmuch 1, we find after a little algebra that to leading order \nd h -1 ( ζ ) d ζ = 3 -1 / 4 2( -g ) 1 / 4 δ 1 / 2 . (50) \nThis divergence is fairly weak however and the PBH initial mass fraction β does not diverge, a result that is confirmed analytically as well as numerically. \nThe constraints on the square root of the power spectrum of the curvature perturbation, P 1 / 2 ζ , which arise from the tightest and weakest constraints on the initial PBH mass fraction, β < 10 -20 and 10 -5 respectively, are \nshown in Figs. 4 and 5. In Fig. 4 we plot the constraints as a function of g NL for small g NL , while in Fig. 5 we plot the constraints as a function of the fraction of the power spectrum which is non-Gaussian, i.e. the ratio of sum of the second and third non-Gaussian terms in Eq. (36) for 〈 ζ 2 〉 to the full expression. We now discuss how the change in the PBH constraints depend on the amount of non-Gaussianity.', 'A. Small | g NL |': 'The constraints on the power spectrum are highly asymmetric between positive and negative g NL . This is because for g NL > 0 an overdensity in the linear ζ regime will be boosted by the cubic term, especially strongly in the tail of the distribution and hence the constraint is tightened. However for mildly negative g NL , the opposite is the case; the two terms tend to cancel each other out and hence the constraints on the power spectrum weaken dramatically in this regime. For very small negative g NL the 2nd term in the expression for β , Eq. (47), the integral of the Gaussian distribution from y 2 to y 3 dominates. As g → g t = -0 . 15 from above y 3 -y 2 → 0 so that this term decreases rapidly and the constraint on the power spectrum rapidly becomes weaker. Only when g t -g /lessmuch 10 -80 does this term become smaller than the first erfc( | y 1 | / √ 2) term though, and then the value of β matches smoothly onto that in the g < g t regime where there is only one real root. As g is decreased below g t the one real root, h -1 ( ζ c ) given by Eq. (42), becomes less negative and the constraint on the power spectrum rapidly tightens again.', 'B. Intermediate | g NL |': 'For g NL > 0 in the intermediate regime, where the nonGaussianity is small enough that P ζ /similarequal σ 2 , i.e. gσ 2 /lessmuch 1, but large enough to satisfy gζ 2 c /greatermuch 1, (hence valid for 1 /lessmuch g /lessmuch σ -2 ), the expression for h -1 ( ζ c ) in Eq. (37) simplifies significantly to h -1 ( ζ c ) /similarequal ( ζ c /g ) 2 / 3 . Using the leading asymptotic expansion for the Gaussian pdf this leads to \nσ ζ c /similarequal ( 1 gζ 2 c ) 1 / 3 √ 1 2 ln(1 /β ) . (51) \nThe term in the square root is the result in the Gaussian case, hence the constraint on σ is tightened by a factor of g -1 / 3 . For comparison, the result for intermediate quadratic non-Gaussianity is given by Eq. (28). \nTo compare the relative size of these changes relative to the Gaussian case, consider a 10% non-Gaussian correction to ζ , and for concreteness assume that σ = 10 -2 . Then gσ 2 = 0 . 1 implies g = 10 3 and the constraint on σ tightens by a factor of 10. For a quadratic nonGaussianity, 3 f NL σ/ 5 = 0 . 1 implies 3 f NL / 5 = 10 and \nhence the constraint only tightens by approximately a factor of 3. However, if instead of considering a fixed ratio of non-Gaussian to Gaussian terms, one considers the non-linearity parameters to have a fixed amplitude then the constraints on σ are tightening by the cube root of g , but only by the square root of f ≡ 3 f NL / 5.', 'C. Large | g NL |': "In the limit of very large | g NL | , ζ ∝ ± ζ 3 G and the constraints don't depend on the sign of g NL . This is because the Gaussian pdf is invariant under a change of sign in ζ G and this is equivalent to changing the sign of g NL , but only in the case that the linear term is absent. As can be seen in Fig. 5 the symmetry between positive and negative g NL only occurs once the modulus of the nonGaussian fraction of the power spectrum becomes close to one, which corresponds to very large values of | g NL | . In the limit of very large | g NL | the constraints on P ζ become significantly tighter than in the case of a large and positive f NL . The bound in this case becomes approximately \nP 1 / 2 ζ /similarequal 5 σ 3 /similarequal 5 ζ c (2 ln(1 /β )) 3 2 , (52) \nwhich is approximately the cube of the bound in the Gaussian case and hence much more stringent (and also tighter compared to the case of a large and positive f NL , as discussed in Sec. III C).", 'V. DISCUSSION': "PBH formation probes the extreme tail of the probability distribution function of the primordial fluctuations. This is the region of the pdf which is most sensitive to the effects of any non-Gaussianity that may be present. We have, for the first time, calculated joint constraints on the amplitude and non-Gaussianity of the primordial perturbations, for arbitrarily large local non-Gaussianity. We have studied both quadratic and cubic local nonGaussianity, parameterised by f NL and g NL respectively. On the scales associated with the cosmic microwave background and large scale structure, the constraints on primordial non-Gaussianity are approximately | f NL | < ∼ 10 2 [34] and | g NL | < ∼ 10 6 [35]. In contrast we have shown that on much smaller scales non-linearity parameters of order unity can have a significant effect on the number of PBH's formed. This is because the non-linearity parameters have a larger effect on the tails of the fluctuation distribution than on the more moderate fluctuations probed by cosmological observations. We expect most other forms of non-Gaussianity to also have a significant effect on PBH production, since in general nonGaussianity generates a skewness which affects the tails of the pdf. \nThe signs of the non-linearity parameters are particularly important. If positive they always make the constraints tighter by acting in the same direction as the linear contribution to ζ . A negative quadratic term tends to cancel the effect of the linear term, thereby reducing the abundance of large PBH forming fluctuations. The constraints on the amplitude of the power spectrum therefore become much weaker, of order unity for f NL < ∼ -0 . 5. In practice this means that the amplitude of fluctuations will either be too small to form any PBH's at all, or so large that almost every horizon region collapses to form a PBH, which is already observationally ruled out. We hence conclude that a future detection of PBH's would rule out a negative value of f NL unless its value is tiny, | f NL | /lessmuch 1. The case of negative g NL is different. For g NL /similarequal -1 the constraints are weakened as in the negative f NL case. However as g NL becomes more negative the constraints quickly become tighter again. In the limit of very large g NL the constraints are independent of its sign and very tight, approximately the cube of the constraints in the Gaussian case. We have also studied and plotted the probability distribution functions, showing that although the pdfs can diverge, all physical quantities, such as the PBH abundance, remain finite. The PBH constraints have previously been calculated for a pure χ 2 pdf [15, 16] and for quadratic non-Gaussianity in the limits that | f NL | /lessmuch 1 and the linear term dominates [11, 13]. We have shown that we recover these limiting cases, however, of particular significance is the fact that our calculations are valid for arbitrarily large quadratic and cubic local non-Gaussianity. \nA bispectrum in the squeezed limit is in general generated by all single field models of inflation, with an amplitude which is related to the spectral index by f NL = -5( n s -1) / 12 [36, 37]. Although this value is too small to be seen on CMB scales, it might be important on PBH scales, since firstly the spectral index might be larger as the slow-roll parameters potentially become larger towards the end of inflation and secondly since the constraints are sensitive to smaller values of f NL . \nAn important issue, which goes beyond the scope of this work, is the calculation of the secondary nonGaussianities generated through the effects of gravity being non-linear and through horizon re-entry after inflation, during which time ζ is no longer conserved. These calculations have been carried out on CMB scales and these effects generally cause an order of unity change to the non-linearity parameters (although the effect is scale dependent) [38-42]. Such small values of the non-linear parameters can have a significant effect on the number of PBH's formed, therefore it would be interesting to carry out a similar analysis valid for the much smaller scales on which PBH's form. If these effects generically lead to f NL ∼ -1 then this would suggest that PBH's are unlikely to have formed, unless inflation generated a larger and positive primordial f NL on the same scales. The current calculation could also be extended by allowing for simultaneous non-zero values of f NL and g NL or by \nstudying the effects of higher order non-linearity parameters. \nThe possibility of PBH formation constrains both the amplitude and degree of non-Gaussianity associated with the primordial density perturbations over a wide range of scales, smaller than those probed by cosmological observations. In this paper we have shown that even relatively small values of the non-linearity parameters, of order unity, can have a significant effect on the PBH constraints on the amplitude of the primordial perturbations. Therefore non-Gaussianity should be taken into account when calculating PBH constraints on inflation models. We have also shown that the observation of \n- [1] B. J. Carr and S. W. Hawking, Mon. Not. Roy. Astron. Soc. 168 , (1974) 399; B. J. Carr Astrophys. J. 201 (1975) 1-19.\n- [2] A. S. Josan, A. M. Green and K. A. Malik, Phys. Rev. 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2017PhRvL.118b1301H
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Superfluid Black Holes
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2017-01-01
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We present what we believe is the first example of a "λ -line" phase transition in black hole thermodynamics. This is a line of (continuous) second order phase transitions which in the case of liquid <SUP>4</SUP>He marks the onset of superfluidity. The phase transition occurs for a class of asymptotically anti-de Sitter hairy black holes in Lovelock gravity where a real scalar field is conformally coupled to gravity. We discuss the origin of this phase transition and outline the circumstances under which it (or generalizations of it) could occur.
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https://arxiv.org/pdf/1609.02564.pdf
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{'Superfluid Black Holes': 'Robie A. Hennigar ∗ and Robert B. Mann † Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1', 'Erickson Tjoa ‡': "Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1 and Division of Physics and Applied Physics, Nanyang Technological University, Singapore, S637371 \nWe present what we believe is the first example of a ' λ -line' phase transition in black hole thermodynamics. This is a line of (continuous) second order phase transitions which in the case of liquid 4 He marks the onset of superfluidity. The phase transition occurs for a class of asymptotically AdS hairy black holes in Lovelock gravity where a real scalar field is conformally coupled to gravity. We discuss the origin of this phase transition and outline the circumstances under which it (or generalizations of it) could occur. \nPACS numbers: 04.50.Gh, 04.70.-s, 05.70.Ce \nThe study of black hole thermodynamics provides valuable insight into quantum properties of the gravitational field. The thermodynamics of anti de Sitter black holes has been of great interest since the pioneering work of Hawking and Page which demonstrated the existence of a thermal radiation/large AdS black hole phase transition [1]. Furthermore, these spacetimes admit a gauge duality description via a dual thermal field theory. \nRecently there has been interest in treating the cosmological constant as a thermodynamic variable [2] which plays the role of pressure in the first law [3-5]. Within this context, the black hole mass takes on the interpretation of enthalpy and a number of connections with ordinary thermodynamics emerge. It was shown that the thermodynamic behaviour of a charged AdS black hole is analogous to the van der Waals liquid/gas system, with the role of the liquid/gas transition played by a small/large black hole phase transition [6]. Subsequent work has revealed examples of triple points [7], (multiple) reentrant phase transitions [8, 9], isolated critical points [10, 11] and a host of other behaviour for black holes (see [12] and references therein for a review). \nHere we present the first example of a line of second order (continuous) black hole phase transitions which strongly resemble those which occur in condensed matter systems, e.g. the onset of superfluidity in liquid helium [13]. The phase transition occurs in a broad class of asymptotically AdS hairy black holes in Lovelock gravity. Lovelock gravity [14] is a geometric higher curvature theory of gravity and is the natural generalization of Einstein gravity to higher dimensions, giving rise to second order field equations for the metric. It provides an excellent testbed for examining the effects of higher curvature corrections which appear in, for example, the low energy effective action of string theory [15]. Recently, it has been shown [16] that a scalar field can be conformally coupled to the Lovelock terms and the resulting theory gives rise to analytic hairy black hole solutions [17] evading no-go results which had been reported previously [18]. These \nsolutions have already been shown to possess interesting thermodynamic properties [19-21] (such as reentrant phase transitions), and are of inherent interest due to the role scalar hair plays in holography, e.g. in descriptions of holographic superconductors and superfluids [22, 23]. \nThe model we consider consists of Lovelock gravity, a Maxwell field, and a real scalar field coupled conformally to the dimensionally extended Euler densities, \nI = 1 16 πG ∫ d d x √ -g ( k max ∑ k =0 L ( k ) -4 πGF µν F µν ) (1) \nwhere \nL ( k ) = 1 2 k δ ( k ) ( a k k ∏ r R α r β r µ r ν r + b k φ d -4 k k ∏ r S α r β r µ r ν r ) (2) \nwith δ ( k ) = δ α 1 β 1 ··· α k β k µ 1 ν 1 ··· µ k ν k the generalized Kronecker tensor. Here the tensor S γδ µν describes how the scalar field couples to gravity, \nS γδ µν = φ 2 R γδ µν -2 δ [ γ [ µ δ δ ] ν ] ∇ ρ φ ∇ ρ φ -4 φδ [ γ [ µ ∇ ν ] ∇ δ ] φ +8 δ [ γ [ µ ∇ ν ] φ ∇ δ ] φ, (3) \nand transforms homogeneously under the conformal transformation, g µν → Ω 2 g µν and φ → Ω -1 φ as S γδ µν → Ω -4 S γδ µν . \nWe take a line element of the form, \nds 2 = -fdt 2 + f -1 dr 2 + r 2 d Σ 2 ( σ ) d -2 (4) \nwhere d Σ 2 ( σ ) d -2 is the line element on a surface of constant curvature σ ( d -2)( d -3) with σ = +1 , 0 , -1 corresponding to spherical, flat and hyperbolic geometries; in the latter cases, the space is compact via identification [24]. For this ansatz, the field equations for the \nmetric reduce to, \nk max ∑ k =0 α k ( σ -f r 2 ) k = 16 πGM ( d -2)Σ σ d -2 r d -1 + H r d -8 πG ( d -2)( d -3) Q 2 r 2 d -4 (5) \nwhere \nα 0 = a 0 ( d -1)( d -2) , α 1 = a 1 , α k = a k 2 k ∏ n =3 ( d -n ) for k ≥ 2 , (6) \nand \nH = k max ∑ k =0 ( d -3)! ( d -2( k +1))! b k σ k N d -2 k (7) \nis the 'hair parameter'. For this configuration the scalar \nfield takes the form, \nφ = N r (8) \nand its equations of motion reduce to the following constraints: \nk max ∑ k =1 kb k ( d -1)! ( d -2 k -1)! σ k -1 N 2 -2 k = 0 , k max ∑ k =0 b k ( d -1)! ( d ( d -1) + 4 k 2 ) ( d -2 k -1)! σ k N -2 k = 0 . (9) \nSince these are two equations in a single unknown ( N ), one equation enforces a constraint on the allowed coupling constants, b k . Computing the temperature by requiring the absence of conical singularities in the Euclidean sector and the entropy using the Iyer-Wald formalism [25], we find the thermodynamic quantities for this solution are \nM = ( d -2)Σ σ d -2 16 πG k max ∑ k =0 α k σ k r d -2 k -1 + -( d -2)Σ σ d -2 H 16 πGr + + Σ σ d -2 Q 2 2( d -3) r d -3 + T = f ' ( r + ) 4 π = 1 4 πr + D ( r + ) [ ∑ k σα k ( d -2 k -1) ( σ r 2 + ) k -1 + H r d -2 + -8 πGQ 2 ( d -2) r 2( d -3) + ] S = Σ ( σ ) d -2 4 G [ k max ∑ k =1 ( d -2) kσ k -1 α k d -2 k r d -2 k + -d 2 σ ( d -4) H ] if b k = 0 ∀ k > 2 . (10) \nwhere D ( r + ) = ∑ k max k =1 kα k ( σr -2 + ) k -1 . It is straightforward to show that they satisfy the extended first law and Smarr formula. As the remaining expressions are quite lengthy we shall report them elsewhere [26]. \nIn what follows we consider α k = 0 ∀ k > 3 and b k = 0 ∀ k > 2. This last condition is for simplicity: the falloff in the metric function is the same for all b k and the contribution to the entropy is always just a constant; so only the first three b k 's are required to see all the physics of the scalar hair. \nIntroducing the dimensionless parameters \nr + = vα 1 / 4 3 , T = tα -1 / 4 3 d -2 , H = 4 πh d -2 α d -2 4 3 Q = q √ 2 α d -3 4 3 , m = 16 πM ( d -2)Σ ( κ ) d -2 α d -3 4 3 p = α 0 ( d -1)( d -2) √ α 3 4 π , α = α 2 √ α 3 , G = M -TS = α -( d -3) 4 3 g . (11) \nThe dimensionless equation of state (obtained by solving the expression for the temperature in Eq. (10) for the pressure) reads \np = t v -σ ( d -3)( d -2) 4 πv 2 + 2 ασt v 3 -α ( d -2)( d -5) 4 πv 4 + 3 t v 5 -σ ( d -7)( d -2) 4 πv 6 + q 2 v 2( d -2) -h v d (12) \nwhere the quantity p represents the pressure and g the dimensionless Gibbs free energy. At equilibrium, the state of the system is that which globally minimizes the Gibbs free energy. \nNoting that the conditions for a critical point are \n∂p ∂v = ∂ 2 p ∂v 2 = 0 (13) \nwe find that for α = √ 5 / 3 if h and q are set to the values \nh = 4(2 d -5)( d -2) 2 v d -6 c πd ( d -4) , q 2 = 2( d -1)( d -2) v 2 d -10 c π ( d -4) (14) \nand σ = -1, Eq. (13) is satisfied by v c = 15 1 / 4 and \np c = [ 8 225 (15) 3 4 ] t c + √ 15(11 d -40)( d -1)( d -2) 900 πd (15) \nfor all temperatures t c ! In other words, this system exhibits infinitely many critical points with critical volume v c = 15 1 / 4 . In the p -v plane, every isotherm is a critical isotherm, i.e. has an inflection point at v = 15 1 / 4 . In the variables ( t, p ) there is no first order phase transition but rather a line of second order phase transitions, characterized by a diverging specific heat c p = -t ∂ 2 g/∂t 2 at the critical values. We show representative thermodynamic behaviour in Fig. 1 for d = 7. \nThe line of second order phase transitions mimics those that occur in condensed matter systems where they correspond to, for example, fluid/superfluid transitions [27], superconductivity [28], and paramagentism/ferromagnetism transitions [29]. Building on the black hole/van der Waals fluid analogy [6], the natural interpretation here is that this second order phase transition between small/large black holes corresponds to a fluid/superfluid type transition. The resemblance to the fluid/superfluid λ -line transition of 4 He (Fig. 2) is striking. In each case, a line of critical points separates the two phases of 'fluid' where specific heat takes on the same qualitative ' λ ' structure. The phase diagram for helium is more complicated, including solid and gaseous states. This is to be expected since helium is a complicated system, while these hairy black hole solutions are comparatively simple being characterized by only four numbers: v, h, q and α . However, it is remarkable that with so few parameters the essence of the λ -line can be captured. Most of the interesting properties of a superfluid are either dynamical or require a full quantum description to understand (see, e.g. [13, 30] for an introduction and review). Since we do not have access to a model of the underlying quantum degrees of freedom it is not possible to explore the black hole analogues of these properties at a deeper level. \nIt is natural to wonder if there are any pathological properties of these black holes. We have examined the Kretchmann scalar near the λ -line and have found it to be finite at all finite temperatures and pressures. We have also studied the explicit solution to the field equations in detail and have found it to be completely regular outside the horizon. Within the horizon and at large enough pressures, the first derivative of the metric function diverges. Such behaviour is neither fatal nor uncommon for Lovelock black holes-similar behaviour occurs for charged black holes in Gauss-Bonnet gravity (cf. Fig. 2 of [9]). The Gibbs free energy and temperature are continuous and differentiable near the critical point and the entropy is positive. The specific heat is positive, indicating thermodynamic stability of these black holes. Furthermore, we have examined the linearized equations of motion for the theory about a maximally symmetric \nbackground and found that for the values of the coupling constants taken here the theory is free from ghost and tachyon instabilities [26]. Thus it seems that there is no underlying pathological behaviour here. \nTo calculate valid critical exponents, the appropriate ordering field, Θ, must be identified. As for liquid helium [27], pressure is no longer the appropriate ordering field for this line of 2nd order phase transitions. In this case there are three options for Θ: q , h or α . The resultant critical exponents are independent of which choice is made, but q is the most natural choice since its variation does not entail any coupling constants. Rearranging the expression for the temperature (10) for the chosen ordering field Θ, it can be expanded near a critical point (in terms of ω = ( v -v c ) /v c and τ = ( t -t c ) /t c ) to give: \nΘ Θ c = 1 -Aτ + Bτω -Cω 3 + O ( τω 2 , ω 4 ) , (16) \nwhere A , B and C are non-zero constants whose numeric value depends on the pressure and choice of ordering field. It is clear from (16) that the critical exponents are \nα = 0 , β = 1 2 γ = 1 , δ = 3 (17) \nand respectively govern the behaviour of the specific heat at constant volume, C V ∝ | τ | -α , the order parameter ω ∝ | τ | β , the susceptibility/compressibility ( ∂ω/∂ Θ) | τ ∝ | τ | -γ and the ordering field Θ ∝ | ω | δ near a critical point. These results coincide with the mean field theory values, in particular for those for a superfluid in d > 5 (cf. Table I of [27]). \nWe conclude by examining under what situations these λ -lines can be expected for black holes. Here the key result was that the conditions for a critical point are satisfied without fixing the temperature. For an equation of state of the form, \nP = a 1 ( r + , ϕ i ) T + a 2 ( r + , ϕ i ) (18) \n(where ϕ i represent additional constants in the equation of state) this condition is satisfied provided a nontrivial solution for the following equations exists: \n∂a i ∂r + = 0 , ∂ 2 a i ∂r 2 + = 0 i = 1 , 2 . (19) \nWith four free parameters ( v, α, q and h ), the hairy black holes permit a non-trivial solution to these 4 equations. This result generalizes to all Lovelock theories cubic and higher: with an appropriate choice of parameters, they possess λ -lines in the presence of conformal hair and charge. However the necessary and sufficient conditions for satisfying (19) for black holes in general remain to be found. We have checked that neither the rotating black hole of 5 d minimal gauged super-gravity [32] nor those in \nFIG. 1. Thermodynamic behaviour near λ transition : Left : A plot of the Gibbs free energy vs. temperature for three distinct pressures chosen so that critical temperatures are t c = 3 , 5 , 7 corresponding to the red, blue and black curves. The dotted lines highlight the points where the second derivative of the Gibbs free energy diverges. Center : A plot of the specific heat c p = -t ∂ 2 g ∂t 2 for the case t c = 3. Right : p -t parameter space. The black line shows the locus of critical points, i.e. a line of second-order phase transitions known as the 'lambda' line in the context of superfluidity. These plots are for d = 7. \n<!-- image --> \nFIG. 2. Thermodynamic properties of 4 He : Top : The P -T phase diagram for 4 He. The λ line corresponds to a line of critical points where a second order phase transition occurs marking the onset of superfluidity . To the left of the λ -line, the liquid begins to exhibit remarkable properties. Bottom : The specific heat of liquid 4 He. As the λ -line is approached, the specific heat spikes taking the shape of the Greek letter ' λ '. These plots have been reprinted with permission from [31]. \n<!-- image --> \nhigher order Lovelock gravity (without hair) admit a nontrivial solution. A particularly interesting case would result if the above equations admitted two (or more) nontrivial solutions. Such a circumstance could give rise to two intersecting λ -lines, a situation that occurs in certain ferromagnetic materials [33]. \nTo summarize, we have presented the first example of a superfluid-like phase transition in black hole thermodynamics. This occurs for hyperbolic black holes with conformal scalar hair in cubic Lovelock gravity for any dimension d ≥ 7. We have examined the black hole solutions and verified they are free from pathological behaviour. We have presented precise conditions (19) that a black hole equation of state must satisfy to display similar behaviour. While satisfaction of these conditions is by no means trivial, we find that all cubic-and-higher Lovelock theories with conformal hair can satisfy these requirements. We suspect there exists a much broader class of gravitational theories containing black holes that will exhibit λ -line phase transitions. Determining further examples of this, or similar, behaviour remains an interesting problem for future research.", 'ACKNOWLEDGMENTS': 'This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. \n- [1] S. Hawking and D. Page, Thermodynamics of Black Holes in Anti-de Sitter Space , Commun.Math.Phys. 87 (1983) 577.\n- [2] J. Creighton and R. B. Mann, Quasilocal thermodynamics of dilaton gravity coupled to gauge fields , Phys.Rev. D52 (1995) 4569-4587, [ gr-qc/9505007 ].\n- [3] M. Caldarelli, G. Cognola and D. Klemm, Thermodynamics of Kerr-Newman-AdS black holes and conformal field theories , Class.Quant.Grav. 17 (2000) 399-420, [ hep-th/9908022 ].\n- [4] D. Kastor, S. Ray and J. Traschen, Enthalpy and the Mechanics of AdS Black Holes , Class.Quant.Grav. 26 (2009) 195011, [ 0904.2765 ].\n- [5] M. Cvetic, G. Gibbons, D. Kubiznak and C. Pope, Black Hole Enthalpy and an Entropy Inequality for the Thermodynamic Volume , Phys.Rev. D84 (2011) 024037, [ 1012.2888 ].\n- [6] D. Kubiznak and R. B. Mann, P-V criticality of charged AdS black holes , JHEP 1207 (2012) 033, [ 1205.0559 ].\n- [7] N. Altamirano, D. Kubiznak, R. B. Mann and Z. Sherkatghanad, Kerr-AdS analogue of triple point and solid/liquid/gas phase transition , Class.Quant.Grav. 31 (2014) 042001, [ 1308.2672 ].\n- [8] N. Altamirano, D. Kubiznak and R. B. Mann, Reentrant Phase Transitions in Rotating AdS Black Holes , Phys.Rev. D88 (2013) 101502, [ 1306.5756 ].\n- [9] A. M. Frassino, D. Kubiznak, R. B. Mann and F. Simovic, Multiple Reentrant Phase Transitions and Triple Points in Lovelock Thermodynamics , JHEP 09 (2014) 080, [ 1406.7015 ].\n- [10] B. P. Dolan, A. Kostouki, D. Kubiznak and R. B. Mann, Isolated critical point from Lovelock gravity , Class. Quant. Grav. 31 (2014) 242001, [ 1407.4783 ].\n- [11] R. A. Hennigar, W. G. Brenna and R. B. Mann, P -v criticality in quasitopological gravity , JHEP 07 (2015) 077, [ 1505.05517 ].\n- [12] D. Kubiznak, R. B. Mann and M. Teo, Black hole chemistry: thermodynamics with Lambda , 1608.06147 .\n- [13] A. J. Leggett, Superfluidity , Rev. Mod. Phys. 71 (Mar, 1999) S318-S323.\n- [14] D. Lovelock, The Einstein tensor and its generalizations , J.Math.Phys. 12 (1971) 498-501.\n- [15] B. Zwiebach, Curvature Squared Terms and String Theories , Phys. Lett. B156 (1985) 315-317.\n- [16] J. Oliva and S. Ray, Conformal couplings of a scalar field to higher curvature terms , Class. Quant. Grav. 29 (2012) 205008, [ 1112.4112 ].\n- [17] G. Giribet, M. Leoni, J. Oliva and S. Ray, Hairy black holes sourced by a conformally coupled scalar field in D dimensions , Phys. Rev. D89 (2014) 085040, \n[ 1401.4987 ]. \n- [18] C. Martinez, Black holes with a conformally coupled scalar field , in Quantum Mechanics of Fundamental Systems: The Quest for Beauty and Simplicity .\n- [19] G. Giribet, A. Goya and J. Oliva, Different phases of hairy black holes in AdS 5 space , Phys. Rev. D91 (2015) 045031, [ 1501.00184 ].\n- [20] M. Galante, G. Giribet, A. Goya and J. Oliva, Chemical potential driven phase transition of black holes in anti de Sitter space , Phys. Rev. D92 (2015) 104039, [ 1508.03780 ].\n- [21] R. A. Hennigar and R. B. Mann, Reentrant phase transitions and van der Waals behaviour for hairy black holes , Entropy 17 (2015) 8056-8072, [ 1509.06798 ].\n- [22] S. A. Hartnoll, C. P. Herzog and G. T. Horowitz, Building a Holographic Superconductor , Phys. Rev. Lett. 101 (2008) 031601, [ 0803.3295 ].\n- [23] Z.-Y. Nie and H. Zeng, P-T phase diagram of a holographic s+p model from Gauss-Bonnet gravity , JHEP 10 (2015) 047, [ 1505.02289 ].\n- [24] R. B. Mann, Topological black holes: Outside looking in , Annals Israel Phys.Soc. 13 (1997) 311, [ gr-qc/9709039 ].\n- [25] V. Iyer and R. M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy , Phys. Rev. D50 (1994) 846-864, [ gr-qc/9403028 ].\n- [26] R. A. Hennigar, E. Tjoa and R. B. Mann, Thermodynamics of hairy black holes in Lovelock gravity , in preparation .\n- [27] P. B. Weichman, A. W. Harter and D. L. Goodstein, Criticality and superfluidity in liquid 4 He under nonequilibrium conditions , Rev. Mod. Phys. 73 (Jan, 2001) 1-15.\n- [28] C. Dasgupta and B. I. Halperin, Phase transition in a lattice model of superconductivity , Phys. Rev. Lett. 47 (Nov, 1981) 1556-1560.\n- [29] R. Pathria and P. D. Beale, Phase transitions: Criticality, universality, and scaling , in Statistical Mechanics (Third Edition) , pp. 401 - 469. Academic Press, Boston, third edition ed., 2011. DOI.\n- [30] T. Guenault, Basic Superfluids . CRC Press, 2002.\n- [31] D. M. Ceperley, Path integrals in the theory of condensed helium , Rev. Mod. Phys. 67 (Apr, 1995) 279-355.\n- [32] Z.-W. Chong, M. Cvetic, H. Lu and C. Pope, General non-extremal rotating black holes in minimal five-dimensional gauged supergravity , Phys.Rev.Lett. 95 (2005) 161301, [ hep-th/0506029 ].\n- [33] M. A. Anisimov, E. E. Gorodetski and V. M. Zaprudski, Phase transitions with coupled order parameters , Soviet Physics Uspekhi 24 (1981) 57.'}
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2016ApJ...823L..25K
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LIGO Gravitational Wave Detection, Primordial Black Holes, and the Near-IR Cosmic Infrared Background Anisotropies
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2016-01-01
| 15 | 0.49 | 163 |
['cosmology dark matter', 'cosmology diffuse radiation', 'cosmology early universe', 'gravitational waves', '-']
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LIGO's discovery of a gravitational wave from two merging black holes (BHs) of similar masses rekindled suggestions that primordial BHs (PBHs) make up the dark matter (DM). If so, PBHs would add a Poissonian isocurvature density fluctuation component to the inflation-produced adiabatic density fluctuations. For LIGO's BH parameters, this extra component would dominate the small-scale power responsible for collapse of early DM halos at z ≳ 10, where first luminous sources formed. We quantify the resultant increase in high-z abundances of collapsed halos that are suitable for producing the first generation of stars and luminous sources. The significantly increased abundance of the early halos would naturally explain the observed source-subtracted near-IR cosmic infrared background (CIB) fluctuations, which cannot be accounted for by known galaxy populations. For LIGO's BH parameters, this increase is such that the observed CIB fluctuation levels at 2-5 μm can be produced if only a tiny fraction of baryons in the collapsed DM halos forms luminous sources. Gas accretion onto these PBHs in collapsed halos, where first stars should also form, would straightforwardly account for the observed high coherence between the CIB and unresolved cosmic X-ray background in soft X-rays. We discuss modifications possibly required in the processes of first star formation if LIGO-type BHs indeed make up the bulk or all of DM. The arguments are valid only if the PBHs make up all, or at least most, of DM, but at the same time the mechanism appears inevitable if DM is made of PBHs.
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[]
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https://arxiv.org/pdf/1605.04023.pdf
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{'LIGO gravitational wave detection, primordial black holes and the near-IR cosmic infrared background anisotropies': 'A. Kashlinsky 1 ,', 'ABSTRACT': "LIGO's discovery of a gravitational wave from two merging black holes (BHs) of similar masses rekindled suggestions that primordial BHs (PBHs) make up the dark matter (DM). If so, PBHs would add a Poissonian isocurvature density fluctuation component to the inflation-produced adiabatic density fluctuations. For LIGO's BH parameters, this extra component would dominate the small-scale power responsible for collapse of early DM halos at z > ∼ 10, where first luminous sources formed. We quantify the resultant increase in highz abundances of collapsed halos that are suitable for producing the first generation of stars and luminous sources. The significantly increased abundance of the early halos would naturally explain the observed source-subtracted near-IR cosmic infrared background (CIB) fluctuations, which cannot be accounted for by known galaxy populations. For LIGO's BH parameters this increase is such that the observed CIB fluctuation levels at 2 to 5 µ m can be produced if only a tiny fraction of baryons in the collapsed DM halos forms luminous sources. Gas accretion onto these PBHs in collapsed halos, where first stars should also form, would straightforwardly account for the observed high coherence between the CIB and unresolved cosmic X-ray background in soft X-rays. We discuss modifications possibly required in the processes of first star formation if LIGO-type BHs indeed make up the bulk or all of DM. The arguments are valid only if the PBHs make up all, or at least most, of DM, but at the same time the mechanism appears inevitable if DM is made of PBHs.", '1. Introduction': "LIGO's recent discovery of the gravitational wave (GW) from an inspiralling binary black hole (BH) system of essentially equal mass BHs ( ∼ 30 M /circledot ) at z ∼ 0 . 1(Abbott et al. 2016b) has led to suggestion that all or at least a significant part of the dark matter (DM) is made up of primordial BHs (PBH) (Bird et al. 2016; Clesse & Garc'ıa-Bellido 2016). In particular, Bird et al. (2016) argue that this PBH mass range is not ruled out by astronomical observations and the observed rate at ∼ (a few) Gpc -3 yr -1 (Abbott et al. 2016a) can be accounted for if DM PBHs are distributed in dense, low velocity-dispersion concentrations which escaped merging. There is abundant motivation \nfor PBHs forming in the very early Universe (Hawking 1971; Carr & Hawking 1974), e.g. during phase transition at the QCD epoch when horizon mass is of the right magnitude (Jedamzik 1997); see a nice overview of possible mechanisms in Mack et al. (2007). If PBHs indeed constitute the bulk or all of DM, they would contribute an additional Poissonian component to the power spectrum of the mass distribution from adiabatic fluctuations from the earlier inflationary era (Afshordi et al. 2003). If so, this component would dominate small scales leading to significant modification of the history of collapse (and possibly formation of the first sources), resulting in greater rates of cosmic infrared background (CIB) production at z> 10. \nCIB contains emissions produced over the entire history of the Universe including from sources inaccessible to direct telescopic studies (see review by Kashlinsky 2005) with early stars and BHs contributing to its near-IR ( ∼ 1 -5 µ m) fluctuation component (Kashlinsky et al. 2004; Cooray et al. 2004). In this context, Kashlinsky (2005); Kashlinsky et al. (2007b) have identified from deep Spitzer data significant CIB fluctuations remaining on sub-degree scales after subtracting individual galaxies to faint levels. The measurement was later extended to degree scales (Kashlinsky et al. 2012) and confirmed in subsequent analyses of Akari and Spitzer data (Matsumoto et al. 2011; Cooray et al. 2012b). It is now established that these fluctuations cannot arise from remaining known galaxy populations (Kashlinsky et al. 2005; Helgason et al. 2012) and it has been suggested that they arise from new populations at early epochs (Kashlinsky et al. 2005, 2007c; Yue et al. 2013a). This proposition is currently a subject of debate (Cooray et al. 2012b; Gong et al. 2015), although the CIB fluctuations on relevant scales appear to be uncorrelated with the diffuse light in the visible produced by sources down to AB mag > 28 (Kashlinsky et al. 2007a) 1 . It was established that the CIB fluctuations are coherent with unresolved soft X-ray CXB (Cappelluti et al. 2013) at levels greater than expected from remaining known populations (Helgason et al. 2014) and that the measured coherence levels require much higher proportions of BHs among the CIB sources than in the known populations. At the same time, Helgason et al. (2016) have argued that if early populations were to produce the measured CIB signal that would require higher than expected efficiencies of early star formation (cf. Kashlinsky et al. (2015b)). \nIn this Letter we point out that if indeed the LIGO discovery is indicative of PBHs making up the DM, the extra Poissonian isocurvature component of the fluctuations would lead to much greater rates of collapse at early times, which would naturally produce the observed levels of the CIB fluctuations. We briefly revisit the required near-IR CIB energetics in Sec. 2 and the effects of the extra power component from PBHs on the collapse of the first halos in Sec. 3. We discuss the effects PBHs and the extra power may have on the formation of first populations in Sec. 4. The discussion below adopts cosmology with ( h, Ω tot , Ω CDM , Ω bar , σ 8 ) = (0 . 7 , 1 , 0 . 23 , 0 . 05 , 0 . 9).", '2. CIB anisotropies vs highz modeling': "The observed CIB fluctuations reflect several aspects of the sources producing them: 1) the measured shot-noise power characterizes the typical flux-magnitude of the sources producing the large-scale (clustering) power; the fact that the arcminute fluctuations arise at very low shot-noise levels means that the individual sources must be very faint consistent with their highz origin (Kashlinsky et al. 2007c). 2) The angular shape of the CIB power spectrum on arcminute scales reflects the epochs spanned by the sources; the shape from the Spitzer data is consistent with highz origin within the current errors and the upcoming Euclid all-sky survey would further probe the epochs and history of emissions much more accurately (Kashlinsky et al. 2015a). 3) Given the angular power template, the amplitude of the fluctuations spectrum at some fiducial scale (we adopt 5 ' below) reflects the overall abundance of the sources with fluxes constrained by 1) and 2) via the corresponding mean CIB flux. \nCIB integrated/bolometric flux levels depend on three efficiency parameters: 1) the efficiency of collapse of halos suitable for forming luminous sources, or the mass-fraction of the Universe in these halos, denoted f Halo , 2) the formation efficiency of conversion of baryons inside each halo into luminous sources, f ∗ , and 3) the radiation efficiency of converting the rest mass into radiation for the luminous sources inside the collapsed halos, /epsilon1 . \nWe now briefly revisit the arguments in Kashlinsky et al. (2015b, Sec. 2 there) for a general set of efficiency requirements for sources at high z to reproduce the observed CIB anisotropies between 2 and 5 µ m 2 . The integrated CIB fluctuation at 5 ' between 2 and 5 µ m from the AKARI to Spitzer bands is δF 2 -5 µ m (5 ' ) /similarequal 0 . 09 nW m -2 sr -1 ; this arises as excess over that from known galaxies remaining in the data (Helgason et al. 2016). Populations at high z are strongly biased, span a short period of cosmic time, and would be expected to produce CIB with relative fluctuation amplitude of ∆ 5 ' ∼ 10% on arcminute scales, which would then require producing F CIB (2 -5 µ m) = δF 2 -5 µ m (5 ' ) / ∆ 5 ' ∼ 1 nW m -2 sr -1 (Kashlinsky et al. 2007c) in the integrated flux at near-IR wavelengths ∼ (2 -5) µ m. The Lyman cutoff would cut the emissions below the observer wavelengths ∼ (1 + z ) / 10 µ m. \nLet us assume that a fraction f Halo of all matter in the Universe collapses, in halos capable of producing luminous sources, at a given redshift converting on average a fraction f ∗ of the halo baryons into luminous sources. The bolometric diffuse flux produced by these populations, after they have converted their mass-energy into radiation with radiation efficiency /epsilon1 , is F tot /similarequal f Halo f ∗ ( c 4 π /epsilon1ρ bar c 2 ) z -1 eff /similarequal 9 . 1 × 10 5 /epsilon1f Halo f ∗ z -1 eff Ω bar h 2 0 . 0227 nW m -2 sr -1 where z eff ≡ 1 / 〈 (1+ z ) -1 〉 is a suitably averaged effective redshift factor which accounts for the radiation energy density de- \nansion as ∝ (1 + z ) -4 vs. the matter density ∝ (1 + z ) -3 . The overall fraction of Universe's baryons needed to explain the CIB is f Halo f ∗ (see Sec. 2.3.2 in Kashlinsky et al. 2015b). For massive stars, which are fully convective and radiate close to the Eddington limit, /epsilon1 /similarequal 0 . 007 for the H-burning phase of a few Myrs per star. Accreting BHs can reach electromagnetic radiation efficiencies /epsilon1 = 0 . 4 for a maximally rotating Kerr BH. If the integrated CIB fluctuation approximates the bolometric flux produced by these sources, the fraction of baryons that on average go into the sources inside each halo, is: \nf ∗ = 0 . 1 × ( f Halo 0 . 01 ) -1 ( /epsilon1 0 . 01 ) -1 ( z eff 10 ) ( ∆ 5 ' 0 . 1 ) -1 [ F CIB (2 -5 µ m) F tot ] -1 (1) \nThus in order to produce the measure CIB at z > 10 with 'reasonable' formation efficiencies ( f ∗ < 10%) one requires a large fraction of matter in collapsed halos capable of producing luminous sources (see next section). \nHelgason et al. (2016) discuss the requirements of highz sources to produce the observed CIB fluctuations within the conventional, if necessarily simplified, framework of gravitational clustering and spherical collapse of adiabatic ΛCDM fluctuations. They conclude that 1) first galaxies, if extrapolated to z > 8 from known UV luminosity functions would produce much less CIB fluctuation power than observed (cf. Cooray et al. 2012a; Yue et al. 2013b), and 2) at still higher z (first) stars would have to form inside the collapsed halos at substantial formation efficiencies (converting f ∗ /greaterorsimilar 5% of the available baryons in collapsing halos) and be very massive ( ∼ 500 M /circledot ) if they are to explain the observed CIB anisotropies. Kashlinsky et al. (2015a) reproduce the observed Spitzer signal with massive early stars forming at the mean formation efficiency f ∗ /similarequal 4% out to z = 10. \nThe 'high-mean-formation-efficiency' difficulty can ultimately be traced to a relative paucity of highz collapsed halos - with the parameters appropriate for star formation - due to the limited amount of power on the relevant scales set by the adiabatic ΛCDM matter fluctuations, which arose from the period of inflation. The next section discusses how the abundances of the highz collapsed halos are dramatically increased if PBHs constitute the DM, and reduce - by large factors - the efficiencies required to produce the observed CIB anisotropies.", '3. PBHs, small scale mass fluctuation power and first object collapse': "LIGO's GW150914 originated at z = 0 . 09 from the merger of two BHs of essentially identical masses at 36 +5 -4 and 29 ± 4 M /circledot (Abbott et al. 2016b). This mass range lies within the horizon mass-scale at ∼ 0 . 01 -0 . 1 Gev where various mechanisms for generating PBHs in the very early Universe operate, such as discussed by e.g. Jedamzik (1997). Bird et al. (2016) discuss how the observed detection rate, inferred from the so far single published event, can be made consistent with that expected from the PBHs making up the DM such that their comoving mean mass density, assumed constant since their formation until at least their possible later evolution (discussed in Sec. \n4 and references therein), is given by \nn PBH = 1 M PBH Ω CDM 3 H 2 0 8 πG /similarequal 10 9 ( M PBH 30 M /circledot ) -1 ( Ω CDM h 2 0 . 1 ) Mpc -3 (2) \nBelow we will assume, for simplicity, that all PBHs have identical mass. The arguments that follow can be generalized to a PBH mass distribution, such as in e.g. Carr (1975); Choptuik (1993), with M PBH being the effective mass leading to the overall n PBH comoving number density. We note that this mass range is allowed by, although close to, the limits from the MACHO microlensing surveys (Alcock et al. 2001). Ricotti (2007); Ricotti et al. (2008) have argued that, if PBHs are of this mass range, accretion onto them may violate COBE /FIRAS constraints on the CMB black-body energy spectrum, but as Bird et al. (2016) discuss, such arguments are model-dependent and subject to complex physics assumptions. Afshordi et al. (2003) limit M PBH < 4 × 10 4 M /circledot from Lyα forest data. \nFig. 1.Black solid line marks the CMBFAST-computed ΛCDM power spectrum at z = 20 vs the mass contained within the comoving radius 2 π/k for the cosmological parameters adopted here. Black dashes show the P ΛCDM ∝ k -3 extrapolation to scales inaccessible to CMBFAST, but relevant for the first halos collapse. Red horizontal solid line shows the Poissonian power from DM PBHs of M PBH = 30 M /circledot , which clearly dominates the scales relevant for halo collapse at this epoch. \n<!-- image --> \nAs pointed out by Afshordi et al. (2003), the DM from PBHs will contain an extra (isocurvature) component due to Poissonian fluctuations with the power component at the time of the PBH formation being P PBH , initial = n -1 PBH in comoving units. From their formation to today ( z = 0) these isocurvature fluctuations would grow, at wavelengths below the horizon at matter-radiation \nequality z eq , by a scale-independent factor of 3 2 (1 + z eq ), so the extra power component at redshift z is given by (Afshordi et al. 2003): \nP PBH ( z ) = 9 4 (1 + z eq ) 2 n -1 PBH [ g ( z )] -2 /similarequal 2 × 10 -2 ( M PBH 30 M /circledot )( Ω CDM h 2 0 . 13 )( 1 g 2 ( z ) ) Mpc 3 (3) \nwhere g ( z ) is the linear growth factor of fluctuations from z to today, with g (0) = 1. Fig. 1 shows the extra power component for M PBH = 30 M /circledot compared to the ΛCDM power spectrum from the purely adiabatic fluctuation component. The power is plotted vs the mass contained in wavelength 2 π/k which is M ( r ) = 1 . 15 × 10 12 ( r/ 1Mpc) 3 M /circledot for the adopted cosmological parameters. This extra power is ∝ M PBH and for M PBH > 1 M /circledot dominates the small scales relevant for collapse of the first halos at z > 10. This isocurvature power component dominates very small scales and has no impact on the observed CMB anisotropies or baryonic-acoustic-oscillations (Eisenstein & Hu 1999) which appear in CIB fluctuations on arcminute scales and can be probed with Lyman-tomography of CIB from the upcoming Euclid survey (Kashlinsky et al. 2015a). Furthermore, unlike the part of the power from clustering, white noise power contributions to the angular CIB power spectrum are not affected by biasing amplification (Kashlinsky et al. 2004). \nFig. 2.Curves show the rms density contrast over the halo mass for M PBH = 0 (thin), 15 (thick), 30 (thickest) M /circledot at z = 30 (red), 20 (green), 15 (blue), 10 (black). Black horizontal line shows δ col , so halos with density contrast > δ col collapse at that z . Vertical dashes with same color notation mark halo mass where T vir > 10 4 K and vertical dash-dotted lines show the same for T vir > 10 3 K (at z > 15 they are to the left of the box). \n<!-- image --> \nThe net power spectrum would be given by P tot ( k, z ) = P ΛCDM ( k, z ) + P PBH ( z ), which we use to evaluate the rms density contrast at z over a sphere of comoving radius r M containing mass \nM ( r M ) as σ M ( z ) = [ 1 2 π 2 ∫ P tot ( k, z ) W TH ( kr M ) k 2 dk ] 1 / 2 after normalizing to σ 8 over 8 h -1 Mpc at z = 0. ( W TH is the top-hat function). Assuming spherical collapse, masses with density contrast > δ col = 1 . 68 at that epoch will have collapsed by z . In general, collapse and subsequent formation of compact objects is driven by balance between pressure and gravity, which in turn is determined by cooling in the collapsing gas. Two modes of halo collapse are relevant here: if enough H 2 forms the gas will have T /similarequal 10 3 K and in the absence of metals the H cooling will in any event keep the gas at T /similarequal 10 4 K (see review by Bromm & Larson 2004). Fig. 2 shows the resultant rms density fluctuation vs mass at various z relevant here for M PBH = (0 , 15 , 30) M /circledot with the vertical lines demarcating where the halo virial temperatures exceed these limits. The strong increase in the rms density contrast, over that in the absence of the PBHs, at masses of the first halos capable of producing luminous objects, is obvious. This increase will lead to substantially more collapsed halos capable for forming luminous sources at z > 10. \n<!-- image --> \nFig. 3.Fraction of collapsed halos (eq. 4) at T vir > 10 4 K (left) and T vir > 10 3 K (right) vs z for standard ΛCDM power spectrum (red filled circles), DM PBHs with M PBH = 15 M /circledot (open black circles) and M PBH = 30 M /circledot (filled black circles). Thick solid curves mark the overall fraction of baryons (effectively f ∗ f Halo ) needed to produce the observed CIB per eq. 1 with f Halo = 1 with the H-burning radiation efficiency /epsilon1 = 0 . 007 (blue) and BH-type efficiency /epsilon1 = 0 . 2 (black). The mean efficiency of the required conversion of baryons into luminous sources inside each halo would be the ratio of the solid curves to the circles. While f ∗ is high (even higher than, or comparable to, 100% at z > ∼ 20), it remains very modest if the PBHs make up the DM. \n<!-- image --> \nWe use the Press-Schechter formalism (Press & Schechter 1974) to compute the fraction of collapsed halos as the probability of a density field region with virial temperature T vir having overdensity > δ col . For Gaussian-distributed density fluctuation the fraction of the halos that \ncollapsed by redshift z is \nf Halo ( M ( T vir ) , z ) = 1 2 erfc ( δ col √ 2 σ M ( T vir ) ( z ) ) (4) \nFig. 3 shows the fraction in halos that collapsed by z having T vir > 10 4 (right) and 10 3 K for M PBH = (0 , 15 , 30) M /circledot . The increase in f Halo is large enough to produce the required CIB flux with very modest baryon conversion efficiencies (eq. 1) of well below ∼ 1% for the H-burning /epsilon1 = 0 . 007 even by z ∼ 20 in halos with H 2 -cooling. Even in halos with T vir > 10 4 K, the required f ∗ remains at a modest few percent level at z /similarequal 12 -15 for M PBH = 30 M /circledot . If the bulk of the CIB comes from BH accretion, the values of the required f ∗ drop by over an order of magnitude. Thus to account for the observed near-IR CIB fluctuation signal with highz emissions, very few baryons would need to be converted into luminous sources inside first collapsed halos at z > 10 -15 if the DM is made of PBHs such as discovered by LIGO.", '4. Discussion': "If PBHs make up DM, luminous sources within the much more abundant early collapsed halos would reproduce the observed Spitzer/Akari source-subtracted CIB fluctuations with modest formation efficiency requirements. This can be demonstrated by taking population models from Helgason et al. (2016) and rescaling them by the collapse-efficiency ratio from Fig. 3. Thus Fig. 2, upper right from Kashlinsky et al. (2015a) would now reproduce the observed CIB signal with only f ∗ < 0 . 5% forming out to z > ∼ 15 (instead of 4% with formation continuing to z /similarequal 10) and the lines in Fig. 5 of Helgason et al. (2016) need to be rescaled down by the corresponding factors. Additionally the measured CIB-CXB coherence (Cappelluti et al. 2013) would require that at least > ∼ (10-15)% of the luminous CIB-producing sources are accreting BHs, broadly consistent with this discussion. \nWe now outline briefly the possible modifications in the early collapse and source formation that the PBHs may require. Two temperature regimes are relevant for description of emitting sources: 1) minihalos where H 2 formation is efficient evolve at T < ∼ 10 3 K and the gas converges toward density of n gas ∼ 10 4 cm -3 (Bromm & Larson 2004, and refs therein), 2) in the absence of H 2 , the metal-free gas will be able to cool to 10 4 K and collapse in halos with larger virial temperature will proceed isothermally. Feedback effects from first sources would affect H 2 formation via a resulting LymanWerner (LW) radiation at [11.2-13.6]eV (see review by Bromm 2013). Gas collapse/evolution in the PBH minihalos may affect the subsequent emitting source formation inside them. \nPBHs will accrete the minihalo gas resulting in both the additional source of emission from PBH accretion as well as the LW radiation feedback. The gas at sound speed c s within the halo of velocity dispersion v d will be accreted within the typical radius r acc = GM PBH /u 2 with u 2 = v 2 d + c 2 s . The total accretion mass will be M acc = 2( n gas / 10 4 cm -3 )( M PBH / 30 M /circledot ) 3 ( u/ 1km sec -1 ) -6 M /circledot . For typical parameters this may be a non-negligible fraction of the minihalo baryons at ∼ M acc /M PBH × \nΩ CDM / Ω bar ∝ M 2 PBH u -6 up to a few percent, but will not increase the PBH mass dramatically. (Note the sensitive dependence on u , so M acc is rapidly decreased when u /greatermuch 1km/sec). The spectrum of the resultant emission may be modeled after Yue et al. (2013a): 1) the multicolor black-body from different parts of the accretion disc with temperatures up to T max /similarequal 0 . 4( M PBH / 30 M /circledot ) -1 / 4 keV shifted mainly into observer's near-IR, after reprocessing by the surrounding medium, and 2) hot corona and reflection emissions, which leave their mark in the observer soft X-rays. The coherence between the near-IR and X-ray emissions would be strong for the PBHs because of the larger value of T max than for DCBHs. This mode of evolution, inevitable if PBHs make up DM, may influence adjacent star formation and DCBH collapse and evolution as discussed e.g. in Bromm & Loeb (2003); Agarwal et al. (2012); Yue et al. (2014). \nThe PBHs in minihalos will evolve via stellar dynamical effects similar to that discussed in Kashlinsky & Rees (1983) and by loss of energy to GW emissions (Bird et al. 2016). Stellar evaporation will lead to a core-halo structure with the isothermal core of radius r c and N PBH PBHs evolving on Gyr-timescales t evap ∼ N PBH / ln N PBH × r c /v d , at constant binding energy, or v d ∝ N -1 / 2 PBH , because evaporating PBHs carry zero energy. At the same time, a fraction of PBHs will become binary when GW emission exceeds their kinetic energy ( ∼ v 2 d ); the cross-section for this process being σ GW /similarequal 10 -8 ( M PBH / 30 M /circledot ) 2 ( v d / 1km sec -1 ) -18 / 7 pc 2 (Bird et al. 2016). The fraction of PBHs that will form binaries before evaporation is then \nf PBH , binary ∼ N 2 PBH ln N PBH 10 -8 pc 2 r 2 c ( M PBH 30 M /circledot ) -2 ( v d 1km sec -1 ) -18 / 7 (5) \nInstead of evaporating the resultant binaries will spiral in to the center due to dynamical friction possibly forming a central large BH contributing to the massive BH formation in early Universe. \nFinally, we note further constraints from reionization by both first stars and BHs as discussed in this context in Atrio-Barandela & Kashlinsky (2014); Helgason et al. (2016). While reionization would be complicated in the presence of X-rays (Ricotti & Ostriker 2004; Ricotti et al. 2005), the Thomson optical depth of τ < ∼ 0 . 1 may imply that the ionizing photons at rest < 0 . 0912 µ m are mostly absorbed in their paternal minihalos, although Atrio-Barandela & Kashlinsky (2014) recover τ < ∼ 0 . 05 from hot gaseous bubbles reionized by first stars producing the observed CIB levels and forming to z = 9. The situation, while important, is clearly model-dependent: emissions probed at observer λ > ∼ 2 µ m, where current CIB data appear established, would translate directly into ionizing, Lyman-continuum, photons only at 1 + z ion ≥ 22( λ 2 µ m ); at these epochs the baryon density is high and harder to reionize and only a small fraction of the CIB is expected to be produced. A possibility, discussed in Yue et al. (2013a) for the DCBH model, whereby the gaseous collapsed halos are Compton thick so the ionizing photons are absorbed and reprocessed into a two-photon continuum, may also appear relevant here. \nThis work was supported by NASA/12-EUCLID11-0003 'LIBRAE: Looking at Infrared Background Radiation Anisotropies with Euclid' project (http://librae.ssaihq.com). I thank my LIBRAE colleagues for comments on the draft of this paper.", 'REFERENCES': "Abbott, B. P., Abbott, R., Abbott, T. 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2021PhRvD.103i5019B
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Black hole superradiance of self-interacting scalar fields
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2021-01-01
| 47 | 0.46 | 163 |
['-', '-', '-', '-']
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[]
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Black hole superradiance is a powerful probe of light, weakly coupled hidden sector particles. Many candidate particles, such as axions, generically have self-interactions that can influence the evolution of the superradiant instability. As pointed out in [A. Gruzinov, arXiv:1604.06422.] in the context of a toy model, much of the existing literature on spin-0 superradiance does not take into account the most important self-interaction-induced processes. These processes lead to energy exchange between quasi-bound levels and particle emission to infinity; for large self-couplings, superradiant growth is saturated at a quasi-equilibrium configuration of reduced level occupation numbers. In this paper, we perform a detailed analysis of the rich dynamics of spin-0 superradiance with self-interactions, and the resulting observational signatures. We focus on quartic self-interactions, which dominate the evolution for most models of interest. We explore multiple distinct regimes of parameter space introduced by a nonzero self-interaction, including the simultaneous population of two or more bound levels; at large coupling, we confirm the basic picture of quasiequilibrium saturation and provide evidence that the "bosenova" collapse does not occur in most of the astrophysical parameter space. Compared to gravitational superradiance, we find that gravitational wave "annihilation" signals and black hole spin-down are parametrically suppressed with increasing interactions, while new gravitational wave "transition" signals can take place for moderate interactions. The novel phenomenon of scalar wave emission is less suppressed at large couplings, and if the particle has Standard Model interactions, then coherent, monochromatic axion wave signals from black hole superradiance may be detectable in proposed axion dark matter experiments.
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https://arxiv.org/pdf/2011.11646.pdf
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{'Black hole superradiance of self-interacting scalar fields': "Masha Baryakhtar ∗ \nCenter for Cosmology and Particle Physics, Department of Physics, New York University, New York, NY 10003, USA \nMarios Galanis, † Robert Lasenby, ‡ and Olivier Simon § Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, USA (Dated: February 18, 2021) \nBlack hole superradiance is a powerful probe of light, weakly-coupled hidden sector particles. Many candidate particles, such as axions, generically have self-interactions that can influence the evolution of the superradiant instability. As pointed out in [1] in the context of a toy model, much of the existing literature on spin-0 superradiance does not take into account the most important self-interaction-induced processes. These processes lead to energy exchange between quasi-bound levels and particle emission to infinity; for large self-couplings, superradiant growth is saturated at a quasi-equilibrium configuration of reduced level occupation numbers. In this paper, we perform a detailed analysis of the rich dynamics of spin-0 superradiance with self-interactions, and the resulting observational signatures. We focus on quartic self-interactions, which dominate the evolution for most models of interest. We explore multiple distinct regimes of parameter space introduced by a non-zero self-interaction, including the simultaneous population of two or more bound levels; at large coupling, we confirm the basic picture of quasi-equilibrium saturation and provide evidence that the 'bosenova' collapse does not occur in most of the astrophysical parameter space. Compared to gravitational superradiance, we find that gravitational wave 'annihilation' signals and black hole spin-down are parametrically suppressed with increasing interactions, while new gravitational wave 'transition' signals can take place for moderate interactions. The novel phenomenon of scalar wave emission is less suppressed at large couplings, and if the particle has Standard Model interactions, then coherent, monochromatic axion wave signals from black hole superradiance may be detectable in proposed axion dark matter experiments. \n| CONTENTS | B. Perturbative calculations of frequency shifts and rates | 43 |\n|-----------------------------------|--------------------------------------------------------------|------|\n| I. Introduction | 1 C. Mixing beyond 211 and 322 | 47 |\n| II. Spin-0 superradiance | 3 D. Equilibrium ratio for moderate | |\n| III. Quartic self-interactions | 4 self-interactions | 49 |\n| IV. Perturbative evolution | 10 E. Boundary of the regime of early equilibrium | 50 |\n| V. Non-perturbative behavior | 23 F. Cloud mass | 51 |\n| VI. Black hole spin-down | 27 G. Self-gravity energy corrections | 52 |\n| VII. Gravitational waves | 29 H. Frequency drifts | 53 |\n| VIII. Axion waves | 36 I. Perturbations from BH companion | 56 |\n| IX. Conclusions | 40 J. Axion wind sensitivity projections | 57 |\n| Acknowledgments | 41 K. Dark matter abundance | 57 |\n| A. Parametric oscillator analysis | 41 References | 58 |", 'I. INTRODUCTION': "As discovered by Penrose [2], it is possible to extract energy and angular momentum from rotating black holes. While the Penrose thought experiments were in terms \nof mechanical scattering, equivalent processes were developed by the Zeldovich group for bosonic waves [3-5]. This phenomenon, termed 'superradiance', is expected to occur in nature and, for certain initial conditions, amplify photon and graviton waves passing near rotating black holes. Moreover, if there exists a new bosonic particle with a small mass, bound states of this particle could be exponentially amplified around astrophysical black holes, forming very high occupation number 'clouds' that could lead to a range of observational signatures. \nBlack hole (BH) superradiance as a probe of new ultralight particles was first proposed in [6], which has given rise to an extensive literature. Superradiance of new particles, including spin-0 [7-16], spin-1 [17-22], and spin2 [23, 24] fields, have been investigated, with observational signatures including black hole spin-down, gravitational wave emission, and modified black hole in-spiral dynamics; see the above for further references and [15] for a review. \nGravitational interactions are all that is necessary for BH superradiance, which makes superradiance a unique window on new particles that are otherwise inaccessible to experimental probes. However, many beyondStandard-Model particle candidates have other interactions. These can include self-interactions, interactions with Standard Model (SM) states, and interactions with other hidden sector states. For some new particles, including the well-motivated QCD axion [25-27], both selfinteractions and interactions with the SM are required by the model. Therefore, it is important to understand the consequences of such interactions for the growth and behavior of superradiant bound states. \nIn this paper, we analyze in detail the consequences of a quartic self-interaction for the superradiance phenomenology of a light scalar around astrophysical black holes. We find that over a large range of parameter space of interest to light axion models, the addition of a quartic coupling leads to rich dynamics in the evolution of the superradiant instability, and new observational consequences. These dynamics include limiting the maximum number of particles in a bound level, populating levels inaccessible through gravitational superradiance alone, saturation to quasi-equilibrium configurations of two or more levels, and emission of non-relativistic and relativistic scalar waves to infinity. As we demonstrate, an effective quartic term is generically the most important effect driving the evolution, for much of the astrophysically relevant parameter space. \nBH superradiance of a self-interacting scalar was first introduced in Ref. [7], which discussed phenomena including relativistic scalar emission, level mixing, and the possibility of a 'bosenova' - a rapid, non-perturbative collapse of the cloud due to attractive self-interactions. The bosenova process was studied numerically in Ref. [28, 29], and these results were used in subsequent phenomenological investigations [30, 31]. However, as we will discuss, these previous analyses did not take into ac- \ncount self-interaction-induced energy transfers between different superradiant levels. This was pointed out (for a toy model) in [1], which showed that these energy transfer processes, along with scalar emission, can result in saturation to a two-level equilibrium configuration before the cloud has had a chance to grow large enough for a bosenova. We provide evidence that during evolution from astrophysical initial conditions, a 'bosenova' does not occur in much of the phenomenologically-relevant parameter space: scalar field values remain small and the cloud size required for collapse is not reached. \nFor small enough self-couplings - including much of the superradiance parameter space for the QCD axion - self-interaction effects are unimportant. Superradiance proceeds as in the purely-gravitational case: a nonrelativistic bound state of scalars is populated by extracting energy and angular momentum from the rotating black hole, and subsequently annihilates to gravitational radiation. \nSlightly larger self-interactions result in nonrelativistic scalar radiation to infinity. This new energy loss mechanism reduces the power emitted over time in gravitational wave 'annihilation' signals. The interactions also populate higher angular momentum levels; the simultaneous occupation of several bound states can give rise to gravitational wave 'transition' signals, in which scalars emit lower frequency gravitational waves by transitioning between two occupied levels. \nLarge enough self-interactions, including those typical of axion dark matter produced through the misalignment mechanism, significantly reduce the occupation number of the cloud. Instead of being limited by angular momentum conservation, superradiant growth is cut off early by self-interactions. The smaller cloud size suppresses the peak gravitational wave signal strains. For even larger self-couplings, the occupation of the cloud reaches quasiequilibrium at parametrically smaller occupation values, as found in [1]. In this regime, the self-interactions parametrically slow the spin-down of the BH compared to the purely-gravitational case. \nThroughout, a new phenomenon of almostmonochromatic, non-relativistic scalar wave emission occurs; for large self-interactions, the signal amplitude is constant on timescales up to the age of the universe. If couplings to Standard Model particles are present in addition to the self-interaction, then this scalar radiation may be detectable in proposed axion dark matter experiments. For a range of models, the self-interaction and SM interactions are controlled by the same scale; consequently, the signal in Earth-based detectors can persist for arbitrarily small occupation numbers, as long as the classical scalar field description holds. \nMany of our analyses in this paper use hydrogenic approximations for bound states around BHs. Consequently, they are valid for scalar Compton wavelengths bigger than a few times the black hole light-crossing time. Understanding the behavior of more massive scalars \nwould require numerical techniques. Since some of the most dramatic superradiance signatures may occur for slightly heavier scalars, further investigations of this kind are strongly motivated. \nWe review purely gravitational superradiance of scalar (spin-0) fields in Sec. II, and discuss the new processes introduced by quartic (and cubic) interactions in Sec. III. In Sec. IV, we explore in detail the evolution of the superradiant cloud in the presence of quartic self-interactions, which lead to several distinct regions in mass-coupling parameter space. In Sec. V, we discuss the maximum amplitude reached by the axion field, and whether this is large enough to cause non-perturbative behavior such as a 'bosenova'. We study the observable signatures of axion superradiance in the presence of self-interactions: spin down of astrophysical black holes (Sec. VI), gravitational wave annihilations and transitions (Sec. VII), and axion waves (Sec. VIII). We provide more detailed calculations related to both self-interactions and gravitational superradiance in App. A -K. We conclude and comment on directions for future investigations in Sec. IX.", 'II. SPIN-0 SUPERRADIANCE': "In this section, we give a brief review of BH superradiance for a scalar with purely gravitational interactions. There is a broad literature on this topic; for a review, see [15]. We take our signature to be -+++, and assume natural units with c = glyph[planckover2pi1] = 1 unless otherwise indicated. We use the convention M pl ≡ 1 / √ G throughout. \nIn the Kerr background, the Killing vector tangent to the horizon, in static (Boyer-Lindquist) coordinates, is ) \nξ = ∂ t +Ω H ∂ φ . Here, Ω H = 1 2 r g a ∗ 1+ √ 1 -a 2 ∗ ) is the angu- \n( \nlar velocity of the horizon and a ∗ = J/GM 2 is the dimensionless spin of the BH, where J is the BH's angular momentum, M is its mass, and r g ≡ GM . Consequently, a wave with frequency ω , and angular momentum m about the BH spin axis, has energy flux ∝ ω ( ω -m Ω H ) across the horizon, relative to distant observers (the energy flux is necessarily ingoing for local observers near the horizon). For ω < m Ω H , there is energy and angular momentum extraction from the BH, as measured at infinity. \nMassive bosonic fields have quasi-bound states around a BH. In a Schwarzschild background, all of these states are unstable to decay. However, in a Kerr background, states with ω < m Ω H are unstable to growth [9, 11, 32, 33]. 1 Exponential growth of these superradiant states, starting either from a pre-existing astrophysical population in the field, or from quantum fluctuations, will occur given enough time. If we start from the vacuum state, then ignoring the BH interior gives effectively \nnon-unitary evolution of the field outside (due to the absorbing boundary conditions at the horizon), producing a mixed state. Interactions with external systems will generally decohere this into an almost-coherent state, with well-defined phase and amplitude. This process is analogous to the growth of a large-occupation-number laser field from quantum fluctuations [34]. \nThe energy flux across the horizon, for a scalar field ϕ , is ˙ E ∞ ∼ A H | ϕ H | 2 ω ( ω -m Ω H ), where | ϕ H | is the amplitude of the field at the horizon (in in-going coordinates, for which ϕ is smooth at the horizon), and A H is the area of the BH horizon. This flux determines the growth rate of a quasi-bound state. For a scalar of mass µ glyph[lessmuch] r -1 g , the lowest energy states are analogous to hydrogenic bound states, since the effect of the BH at large radii is that of a point source with a 1 /r potential. The hydrogenic level with principal quantum number n , total angular momentum l , and azimuthal angular momentum m (around the BH spin axis) has frequency ω = ω r + iω i , where \nω r glyph[similarequal] µ ( 1 -α 2 2 n 2 + O ( α 4 ) ) (1) \nwith α ≡ GMµ acting as the equivalent of the finestructure constant [13, 22]. The imaginary part of the frequency is \nω i ∝ α 4 l +5 ( m Ω H -ω r )(1 + O ( α )) (2) \nStrictly speaking, for m glyph[negationslash] = 0, the leadingα form of this expression is simply α 4 l +5 m Ω H . However, if m Ω H is also small relative to r -1 g , then the expression in Eq. (2) is appropriate (and more generally, changes sign at the correct ω r ). The 211 ( n = 2 , l = 1 , m = 1) level, which has the fastest growth rate at small α , has ω i = a ∗ 48 α 8 µ at leading order in α . The 'superradiance rate', which is usually defined as the growth rate of the occupation number, is Γ SR ≡ 2 ω i . The α 4 l +5 scaling for the growth rate corresponds to the field amplitude at the BH horizon - for higherl modes, the amplitude is suppressed by the angular momentum barrier, leading to exponentially smaller growth rates for higher l modes [7, 9, 10, 22]. \nWhile the expansions above were phrased in terms of α being small, it is actually the case that α/l is a good expansion parameter. Whenever a level is superradiant, we must have α < m/ 2, so α/l < 1 / 2, and the hydrogenic approximation can be used. \nIf the Compton wavelength of the particle is very large, i.e. α glyph[lessmuch] 1, then all of the superradiance rates are suppressed by a high power of α , Γ ∝ α 4 l +4 µ , so are very small. Conversely, if the Compton wavelength of the particle is significantly smaller than the size of the BH, i.e. α glyph[greatermuch] 1, then only modes with m glyph[greatermuch] 1 can be superradiant; however, these have exponentially suppressed growth rates. Consequently, for observationally-relevant superradiance rates, the Compton wavelength of the particle should approximately match the size of the BH. For stellar-mass black holes, M BH ∼ 10 M glyph[circledot] , this corresponds to µ ∼ 10 -13 -10 -11 eV. While the superradiant growth \nrates around such BHs are rather slow on particle physics scales - with e -folding times a few minutes or longer they can still be much faster than other astrophysical processes and timescales, allowing superradiance to occur in realistic astrophysical environments. \nOnce a Kerr BH is 'born' , e.g. in a binary merger or a supernova, the superradiant bound states start growing in amplitude. The fastest-growing level, which usually has the minimum m satisfying the superradiance condition (except close to the ω r = m Ω H threshold), is the first to extract a significant amount of angular momentum from the BH, spinning it down to Ω H glyph[similarequal] ω/m . 2 For modes with the same m , the most tightly bound mode is often (for small m ) the one with the largest growth rate, since it has larger amplitude at the horizon. Consequently, if ω = m Ω H for that mode, then ω > m Ω H for the other modes, and they are not growing (this is not always true for m ≥ 3; see Sec. VII). \nSince the angular momentum of an astrophysical BH is very large, \nJ = a ∗ GM 2 = a ∗ M 2 M pl 2 glyph[similarequal] 10 78 a ∗ ( M 10 M glyph[circledot] ) 2 , (3) \nit takes ∼ log( J/m ) ∼ 180 e-folds of superradiant growth to cause O (1) BH spin-down. Correspondingly, the fullygrown superradiant cloud has an extremely high occupation number ∼ O (1) J . This corresponds to an energy density which is significantly higher than astrophysical DM densities (assuming that DM is not in extremely dense clumps), App. K. Consequently, the presence or absence of an astrophysical scalar field abundance makes little difference to its superradiant growth. \nThe oscillating scalar field sources gravitational wave (GW) radiation, at a frequency glyph[similarequal] 2 µ - on a particle level, this corresponds to scalars annihilating to gravitons in the black hole background. The emitted power scales as P ∝ GN 2 µ 4 α 16+4 l , where N is the occupation number of the mode [12-14]. The smallness of G , and the high power of α , mean that this process is slow; in particular, it is always too slow to disrupt the initial superradiant growth of the level [12]. \nThe superradiant growth of higher l levels will also take place. Once lowerl modes have grown to saturation, higherl modes can still be superradiant, but their growth rate is slower, so there is a parametric separation between the growth times of successive levels. The annihilation process generally depletes the majority of the scalar cloud before the next level grows. Once the next level significantly spins down the BH, the first mode now has ω > m Ω H , so is decaying with a rate comparable \nto its initial growth rate, and its remaining density falls back into the BH. Over sufficiently long times, a similar process will repeat for the next level. \nThere are a number of observational signatures of purely gravitational scalar superradiance. The first is a lack of old, fast-spinning BHs, at masses for which the scalar would have spun them down in the time available. There have been ∼ 10 measurements of stellar-mass BH spins in X-ray binary systems [35]; for high-spin BHs, these measurements can be accurate to a few percent, and have been used to set constraints the mass of weaklyinteracting scalars [13]. LIGO observations of binary BH mergers also enable spin measurements of the pre-merger BHs [16, 36]. While most of these measurements are currently too imprecise to provide evidence for existence of a scalar [16, 36, 37], initial bounds are already possible [37] (see section VI for a more detailed discussion). \nAnother possibility is the observation of gravitational radiation from the scalar cloud. For stellar-mass black holes, this radiation could potentially be observed at LIGO [13, 16, 38-40]; for heavier BHs, lower-frequency observatories such as LISA or atom interferometers [41] could have sensitivity [13, 38, 39]. The presence of a scalar cloud during a binary merger could also change inspiral dynamics, yielding further gravitational wave signatures [42-45]. While LIGO only observes the last few periods of BBH mergers, making such observations difficult, lower-frequency detectors will observe many more cycles, which will likely improve their chances of observing such effects.", 'III. QUARTIC SELF-INTERACTIONS': "For a spin-0 particle, the simplest non-gravitational interaction is a quartic self-interaction. This is generic in the sense that, if we expand a potential about a symmetric minimum, then the quartic is the most important interaction term for small amplitudes. \nMore specifically, a naturally small mass for a scalar field, as required for superradiance around astrophysical black holes, can be achieved through the breaking of a shift symmetry at some high energy scale f a . A potential of the form V ( ϕ ) = Λ 4 g ( ϕ/f a ) can be generated from non-perturbative physics, so that Λ glyph[lessmuch] f a . For the case of a generic potential g , expanding around the minimum of the potential gives a mass scale µ 2 = g '' Λ 4 /f 2 a and a self-interaction term of order λ = g (4) Λ 4 /f 4 a . \nA well-known example is the QCD axion; given a coupling L ⊃ ϕ f a g 2 s 32 π 2 G a µν ˜ G a,µν of the axion ϕ to the QCD pseudoscalar field strength, it acquires a potential of the form [46] \nV ( ϕ ) glyph[similarequal] -m 2 π f 2 π √ 1 -4 m u m d ( m u + m d ) 2 sin 2 ( ϕ/ (2 f a )) . (4) \nresulting in a mass µ glyph[similarequal] 6 × 10 -12 eV 10 18 GeV f a , and quartic \nself-interaction [46], \nλ glyph[similarequal] 0 . 3 µ 2 /f 2 a glyph[similarequal] 10 -80 ( µ 10 -12 eV ) 4 (5) \nFor more general axion-like particles, the natural parametric value of the quartic coupling is \nλ ∼ µ 2 f 2 a glyph[similarequal] 10 -74 ( µ 10 -12 eV ) 2 ( 10 16 GeV f a ) 2 , (6) \nwhere we chose the nominal value of µ to be in the range of interest for stellar-mass BHs, and f a to be around the Grand Unification (GUT) scale, for illustration. For example, a motivated target model is an axion-like particle which makes up O (1) of the dark matter abundance. If it is produced in the early universe by the misalignment mechanism, and starts out with a field value that is ∼ O (1) f a , then the scale for which we obtain the correct DM abundance is f a glyph[similarequal] 3 × 10 14 GeV(10 -12 eV /µ ) 1 / 4 (assuming a time-independent potential, unlike the QCD axion case). This gives a typical quartic coupling of \nλ ∼ 10 -71 ( µ 10 -12 eV ) 5 / 2 . (7) \nWe will see that even such tiny self-coupling values can have important consequences for the dynamics and phenomenology of spin-0 superradiance. \nThe Lagrangian for a scalar field ϕ with a quartic coupling λ in a fixed background spacetime is given by \nL = -1 2 ( D µ ϕ )( D µ ϕ ) -1 2 µ 2 ϕ 2 + 1 4! λϕ 4 , (8) \nwhere D µ is the covariant derivative and µ is the mass of ϕ . This gives the equation of motion \n( D 2 -µ 2 ) ϕ = -λ 6 ϕ 3 . (9) \nThe quartic interaction strength λ can have either sign; λ > 0 corresponds to an attractive self-interaction, as is the case for axion-like-particles, while λ < 0 is repulsive. For future convenience, we also define an energy scale f such that the quartic λ ≡ µ 2 /f 2 ; for an axion-like particle, we expect f ∼ f a , where f a is the symmetrybreaking scale. \nThe states that dominate the evolution of superradiance are generally non-relativistic, hydrogen-like wavefunctions; these have the fastest growth rates and so obtain the largest amplitudes. Consequently, it is helpful to perform a non-relativistic reduction, writing \nϕ = 1 √ 2 µ ( ψe -iµt +c . c) . (10) \nHere, the 'wavefunction' ψ is a complex scalar field, with ∫ dV | ψ | 2 glyph[similarequal] N the occupation number. The equation of motion is \n( D 2 -µ 2 ) ψe -iµt +c . c . = -λ 12 µ ( ψ 3 e -3 iµt +3 ψ 2 ψ ∗ e -iµt ) +c . c . (11) \nIf ψ changes slowly with time, compared to µ -1 , then we can ignore the ∂ 2 t ψ terms, and extract the e -iµt part of the EoM to obtain the Gross-Pitaevskii equation [7], \n( i∂ t + ∇ 2 2 µ + α r ) ψ glyph[similarequal] -3 24 µ 2 λψ 2 ψ ∗ . (12) \nThe ψ 3 e -3 iµt term in Eq. (11) leads to additional subdominant processes, such as the emission of relativistic ϕ waves, that are not captured by Eq. (12) (see Sec. III A and App. B4). \nAs a visual aid for understanding the λ -induced interactions, we can use a diagrammatic notation for the terms of \nλ 4! ϕ 4 = λ 96 µ 2 ( ψe -iµt + ψ ∗ e iµt ) 4 (13) \nin close analogy to Feynman diagrams. If we expand ψ = ∑ α i ψ i in some basis { ψ i } , then legs on the lefthand-side of the diagram will correspond to ψ i terms in Eq. (13), while legs on the right-hand-side will correspond to ψ ∗ i terms. For example, relativistic emission sourced by the 211 hydrogenic level corresponds to the diagram \n211 211 211 l = 3 , m = 3 \nin the sense that the relevant terms in the equation of motion are obtained from terms involving ψ 3 211 in the Lagrangian, which source a l, m = 3 , 3 relativistic mode. We will make use of these diagrams throughout this section. \nThe (typically tiny) values of λ introduced in Eq. (6) have very little effect on processes involving only a few ϕ quanta. In particular, if we start in a vacuum (or nearvacuum) state, the first process of interest is the superradiant growth of the most unstable hydrogenic levels, exactly as in the purely-gravitational case. However, since the occupation number N of a superradiant level can reach exponentially large values (Eq. (3)), the large field amplitude can compensate for a small self-interaction, and the quartic term's effects can qualitatively alter the dynamics of superradiance. We investigate these effects below. \nHigher-dimensional interactions, corresponding to higher powers of the field, will be present in general. However, we will see that, in much of the astrophysicallyrelevant parameter space, the field never reaches large enough amplitudes for them to be important, for natural hierarchies between the mass, quartic, and higher-order terms (see section V A). The case of an additional cubic coupling leads to qualitatively similar dynamics as for the quartic alone, as discussed in section III D. \nIn the presence of a quartic interaction, three types of perturbative processes affect the evolution of the levels (here, perturbative is meant in the sense that dynamics can be treated as involving approximately hydrogenic modes, interacting on timescales long compared to \ntheir oscillation times). These are relativistic emission of axions to infinity (Sec. III A), non-relativistic emission of axions to infinity (Sec. III B), and bound-state interactions leading to energy exchange between levels (Sec. III C). We will see in the following sections that the latter two processes will be most important for determining the dynamics of the scalar cloud.", 'A. Relativistic scalar emission': "One of the simplest kinds of process arising from the equation of motion (Eq. (9)) is the 3 → 1 process in which bound-state particles 'annihilate' into a relativistic ϕ . In terms of the non-relativistic reduction, the relativistic mode ϕ ∞ is sourced by \n( D 2 -µ 2 ) ϕ ∞ glyph[similarequal] -λ/ 6 (2 µ ) 3 / 2 ψ 3 e -3 iµt +c . c . (14) \nThis can be solved via Green's function methods, using the solution of ( D 2 -µ 2 ) ϕ = 0 in the Kerr background. For small α , when the wavelength ∼ µ -1 of the emitted radiation is much larger than the horizon scale r g , we can ignore the near-horizon structure of the Kerr metric, and consider only its 1 /r behavior. These calculations are discussed in more detail in App. B. \nFor radiation sourced by the 211 hydrogenic level, which we write as 211 × 211 × 211 → ∞ , the emitted power to infinity is \nP glyph[similarequal] 1 . 5 × 10 -8 α 17 µ 2 λ 2 N 3 211 , (15) \nat leading order in α . The corresponding diagram is \n<!-- image --> \nIn principle, the emitted mode has ω < m Ω H when the 211 level is superradiant, and so will extract additional energy from the BH. However, like the SR rate of bound states, this horizon flux is suppressed by the small overlap between the BH and the radiation, and is consequently a subleading effect in the smallα limit. \nEq. (15) is ∼ 15 times larger than the estimate in [7]. The latter effectively solved the equation ∂ 2 ϕ ∞ = -λ/ 6 (2 µ ) 3 / 2 ψ 3 e -3 iµt + c . c . ; that is, they approximated the emitted radiation as being massless, and propagating on a flat-space background. \nIf there is some occupation number in states other than 211, then any combination of three initial states can result in relativistic radiation. If the bound states have orbital angular momenta l, l ' , l '' , then the emitted power scales as P ∝ α 11+2( l + l ' + l '' ) µ 2 NN ' N '' , where N,N ' , N '' are the respective occupation numbers. In particular, as we will see below, populations in multiple superradiant \nlevels can lead to forced oscillations in the l = 0 , m = 0 mode. This might lead us to wonder whether the less severe α suppression in the \n<!-- image --> \nprocess, as compared to 211 × 211 × 211 →∞ , can compensate for the smaller amplitude of the 00 mode in comparison to 211. However, for the 211 and 322 occupation numbers attained in the evolution of the cloud (see section IV), the emitted power via 211 × 211 × 211 → ∞ , Eq. (15), is suppressed by fewer powers of α , and numerically always much larger.", 'B. Non-relativistic scalar emission': "Emission to unbound states can also occur in the nonrelativistic regime. Suppose that we have bound oscillations ψ j ( t ) = ψ j e -i ˜ ω j t , where j labels a particular bound state, with frequencies ˜ ω j,j ' ,j '' < 0 (i.e. the physical frequencies are ω = µ + ˜ ω < µ ). If ˜ ω j + ˜ ω j ' -˜ ω j '' > 0, then the ψ j ψ j ' ψ ∗ j '' term in the equation of motion will source unbound, non-relativistic radiation, corresponding to the diagram \n<!-- image --> \nSince the emitted state is also non-relativistic, we can consistently use the Gross-Pitaevskii equation (Eq. (12)). Writing ψ for the radiated wave, we want to solve \n( ˜ ω + ∇ 2 2 µ + α r ) ψ = -3 12 µ 2 λψ j ψ j ' ψ ∗ j '' (16) \n(with the appropriate multiplicity factors). For each of the different spherical harmonic components in the right hand side of Eq. (16), we can write a one-dimensional radial equation for the part of Ψ with the corresponding angular dependence. These radial equations can be nondimensionalised [1], showing that the power emitted in non-relativistic modes is given by P ∝ α 4 λ 2 N j N j ' N j '' µ 2 at leading order in α , where N j , N j ' , N j '' are the occupation numbers of the bound modes. The constant factors can be found by numerically solving the radial equations, as reviewed in App. B 3. \nConsidering an example which will, in many circumstances, be very important for the cloud's evolution, suppose that we have some population in the 211 and 322 modes. Taking ψ j,j ' = ψ 322 and ψ j '' = ψ 211 , we have \nFIG. 1. Processes relevant to the evolution of the 211 and 322 hydrogenic modes. The first row corresponds to the interactions between non-relativistic modes (Sec. III B and III C) and the second corresponds to the emission of relativistic scalar radiation (Sec. III A), both mediated by the quartic self-interaction. The third row corresponds to the emission of gravitational radiation (indicated by wavy legs), also present in gravitational superradiance. \n<!-- image --> \n2˜ ω 322 -˜ ω 211 glyph[similarequal] α 2 µ 72 > 0, so emission to infinity is possible. As reviewed in App. B 3, this emission is dominantly sourced at radii r ∼ r c ≡ r g /α 2 , i.e. where most of the cloud's mass sits. Since the dominant part of the BH potential is ∼ 1 /r at large distances, which is spherically symmetric, both the bound modes and the emitted wave will have have approximately spherical harmonic angular dependence. For this particular case, Y 2 22 Y ∗ 11 = √ 5 42 π 2 Y 33 -√ 5 1848 π 2 Y 53 , so the emitted quanta are in the l = 3 , m = 3 and l = 5 , m = 3 modes. At leading order in α , the total emitted power for the \n322 322 211 (3 , 3) , (5 , 3) (17) \nprocess is \nP glyph[similarequal] 10 -8 α 4 λ 2 µ 2 N 2 322 N 211 (18) \nwith the ( l, m ) = (3 , 3) radiation dominating the emitted power. 3 This is a factor 4 smaller than the rate given in [1], due to the hydrogenic wavefunctions used in the latter having a normalization that is a factor √ 2 too large. The rates for processes involving different bound states are discussed in App. B 3, and tabulated in Table VI. \nAt larger α , deviations from the non-relativistic approximation become more important. However, at small enough α such that 211 is still superradiant, the ψ 211 and ψ 322 wavefunctions are still well-approximated by \nthe hydrogenic form, except near the origin. Since the source term ψ 2 322 ψ ∗ 211 for the non-relativistic radiation is largest at the characteristic radius of the bound states, a ∼ r g /α 2 , where the potential is dominantly ∼ 1 /r , we would expect the corrections to the non-relativistic calculation to be small. This can be confirmed by performing a numerical computation in the Kerr background, the results of which match the leading-order formula for the emitted power (Eq. (18)) at the few percent level. \nAs well as relativistic effects, there will also be higherorder effects of λ ; for example, self-interaction-induced distortions to the bound state wavefunctions, and to the radiated wave. For ϕ/f glyph[lessmuch] 1, these effects will be small. In much of the astrophysically-relevant parameter space, this condition holds, as we discuss in section V.", 'C. Bound state interactions': "If we have bound oscillations ψ j,j ' ,j '' for which ˜ ω = ˜ ω j + ˜ ω j ' -˜ ω j '' < 0, then the oscillation that they source is also bound. For example, the ψ 2 211 ψ ∗ 322 term has frequency 2˜ ω 211 -˜ ω 322 glyph[similarequal] -7 α 2 µ 36 < 0. In general, ˜ ω will not be very close to the frequency of any of the hydrogenic bound levels (with some exceptions that we review below) so the oscillation that they source will be forced. \nDepending on the angular properties of the driving modes, the forced oscillation may gain or lose energy from the BH. If it loses energy to the BH, then for a forcing term ψ j ψ j ' ψ ∗ j '' , this corresponds to energy loss from the ψ j , ψ j ' modes, but energy gain for the ψ j '' mode. The example that will be the most important for us is when ψ j , ψ j ' = ψ 211 , and ψ j '' = ψ 322 : \n<!-- image --> \nThe forced oscillation has m = 0, so loses energy through the BH horizon. Given some amplitude in the 211 and 322 modes, each ∼ µ of energy lost from the forced oscillation into the BH corresponds to ∼ 2 µ loss from the 211 mode, and ∼ µ gain in the 322 mode. The energy loss rate is proportional to the squared amplitude of the forced oscillation, which is ∝ N 2 211 N 322 . Consequently, if we have a large initial occupation number in 211, and a small initial occupation number in 322, then this process will lead to the exponential growth of N 322 , at the expense of 211. \nThis picture makes intuitive sense when the amplitudes of the 'forcing' modes (211 and 322 in the above example) are large. However, if we are interested in e.g. the growth of 322 from quantum fluctuations, we might worry about the validity of treating it as a forcing for the m = 0 oscillation. A more systematic approach (reviewed in App. A) is to assume that we have some largeamplitude ψ c , and treat this as the source for only two of the 'legs', i.e. to solve \n( i∂ t + M ) ψ = -3 λ 24 µ 2 ( ψ 2 c ψ ∗ + | ψ c | 2 ψ ) (20) \n(here, M represents the other terms in the nonrelativistic Hamiltonian, including an absorbing term corresponding to the BH horizon) with ψ c acting as a parametric driving term, rather than a simple forcing. When the amplitude of this driving term is small, its effects can be described as perturbations to the usual modes, 'mixing' them with others. The key point is that, if the ψ 2 c ψ ∗ term induces a mixing with a decaying mode, then this contributes a growing term to the original ψ mode. In our 211 × 211 → 322 × BH example, if we take ψ c = ψ 211 , then this acts as a parametric driving, which mixes 322 with decaying modes such as 100. This results in the same growth rate for 322 as we would calculate from the forced oscillation picture above. Quantitatively, the energy flux into the BH is, at leading order in α , \nP glyph[similarequal] 4 × 10 -7 α 7 λ 2 (1 + √ 1 -a 2 ∗ ) µ 2 N 2 211 N 322 (21) \n√ \nMore generally, for ψ j,j ' ,j '' such that the forced oscillation has a m = 0 component, the energy flux through the BH horizon is P ∝ α 7 λ 2 (1 + 1 -a 2 ∗ ) µ 2 N j N j ' N j '' . \n∗ These calculations are discussed in App. B 2, and rates for different processes are tabulated in Table V. The listed processes all correspond to forced oscillations with m = 0. Forced oscillations with larger | m | have smaller energy fluxes into (or out of) the horizon, corresponding to bound state interaction rates that are suppressed by higher powers of α . \nAt larger α , there will be deviations from the leading power-law behavior of Eq. (21). Since the energy lost through the forced oscillation depends on its value at the horizon, i.e. on the behavior at small distances, we would expect these deviations to be relatively greater than those for non-relativistic radiation in the previous subsection. \nAs we discuss in App. A, the behavior is similar to that of the 100 level's decay rate, with the rate a factor few larger than the leading-order value at α ∼ 0 . 2. While we provide leadingα expressions in the text, the semianalytic and numerical results from App. A are used for our results. \nIf all four legs of the interaction are almost on-shell, then the α scaling of the energy flux can be different from that of Eq. (21). An example, that will be of interest in section IV, is \n211 311 322 m = 0 (22) \nSince ω r = µ (1 -α 2 / (2 n 2 ) + O ( α 4 )), we have ω 211 + ω 311 -ω 322 = ω 200 + O ( α 4 ) (whereas for 211 × 211 → 322 × BH, 2 ω 211 -ω 322 is O ( α 2 ) away from the frequency of any quasi-bound level). Consequently, the 200 forced oscillation dominates the energy flux into the BH, and we obtain \nP glyph[similarequal] 3 × 10 -10 α 3 λ 2 (1 + 1 -a 2 ∗ ) µ 2 N 211 N 311 N 322 (23) \nThis parametrically faster rate means that any 311 occupation can be quickly depleted by this process, as we will see in section IV C 2. \n√", 'D. Cubic couplings': "In the above, we assumed that the self-interactions consist of a quartic λϕ 4 interaction. A generic scalar can also have a cubic term, \nL ⊃ 1 2 µ 2 ϕ 2 + g 3! ϕ 3 + λ 4! ϕ 4 . (24) \nIf we write λ = µ 2 /f 2 , then a natural value for the cubic is g = Cµ 2 /f , C ∼ O (1). For example, if we take a cosine potential and add a slope \nV ( ϕ ) = µ 2 f 2 (1 -cos( ϕ/f ) -Cϕ/f ) , (25) \nthen the expansion of the potential around its minimum is \nV ( ϕ 0 + δϕ ) = µ 2 2 δϕ 2 -C 3! µ 2 f δϕ 3 -1 4! µ 2 f 2 δϕ 4 + . . . (26) \nto leading order in small C and δϕ . \nAt leading order in g , the only relevant process is relativistic 2 → 1 emission, in analogy to the relativistic 3 → 1 emission discussed in section III A. For definiteness, consider again the situation for the level with the fastest superradiant rate, 211. The leading order cubic process is \n<!-- image --> \nwith power: \nP glyph[similarequal] 10 -4 α 12 C 2 ( µ 4 /f 2 ) N 2 211 . (28) \nMore generally, for radiation sourced by quasi-bound levels with orbital angular momentum l and l ' , the emitted power scales as P ∝ α 8+2( l + l ' ) C 2 ( µ 4 /f 2 ) NN ' . Unlike for the case of relativistic 3 → 1 emission via a quartic coupling (section III A), the leadingα contribution can be obtained by treating the radiation as propagating in flat space, i.e. by solving ( ∂ 2 -µ 2 ) ϕ = source. \nSimilarly to the discussion in section III A, we can ask whether the smaller α suppression of the \n∞ \n<!-- image --> \nprocess, sourced by forced oscillations in the l = 0 , m = 0 mode, can compensate for its smaller source amplitude compared to 211 × 211 → ∞ . For the 211 and 322 occupation numbers attained (section IV), the power from the latter process is again parametrically and numerically larger. \nIn the next section, we will show that, at the very least for large parts of parameter space, relativistic processes in general (from cubic or quartic vertices) are less important than quartic self-interactions between nonrelativistic states. \nAs well as these leading-order processes, interactions between non-relativistic modes are generated at order g 2 : \n<!-- image --> \nIn terms of interactions between non-relativistic modes, these are equivalent to a quartic interaction λ eff = 5 3 g 2 µ 2 = 5 C 2 3 µ 2 f , which is always attractive. 4 It should be noted that this is only true for non-relativistic modes; other processes induced at order g 2 , such as 3 → 1 emissions, will not be captured by the same effective quartic. Nevertheless, as we will discuss in section IV, in many circumstances, only processes involving non-relativistic states are important for the evolution of the field around the BH. \nSince the most important behavior can generally be captured by an effective quartic coupling, we will ignore the cubic coupling for most of this paper, setting C = 0. For C glyph[negationslash] = 0, one can use the replacement rule \n1 f 2 → 1 f 2 eff = ( 1 + 5 3 C 2 ) 1 f 2 (29) \nfor processes involving only non-relativistic states.", 'E. Summary': "In 'gravitational' superradiance, there are two generic ways for bound states to gain or lose energy and thus particle number: superradiance itself, in which the black hole acts as an energy and angular momentum source, and gravitational radiation, which carries energy and angular momentum to infinity. We have seen that in the presence of quartic self-interactions, three new classes of processes are introduced: emission of relativistic axion waves to infinity, emission of non-relativistic axion waves to infinity, and excitation of forced oscillations which typically are absorbed back into the black hole. \nA non-zero cubic self-interaction can act as an additional source of relativistic emission, as well as contributing to an effective quartic term. We will see that, unless the cubic coupling is tuned so as to suppress the effective quartic coupling, or the cubic is rather large compared to its natural value ( | C | glyph[greatermuch] 1), relativistic emission generally does not have an important effect on the dynamics. \nThe first investigation of scalar self-interactions in BH superradiance was in Ref. [7], which carried out a very similar analysis to ours; for example, Eq. (50) in Ref. [7] corresponds to our Eq. (20) describing bound-state interactions. However, in considering whether a perturbation grows or shrinks, Ref. [7] focused on the energy flux through the BH horizon, and did not take into account energy transfer, through the parametric forcing term, between bound states. Since the BH absorbs energy in e.g. the 211 × 211 → 322 × BHprocess, the conclusion was that interaction between modes suppresses occupation number growth. This seems to account for the discrepancy between our analysis and the conclusions of Ref. [7]. \nThe processes outlined in this section create new energy loss mechanisms for bound states, thereby typically limiting their occupation numbers below those of gravitational superradiance. They also create the ability to exchange particles efficiently between bound states with different energy and angular momentum, enabling the growth of high angular momentum states on timescales much faster than the growth possible through gravitational superradiance alone. In the following section, we will discuss in detail the new dynamics for a range of self-interaction strengths. \nFinally, similarly to the emission processes discussed above, there will also be effects that are higher order in λ . In particular, if the amplitude of the cloud becomes too large, then the attractive self-interactions will lead to a rapid, non-perturbative collapse, the 'bosenova' [7]. However, we will see that, for most parts of parameter space, the leading order in λ processes that we have described will prevent the field from reaching such large amplitudes. We discuss such non-perturbative behavior in more detail in section V. \nTABLE I. Rates for the most important processes involved in the evolution of the 211 and 322 hydrogenic levels, at leading order in α . The second column shows the rate constants appropriate for occupation numbers N 211 etc, as per equations (30) and (31), while the third column shows the rate constants for normalized occupation numbers ε 211 ≡ N 211 / ( GM 2 BH ) etc, as per Eq. (32). \n| process | Rate constant (occupation numbers N ) | Rate constant (normalized occupation numbers ε ) |\n|-------------------------|----------------------------------------------------------------------------------|--------------------------------------------------------------------------------------------|\n| 211 superradiance | Γ SR 211 glyph[similarequal] 4 × 10 - 2 α 8 ( a ∗ - 2 α (1 + √ 1 - a 2 ∗ ) µ γ | SR 211 = Γ SR 211 |\n| 211 ∞ | Γ GW 211 × 211 glyph[similarequal] 10 - 2 α 12 ( µ M pl ) 2 µ | γ GW 211 × 211 glyph[similarequal] 10 - 2 α 14 µ |\n| 211 211 322 BH | Γ 322 × BH 211 × 211 glyph[similarequal] 4 × 10 - 7 α 7 λ 2 (1 + √ 1 - a 2 ∗ ) µ | γ 322 × BH 211 × 211 glyph[similarequal] 4 × 10 - 7 α 11 ( M pl f ) 4 (1 + √ 1 - a 2 ∗ ) µ |\n| 322 211 | Γ 211 ×∞ 322 × 322 glyph[similarequal] 10 - 8 α 4 λ 2 µ | γ 211 ×∞ 322 × 322 glyph[similarequal] 10 - 8 α 8 ( M pl f ) 4 µ |\n| 322 ∞ 322 superradiance | Γ SR 322 glyph[similarequal] 8 × 10 - 5 α 12 ( a ∗ - α (1 + √ 1 - a 2 ∗ ) | µ γ SR 322 = Γ SR 211 |", 'IV. PERTURBATIVE EVOLUTION': 'In this section, we study the evolution of the cloudBH system, when the new dynamics introduced by selfinteractions can be treated perturbatively. That is, we treat the cloud as consisting of approximately hydrogenic levels, interacting on timescales long compared to their oscillation timescales. Although the processes are individually simple, the number of them involved can make the narrative hard to follow. Accordingly, we have collated some of the most important information into a number of tables and figures. Table I lists the most important processes affecting level evolution, and gives their rates. Fig. 3 is an important guide to how our discussion is structured, showing the four qualitatively different regimes of parameter space that we analyze. Table II gives approximate expressions for the boundaries of these regions, and points to their definitions in the text. Fig. 4 shows examples of the time evolution of the cloud-BH system, drawn from the four different regions. Table III summarizes the level occupation numbers, observational signatures, and characteristic timescales associated with each region.', 'A. Evolution of occupation numbers': "The evolution of the scalar field around the BH is driven by the gravitational processes discussed in Sec. II - superradiant growth or decay, and GW emission and by the interaction-mediated processes discussed in Sec. III. As we have seen, when these processes can be treated perturbatively, they can be viewed as transferring energy to and from the quasi-bound states of the field (which are themselves only slightly perturbed from their hydrogenic forms). Putting everything together, we can write down a set of coupled differential equations, gov- \nerning the evolution of the occupation numbers of the modes. \nSchematically, if we write the occupation number of level j as N j (where we index the different quasi-bound states by a single index j ), then \n˙ N j = Γ SR j N j (30) + ∑ j ' ( -c Γ GW j × j ' +Γ GW j ' → j -Γ GW j → j ' ) N j N j ' + ∑ j ' ,j '' (Γ j × k j ' × j '' -c Γ j '' × k j × j ' -c Γ j ' × k j × j '' -c Γ ∞ j × j ' × j '' ) N j N j ' N j '' \nwhere the c Γ notation encodes the appropriate multiplicity factors, and \n- · Γ SR j is the growth(/decay) rate corresponding to the mode's flux across the BH horizon\n- · Γ GW j × j ' is the annihilation rate of j × j ' to gravitational radiation.\n- · Γ GW j → j ' is the rate of transitions, via gravitationalwave emission, from j ' to j .\n- · Γ j × k j ' × j '' is the rate of the \nj ' j '' j k \nprocess, where the k leg corresponds to nonrelativistic scalar emission, or to bound forced oscillation damped by the BH. For emission to infinity, we will sometimes write Γ j '' ×∞ j × j ' , while for a bound forced oscillation, we will write Γ j '' × BH j × j ' . \n- · Γ ∞ j × j ' × j '' is the rate of the \n<!-- image --> \nrelativistic emission process. Repeated indices will sometimes be abbreviated using an exponential (i.e. Γ ∞ j × j × j = Γ ∞ j 3 ) \nFor example, the evolution of the fastest-growing level is given by \n˙ N 211 = Γ SR 211 N 211 (31) -2Γ GW 211 × 211 N 2 211 -Γ GW 211 → 322 N 211 N 322 + . . . -2Γ 322 × BH 211 × 211 N 2 211 N 322 +Γ 211 ×∞ 322 × 322 N 211 N 2 322 + . . . -3Γ ∞ (211) 3 N 3 211 -2Γ ∞ (211) 2 × 322 N 2 211 N 322 + . . . \nSome of the key rates, at leading order in α , are listed in Table I. \nWhile, as we observed above, λ is often extremely small, the N j can become extremely large. From Eq. (3), the angular momentum of a BH is J = a ∗ GM 2 glyph[similarequal] 10 78 a ∗ ( M 10 M glyph[circledot] ) 2 . To spin it down by O (1), as is necessary to saturate the superradiant instability, we need N j to be of this order. Consequently, it is often more convenient to work in terms of 'normalized' occupation numbers, ε j ≡ N j / ( GM 2 BH ) < 1, and normalized rates γ such that \n˙ ε j = γ SR j ε j (32) + ∑ j ' ( -cγ GW j × j ' + γ GW j ' → j -γ GW j ' → j ) ε j ε j ' + ∑ j ' ,j '' ( γ j × k j ' × j '' -cγ j '' × k j × j ' -cγ j ' × k j × j '' -cγ ∞ j × j ' × j '' ) ε j ε j ' ε j '' \nSimilarly, it is helpful to write λ ≡ µ 2 /f 2 , as motivated around Eq. (6). In terms of these, the scalings with α and f of the different γ are: \n- · For growth (or decay) of a bound oscillation via the BH horizon γ SR j ∝ α 4 l +4\n- · For non-relativistic scalar emissions to infinity, γ j ×∞ j ' × j '' ∝ α 8 ( M pl /f ) 4\n- · For the absorption of energy from a forced bound oscillation with angular momentum l damped by the BH, γ j × BH j ' × j '' ∝ α 11+4 l ( M pl /f ) 4 (except in the case of 'resonant' processes, as discussed in section III C).\n- · For 3-to-1 relativistic scalar emissions to infinity γ ∞ j × j ' × j '' ∝ α 2( l + l ' + l '' )+15\n- · For annihilation to gravitational waves γ GW j × j ' ∝ α 10+2( l + l ' )\n- · For transitions between bounds states with gravitational wave emission, γ GW j → j ' , see section VII B. \nIn addition, a non-zero cubic interaction contributes to the evolution equations (32) as \n˙ ε j = -∑ j ' cγ ∞ j × j ' ε j ε j ' + . . . (33) \nwith rate γ ∞ j j ' ∝ α 2( l + l ' )+10 | C | 2 ( M pl /f ) 2 µ . \nThe rates that determine the evolution in large parts of the parameter space are listed in Table I, at leading order in α . As discussed above, for some of these processes, this approximation can be quite poor at α values of interest, and for the computations involved in producing our plots, we use more accurate numerical or semi-analytic expressions. \n× \nWhen all of the ε j are very small, then only the γ SR j are important, and evolution proceeds as in the purelygravitational case, with the fastest-growing level increasing exponentially in amplitude. Since the ε j for this level will usually dominate exponentially over the other ε j ' , other levels can only be built up (faster than their superradiance rates) through 5 \n<!-- image --> \nwhere the BH leg corresponds to a bound oscillation. If interaction processes are strong enough to significantly affect the evolution, then the j ' for which this growth rate is fastest will be the next level to become important. \nFor small α , the fastest superradiant growth is for the 211 level, and the fastest quartic process, given a 211 amplitude, is \n211 211 322 BH \nas discussed in section III C. It turns out that, similarly to the toy model discussed in [1], there is a large parameter space for which only the (perturbed) 211 and 322 levels are ever significantly populated. This regime will be the main focus of our paper. \nSituations in which 211 is the first superradiant level generally lead to the strongest radiative signals, either in gravitational or scalar waves. However, superradiance \ninto higher levels can be important for other phenomenological signatures, such as BH spin-down. In such circumstances, levels other than 211 and 322 will be important. For example, if 322 is the first level to grow through superradiance, then 544 will generally be the next level to be built up through self-interactions. Though we do not investigate such scenarios in detail in this paper, they represent an important subject for future work.", 'B. Two-level system': 'If the (suitably perturbed) 211 and 322 modes are the only ones with significant occupation numbers, then the relevant processes are illustrated in Fig. 1. Given this multitude of processes, the behavior of the system seems potentially very complicated. However, we will see that, because the relativistic emission rates are suppressed by high powers of α (and the gravitational radiation rates have an additional relative suppression of ( f/M pl ) 4 , which will turn out to be small when selfinteractions are important), only the two non-relativistic processes (along with superradiance) are generally significant. \nAssuming that 211 is the fastest-growing mode at the start of the evolution, these give rise to fairly simple qualitative behavior, for large enough couplings λ . Initially, the 211 mode grows through superradiance. Once its occupation number is large enough, the growth rate of the 322 mode, through the 211 × 211 → 322 × BH process, becomes significant. This stops the growth of the 211 mode. Since 322 is depleted via the 322 × 322 → 211 ×∞ process, but built up via 211 × 211 → 322 × BH (and vice versa for 211), the 211 and 322 modes reach a quasi-equilibrium configuration, in which their occupation numbers are almost constant. This evolution is illustrated schematically in Fig. 2, and is the regime that was studied in the toy model of [1]. \nThe above picture holds for the case of large enough self-couplings; in the opposite limit of very small selfcouplings, the evolution will be almost the same as the purely gravitational case. For intermediate values of λ , there can be more complicated behaviors. In the rest of this section, we will make all of these statements precise, by investigating in detail the evolution of the cloud, for different µ and f . Fig. 3, and Tables II and III, serve as guides to this discussion. Readers more interested in the observational effects of superradiance around astrophysical BHs can skip ahead to sections VI and VII, referring back to this section when necessary.', '1. Evolution equations': "As discussed above, only the processes in Table I are generally important in the evolution of the 211/322 system. We highlight these rates (which are presented outside the parentheses) in the full evolution equations for \nand \n˙ M µ 2 GM 2 glyph[similarequal] -κ SR 211 α 8 ( a ∗ -2 α ˜ r + ) ε 211 (37) -κ SR 322 α 12 ( a ∗ -α ˜ r + ) ε 322 + κ 322 × BH 211 × 211 α 11 ( M pl /f ) 4 ˜ r + ε 2 211 ε 322 . \nA simplifying assumption at small α is to neglect the change in the mass of the black hole; we will often use this approximation in the text. This is equivalent to setting the maximum 211 fractional occupation value attained through purely gravitational evolution, ε max 211 , to | ∆ a ∗ | = a ∗ ( t 0 ) -4 α/ (1 + 4 α 2 ). At larger α , the mass of the BH changes more significantly and ε max 211 > | ∆ a ∗ | . Our expressions can still be used, however, with the correct value of ε max 211 , for which we derive good analytic approximations in App. F. \nthe occupation numbers of the 211 and 322, which are (at leading order in α ) \n˙ ε 211 µ = κ SR 211 α 8 ( a ∗ -2 α ˜ r + ) ε 211 (34) -2 κ 322 × BH 211 × 211 α 11 ( M pl /f ) 4 ˜ r + ε 2 211 ε 322 + κ 211 ×∞ 322 × 322 α 8 ( M pl /f ) 4 ε 2 322 ε 211 -2 κ GW 211 × 211 α 14 ε 2 211 + ( -κ GW 211 × 322 α 16 ε 211 ε 322 + κ GW 322 → 211 α 10 ε 211 ε 322 -3 κ ∞ (211) 3 α 21 ( M pl /f ) 4 ε 3 211 -2 κ ∞ (211) 2 × (322) α 23 ( M pl /f ) 4 ε 2 211 ε 322 -κ ∞ (211) × (322) 2 α 25 ( M pl /f ) 4 ε 211 ε 2 322 ) , \n˙ ε 322 µ = κ SR 322 α 12 ( a ∗ -α ˜ r + ) ε 322 (35) + κ 322 × BH 211 × 211 α 11 ( M pl /f ) 4 ˜ r + ε 2 211 ε 322 -2 κ 211 ×∞ 322 × 322 α 8 ( M pl /f ) 4 ε 2 322 ε 211 + ( -2 κ GW 322 × 322 α 18 ε 2 322 -κ GW 211 × 322 α 16 ε 211 ε 322 -κ GW 322 → 211 α 10 ε 211 ε 322 -3 κ ∞ (322) 3 α 27 ( M pl /f ) 4 ε 3 322 -κ ∞ (211) 2 × (322) α 23 ( M pl /f ) 4 ε 2 211 ε 322 -2 κ ∞ (211) × (322) 2 α 25 ( M pl /f ) 4 ε 211 ε 2 322 ) , \nwhere ˜ r + ≡ r + /r g = 1 + √ 1 -a 2 ∗ , and the κ values correspond to the γ rates, with the leading α , f and a ∗ dependence factored out (e.g. γ 322 × BH 211 × 211 = κ 322 × BH 211 × 211 α 11 ( M pl /f ) 4 ˜ r + µ , etc). We also need to keep track of the BH's mass and spin, for which \n˙ a ∗ µ = -κ SR 211 α 8 ( a ∗ -2 α ˜ r + ) ε 211 (36) -2 κ SR 322 α 12 ( a ∗ -α ˜ r + ) ε 322 , \nBH \n211 \n322 \nFIG. 2. Schematic illustration of the effects of a large quartic self-interaction on the growth of scalar fields around a spinning BH. The left-hand figure shows the energy densities of the 211 (blue) and 322 (red) modes in the ( x, z ) plane, taking the BH spin to be in the z direction. The right-hand panel shows the evolution of the 211 (blue) and 322 (red) occupation numbers with time (where the N axis is taken to be logarithmic). We assume that the initial BH spin is high enough that the first process to occur is superradiant growth of 211. In the absence of self-interactions, this growth would continue until the BH was spun down to the m = 1 threshold (as indicated by the dashed blue line). When sufficiently large self-interactions are present, the 322 mode is built up from the 211 mode, via the non-linear pumping process described in section III C. This stops the growth of 211, and the levels quickly reach a quasi-equilibrium configuration, in which the processes of 211 superradiance, 211 × 211 → 322 × BH and 322 × 322 → 211 ×∞ emission (section III B) keep the 211 and 322 occupation numbers almost constant. \n<!-- image --> \nTABLE II. Approximate expressions for the boundaries between different regions in µ, f parameter space, as diagrammed in the bottom-right panel of Fig. 3. The first column identifies the section in the text discussing the particular parameter space region, while the third column presents the f range (for given µ ) corresponding to that region, along with references to the relevant equations in the text. The expressions given are to leading order in small α , and numerical coefficients are approximate; the reader should refer to the text for more precise expressions. \n| Coupling strength | Fig 3 | Boundary in parameter space |\n|----------------------|---------|----------------------------------------------------------------------------------------------------------------------------------------------------------------------|\n| Small (IV B 2) | A | f > f AB ≈ min [ 3 × 10 16 GeV ( T BH 10 10 yr ) 1 4 ( µ 10 - 13 eV ) 1 4 ( α 0 . 01 ) 11 4 , 8 × 10 18 GeV ( 0 . 01 α ) 3 4 ( a ∗ 0 . 9 ) 1 4 ] (Eqs. (41), (42)) 1 |\n| Moderate (IV B 3) | B | f AB > f > f BC ≈ 2 × 10 16 GeV ( a ∗ ( t 0 ) 0 . 9 ) 4 min [ ( α 0 . 04 ) 3 4 , ( α 0 . 04 ) 3 2 ] (Eqs. (53), (54), (56)) |\n| Large (IV B 4) | C | f BC > f > f CD ≈ 3 × 10 14 GeV ( 10 10 yr T BH ) 1 2 ( 10 - 13 eV µ ) 1 2 ( 0 . 01 α ) 5 2 ( 0 . 9 a ∗ ( t 0 ) ) 3 4 (Eqs. (62)) |\n| No spindown (IV B 5) | D | f CD > f glyph[greatermuch] µ |", '2. Small self-coupling: gravitational superradiance': "In the limit of very small coupling, f →∞ , the system evolves under purely gravitational dynamics, as summarized in section II. As long as the fastest and secondfastest growing superradiant levels have sufficiently different growth rates, the former will grow first, and attain exponentially larger occupation numbers than other modes. For most of this paper, we focus on situations where the initially fastest-growing mode is the 211 level. This grows to maximum size, and spins the BH down to the m = 1 superradiance threshold, in a time \nlog GM 2 Γ SR 211 glyph[similarequal] ( M BH 10 M glyph[circledot] ) × { 9 hour α = 0 . 4 6 × 10 3 yr ( 0 . 05 α ) 9 α glyph[lessorsimilar] 0 . 2 (38) \nfor high spin ( a ∗ = 0 . 99). On a timescale that, for small α , is parametrically larger, the 211 level is depleted through gravitational wave annihilations, with a decay time of \nτ ann ≈ 1 2Γ GW 211 × 211 N 211 , max (39) glyph[similarequal] ( M BH 10 M glyph[circledot] ) × { 4 hour α = 0 . 4 , 3 × 10 9 yr ( 0 . 05 α ) 15 α glyph[lessorsimilar] 0 . 2 . \nOn even longer timescales, the fastest-growing m = 2 level (i.e. 322) spins down the BH via superradiance, \nlog GM 2 Γ SR 322 glyph[similarequal] ( M BH 10 M glyph[circledot] ) × { 4 yr α = 0 . 4 , 10 11 yr ( 0 . 05 α ) 13 α glyph[lessorsimilar] 0 . 5 . (40) \nFIG. 3. Parameter space for superradiance of a scalar with mass µ and quartic coupling λ = µ 2 /f 2 , around a BH with M BH = 10 M glyph[circledot] and a ∗ = 0 . 9 (initially), given a total evolution time of 10 10 yr. Top-left: parameter space in which the 211 level grows to saturation through superradiance. Top-right: parameter space in which the 322 level grows faster due to selfinteractions than it would have through superradiance alone. Bottom-left: parameter space in which the BH is spun down to the threshold of 211 superradiance. For µ glyph[greaterorsimilar] 4 × 10 -12 eV (i.e. past the threshold for 211 superradiance), we show the parameter space region in which 322 superradiance is not cut off by self-interactions, and we can be confident that the BH is spun down to the threshold of 322 superradiance. The gray hatched region corresponds to parameter space in which levels other than 211 and 322 are expected to grow; we have not fully analyzed the behavior in these regimes. The blue dashed line corresponds to the quartic coupling for the QCD axion. The 'ALP DM' band corresponds to the range of quartic couplings that, for an axion with a time-independent cosine potential, allow the observed DM abundance to be produced by the early-universe misalignment mechanism. The darker middle band corresponds to O (1) values of the initial misalignment angle, while the lighter bands above and below correspond to 'tuned' initial values (see Sec. VI A for details). Bottom-right: parameter space regions discussed in the text. (A) corresponds to the 'small self-coupling' regime discussed in section IV B 2, (B) corresponds to the 'moderate self-coupling' regime discussed in section IV B 3, (C) corresponds to the 'large self-coupling' regime discussed in section IV B 4, and (D) corresponds to the 'lack of BH spindown' regime discussed in section IV B 5. The '322 SR' region is where 322 superradiance is not cut off by self-interactions, while the gray parameter space above this is when this does occur, and further analysis would be required. \n<!-- image --> \nBy this point, only a small fraction of the initial 211 occupation generally remains (for α large enough that growth occurs on relevant timescales), so gravitational wave transition signals from 322 → 211 × GW events are small. The upper panels of Fig. 4 illustrate this evolution, for f glyph[similarequal] M pl . For BHs with long enough lifetimes, a \nsimilar story applies to the growth of higherm levels. \nAs we discuss below, the purely gravitational story describes the evolution well if the self-interaction-induced 211 × 211 → 322 × BH process is always slow compared to superradiant growth processes. The parameter space for which this is true is plotted as region (A) in the bottom- \nTABLE III. Summary of important quantities in the parameter space regimes A-D (Fig 3, Table II). The second column lists the ratio of the peak value ε peak 211 attained in the corresponding region to the maximum value attained through gravitational superradiance ε max 211 . The fourth column describes the most important observational signatures of superradiance in each regime. For regions A and B, these are BH spindown (see Sec. VI), the emission of gravitational radiation (see Sec. VII) from 211 × 211 → GW annihilations and from 322 → 211 × GW transitions (only in region B). For regions C and D, gravitational radiation is suppressed, but non-relativistic scalar radiation ('AW', for 'axion waves') from the 322 × 322 → 211 × ∞ process may be detectable, if the scalar field couples to SM states (see Sec. VIII). The right-most column gives approximate expressions for the relevant dynamical timescales, which also correspond to typical signal timescales of GW radiation (for A and B) and scalar radiation (for C and D). The expressions given are to leading order in small α , and numerical coefficients are approximate; the reader should refer to the text for more precise expressions. \n| Coupling strength | ε peak 211 /ε max 211 | η = ε 322 /ε 211 | Signatures | Timescales |\n|-------------------------|-------------------------|-----------------------|--------------------------------------|----------------------------------------------------------------------------|\n| Small (IV B 2), A | 1 | glyph[similarequal] 0 | spindown, GW | τ ann ≈ 10 5 yr ( 0 . 1 α ) 14 ( 10 - 12 eV µ ) (Eq. (39)) |\n| Moderate (IV B 3), B | 1 | 10 - 5 ( α 0 . 01 ) 3 | spindown, GW τ scalar | ≈ 10 - 1 yr ( 0 . 1 α ) 14 ( 10 - 12 eV µ ) ( f 10 17 GeV ) 4 (Eq. (49)) 3 |\n| Large (IV B 4), C | ( f f BC ) 2 | 10 - 5 ( α 0 . 01 ) 3 | slow spindown, AW τ sd ≈ 10 7 yr ( 0 | . 01 α ) 5 ( 10 - 12 eV µ )( 0 . 9 a ∗ ) 2 ( 10 15 GeV f ) 2 (Eq. (60)) |\n| No spindown (IV B 5), D | ( f f BC ) 2 | 10 - 5 ( α 0 . 01 ) 3 | no spindown, AW | τ sd glyph[greaterorsimilar] T BH (Eq. (63)) | \nε eq 211 ≈ 2 √ 3 √ κ ∞ κ SR ( a ∗ -2 α ˜ r + ) α 3 κ BH ˜ r + ( f M pl ) 2 = 2 . 5 × 10 -1 ( 0 . 01 α ) 3 ( a ∗ 0 . 9 ) 1 / 2 ( f 10 15 GeV ) 2 (Eq. (55a)) ; ε eq 322 ≈ √ 1 3 κ SR ( a ∗ -2 α ˜ r + ) κ ∞ ( f M pl ) 2 = 6 . 9 × 10 -6 ( a ∗ 0 . 9 ) 1 / 2 ( f 10 15 GeV ) 2 (Eq. (55b)) \nright panel of Fig. 3.", '3. Moderate self-coupling: early growth of 322 and late equilibrium': "If we decrease f , while holding other parameters fixed, the first significant difference from purely-gravitational evolution that arises is earlier growth of the 322 level. We label this regime, where 211 still grows to saturation, but 322 grows sooner than it would have if λ = 0, the 'moderate self-coupling' regime. The upper-left panel of Fig. 4 illustrates the evolution of the 211 and 322 occupation numbers for an f value in this regime (as well as for a larger f in the small self-coupling regime). \nThe parameter space for moderate self-coupling is plotted as region (B) in the bottom right-hand panel of Fig. 3, and corresponds to the intersection of the shaded regions in the upper two panels. In this subsection, we will focus on the threshold between the small self-coupling and moderate self-coupling regimes, deferring the smallf boundary of the moderate regime (i.e. the point at which 211 no longer grows to saturation) to the next subsection. \nFor the 211 × 211 → 322 × BH process to build up 322 within the lifetime of the BH, we need \nγ 322 × BH 211 × 211 ( ε max 211 ) 2 glyph[greaterorsimilar] log( ε final 322 /ε initial 322 ) T BH glyph[similarequal] log( GM 2 BH ) T BH (41) thresh \nwhere ε max 211 ≈ a ∗ ( t 0 ) -a thresh ∗ ≈ a ∗ ( t 0 ) -4 /α (1 + 4 α 2 ) is the occupation number of the saturated 211 level. Parametrically, if we start from very small fluctuations in the 322 level, and ε final 322 is not exponentially small, then \nε final 322 /ε initial 322 ∼ GM 2 . For this growth to be faster than 322 superradiance, we need γ 322 × BH 211 × 211 ( ε max 211 ) 2 glyph[greaterorsimilar] γ SR 322 . \n× The condition (41) is necessary for early 322 growth to occur, but not sufficient, since annihilations to gravitational waves may deplete 211 before 322 can grow. In order for this not to happen, we need \nγ 322 × BH 211 × 211 ( ε max 211 ) 2 log( ε final 322 /ε initial 322 ) glyph[greaterorsimilar] 2 γ GW 211 × 211 ε max 211 (42) \nReplacing the rates by their smallα expansions, this is equivalent to \nκ BH ˜ r + ( M pl /f ) 4 ε max 211 log ( GM 2 ) glyph[greaterorsimilar] 2 κ GW 211 × 211 α 3 . (43) \nThe combination of the conditions (41) and (42) is responsible for the shape of the (A)-(B) boundary in Fig. 3. At small α , (41) is more constraining, while at larger α , (42) takes over. The parametric form of this threshold value f AB is given in table II. \nEvolution of levels : Unlike in the gravitational scenario, where the growth of 322 via superradiance is accompanied by a rapid drop in 211 occupation, here both levels eventually reach roughly-comparable occupation numbers. Subsequently, the joint cloud is slowly depleted by the combination of non-relativistic scalar emission and damping by the BH. Other processes, including gravitational annihilations and transitions as well as relativistic scalar emission, are small perturbations to this overall evolution. \nAs discussed above, only a few rates drive the dynamics in the regions of parameter space for which selfinteractions modify the purely gravitational scenario. \nFIG. 4. Left panel: fractional occupation numbers of 211 (solid lines) and 322 (dashed lines) levels, and Right panel: BH spin, as a function of time, for a BH of mass 10 M glyph[circledot] and initial spin a ∗ = 0 . 9, given a scalar of mass µ = 1 . 5 × 10 -12 eV. The different colors correspond to the different self-interaction strengths indicated in the right-hand plots (see section IV for explanations of the behaviours at different couplings). \n<!-- image --> \nThese are κ SR 211 , κ 322 × BH 211 × 211 , and κ 211 ×∞ 322 × 322 (and κ SR 322 , in some circumstances). To streamline our notation, we will refer to them as κ SR , κ BH , and κ ∞ respectively. \nIn the regime of moderate self-coupling, the growth of the 211 level occurs as in the purely-gravitational case; both the occupation number and the BH angular momentum change 'suddenly', with almost all of the change happening in the last few e-folds of superradiant growth. This is illustrated in the top panels of Fig. 4. The BH spin decreases to a ∗ ≈ 4 α/ (1 + 4 α 2 ), and ε 211 stays at ≈ ε max 211 for a long time. In the purely gravitational scenario, the cloud would then slowly self-annihilate to gravitational waves until ∼ 200 e-folds of 322 superradiance have passed. Here, however, the quartic process dominates, and the 322 growth rate is higher: \n˙ ε 322 µ ≈ κ BH ˜ r + α 11 ( M pl /f ) 4 ( ε max 211 ) 2 ε 322 . (44) \nEventually, the 322 occupation number becomes large enough that the quartic vertex 322 × 322 → 211 ×∞ becomes important and a quasi-equilibrium is established, \nroughly after time \nt ∗ glyph[similarequal] GM log( GM 2 ) κ BH ˜ r + α 12 ( M pl /f ) 4 ( ε max 211 ) 2 (45) \nhas passed. \nAt this point, superradiance to 211 has effectively shut down, and 322 superradiance is too slow to be significant. Particles are leaving the combined cloud, going back to the BH (via 211 × 211 → 322 × BH) and to infinity (via 322 × 322 → 211 × ∞ ). Gravitational and relativistic scalar processes are suppressed by high powers of α . Accordingly, the coupled dynamics of the two-level system simplifies to \n˙ ε 211 µ ≈ -2 κ BH ˜ r + α 11 ( M pl /f ) 4 ε 2 211 ε 322 + κ ∞ α 8 ( M pl /f ) 4 ε 2 322 ε 211 , (46a) \n˙ ε 322 µ ≈ κ BH ˜ r + α 11 ( M pl /f ) 4 ε 2 211 ε 322 -2 κ ∞ α 8 ( M pl /f ) 4 ε 2 322 ε 211 , (46b) \na ∗ ≈ 4 α 1 + 4 α 2 . (46c) \nSince there are no processes (except for the negligible superradiance of 322) which contribute particles to the cloud, particles are only leaving. Accordingly, the system has no true equilibrium occupations. However, (46) still admits a time-independent equilibrium ratio of occupation numbers, ε 322 /ε 211 = η B to which the system flows, \nη B glyph[similarequal] 1 2 κ BH α 3 ˜ r + κ ∞ glyph[similarequal] 4 × 10 -5 ( α 0 . 01 ) 3 . (47) \nFor the regime of moderate self-coupling, the scalings in (47) are only representative at leading orders in α . A more accurate expression is derived in App. D. \nWhen the equilibrium ratio is obtained at time t ∗ , the occupations evolve as \nwhere \nε 211 ( t ) glyph[similarequal] ε 211 ( t ∗ ) √ 1 + 2 ε 2 211 ( t ∗ )( t -t ∗ ) /τ scalar , (48) \nτ scalar ≡ 4 3 µ κ ∞ ( κ BH ˜ r + ) 2 α 14 ( f M pl ) 4 ≈ 10 -1 yr ( 0 . 1 α ) 14 ( 10 -12 eV µ )( f 10 17 GeV ) 4 , (49) \nand ε 322 ( t ) = ε 211 ( t ) η B . \nThe joint cloud continues to deplete until the occupation of 211 has diminished enough that the superradiance rate of 322 outcompetes the 'stimulated' emission process 322 × 322 → 211 ×∞ , and the cloud starts growing again. A large occupation builds up in 322, causing rapid 211 depletion via 211 × 211 → 322 × BH. Moreover, as superradiance extracts angular momentum from the BH to 322, the BH's spin decreases further, making 211 (and other m = 1 states) damped . This sequence of events is illustrated in the top panels of Fig. 4 (where the green curves correspond to moderate self-coupling, and the blue to small self-coupling). \nIn the λ = 0 case, m = 2 superradiance must proceed from zero-point quantum fluctuations, or from a small pre-existing astrophysical density. Here, superradiance gets to act on the pre-existing occupation ε 322 , since 322 has already been populated by self-interaction-mediated processes. In this way, self-interactions 'assist' superradiance, sometimes leading to more rapid saturation of the m = 2 instability than allowed in the purely gravitational story. The f = 5 × 10 17 GeV curves in the upper panels of Fig. 4 show an example of this, with 322 spindown occurring after only ∼ few × 10 6 yr, compared to almost 10 8 yr in the purely-gravitational case. \nThe above discussion summarizes the evolution of the cloud in the moderate self-coupling regime. Before moving on, we will discuss the effects of processes other than \n211 × 211 → 322 × BH, 322 × 322 → 211 × ∞ , and superradiance, and review why they are (in most cases) subdominant. \nAnnihilations to GWs: An important point is that, to be in the moderate self-coupling regime for astrophysical BH masses, we need f glyph[lessorsimilar] M pl (as illustrated in Fig. 3). This is evident from the form of the threshold f AB given in table II, f AB = min( f 1 , f 2 ). The first term f 1 comes from the condition γ 322 × BH 211 × 211 ( ε max 211 ) 2 glyph[greaterorsimilar] log( GM 2 BH ) T BH ; to make f 1 ≥ M pl , we need to take α glyph[greaterorsimilar] 0 . 07 (for M BH = O (10 M glyph[circledot] )). Such large values of α make the f 2 , coming from the condition that GW annihilations are not too fast (42), much less than M pl . Consequently, gravitational wave emission processes suffer a suppression ∼ ( f/M pl ) 4 , relative to self-interactionmediated quartic processes. This means that, once 322 has reached its equilibrium ratio with 211 (Eq. (47)), even the fastest GW emission process, 211 × 211 → GW, is generally slower than 211 × 211 → 322 × BH and 322 × 322 → 211 ×∞ (at least until the levels have depleted significantly). \nGW transitions: From table IV, gravitational wave transitions 322 → 211+ GW contribute a term ˙ ε 322 glyph[similarequal] -3 × 10 -6 α 10 ε 322 ε 211 µ + . . . to the evolution equations. If we take ε 322 = η B ε 211 (Eq. (47)), this gives \n˙ ε 322 /µ glyph[similarequal] -3 α 13 ε 2 211 +0 . 4 α 14 ε 3 211 ( M pl f ) 4 + . . . (50) \nwhere we have also included the 211 × 211 → 322 × BH term for comparison. While the GW transition term is suppressed by one less power of α , Fig. 3 illustrates that, as α decreases, the maximum f for the moderate selfcoupling regime decreases (from table II, f AB ∝ α 11 / 4 for small α ). Consequently, the relative ( M pl /f ) 4 enhancement of the quartic self-interaction terms always wins out. \nEven though gravitational wave emission no longer dominates the evolution compared to the small selfinteractions regime of gravitational superradiance, GW annihilation signals can still be strong enough for detection in this regime. In addition, the simultaneous occupation of the two levels allows for the possibility of GW signals from transitions. We explore potential signatures in more detail in Sec. VII. \nRelativistic 3 → 1 emission: As discussed in section III A, quartic self-interactions also lead to processes emitting relativistic scalar waves, such as 211 × 211 × 211 →∞ . This contributes \n˙ ε 211 /µ glyph[similarequal] -5 × 10 -9 α 21 ( M pl f ) 4 ε 3 211 + . . . (51) \nBecause of the high power of α this is suppressed by, its effect is small compared to the non-relativistic quartic processes. \nRelativistic cubic emission: In section III D, we discussed how, in addition to a quartic self-interaction, there \nmay also be a cubic interaction term, L ⊃ 1 6 C µ 2 f ϕ 3 , which can lead to relativistic emission processes such as 211 × 211 →∞ . This contributes \n˙ ε 211 /µ glyph[similarequal] -2 × 10 -4 α 14 | C | 2 ( M pl f ) 2 ε 2 211 + . . . (52) \nCompared to the quartic-induced term in Eq. (50), the lower power of M pl /f , and the smaller constant factor, mean that unless | C | glyph[greatermuch] 1, relativistic emission from the cubic coupling will be a subdominant effect.", '4. Large self-coupling: early equilibrium and halted extraction of angular momentum': "If we further decrease f , we reach a point where 322 grows large enough, early enough, that 211 superradiance is disrupted, and 211 does not reach its saturation value. We call this the regime of 'large self-coupling'; it corresponds to regions (C) and (D) in the bottom-right panel of Fig. 3, and to the bottom panels in Fig. 4. \nFor the 211 × 211 → 322 × BH process to disrupt 211 superradiance, we need that 2 γ 322 × BH 211 × 211 ε 211 ε 322 glyph[greaterorsimilar] γ SR 211 before ε 211 has grown to its saturation value. This does not necessarily preclude 211 reaching ε max 211 ( ε 211 can still grow after that point, albeit more slowly than it would have with λ = 0), but it is necessary to have a significant effect. Parametrically, this condition is approximately equivalent to \nγ 322 × BH 211 × 211 ( ε max 211 ) 2 glyph[greaterorsimilar] 2 log GM 2 ) γ SR 211 , (53) \n( \n) where we neglect the dependence of the rates on the BH spin (i.e. set a ∗ ( t ) = a ∗ ( t 0 )). A more precise condition is derived in App. E. \nThe condition (53) can be expressed as a condition on f . 211 superradiance is basically unaffected if f glyph[greaterorsimilar] f thresh , where \nf thresh ≈ M pl ( α 3 2 log( GM 2 ) κ BH ˜ r + ( ε max 211 ) 2 κ SR a ∗ ( t 0 ) ) 1 / 4 ≈ 6 × 10 15 GeV ( α 0 . 01 ) 3 / 4 ( a ∗ ( t 0 ) 0 . 9 ) 1 / 4 . (54) \nThe scalings in (54) are only representative when α glyph[lessmuch] a ∗ ( t 0 ). For larger values of α , rates obtained numerically, and a more precise version of (53) (App. E), can be used. \nAs pointed out in [1], if a ∗ is held fixed, the system admits equilibrium occupations for which ˙ ε 211 = ˙ ε 322 = 0 : \nε eq 211 ( a ∗ ) ≈ 2 √ 3 √ κ ∞ κ SR ( a ∗ -2 α ˜ r + ) α 3 κ BH ˜ r + ( f M pl ) 2 ≡ ( f f eq ) 2 ε max 211 = 2 . 5 × 10 -1 ( 0 . 01 α ) 3 ( a ∗ 0 . 9 ) 1 / 2 ( f 10 15 GeV ) 2 , (55a) \nε eq 322 ( a ∗ ) ≈ √ 1 3 κ SR ( a ∗ -2 α ˜ r + ) κ ∞ ( f M pl ) 2 = 6 . 9 × 10 -6 ( a ∗ 0 . 9 ) 1 / 2 ( f 10 15 GeV ) 2 , (55b) \nwhere \nf eq ≈ M pl ( √ 3 2 α 3 κ BH ˜ r + ε max 211 √ κ SR κ ∞ ( a ∗ -2 α ˜ r + ) ) 1 / 2 ≈ 2 × 10 15 GeV ( α 0 . 01 ) 3 / 2 ( a ∗ ( t 0 ) 0 . 9 ) 1 / 4 . (56) \nNote that the ratio η eq ≡ ε eq 322 /ε eq 211 is \nη eq = γ BH 2 γ ∞ ≈ ( η B ) small α , (57) \naccording to the approximation (47) valid for small α . At larger values of α , η B > η eq . See App. D for more details. \nWe now consider what happens in the physical case, where a ∗ can change. If ε eq 211 is much less than its saturation value, then the timescale to extract an O (1) fraction of the BH's spin is much longer than the characteristic timescale of the processes maintaining the equilibrium. Consequently, we expect the quasi-equilibrium to be maintained to a good approximation, as a ∗ undergoes a slow descent. The equilibrium occupation numbers ε eq 211 ( a ∗ ) and ε eq 322 ( a ∗ ) stay almost constant, with the angular momentum extracted from the BH via 211 superradiance being emitted to infinity via the 322 × 322 → 211 × ∞ process. This is in contrast to the regimes of small and moderate self-interactions, where the angular momentum lost from the BH builds up in the cloud. \nClose to the transition from moderate to large selfinteractions, there is a sliver of parameter space for which the exponential growth of 211 is maintained for some time and O (1) of the maximum spin extraction occurs, before getting cut short by the equilibrium. Deep inside the region of small f , however, the spin of the BH is essentially unchanged at the time the equilibrium is established, and most of the extraction of angular momentum happens adiabatically. \nAlthough (55) is valid at equilibrium, if α is large enough then ε 211 will 'overshoot' its equilibrium value before ε 322 has caught up with it. Before equilibrium, if we neglect the dependence of γ SR 211 on the BH spin, ε 211 ∝ exp( γ SR 211 t ). In App. E, we derive an estimate for the value of the exponent γ SR 211 t at the time when 211 × 211 → 322 × BH is comparable to SR. To a good approximation \nε thresh 211 ≈ √ 2 γ SR 211 log( GM 2 ) γ 322 × BH 211 × 211 ≈ ( f f thresh ) 2 ε max 211 , (58) \nwhere we set a ∗ ( t ) = a ∗ ( t 0 ) in both rates. \nAccordingly, the evolution towards equilibrium can happen in two qualitatively different ways. When α glyph[greaterorsimilar] 0 . 04, f thresh < f eq and ε thresh 211 > ε eq 211 . In this case, the occupation ε 211 overshoots its equilibrium value and subsequently evolves toward it from above. This is illustrated in the bottom-left panel of Fig. 4 (for which α = 0 . 11). Conversely, when α glyph[lessorsimilar] 0 . 04, then ε thresh 211 < ε eq 211 . There is no overshoot, and ε 211 evolves toward its equilibrium occupation from below. \nGiven this, the boundary between the moderate selfcoupling regime, where ε 211 reaches ε max 211 , and large selfcoupling, where it does not, is set by \nf glyph[lessorsimilar] f BC ≡ min[ f thresh , f eq ] . (59) \nTo review, the evolution of the superradiant cloud, in the regime of large self-coupling, occurs in different stages: \n- 1. An initial stage of exponential 211 growth, during which ε 322 is too small to significantly affect the evolution of ε 211 .\n- 2. A 'non-equilibrium' stage in which ε 211 and ε 322 evolve towards their equilibrium values. The timescale to approach the equilibrium values is at most a logarithmic multiple of 1 /γ SR 211 , since the relevant self-interaction processes are at least as fast as γ SR 211 .\n- 3. Once ε 211 and ε 322 are close to their equilibrium values, there is a long period of quasi-adiabatic evolution. The spin-down of the BH due to spin extraction through 211 superradiance, which changes a ∗ on a timescale (˙ a ∗ /a ∗ ) -1 ∼ ( ε max 211 /ε eq 211 ) /γ SR 211 , leads to the slow evolution of the equilibrium occupation numbers.\n- 4. If the BH lifetime is long enough that spin-down to the m = 1 threshold occurs, then similar behavior to the moderate self-coupling regime will result. The 211 and 322 levels will maintain a quasiequilibrium ratio, but with decreasing occupation numbers, as scalars are emitted to infinity. Eventually, the occupation numbers will become small enough that 322 superradiance starts to dominate, at which point the 322 occupation number starts growing again (e.g. the f = 10 15 GeV curves in the bottom-left panel of Fig. 4). \nConsequently, when f is appreciably smaller than f BC , the first and second stages change a ∗ by only a small amount, and the majority of the BH's spin-down to the m = 1 threshold happens during the period of almost adiabatic, quasi-equilibrium evolution. \nWhen the equilibrium occupations (55) are obtained, the angular momentum of the BH decreases according to (36), with ε 211 = ε eq 211 ( a ∗ ) (and we can ignore κ SR 322 ). The \ntimescale for spindown is therefore set by \nτ sd ( a ∗ ) ≈ √ 3 2 α 5 µ κ BH ˜ r + ( M pl /f ) 2 √ κ ∞ ( κ SR ( a ∗ -2 α ˜ r + )) 3 / 2 ≈ 10 7 yr ( 0 . 01 α ) 5 ( 10 -12 eV µ ) × ( 0 . 9 a ∗ ) 3 2 ( 10 15 GeV f ) 2 . (60) \nWhile in (slowly-varying) equilibrium, the cloud emits non-relativistic axion waves through the 322 × 322 → 211 ×∞ process. These could, in the presence of axionSM interactions, be detected by experiments on Earth. Even though the occupation number of the cloud decreases ∝ f 2 for small f , the coupling strength of axionSM interactions will generically scale as ∼ 1 /f . Consequently, the interaction rate of the emitted radiation with a laboratory target can be independent of f in the smallf regime. This in contrast to gravitational wave signals, which are suppressed at small f . We discuss this possibility more fully in section VIII. \nIn the previous subsection on the moderate selfcoupling regime, we discussed how interaction processes, other than non-relativistic quartic interactions and superradiance, are generally subdominant in their effects on the evolution of the cloud. Very similar calculations apply to the large self-coupling regime; the equilibrium ratio of ε 322 /ε 211 is the same, with the difference being that the equilibrium occupation numbers are suppressed, scaling ∝ f 2 . \nThis scaling only makes a difference to comparisons between processes with different multiplicities. For annihilation to GWs, the ( f/M pl ) 2 scaling of the occupation number is not enough to make up for the ( M pl /f ) 4 relative enhancement of the quartic interaction rates, so GW annihilation processes are even less important than they are in the moderate self-coupling regime. \nFor relativistic cubic emissions, the fastest of which is 211 × 211 →∞ , we can compare the contribution to the evolution rate to that from 211 × 211 → 322 × BH: \n˙ ε 211 /µ glyph[similarequal] -2 × 10 -4 α 14 | C | 2 ( M pl f ) 2 ε 2 211 -8 × 10 -7 α 11 ( M pl f ) 4 ε 2 211 ε 322 glyph[similarequal] ( -2 × 10 -4 α 14 | C | 2 -10 -3 α 11 ) × ( M pl f ) 2 ( ε eq 211 ) 2 (61) \nwhere the second equality applies for the equilibrium occupation numbers (55). Consequently, if | C | glyph[lessorsimilar] 16(0 . 2 /α ) 3 / 2 , then the effect of the cubic emission term is small compared to that of the non-relativistic quartic processes. \nFor α glyph[greaterorsimilar] 0 . 04, the equilibrium values of ε 211 and ε 322 are smaller than the 'overshoot' values at which self- \neractions first affect the evolution of 211. Consequently, if the relativistic cubic processes are unimportant in equilibrium, then they are always less important than the quartic 211 × 211 → 322 × BH process, whenever the latter has a significant effect on 211 evolution. \nFor smaller α , the 2 → 1 process will be relatively most important around the initial time at which 211 growth is slowed down, since the equilibrium occupation numbers are approached from below. Still, even without calculating the thresholds carefully, we can see that as long as | C | glyph[lessorsimilar] 16(0 . 2 / 0 . 04) 3 / 2 glyph[similarequal] 180, cubic emission will be insignificant in that regime (since decreasing α decreases the relative importance of cubic emission). Overall, we can see that, unless | C | glyph[greatermuch] 1, relativistic emission through the cubic coupling should always be a subdominant effect on the evolution of the 211 level (cubic emission for higherl levels is suppressed by higher powers of α , so should generally be less significant again).", '5. Large self-coupling: lack of BH spindown': 'Since ε eq 211 ∝ f 2 , and the rate of spin extraction from the BH is ∝ ε 211 , the spin-down rate for small enough f will be so slow that the m = 1 threshold spin is not reached within the BH lifetime. The f = 10 12 GeV curves in the bottom panels of Fig. 4 show an example, if we take the BH lifetime to be < 10 10 yr. This affects BH spin-down signatures of superradiance, as we discuss in section VI. \nThe timescale for spin extraction in the large selfcoupling regime is set by τ sd (Eq. (60)). Setting this equal to the age T BH of the BH gives the threshold value of f \nf CD ≈ 3 × 10 14 GeV ( 10 10 yr T BH ) 1 2 ( 10 -13 eV µ ) 1 2 (62) × ( 0 . 01 α ) 5 2 ( 0 . 9 a ∗ ( t 0 ) ) 3 4 \ni.e. if f glyph[lessorsimilar] f CD , then the BH does not have time to fully spin down. The parameter space in which this is the case is plotted as region (D) in the bottom-right panel of Fig. 3, and is illustrated by the smallestf curve in Fig. 9. For f glyph[lessmuch] f CD , which gives T BH glyph[lessmuch] τ sd , the amount of angular momentum extracted is \n| ∆ a ∗ | glyph[similarequal] T BH τ sd ( a ∗ ( t 0 )) . (63)', 'C. Beyond the two-level system': "So far, we have focussed on BH-cloud systems which are dominated by the 211 and 322 hydrogenic levels. In this subsection, we consider the effect of other levels on the dynamics, including higher principal number n and higher angular momentum numbers l, m . We continue \nto assume that the initial conditions are such that 211 satisfies the superradiance condition and is the first level to grow; this is the regime of fastest black hole spindown and the largest gravitational and scalar emission rates, and is thus the most relevant from an observational perspective. \nWe find that, for α glyph[lessorsimilar] 0 . 2, the two-level picture discussed so far is probably sufficient, with only 211 and 322 growing to large occupation numbers. For α glyph[greaterorsimilar] 0 . 2, we expect that self-interactions would cause other levels to grow; we leave a full analysis of this regime to future work. \nOur analysis in this section focusses on perturbative processes, assuming that evolution is well-approximated by a combination of approximately hydrogenic levels. In section V, we investigate whether non-perturbative processes, such as 'bosenova', could change this picture; we find that, for α glyph[lessorsimilar] 0 . 2, this seems rather unlikely.", '1. Growth mechanisms in the presence of self-interactions': 'As discussed in section IV A, if 211 is initially the only state with appreciable occupation number, then other states j can be built up through processes of the form \n211 211 j BH \nTaking j = 322 gives the fastest growth rate, since the forced oscillation damped by the BH has m = 0 (maximizing the damping rate), and the overlap factors are large. \nIf a 322 and 211 abundance are both present, then other states can also be built up through \n211 211 j BH 211 322 j BH 322 322 j BH \nHowever, as well as these processes building up new states, there are also processes reducing their abundance; \n322 j 211 ∞ 211 j 322 BH j j 211 ∞ . . . \nTo determine whether, starting from very small fluctuations, another level j will start growing, we can look at the linear-inε j evolution terms (i.e. ignore processes such as the last diagram), and see whether the growth rate is positive or negative.', '2. n 11 levels': "For a state j with m = 1, the quartic processes with j in the final state all have forced oscillations with m ≥ 1, which are growing rather than decaying (in the parameter \nspace where 211 is superradiant). Consequently, they contribute a negative term to j 's growth rate. Hence, growth of j can only come about through superradiance. \nIn the large self-coupling regime, a quasi-equilibrium for 211 and 322 can be reached with very little effect on the BH spin, so the superradiance rates for m = 1 states are still positive. The fastest such rates are for the n 11 states. The linear-order evolution of the occupation number is set by \n˙ ε n 11 ε n 11 = γ SR n 11 -( γ 322 × BH 211 × n 11 + γ 211 ×∞ n 11 × 322 ) ε 211 ε 322 . (64) \nSubstituting in the equilibrium values for ε 211 and ε 322 , we have \n˙ ε n 11 γ SR n 11 ε n 11 glyph[similarequal] 1 -2 3 γ SR 211 γ SR n 11 γ 322 × BH 211 × n 11 + γ 211 ×∞ n 11 × 322 γ 322 × BH 211 × 211 (65) \nIt is useful to analyse the largen behaviour of this expression. At leading order in small α , the ratio γ SR 211 γ SR n 11 γ 322 × BH 211 × n 11 γ 322 × BH 211 × 211 is independent of α and a ∗ ; it exceeds 1 for n glyph[greaterorsimilar] 10, and approaches 1.27 at large n (see App. C 2 a and Fig. 22). As discussed in section III C, the most important finiteα effects on the quartic BH rates arise via the horizon flux of the associated forced oscillation. Since they are driven by near-horizon behaviour, these do not have large effects on ratios of rates (Fig. 21). Consequently, the ratio of analytic superradiance rates should be accurate at the few-percent level, except close to the superradiance boundary. \nThe ratio γ SR 211 γ SR n 11 γ 211 ×∞ n 11 × 322 γ 322 × BH 211 × 211 scales as α -3 at small α . For n large, it approaches \n2 γ SR n 11 3 γ SR 211 γ 211 ×∞ n 11 × 322 γ 322 × BH 211 × 211 → ( 0 . 29 α ˜ r 1 / 3 + ) 3 , n →∞ , α glyph[lessmuch] 1 (66) \n(see App. C 2 a and Fig. 23). \nThe combination of these negative contributions means that no n 11 level with n glyph[greaterorsimilar] 6 gets populated, at least for α ˜ r + glyph[lessorsimilar] 0 . 3. 6 For n = 3, the process 211 × 311 → 322 × BH is resonant, as discussed in section III C; this makes it more difficult to populate 311. However, for α glyph[greaterorsimilar] 0 . 2, we expect that the 411 level will grow, given enough time. This is illustrated in Fig. 5. \nSince the 411 superradiance rate is O (10) smaller than that of 211, the evolution of the 211/322 two-level system should proceed, at first, without modifications. Therefore, in the moderate and large self-coupling regimes we are considering, 211 and 322 will reach their two-level \nquasi-equilibrium occupation numbers, as described in section IV B. After two-level quasi-equilibrium is reached, we can initially treat 211 and 322 as constant sources while 411 grows (since the BH spin-down timescale is relatively very long). As a result, 411 grows with an 'effective' superradiance rate which is smaller than its usual superradiance rate, \nγ SR-eff 411 ≡ γ SR 411 -( γ 322 × BH 211 × 411 + γ 211 ×∞ 411 × 322 ) ε eq 211 ε eq 322 (67) \nwhere the quasi-equilibrium concentrations are given by Eqs. (55a) & (55b). \nAfter O (100) e-folds, the occupation number of 411 will become comparable to those of 211 and 322, and the three levels reach a new quasi-equilibrium. The most striking feature of this is that the equilibrium 411 occupation number is significantly higher than the equilibrium occupation numbers in the two-level 211/322 equilibrium. The 411 evolution equation is \n˙ ε 411 ε 411 glyph[similarequal] γ SR 411 -( γ 322 × BH 211 × 411 + γ 211 ×∞ 411 × 322 ) ε 211 ε 322 -γ 322 × BH 411 × 411 ε 322 ε 411 = γ SR -eff 411 -γ 322 × BH 411 × 411 ε 322 ε 411 \nSince the numerical coefficient of the γ 322 × BH 411 × 411 rate is significantly smaller than e.g. that of γ 322 × BH 211 × 411 (see Table V), then unless γ SR -eff 411 is significantly smaller than the components of Eq. (67), we need ε eq 411 glyph[greatermuch] ε eq 211 , 322 to compensate. This is illustrated in Fig. 6, which shows the growth of 411, and development of a new three-level equilibrium, for α glyph[similarequal] 0 . 22. From numerical calculations, 411 grows to be up to ∼ 50 times larger than the benchmark two-level quasi-equilibrium value of 211 (Eq. (55a)). \nGiven this enhanced occupation number, it is natural to ask whether higher-order or non-perturbative processes could occur, even if they do not for the two-level system. As discussed in section V, the more spread-out wavefunction of the 411 level makes this unlikely. The emission of scalar radiation will also be enhanced, as discussed in section VIII. \nThis three-level quasi-equilibrium is unlikely to be the full story. As we discuss in the next section, within the two-level equilibrium, we do not expect n 22 levels to grow. However, the large value of ε eq 411 can change this conclusion. For example, the dominant processes building up and depleting the 422 level, in the presence of equilibrium 211, 322 and 411 occupations, are \n411 411 422 BH 411 422 211 ∞ \nThe first diagram is almost on-shell for a 400 forced oscillation, so the 411 × 411 → 422 × BHprocess is 'resonant', like the 211 × 311 → 322 × BH process discussed in section III C. Consequently, its rate is suppressed by a lower power of α . Along with the large value of ε 411 relative to ε 211 , this means that the growth rate of 422 is positive \nfor the three-level equilibrium occupation numbers. As a result, after O (100) e-folds of this new growth time, the three-level equilibrium would be disrupted by the growth of the 422 level. \nWe leave a more detailed analysis of evolution in this largeα regime to future work (as well as the evolution being complicated, our hydrogenic approximations are less reliable here). It is possible that further levels will grow after 422 does, leading to a complicated, multistate superradiant cloud. In particular, is is possible that the cloud could reach large enough field amplitudes that higher-order or non-perturbative processes become important, as we discuss in section V.", '3. n 22 levels': 'n 22 states grow and are depleted similarly to the 322 level, via the processes \n211 211 n 22 BH 322 n 22 211 ∞ \nat linear order in ε n 22 (the superradiance rate of n 22 states is small enough not to be important, for parameters of interest). The linear-order growth rate is \n˙ ε n 22 = γ n 22 × BH 211 × 211 ( 1 -γ 211 ×∞ n 22 × 322 γ n 22 × BH 211 × 211 η ) ε 2 211 ε n 22 , (68) \nwhere η ≡ ε 322 /ε 211 . \nAt early times, ε 322 /ε 211 glyph[lessmuch] 1, and n 22 is sourced in the same way as 322. However, since the 322 growth rate is at least O (1) larger, it has an exponentially larger occupation number than the other n 22 levels by the time quasi-equilibrium is established. For example, \nγ 422 × BH 211 × 211 γ 322 × BH 211 × 211 glyph[similarequal] 0 . 36; γ n 22 × BH 211 × 211 γ 322 × BH 211 × 211 ∝ n -3 . (69) \n(see App. C 2 a and Fig. 24 for further details). For the quasi-equilibrium abundances of 211 and 322, the negative term in Eq. (68) dominates, reaching a value of 1 . 96 for n = 4 (1 . 69 for n →∞ ), \nγ 211 ×∞ n 22 × 322 γ n 22 × BH 211 × 211 η glyph[greaterorsimilar] 1 2 κ 211 ×∞ n 22 × 322 κ 211 ×∞ 322 × 322 κ 322 × BH 211 × 211 κ n 22 × BH 211 × 211 glyph[greaterorsimilar] 1 . 69 . (70) \nIncluding higher order corrections to the equilibrium ratio of 322 to 211, as well as the superradiance of 322, increases the ratio further. Thus the time derivative of n 22 becomes negative at leading order in α , independently of α, n, and a ∗ . \n4. n 33 levels \nn 33 states grow and are depleted by \n211 322 n 33 BH 322 n 33 211 ∞ \ngiving \n˙ ε n 33 = ( γ n 33 × BH 211 × 322 -γ 211 ×∞ 322 × n 33 ) ε 211 ε 322 ε n 33 glyph[similarequal] ( κ n 33 × BH 211 × 322 ˜ r + α 11 -κ 211 ×∞ 322 × n 33 α 8 ) ( M pl f ) 4 ε 211 ε 322 ε n 33 (71) \nat linear order in ε n 33 . Due to the different α scaling, the grow rate is negative at small enough α . Quantitatively, \n( κ 211 ×∞ 322 × n 33 κ n 33 × BH 211 × 322 ) 1 / 3 = { 0 . 31 n = 4 0 . 5 n →∞ (72) \nso at high spin, where ˜ r + glyph[similarequal] 1, the growth rate is always negative for α glyph[lessorsimilar] 0 . 3 (see App. C 2 a and Fig. 25). \n5. n 44 levels \nFor n 44, we have \n322 322 n 44 BH 322 n 44 211 ∞ \ngiving \n˙ ε n 44 = ( γ n 44 × BH 322 × 322 ε 322 ε 211 -γ 211 ×∞ n 44 × 322 ) ε 211 ε 322 ε n 44 glyph[similarequal] ( κ n 44 × BH 322 × 322 ˜ r + α 3 η -κ 211 ×∞ n 44 × 322 ) α 8 ( M pl f ) 4 ε 211 ε 322 ε n 44 (73) \nat linear order in ε n 44 . \nWith quasi-equilibrium occupations for 211 and 322, the growth of n 44 states occurs when α is large enough that \nκ n 44 × BH 322 × 322 κ 211 ×∞ n 44 × 322 ˜ r + α 3 η ≈ 1 2 κ n 44 × BH 322 × 322 κ 211 ×∞ n 44 × 322 κ 322 × BH 211 × 211 κ 211 ×∞ 322 × 322 α 6 ˜ r 2 + glyph[greaterorsimilar] 1 , (74) \nor equivalently \nα ˜ r 1 / 3 + glyph[greaterorsimilar] 0 . 3 (75) \nwhere the right hand side is as large as 0 . 34 for n = 5 (0 . 3 for n →∞ ) (see App. C 2 a and Fig. 26).', '6. Other levels': 'The n 22 , n 33 and n 44 levels considered above are the only ones which can be built up via quartic processes \nFIG. 5. Growth rates of n 11 levels once 211/322 quasiequilibrium has been reached, relative to their superradiance rates. At α glyph[lessorsimilar] 0 . 2 none of the levels have positive growth rates; levels with n glyph[greaterorsimilar] 10 have negative growth rates for all α , within our hydrogenic approximation. \n<!-- image --> \nwhere the forced oscillation has l = m = 0. 7 To build up other processes via self-interactions, starting from 211 and 322, we need to use forced oscillations with l > 0, which have a parametrically smaller flux through the BH horizon. They therefore stand even less chance of having positive growth rates. For l ≥ 2, we can often rule out these processes being relevant on astrophysical timescales, simply by estimating the magnitude of the growth rate. For example, for l = 2, we have \nγ 766 × BH(2 , -2) 322 × 322 ( ε eq 322 ) 2 ∼ 10 -2 ( α 0 . 3 ) 19 ( M glyph[circledot] M ) Myr -1 , (76) \nwhere the superscript BH( l, m ) indicates the angular momentum numbers of the damped leg. \nTaking an l = 1 example, \n˙ ε 655 = \nγ 655 × BH(1 , -1) 322 × 322 ( 1 -γ 211 ×∞ 655 × 322 γ 655 × BH(1 , -1) 322 × 322 ε 211 ε 322 ) ε 2 322 ε 655 . \n(77) \nThe depletion term dominates at equilibrium as long as \nα ˜ r 1 / 9 + glyph[lessorsimilar] ( κ 211 ×∞ 655 × 322 κ 655 × BH(1 , -1) 322 × 322 1 η B ) 1 / 9 ≈ 0 . 7 . (78) \nSimilar checks can be performed for other processes involving mixing with an l = 1 damped state (see \nFIG. 6. Example of 411 level growth after a period of 211/322 quasi-equilibrium. This plot assumes a 10 M glyph[circledot] BH, with α glyph[similarequal] 0 . 22, and an initial BH spin of 0 . 9. As discussed in section IV C 2, the three levels reach a new quasi-equilibrium state, in which we expect the 422 level to grow, becoming large at later times than those shown here. \n<!-- image --> \nApp. C2a). One finds that, for all of them, the depletion process to infinity dominates over the pumping process for the entire range of α for which m = 1 states can be superradiant ( α glyph[lessorsimilar] 0 . 5).', 'V. NON-PERTURBATIVE BEHAVIOR': "So far, our analysis has assumed that the scalar field is always well-approximated by a combination of approximately hydrogenic bound states, and that quartic interactions result in the slow transfer of energy to and from these bound states. However, if the field amplitude becomes large enough, we expect this picture to break down. Most directly, for a generic potential, higher-order field interactions can become important. In addition, for large enough amplitudes, attractive interactions would make hydrogenic bound states unstable to collapse, in a 'bosenova' [7, 28, 29]. \nAs we explored in section IV, for large self-couplings, the quartic interactions lead to the saturation of the cloud to a quasi-equilibrium configuration (for much of the parameter space of interest), with field amplitude ∝ f . For a potential of the form V ( ϕ ) ∝ g ( ϕ/f ), this means that the relative importance of higher-dimensional interactions becomes independent of f (for small enough f ). As we will show below, for small α , the maximum value of θ ≡ ϕ/f is small, and the quartic-driven behaviour we have investigated should be a good approximation. Similarly, for small α , the cloud is always far from the non-perturbative 'bosenova' regime. For α glyph[greaterorsimilar] 0 . 2, we expect levels beyond 211 and 322 to grow in the smallf regime, as discussed in the previous section, so their behaviour would need to be analysed to draw conclusions about non-perturbative behaviour.", 'A. Maximum field amplitude': 'When a single hydrogenic level dominates the energy stored in the cloud, the dimensionless field amplitude θ = ϕ/f is related to the occupation number of that level by | θ | ∝ α 5 / 2 √ εM pl /f . In the small and moderate self-coupling regimes, where 211 reaches its saturation occupation number, | θ | increases ∝ 1 /f as f decreases. However, once we are in the large-self-coupling regime, the occupation numbers reached are ∝ f 2 , so θ becomes independent of f . \nIf 211 is the dominant level, then the maximum value of θ is attained at r = 2 a 0 and θ = π/ 2, with \n| θ max | ≈ α 5 / 2 √ ε 211 ( M pl f ) √ 1 8 π e -1 . (79) \nAs we decrease f , this increases until f glyph[similarequal] f BC (Eq. (59)). For α glyph[greaterorsimilar] 0 . 04, f BC = f thresh and \n| θ max ( f BC ) | ≈ α 7 / 4 ( log( GM 2 ) κ SR a ∗ ( t 0 ) κ BH ) 1 / 4 e -1 2 √ √ 2 π ≈ 0 . 03 ( α 0 . 05 ) 7 / 4 . (80) \nThe scalings in (80) are only representative when α glyph[lessmuch] a ∗ ( t 0 ) (see App. E). For α glyph[lessorsimilar] 0 . 04, f BC = f eq and the maximum value of θ is equal to its value at equilibrium: \n| θ eq max | ≈ α ( √ κ SR a ∗ ( t 0 ) κ ∞ κ BH ) 1 / 2 √ 1 √ 24 π e -1 ≈ 0 . 005 ( α 0 . 01 ) ( a ∗ ( t 0 ) 0 . 99 ) 1 / 4 . (81) \n(again, these scalings are valid when α glyph[lessmuch] a ( t 0 )). 8 \nThese equations suggest that, for small α , the value of | θ | never becomes large, so we would generically expect higher-dimensional interactions to remain unimportant. To see this more quantitatively, Fig. 7 shows the maximum value of | θ | attained during the evolution of the two-level 211/322 system, for different values of α and \n∗ \nFIG. 7. Maximum value of | θ | ≡ | ϕ/f | attained during the evolution of the two-level 211/322 system, for a BH with initial spin a ∗ = 0 . 99 and initial mass 10 M glyph[circledot] (the BH mass only affects this plot via the number of e -folds ∼ log( GM 2 ) a level can grow). The dashed orange line indicates the boundary between the moderate and large self-coupling regimes (corresponding to f BC as defined in section IV). | θ max | is computed by numerically solving the evolution equations for the 211 and 322 occupation numbers. \n<!-- image --> \nf . This has the expected behaviour, increasing with decreasing f for f glyph[greaterorsimilar] f BC , and reaching a constant value for smaller f (at a given α ). \nAs discussed in section IV, we expect that, for small f and α glyph[greaterorsimilar] 0 . 2, levels other than 211 and 322 will grow. At these parameters, the | θ max | values in Fig. 7 represent a lower bound (since the initial 211 overshoot value is still set by 211/322 dynamics). For the 411 level, which we expect to be the first to grow after the 211/322 quasiequilibrium (section IV C 2), the maximum occupation reached is only around twice the maximum occupation number of 211. Consequently, the more spread-out wavefunction of 411 means that it does not attain a larger | θ | value. However, a more careful analysis would be required to determine | θ max | once other levels grow.', 'B. Bosenova': "As well as higher-dimensional interactions becoming important, another possible issue arising at large occupation numbers is that the cloud may undergo a sudden collapse due to attractive self-interactions, known as a 'bosenova' [7]. Here, we estimate the occupation number threshold for a bosenova to occur, using a variational approach. \nThe wavefunction for the hydrogenic 211 level is \nψ 211 = √ N 211 2 √ 6 a -5 / 2 0 re -r/ (2 a 0 ) Y 11 ( θ, φ ) (82) \nwhere a 0 ≡ 1 / ( αµ ) is the Bohr radius. As our variational ansatz, we will take a wavefunction of this form, but with \na modified radius, \nψ = √ N 2 √ 6 R -5 / 2 re -r/ (2 R ) Y 11 ( θ, φ ) (83) \nFor convenience, we will define a dimension-2 wavefunction ˜ ψ = √ µψ . Then, the non-relativistic action for ˜ ψ interacting with a gravitational field, sourced both by the central BH and by itself, is given by \nS glyph[similarequal] ∫ d 3 r d t i 2 µ ( ˜ ψ ∗ ∂ t ˜ ψ -˜ ψ∂ t ˜ ψ ∗ ) -1 2 µ 2 |∇ ˜ ψ | 2 -Φ | ˜ ψ | 2 + λ 16 µ 4 | ˜ ψ | 4 -1 8 πG |∇ Φ | 2 -ρ BH Φ (84) \nThe gravitational potential Φ obeys the Poisson equation, \n∇ 2 Φ = 4 πG ( ρ BH + | ˜ ψ | 2 ) (85) \nwhere we take ρ BH = Mδ 3 ( r ) and M is the mass of the BH. Using this potential, and integrating the action of Eq. (84) over space, we obtain an effective potential for R . Ignoring self-gravity of ψ , this is \nV ( ˜ R ) = α 4 M pl 2 ε µ ( 1 8 ˜ R 2 -1 4 ˜ R -3 α 3 εM pl 2 16384 π ˜ R 3 f 2 ) , (86) \nwhere ˜ R ≡ R/a 0 . The first two terms correspond to kinetic and gravitational energy, and set the radius of small-amplitude hydrogenic levels - the last terms arises from attractive self-interactions. The extrema of the potential V ( ˜ R ) are at \n˜ R ± extrema = 1 2 ± √ 1 4 -9 α 3 εM pl 2 4096 πf 2 . (87) \nIf we decrease f , at some point these extrema will coincide, and the potential will no longer have a stable minimum. This leads to a 'bosenova', with the cloud collapsing. The critical occupation number for this to occur is \nε crit = 1024 πf 2 9 α 3 M pl 2 (88) \nIncorporating the effects of self-gravity, this becomes \nε crit = 32 711 α 2 √ 75840 π ( f M Pl ) 2 +225 α 2 -160 237 α (89) \nwhich reduces to Eq. (88) for small f , i.e. for small clouds. Given this, we can ask whether the 211 occupation number reaches ε crit during its perturbative evolution. If it does not, then our assumption of perturbative evolution can be self-consistent. Fig. 8 shows the maximum value of ε 211 /ε crit 211 attained during the evolution of the two-level 211/322 system. For α small enough that other \nFIG. 8. Maximum value of ε 211 /ε crit 211 attained during the evolution of the two-level 211/322 system, for a BH with initial spin a ∗ = 0 . 99. ε crit 211 is the critical occupation number above which a rapid collapse of the cloud (a 'bosenova') is expected to occur (section V B). The dashed orange line indicates the boundary between the moderate and large selfcoupling regimes (corresponding to f BC as defined in section IV). ε 211 is computed by numerically solving the evolution equations for the 211 and 322 occupation numbers. The plot is roughly independent of the BH mass, within the range of astrophysical BHs. \n<!-- image --> \nlevels do not grow ( α glyph[lessorsimilar] 0 . 2), we can see that this ratio is always glyph[lessorsimilar] 0 . 3, so we do not expect a bosenova to occur. This is in contrast to the conclusions of much of the existing literature. As emphasized previously, other papers neglect the perturbative processes that lead to energy exchange between hydrogenic levels, causing the cloud to saturate to a quasi-equilibrium configuration before its amplitude becomes large enough for a bosenova. \nFor α glyph[greaterorsimilar] 0 . 2, we expect that levels other than 211 and 322 will grow. This means that the ε/ε crit values in Fig. 8 represent a lower bound. As we discussed in the previous subsection, the more spread-out wavefunction of the 411 level means that it is unlikely to get closer to the critical occupation number than the 211 level; we leave an analysis of the situation once other levels have grown to future work.", '1. Sub-leading effects': "As discussed in App. F, superradiance extracts mass from the BH in addition to angular momentum. As such, the cloud can actually grow to be somewhat larger than we have assumed so far. The modified equations for purely gravitational superradiance can be found in App. F. In deriving Fig. 7 and 8 we have included the correction coming from the change of the BH mass or, equivalently, from the time-dependence of α . As expected, we find that this correction can become quite large near the superradiance boundary, as the final spin is slightly modified (see Eq. (F10)). However, for strong \nself-interactions, where the bosenova might be relevant, there is practically no significant correction, as the cloud does not grow appreciably and thus does not extract a significant amount of spin or mass from the BH. \nOne might also ask how the inclusion of another level, say 322, changes the above picture. Assuming that its fractional occupation number is small compared to our primary level (e.g. 211), we can treat such a level as a small perturbation and check whether our results are consistent. In what follows we will neglect self-gravity for clarity or, equivalently, we will work in the small f (large self-interactions) limit, where Eq. (89) coincides with Eq. (88). We add a contribution from 322 to our variational ansatz \n˜ ψ ⊃ M 1 / 2 c2 4 a 3 / 2 0 4 81 √ 30 ( r a 0 ) 2 exp ( -r 3 a 0 ) Y 2 2 ( θ, φ ) (90) \nwhere M c2 is the mass of the 322 cloud. Note that we treat 322 as rigid, i.e. we do not allow its radius to change. Following the same procedure as before, we get an effective potential for 211 with an additional attractive term, stemming from its interaction with 322 \nV ( ˜ R ) = α 4 M pl 2 ε µ ( 1 8 ˜ R 2 -1 4 ˜ R -3 α 3 εM pl 2 16384 π ˜ R 3 f 2 -27 ˜ R 4 α 3 ε 2 M pl 2 2 π (3 + 2 ˜ R ) 9 f 2 ) (91) \nwhere ε 2 is the fractional occupation number of 322. Expanding around the critical values as ˜ R = 1 2 + √ ε 2 δ ˜ R and ε = ε crit + ε 2 δε , we find the correction δε = 21 / 16384, giving \nε 2 δε ε = 21 16384 ε 2 ε glyph[lessmuch] 1 . (92) \nThe result is indeed small and, thus, it does not change our conclusions about the bosenova. In particular, the correction to ε crit is positive . Since the interaction is attractive, as seen from the potential in Eq. (91), the 322 cloud attracts the 211 one and, since it resides at a larger radius, it effectively dilutes it. \nIn Fig. 8, we compared the ε 211 value attained during the perturbative level evolution to ε crit . However, the rates of the different processes involved in the evolution were calculated for the unperturbed hydrogenic wavefunctions. Consequently, we should ask whether self-interaction-induced perturbations to the wavefunctions make a significant difference to the rates, and so the occupation numbers attained. From Eq. (87), we can see that if ε 211 /ε crit is always small, then the corrections to the wavefunctions will always be small, and our calculations should be self-consistent. Since ε 211 /ε crit only becomes large for larger α , where (as discussed previously) our perturbative evolution calculations are already incomplete, we leave a full analysis to future work. \nIn plotting | θ max | , we have used the field defined using Eq. (83), that is, by taking into account the corrected \nradius of Eq. (87). This amounts to multiplying Eq. (79) by a factor of ( ˜ R + ext ) -3 / 2 (Eq. (87)), giving \n| θ | ≈ α 5 / 2 √ ε 211 ( 1 ˜ R + ext ) 3 / 2 ( M pl f ) √ 1 8 π e -1 . (93) \nWe have determined numerically that the radius change is at most 15% and introduces at most a 25% change in the region where | θ | grows to be the largest possible, driving to a value of ∼ 0 . 5, whereas the change is much smaller everywhere else. \nAnother possible issue with our variational analysis is that the evolution is not adiabatic during the last few e-folds before 211 reaches its maximum occupation number. As a result, the cloud might not trace the minimum of the potential of Eq. (86) but rather oscillate around it, in the manner of an 'excited state'. In this case, the cloud could overcome the barrier at ˜ R -(Eq. (87)) and collapse. We note that oscillations of the radius of the peak seem consistent with the results of ref. [47]. The minimum of the potential would need to be fairly close to critical for this to be an issue, but we leave detailed investigation of this point to future work.", '2. Comparison to simulations': "While we expect our hydrogenic ansatz to be a good approximation, properly understanding the dynamics of a bosenova requires numerical simulations. In [28, 29], the authors numerically simulate the evolution of a selfinteracting scalar field around a high-spin Kerr BH, starting from a hydrogenic bound state profile with θ ∼ O (1). These simulations effectively operate in the large selfcoupling regime, taking the cloud's mass to be very small compared to the BH. In simulations with a ∗ = 0 . 99 and α = 0 . 3 [29], they find that a 211 bound state with initial amplitude such that | θ max | = 0 . 4 does not undergo a bosenova, but one with | θ max | = 0 . 45 does. \nComparing these to our variational calculations, we can convert the critical occupation number (88) to a field amplitude, giving the leadingα expression | θ crit max | = 8 √ 2 3 e α glyph[similarequal] 0 . 42 α 0 . 3 . This is highly compatible with the threshold behaviour observed in the simulations. \nThe simulations in [28, 29] were evolved forward for t glyph[similarequal] 2000 r g . This is much shorter than the timescales for any of the perturbative processes studied in section IV, including 211 superradiance, and the growth of 322 through self-interactions. A simulation would have to be run for much longer times to observe these effects. In particular, the fact that a bosenova was observed for the initial state | θ max | = 0 . 45 is not evidence that a bosenova would occur around an astrophysical black hole. In the latter case, the true initial conditions are at an exponentially smaller amplitude, and according to our estimates, the maximum 211 amplitude reached during the evolution is | θ max | glyph[similarequal] 0 . 3 (Fig. 7), at which point interactions with 322 cut off its growth. \nIn [1], it is claimed that if self-interactions are repulsive, they can completely suppress the growth of 322, by spreading out the 211 cloud and reducing the rate of the 211 × 211 → 322 × BH process. We can estimate the effect of repulsive self-interactions by looking at how they shift the 211 wavefunction radius in our variational ansatz. This gives \n˜ R rep = 1 2 ( 1 + √ 1 + ε 211 ε crit 211 ) (94) \nwith ε crit 211 from Eq. (88). Since the perturbative evolution processes from section IV all depend on λ 2 , they are the same for attractive and repulsive self-interactions. Consequently, the maximum value of ε 211 attained through perturbative evolution should be the same. As a result, we expect that, unless ε 211 /ε crit 211 becomes large (which we cannot rule out for α glyph[greaterorsimilar] 0 . 2 and small f ), the effects of repulsion should be small.", 'VI. BLACK HOLE SPIN-DOWN': "One of the observational signatures of superradiance is the spin-down of initially fast-spinning BHs [6, 7]. In the absence of non-gravitational interactions, if a BH is born with spin high enough that a mode is superradiant, and the mode's growth time is much shorter than the lifetime of the BH, then a superradiant cloud will form around the BH. This spins down the BH to the point where the mode is stable, rather than growing. Consequently, observing a sufficiently old, sufficiently fast-spinning BH is good evidence against the existence of a light boson with such properties. Constraints of this kind have been placed on spin-0 [13, 48] and spin-1 [21] particles from measurements of BH spins in X-ray binaries [35, 49] (higher-spin particles have also been considered [50, 51], though such models encounter theoretical issues, as we discuss in the conclusions). \nIn contrast, if self-interactions are large, then as discussed in section IV, the occupation numbers in the quasi-equilibrium state are suppressed. Consequently, the rate of energy and angular momentum extraction from the BH is suppressed, and the spin-down constraints described in the previous paragraph will not apply directly. \nInstead, for small enough f , the time-averaged spin extraction rate will be approximately set by the equilibrium occupation number of the 211 level (at least in the case of 211 superradiance), as discussed in section IV B 4. Since ε eq 211 ∝ α -3 f 2 M 2 pl (Eq. (55a)), the time taken to fully spin down the BH (to the point where 211 superradiance is saturated) scales ∝ f -2 . Consequently, as reviewed in section IV B 5, there is some minimum f below which the BH is not significantly spun down in the time available. \nThis behaviour is illustrated, for particular initial BH parameters, in Fig. 9. The figure shows how, for f glyph[lessorsimilar] f BC (table II; f BC glyph[similarequal] 3 × 10 16 GeV for the left-hand panel, and glyph[similarequal] 2 × 10 17 GeV for the right-hand panel), spin-down to the m = 1 superradiance threshold takes longer as f is decreased, until it no longer occurs within the lifetime of the BH for f glyph[lessorsimilar] f CD . The region of ( µ, f ) parameter space in which the BH is spun down to the m = 1 superradiance threshold is shown in the bottom-left panel of Fig. 3. \nWe have only performed a detailed analysis (at all f ) of situations in which 211 is the first superradiant level to grow, and levels beyond 211 and 322 do not grow. From section IV, this corresponds to α glyph[lessorsimilar] 0 . 2. Nevertheless, we can be confident that, when interactions are weak enough that superradiant growth of the 322 level is unaffected, the black hole is spun down as in the purely gravitational case. This is indicated in the bottom right of the lower panels in Fig. 3. \nApplying this physics to observations of astrophysical BHs, Fig. 10 shows the regions in the ( µ, f ) plane for which sufficient spin-down occurs, so that spin measurements from BHs in X-ray binaries constrain an axion with that mass and coupling. For each black hole, the solid line of the corresponding color indicates the region in which spin-down would occur with high confidence, given the uncertainties on the measured BH parameters. The larger shaded regions are those in which spin-down may occur, given BH parameter values within the confidence intervals; these represent the regions of parameter space which may be constrained by future, better observations of these BHs. Given the uncertainties in our analyses when α glyph[greaterorsimilar] 0 . 2 and f is small, the constraints in those parts of parameter space should be treated as estimates requiring further study. \nFig. 10 can be compared to Fig. 11 of [13]. The latter assumed that the dominant effect of quartic selfinteractions was to cause periodic bosenova events when the cloud became too large; parametrically, when \nN glyph[greaterorsimilar] 16 π glyph[lscript] 4 α f 2 µ 2 (95) \nfor an l, m = glyph[lscript] superradiant level, as discussed in [7]. From the previous section, we know that, at small α and small f , the critical occupation number for a bosenova to occur has the same parametric scaling as the equilibrium 211 occupation number, but is numerically larger, ε eq 211 /ε crit 211 ∼ 0 . 1 (Eq. (89) and Fig. 8). Consequently, we expect the time-averaged 211 occupation number in our picture to be parametrically the same as that assumed in [13]. Numerically, since [13] assumes that a bosenova completely destroys the cloud, which then takes O (100) e-folds to be rebuilt, our time-averaged 211 occupation number is actually slightly larger, for the same parameters, resulting in slightly stronger spin-down constraints. \nThe age (or accretion timescale) of the BH limits how small a particle mass µ can be constrained by spin-down measurements - if µ is too small, then superradiance is not fast enough to spin down the BH. A separate effect \n<!-- image --> \nFIG. 9. Black hole spin-down as a function of time for µ = 8 × 10 -13 eV ( left panel ) and µ = 2 . 5 × 10 -12 eV ( right panel ) for a range of self-interactions strengths, and a 10 M glyph[circledot] black hole. These axion masses correspond to α glyph[similarequal] 0 . 06 and α glyph[similarequal] 0 . 19 respectively. The dashed horizontal lines show the superradiance boundary for levels 211 (upper) and 322 (lower). The dashed vertical lines show the expected spindown time in the limit of no self-interactions for levels 211 (smaller t ) and 322 (larger t ). \n<!-- image --> \nis that, for small µ , the cloud is more dilute, and can be disrupted by tidal forces from the companion star [45]. These gravitational perturbations mix superradiant levels with decaying ones (e.g. 211 with 21 -1), which can inhibit their growth. We do not attempt a careful analysis of the effects on the evolution of the cloud, but adopt the conservative approach of not placing constraints when the companion is closer than the maximum radius for the resonant depletion processes identified in [45] (see App. I). This sets the smallµ boundary of the constrained region in Fig. 10. We are able to constrain axion masses a factor ∼ 2 lighter than the limits from [13], which included an unphysical dipole gravitational potential effect from the companion. \nIn most of this paper, we have taken our nominal BH mass to be O (10 M glyph[circledot] ). However, our analyses can be easily rescaled to different BH masses; the most important dimensionless parameter that changes is the ratio of the BH lifetime to the light-crossing time. Fig. 11 shows the spin-down parameter space for a supermassive BH (SMBH), with M = 10 7 M glyph[circledot] . This parameter space sits at smaller µ (due to the larger BH size) and larger f (due to the smaller T BH µ parameter) than for a stellar-mass BH. There do exist spin measurements for some SMBHs [5254], and these could be used to place constraints on very-low-mass bosons (see e.g. [31, 55, 56]). However, the galactic center environments in which SMBHs live are rather complicated, and understanding environmental effects on the evolution of a superradiant cloud (e.g. due to the occasional infall of compact objects) would be necessary to place robust constraints. We leave such an analysis to future work, but include Fig. 11 as a guide to the kind of region that might be constrained by these measurements. \nAs well as spin measurements for BHs in X-ray binaries, there are also spin measurements for O (10 M glyph[circledot] ) BHs from gravitational wave observations of binary BH mergers at LIGO and Virgo [57-62]. The statistical uncer- \ntainty of these measurements is generally much greater than the estimated errors of X-ray binary spin measurements - for most of the binary BH mergers observed so far, the spins of the primary BHs could lie in an O (1) range, and are consistent with zero. However, there were two events in recent observing runs for which one of the primary BHs was measured to have high spin (significantly different from zero); GW190412 and GW190517 [37]. The inferred masses of these BHs were ∼ 30 M glyph[circledot] , which is significantly heavier than the BHs observed in X-ray binary systems. Consequently, if one assumes that the history of the system would have allowed a superradiant cloud to grow around the BH, one can constrain smaller boson masses, in the range µ ∼ 1 . 3 × 10 -13 eV - 2 . 7 × 10 -13 eV [37]. \nGiven that we have no reliable information about the pre-merger history of these BHs, we do not include them in Fig. 10. However, with better understanding of such systems, gravitational wave observations of binary BH mergers could become a valuable tool for constraining (or providing evidence for) light bosons. In addition, while mergers other than the two mentioned above do not provide strong evidence regarding superradiance [36, 37], 9 future data from many such mergers may provide \nFIG. 10. Constraints on axion parameter space from black hole spin measurements in X-ray binaries. For each black hole, the region enclosed by the solid line of the corresponding color (see key at top left) is the intersection of the m = 1 spin-down regions for different BH parameters (mass, spin, lifetime, binary period, and mass of the binary companion) within the observational error intervals. This corresponds to the parameter space region in which we can be confident that spin-down occurs, so is constrained by observations of that BH. The light shaded regions of each color are the unions of the spin-down regions for different BH parameters and could be constrained by improved measurement and analysis of these BHs. Higher axion masses could potentially be constrained using higherm levels; we include only the analog of the small and moderate self-coupling regimes A and B (for which self-interactions do not affect the extraction of angular momentum to the level with the largest SR rate) for m = 2, where the analysis in this work applies. The 'ALP DM' band corresponds to the range of quartic couplings that allow the observed DM abundance to be produced by the misalignment mechanism. The darker middle band corresponds to O (1) values of the initial misalignment angle ( θ ∈ (1 , π -1)), while the lighter bands above and below correspond to 'tuned' initial values ( θ ∈ (10 -1 , π -10 -6 )). \n<!-- image --> \nstatistical evidence for or against superradiant BH spindown [16, 64].", 'A. Axion models': "Understanding the parameter space in which spindown constraints apply is important in determining the consequences for motivated particle physics models. For the QCD axion, Fig. 10 confirms that, at least for 211 and 322 superradiance, self-interactions are small enough not to affect spin-down constraints. \nAnother motivated target model is an axion with a fixed (rather than temperature-dependent) potential. An \nexcluded at the 90% level, despite obtaining no new information; to set constraints a more complete analysis is needed. \ninitial 'misalignment' axion field value in the early universe will lead to a dark matter density at late times, depending on the axion mass, the shape of the potential, and the initial field value. Consequently, while the mass and self-couplings of a generic axion can vary independently, imposing that the misalignment mechanism must generate the observed DM density gives the 'ALP DM' band in Fig. 10 (for a cosine potential V ∝ cos( ϕ/f )). \nThe darker central part of this band corresponds to masses and self-couplings for which a 'generic', O (1) misalignment angle, θ init = a initial /f ∈ (1 , π -1), gives the correct dark matter density. For the same µ and θ init , but larger f , we would obtain too large a dark matter density. However, this can be fixed by 'tuning' the initial field value to be close to the bottom of the potential. Since ρ DM ∝ µ 1 / 2 θ 2 init f 2 for small θ init , the tuning required is simply θ init ∝ 1 /f . The lower edge of the band in Fig. 10 corresponds to θ init = 0 . 1. \nAt smaller f , we have the opposite problem of not producing enough DM. For a cosine-type potential, this can be solved by tuning the initial field value to be close to the top of the potential, so that its transition to matterlike oscillations around the bottom of the potential is delayed. This 'large-misalignment mechanism' [65] can lead to significant enhancements of dark matter density perturbations, resulting in a range of phenomenological signatures. In Fig. 10, the top edge of the band corresponds to θ init = π -10 -6 (see App. K for formulae), illustrating that, apart from the lower end of the µ range, BH spin-down constraints still apply to such models. \nAs well as affecting dark matter in the early universe, self-interactions could have effects at late times, leading to DM-DM scattering in halos. The associated relaxation rate is, parametrically [66, 67], \nΓ ∼ ρ 2 DM f 4 µ 3 v 2 ∼ 3 × 10 -26 yr -1 ( ρ DM GeVcm -3 ) 2 ( 10 11 GeV f ) 4 × ( 10 -12 eV µ ) 3 ( 10 -3 v ) 2 (96) \nwhere v is the halo's virial velocity (this should be compared to the relaxation rate Γ ∼ ρ 2 DM M 4 pl µ 3 v 6 for gravitational interactions [68-70]). Consequently, unless DM forms very dense structures, quartic self-interactions will not be significant in halos, for the parameter space we have been considering.", 'VII. GRAVITATIONAL WAVES': "Gravitational waves emitted by the superradiant cloud are a unique signal of ultralight bosons, turning gravitational wave observatories into indirect particle detectors [6, 7]. The superradiant cloud can grow to up to several percent of the black hole's mass, and sources \nFIG. 11. Parameter space for which the 211 level of a supermassive BH ( M BH = 10 7 M glyph[circledot] ), with initial spin a ∗ = 0 . 9, spins the BH down to saturation within an Eddington accretion timescale, t Edd glyph[similarequal] 4 × 10 8 yr. The 'ALP DM' band is defined as in Fig. 10. \n<!-- image --> \ngravitational waves through its oscillating stress-energy tensor. These are almost-monochromatic, coherent, and long-lasting. Such emission occurs in two parametricallydifferent frequency ranges; higher-frequency 'annihilation' signals, with ω glyph[similarequal] 2 µ , and lower-frequency 'transitions', with ω = ω j -ω j ' set by the frequency difference between different bound levels. \nConceptually, annihilation signals are sourced by the annihilation of two axions into a graviton. Consequently, they are emitted by any level populated by a single real scalar field. The timescale over which such emission lasts is parametrically longer than the superradiant growth time (Sec. IV B 2), making them promising for detection at gravitational wave observatories. Up to thousands of potential annihilation signals could be detectable, from black holes in the Milky Way, at Advanced LIGO and Virgo [13, 16, 38-40]. Such signals, and their detectability, have been studied in the context of continuous wave searches [13, 16], stochastic searches [38, 39, 71], directed searches for clouds around products of binary mergers [16, 72], and directed searches for clouds around BHs in X-ray binaries [73, 74]. Searches with LIGO/Virgo data are ongoing; so far, no signals have been observed [40, 75, 76], though using this nonobservation to constrain superradiance relies on poorly measured black hole population properties, and may suffer from down-weighting of the signal [40]. Searches at space-based, lower-frequency gravitational wave detectors such as LISA will be sensitive to lighter axions [13, 38, 39], while heavier axions may be observable with future higher-frequency detectors [77, 78]. \nTransition signals correspond to axions dropping into a more deeply bound level, emitting gravitational radiation at the frequency set by the level splitting. Attaining a significant emission rate requires both levels to have large occupation numbers simultaneously. For the case of purely-gravitational superradiance, these circumstances \nonly arise for higherl levels and for short times, leading to limited observational prospects at current gravitational wave observatories [13]. \nMore specifically, for a given m < 3, the fastestgrowing superradiant level is also the most tightly bound one, so other modes with the same m have exponentially smaller occupation numbers. For m ≥ 3, this is not always the case - for example, at large a ∗ and nearthreshold α , the growth rate of 433 becomes smaller than that of 533 and higher levels. This can lead to multiple m = 3 levels having large occupation numbers simultaneously. Similar crossings happen for m = 4 and higher levels, as illustrated in Fig. 12. \nThese circumstances allow gravitational wave transition signals of non-negligible amplitude to occur around astrophysical BHs. Even so, compared to annihilation signals, they offer less promising observational prospects. The total energy released, if the occupation number of the higher level transitions entirely to the lower one, is E = ∆ ωN glyph[lessorsimilar] α 2 µN , whereas annihilations can emit the entire energy stored in a cloud, E ∼ µN . In addition, signal durations for transitions are typically of order a superradiance time, compared to the parametrically longer annihilation signals [13]. Nevertheless, transition signals could probe interesting parts of parameter space, providing sensitivity to heavier axions than annihilation signals do (for a given BH mass). \nCompared to the purely-gravitational behavior summarized in the preceding paragraphs, the presence of selfinteractions can have a significant effect on the gravitational wave signatures of superradiance. For annihilations, self-interactions suppress the potential signals due to two main effects: the gravitational wave power emitted is reduced due to the smaller cloud size, and the new energy loss mechanisms via scalar radiation reduce the total energy emitted in GWs. On the other hand, selfinteractions provide a mechanism to populate multiple levels simultaneously, potentially increasing the parameter space for transition signals (though the cloud size and scalar radiation caveats still apply). In the rest of this section, we discuss annihilation and transition signals and their observational prospects in more detail. We focus on continuous wave searches for such signals, which are well-suited to louder signals from within our galaxy, and can provide a wealth of information about the detected signal properties. Stochastic searches to look for excess power in a narrow frequency range could potentially be performed more (computationally) cheaply and would also be interesting to study in future work.", 'A. Annihilations': "In this subsection, we focus on the prospects for observing annihilation signals from the 211 level, for a range of self-couplings, at current gravitational wave observatories. We also comment briefly on other types of annihilation signals, including annihilation signals from complex \nFIG. 12. Superradiance rates for the n 33 and n 44 hydrogenic bound states, computed numerically on the full Kerr background (using the continued fraction method of [79]). The left-hand plot shows rates for a ∗ = 0 . 9, and right-hand plot those for a ∗ = 0 . 99. The red curves correspond to the levels with smallest n ; levels with larger n have cutoffs at progressively smaller α . These plots illustrate how, at some α parameters, different hydrogenic levels can have the same superradiance rates. As discussed in Sec. VII, this can give rise to gravitational wave transition signals. \n<!-- image --> \nscalar fields. \nFigure 13 illustrates the effects of self-interactions on gravitational signatures of 211 superradiance, showing the peak signal amplitude, signal duration and sensitivity reach for different axion masses and self-couplings. To estimate the projected reach, we take the design strain sensitivity of Advanced LIGO [80], and assume all-sky semi-coherent continuous wave (CW) search strategies, with coherent integration times of 240 hours, and sensitivity depth D c ( f ) ∼ 50 / √ Hz. The sensitivity depth is defined by D c ( f ) ≡ √ S h ( f ) /h c 0 ( f ), where √ S h ( f ) is the noise spectral density and h c 0 ( f ) is the strain limit at the desired confidence level c . It allows comparisons of different searches, independently of the data used, and depends on the detailed search technique, coherent integration time, total integration time, etc. [81]. The latest searches with O2 data have used coherence times of up to T coh = 60hrs with N seg = 64 segments in the first analysis stage [82], and have reached sensitivity depths of ∼ 30 / √ Hz [83] to ∼ 50 / √ Hz [82] for c = 90% exclusion limits. Since the CW searches assumes a constant signal amplitude over the entire integration time, while our signals may change on times shorter than the coherent search time, we conservatively penalize our reach by √ τ sig /T coh (though the searches could be improved to take into account the time dependence of the signal, alleviating this penalty). \nWhile the sensitivity reach is a useful quantity for a search targeting a specific BH, standard CW searches are 'blind', and look for signals from sources anywhere in the sky. Figure 14 shows the expected number of events in such a search at Advanced LIGO, given assumptions about the galactic BH population, for different \nself-couplings. 10 We assume a power-law BH mass distribution, dN/dM ∝ M -2 . 35 , with a minimum black hole mass of 5 M glyph[circledot] , and vary the maximum black hole mass from 20 to 45 M glyph[circledot] [84]. For the BH spatial distribution, we take a combination of the disk and bulge distributions as in [40], with a total number of 10 8 BHs, born at a uniform rate throughout the age of the galaxy. We vary the BH spin distribution, with our extreme cases having 10% and 0 . 2% of BHs with initial spin a ∗ ( t 0 ) ≥ 0 . 9, respectively. The 10% figure is consistent with spin measurements from X-ray binaries [85, 86], and 0 . 2% with models of rare high spin BHs associated with gamma ray bursts [87, 88], making them reasonable upper and lower bounds. \nThe shaded bands in Fig 14 correspond to this range of BH population assumptions. While these unknowns do give rise to orders of magnitude uncertainty in the expected event rate, we can see that, for particle masses just below the spin-down threshold, even the pessimistic distributions give a promising number of events for purelygravitational superradiance. Conversely, the very large number of events (at design sensitivity) predicted by the optimistic distributions means that some of this parameter space is already ruled out by existing observations; axions with gravitational interactions and mass between 3 -7 × 10 -13 eV would yield more than 10 signals in current LIGO data for all the BH mass and spin distributions considered here; masses between 2 × 10 -13 -2 × 10 -12 eV would yield 10 or more signals for the most optimistic \nFIG. 13. Upper left: Peak strain from 211 × 211 → GW annihilations for an observer at 1 kpc from a 10 M glyph[circledot] BH, with initial spin 0.9. Upper right: Typical duration τ peak of peak signal, log 10 ( τ peak / sec). In the large self-interactions regime, we show the time-scale of the overshoot regime, corresponding to the peak signal strain. Lower left: Sensitivity reach in kpc to a 10 solar mass BH, for continuous wave searches at Advanced LIGO design sensitivity [80]. Lower right: Reach in kpc to a 100 solar mass BH. The dashed orange line indicates the boundary between the moderate and large self-coupling regimes (corresponding to f BC , Sec. IV), while the dotted black line indicates the boundary of the regime in which the 322 level grows appreciably ( f AB ). \n<!-- image --> \nspin distribution considered here [40]. An analysis of existing data taking into account the reduced event rates at larger self-interactions has not been performed and would be very valuable. \nOnce we incorporate self-interactions, there are three different parameter space regimes, with distinct behavior (as per Sec. IV). In the small self-coupling regime, f > f AB , the 322 level does not grow through selfinteractions, and the dynamics proceeds as in the purely gravitational case. Consequently, the annihilation signal properties are independent of the self-coupling, and existing analyses of gravitational wave signals will apply without modification. This regime, which (for stellar mass BHs) includes f ∼ M pl as well as QCD axion selfcouplings, can lead to as many as thousands of signals at LIGO/Virgo, as shown in Fig. 14. \nIn the moderate self-coupling regime, f AB > f > f BC , the growth of the 211 level is unaffected, but 322 grows earlier than it would otherwise have done. The main effect on the annihilation signal is through the addition of another energy loss process for the cloud, via 322 × 322 → 211 × ∞ emission. Consequently, while the peak emission amplitude is unaffected, the signal duration is reduced. This corresponds to the parameter space region between the orange and black dashed lines in the upper-right panel of Fig. 13. More specifically, when 211 is primarily depleted through gravitational waves, the signal strain as a function of time is given by, \nh GW,ann ( t ) = h peak 1 + t/τ ann (97) \nwith τ ann defined in Eq. (39). However, due to the \nself-interaction processes, there is additional energy lost from the cloud, changing the time-evolution to that in Eq. (48), with \nh GW,ann ( t ) ∝ τ scalar /t (98) \nat late times, where τ scalar ∝ ( f/M pl ) 4 , Eq. (49). For f in the moderate self-coupling regime, τ scalar can be significantly less than τ ann . Given the typical assumptions on black hole formation rates and distributions, the shortest signals that are likely to be observable in an all-sky continuous wave search have signal times on the order of 10 4 years or more [40]. \n√ \nSince, for moderate self-couplings, the peak signal strain is not affected, the sensitivity reach of gravitational wave detectors for signals observed around the optimum time is only moderately affected, as illustrated in the bottom panels of Fig. 13. One effect is that, especially for lighter black holes, the signal duration can become comparable to the typical coherent integration times used in continuous wave searches (e.g. [82]), which degrades the signal to noise. \nFor blind searches, the faster decrease of signal strain with time leads to less chance of seeing a signal, as illustrated in Figure 14. The expected number of observable signals at f ∼ 10 18 GeV, which is in the moderate selfinteractions regime for µ ∼ 10 -12 eV, is around an order of magnitude lower than in the purely gravitational case. For larger and smaller µ , this value of f falls back into the weak self-interactions regime, so the difference is reduced. At f ∼ 10 17 GeV, which is in the moderate self-interactions regime for the whole µ range, the signal durations are much shorter, and the expected number of observable signals is less than 1. As a result, such signals are unlikely to observed with current detectors, in a blind search. In addition, the faster time-evolution can lead to larger frequency drifts, which could degrade search sensitivity further (see Sec. VII C). \nFor strong self-couplings, f > f BC , the peak signal amplitude drops with increasing coupling as ( f/f BC ) 2 (Fig. 13). In particular, this drop-off starts at larger f than for the suppression of BH spin-down, since f BC > f CD . Consequently, with current detectors, selfinteractions strong enough to avoid BH spin-down constraints (Sec. VI) also render GW annihilation signals undetectable, for any plausible BH spin and mass distributions. For f glyph[lessorsimilar] f BC , i.e. f glyph[lessorsimilar] 10 16 GeV for stellar-mass BHs, the expected number of events in a blind search is glyph[lessorsimilar] 10 -3 , while for f glyph[lessorsimilar] 10 15 GeV, where signal durations become comparable to those in the small self-interaction regime, signals beyond 10 -100 pc are unlikely to be visible at Advanced LIGO sensitivities. \nNevertheless, it is possible that advanced future detectors, such as the Cosmic Explorer [89, 90] or Einstein Telescope [91-94], may be able to probe this parameter space. The signal strain in the quasi-equilibrium regime is a factor O (1 -5) below the overshoot peak shown in the left panel, but the quasi-equilibrium regime lasts parametrically longer than in the moderate self-interaction \nFIG. 14. Projections for the number of observable 211 × 211 → GW annihilation signals, using continuous wave searches at Advanced LIGO (with design sensitivity), for a range of selfinteraction strengths (see text for details). The width of the bands results from varying the BH spin distribution and maximum BH mass as described in the text. The highest number of observable signals is in the small self-interactions regime, which includes gravitational superradiance and QCD axion parameter space. Increasing self-interactions reduces the number of signals expected. At high masses, the signal frequency falls above the band of typical CW searches ( ν glyph[greaterorsimilar] 2 kHz). The darker (lighter) shaded regions are disfavored by black hole spin down for initially superradiating levels with m = 1 ( m = 2) (see Sec. VI). \n<!-- image --> \nregime, τ sig ∝ ( f BC /f ) 2 (see Fig. 18). If smaller strains come within reach of future detectors, the long-lasting signals would have an increased chance of being observed in the quasi-equilibrium regime. \nAdditional annihilation channels. In addition to 211 × 211 → GW annihilations, as occur in the purelygravitational case, the presence of the 322 level allows 211 × 322 → GWand 322 × 322 → GWprocesses. These GWs will still have frequency ω glyph[similarequal] 2 µ , but due to the larger angular momentum of the 322 level, their rates are suppressed by higher powers of α , P l,l ' GW ∝ α 16+2( l + l ' ) , where l and l ' are the angular momentum numbers of the two levels. These powers are significantly smaller than the primary 211 × 211 → GW annihilation channel, and are further suppressed by the smaller occupation number of 322 at small α (App. D). For example, the 322 × 322 → GWprocess would lead to signals strains O (10 -4 ) weaker than the primary signal at α ∼ 0 . 3. 'Cross-annihilation' signals between two levels, 211 × 322 → GW, may be observable for the closest black holes; further study would require numerical GW power calculations which have not yet been performed for cross-annihilation signals. \nAnnihilation signals from complex fields. In this section, and throughout the rest of this paper, we have considered superradiance of a single, real spin-0 field. As has been pointed out in a number of papers [95-98], for the case of two scalar fields of degenerate masses (equiva- \nFIG. 15. Left panel: peak strain of the 322 → 211+ GW transition signal at 1 kpc from a BH of mass 3 M glyph[circledot] , as a function of the mass µ and self-coupling scale f of the scalar particle. Right panel: sensitivity reach for the detection of such signals, using the Advanced LIGO detector, or with the MAGIS proposal for a future space-based atom interferometer [41]. The dashed orange and dotted black lines are the f BC and f AB curves, respectively, as in Fig. 13. \n<!-- image --> \nlently, a single complex scalar field), there are cloud configurations with a time-independent stress-energy tensor, which consequently do not emit any gravitational radiation. In complex field terms, these correspond to allparticle or all-antiparticle field configurations, whereas gravitational waves arise from particle-antiparticle annihilation. This has sometimes been interpreted [99] as indicating that annihilation radiation, of the type considered in this section, is not expected from superradiance of complex fields. \nHowever, as per the discussion in Sec. II, the initial conditions for the growth of superradiant modes are either vacuum fluctuations, or whatever pre-existing astrophysical fields are present. In the former case, we can view the growth of the particle and antiparticle field modes as effectively separate, and generically, they will obtain O (1)-similar occupation numbers. For preexisting astrophysical fields, a generic expectation in many circumstances is for O (1)-similar initial conditions for particle and antiparticle fields. Consequently, unless some mechanism drives us to an all-particle or allantiparticle state, we expect that the particle and antiparticle fields generically attain roughly comparable occupation numbers. Compared to a real scalar field, this results in a total GW annihilation signal energy that is only O (1) smaller.", 'B. Transitions': "For large enough self-interactions (regions B,C,D in Fig. 3), the 322 level grows earlier than it would have done otherwise, and both 211 and 322 can have significant occupation numbers at the same time. This gives rise to GW transition signals. \nThe transition quadrupole moment for the 322 → 211 + GW process vanishes at leading order, so its rate is suppressed by a larger power of α than other gravitational transition processes (such as the 644 → 544 process considered in [7, 13, 16]). At leading order in α , the emitted power, as a function of polar angle θ , is \ndP d Ω = GN 322 N 211 πr 4 g α 14 × (99) \n( 2 5 3 6 5 8 (1 -cos 4 θ ) + (27 + 28 cos(2 θ ) + 9 cos(4 θ )) sin 2 θ 2 2 3 6 5 10 7 2 ) \nwhere the first term corresponds to l, m = 2 , 1 emission, and the second to l, m = 3 , 1. This gives a total emitted power of [7] \nP = 2 8 × 5717 3 5 5 11 7 3 GN 322 N 211 r 4 g α 14 . (100) \nThe emitted radiation is at a frequency ω = ω 322 -ω 211 glyph[similarequal] 5 72 α 2 µ . In terms of the normalized occupation numbers, it contributes a term \n˙ ε 322 glyph[similarequal] -5 × 10 -6 α 10 ε 211 ε 322 + . . . (101) \nto the equations of motion. \nCompared to the processes discussed in Sec. IV, which drive the evolution of the superradiant cloud, the effects of GW transitions are always subdominant. While this does reduce the peak signal amplitude, it also means that signal timescales can be longer compared to the transitions in the purely gravitational regime, which is helpful for detection. \nFig. 15 shows projections for the peak signal strain, and sensitivity reach, for transition signals from a fairly light BH, M BH = 3 M glyph[circledot] . The signal durations (for a given \nBHmass) are the same as those for annihilations (Fig. 13) in the region where 322 grows, f < f AB , as the two levels evolve together over time. Given the lower frequency compared to annihilations, the signal strains are typically larger (Fig. 15 left). However, transition signals only occur in the moderate and large self-interaction regimes, where much of the energy loss is through scalar radiation. Furthermore, for given BH mass, the frequency decreases ∝ µ 3 with decreasing µ , rapidly falling out of the sensitivity band of current detectors such as Advanced LIGO. For heavier BHs, the frequency of transition signals would always be too low for ground-based GW detectors, due to overwhelming seismic and gravity-gradient noise. \nFor a narrow range of axion masses above 10 -11 eV, current detectors could potentially probe signals in the moderate self-interaction regime (Fig. 15, right). Although the reach is poor at small f , there is a roughly order-of-magnitude range in f for which sensitivity to signals from the galactic centre would be possible. The signal times in this region last on the order of minutes to hours, and the expected number of signals in a blind search is heavily dependent on the poorly-measured black hole distribution in the 'mass gap' below 5 M glyph[circledot] [100-103] (although evidence for compact objects in this mass range is emerging [104-106]). Consequently, blind searches with current detectors are unlikely to lead to observable signals. \nHowever, future space-based detectors such as LISA [107, 108] and atom interferometer missions [41], could have promising sensitivity to such signals. For illustration we show the reach of the MAGIS proposal [41] in the right panel of Fig. 15, which can achieve a reach of 10 kpc for axions around 3 M glyph[circledot] black holes, and up to 10 3 kpc for 100 M glyph[circledot] black holes. Some of the more promising signals fall in the 0 . 1 -10 Hz range, where future proposals such as DECIGO [109] could improve transition detection prospects.", 'C. Frequency drifts': "While the frequency of gravitational wave annihilation signals is almost constant at ν ann ≡ 2 ω/ (2 π ) glyph[similarequal] 2 µ/ (2 π ) (we will use frequency rather than angular frequency in this section, to match the GW literature), the potentially long signal durations mean that even very small frequency drifts can be measured. Moreover, the search algorithms employed in continuous wave detection analyses can be strongly affected by these small frequency drifts, so it is important to quantify them to determine the appropriate search strategy and sensitivity [110]. \nThe self-energy of the cloud, from both gravity and self-interactions, affects the frequency of the bound axions, and therefore the frequency of the GWs emitted [13]. As the occupation numbers of the levels evolve, the self-energy contribution to the binding energy ∆ ω and thus the emitted frequency ν change over time. \nThe gravitational and self-interaction contributions to the energy of axions in level 211 are, respectively, (see App. G and App. B1 ) \n∆ ω g glyph[similarequal] -0 . 19 µα 3 ε 211 (102) \n∆ ω λ glyph[similarequal] -3 . 5 × 10 -5 µα 5 ε 211 ( M pl f ) 2 , (103) \nwhere the energy is decreased (increased) in the presence of an attractive (repulsive) self-interaction. These corrections are always small compared to the axion mass, as well as the energy splitting between levels (for occupation numbers below the non-perturbative regime - see Sec. V). \nAs the cloud is growing through superradiance, the frequency changes relatively rapidly as ∝ µα ˙ α on the order of the superradiance time due to the changing BH mass. However this period is short, and generally does not contribute much of the detectable signal. At late times, the cloud size is depleted over time, and the level's frequency drift is positive (assuming negligible or attractive selfinteractions). This is in contrast to standard astrophysical sources of continuous gravitational radiation, such as spinning neutron stars, and may provide a hint that a detected signal arises from superradiance. We describe the main contributions to these frequency drifts, at leading order in α , below. For a more complete discussion of frequency drifts we refer the reader to App. H. \nAt small self-interactions, the frequency drift is dominated by the depletion of the gravitational self-binding energy through annihilations, resulting in a frequency drift of order \n˙ ν ann glyph[similarequal] 7 × 10 -15 Hz s ( α 0 . 1 ) 17 ( µ 10 -12 eV ) 2 , (104) \nto leading order in α . Throughout the small selfinteraction regime f > f AB (see also Fig. 3), the gravitational frequency drift dominates any contribution from the self-interactions. \nAs self-interactions increase, the frequency drift from the gravitational binding energy is increased due to the faster depletion of the cloud from axion emission, \n˙ ν g glyph[similarequal] 10 -10 Hz s ( 10 17 GeV f ) 4 ( µ 10 -12 eV ) 2 ( α 0 . 1 ) 17 , (105) \nand there is an additional frequency drift from the change of self-interaction energy, \n˙ ν λ glyph[similarequal] 10 -10 Hz s ( 10 17 GeV f ) 6 ( µ 10 -12 eV ) 2 ( α 0 . 1 ) 19 . (106) \nThe latter dominates when f glyph[lessorsimilar] 8 . 5 × 10 16 GeV( α/ 0 . 1). Finally, in the strong self-interactions regime f < f BC , the cloud reaches a long-lived quasi-equilibrium configuration, and the dominant source of frequency drifts comes from the slow spindown of the BH. \nGravitational wave signals from 322 → 211+ GW transitions have frequency ν 322 -ν 211 , so the changing contributions to the 211 and 322 frequencies partially cancel, making frequency drifts a factor of a few smaller than for annihilations, and negative in most parts of the parameter space. Similarly to annihilations, for moderate self-couplings, self-interactions dominate the frequency drifts for f glyph[lessorsimilar] 10 17 GeV( α/ 0 . 1). \nAt small α , the frequency drift can be small enough so as to be unobservable. Over a year, the minimum frequency change that can be measured is ∼ yr -1 ∼ 3 × 10 -8 Hz, so if the frequency drift is glyph[lessorsimilar] yr -2 glyph[similarequal] 10 -15 Hzs -1 , it has no observational effect. At the other extreme, too large a frequency drift can be problematic for the search algorithms employed. Current LIGO/Virgo continuous wave searches cover a range of positive to negative frequency derivatives of e.g. 2 × 10 -9 Hz/s through -1 × 10 -8 Hz/s [83]. More sensitive searches, using longer coherent integration times, may require even smaller frequency drifts [75]. In the small coupling regime, the drift of the signal becomes larger than this threshold at α ∼ 0 . 25. In the moderate selfinteractions regime, both annihilation and transition signals have drifts large compared to the current search range for f glyph[lessorsimilar] 5 × 10 16 GeV( α/ 0 . 1) 17 / 4 . However, as discussed above, the observational prospects for GW signals at such small f are not promising, with currentgeneration experiments.", 'VIII. AXION WAVES': 'As well as emitting gravitational radiation, the cloud also emits both relativistic (section III A) and nonrelativistic (section III B) scalar waves. If the scalar ϕ has non-gravitational interactions 11 with the SM, such ϕ radiation could be detected in laboratory experiments. For an axion-like particle, a natural assumption is that interactions with the SM are suppressed by parametrically the same symmetry breaking scale f that sets the axion potential. If this is the case, then we have the unusual feature that, in the large self-coupling regime f < f BC , the signal does not decouple: while the power in axion radiation decreases as the quasi-equilibrium size of the cloud decreases, this is compensated for by the increased interaction strength from the smaller f . In addition, the BH spin-down time increases with decreasing f , so such signals can last for very long times, increasing the chance of observing them. Consequently, axion waves could be a probe of the smallf regime, in which both GW and spin-down signatures are suppressed. \nQuantitatively, if we take the 211 and 322 quasiequilibrium occupation numbers (55), then the emitted power is dominated by non-relativistic 322 × 322 → 211 × ∞ radiation. At large distances r from the BH, this radiation has energy density \nρ rad ∼ µ 4 πr 2 GM 2 γ 211 ×∞ 322 × 322 ( ε eq 322 ) 2 ε eq 211 v glyph[similarequal] 10 -6 GeV / cm 3 ( α 0 . 1 ) 6 ( 10 kpc r ) 2 × ( f 10 16 GeV ) 2 , (107) \nwhere v = α/ 6 is the velocity of the non-relativistic axions emitted. The energy density ρ rad depends only on α , and not on µ and M BH independently. For given f , the emitted power is maximized when the superradiance rate is largest, at high a ∗ and α . The corresponding dimensionless amplitude θ of the axion waves is \nθ glyph[similarequal] 10 -19 ( 10 -12 eV µ ) ( α 0 . 1 ) 3 ( 10 kpc r ) , (108) \nindependent of f . This is in contrast to GW signals, for which the amplitude at Earth decreases as f 2 in the quasi-equilibrium regime. Relativistic axion radiation from the 3 → 1 process (section III A), and 2 → 1 cubic emission, also have f -independent θ , but are suppressed by higher powers of α , and are smaller than the non-relativistic radiation for the parameter space we are interested in. \nAs we discussed in section IV C, for α glyph[greaterorsimilar] 0 . 2 and small f we expect additional hydrogenic levels, other than 211 and 322, to be populated. While we have not performed a full analysis in this regime, a example of the possible effects can be seen from the 411 build-up studied in section IV C 2, which for α not too far above 0 . 2 is expected to be the first additional level to grow. The 211, 322, and 411 levels form a new quasi-equilibrium, with the 411 level having enhanced occupation number relative to those of the 211/322 equilibrium. Consequently, the rate of scalar radiation during this equilibrium is enhanced; numerically, we find that ρ rad 3 -level ∼ 25 ρ rad 2 -level for α glyph[similarequal] 0 . 3. While this equilibrium will be disrupted in turn by the growth of further levels, this illustrates that, while the parametric behaviour in f should remain the same, additional levels may change the numerical factors affecting the scalar radiation power. As discussed in section V, if the growth of additional levels leads to large enough field amplitudes in the cloud, then higher-order processes or a non-perturbative collapse of the cloud may become possible, significantly altering the behaviour. \nSince the axion radiation is non-relativistic and narrow-bandwidth, its effects on a laboratory system are similar to those of axion dark matter at the same mass. The masses of interest correspond to rather low frequencies, e.g. 10 -12 eV glyph[similarequal] 2 π × 200 Hz. For this parameter \nspace, the axion-SM couplings most amenable to laboratory detection experiments are those to nuclear spins and to photons, which we discuss below. \nSearches for axion DM via the axion-gluon coupling L int ∝ ( ϕ/f ) G µν ˜ G µν have promising sensitivity reach at low axion masses [111]. However, if an axion-like particle has the same G ˜ G coupling, but a smaller mass than the QCD axion (or equivalently, a larger G ˜ G coupling for the same mass), then it is strongly constrained by its behaviour in dense environments such as the early universe and stellar cores [112, 113]. For superradiance-sourced signals, G ˜ G couplings significantly higher than the QCD axion value (for a given axion mass) are needed to have experimental sensitivity, and are affected by these constraints.', 'A. Nucleon spin coupling': "The axion coupling to fermion spins is L ⊃ g N ( ∂ µ ϕ ) ¯ ψγ µ γ 5 ψ , where we generically expect g N ∼ 1 /f a . For a non-relativistic fermion, this gives an axiondependent term in the fermion Hamiltonian, \nH ⊃ g N glyph[vector]σ · ( ∇ ϕ + ˙ ϕglyph[vector]v ) (109) \nwhere glyph[vector]σ is the fermion's spin, and glyph[vector]v is its velocity. We will focus on couplings to nucleons, which for low axion frequencies are easier to detect than couplings to electrons. \nSince the 322 × 322 → 211 ×∞ axion radiation from the BH has v ∼ α/ 6 (Sec. III B), while the nucleon velocity changes associated to low-energy laboratory processes are much smaller, the 'axion wind' term H wind = g N glyph[vector]σ · ∇ ϕ dominates. Due to the ∼ α/ 6 velocity being significantly larger than the virial velocity of DM in the galaxy, ∼ 10 -3 , and because of the coherent nature of the emitted radiation, an experiment searching for the axion wind coupling will have better sensitivity to BH-sourced radiation than it would for DM for an equivalent axion energy density. \nThe best-developed experimental proposal aiming to detect the axion wind coupling is CASPEr-Wind [111], which employs Nuclear Magnetic Resonance (NMR) technologies. This uses a liquid xenon target, whose nuclear spins are polarized in a strong magnetic field. The axion wind coupling acts on the nuclei like an effective magnetic field, H wind = g N glyph[vector]σ · ∇ ϕ ≡ B a · glyph[vector] µ n , where µ n is the nuclear magnetic moment and B a is the effective axion 'magnetic field'. If this effective magnetic field oscillates at close to the Larmor frequency of the nucleons in the external magnetic field, then the resulting spin precession of the nuclei is resonantly enhanced. This spin precession can then be picked up by a sensitive magnetometer. \nIn App. J, we review the sensitivity of such experimental setups to a monochromatic axion oscillation. If we are uncertain about the axion mass, and want to experimentally probe an O (1) axion mass range around an angular \nFIG. 16. Projected detectability of non-relativistic axion radiation, assuming an axion-nucleon coupling. The signal strength is expressed in terms of the equivalent pseudomagnetic field felt by nuclei. The blue dotted lines correspond to sensitivity estimates for NMR axion-wind detection experiments [111] with the indicated parameters. The bands correspond to signals from three astrophysical BHs and two nominal BHs with the indicated parameters. The widths of the bands correspond to the uncertainty on the BH parameters (for the nominal BHs, to the distance range indicated). The darker bands bounded by solid contours correspond to the signal emitted during two-level quasi-equilibrium (Sec. IV). The lighter-shaded extensions above represent the enhanced signal from the three-level equilibrium with 411 (Sec. IV C 2), illustrating the potential range of signals. \n<!-- image --> \nFIG. 17. Projected sensitivity reach (SNR = 1) for the detection of non-relativistic axion waves from a BH-cloud system at large self-interactions, f < f BC , in a NMR-based axionwind detection experiment. The bands show the reach to a BH-cloud system, ranging from a two-level quasi-equilibrium with parameters a ∗ = 0 . 9 and α = 0 . 2 (lower edge, solid), to that of a three-level quasi-equilibrium system with a ∗ = 0 . 99 and α = α optimal (0 . 99) ≈ 0 . 41 (upper edge). The reach for a BH-cloud system with a ∗ = 0 . 9 and α = α optimal (0 . 9) ≈ 0 . 28 is also indicated for a two-level equilibrium (dotted line), and a three-level equilibrium (dashed line) system. \n<!-- image --> \nfrequency ω 0 , then a signal can be detected for \nB 2 a glyph[greaterorsimilar] few × ω 0 µ 2 n N n T tot , (110) \nwhere T tot is the total experimental running time, and N n is the number of aligned spins in our spin-polarized sample. 12 This is a best-case sensitivity estimate, limited by the fundamental spin-projection noise of the sample -to achieve it, a well-shielded sample and a sufficiently sensitive magnetometer would be required. Experiments capable of sensing nuclear spin projection noise have been carried out [114], and such sensitivities are a goal for the CASPEr-Wind experimental program [111]. \nA fully polarized liquid 129 Xe sample has ∼ 10 22 spins/cm 3 [111], so the sensitivity limit for a relatively small target volume is \nB a glyph[greaterorsimilar] 10 -20 T √ ν kHz 10 22 N n yr T tot (111) \nFor comparison, an axion DM signal at the sensitivity threshold estimated in [111], for these parameters, has an effective magnetic field of ∼ few × 10 -20 T. The effective magnetic field from axion radiation emitted by a superradiant cloud is \nB a glyph[similarequal] 3 × 10 -24 T ×C N ( α 0 . 1 ) 4 ( 1 kpc r ) , (112) \nfor a high-spin BH, where C N ≡ g N f . Consequently, some combination of larger experimental volumes (as planned for CASPEr-Wind phase II [111]), larger C N , larger α and a closer BH would enable laboratory experiments to be sensitive to axion waves. \nThis is illustrated in Fig. 16, which shows projected signal strengths for a selection of astrophysical BHs (both nominal and observed), along with sensitivity thresholds for different experimental configurations. While C N ∼ O (1) is the 'natural' expectation in many models, larger values of C N are possible. In particular, it is interesting to consider how large a reach can be obtained in as-yet-unconstrained parameter space, below the existing astrophysical limits of g N glyph[lessorsimilar] (few × 10 8 GeV) -1 [115118]. While much of the axion mass range in Fig. 16 is excluded for large f by BH spin measurements (Fig. 10), these constraints do not apply for f glyph[lessorsimilar] 10 12 -10 13 GeV, where the BH spin-down is too slow. The astrophysical bounds translate into |C N | glyph[lessorsimilar] 10 3 ( f/ 10 12 GeV); the C N = 10 3 line in Fig. 16 illustrates that such couplings can give good detection prospects for a wide range of BHs and axion masses. \nFIG. 18. Typical duration log 10 ( τ sig / sec) the axion wave signal for a 10 M glyph[circledot] BH with initial spin 0.9. In the large selfinteractions regime, we show the time-scale corresponding to the quasi-equilibrium evolution. For f glyph[lessorsimilar] 10 12 GeV the signals can last longer than the age of the universe (note that 10 Gyr glyph[similarequal] 3 × 10 17 s). The dashed orange and dotted black lines are the f BC and f AB curves, respectively, as in Fig. 13. \n<!-- image --> \nTo reflect the uncertain behavior of the superradiant cloud at α glyph[greaterorsimilar] 0 . 2, Fig. 16 displays the signal resulting from the radiation power during the three-level quasiequilibrium phrase, as a shaded area above the signal from the two-level equilibrium. The signal curves illustrate that, with larger-volume experiments, sensitivity to astrophysical BHs may be possible for C N ∼ O (1). They also strongly motivate detailed numerical analyses of the highα regime, where the strongest signals would arise. \nFig. 17 displays the sensitivity reach to an optimal BH for a given axion mass. Again, we see that for larger experimental volumes, astrophysically-relevant reaches in particular, to the Galactic Center ∼ 8 kpc away may be possible for fairly natural C N values. \nIf we are interested in the signal from a specific, known BH, then the sensitivity reach is the most important parameter. However, as is the case for gravitational wave searches, many signals are expected to arise from asyet-unobserved BHs, and could only be detected via a 'blind', all-sky search. In this situation, another important factor is the typical duration of signals, which affects the probability that a given BH is still emitting today. Figure 18 shows the duration of the peak axion signal (which contributes most of the detectable SNR) from a nominal BH, as a function of axion mass and coupling. Lower f values lead to slower BH spin-down, and so to longer durations of quasi-equilibrium signal emission; this is relevant down to f ∼ 10 11 -10 12 GeV, below which signals can last longer than the age of the universe. \nSince, in the quasi-equilibrium regime, the peak signal strength at Earth is independent of f for fixed C N , decreasing f down to ∼ 10 11 GeV increases the expected number of events in a blind search. This is illustrated \nFIG. 20. Number of observable signals expected in an NMR axion wind experiment with V = 10cm 3 , for f = 10 12 GeV and different couplings to nuclear spins, as shown. We require observable signals to have SNR ≥ 10, given the blind search strategy required for these events. The width of the bands results from varying the assumed BH distribution as in Fig. 19. For a fixed self-interaction strength, the highest number of observable signals is for the largest coupling strength to nuclei. \n<!-- image --> \nFIG. 19. Number of observable signals expected in an NMR axion wind experiment with V = 10cm 3 and C N = 100, with different bands corresponding to different quartic coupling scales f . We require observable signals to have SNR ≥ 10, given the blind search strategy required for these events. The width of the bands results from varying the assumed BH spin distribution and maximum BH mass (see Section VII A). For a fixed C N , the number of observable signals increases for smaller f , due to longer signal durations, saturating at f ∼ 10 11 GeV. \n<!-- image --> \nin Fig. 19. If, rather than fixing C N , we require that g N is below the astrophysical bounds, then as shown in Fig. 20, there is a wide range of axion masses over which we might expect visible signals in an all-sky search (depending on the mass and spin distribution of astrophysical BHs). In both the Fig. 19 and Fig. 20 projections we assume the reach to the axion waves from the twolevel equilibrium, not taking into account the possible enhancements in power from additional levels; on the other hand, the dynamics of additional levels could shorten the signal lifetime at large α values. In the blind search, an analysis similar to the techniques employed by Continuous Waves searches at LIGO/Virgo (Sec. VII A) would be required, to make use of the extremely long signal coherence times while at the same time taking into account the Doppler shifts from the many relative motions between the experiment and the unknown black hole positions. \nUnless C N is extremely large, the effects of the axion field on spins in the vicinity of the black hole, and the effect of these spins on the axion field, are always small. The largest effective magnetic field obtained in the cloud is ∼ |C N | 10 -6 T µ 10 -12 eV , which would not have any significant affect on accretion disk behavior. Similarly, the axion field sourced by a coherent nuclear spin density, if any exists in the accretion disk, is tiny compared to the fields of a superradiant cloud. For any reasonable nuclear spin response to small magnetic field perturbations, the effect of spin response on the dynamics of quasi-bound axion levels will be extremely small, so the growth of the cloud will not be affected. Similar considerations apply to the propagation of scalar waves through interstellar \nspace; these will be undisturbed to a very good approximation.", 'B. Photon coupling': 'The axion coupling to photons is L ⊃ -g aγγ 4 ϕF µν ˜ F µν = g aγγ ϕE · B . Generically, we expect the coupling constant to be g aγγ = C γ α EM 2 πf a , where C γ ∼ O (1) is related to the charged matter content of the UV theory [119]. \nAn axion oscillation sources EM fields through the effective current density J a = g aγγ ( ˙ ϕB + ∇ ϕ × E ) (and the corresponding effective charge density ρ a = -g aγγ ∇ ϕ · B ). Axion DM, which is non-relativistic, has | ˙ ϕ | glyph[greatermuch] |∇ ϕ | , so detection experiments use strong magnetic fields to maximize J a . Searches for low-frequency ( ∼ kHz) axions have been proposed using static background magnetic fields [120, 121], or GHz-frequency fields in superconducting cavities [122-125]. 13 \nIf they can be realized in the future, quantum-limited meter-scale experiments could probe axion DM couplings as small as g aγγ ∼ 10 -17 GeV -1 at ∼ kHz frequencies (unfortunately, this is still far from QCD axion sensitivity). With a monochromatic signal, as opposed to virialized axion DM, this would correspond to a sensitivity of \ng aγγ ∼ 10 -18 GeV -1 ( ρ ρ DM ) -1 / 2 . 14 For non-relativistic emission from a superradiant cloud, we would obtain a reach of \nr kpc ≈ (2 × 10 -3 ) |C γ | ( µ 10 -12 eV )( α 0 . 1 ) 3 , (113) \nConsequently, signals from an superradiant cloud via the axion-photon coupling could only be seen for an exceptionally close, fast-spinning BH, and/or in models where |C γ | is large. \nAt the small axion masses we are interested in, SN1987A observations constrain the axion-photon coupling to be | g aγγ | glyph[lessorsimilar] 5 × 10 -12 GeV -1 [129]. This translates to |C γ | glyph[lessorsimilar] 500 ( f/ 10 11 GeV), which allows for somewhat smaller expected blind-search event rates than the nucleon-coupling case shown in Fig. 20. \nSimilarly to the case of nucleon couplings, the effects of astrophysical EM fields on the SR cloud will be tiny unless |C γ | glyph[greatermuch] 1. In addition, the naive ϕ → γγ decay rate, Γ ϕ → γγ glyph[similarequal] g 2 aγγ µ 3 64 π , is much longer than the age of the universe for couplings of interest. However, in some circumstances it is possible for parametric resonance to greatly enhance the photon emission rate [130]. Parametrically, in the limit where g aγγ is arbitrarily small, and taking L to be the approximate spatial extent of the axion profile, the total decay rate into a particular mode within the ∼ L 3 volume is Γ ∼ g 2 aγγ ϕ 2 µ 2 L , where ϕ is the typical field amplitude. Consequently, the number of photons emitted into that mode, in the light-crossing time ∼ L , is ∼ Γ L ∼ g 2 aγγ ϕ 2 ( µL ) 2 . This tells us that for finite g aγγ , if Γ L glyph[greaterorsimilar] 1, then stimulated emission will become important; for Γ L glyph[greatermuch] 1, the emission rate will be exponentially enhanced. \nThis parametric argument agrees with the conclusions of [130], which analyses the growth of electromagnetic perturbations using Floquet theory, and finds that parametric resonance occurs if \n| g aγγ µϕL | glyph[greaterorsimilar] few . (114) \nSince g aγγ ϕ = C γ α 2 π θ , the LHS is maximized (for given C γ ) by maximizing θ . For an axion of mass µ , this occurs at f glyph[similarequal] f BC (for the 211 level). Using Eq. (80), we find that for parametric resonance to occur, we need \n|C γ | glyph[greaterorsimilar] (9 × 10 2 ) ( 0 . 1 α ) 3 / 4 , (115) \nfor a ∗ ( t 0 ) = 0 . 99. Consequently, if |C γ | glyph[lessmuch] 10 3 , then photon emission will be unimportant. \nIt should be noted that the above is a best-case estimate, which will only hold if the BH is in a sufficiently pristine environment. The plasma frequency in the interstellar medium is ω p ∼ 10 -12 -10 -10 eV, which is comparable to the mass range for a superradiant axion around a stellar-mass BH. Moreover, one expects the plasma density in the vicinity of the BH to be greater, due to accretion [131]. Consequently, it is likely that plasma effects suppress the parametric resonance process, even at large |C γ | [132].', 'IX. CONCLUSIONS': "In this paper, we have investigated some of the most important consequences of scalar self-interactions for superradiance around astrophysical BHs. As we have showed, self-interactions can result in very rich and complicated dynamics, and there are a number of aspects which would benefit from further study. In particular, we have not systematically treated situations in which the initially fastest-growing level has m ≥ 2. While we generally expect gravitational (and scalar) wave signatures to be dominated by cases where 211 grows first, BH spindown constraints for higher-mass axions will depend on higherm superradiance. \nIn addition, even for the 211 case, our calculations have been at the (semi-)analytic level, and may not be reliable for large enough α . In particular, we found that for α glyph[greaterorsimilar] 0 . 2 and small f , levels other than 211 and 322 might play an important role in the dynamics. One route to properly understanding the highα regime might be to perform numerical simulations of the (self-interacting) field equations themselves, rather than of the occupation numbers of hydrogenic modes. Such approaches have been used to study purely-gravitational superradiance in a number of papers [133-137]. As mentioned in Sec. V, numerical methods were applied to a self-interacting scalar field on the Kerr background by [28, 29], but they did not evolve the system for long enough to observe the perturbative effects we have studied. Since the highα regime is where observational signatures may be the strongest, and in which there is the possibility of phenomena such as bosenova, a fuller treatment would be valuable. \nOur analyses focussed on the simplest form of selfinteractions for a spin-0 particle; the lowest-order (renormalizable) potential terms. In more complicated hidden sector models, other forms of interactions, or extra hidden sector states, could affect the superradiance behavior. For example, [138] discusses a model in which the QCD axion couples to a hidden-sector photon, and there are hidden-sector fermions which interact with this photon. Such models illustrate that, while the minimal DM models we considered in Figures 10 and 11 are often still subject to BH spin-down constraints, others may not be. \nBeyond the spin-0 particle candidates we considered, \nsuperradiance of massive vectors is also of interest. Vector self-interactions are somewhat more complicated than those for scalars, since renormalizable interactions between vectors must take the form of Yang-Mills theory. For abelian theories, 'light-by-light' scattering could lead to qualitatively similar dynamics to those discussed here, but has to be investigated in the context of a low cutoff and potential production of the charged particles which give rise to the vector self-interaction. Beyond self-interactions, a simple example of both theoretical and phenomenological interest is a light vector interacting with the SM via a kinetic mixing with the SM photon (though plasma dynamics may make the behavior around astrophysical black holes very complicated). A vector may also have interactions with other hidden sector states - for example, its mass may come from a Higgs mechanism, or it may mediate interactions between hidden sector matter. For the purely gravitational story to hold, such states must be sufficiently heavy, and/or sufficiently weakly coupled [21]. We leave investigations of such scenarios to future work. \nSuperradiance of spin-2 particles has also been investigated in the literature [23, 24]. An issue with such models is that an effective field theory with a spin-2 particle of mass µ , along with the massless graviton (a 'bigravity' theory), has a cutoff scale at or below Λ 3 = ( M P µ 2 ) 1 / 3 [139, 140]. Here, M P ∼ min( M pl , Λ) is an effective mass scale set by the mass scales M pl , which suppresses massless graviton interactions, and Λ, which suppresses massive spin-2 interactions. At the small masses µ we are interested in for BH superradiance, Λ 3 glyph[lessorsimilar] 10 eV ( µ 10 -12 eV ) 2 / 3 is small compared to energy scales of interest. For example, the energy density in a fully-occupied superradiant cloud is ρ ∼ (6 MeV) 4 ( α 0 . 2 ) 5 ( µ 10 -12 eV ) 2 . Consequently, it is unclear whether there are theories for which reliable calculations can be carried out in the regimes of interest. \nReturning to the topic of spin-0 superradiance; as well as exploring the new observational signatures that may arise from self-interactions, our analyses clarify when selfinteractions are small enough not to affect the usual gravitational dynamics of superradiance. As illustrated in Figures 10 and 11, this is important for understanding when constraints and signatures from motivated models, such as the QCD axion or misalignment DM, can be trusted. \nAs we have demonstrated, adding a simple quartic interaction can dramatically change the dynamics of scalar superradiance. The additional interaction inevitably reduces the efficiency of black hole spindown as well as the strength and timescale of gravitational wave annihilation signals. Nevertheless, the new dynamics can lead to simultaneous population of multiple levels giving rise to gravitational wave transition signals, a narrow range of which may be observable at LIGO/Virgo. Given that the transition signals are at parametrically lower frequencies corresponding to the energy splitting between different levels, signals from scalars around stellar mass black holes \ngenerally fall below the LIGO/Virgo sensitivity band in frequency and present new targets for future mid-band detectors. \nPerhaps the most novel signature is the emission of particles to infinity: a light, self-coupled axion can extract the energy of rotating black holes and populate our galaxy with axion waves, without the need for a cosmological abundance or a coupling to Standard Model matter. In the presence of such a coupling, these axion waves could be detected in the lab. While current experiments are not yet sensitive to this population of light axions, this mechanism further motivates the development of light axion direct detection experiments, as well as numerical work on self-interactions in superradiance to better characterize the signal from compact, semi-relativistic axion clouds.", 'ACKNOWLEDGMENTS': "We thank Asimina Arvanitaki, Savas Dimopoulos, Sergei Dubovsky, Peter Graham, Kurt Hinterbichler, Junwu Huang, Ken Van Tilburg, and Sylvia Zhu for helpful discussions. We thank Perimeter Institute for warm hospitality during the completion of part of this work. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development & Innovation. MB is supported by the James Arthur Postdoctoral Fellowship. MG and RL are supported in part by the National Science Foundation under Grant No. PHYS-1720397, and the Gordon and Betty Moore Foundation Grant GBMF7946. OS is supported by the Fonds de recherche du Qu'ebec Nature et Technologies and by a DARE Fellowship from the Vice Provost for Graduate Education at Stanford University.", 'Appendix A: Parametric oscillator analysis': "As discussed in section III C, a useful way to analyse the growth of bound levels is to assume that we have some large-amplitude ψ c , and to treat this as a parametric forcing in the Gross-Pitaevskii (GP) equation (Eq. (12)), i.e. to solve \n( i∂ t + M ) ψ = -3 λ 24 µ 2 ( ψ 2 c ψ ∗ + | ψ c | 2 ψ ) (A1) \n(here, M represents the terms in the non-relativistic Hamiltonian, including an absorbing term corresponding to the BH horizon). As compared to the forced oscillation analysis in section III C, we ignore back-action for only two of the 'legs' in diagrams such as Eq. (19), rather than for three of them. \nTo simplify our discussion, we will take ψ c ∝ ψ 211 (so we are interested in processes such as Eq. (19)). It is helpful to extract the time dependence corresponding to the 211 oscillation, and write ψ = Ψ e -i ˜ ω c t , where ˜ ω c ≡ \n˜ ω 211 (for simplicity, we will assume that ω 211 is real, as it is when 211 has reached its saturation value). Then, if we take a harmonic ansatz, Ψ = Ae -i ˆ ωt + Be i ˆ ω ∗ t , the GP equation \n( i∂ t + ˜ ω c + M ) Ψ = ˜ λ (Ψ 2 211 Ψ ∗ + | Ψ 211 | 2 Ψ) (A2) \n(where ˜ λ ≡ -3 λ 24 µ 2 ) implies that \n(ˆ ω + ˜ ω c + M ) A = ˜ λ (Ψ 2 211 B ∗ + | Ψ 211 | 2 A ) (A3) \nand \n( -ˆ ω ∗ + ˜ ω c + M ) B = ˜ λ (Ψ 2 211 A ∗ + | Ψ 211 | 2 B ) (A4) \nIf we take the complex conjugate of Eq. (A4), then together with Eq. (A3), we have a linear eigenvalue problem that we can solve for ˆ ω . For λ = 0, the solutions correspond to usual hydrogenic (quasi-bound) states. \nFor non-Hermitian Hamiltonians, the eigenstates are generally non-orthogonal [141]. However, in our case, we can write M = M R + i M I , and treat M I as being diagonal in the basis of M R eigenstates (that is, we ignore the detailed dynamics behind the absorption, since this is outside the regime of the non-relativistic approximation). In this case, the ( λ = 0) quasi-bound states Ψ k are orthogonal [141], and we will assume the normalization dV Ψ ∗ k Ψ j = δ jk . \nTo linear order in λ , if we start with the unperturbed solution A = Ψ i , B = 0, then we can write the perturbed solution as A = Ψ i + ∑ k α k Ψ k , B = ∑ k β k Ψ k (expanding in the unperturbed basis). Using equations (A3) and (A4), \n∫ \n(ˆ ω i -ˆ ω k ) α k = ˜ λ ∫ dV Ψ ∗ k | Ψ 211 | 2 Ψ i (A5) \n( -ˆ ω ∗ i -ˆ ω k ) β k = ˜ λ ∫ dV Ψ ∗ k Ψ 2 211 Ψ ∗ i (A6) \nAs well as these perturbations to the wavefunction, we are interested in finding the perturbation to the frequency ˆ ω of the state. Writing ˆ ω = ˆ ω i + δ ˆ ω , we have \n(ˆ ω i +˜ ω c + M ) A = -δ ˆ ωA + ˜ λ (Ψ 2 211 B ∗ + | Ψ 211 | 2 A ) (A7) \nIf we take A to be normalized so that ∫ dV Ψ ∗ i A = 1 even for non-zero λ , then this implies that \n= ˜ λ ∫ dV Ψ ∗ i | Ψ 211 | 2 Ψ i (A9) \nδ ˆ ω = ˜ λ ∫ dV Ψ ∗ i (Ψ 2 211 B ∗ + | Ψ 211 | 2 A ) (A8) \n-˜ λ 2 ∑ k 1 ˆ ω i + ˆ ω ∗ k ∣ ∣ ∣ ∫ dV Ψ ∗ k Ψ 2 211 Ψ ∗ i ∣ ∣ ∣ 2 (A10) \n∣ \n∣ \n∣ ∣ + ˜ λ 2 ∑ k 1 ˆ ω i -ˆ ω k ∣ ∣ ∣ ∣ ∫ dV Ψ ∗ k | Ψ 211 | 2 Ψ ∗ i ∣ ∣ ∣ ∣ 2 (A11) \nThe second and fourth lines of this expression give behaviour similar to standard perturbation theory. However, the -1 / (ˆ ω ∗ i +ˆ ω k ) factor in the second line gives rise to qualitatively different effects. If the Ψ i mode is decaying, but the Ψ k mode is damped sufficiently strongly that Im(ˆ ω i + ˆ ω ∗ k ) > 0, then Im ( -1 ˆ ω i +ˆ ω ∗ k ) > 0. Consequently, the 'mixing' with the Ψ k mode contributes a growing term to the perturbed Ψ i mode. In our case, the 211 parametric forcing gives the 322 mode a 'mixing' with the decaying 100 mode (and the n 00 modes, etc), contributing a growing term for 322. The perturbations to the 322 wavefunction correspond to the forced oscillation discussed in section III C. Using Eq. (A11), we obtain the same 322 growth rate as calculated from the forced-oscillation picture. \nSimilarly, mixing with superradiant (rather than decaying) modes contributes a negative imaginary part to δ ˆ ω . This is again as we'd expect from the forced oscillation picture. Including a growing 211 occupation number, as is appropriate when 211 is still superradiant, leads to more complicated expressions. However, since the superradiant growth timescale is always much longer than the oscillation period of ψ 211 , we can separate these timescales, with 322 growth at a particular time being driven by the 211 amplitude at that time. \nThis kind of perturbative analysis can be applied in the hydrogenic approximation (at leading order in α ), or using numerical wavefunctions for the bound states, which will be more accurate at higher α . In Fig. 21, we plot the decay rate of the 100 level, relative to its leadingα power-law behavior (the n 00 levels have very similar behavior). The lower panel of the figure also shows a numerical approximation to the rate of the 211 × 211 → 322 × BH process, computed by numerically integrating the forced equation of motion in the Kerr background (for practical reasons, over a restricted range in α ). The close correspondence between the behaviours of these two rates illustrates that, for the 211 × 211 → 322 × BH process, the most significant highα corrections come from shortdistance effects that affect the flux across the horizon; at long distances from the BH, the forcing term, and the forced oscillation, are not strongly affected (at the α of interest). \nThe parametric forcing analysis above is not specific to black hole superradiance. In the simplest case, if we had two oscillators, with an oscillating coupling between them, \nx + ω 2 0 x = f cos(2 ω c t ) y (A12) \ny + γ ˙ y + ω 2 1 y = f cos(2 ω c t ) x (A13) \nthen the same kind of analysis would apply. In the absence of the damping term γ , if ω 0 + ω 1 is detuned from 2 ω c , then the system is not unstable to growth. Introducing γ leads to the exponential growth of x , as per above. \nWhile the above analyses were at the level of classical equations, a similar analysis could be done in terms of quantum master equations. The most important physical difference is that, while the classical ground state is stationary, quantum fluctuations are amplified by the instability, so the ground states evolves into a probability mixture of coherent states. This is precisely analogous to the amplification of quantum fluctuations by superradiance, as discussed in section II. \nFrom Eq. (A6), our perturbative treatment breaks down when \n˜ λ ∫ dV Ψ ∗ k Ψ 2 211 Ψ ∗ i ˆ ω ∗ i + ˆ ω k glyph[greaterorsimilar] 1 (A14) \nIn terms of the physical mode frequencies, ˆ ω ∗ i + ˆ ω k = ω ∗ i + ω k -2Re ω 211 . For generic hydrogenic modes, this is O ( α 2 ) µ , and in this case, the LHS of Eq. (A14) is parametrically ∼ α 3 M 2 pl f 2 ε 211 , similarly to the self-energy corrections (Eq. (H1a)). As we discuss in section V, the largest value that ε 211 attains decreases as we decrease f , and the numerical value of this quantity is always small. \nIn special cases, the source term for the k oscillation can be almost on resonance, and the denominator can become smaller. We discuss a specific example in section III C (Eq. (22)), where it is O ( α 4 ) µ for the 211 × 311 → 322 × BH process. However, in this case, the source term does not appear to grow large enough for there to be a problem, in most of the parameter space of interest. \nIt is also possible to treat emission to infinity, e.g. through the 322 × 322 → 211 × ∞ process discussed in section III B, in terms of a parametric forcing, with loss to infinity acting as like a damping term.", 'Appendix B: Perturbative calculations of frequency shifts and rates': 'In this appendix, we will provide more detailed derivations of the leadingα rates for quartic self-interaction processes involving hydrogenic levels. \nUp to corrections from self-gravity, the system obeys the classical equation of motion \n( D 2 -µ 2 ) ϕ = -λ 6 ϕ 3 , (B1) \nwhere D 2 = D ν D ν and D ν is the covariant derivative of the Kerr geometry. Expending D 2 to first order in r g /r , this becomes \n( ∂ 2 ∂t 2 -glyph[vector] ∇ 2 + µ 2 ) ϕ -2 α r ( µ + ˆ K ) = λ 6 ϕ 3 . (B2) \nThe term \nˆ K = 1 r ∂ ∂r -2 ( µ 2 + ∂ 2 ∂t 2 ) -ˆ L 2 r 2 (B3) \nwhere \nis parametrically suppressed relative to µ for nonrelativistic components of ϕ , and we drop ˆ K except for calculations of relativistic emissions. Here ˆ L 2 denotes the total angular momentum operator: ˆ L 2 Y m l = l ( l +1) Y m l for the Laplace spherical harmonics Y m l ( θ, ϕ ). \nWe seek a perturbative solution in the self-interaction parameter λ , \nϕ = ϕ (0) + λϕ (1) + . . . (B4) \nAt zeroth order, \n( ∂ 2 ∂t 2 -glyph[vector] ∇ 2 + µ 2 -2 αµ r ) ϕ (0) = 0 . (B5) \nThis equation admits non-relativistic (quasi-)bound states with hydrogenic waveforms and energies which we identify with the superradiant cloud: \nϕ (0) ≡ ∑ nlm ϕ (0) nlm = ∑ nlm √ N nlm 2 µ e -iω nlm t ψ nlm +c.c. , (B6) \nup to phases, where ψ nlm are the normalized hydrogenic wavefunctions | ψ nlm | 2 d 3 glyph[vector]r = 1 . \nTo avoid secular terms at the next perturbative order, we must also introduce a perturbation series for the normal frequencies: \n∫ \nω nlm = ω (0) nlm + λω (1) nlm + . . . , (B7) \nω (0) nlm = ω n + i Γ SR nlm , (B8) \nω n ≈ µ ( 1 -α 2 2 n 2 ) , (B9) \nand Γ SR nlm is the superradiance rate. We call the energy corrections ∆ ω nlm ≡ λω (1) nlm \nAt first order in perturbation theory, this gives a driven massive Coulomb wave equation, \n( ∂ 2 ∂t 2 -glyph[vector] ∇ 2 + µ 2 -2 αµ r ) ϕ (1) = 1 6 ( ϕ (0) ) 3 + ∑ nlm 2 µω (1) nlm ϕ (0) nlm . (B10) \nPlugging (B6) into (B10), and expanding the driving term as a sum of harmonic driving terms gives \n( ϕ (0) ) 3 ∼ ∑ Ω f ( glyph[vector]r , Ω) e -i Ω t +c.c. , Ω > 0 . (B11) \nSince ϕ (0) ∼ a -3 / 2 0 µ -1 / 2 ∼ α 3 / 2 µ , the source f ( glyph[vector]r ) ∼ ( ϕ (0) ) 3 scales as ∼ α 9 / 2 µ 3 . The physical intuition behind that scaling is that a cloud with larger α has a smaller characteristic size a 0 and therefore larger densities, enhancing the rate of many-body processes. \nThe physical nature of the process associated to each summand depends on the value of Ω: \n- 1. Ω -µ > 0 corresponds to free radiation emitted in the continuum and travelling to infinity either with non-relativistic or relativistic velocities,\n- 2. Ω -µ < 0 and Ω glyph[negationslash] = ω n for all n is off-resonant driving of discrete bound modes, i.e. the production of off-shell particles trapped in the gravitational well\n- 3. Ω = ω n for some n is resonant driving, which either corresponds to resonant (on-shell) production of particles inside the cloud, or to a correction to the frequencies (one-particle energies) and waveforms (one-particle states) of the zeroth-order normal modes. \nFor clarity, we focus on the source \nϕ (0) ( glyph[vector]r , t ) = √ N 211 2 µ e -iω 2 t ψ 211 ( glyph[vector]r ) + √ N 322 2 µ e -iω 3 t ψ 322 ( glyph[vector]r ) + c.c. (B12) \nfor the remainder of this appendix. The source (B12) represents the only two levels of the cloud relevant to the intra-cloud dynamics at small enough α , as argued in Sec. IV C and App. C 2 a.', '1. Frequency corrections': 'The source term includes components at the frequency ω 2 of the 211 bound state, \n1 6 ( ϕ (0) ) 3 ⊃ e -iω 2 t (2 µ ) 3 / 2 × (B13) ( 1 2 N 3 / 2 211 ( ψ 211 ) 2 ψ ∗ 211 + N 322 √ N 211 ψ 322 ψ ∗ 322 ψ 211 ) \nThis source contains components in resonance with the normal mode ψ 211 e -iω 2 t which would drive ϕ (1) to very large amplitudes, preventing a perturbative treatment. The frequency correction ω (1) 211 is therefore determined by demanding that those resonant components be exactly cancelled: \nω (1) 211 = -1 4 µ 2 × ∫ ( N 211 2 | ψ 211 | 4 + N 322 | ψ 322 | 2 | ψ 211 | 2 ) d 3 glyph[vector]r . (B14) \nThe two terms in Eq. (B14) correspond to self-energy corrections of the level 211 from its interaction with itself and with 322, respectively. The integral can be computed analytically by using the explicit form of the hydrogenic waveforms ψ 211 and ψ 322 . \nSince bound state wavefunctions scale as ψ nlm ∝ 1 /a -3 / 2 0 ∝ ( αµ ) 3 / 2 and only depend on glyph[vector]r through r/a 0 , \nthe frequency correction scales with α as ω (1) nlm ∼ α 3 µ . A denser cloud gives larger frequency corrections. \nWe calculated the integral of Eq. (B14) and the equivalent for ω (1) 322 and we found the corrections: \n∆ ω 211 glyph[similarequal] -λα 3 µ ( 1 . 2 × 10 -4 N 211 +3 . 5 × 10 -5 N 322 ) = -α 5 µ ( M pl f ) 2 ( 1 . 2 × 10 -4 ε 211 +3 . 5 × 10 -5 ε 322 ) (B15) \n∆ ω 322 glyph[similarequal] -λα 3 µ ( 3 . 5 × 10 -5 N 211 +1 . 4 × 10 -5 N 322 ) = -α 5 µ ( M pl f ) 2 ( 3 . 5 × 10 -5 ε 211 +1 . 4 × 10 -5 ε 322 ) (B16)', '2. l = 0 damped-driven oscillation': "When Ω -µ < µ and Ω glyph[negationslash] = ω n for any n , the source generates a forced bound oscillation which is damped by the BH. For example, when the cloud consists of particles in the 211 and 322 levels (B12), the frequency of the forced oscillation is ω ind = 2 ω 2 -ω 3 = µ (1 -7 α 2 / 36) < µ , so the oscillation is bound. \nThe bound state ϕ (1) ⊃ e -iω ind t Ψ (1) ( glyph[vector]r ) + c.c. satisfies the time-independent equation for the complex field Ψ (1) , \n( k 2 ind -glyph[vector] ∇ 2 -2 αµ r ) Ψ (1) ( glyph[vector]r ) = 1 2 N 211 √ N 322 (2 µ ) 3 / 2 ( ψ 211 ) 2 ψ ∗ 322 , (B17) \nwhere k 2 ind = µ 2 -ω 2 ind ≈ (7 / 18) α 2 µ 2 . \nWe expand Ψ (1) ( glyph[vector]r ) in the complete basis of the hydrogenic differential operator -∇ 2 -2 αµ/r , \n. \nΨ (1) ( glyph[vector]r ) = ∑ nlm c nlm ψ nlm + ∑ lm ∫ dkc ( k ) ψ klm , (B18) \nwhere the eigenfunctions nlm of the discrete spectrum satisfy \n( -∇ 2 -2 αµ r ) ψ nlm = -k 2 n ψ nlm , k 2 n = α 2 µ 2 n 2 , (B19) \nwith n a positive integer, and eigenfunctions klm of the continuous spectrum obey \n( -∇ 2 -2 αµ r ) ψ klm = k 2 ψ klm , k αµ ∈ (0 , + ∞ ) . (B20) \nMoreover, the eigenfunctions obey orthonormality conditions: \n∫ d 3 glyph[vector]r ψ ∗ klm ψ k ' l ' m ' = 2 πδ ( k ' -k ) δ m,m ' δ l,l ' , (B21b) \n∫ d 3 glyph[vector]r ψ ∗ nlm ψ n ' l ' m ' = δ n,n ' δ m,m ' δ l,l ' , (B21a) \n∫ d 3 glyph[vector]r ψ ∗ klm ψ nl ' m ' = 0 . (B21c) \nExplicitly, the states of the discrete spectrum are the usual bound hydrogenic wavefunctions, \nψ nlm ( r, θ, ϕ ) = R nl ( r ) Y m l ( θ, ϕ ) , (B22) \nwith the radial part \nR nl ( r ) = √ ( 2 na 0 ) 3 ( n -l -1)! 2 n ( n + l )! (B23) × exp [ -r na 0 ]( 2 r na 0 ) l L 2 l +1 n -l -1 [ 2 r na 0 ] , \nwhere L 2 l +1 n -l -1 ( x ) is the generalized Laguerre polynomial of degree n -l -1. \nThe states of the continuous spectrum are stationary Coulomb waves [142], 15 \nψ klm ( r, θ, ϕ ) = R kl ( r ) Y m l ( θ, φ ) (B24) \nwith the radial part \nR kl ( r ) = 2 ke π/ (2 ka 0 ) | Γ( l +1 -i/ ( ka 0 )) | (2 l +1)! × (2 kr ) l e -ikr 1 F 1 ( i/ ( ka 0 ) + l +1 , 2 l +2 , 2 ikr ) , (B25) \nwhere 1 F 1 is the confluent hypergeometric function of the first kind. \nTo obtain the coefficients c nlm , we put (B18) in (B17) and integrate both sides against ψ ∗ n ' l ' m ' . We can then use the Hermiticity of ( -∇ 2 -2 αµ/r ) (which in this case amounts to integrating by parts, so that -∇ 2 -2 αµ/r acts on ψ ∗ n ' l ' m ' ), along with (B19) and (B21) to find \nc nlm = 1 k 2 ind -k 2 n × ∫ 1 2 N 211 √ N 322 (2 µ ) 3 / 2 ( ψ 211 ) 2 ψ ∗ 322 ψ ∗ nlm d 3 glyph[vector]r . (B26) \nSimilarly, the values of the transform c ( k ) are obtained by integrating both sides of (B17) against ψ k ' l ' m ' . The analogue procedure then yields \nc ( k ) = 1 2 π 1 k 2 ind + k 2 × ∫ 1 2 N 211 √ N 322 (2 µ ) 3 / 2 ( ψ 211 ) 2 ψ ∗ 322 ψ ∗ klm d 3 glyph[vector]r . (B27) \nIt is appropriate to do these integrals in units of the Bohr radius a 0 = ( αµ ) -1 to reconstitute the dependence on α . The prefactors of k ind , k n and k are naturally in units of a -1 0 , while bound state wavefunctions are in units of \nwith \na -3 / 2 0 and continuum wavefunctions are in units of a -1 0 . The c nlm 's then have dimension a -1 0 µ -3 / 2 and c ( k ) has units of a -1 / 2 0 µ 3 / 2 . The amplitude of the induced oscillation ϕ (1) therefore has units of a -5 / 2 0 µ -3 / 2 = α 5 / 2 µ . \nThese overlap integrals are non-vanishing for l = 0 , 2 , 4 and m = 0. For l > 0 however, the angular momentum barrier suppresses the field amplitude at the horizon, and therefore the corresponding rates of absorption are smaller. This in turn leads to a smaller induced growth rate, as discussed previously. We therefore focus on l = m = 0 and ignore the l = 2 , 4 terms. \nFor l = 0 states, the power absorbed at the horizon in terms of complex field Ψ (1) goes as the square of the norm at the origin: \nP abs ≈ 4 α 2 (1 + 1 -a 2 ∗ ) λ 2 | Ψ(0) (1) | 2 . (B28) \nIn terms of particles in the cloud carrying energy ≈ µ , this contributes \n√ \n˙ N c ⊃ -Γ damp N 2 211 N 322 , (B29) \nΓ damp = P abs µN 2 211 N 322 . (B30)", '3. Non-relativistic emission': "Generally, the source term ( ϕ (0) ) 3 will generate some driving terms oscillating at the frequency ω NRE r = ω n + ω n ' -ω n ' . When ω NRE r > µ , the driven oscillation is free. These free emissions are non-relativistic because ω NRE r ≈ µ + O ( α 2 ) for the constituents of the superradiant cloud. The superscript NRE ('non-relativistic emissions') will be suppressed will be suppressed for the remainder of this section. \nGenerically, we seek to solve \n( ∂ 2 ∂t 2 -glyph[vector] ∇ 2 + µ 2 -2 αµ r ) ϕ r = e -iω r t f ( glyph[vector]r ) + c.c. , (B31) \nwhere e -iω r t f ( glyph[vector]r ) is a localized source of radiation with harmonic time-dependence, and ϕ r ⊂ ϕ (1) is the 'radiation' part of the field. The time-averaged differential power per solid angle that such a source emits in the radiation zone at infinity in the direction ( θ glyph[vector] k , ϕ glyph[vector] k ) is \nd 〈 P 〉 d Ω ( θ glyph[vector] k , ϕ glyph[vector] k ) = 2 ω r | glyph[vector] k | (4 π ) 2 λ 2 | ˜ f ( glyph[vector] k ) | 2 , (B32) \nwhere glyph[vector] k = ( √ ( ω r ) 2 -µ 2 )ˆ r is the momentum at spatial infinity, ˆ r is a radial unit vector pointing in the direction ( θ k , ϕ k ) and ˜ f ( glyph[vector] k ) is the 'Coulomb' transform \n˜ f ( glyph[vector] k ) = ∑ lm Y m l ( θ k , ϕ k ) ∫ d 3 glyph[vector]r (4 π )( -i ) l f ( glyph[vector]r ) ψ ∗ klm ( glyph[vector]r ) 2 k . (B33) \nFIG. 21. Top panel: decay rates of the n 00 hydrogenic levels, for n = 1 to 5, relative to their leading-order power-law behaviour as a function of α (for a BH with a ∗ = 0 . 9). Bottom panel: rate of the 211 × 211 → 322 × BH process, for a BH with a ∗ = 0 . 9, relative to its leading power-law behaviour Γ l . o . as a function of α (see Table V). As discussed in section III C, the deviation of this rate from its leading-order form is mostly driven by the same short-distance effects that modify the n 00 decay rates. \n<!-- image --> \nThis is analogous to the usual Fourier transform that one would compute for the emission rate in flat spacetime, with the regular spherical Bessel functions having been replaced by the appropriate regular Coulomb waves. \nFor non-relativistic emissions, k ∼ O ( a -1 0 ). It was noted earlier that f ( glyph[vector]r ) ∼ α 9 / 2 µ 3 . Furthermore, since it is a product of hydrogenic wavefunctions, f ( glyph[vector]r ) depends on glyph[vector]r only through the combination glyph[vector]r /a 0 . On the other hand, ψ klm ( glyph[vector]r ) / 2 k is dimensionless and depends on glyph[vector]r only through the combination kr ∼ r/a 0 , for non-relativistic k . Therefore, all the dependence of (B33) on α can be extracted by evaluating the intgeral in units of the Bohr radius a 0 = ( αµ ) -1 . Thus ˜ f ( glyph[vector] k ) ∼ α 3 / 2 and d 〈 P 〉 /d Ω ∼ λ 2 α 4 µ 2 . \nThe total radiated power is determined by integrating \n(B32) over solid angles: \nP NRE r = ∫ d Ω d 〈 P 〉 d Ω (B34) \nIn terms of particles in the cloud, and particles radiated to infinity with energy ω r ≈ µ , we have \n˙ N c ⊃ -Γ NRE r N nlm N n ' l ' m ' N n ' l ' m ' , (B35) \nwith the rate \nΓ NRE r = P NRE r µN nlm N n ' l ' m ' N n ' l ' m ' . (B36) \nA particularly important process is 211 × 211 → 322 × ∞ . This is sourced by \n1 6 ( ϕ (0) ) 3 ⊃ e -i (2 ω 3 -ω 2 ) t × 1 2 N 322 √ N 211 (2 µ ) 3 / 2 ( ψ 322 ) 2 ψ ∗ 211 +c.c. (B37) \nBy substituting \nf ( glyph[vector]r ) → 1 2 N 322 √ N 211 (2 µ ) 3 / 2 ( ψ 322 ) 2 ψ ∗ 211 (B38) \nin the above, we obtain the rate in table II.", '4. Relativistic emission': "The source term ( ϕ (0) ) 3 will also contain terms oscillating at the frequency ω RE r = ω n + ω n '' + ω n ''' . When ω RE r > µ , the driven oscillation is free. These free emissions are relativistic because ω r ≈ 3 µ + O ( α 2 ) for the constituents of the superradiant cloud. Cubic self-interactions would also generate relativistic emissions through ( ϕ (0) ) 2 in the equations of motion. In this case ω RE r ≈ 2 µ + α 2 . \nFor the remainder of this section, the superscript RE ('relativistic emissions') will be suppressed. As is the case for non-relativistic emissions, the radiated power is controlled by the integral (B33) which projects the source onto the Coulomb scattering state with outgoing momentum glyph[vector] k . The source f ( glyph[vector]r ) is a product of hydrogenic wavefunctions, \nf ( glyph[vector]r ) glyph[similarequal] -λ/ 6 (2 µ ) 3 / 2 N 3 / 2 211 ψ 3 211 = λ 768 π √ 70 α 3 / 2 a -3 0 ( r/a 0 ) 3 e -3 r/ (2 a 0 ) Y 33 N 3 / 2 211 (B39) \nFor non-relativistic emission, k ∼ a -1 0 so we need to use the full form of the Coulomb scattering state. In contrast, for relativistic emission, k ∼ µ ∼ α -1 a -1 0 , so ka 0 ∼ α -1 is a large parameter. As a result, we can expand the \nradial part of the Coulomb wavefunction around its flatspace, spherical Bessel function form. \nIt turns out that the contributions to ˜ f ( glyph[vector] k ) from the spherical Bessel function, and from the leadingα correction, are at the same order in α . This effectively occurs due to the contribution from the spherical Bessel function suffering a 'cancellation', making it higher-order in α than a naive guess based on the behaviour of f ( glyph[vector]r ) near the origin would have indicated. The integral against the leadingα correction term does not suffer this kind of cancellation, making the contributions from both of the same order. This is why our result for the emitted power (Eq. (15)) has the same α dependence as that derived in [7] using a flat-space approximation, but has a larger constant factor ([7] also treats emission as light-like, taking ω 2 r = k 2 rather than ω 2 = k 2 + µ 2 ). \nHigher-order corrections, and effects from working in the full Kerr metric instead of just a 1 /r potential, all contribute to the emitted power at higher order in α .", '1. Selection rules for mixing with damped states': "As explained in App. A, in the presence of a quartic self-coupling λ , one can view a background SR cloud ϕ c ( glyph[vector]r , t ) ∼ e -iµt ψ 211 ( t ) + e -iµt ψ 322 ( t ) + c.c as providing a time-dependent mixing potential V mixing ∼ λe -i 2 µt ( ψ 211 ( t ) 2 + ψ 211 ( t ) ψ 322 ( t ) + ψ 322 ( t ) 2 ) between states. In particular, if the mixing matrix element 〈 ψ n ' l ' m ' | V mixing | ψ ∗ nlm 〉 between a superradiant state ψ nlm and a decaying state ψ n ' l ' m ' is non-vanishing, then a forced oscillation ∝ ψ n ' l ' m ' is sustained and a growth instability is induced for ψ nlm . \nWe are therefore interested in the selection rules when V mixing ∼ ψ 2 211 ∼ Y 2 2 , V mixing ∼ ψ 211 ψ 322 ∼ Y 3 3 , and V mixing ∼ ψ 2 322 ∼ Y 4 4 . In each case, V mixing ∼ Y m '' l '' can be viewed as an element of an irreducible tensor operator representation of the rotation group with angular momentum numbers ( l '' , m '' ). 16 Considering further that, ψ ∗ nlm ∝ Y -m l , then by the Wigner-Eckart theorem 〈 ψ n ' l ' m ' | V mixing | ψ ∗ nlm 〉 ∝ ( l, l '' , -m,m '' | l ' m ' ), where ( j 1 , j 2 , m 1 , m 2 | J, M ) ≡ 〈 j 1 , j 2 , m 1 , m 2 | j 1 , j 2 , J, M 〉 is the Clebsch-Gordon (CG) coefficient for the addition of two irreducible angular momentum representations j 1 and j 2 . Furthermore, since the parity of a spherical harmonic Y m l is ( -1) l , inserting parity transformations inside the matrix element yields 〈 ψ n ' l ' m ' | Y m '' l '' | ψ ∗ nlm 〉 = ( -1) l + l ' + l '' 〈 ψ n ' l ' m ' | Y m '' l '' | ψ ∗ nlm 〉 . \nFrom this we get the selection rules for an induced growth instability to develop: \n- 1. Mixing with a damped state: m ' ≤ 0,\n- 2. CG coefficient: m '' = m ' + m ,\n- 3. CG coefficient: | l -l '' | ≤ l ' ≤ l + l '' ,\n- 4. Invariance under parity: l + l ' + l '' = even. \nThe first rule assumes that the spin in the BH is such that m ≥ 1 states are SR.", '2. Dependence of rates on the quantum numbers': '- a. Dependence of rates on overtone number n \nThe sources components ψ 211 and ψ 322 are peaked within a few Bohr radii, while hydrogenic wavefunctions in general are peaked further and further away from the origin as the quantum numbers are taken to be larger and larger. Thus, the interaction of a level nglyph[lscript]m with a combination of 211 and 322 will depend on the behavior of R nl near the a 0 : \nR nl ( r ∼ a 0 ) ∼ ( 2 ( n r + l +1) a 0 ) 3 / 2 ( 1 2( n r + l +1) ) 1 / 2 × ( ( n r +2 l +1)! n r ! ) 1 / 2 1 (2 l +1)! ( 2 r ( n r + l +1) a 0 ) l , (C1) \nwhere n r = n -l -1 is the radial quantum number. If n r →∞ , while l is held fixed, \nR nl ( r ∼ a 0 ) ∼ ( 1 n r a 0 ) 3 / 2 ∼ ( 1 na 0 ) 3 / 2 . (C2) \nThus, any overlap integral with R nl decreases as ∼ n -3 / 2 r ∼ n -3 / 2 . This is simply saying that as n r is taken larger, the characteristic volume of the driving wavefunction ψ nlm gets larger as ∼ ( na 0 ) 3 , and so the driving is uniformly diluted by that same factor. A forced oscillation with a ψ nl component as a source term therefore suffers the same suppression. \nRates (whether emission rates or rates of absorption into the BH) depend on the square of the forced oscillation and therefore behave as ∝ n -3 in the limit of large n . This means that ratios of emissions and absorption processes become independent of n . \nThe discussion in section IV C relied on the behavior of various ratios of rates at large n . To assess how fast the relevant ratios converge to the expected scaling in n , we plot them for the first 200 n (Figs. 22, 23, 24, 25, 26). \nFIG. 22. Behavior of the first term in the ratio in (65) as n → ∞ . As discussed in the paragraph below (65), and as expected from C 2 a, the ratio rapidly becomes independent of n and is > 1 for n glyph[greaterorsimilar] 10. \n<!-- image --> \nFIG. 23. Behavior of the ratio in (66) as n →∞ . As expected from C2a, the ratio rapidly becomes independent of n . \n<!-- image -->', "b. Mixing with l ' = 0 damped states": "The analysis of levels that can grow from 211 and 322 mixing with an l ' = 0 forced oscillation is done in the main text (IV C).", "c. Mixing with l ' > 0 damped states": "We give an exhaustive list of the possible processes involving mixing with l ' = 1 and l ' = 2 damped states. \nFor l ' = 1, \nγ 655 × BH(1 , -1) 322 × 322 ( ε eq 322 ) 2 ∼ 10 5 ( α 0 . 3 ) 15 ( M glyph[circledot] M ) Myr -1 , (C3a) \nFIG. 24. Behavior of the growth ratio 211 × 211 → n 22 × BH normalized to its value at n = 200. As stated in (69), the ratio scales as n -3 . \n<!-- image --> \nFIG. 25. Behavior of the ratio in (72) as n →∞ . As expected from C2a, the ratio rapidly becomes independent of n . \n<!-- image --> \nγ 654 × BH(1 , 0) 322 × 322 ( ε eq 322 ) 2 ∼ 10 4 ( α 0 . 3 ) 16 ( M glyph[circledot] M ) Myr -1 , (C3b) \nγ 543 × BH(1 , 0) 211 × 322 ε eq 211 ε eq 322 ∼ 10 6 ( α 0 . 3 ) 13 ( M glyph[circledot] M ) Myr -1 , (C3c) \nγ 432 × BH(1 , 0) 211 × 211 ( ε eq 211 ) 2 ∼ 10 7 ( α 0 . 3 ) 10 ( M glyph[circledot] M ) Myr -1 . (C3d) \nFor l ' = 2, \nγ 766 × BH(2 , -2) 322 × 322 ( ε eq 322 ) 2 ∼ 10 -2 ( α 0 . 3 ) 19 ( M glyph[circledot] M ) Myr -1 , (C4a) \nγ 765 × BH(2 , -1) 322 × 322 ( ε eq 322 ) 2 ∼ 10 -3 ( α 0 . 3 ) 19 ( M glyph[circledot] M ) Myr -1 , (C4b) \nFIG. 26. Behavior of the ratio in (74). As expected from C2a, the ratio rapidly becomes independent of n . \n<!-- image --> \nγ 764 × BH(2 , 0) 322 × 322 ( ε eq 322 ) 2 ∼ 10 -4 ( α 0 . 3 ) 20 ( M glyph[circledot] M ) Myr -1 , (C4c) \nγ 653 × BH(2 , 0) 211 × 322 ε eq 322 ε eq 211 ∼ 10 -3 ( α 0 . 3 ) 17 ( M glyph[circledot] M ) Myr -1 , (C4d) \nγ 542 × BH(2 , 0) 211 × 211 ( ε eq 211 ) 2 ∼ 10 -4 ( α 0 . 3 ) 14 ( M glyph[circledot] M ) Myr -1 . (C4e) \nClearly, rates for processes involving l ' ≥ 2 are too small to be relevant on astrophysical timescales. Rates from mixing with l ' = 1 states however can become quite large for α = O (0 . 1), but, similarly to processes with l ' = 0, they should be compared to depletion processes of the form nlm × 322 → 211 ×∞ . \nFirst, \n˙ ε 655 = γ 655 × BH(1 , -1) 322 × 322 ( 1 -γ 211 ×∞ 655 × 322 γ 655 × BH(1 , -1) 322 × 322 ε 211 ε 322 ) ε 2 322 ε 655 . (C5) \nThe depletion term dominates as long as \nα ˜ r 1 / 9 + glyph[lessorsimilar] ( κ 211 ×∞ 655 × 322 κ 655 × BH(1 , -1) 322 × 322 2 κ 211 ×∞ 322 × 322 κ 322 × BH(0 , 0) 211 × 211 ) 1 / 9 ≈ 0 . 7 . (C6) \nNext, \n˙ ε 654 = \nγ 654 × BH(1 , 0) 322 × 322 ( 1 -γ 211 ×∞ 654 × 322 γ 654 × BH(1 , 0) 322 × 322 ε 211 ε 322 ) ε 2 322 ε 654 . (C7) \nThe depletion term dominates as long as \nα ˜ r 1 / 10 + glyph[lessorsimilar] ( κ 211 ×∞ 654 × 322 κ 654 × BH(1 , 0) 322 × 322 2 κ 211 ×∞ 322 × 322 κ 322 × BH(0 , 0) 211 × 211 ) 1 / 10 ≈ 0 . 7 . (C8) \nTABLE IV. Rates for gravitational processes involved in the evolution of the scalar cloud. \n| Process | Rate ( γ/µ , Eq. (32)) |\n|-----------------------------------------|---------------------------------------------------|\n| Γ SR 211 | 4 × 10 - 2 α 8 ( a ∗ - 2 α (1 + √ 1 - a 2 ∗ )) |\n| Γ SR 322 | 8 × 10 - 5 α 12 ( a ∗ - α (1 + √ 1 - a 2 ∗ )) |\n| Γ SR 433 | 2 × 10 - 8 α 16 ( a ∗ - 2 3 α (1 + √ 1 - a 2 ∗ )) |\n| Γ SR 544 | 2 × 10 - 12 α 20 ( a ∗ - 1 α (1 + √ 1 - a 2 ∗ )) |\n| Γ GW,ann 211 Γ GW,ann 322 Γ GW,tr 322 → | 1 × 10 - 2 α 14 3 × 10 - 8 α 18 |\n| | 5 × 10 - 6 α 10 |\n| | 2 |\n| 211 | | \nNext, \n˙ ε 543 = γ 543 × BH(1 , 0) 211 × 322 ( 1 -γ 211 ×∞ 543 × 322 γ 543 × BH(1 , -1) 211 × 322 ) ε 322 ε 211 ε 543 . (C9) \nThe depletion term dominates as long as \nα ˜ r 1 / 7 + glyph[lessorsimilar] ( κ 211 ×∞ 543 × 322 κ 543 × BH(1 , 0) 211 × 322 ) 1 / 7 ≈ 1 . (C10) \nFinally, \n˙ ε 432 = γ 432 × BH(1 , 0) 211 × 211 ( 1 -γ 211 ×∞ 432 × 322 γ 432 × BH(1 , 0) 211 × 211 ε 322 ε 211 ) ε 322 ε 211 ε 432 . (C11) \nThe depletion term dominates as long as \nα glyph[lessorsimilar] ( κ 211 ×∞ 432 × 322 κ 432 × BH(1 , 0) 211 × 211 1 2 κ 322 × BH(0 , 0) 211 × 211 κ 211 ×∞ 322 × 322 ) 1 / 4 ≈ 1 . (C12) \nSince 211 SR stops for α ≥ 0 . 5, we conclude that the net growth rate of all four levels (and their radial overtones) is negative over the whole range of relevant parameter space.", 'Appendix D: Equilibrium ratio for moderate self-interactions': "We derive a more precise formula for the value of the time-independent equilibrium ratio by the system of equations (46). In terms of the γ rates, \n˙ ε 211 = γ 211 ×∞ 322 × 322 ε 211 ε 2 322 -2 γ 322 × BH 211 × 211 ε 2 211 ε 322 , (D1a) \n˙ ε 322 = -2 γ 211 ×∞ 322 × 322 ε 211 ε 2 322 + γ 322 × BH 211 × 211 ε 2 211 ε 322 . (D1b) \nTABLE V. Rates for quartic processes involving nonrelativistic bound states. \n| Process | Rate ( γ/µ , Eq. (32)) |\n|-----------------------|------------------------------------------------------|\n| Γ 322 × BH 211 × 211 | 4 . 3 × 10 - 7 α 11 ( M pl f ) 4 (1 + √ 1 - a 2 ∗ ) |\n| Γ 422 × BH 211 × 211 | 1 . 5 × 10 - 7 α 11 ( M pl f ) 4 (1 + √ 1 - a 2 ∗ ) |\n| Γ 322 × BH 211 × 411 | 2 . 5 × 10 - 8 α 11 ( M pl f ) 4 (1 + √ 1 - a 2 ∗ ) |\n| Γ 322 × BH 411 × 411 | 9 . 8 × 10 - 11 α 11 ( M pl f ) 4 (1 + √ 1 - a 2 ∗ ) |\n| Γ 433 × BH 211 × 322 | 9 . 1 × 10 - 8 α 11 ( M pl f ) 4 (1 + √ 1 - a 2 ∗ ) |\n| Γ 544 × BH 322 × 322 | 1 . 9 × 10 - 9 α 11 ( M pl f ) 4 (1 + √ 1 - a 2 ∗ ) |\n| Γ 544 × BH 211 × 433 | 1 . 1 × 10 - 9 α 11 ( M pl f ) 4 (1 + √ 1 - a 2 ∗ ) |\n| Γ 655 × BH 322 × 433 | 2 . 8 × 10 - 10 α 11 ( M pl f ) 4 (1 + √ 1 - a 2 ∗ ) |\n| Γ 655 × BH 211 × 544 | 3 . 6 × 10 - 12 α 11 ( M pl f ) 4 (1 + √ 1 - a 2 ∗ ) |\n| Γ 766 × BH 433 × 433 | 2 . 1 × 10 - 10 α 11 ( M pl f ) 4 (1 + √ 1 - a 2 ∗ ) |\n| Γ 877 × BH 433 × 544 | 5 . 2 × 10 - 12 α 11 ( M pl f ) 4 (1 + √ 1 - a 2 ∗ ) |\n| Γ 988 × BH 544 × 544 | 1 . 6 × 10 - 12 α 11 ( M pl f ) 4 (1 + √ 1 - a 2 ∗ ) |\n| Γ 1099 × BH 544 × 655 | 5 . 6 × 10 - 13 α 11 ( M pl f ) 4 (1 + √ 1 - a 2 ∗ ) |\n| Γ 433 × 200 211 × 422 | 1 . 1 × 10 - 9 α 7 ( M pl f ) 4 (1 + √ 1 - a 2 ∗ ) | \nTherefore, \n1 ε 211 ε 322 d dt ( ε 322 ε 211 ) = γ 322 × BH 211 × 211 +2 ( γ 322 × BH 211 × 211 -γ 211 ×∞ 322 × 322 ) ( ε 322 ε 211 ) -γ 211 ×∞ 322 × 322 ( ε 322 ε 211 ) 2 . (D2) \nThe zeros of the right-hand side are \nη B = 1 γ 211 ×∞ 322 × 322 ( γ 322 × BH 211 × 211 -γ 211 ×∞ 322 × 322 ± √ ( γ 322 × BH 211 × 211 ) 2 -γ 322 × BH 211 × 211 γ 211 ×∞ 322 × 322 +( γ 211 ×∞ 322 × 322 ) 2 ) . (D3) \nSince the right-hand side is an inverted parabola, the '+' solution is dynamically stable (attractive), while the ' -' solution is unstable. Parametrically, γ 322 × BH 211 × 211 ∝ α 11 and γ 211 ×∞ 322 × 322 ∝ α 8 . Therefore at small α , γ 322 × BH 211 × 211 < γ 211 ×∞ 322 × 322 , and so the ' -' root is negative. Moreover, the '+' root is \n( η B ) small α ≈ 1 2 γ 322 × BH 211 × 211 γ 211 ×∞ 322 × 322 = ε eq 322 ε eq 211 . (D4) \nTABLE VI. Rates for quartic processes leading to nonrelativistic emission. \n| Process | Rate ( γ/µ , Eq. (32)) |\n|-----------------------|-------------------------------|\n| Γ 100 ×∞ 211 × 211 | 1 . 3 × 10 - 7 α 8 ( M pl f ) |\n| Γ 100 ×∞ 211 × 322 | 8 . 5 × 10 - 9 α 8 ( M pl f ) |\n| Γ 100 ×∞ 322 × 322 | 1 . 1 × 10 - 10 α 8 ( M pl f |\n| Γ 211 ×∞ 322 × 411 | 3 . 8 × 10 - 9 α 8 ( M pl f ) |\n| Γ 211 ×∞ 322 × 322 | 1 . 1 × 10 - 8 α 8 ( M pl f ) |\n| Γ 211 ×∞ 322 × 433 | 2 . 6 × 10 - 9 α 8 ( M pl f ) |\n| Γ 211 ×∞ 433 × 433 | 9 . 2 × 10 - 11 α 8 ( M pl f |\n| Γ 211 ×∞ 322 × 544 | 6 . 1 × 10 - 11 α 8 ( M pl f |\n| Γ 211 ×∞ 433 × 544 | 1 . 9 × 10 - 11 α 8 ( M pl f |\n| Γ 211 ×∞ 544 × 544 | 4 . 2 × 10 - 13 α 8 ( M pl f |\n| Γ 322 ×∞ 544 × 544 | 4 . 4 × 10 - 11 α 8 ( M pl f |\n| Γ 322 ×∞ 433 × 544 | 7 . 8 × 10 - 10 α 8 ( M pl f |\n| Γ 21 - 1 ×∞ 322 × 322 | 2 . 3 × 10 - 10 α 8 ( M pl f |\n| Γ 211 ×∞ 655 × 322 | 7 . 3 × 10 - 13 α 8 ( M pl f |\n| Γ 211 ×∞ 655 × 433 | 4 . 6 × 10 - 13 α 8 ( M pl f |\n| Γ 211 ×∞ 655 × 544 | 6 . 9 × 10 - 14 α 8 ( M pl f |\n| Γ 211 ×∞ 655 × 655 | 1 . 1 × 10 - 15 α 8 ( M pl f |\n| Γ 322 ×∞ 655 × 433 | 3 . 7 × 10 - 11 α 8 ( M pl f |\n| Γ 322 ×∞ 655 × 544 | 1 . 6 × 10 - 11 α 8 ( M pl f |\n| Γ 322 ×∞ 655 × 655 | 6 . 2 × 10 - 13 α 8 ( M pl f |\n| Γ 433 ×∞ 766 × 766 | 5 . 6 × 10 - 13 α 8 ( M pl f | \nTABLE VII. Rates for self-interaction induced relativistic emission processes. \n| Process | Rate ( γ/µ , Eq. (32)) |\n|-----------------------|--------------------------------------|\n| Γ 2 → 1 211 (cubic) 1 | . 9 × 10 - 4 α 14 | C | 2 ( M pl f ) |\n| Γ 3 → 1 211 | 5 × 10 - 9 α 21 ( M pl f ) 4 ) |\n| Γ 3 → 1 322 | 6 × 10 - 14 α 27 ( M pl f 4 |", 'Appendix E: Boundary of the regime of early equilibrium': 'We derive a more precise formula for the value of f/M pl such that the SR growth of 211 is halted before O (1) of the spin is extracted. At early times, if we neglect the dependence of γ SR 211 on the BH spin a ∗ , \nε 211 ( t ) ≈ 1 GM 2 e γ SR 211 t . (E1) \nWe use this into \n˙ ε 322 = γ 322 × BH 211 × 211 ε 2 211 ε 322 , (E2) \nwhere we neglect the dependence of γ 322 × BH 211 × 211 on a ∗ . Therefore \nε 322 ( t ) ≈ 1 GM 2 exp [ γ 322 × BH 211 × 211 2 γ SR 211 1 G 2 M 4 ( e 2 γ SR 211 t -1 ) ] . (E3) \nThe condition for SR to be impeded is that \nγ SR 211 glyph[similarequal] 2 γ 322 × BH 211 × 211 ε 211 ( t ) ε 322 ( t ) . (E4) \nUsing the approximations (E1) and (E3), one finds that (E4) is satisfied at the time t eq such that \nγ SR 211 t eq ≈ 1 + 4 β log β -2 βW ( 1 2 βe 1 / 2 β ) 4 β , (E5) \nwhere \nβ ≡ G 2 M 4 γ SR 211 2 γ 322 × BH 211 × 211 , (E6) \nand W ( z ) is the product logarithm (sometimes called the Lambert W function). \nWhen γ SR 211 t glyph[similarequal] log( GM 2 ∆ a ∗ ), then SR has happened completely. So, in order for (E4) to be obtained before SR has run its course, we must have \n1 + 4 β log β -2 βW ( 1 2 βe 1 / 2 β ) 4 β glyph[lessorsimilar] log ( GM 2 ε max 211 ) . (E7) \n(E7) implicitly defines f thresh . Note that since M glyph[greatermuch] M pl , β glyph[greatermuch] 1 for much of parameter space. One can then approximate W ( z ) with the leading terms of its expansion around a large argument: W ( z ) → log z -log(log z ) as z → + ∞ . In this approximation, the left-hand side of (E7) becomes ≈ log √ 2 β log β , and the condition for SR to be halted early simplifies to \n2 γ SR 211 log( GM 2 ) glyph[lessorsimilar] γ 322 × BH 211 × 211 ( ε max 211 ) 2 . (E8)', 'Appendix F: Cloud mass': "Here we calculate the mass of the cloud in the case f → ∞ , i.e. in the purely gravitational case. We will do the computation for the 211 level for clarity, but it is straightforward to generalize the formalism to any nglyph[lscript]m level. To simplify notation, we drop the level subscripts for the rest of our discussion here. The cloud parameters are referring to 211, unless stated otherwise. \nSince the BH loses < 0 . 1% of its mass due to SR, we usually treat its mass to be constant, or, equivalent, that α is just a parameter. In the case of self-interactions, in particular, the cloud tends to grow to a smaller occupation number, which strengthens this assumption. A \nwhere \nω 211 = µ ( 1 -α 2 8 ) (F2e) \nWe define the ε with respect to the initial BH mass M i , i.e. ε = N/G ( M i ) 2 and Eqs. (F2) become: \n˙ ε = γ SR ε (F3) \n˙ a ∗ = -˙ α α [ 2 a ∗ -1 α (1 -α 2 / 8) ] (F5) \n˙ α = -α 2 i ( 1 -α 2 8 ) ˙ ε (F4) \nwhere α i = α ( t = 0), given by the initial BH mass. The usual treatment is to expand these equations for small α , which is equivalent to neglecting terms of order O ( α ). This reduces Eq. (F4) to ˙ α = 0. However, the expansion in Eq. (F5) has to be taken more carefully because the denominator is also small in this limit. By substituting Eq. (F4) it becomes evident that the first term is of order O ( α ), whereas the second is independent of α . Therefore, we can neglect the former, which gives the standard result ˙ a ∗ = -˙ ε . \nEq. (F4) has the following solution \nα = -2 √ 2 tanh [ 1 4 ( √ 2 α 2 i ε -4 arctanh [ α i 2 √ 2 ])] (F6) \nNow, Eq. (F5) can also be solved analytically. The result is \na ∗ = 1 α 2 [ a 0 α 2 i -2 √ 2arctanh ( 2 √ 2( α -α i ) -8 + αα i )] (F7) \nThe final spin of the BH is that which saturates the SR condition ω -m Ω H is \na fin ∗ = -8( -8 α fin + α 3 fin ) 16 + 64 α 2 fin -16 α 4 fin + α 6 fin , (F8) \nfurther simplification comes from setting ω glyph[similarequal] µ . By noting that ε = -˙ a ∗ in this regime, we get that ε max = ∆ a ∗ . The final a ∗ can be found by setting the SR rate equal to zero. Eventually, the maximum occupation number one gets is \nε max = a ∗ (0) -4 α 1 + 4 α 2 . (F1) \nIn general, the equations we need to solve are: \n˙ N = γ SR N (F2a) \n˙ M = -ω 211 γ SR N (F2b) \n˙ J = -γ SR N (F2c) \na ∗ ≡ J GM 2 (F2d) \nwhere the 'fin' superscript denotes final quantities, after the 211 cloud has been saturated and the BH has spun down. \nNow we can use Eqs. (F6), (F7) and (F8) to numerically solve for ε max , the final occupation number of the cloud. The mass of the cloud is then M c = ε max G M i ) 2 ω . \n) By neglecting the α 2 term in Eq. (F2e), i.e. by approximating ω glyph[similarequal] µ , we can get a simpler analytic result for the final BH mass. In this case, the equivalents of Eqs. (F6), (F7) are \nα = α i (1 -α i ε ) (F9) \na ∗ = a 0 -ε (1 -α i ε ) 2 (F10) \nwhich can be used along with Eq. (F8), truncated to O ( α 2 ), to give the final occupation number of the cloud. We find that \nε max = 1 -8 α 2 i +8 α 3 i a 0 -√ 1 -16 α 2 i +32 a 0 α 3 i -16 a 2 0 α 4 i 8( -α 3 i + a 0 α 4 i ) (F11) \nwhere a 0 = a ∗ ( t = 0). \nIn Fig. 27 we plot the ratio of the final cloud mass over the initial BH mass. We solve numerically Eq. (F8) with respect to ε max and compare it to the numerical evolution of Eqs. (F3)-(F5). We also plot the results of Eq. (F1) and Eq. (F11) for comparison. We find that the mass of the cloud can grow up to 7% of the initial BH mass. \nFIG. 27. Ratio of the final mass of the cloud to the initial BH mass. We plot points from the numerical evolution of Eqs. ((F3)-(F5)), the full analytic result of Eqs. ((F6)-(F8)), as well as the ˙ α = 0 approximation of Eq. (F1) and the ω 211 glyph[similarequal] µ, ˙ α glyph[negationslash] = 0 approximation of (F11). The cloud can grow to have a mass of up to 7% of the initial BH mass. This plot assumes an initial spin a ∗ ( t 0 ) = 0 . 9. \n<!-- image --> \n(", 'Appendix G: Self-gravity energy corrections': "The Poisson equation for the gravitational potential sourced by the cloud is \n∇ 2 Φ SG = 4 πGµ | ψ | 2 (G1) \nwhere ψ is the wavefunction of the cloud, i.e. \nψ ( r ) = ∑ nlm √ N nlm ψ nlm (G2) \nwhere N nlm are the occupation numbers of the levels and ψ nlm the hydrogenic wavefunctions. Treating Φ SG as a small perturbation, the energy correction of the ( n, l, m ) level is \n∆ ω nlm = 〈 nlm | µ Φ SG | nlm 〉 = -Gµ 2 ∫ | ψ nlm ( r ) | 2 ∫ | ψ ( r ' ) | 2 | r -r ' | d 3 r ' d 3 r (G3) \nExpanding 1 / | r -r ' | in spherical harmonics we get \n1 | r -r ' | = 4 π ∞ ∑ l ' =0 l ' ∑ m ' = -l ' 1 2 l ' +1 r l ' < r l ' +1 > Y m ' ∗ l ' ( θ ' , φ ' ) Y m ' l ' ( θ, φ ) , (G4) \nwhere r < ( > ) is the smallest (largest) of r and r ' . We can perform the integration over θ and φ , since ψ nlm ∝ Y m l . By the selection rules of the spherical harmonics we can write \nY m l Y m ∗ l = l ∑ k =0 c k,lm Y 0 2 k , c k,lm = ∫ | Y m l | 2 Y 0 ∗ 2 k dΩ (G5) \nTherefore, the integral over θ and φ selects m ' = 0 and l ' = 2 k , giving \n∆ ω nlm = -4 πGµ 2 l ∑ k =0 c k,lm 4 k +1 ∫ R nl ( r ) r 2 × ∫ | ψ ( r ' ) | 2 r 2 k < r 2 k +1 > Y 0 ∗ 2 k ( θ ' , φ ' )d 3 r ' d r (G6) \nwhere R nl are the hydrogenic radial wavefunctions. We will now make the simplifying assumption that the ψ given by Eq. (G2) is a sum of levels such that ( n, l, m ) = ( l +1 , l, l ), which is the case treated in this work. Since | ψ | 2 is integrated against Y 0 2 k , only the terms consisting of products of complex conjugates will survive. Thus, we can substitute the integrand as follows: \n| ψ ( r ' ) | Y 0 ∗ 2 k ( θ ' , φ ' ) → ∑ l ' =0 ∣ ∣ ∣ N 1 / 2 l ' +1 ,l ' ,l ' R l ' +1 ,l ' ( r ' ) ∣ ∣ ∣ 2 ∣ ∣ ∣ Y l ' l ' ( θ ' , φ ' ) ∣ ∣ ∣ 2 Y 0 ∗ 2 k ( θ ' , φ ' ) . (G7) \nThen the integral over θ ' and φ ' is just c k,l ' l ' , as defined in Eq. (G5). Note that this integral is non-zero only for l ' > k . Thus, we can re-write the sum ∑ l ' =0 → ∑ l ' = k . The coefficients c k,l ' l ' have a simple analytic form \nc k,l ' l ' = ( -1) k 2 l ' +1 √ 4 π √ 4 k +1 (2 l ' )!(2 k )!( l ' + k )! ( k !) 2 (2 l ' +2 k +1)!( l ' -k )! , for k < l ' . \n(G8) \nThe energy corrections are then \n∆ ω nlm = -4 πGµ 2 l ∑ k =0 c k,lm 4 k +1 ∑ l ' = k N l ' +1 ,l ' ,l ' c k,l ' ,l ' I kl ' nl (G9) \nwhere the last quantity is the radial integral given by \nI kl ' nl = ∫ R 2 nl ( r ) ∫ R 2 l ' +1 ,l ' ( r ' ) r 2 k < r 2 k +1 > r ' 2 r 2 d r ' d r, (G10) \nwhich can be calculated analytically. Assuming a simultaneous occupation of just 211 and 322, the corrections are \n∆ ω 211 glyph[similarequal] -α 3 µ GM 2 (0 . 19 N 211 +0 . 11 N 322 ) (G11) \n∆ ω 322 glyph[similarequal] -α 3 µ GM 2 (0 . 11 N 211 +0 . 09 N 322 ) (G12)", 'Appendix H: Frequency drifts': "The corrections to the energy of the 211 and 322 levels from self-interactions and self-gravity were calculated in Apps. B1 and G respectively. The angular frequency of a particle occupying 211 or 322 is: \nω 211 = µ ( 1 -α 2 8 ) -µα 5 ( M pl f ) 2 ( κ λ 1 ε 211 + κ λ 2 ε 322 ) -µα 3 ( κ gr 1 ε 211 + κ gr 2 ε 322 ) , (H1a) ω 322 = µ ( 1 -α 2 36 ) -µα 5 ( M pl f ) 2 ( κ λ 3 ε 211 + κ λ 4 ε 322 ) -µα 3 ( κ gr 3 ε 211 + κ gr 4 ε 322 ) , (H1b) \nwhere α = µMM pl -2 and κ λ 1 = 1 . 2 × 10 -4 , κ λ 2 = 3 . 5 × 10 -5 , κ gr 1 = 0 . 19, κ gr 2 = 0 . 11, κ λ 3 = 3 . 5 × 10 -5 , κ λ 4 = 1 . 4 × 10 -5 , κ gr 3 = 0 . 11 and κ gr 4 = 0 . 09 are numerical coefficients. \nIn what follows, we define the frequency ν as \nν ≡ ω 2 π . (H2) \nSo the frequency drifts ˙ ν are given by, \n˙ ν 211 = -µα 2 2 π [ 1 4 ˙ α α + α 2 ( M pl f ) 2 (H3a) × [ α ( κ λ 1 ˙ ε 211 + κ λ 2 ˙ ε 322 ) +5 ( κ λ 1 ε 211 + κ λ 2 ε 322 ) ˙ α ] + +[ α ( κ gr 1 ˙ ε 211 + κ gr 2 ˙ ε 322 ) + 3 ( κ gr 1 ε 211 + κ gr 2 ε 322 ) ˙ α ] ] \n˙ ν 322 = -µα 2 2 π [ 1 18 ˙ α α + α 2 ( M pl f ) 2 (H3b) × [ α ( κ λ 3 ˙ ε 211 + κ λ 4 ˙ ε 322 ) +5 ( κ λ 3 ε 211 + κ λ 4 ε 322 ) ˙ α ] + +[ α ( κ gr 3 ˙ ε 211 + κ gr 4 ˙ ε 322 ) + 3 ( κ gr 3 ε 211 + κ gr 4 ε 322 ) ˙ α ] ] \nto leading order in α for every term. \nThe mass of the BH evolves according to (37), which can be written equivalently as an equation for α as \n˙ α glyph[similarequal] -α 2 ( γ SR 211 ε 211 + γ SR 322 ε 322 -γ 322 × BH 211 × 211 ε 2 211 ε 322 ) , (H4) \nAs a result, the last terms in the second and third row of Eqs. (H3a) and (H3b) are parametrically suppressed by an additional power of α and ε i compared to the respective first term and thus will be neglected in what follows. In addition, all drifts are given to leading order in α and are the maximum possible for each individual regime. \nIn what follows we calculate the frequency drifts of the GWs coming from annihilations of two 211 particles and from transitions from 322 to 211. These are given by the relations ˙ ν ann ≡ 2 ˙ ν 211 and ˙ ν tr ≡ ˙ ν 322 -˙ ν 211 We separate the sources of frequency drifts in the following categories: \n- 1. Due to the change of the mass of the BH, given by the first terms of (H3a) and (H3b), denoted as ν α .\n- 2. Due to the change in the self-interaction energy, given by the second term of (H3a) and (H3b), denoted as ν λ .\n- 3. Due to the change in the self-gravitational energy, given by the third term of (H3a) and (H3b), denoted as ν gr . \nIn the regime of small self-interactions we treat the depletion due to gravitational radiation (annihilations and transitions) separately for points 2 and 3 above, and we denote by the superscript 'GW'. \nWe also note that there is an additional source of frequency drift coming from the change of the radial velocity of the BH to the observer, but for isolated black holes it is ˙ ν Doppler < 10 -19 Hz / s [40], which is negligible. \nFor reference, LIGO/Virgo continuous wave searches currently cover a range of positive to negative frequency derivatives of [83] \n2 × 10 -9 Hz/s through -1 × 10 -8 Hz/s . (H5) \nAll drift calculations carried out here are to leading approximation in α (which is accurate only for α glyph[lessmuch] a ∗ (0)) but the formalism includes in principle all higher-order corrections. At higher α the calculations can be carried out numerically using the full expressions and the numerical rates, but at α glyph[greaterorsimilar] 0 . 2 the approximation of the twolevel system essentially breaks down. We have verified that, for our purposes, the leading order approximation gives accurate results.", '1. Small self-coupling': 'Here we revisit the frequency drifts from purelygravitational interactions, i.e. f → ∞ as described in [13], which corresponds to region (A) of Fig. 3. There is a clear separation of times when different levels grow, so whenever a higher level gets populated, the lower ones have already fallen back into the BH, as their SR rates have become negative. In what follows, we will consider only 211, from which comes the stronger signal. \nThe interesting region for signatures is when the BH has spun down, the level has saturated and slowly gets depleted by radiating GWs. The only source of a frequency drift then comes from the gravitational self-energy of the cloud, given by the last line of Eqs. (H3a) and (H3b). In particular, the last term is exactly zero, since ˙ α = 0. \nThe 211 cloud obeys the equation ˙ ε 211 = -2 γ GW 211 × 211 α 14 ε 2 211 . The maximum drift comes about when ε 211 = ε max 211 glyph[similarequal] ∆ a ∗ (for a better estimate, see App. F), when SR shuts just off. For a ∗ (0) = 0 . 9 we find the drifts to be \n˙ ν λ, GW ann glyph[similarequal] 4 × 10 -22 Hz sec ( α 0 . 075 ) 19 ( µ 10 -12 eV ) 2 ( 10 19 GeV f ) 2 (H6) \n˙ ν gr,GW ann glyph[similarequal] 8 × 10 -17 Hz sec ( α 0 . 075 ) 17 ( µ 10 -12 eV ) 2 (H7) \nIn the small self-interactions regime, the drift coming from self-interactions is always subdominant to that of self-gravity in the parameter space of interest. \nThe drift can become larger than the range LIGO/Virgo cover (Eq. (H5)) only for α around 0 . 27, taking higher order α contributions into account.', '2. Moderate self-coupling': 'Here we are interested in the region where both levels are occupied and they drift away slowly, which corresponds to region (B) of Fig. 3. In this regime 211 reaches its maximum occupation ∆ a ∗ and we can use Eq. (47) to relate the ε 322 to ε 211 . Note that even though the BH has spun down due to the growth of 211, ˙ α glyph[negationslash] = 0, since particles fall back into the BH, as described by the last term of Eq. (H4). The resulting frequency drifts are as follows: \nDue to the change of the BH mass: \n˙ ν α ann glyph[similarequal] -10 -11 Hz sec ( 10 17 GeV f ) 4 ( µ 10 -12 eV ) 2 ( α 0 . 075 ) 17 (H8a) \n˙ ν α tr glyph[similarequal] 3 × 10 -12 Hz sec ( 10 17 GeV f ) 4 ( µ 10 -12 eV ) 2 ( α 0 . 075 ) 17 (H8b) \nThe negative sign in Eq. (H8a) comes from the fact that the SR rates are zero, so the BH is actually gaining mass by the depletion of 211, from the last term of Eq. (H4). \nDue to self-interactions: \n˙ ν λ ann glyph[similarequal] 6 × 10 -13 Hz sec ( 10 17 GeV f ) 6 ( µ 10 -12 eV ) 2 ( α 0 . 075 ) 19 (H9a) \n˙ ν λ tr glyph[similarequal] -2 × 10 -13 Hz sec ( 10 17 GeV f ) 6 ( µ 10 -12 eV ) 2 ( α 0 . 075 ) 19 (H9b) \nDue to self-gravity: \n˙ ν gr ann glyph[similarequal] 10 -11 Hz sec ( 10 17 GeV f ) 4 ( µ 10 -12 eV ) 2 ( α 0 . 075 ) 17 (H10a) \n˙ ν gr tr glyph[similarequal] -2 × 10 -12 Hz sec ( 10 17 GeV f ) 4 ( µ 10 -12 eV ) 2 ( α 0 . 075 ) 17 (H10b) \nThese are calculated for a ∗ (0) = 0 . 9. Note that α scalings of Eqs. (H8) and (H10) are the same, which comes from the fact that SR has shut off and the scalings in both ˙ α and ˙ ε i of Eqs. (H4) and of (H3) are set by the same term, i.e. γ 322 × BH 211 × 211 ε 2 211 ε 322 . This is why the numerical coefficients of both the annihilation and transition drifts are very close. In particular, for the annihilation drift we find more precisely that \n˙ ν α ann + ˙ ν gr ann (H11) \nglyph[similarequal] 1 . 4 × 10 -12 Hz sec ( 10 17 GeV f ) 4 ( µ 10 -12 eV ) 2 ( α 0 . 075 ) 17 \nWithin the moderate self-interactions regime, we find that self-interactions are the dominant source of frequency drift for f glyph[lessorsimilar] 8 . 5 × 10 16 ( α/ 0 . 1) GeV. The drift can become larger than the range LIGO/Virgo cover (Eq. (H5)) for f glyph[lessorsimilar] 5 . 6 × 10 16 ( α/ 0 . 1) 17 / 4 GeV. In Fig. 28 we plot the full annihilation frequency drift stemming from Eq. (H3a) in this regime. \nAnalogously, for transitions, self-interactions are the dominant source of frequency drift for f glyph[lessorsimilar] 10 17 ( α/ 0 . 1) GeV. The drift can become larger than the range LIGO/Virgo cover (Eq. (H5)) for f glyph[lessorsimilar] 4 × 10 16 ( α/ 0 . 1) 17 / 4 GeV. In Fig. 29 we plot the full annihilation frequency drift stemming from Eqs. (H3a) & (H3b), in this regime. \nFIG. 28. Frequency drift contours for annihilations of axions to GWs, given by twice the quantity in Eq. (H3a), in the moderate self-coupling regime. The gray shaded region above the dashed black contour is where the drift due to selfinteractions (second line of Eq. (H3a)) dominates. The red contour corresponds to the largest positive drift covered by LIGO/Virgo continuous searches, taken here to be 2 × 10 -9 Hz/s [83]. \n<!-- image -->', '3. Large self-coupling': 'We are interested in the part of the evolution where the levels have reached their equilibrium values, given by Eqs. (55a) and (55b), which corresponds to region (C) in Fig. 3. These are slowly drifting because of the slow spindown of the BH and the change of its mass. Neglecting the SR of ε 322 , which is subdominant, the spin evolves according to \n˙ a ∗ = -γ SR 211 ε eq 211 , (H12) \nand its mass changes according to Eq. (H4). By plugging in the equilibrium values of Eq. (55) we get \n˙ α = -2 α 2 γ SR 211 3 √ 3 √ γ SR 211 γ 211 ×∞ 322 × 322 γ 322 × BH 211 × 211 . (H13) \nFIG. 29. Frequency drift contours for GWs sourced by axion transitions from 322 to 211, given by the difference of Eq. (H3a) and Eq. (H3b), in the moderate self-coupling regime. The gray shaded region above the black contour (solid and dashed) is where the drift due to self-interactions (second line of Eqs. (H3)) dominates. The frequency drift is negative to the right (i.e. to the largeα side) of the solid black line. Note that here we are plotting the absolute value of the frequency drift. The red contour corresponds to the largest negative drift covered by LIGO/Virgo continuous searches, taken here to be -1 × 10 -8 Hz/s [83]. \n<!-- image --> \nThen, the equilibrium values evolve according to \n˙ ε = ˙ ε eq = ∂ε eq ∂a ∗ ˙ a ∗ + ∂ε eq ∂α ˙ α. (H14) \nThe second term of Eq. (H14) gives a subdominant contribution and is further suppressed by another power of α compared to the first term. The signal is maximum at the beginning when a ∗ glyph[similarequal] a ∗ (0). The resulting drifts are given below. \nDue to the change of the BH mass: \n˙ ν α ann glyph[similarequal] 2 × 10 -13 Hz sec ( f 10 15 GeV ) 2 ( µ 10 -12 eV ) 2 ( α 0 . 075 ) 8 (H15a) \n˙ ν α tr glyph[similarequal] -6 × 10 -14 Hz sec ( f 10 15 GeV ) 2 ( µ 10 -12 eV ) 2 ( α 0 . 075 ) 8 (H15b) \nDue to self-interactions: \n˙ ν λ ann glyph[similarequal] 3 × 10 -13 Hz sec ( f 10 15 GeV ) 2 ( µ 10 -12 eV ) 2 ( α 0 . 075 ) 7 (H16a) \n˙ ν λ tr glyph[similarequal] -10 -13 Hz sec ( f 10 15 GeV ) 2 ( µ 10 -12 eV ) 2 ( α 0 . 075 ) 7 (H16b) \nDue to the self-gravity: \n˙ ν gr ann glyph[similarequal] 5 × 10 -16 Hz sec ( f 10 15 GeV ) 4 ( µ 10 -12 eV ) 2 ( α 0 . 075 ) 5 (H17a) \n˙ ν gr tr glyph[similarequal] -10 -16 Hz sec ( f 10 15 GeV ) 4 ( µ 10 -12 eV ) 2 ( α 0 . 075 ) 5 (H17b) \nThese are calculated for a (0) = 0 . 9 as well. \nIn the large self-interactions regime, for α glyph[greaterorsimilar] 0 . 1 the change of the mass of the BH is the dominant source of frequency drift for annihilations. For α glyph[lessorsimilar] 0 . 1 selfinteractions are dominant. The drift can become larger than the range LIGO/Virgo cover (Eq. (H5)) for f glyph[greaterorsimilar] 3 × 10 16 ( α/ 0 . 1) -4 GeV, which is relevant above α glyph[similarequal] 0 . 1. \n∗ \nAnalogously for transitions, for α glyph[greaterorsimilar] 0 . 13, the change of the mass of the BH dominates and for α glyph[lessorsimilar] 0 . 13 self-interactions are dominant. The drift can become larger than the range LIGO/Virgo cover (Eq. (H5)) for f glyph[greaterorsimilar] 2 . 5 × 10 16 ( α/ 0 . 15) -4 GeV, which is relevant above α glyph[similarequal] 0 . 13.', 'Appendix I: Perturbations from BH companion': "When the primary BH has a companion, the perturbation in the gravitational potential induces mixing of different levels. In particular, SR levels can mix with non-SR ones, resulting in the depletion of the cloud. According to [45], the perturbation δV gr mixes the levels ψ i and ψ j according to \n〈 ψ j | δV c | ψ i 〉 = -αM c M ∑ l ≥ 2 ∑ | m |≤ l 4 π 2 l +1 Y m ∗ l ( θ c , φ c ) R l +1 c I ¯ r I Ω (I1) \nwhere the subscript M c is the mass of the companion, θ c , φ c its angular coordinates and R c its distance from the primary BH of mass M , whereas the constant α = GµM . We have also defined \nI r ≡ ∫ ∞ 0 d r r 2+ l R n j l j ( r ) R n i l i ( r ) , (I2) \nI Ω ≡ ∫ dΩ Y m j ∗ l j ( θ, φ ) Y m i l i ( θ, φ ) Y m l ( θ, φ ) (I3) \nwhere R nl is the radial part of the hydrogenic wavefunction and Y l m are the spherical harmonics. \nNote that the first sum in Eq. (I1) starts from l = 2, which demonstrates the fact that the first non-zero correction from gravity comes from the quadrupole term, as expected from the equivalence principle. 17 \nWe are interested in the mixing of the 211 level with non-SR levels of the BH, which can lead, in principle, to the depletion of our cloud. The dominant contribution comes from n = 2 , l = 1 , m = -1, and it is largest when the companion lies on the plane perpendicular to the spin of the primary BH, i.e. when θ c = π/ 2. \nThe horizon flux becomes positive , i.e. more axions fall back into the BH than are extracted due to SR [13], when \n∣ ∣ ∣ ∣ ∣ Γ j dump Γ i ∣ ∣ ∣ ∣ 1 / 2 ∣ ∣ ∣ ∣ 〈 ψ j | δV c | ψ i 〉 ∆ ω ji ∣ ∣ ∣ > 1 (I4) \n∣ ∣ where 'dump' denotes the non-SR level that mixes with the SR one, Γ are the superradiance rates, and ∆ ω ji is the difference of the energies between the two levels, which are given by [45] \n∣ \n∣ \nω nlm = µ ( 1 -α 2 2 n 2 -α 4 8 n 4 + (2 l -3 n +1) α 4 n 4 ( l +1 / 2) + 2 a ∗ mα 5 n 3 l ( l +1 / 2)( l +1) ) (I5) \nThe physical quantities measured for BH binaries are the BH masses, their spins and the orbital period. We assume that the companion is far away (which is where Eq. (I1) is valid), so we relate the distance to the orbital period using Kepler's 3rd Law: R 3 /T 2 = G ( M + M c ) / (4 π 2 ), where T is the orbital period. Then, the condition (I4) becomes parametrically: \nM c M 144 π 2 √ 3 a ∗ α 7 ( 1 + M c M ) ( µT ) 2 glyph[greaterorsimilar] 1 (I6) \nwhere we have omitted an O (1) factor in the α region of interest. \nThe cloud may also be depleted by resonances that can occur when the period of the companion hits the energy difference between two levels, as shown in [45]. To estimate when this happens, we can compare the period to the energy splitting of the two mixing levels. As the companion spirals closer to the primary BH, its orbital period increases. When it crosses the value ∆ ω -1 ji , we expect that the cloud will be significantly depleted. A \nmore careful analysis can be found in [45]. The condition, therefore, is \n1 6 a ∗ α 5 µT glyph[similarequal] 1 (I7) \nIn deriving the BH spin bounds in Sec. VI, we take into account both Eqs. (I6) and (I7).", 'Appendix J: Axion wind sensitivity projections': 'As discussed in section VIII A, given an axion coupling to nucleon spins, an axion oscillation ϕ ( t ) = ϕ 0 cos ωt will act on nuclei as an effective magnetic field B a ( t ) = B a cos ωt . For nuclei which are spin-polarized in an external magnetic field, with Larmor frequency ω 0 glyph[similarequal] ω , a transverse B a will induce a transverse magnetic moment \nµ a glyph[similarequal] µ 2 n N n B a ω 0 ω 2 -ω 2 0 + iωγ (J1) \nwhere µ n is the nuclear magnetic moment, N n is the total number of nuclei, and γ is the damping rate (in terms of the spin coherence time T 2 , γ = 2 /T 2 [114]). \nIn the absence of an axion forcing, the fluctuation spectrum for the transverse magnetic momentum is \nS µµ glyph[similarequal] µ 2 n N n γ 1 1 + T 2 2 ( ω -ω 0 ) 2 (J2) \nwhich is related to the response function (Eq. (J1)) by the fluctuation-dissipation relation [114]. \nIf we read out the transverse magnetic moment using a sufficiently sensitive magnetometer (e.g. a SQUID [111, 114]), then it is possible to detect fluctuations as small as the quantum fluctuations from Eq. (J2). With a sensor that is bounded by the Standard Quantum Limit [143, 144], this is possible over a bandwidth ∼ 1 /T 2 . Consequently, for an integration time of T glyph[greaterorsimilar] T 2 , we need \nµ 2 a S µµ /T glyph[similarequal] 1 2 µ 2 n N n B 2 a TT 2 glyph[greaterorsimilar] few (J3) \nin order to reliably detect an axion signal. \nTo cover an O (1) axion mass range, we need to operate in ∼ ω 0 T 2 different resonant configurations (we will not be careful about constant factors). Consequently, if our total experimental time is T tot , the time we spend in each configuration is T ∼ T tot / ( ω 0 T 2 ), and our sensitivity limit is 18 \nB 2 a glyph[greaterorsimilar] few × ω 0 µ 2 n N n T tot (J4) \nNote that, while this naive form does not depend on T 2 , the signal amplitude from Eq. (J1) is ∝ T 2 ; consequently, achieving a sensitive enough magnetometer may be easier for larger T 2 . As discussed in section VIII A, the CASPEr-Wind project aims to achieve spin-noiselimited sensitivities at frequencies in the kHz -30 kHz range [111].', 'Appendix K: Dark matter abundance': "In section VI A, we reviewed models in which an axion dark matter abundance is generated via the earlyuniverse misalignment mechanism. For attractive potentials, if the initial value of the axion field is tuned close to the top of its potential, then the generated dark matter abundance can be enhanced through the 'largemisalignment mechanism' [65]. In this appendix, we give formulae for the DM density obtained in this way. \nFor a general cosine potential of the form V ( ϕ ) = m 2 f 2 [1 -cos( ϕ/f )], the enhanced final density for a large initial misalignment is given by \nρ ρ π/ 2 glyph[similarequal] 0 . 2 [ t osc µ +4log t osc µ ] 2 (K1) \nt osc µ ≡ log [ 1 π -| θ 0 | 2 1 / 4 π 1 / 2 Γ(5 / 4) ] (K2) \nwhere ρ π/ 2 is the final density when the initial amplitude of the field is θ 0 = φ 0 /f = π/ 2, and t osc µ marks the onset of the oscillation in units of µ . \nFixing the final density to be the observed DM abundance today, we arrive at the relation [65] \nf DM m pl glyph[similarequal] 3 1 / 2 2 5 / 4 C 1 / 2 π/ 2 ( ρ ρ π/ 2 ) -1 / 2 ( H eq µ ) 1 / 4 (K3) \nwhere C π/ 2 glyph[similarequal] 1 . 15, H eq is the Hubble parameter at matter-radiation equality and m pl is the reduced Planck mass. We plot Eq. (K3) for different initial misalignments in Fig. 30, as a function of α = GMµ , for a 10 M glyph[circledot] BH. \nSomewhat separately, we can compare the energy density in a superradiant cloud to the DM energy density. The energy density of the cloud is ρ c ∼ θ 2 f 2 µ 2 , up to an O (1) prefactor, which can be found to be \nρ c = 1 2 ˙ ϕ 2 + 1 2 ( ∇ ϕ ) 2 + 1 2 µ 2 + 1 4! µ 2 f 2 ϕ 4 ∼ ( 1 + α 2 2( ˜ R + ) 2 + θ 2 4! ) θ 2 ( fµ ) 2 (K4) \nFIG. 30. The decay constant f of Eq. (K3) that gives the observed DM abundance today, as a function of α = GMµ for a 10 M glyph[circledot] BH. Note that the vertical axis is reversed. We are assuming a general cosine potential of the form V ( ϕ ) = µ 2 f 2 (1 -cos( ϕ/f )) and plot for different large initial misalignments from the top of the potential, following the results of [65]. We also plot for usual misalignment values of | θ 0 | = 1 , π -1. \n<!-- image --> \n- [1] A. Gruzinov, (2016), arXiv:1604.06422 [astro-ph.HE].\n- [2] R. Penrose and R. 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1998PhRvD..57.2436B
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(Anti-)evaporation of Schwarzschild-de Sitter black holes
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1998-01-01
| 13 | 0.45 | 162 |
['-', '-', '-', '-', '-', '-', '-', '-', 'astrophysics', '-']
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[]
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We study the quantum evolution of black holes immersed in a de Sitter background space. For black holes whose size is comparable to that of the cosmological horizon, this process differs significantly from the evaporation of asymptotically flat black holes. Our model includes the one-loop effective action in the s-wave and large N approximation. Black holes of the maximal mass are in equilibrium. Unexpectedly, we find that nearly maximal quantum Schwarzschild-de Sitter black holes anti-evaporate. However, there is a different perturbative mode that leads to evaporation. We show that this mode will always be excited when a pair of cosmological holes nucleates.
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https://arxiv.org/pdf/hep-th/9709224.pdf
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{'(Anti-)Evaporation of Schwarzschild-de Sitter Black Holes': 'Raphael Bousso ∗ and Stephen W. Hawking † \nDepartment of Applied Mathematics and Theoretical Physics University of Cambridge Silver Street, Cambridge CB3 9EW \nDAMTP/R-97/26 hep-th/9709224 submitted to Phys. Rev. D', 'Abstract': 'We study the quantum evolution of black holes immersed in a de Sitter background space. For black holes whose size is comparable to that of the cosmological horizon, this process differs significantly from the evaporation of asymptotically flat black holes. Our model includes the one-loop effective action in the s-wave and large N approximation. Black holes of the maximal mass are in equilibrium. Unexpectedly, we find that nearly maximal quantum Schwarzschildde Sitter black holes anti-evaporate. However, there is a different perturbative mode that leads to evaporation. We show that this mode will always be excited when a pair of cosmological holes nucleates. \n† [email protected]', '1 Introduction': 'Of the effects expected of a quantum theory of gravity, black hole radiance [1] plays a particularly significant role. So far, however, mostly asymptotically flat black holes have been considered. In this work, we investigate a qualitatively different black hole spacetime, in which the black hole is in a radiative equilibrium with a cosmological horizon. \nThe evaporation of black holes has been studied using two-dimensional toy models, in which one-loop quantum effects were included [2, 3, 4]. We have recently shown how to implement quantum effects in a more realistic class of two-dimensional models, which includes the important case of dimensionally reduced general relativity [5]. The result we obtained for the trace anomaly of a dilaton-coupled scalar field will be used here to study the evaporation of cosmological black holes. \nWe shall consider the Schwarzschild-de Sitter family of black holes. The size of these black holes varies between zero and the size of the cosmological horizon. If the black hole is much smaller than the cosmological horizon, the effect of the radiation coming from the cosmological horizon is negligible, and one would expect the evaporation to be similar to that of Schwarzschild black holes. Therefore we shall not be interested in this case. Instead, we wish to investigate the quantum evolution of nearly degenerate Schwarzschild-de Sitter black holes. The degenerate solution, in which the black hole has the maximum size, is called the Nariai solution [6]. In this solution the two horizons have the same size, and the same temperature. Therefore they will be in thermal equilibrium. Intuitively, one would expect any slight perturbation of the geometry to cause the black hole to become hotter than the background. Thus, one may suspect the thermal equilibrium of the Nariai solution to be unstable. The initial stages of such a run-away would be an interesting and novel quantum gravitational effect quite different from the evaporation of an asymptotically flat black hole. In this paper we will investigate whether, and how, an instability develops in a two-dimensional model derived from fourdimensional general relativity. We include quantum effects at the one-loop level. \nThe paper is structured as follows: In Sec. 2 we review the Schwarzschildde Sitter solutions and the Nariai limit. We discuss the qualitative expectations for the evaporation of degenerate black holes, which motivate our one-loop study. The two-dimensional model corresponding to this physical situation is presented in Sec. 3, and the equations of motion are de- \nrived. In Sec. 4 the stability of the quantum Nariai solution under different types of perturbations is investigated. We find, quite unexpectedly, that the Schwarzschild-de Sitter solution is stable, but we also identify an unstable mode. Finally, the no-boundary condition is applied in Sec. 5 to study the stability of spontaneously nucleated cosmological black holes.', '2.1 Metric': 'The neutral, static, spherically symmetric solutions of the Einstein equation with a cosmological constant Λ are given by the Schwarzschild-de Sitter metric \nwhere \nds 2 = -V ( r ) dt 2 + V ( r ) -1 dr 2 + r 2 d Ω 2 , (2.1) \nV ( r ) = 1 -2 µ r -Λ 3 r 2 ; (2.2) \nd Ω 2 is the metric on a unit two-sphere and µ is a mass parameter. For 0 < µ < 1 3 Λ -1 / 2 , V has two positive roots r c and r b , corresponding to the cosmological and the black hole horizons, respectively. The limit where µ → 0 corresponds to the de Sitter solution. In the limit µ → 1 3 Λ -1 / 2 the size of the black hole horizon approaches the size of the cosmological horizon, and the above coordinates become inappropriate, since V ( r ) → 0 between the two horizons. Following Ginsparg and Perry [7], we write \n9 µ 2 Λ = 1 -3 /epsilon1 2 , 0 ≤ /epsilon1 /lessmuch 1 . (2.3) \nThen the degenerate case corresponds to /epsilon1 → 0. We define new time and radial coordinates ψ and χ by \nτ = 1 /epsilon1 √ Λ ψ ; r = 1 √ Λ [ 1 -/epsilon1 cos χ -1 6 /epsilon1 2 ] . (2.4) \nIn these coordinates the black hole horizon corresponds to χ = 0 and the cosmological horizon to χ = π . The new metric obtained from the transformations is, to first order in /epsilon1 , \nds 2 = -1 Λ ( 1 + 2 3 /epsilon1 cos χ ) sin 2 χ dψ 2 + 1 Λ ( 1 -2 3 /epsilon1 cos χ ) dχ 2 (2.5) + 1 Λ (1 -2 /epsilon1 cos χ ) d Ω 2 2 . \nThis metric describes Schwarzschild-de Sitter solutions of nearly maximal black hole size. \nIn these coordinates the topology of the spacelike sections of Schwarzschildde Sitter becomes manifest: S 1 × S 2 . In general, the radius, r , of the two-spheres varies along the S 1 coordinate, χ , with the minimal (maximal) two-sphere corresponding to the black hole (cosmological) horizon. In the degenerate case, the two-spheres all have the same radius.', '2.2 Thermodynamics': 'The surface gravities of the two horizons are given by [8] \nκ c , b = √ Λ ( 1 ∓ 2 3 /epsilon1 ) + O ( /epsilon1 2 ) , (2.6) \nwhere the upper (lower) sign is for the cosmological (black hole) horizon. In the degenerate case, the two horizons have the same surface gravity, and, since T = κ/ 2 π , the same temperature. They are in thermal equilibrium; one could say that the black hole loses as much energy due to evaporation as it gains due to the incoming radiation from the cosmological horizon. Away from the thermal equilibrium, for nearly degenerate Schwarzschild-de Sitter black holes, one could make the simplifying assumption that the horizons still radiate thermally, with temperatures proportional to their surface gravities. This would lead one to expect an instability: By Eq. (2.6), the black hole will be hotter than the cosmological horizon, and will therefore suffer a net loss of radiation energy. To investigate this suspected instability, a two-dimensional model is constructed below, in which one-loop terms are included.', '3 Two-dimensional Model': "The four-dimensional Lorentzian Einstein-Hilbert action with a cosmological constant is \nS = 1 16 π ∫ d 4 x ( -g IV ) 1 / 2 [ R IV -2Λ -1 2 N ∑ i =1 ( ∇ IV f i ) 2 ] , (3.1) \nwhere R IV and g IV are the four-dimensional Ricci scalar and metric determinant, and the f i are scalar fields which will carry the quantum radiation. \nWe shall consider only spherically symmetric fields and quantum fluctuations. Thus, we make a spherically symmetric metric ansatz, \nds 2 = e 2 ρ ( -dt 2 + dx 2 ) + e -2 φ d Ω 2 , (3.2) \nwhere the remaining two-dimensional metric has been written in conformal gauge; x is the coordinate on the one-sphere and has the period 2 π . Now the spherical coordinates can be integrated out, and the action is reduced to \nS = 1 16 π ∫ d 2 x ( -g ) 1 / 2 e -2 φ [ R +2( ∇ φ ) 2 +2 e 2 φ -2Λ -N ∑ i =1 ( ∇ f i ) 2 ] , (3.3) \nwhere the gravitational coupling has been rescaled into the standard form. Note that the scalar fields have acquired an exponential coupling to the dilaton in the dimensional reduction. In order to take quantum effects into account, we will find the classical solutions to the action S + W ∗ . W ∗ is the scale-dependent part of the one-loop effective action for dilaton coupled scalars, which we derived in a recent paper [5]: \nW ∗ = -1 48 π ∫ d 2 x ( -g ) 1 / 2 [ 1 2 R 1 ✷ R -6( ∇ φ ) 2 1 ✷ R -2 φR ] . (3.4) \nThe ( ∇ φ ) 2 term will be neglected; we justify this neglect at an appropriate place below. \nFollowing Hayward [9], we render this action local by introducing an independent scalar field Z which mimics the trace anomaly. The f fields have the classical solution f i = 0 and can be integrated out. Thus we obtain the action \nS = 1 16 π ∫ d 2 x ( -g ) 1 / 2 [( e -2 φ + κ 2 ( Z + wφ ) ) R -κ 4 ( ∇ Z ) 2 +2+2 e -2 φ ( ∇ φ ) 2 -2 e -2 φ Λ ] , (3.5) \nwhere \nκ ≡ 2 N 3 . (3.6) \nThere is some debate about the coefficient of the φR term in the effective action. Our result [5] corresponds to the choice w = 2; the RST coefficient [3] corresponds to w = 1, and the result of Nojiri and Odintsov [10] can be represented by choosing w = -6. In Ref. [9], probably erroneously, w = -1 \nwas chosen. We take the large N limit, in which the quantum fluctuations of the metric are dominated by the quantum fluctuations of the N scalars; thus, κ /greatermuch 1. In addition, for quantum corrections to be small we assume that b ≡ κ Λ /lessmuch 1. To first order in b , we shall find that the behaviour of the system is independent of w . \nFor compactness of notation, we denote differentiation with respect to t ( x ) by an overdot (a prime). Further, we define for any functions f and g : \n∂f ∂g ≡ -˙ f ˙ g + f ' g ' , ∂ 2 g ≡ -g + g '' , (3.7) \nand \nwhere η satisfies \n∂ 2 η = 0 . (3.15) \nThe remaining freedom in η can be used to satisfy the constraint equations for any choice of ρ , ˙ ρ , φ and ˙ φ on an initial spacelike section. This can be seen most easily by decomposing the fields and the constraint equations into Fourier modes on the initial S 1 . By solving for the second term on \nδf δg ≡ ˙ f ˙ g + f ' g ' , δ 2 g ≡ g + g '' . (3.8) \nVariation with respect to ρ , φ and Z leads to the following equations of motion: \n-( 1 -wκ 4 e 2 φ ) ∂ 2 φ +2( ∂φ ) 2 + κ 4 e 2 φ ∂ 2 Z + e 2 ρ +2 φ ( Λ e -2 φ -1 ) = 0; (3.9) \n( 1 -wκ 4 e 2 φ ) ∂ 2 ρ -∂ 2 φ +( ∂φ ) 2 +Λ e 2 ρ = 0; (3.10) \n∂ 2 Z -2 ∂ 2 ρ = 0 . (3.11) \nThere are two equations of constraint: \n( 1 -wκ 4 e 2 φ ) ( δ 2 φ -2 δφδρ ) -( δφ ) 2 = κ 8 e 2 φ [ ( δZ ) 2 +2 δ 2 Z -4 δZδρ ] ; (3.12) ( 1 -wκ 4 e 2 φ ) ( ˙ φ ' -˙ ρφ ' -ρ ' ˙ φ ) -˙ φφ ' = κ 8 e 2 φ [ ˙ ZZ ' +2 ˙ Z ' -2 ( ˙ ρZ ' + ρ ' ˙ Z )] . (3.13) \nFrom Eq. (3.11), it follows that \nZ = 2 ρ + η, (3.14) \nthe right hand side of Eq. (3.12), and by using Eqs. (3.14) and (3.15), the first constraint yields one algebraic equation for each Fourier coefficient of η . Similarly, the second constraint yields one algebraic equation for the time derivative of each Fourier coefficient of η . If the initial slice was noncompact, this argument would suffice. Here it must be verified, however, that η and ˙ η will have a period of 2 π . The problem reduces to the question whether the two constant mode constraint equations can be satisfied. Indeed, while for each oscillatory mode of η , there are two degrees of freedom (the Fourier coefficient and its time derivative), the second time derivative of the constant mode coefficient, ¨ η 0 , must vanish by Eq. (3.15). Thus there is only one degree of freedom, ˙ η 0 , for the two constant mode equations. However, since we have introduced no odd modes (i.e., modes of the form sin kx ) in the perturbation of φ , none of the fields will contain any odd modes. Since each term in Eq. (3.13) contains exactly one spatial derivative, each term will be odd. Therefore all even mode components of the second constraint vanish identically. In particular the constant mode component will thus be automatically satisfied. Then the freedom in ˙ η 0 can be used to satisfy the constant mode component of the remaining constraint, Eq. (3.12), through the first 1 term on the right hand side.", '4.1 Perturbation Ansatz': 'With the model developed above we can describe the quantum behaviour of a cosmological black hole of the maximal mass under perturbations. The Nariai solution is still characterised by the constancy of the two-sphere radius, e -φ . Because of quantum corrections, this radius will no longer be exactly Λ -1 / 2 . Instead, the solution is given by \ne 2 ρ = 1 Λ 1 1 cos 2 t , e 2 φ = Λ 2 , (4.1) \nwhere \n1 Λ 1 = 1 8Λ [ 4 -( w +2) b + √ 16 -8( w -2) b +( w +2) 2 b 2 ] ; (4.2) \nΛ 2 = 1 2 wκ [ 4 + ( w +2) b -√ 16 -8( w -2) b +( w +2) 2 b 2 ] . (4.3) \nExpanding to first order in b , one obtains: \n1 Λ 1 ≈ 1 Λ ( 1 -wb 4 ) ; (4.4) \nΛ 2 ≈ Λ ( 1 -b 2 ) . (4.5) \nLet us now perturb this solution so that the two-sphere radius, e -φ , varies slightly along the one-sphere coordinate, x : \ne 2 φ = Λ 2 [1 + 2 /epsilon1σ ( t ) cos x ] , (4.6) \nwhere we take /epsilon1 /lessmuch 1. We will call σ the metric perturbation . A similar perturbation could be introduced for e 2 ρ , but it does not enter the equation of motion for σ at first order in /epsilon1 . This equation is obtained by eliminating ∂ 2 Z and ∂ 2 ρ from Eq. (3.9) using Eqs. (3.11) and (3.10), and inserting the above perturbation ansatz. This yields \n¨ σ σ = a cos 2 t -1 , (4.7) \nwhere \nTo first order in b , one finds that \na ≡ 2 √ 16 -8( w -2) b +( w +2) 2 b 2 4 -wb (4.8) \na ≈ 2 + b, (4.9) \nwhich means that w , and therefore the φR term in the effective action, play no role in the horizon dynamics at this level of approximation. This is also the right place to discuss why the term √ -g ( ∇ φ ) 2 1 ✷ R in the effective action can be neglected. In conformal coordinates this term is proportional to ( ∂φ ) 2 ρ . Thus, in the ρ -equation of motion, Eq. (3.9), it will lead to a ( ∂φ ) 2 term, which is of second order in /epsilon1 and can be neglected. In the φ -equation of motion, Eq. (3.10), it yields terms proportional to κ that are of first order in /epsilon1 . They will enter the equation of motion for σ via the κe 2 φ ∂ 2 Z term in Eq. (3.10). Thus they will be of second order in b and can be dropped. The neglect of the log µ 2 term [5] can be justified in the same way.', '4.2 Horizon Tracing': 'In order to describe the evolution of the black hole, one must know where the horizon is located. The condition for a horizon is ( ∇ φ ) 2 = 0. Eq. (4.6) yields \n∂φ ∂t = /epsilon1 ˙ σ cos x, ∂φ ∂x = -/epsilon1σ sin x. (4.10) \nTherefore, the black hole and cosmological horizons are located at \nx b ( t ) = arctan ∣ ∣ ∣ ˙ σ σ ∣ ∣ ∣ ∣ , x c ( t ) = π -x b ( t ) . (4.11) \n∣ \n∣ To first order in /epsilon1 , the size of the black hole horizon, r b , is given by \nr b ( t ) -2 = e 2 φ [ t,x b ( t )] = Λ 2 [1 + 2 /epsilon1δ ( t )] , (4.12) \nwhere we define the horizon perturbation \nδ ≡ cos x b = σ ( 1 + ˙ σ 2 σ 2 ) -1 / 2 . (4.13) \nWe will focus on the early time evolution of the black hole horizon; the treatment of the cosmological horizon is completely equivalent. \nTo obtain explicitly the evolution of the black hole horizon radius, r b ( t ), one must solve Eq. (4.7) for σ ( t ), and use the result in Eq. (4.13) to evaluate Eq. (4.12). If the horizon perturbation grows, the black hole is shrinking: this corresponds to evaporation. It will be shown below, however, that the behaviour of δ ( t ) depends on the initial conditions chosen for the metric perturbation, σ 0 and ˙ σ 0 .', '4.3 Classical Evolution': 'As a first check, one can examine the classical case, κ = 0. This has a = 2, and Eq. (4.7) can be solved exactly. From the constraint equations, Eq. (3.12) and (3.13), it follows that \n˙ σ = σ tan t. (4.14) \nTherefore the appropriate boundary condition at t = 0 is ˙ σ 0 = 0. The solution is \nσ ( t ) = σ 0 cos t . (4.15) \nThen Eq. (4.13) yields \nδ ( t ) = σ 0 = const . (4.16) \nSince the quantum fields are switched off, no evaporation takes place; the horizon size remains that of the initial perturbation. This simply describes the case of a static Schwarzschild-de Sitter solution of nearly maximal mass, as given in Eq. (2.6).', '4.4 Quantum Evolution': "When we turn on the quantum radiation ( κ > 0) the constraints no longer fix the initial conditions on the metric perturbation. There will thus be two linearly independent types of initial perturbation. The first is the one we were forced to choose in the classical case: σ 0 > 0, ˙ σ 0 = 0. It describes the spatial section of a quantum corrected Schwarzschild-de Sitter solution of nearly maximal mass. Thus one might expect the black hole to evaporate. For a > 2, Eq. (4.7) cannot be solved analytically. Since we are interested in the early stages of the evaporation process, however, it will suffice to solve for σ as a power series in t . Using Eq. (4.13) one finds that \nδ ( t ) = σ 0 [ 1 -1 2 ( a -1)( a -2) t 2 + O ( t 4 ) ] ≈ σ 0 [ 1 -1 2 bt 2 ] . (4.17) \nThe horizon perturbation shrinks from its initial value. Thus, the black hole size increases , and the black hole grows, at least initially, back towards the maximal radius. One could say that nearly maximal Schwarzschild-de Sitter black holes 'anti-evaporate'. \nThis is a surprising result, since intuitive thermodynamic arguments would have led to the opposite conclusion. The anti-evaporation can be understood in the following way. By specifying the metric perturbation, the radiation distribution of the Z field is implicitly fixed through the constraint equations, (3.12) and (3.13). Our result shows that radiation is heading towards the black hole if the boundary condition σ 0 > 0, ˙ σ 0 = 0 is chosen. \nLet us now turn to the second type of initial metric perturbation: σ 0 = 0, ˙ σ 0 > 0. Here the spatial geometry is unperturbed on the initial slice, but it is given a kind of 'push' that corresponds to a perturbation in the radiation bath. Solving once again for σ with these boundary conditions, and using \nEq. (4.13), one finds for small t : \nδ ( t ) = ˙ σ 0 t 2 . (4.18) \nThe horizon perturbation grows. This perturbation mode is unstable, and leads to evaporation. \nWe have seen that the radiation equilibrium of a Nariai universe displays unusual and non-trivial stability properties. The evolution of the black hole horizon depends crucially on the type of metric perturbation. Nevertheless, one may ask the question whether a cosmological black hole will typically evaporate or not. Cosmological black holes cannot come into existence through classical gravitational collapse, since they live in an exponentially expanding de Sitter background. The only natural way for them to appear is through the quantum process of pair creation [7]. This pair creation process can also occur in an inflationary universe, because of its similarity to de Sitter space [8, 11, 12]. The nucleation of a Lorentzian black hole spacetime is described as the analytic continuation of an appropriate complex solution of the Einstein equations, which satisfies the no boundary condition [13]. We will show below that the no boundary condition selects a particular linear combination of the two types of initial metric perturbation, thus allowing us to determine the fate of the black hole.", '5 No Boundary Condition': 'To obtain the unperturbed Euclidean Nariai solution in conformal gauge, one performs the analytic continuation t = iτ in the Lorentzian solution, Eq. (4.1). This yields \nand \n( ds IV ) 2 = e 2 ρ ( dτ 2 + dx 2 ) + e -2 φ d Ω 2 , (5.1) \ne 2 ρ = 1 Λ 1 1 cosh 2 τ , e 2 φ = Λ 2 . (5.2) \nIn four dimensions, this describes the product of two round two-spheres of slightly different radii, Λ -1 / 2 1 and Λ -1 / 2 2 . The analytic continuation to a Lorentzian Nariai solution corresponds to a path in the τ plane, first along the real τ axis, from τ = -∞ to τ = 0, and then along the imaginary axis from t = 0 to t = π/ 2. This can be visualised geometrically by cutting \nthe first two sphere in half, and joining to it a Lorentzian 1 + 1-dimensional de Sitter hyperboloid. Because the ( τ, x ) sphere has its north (south) pole at τ = ∞ ( τ = -∞ ), it is convenient to rescale time: \nsin u = 1 cosh τ , (5.3) \nor, equivalently, \ncos u = -tanh τ, cot u = -sinh τ, du = dτ cosh τ . (5.4) \nWith the new time coordinate u , the solution takes the form \n( ds IV ) 2 = 1 Λ 1 ( du 2 +sin 2 udx 2 ) + 1 Λ 2 d Ω 2 . (5.5) \nNow the south pole lies at u = 0, and the nucleation path runs to u = π/ 2, and then parallel to the imaginary axis ( u = π/ 2 + iv ) from v = 0 to v = ∞ . The perturbation of e 2 φ , Eq. (4.6) introduces the variable σ , which satisfies the Euclidean version of Eq. (4.7): \nsin 2 u d 2 σ du 2 +sin u cos u dσ du -( 1 -a sin 2 u ) σ = 0 . (5.6) \nIn addition, the nature of the Euclidean geometry enforces the boundary condition that the perturbation vanish at the south pole: \nσ ( u = 0) = 0 . (5.7) \nOtherwise, e 2 φ would not be single valued, because the coordinate x degenerates at this point. This leaves ˙ σ as the only degree of freedom in the boundary conditions at u = 0. \nIt will be useful to define the parameter c by the relation c ( c + 1) ≡ a . The classical case, a = 2, corresponds to c = 1; for small b , they receive the quantum corrections a = 2 + b and c = 1 + b/ 3. With the boundary condition, Eq. (5.7), the equation of motion for σ , Eq. (5.6), can be solved exactly only for integer c ( a = 2, 6, 12, 20,. . . ). The solution is of the form \nσ ( u ) = ∑ 0 ≤ k<c/ 2 A k sin( c -2 k ) u, (5.8) \nwith constants A k . Even for non-integer c , however, this turns out to be a good approximation in the region 0 ≤ u ≤ π/ 2 of the ( u, v ) plane. Since we are interested in the case where b /lessmuch 1, the sum in Eq. (5.8) contains only one term, and we use the approximation 2 \nσ ( u ) ≈ ˜ A sin cu. (5.9) \nIt is instructive to consider the classical case first. (Physically, this is questionable, since the no boundary condition violates the constraints at second order in /epsilon1 .) For b = 0, the solution σ ( u ) = ˜ A sin u is exact. Along the Lorentzian line ( u = π/ 2 + iv ), this solution becomes σ ( v ) = ˜ A cosh v . By transforming back to the Lorentzian time variable t , one can check that this is the stable solution found in the previous section, with σ 0 = ˜ A , ˙ σ 0 = 0. For real ˜ A , it is real everywhere along the nucleation path. Thus, when the quantum fields are turned off, the Euclidean formalism predicts that the unstable mode will not be excited. This is a welcome result, since there are no fields that could transport energy from one horizon to another. \nOnce b is non-zero, however, it is easy to see that ∂σ/∂u no longer vanishes at the origin of Lorentzian time, u = π/ 2. This indicates that the unstable mode, ˙ σ 0 /negationslash = 0, will be excited. Unfortunately, checking this is not entirely straightforward, because σ is no longer real everywhere along the nucleation path. One must impose the condition that σ and ˙ σ be real at late Lorentzian times. We will first show that this can be achieved by a suitable complex choice of A . One can then calculate the horizon perturbation, δ , from the real late-time evolution of the metric perturbation, σ , to demonstrate that evaporation takes place. \n∂σ/∂v σ , \nusing the approximation, agrees with the numerical result to machine accuracy. Therefore the relative error in Eq. (5.15) is the same as in Eq. (5.9); in both equations it is located practically entirely in the magnitude of the prefactor. - These statements hold for 0 ≤ b ≤ 1, which really is a wider interval than necessary. \nFrom Eq. (5.9) one obtains the Lorentzian evolution of σ , \nσ ( v ) = ˜ A sin c ( π 2 + iv ) (5.10) \n= ˜ A ( sin cπ 2 cosh cv + i cos cπ 2 sinh cv ) . (5.11) \nFor late Lorentzian times (i.e., large v ), cosh cv ≈ sinh cv ≈ e cv / 2, so the equation becomes \nσ ( v ) ≈ 1 2 ˜ A ( ie -icπ/ 2 ) e cv . (5.12) \nThis can be rendered purely real by choosing the complex constant ˜ A to be \nwhere A is real. \n˜ A = A ( -ie icπ/ 2 ) , (5.13) \nNow we can return to the question whether the Euclidean boundary condition leads to evaporation. After transforming the time coordinate, the expression for the growth of the horizon perturbation, Eq. (4.13), becomes \nδ ( v ) = σ 1 + cosh 2 v ( ∂σ/∂v σ ) 2 -1 / 2 . (5.14) \nThe late time evolution is given by σ ( v ) = A 2 e cv . This yields, for large v , \nδ ( v ) ≈ A 2 e cv ( 1 + c 2 e 2 v ) -1 / 2 ≈ A 2 c exp( b 3 v ) . (5.15) \nThis result confirms that pair created cosmological black holes will indeed evaporate. The magnitude of the horizon perturbation is proportional to the initial perturbation strength, A . The evaporation rate grows with κ Λ. This agrees with intuitive expectations, since κ measures the number of quantum fields that carry the radiation.', '6 Summary': 'We have investigated the quantum stability of the Schwarzschild-de Sitter black holes of maximal mass, the Nariai solutions. From four-dimensional spherically symmetric general relativity with a cosmological constant and \nN minimally coupled scalar fields we obtained a two-dimensional model in which the scalars couple to the dilaton. The one-loop terms were included in the large N limit, and the action was made local by introducing a field Z which mimics the trace anomaly. \nWe found the quantum corrected Nariai solution and analysed its behaviour under perturbations away from degeneracy. There are two possible ways of specifying the initial conditions for a perturbation on a Lorentzian spacelike section. The first possibility is that the displacement away from the Nariai solution is non-zero, but its time derivative vanishes. This perturbation corresponds to nearly degenerate Schwarzschild-de Sitter space, and somewhat surprisingly, this perturbation is stable at least initially. The second possibility is a vanishing displacement and non-vanishing derivative. These initial conditions lead directly to evaporation. The different behaviour of these two types of perturbations can be explained by the fact that the initial radiation distribution is implicitly specified by the initial conditions, through the constraint equations. \nIf neutral black holes nucleate spontaneously in pairs on a de Sitter background, the initial data will be constrained by the no boundary condition: it selects a linear combination of the two types of perturbations. By finding appropriate complex compact instanton solutions we showed that this condition leads to black hole evaporation. Thus neutral primordial black holes are unstable.', 'References': "- [1] S. W. Hawking: Particle creation by black holes . Commun. Math. Phys. 43 , 199 (1974).\n- [2] C. G. Callan, S. B. Giddings, J. A. Harvey and A. Strominger: Evanescent black holes . Phys. Rev. D 45 , 1005 (1992), hep-th/9111056.\n- [3] J. G. Russo, L. Susskind and L. Thorlacius: Black hole evaporation in 1 + 1 dimensions . Phys. Lett. B292 , 13 (1992), hep-th/9201074.\n- [4] J. G. Russo, L. Susskind and L. Thorlacius: The endpoint of Hawking radiation . Phys. Rev. D 46 , 3444 (1992), hep-th/9206070. \n- [5] R. Bousso and S. W. Hawking: Trace anomaly of dilaton coupled scalars in two dimensions . Submitted to Phys. Rev. Lett.; preprint no. DAMTP/R-97/25, hep-th/9705236.\n- [6] H. Nariai: On a new cosmological solution of Einstein's field equations of gravitation . Sci. Rep. Tohoku Univ. Ser. I 35 , 62 (1951).\n- [7] P. Ginsparg and M. J. Perry: Semiclassical perdurance of de Sitter space . Nucl. Phys. B222 , 245 (1983).\n- [8] R. Bousso and S. W. Hawking: Pair creation of black holes during inflation . Phys. Rev. D 54 , 6312 (1996), gr-qc/9606052.\n- [9] J. D. Hayward: Entropy in the RST model . Phys. Rev. D 52 , 2239 (1995), gr-qc/9412065.\n- [10] S. Nojiri and S. D. Odintsov: Trace anomaly and non-local effective action for 2D conformally invariant scalar interacting with dilaton . Eprint no. hep-th/9706009.\n- [11] R. Bousso and S. W. Hawking: The probability for primordial black holes . Phys. Rev. D 52 , 5659 (1995), gr-qc/9506047.\n- [12] R. Bousso: Charged Nariai black holes with a dilaton . Phys. Rev. D 55 , 3614 (1997), gr-qc/9608053.\n- [13] J. B. Hartle and S. W. Hawking: Wave function of the Universe . Phys. Rev. D 28 , 2960 (1983)."}
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2000ApJ...532L..29G
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Binary Black Hole Mergers from Planet-like Migrations
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2000-01-01
| 7 | 0.47 | 162 |
['accretion', 'accretion disks', 'stars binaries close', 'black hole physics', 'galaxies quasars', 'astrophysics']
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[]
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If supermassive black holes (BHs) are generically present in galaxy centers, and if galaxies are built up through hierarchical merging, BH binaries are at least temporary features of most galactic bulges. Observations suggest, however, that binary BHs are rare, pointing toward a binary lifetime far shorter than the Hubble time. We show that, almost regardless of the detailed mechanism, all stellar dynamical processes are too slow in reducing the orbital separation once orbital velocities in the binary exceed the virial velocity of the system. We propose that a massive gas disk surrounding a BH binary can effect its merger rapidly, in a scenario analogous to the orbital decay of super-Jovian planets due to a proto-planetary disk. As in the case of planets, gas accretion onto the secondary (here a supermassive BH) is integrally connected with its inward migration. Such accretion would give rise to quasar activity. BH binary mergers could therefore be responsible for many or most quasars.
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[]
| 2 |
https://arxiv.org/pdf/astro-ph/9912111.pdf
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{'Binary Black Hole Mergers from Planet-like Migrations': 'Andrew Gould \nOhio State University, Department of Astronomy, Columbus, OH, USA E-mail: [email protected] \nHans-Walter Rix \nMax-Planck-Institut fur Astronomie, Heidelberg, Germany E-mail: [email protected]', 'ABSTRACT': 'If supermassive black holes (BHs) are generically present in galaxy centers, and if galaxies are built up through hierarchical merging, BH binaries are at least temporary features of most galactic bulges. Observations suggest, however, that binary BHs are rare, pointing towards a binary lifetime far shorter than the Hubble time. We show that, regardless of the detailed mechanism, all stellar-dynamical processes are insufficient to reduce significantly the orbital separation once orbital velocities in the binary exceed the virial velocity of the system. We propose that a massive gas disk surrounding a BH binary can effect its merger rapidly, in a scenario analogous to the orbital decay of super-jovian planets due to a proto-planetary disk. As in the case of planets, gas accretion onto the secondary (here a supermassive BH) is integrally connected with its inward migration. Such accretion would give rise to quasar activity. BH binary mergers could therefore be responsible for many or most quasars. \nSubject headings: accretion disks - binaries: close - black hole physics - quasars', '1. Introduction': "Supermassive black holes (BH) are nearly ubiquitous in nearby galaxy nuclei (e.g. Ho 1999). These BHs formed very early, probably during the epoch of quasars, z ∼ > 2, and are now largely dormant remnants of quasars. In the hierarchical picture \nof structure formation, present day galaxies are the product of successive mergers (e.g. White 1996), and indeed there is evidence for many mergers in the highz universe (Abraham et al. 1996). Hence, it appears almost inevitable that modern galaxies should harbor, or at least should have once harbored, multiple BHs that were collected during their merger history (Kauffman & Haehnelt 1999). \nBHs of mass M ∼ > 10 7 M /circledot will quickly find their way to the center of a merger remnant by dynamical friction. Logically, there are only three possibilities. First, BH pairs could merge to form a single, larger BH. Second, the pairs of BHs could form binaries that would remain at galaxy centers to this day. Finally, a third BH could also fall in, leading to a three-body interaction violent enough to expel any number of the three BHs from the galaxy (Begelman, Blandford, & Rees 1980). While in principle this means that all three holes could be ejected, in practice such a violent ejection event is unlikely unless the binary's internal velocity is much higher than the escape velocity from the galaxy ( ∼ > 2000 kms -1 ); in this case, the binary would be in the late stages of merging anyway (see § 2). Since the broad lines of quasars are not often observed to be displaced from the narrow lines by such high velocities, the fraction of binaries with such high internal velocities cannot be large, and therefore triple ejection cannot be common. Hence, mergers generically produce BH binaries, and these binaries either merge on timescales short compared to a Hubble time, or they are present in galaxies today. \nObservationally, there is evidence only for a few massive BH binaries (e.g., Lehto & Valtonen 1996) and in none of these cases is the evidence absolutely compelling. Theoretically, it has proven difficult to construct viable merger scenarios for these BH binaries. Here we first review this difficulty of driving the merger by the stellar-dynamical means that are discussed in the literature. We then propose a gas-dynamical alternative.", '2. Near Impossibility of Stellar Dymnamics-Driven Mergers': "If a BH binary could (somehow) be driven to a sufficiently small orbit, then gravitational radiation would increasingly sap energy from the system and so engender a merger. For a circular orbit with an initial velocity v gr , the time T to a \nmerger due to gravitational radiation is given by \nv gr = c ( 5 256 GM 2 tot µTc 3 ) 1 / 8 = 3400 kms -1 ( M 2 tot /µ 8 × 10 8 M /circledot ) 1 / 8 ( T 10 Gyr ) -1 / 8 (1) \nwhere M tot = M 1 + M 2 is the total mass, µ = M 1 M 2 /M tot is the reduced mass, and where we have normalized to the case M 1 = M 2 = 10 8 M /circledot . Note that for fixed total mass, the equal-mass case gives a lower limit on this required velocity, and that the result depends only very weakly on the total mass. \nHowever, as we now show it is almost impossible to achieve this velocity by any conceivable stellar-dynamical process. The basic problem is that when the orbital velocity v orb is about equal to the stellar velocity dispersion σ ∼ 200 km s -1 , the total mass in stars within a volume circumscribed by the BH orbital radius ( a ∼ 5 pc M tot / 10 8 M /circledot ) is about M tot . If all of these stars were expelled from the BH binary at speed v orb ( M 2 /M ) 1 / 2 (Rajagopal & Romani 1995 and references therein) the binding energy of the binary would increase by only a factor ∼ e . However, to get from a virial velocity of ∼ 200km/s to v gr (eq. 1), would require N e ∼ 6 e -foldings in binding energy. Hence, the binary will clear out a hole in the stellar distribution, and dynamical friction will be shut down (Quinlan 1996; Quinlan & Hernquist 1997). \nThe most efficient conceivable process to rejuvenate the orbital decay would be to equip the binary with an intelligent 'captain'. Like a fisherman working in over-fished waters, whenever the captain saw that the binary was running out of stars to expel, she would steer the binary to the densest unexploited region of the galaxy. To effect the merger, this would mean systematically moving through and expelling all the stars within a region containing about N e M tot in stars. For a galaxy with an r -2 density profile, this implies expelling all the stars within a radius r = N e GM tot / 2 σ 2 ∼ 60 pc, where we have made the evaluation for M tot = 2 × 10 8 M /circledot and σ = 200 kms -1 . \nThe real difficulty of the captain's work is best understood by considering the last e -folding before gravitational radiation can take over. For v orb /greatermuch σ , the cross section for hard interactions (including gravitational focusing) is ∼ < πa 2 v orb /σ . If each incident particle is expelled with speed v orb ( M 2 /M ) 1 / 2 (Rajagopal & Romani 1995), then the binding energy E b decays at d ln E b /dt ∼ 2 πa 2 v orb ρ/M = GρP , where P is the period, and ρ is the local density. The last e -folding alone would require a time t ∼ [ Gρ ( r ) P ] -1 ∼ 2 π ( r/σ ) 2 /P ∼ 2 Gyr, where we have assumed r ∼ 30 pc and our other canonical parameters. Thus, even with the captain's careful guidance, the full \nmerger requires a large fraction of a Hubble time. Moreover, comparing this decay rate with the standard formula for the decay of translational energy E t (Binney & Tremaine 1987) yields, \nd ln E b dt ∼ < 0 . 1 ( σ v orb ) 3 d ln E t dt . (2) \nThat is, d ln E b /d ln E t ∼ < 10 -4 , so that the binary would be driven by dynamical friction back to the center of the Galaxy before it had completed 10 -4 of an e -folding of energy loss. Hence, the captain would have to initiate 10 4 'course changes' in the last e -folding alone. Since the 'captain' must in fact be some random process, the only source of such 'course changes' is brownian motion due to continuous interaction with other compact objects. However, for stars of mass m in an r -2 profile, the range of such Brownian motion is ∆ ln r ∼ m/M tot , i.e., too small by several orders of magnitude. In contrast to ordinary Brownian motion, the present system has an 'external' energy source, the binary's binding energy. However, it follows from equation (2) that even if all of this donated energy were acquired by the binary's transverse motion, the brownian motion would be only slightly augmented. In any event, most of the donated energy goes to the stars, not the binary. Infall of globular clusters might well give the binary an occassional jolt, but these would be far too infrequent to drive the merger. In brief, any sort of mechanism to drive a merger by ordinary dynamical friction, no matter how contrived, is virtually ruled out. \nThe only loophole to this argument is that we have assumed circular binary orbits. If an instability existed that systematically drove the BH binaries toward eccentricity e → 1 orbits, then either the binaries would suffer enhanced gravitational radiation (for a fixed semi-major axis) or could even merge in a head-on collision. Fukushige, Ebisuzaki, & Makino (1992) first suggested such an instability based on the following qualitative argument: dynamical friction is more effective at low speeds than high speeds and hence, in the regime where the ambient particles interact with the binary mainly by encounters with its individual members ( v orb ∼ < σ ), the binary would suffer more drag at apocenter than pericenter, tending to make the orbit more eccentric. Fukushige et al. (1992) presented numerical simulations that gave initial support to this conjecture. There are, however, two reasons for believing that this effect cannot drive mergers. First, several groups have conducted more sophisticated simulations, and these do not show any strong tendency for e → 1 (Makino et al. 1994; Rajagopal & Romani 1995; Quinlan & Hernquist 1997). Second, once the binary entered the regime v orb /greatermuch σ , the ambient particles would interact with \nthe binary as whole, and so there is no reason to expect any drive toward high eccentricities. Hence, while this loophole is not definitively closed, neither does it look particularly promising.", '3. Gas Dynamical Solution': "Begelman, Blandford & Rees (1980) were the first to suggest that gas infall may 'lead to some orbital evolution'. But, at the time it was not clear that all other mechanism to overcome the BH hangup would most likely fail. \nTo resolve the above dilemma, we suggest that gas dynamics play the decisive role in orbital decay, forcing the secondary BH to 'migrate' in toward the primary in a manner analogous to the migration of planets. Such migration has been proposed to account for the discovery of jovian-mass and superjovian-mass planets at ∼ < 1 AU from solar-type stars, while it is generally believed that such massive planets can only be created several AU from the stars (Trilling et al. 1998). Artymowicz & Lubow (1994, 1996) simulated interactions between moderately unequal-mass binaries and accretion disks, which is more directly relevant to the present case than extreme-ratio (planetary) systems. They did not follow the orbital evolution as has been done in more recent work on planets, but only evaluated the instantaneous effect of the torques. They found a migration to higher eccentricities was a larger effect than migration to smaller orbits. Regardless of which effect dominates, one would expect the final merger to be from circular rather than radial orbits: if the binary is driven toward radial orbits, its emission of gravitational radiation near pericenter will eventually pull in the apocenter of the orbit, decoupling the binary from the disk and allowing the gravitational radiation to circularize the orbit before final coalescence. \nFor migration to work, the galaxy merger that creates the BH binary must eventually dump at least M 2 worth of gas into the inner ∼ 5 pc of the merger remnant where the binary coalescence has gotten 'hung up'. Whether this happens on timescales short compared to a dynamical time at 5 pc ( ∼ 10 7 yr), leading to tremendous gas densities and ensuing rapid star formation (Taniguchi & Wada 1996), or whether the gas accumulates over a longer timescale and so does not trigger a starburst, the basic scenario will be the same. \nThere is every reason to expect mergers effect such a gas accumulation. First, \nquasars must gorge themselves on gas to reach their present size. Hence, regardless of whether our picture of binary mergers is correct, this much gas must find its way to central BHs. Second, there is substantial evidence that many quasars are in either recent merger remnants or at least significantly disturbed galaxies (Kirhakos et al. 1999 and references therein). Hence, it seems likely that mergers are the most efficient means to drive gas to the center. Third, many spiral bulges and ellipticals have cuspy profiles populated by metal rich stars whose total mass is comparable to that of their massive BHs (van der Marel 1999). Thus, it must be possible to funnel huge amounts of gas to the centers of galaxies. \nIn planet migration, the migration timescale is similar to the accretion timescale for growing the planet because the two processes are governed by the same phenomena, gravitational torques and dissipation (Trilling et al. 1998; A. Nelson 1998, private communication). We expect the same to be true of migration of BH binaries. Thus, there should be a grand accretion disk around the primary with a 'gap' opened up by the secondary. Material should be transported across this gap to a second, smaller accretion disk surrounding the secondary BH. The total energy liberated by this smaller accretion disk should be ∼ /epsilon1M 2 c 2 ∼ 2 × 10 61 ( M 2 / 10 8 M /circledot ) ergs, where we have taken the efficiency to be /epsilon1 = 0 . 1, producing a quasar-like appearance during this phase.", '4. Discussion': "While our suggestion, driven by the lack of alternatives, makes few unambigous predictions, it does open several lines of investigation that could help test and flesh out our picture. \nFirst, merging binaries would appear very much like quasars, since our picture of the migrating secondary is essentially identical to the standard picture of a quasar. The one difference is that the jet from a migrating BH could precess if the orbit of the secondary were substantially misaligned relative to the accretion disk. At present, however, we have no method of estimating how often significant misalignment should occur. \nSecond, the redshift of the broad lines from a migrating BH's accretion disk should be offset from the redshift of the host galaxy (as traced perhaps by the narrow lines). Since the migration probably accelerates with time, most migrating \nquasars should have v orb ∼ σ . Nevertheless, some should have substantially higher offsets, and measuring the distribution of these offsets would allow one to trace the migration process. However, if no offsets were observed, this would not in itself rule out our hypothesis. It could be, for example, that migrating binaries in merger remnants are preferentially buried in a larger, roughly spherical cloud of dusty gas. In this case, they would have more similarity to ultra-luminous infrared galaxies (ULIRGs) than to quasars, and the line centers of their emission would be at the galaxy velocity, not that of the secondary. \nThird, it is at least possible that one would see two broad-line systems, one from the primary and one from the secondary. Since broad lines are by definition broad ( ∼ > 3000 kms -1 ), the existence of two systems would not easily be recognized for v orb ∼ σ . However, distinct peaks might be discernible when the BHs were closer to merger. On the other hand, it may be that the major supply of gas lies outside the orbit of the secondary, and hence the primary does not generate a significant broad-line region. \nFourth, it will be important to carry out simulations to determine whether the gas dissipation timescale is short enough for the acceting material to follow the binary inward. This is certainly the case for the simulations that have been done for extreme mass-ratio (planetary) systems, but needs to be checked for the less extreme case also. \nFinally, we suggest that migrating BH binaries may simply be the quasars, or at least most of them. They have the same integrated energy output as quasars, they have the same accretion-disk fuel source as quasars, and like quasars, they turn on in the wake of mergers. It may be easier to move gas inward from ∼ 5 pc scales for a binary BH than for a single BH because the binary would excite spiral density waves in the grand accretion disk and so augment viscous drag. Accretion in the inner disk around the secondary might also be easier than for an isolated BH because of the tidal effects of the primary. If this hypothesis is correct, then quasars should generically show offsets between the centers of their broad and narrow lines with a root mean square of ∼ (2 / 3) 1 / 2 σ . \nAcknowledgements : We thank Andy Nelson for valuable discussions. A.G. thanks the Max-Planck-Institut fur Astronomie for its hospitality during a visit when most of work of this Letter was completed. His work was supported in part by grant AST 97-27520 from the NSF.", 'REFERENCES': 'Abraham, R.G., van den Bergh, S., Glazebrook, K., Ellis, R.S., Santiago, B.X., Surma, P., & Griffiths, R.E. 1996, ApJS, 107, 1 \nArtymowicz, P., & Lubow, S. 1994, ApJ, 421, 651 \nArtymowicz, P., & Lubow, S. 1996, ApJ, 427, 77 \nBegelman, M.C., Blandford, R.D., & Rees M.J. 1980, Nature, 287, 307 \nBinney, J., & Tremaine, S. 1987, Galactic Dynamics, (Princeton: Princeton University Press) \nFukushige, T., Ebisuzaki, T., & Makino, J. 1992, PASP, 44, 281 \nHo, L.C. 1999, ApJ, 516, 672 \nKauffmann, G. & Haehnelt, M. 1999, astro-ph/9906493 \nKirhakos, S., Bahcall, J.N., Schneider, D.P., Kristian, J. 1999, ApJ, 520, 67 \nLehto, H., & Valtonen, M.J. 1996, ApJ, 460, 207 \nMakino, J., Fukushige, T., Okumura, S.K., & Ebisuzaki, T. 1993, PASJ, 45, 303 \nQuinlan, G.D. 1996, New Astronomy, 1, 35 \nQuinlan, G.D., & Hernquist, L. 1997, New Astronomy, 2, 533 \nRajagopal, M., & Romani, R.W. 1995, ApJ, 446, 543 \nTaniguchi, Y., & Wada, K. 1996, ApJ, 469, 581. \nTrilling, D.E., Benz, W., Guillot, T., Lunine, J.I., Hubbard, W.B., & Burrows, A. 1998, ApJ, 500, 428 \nvan der Marel, R.P. 1999, AJ, 117, 744 \nWhite, S.D.M. 1996 in Cosmology and Large Scale Structure, Les Houches Session LX, eds. R. Schaeffer et al. North Holland Publishing'}
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2007PASJ...59S.315M
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Suzaku Observations of the Hard X-Ray Variability of MCG -6-30-15: the Effects of Strong Gravity around a Kerr Black Hole
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2007-01-01
| 6 | 0.47 | 162 |
['galaxies active', 'galaxies', 'galaxies seyfert', 'astronomy x rays', 'astrophysics']
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[]
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Suzaku has, for the first time, enabled the hard X-ray variability of the Seyfert 1 galaxy MCG -6-30-15 to be measured. The variability in the 14-45keV band, which is dominated by a strong reflection hump, is quenched relative to that at a few keV. This directly demonstrates that the whole reflection spectrum is much less variable than the power-law continuum. The broadband spectral variability can be decomposed into two components -- a highly variable power-law and constant reflection -- as previously inferred from other observations in the 2-10keV band. The strong reflection and high iron abundance give rise to a strong broad iron line, which requires the inner disc radius to be at about 2 gravitational radii. Our results are consistent with the predictions of the light bending model which invokes the very strong gravitational effects expected very close to a rapidly spinning black hole.
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| 24 |
https://arxiv.org/pdf/astro-ph/0609521.pdf
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{'Suzaku observations of the hard X-ray variability of MCG-6-30-15: the effects of strong gravity around a Kerr black hole': "Giovanni Miniutti 1 ∗ , Andrew C. Fabian 1 , Naohisa Anabuki 2 , Jamie Crummy 1 , Yasushi Fukazawa 3 , Luigi Gallo 4 , 5 , Yoshito Haba 6 , Kiyoshi Hayashida 2 , Steve Holt 7 , Hideyo Kunieda 8 , Josefin Larsson 1 , Alex Markowitz 9 , Chiho Matsumoto 8 , Masanori Ohno 3 , James N. Reeves 9 , 10 Tadayuki Takahashi 5 , Yasuo Tanaka 4 , Yuichi Terashima 5 , 11 , Ken'ichi Torii 2 , Yoshihiro Ueda 12 , Masayoshi Ushio 5 , \nShin Watanabe 5 , Makoto Yamauchi 13 , Tahir Yaqoob 9 , 10 1 Institute of Astronomy, University of Cambridge, Madingley Road, CB3 0HA Cambridge, UK 2 Department of Earth and Space Science, Osaka University, 1-1 Machikaneyama, Toyonaka, 560-0043 Osaka, Japan \n- 3 Department of Physics, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8526, Japan 4 Max-Planck-Institut fur extraterrestrische Physik, Postfach 1312, Garching, Germany\n- 5 Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, Yoshinodai 3-1-1, Sagamihara, Kanagawa 229-8510, Japan \n6 Department of Astrophysics, Nagoya University, Nagoya 464-8602, Japan 7 F. W. Olin College of Engineering, 1735 Great Plain Avenue, Needham, MA 02492, USA 8 Department of Physics, Nagoya University, Furo-cho, Chikusa, Nagoya 464-8602, Japan \n9 Exploration of the Universe Division, Code 662, NASA Goddard Space Flight Center, Greenbelt, MD 20771, USA 10 Department of Physics and Astronomy, John Hopkins University, 3400 N Charles Street, Baltimore, MD 21218, USA 11 Department of Physics, Ehime University, Matsuyama 790-8577, Japan \n12 Department of Astronomy, Kyoto University, Kyoto 606-8502, Japan \n13 Department of Applied Physics, University of Miyazaki, 1-1, Gakuen-Kibanadi-Nishi, Miyazaki 889-2192, Japan \n(Received 2000 December 31; accepted 2001 January 1)", 'Abstract': 'Suzaku has, for the first time, enabled the hard X-ray variability of the Seyfert 1 galaxy MCG-6-30-15 to be measured. The variability in the 14-45 keV band, which is dominated by a strong reflection hump, is quenched relative to that at a few keV. This directly demonstrates that the whole reflection spectrum is much less variable than the power-law continuum. The broadband spectral variability can be decomposed into two components - a highly variable power-law and constant reflection - as previously inferred from other observations in the 2-10 keV band. The strong reflection and high iron abundance give rise to a strong broad iron line, which requires the inner disc radius to be at about 2 gravitational radii. Our results are consistent with the predictions of the light bending model which invokes the very strong gravitational effects expected very close to a rapidly spinning black hole. \nKey words: galaxies: individual (MCG-6-30-15); galaxies: active; galaxies: Seyfert; X-rays: galaxies', '1. Introduction': 'The Seyfert 1 galaxy MCG-6-30-15 at z =0 . 00775 has been the centre of much interest since a broad iron line was discovered in its X-ray spectrum with the ASCA satellite (Tanaka et al 1995). Iron line emission is part of the reflection spectrum produced by the hard X-ray powerlaw continuum in the source irradiating the accretion disc (Guilbert & Rees 1988; Lightman & White 1988; Ross & Fabian 1993) and is broadened and skewed by Doppler, gravitational redshift effects, and light aberration and bending (Fabian et al 1989). The low energy extent of the line can reveal the inner radius of the accretion disc and thus the black hole spin (for reviews see Fabian et al 2000; Reynolds & Nowak 2003 and Fabian & Miniutti 2006). The iron abundance in MCG-6-30-15 appears to \nbe about 2-3 times the Solar value (Fabian et al 2002), making the iron line particularly strong. \nMCG-6-30-15 was observed several more times with ASCA (Iwasawa et al 1996, 1999; Matsumoto et al 2003; Shih et al 2002), by BeppoSAX (Guainazzi et al 1999), RXTE (Lee et al 1999; Vaughan & Edelson 2001) and by XMM-Newton (Wilms et al 2001; Fabian et al 2002, 2003; Vaughan & Fabian 2004). All of these observations confirmed the presence and general broad shape of the iron line. Evidence that the black hole in MCG-6-30-15 is rapidly spinning has been obtained from the extreme breadth of the line by Iwasawa et al (1996), Dabrowski et al (1997), Wilms et al (2001), Fabian et al (2002) and most recently Reynolds et al (2005), who determine a spin parameter a =0 . 989. \nThe XMM-Newton work emphasizes that much of the emission arise from smaller radii, between 2 -6 r g (where r g = GM/c 2 ). This may help to explain the otherwise puz- \nzling spectral variations shown by the source. Shih et al (2002), Fabian et al (2002), Fabian & Vaughan (2003) and Taylor, Uttley & McHardy (2003) found that the spectral variability can be explained by a simple two component model consisting of a highly variable power-law continuum (of almost fixed spectral slope) and a much less variable reflection spectrum, which of course includes the iron line. The reflection (and iron line) strength do not follow the power-law continuum intensity, as expected in a simple reflection picture. The small radius of much of the emission can however explain this behaviour when it is recalled that as well as strong gravitational red-shifting occurring in this region, there is also strong gravitational light bending. This can disconnect variations in the continuum from those of the reflection and has led to the development of the light-bending model (Fabian & Vaughan 2003; Miniutti et al 2003, 2004), which is a generalization of earlier work on the strong field regime (Martocchia & Matt 1996; Reynolds & Begelman 1997). \nIf the source of the continuum emission (assumed to be an isotropic emitter) changes location close to the black hole, then, even if the continuum has a constant intrinsic luminosity, it appears to the outside observer to change in brightness. This is just due to gravity bending the light rays out of the line of sight by different amounts depending upon the precise location of the source. Much of the radiation is bent down onto the disc, so the observed reflection intensity changes little. The two-component variability pattern and its light bending interpretation have recently found application in many other accreting black hole sources (Fabian et al 2004, 2005; Miniutti, Fabian & Miller 2004; Ponti et al 2006). \nOne striking feature of the model reflection spectrum is a large reflection hump peaking at 20-40 keV. This is where the disc albedo is highest; at lower energies the albedo is reduced by photoelectric absorption and at higher energies it is reduced (at a given energy) by Compton down-scattering of the photons (see e.g. George & Fabian 1991). The presence of the Compton hump has been confirmed in the spectrum of MCG-6-30-15 by BeppoSAX and RXTE observations (Guainazzi et al 1999; Lee et al 2000; Fabian et al 2002). What those observations have not done is to determine the variability of the Compton hump and show that it varies in the same way as the rest of the reflection spectrum 1 . Here we carry out this important step with the Hard X-ray Detector (HXD) on Suzaku . \nWe concentrate on the Suzaku data above 3 keV since there is a complex warm absorber in MCG-6-30-15 (Otani et al 1996; Lee et al 2001; Turner et al 2003,2004). The difference spectrum between low and high states of the source shows that absorption is minimal above 3 keV (Fabian et al 2002; Turner et al 2004) and High Energy Transmission Grating (HETG) spectra from Chandra \n1 \nshow no absorption feature around 6.5 keV which would indicate absorption by species of intermediate ionization which could affect our fits (Young et al 2005).', '2. The Suzaku observations': "MCG-6-30-15 was observed four times by Suzaku (Mitsuda et al 2006), once between August 17-19 in 2005 for about 45 ks and three times in January 2006 with longer exposures. Here we focus on the three 2006 observations performed between 9-14 (150 ks), 23-26 (99 ks)and 27-30 (97 ks) January 2006. We use event files from revision 0.7 of the Suzaku pipeline. Version 0.7 processing is an internal processing applied to the Suzaku data obtained during the Suzaku Working Group phase, for the purpose of establishing the detector calibration as quickly as possible. Some processes that are not critical for most of the initial calibration and scientific studies, e.g., aspect correction, fine tuning of the event time tagging of the XIS data, are skipped in version 0 processing, making the quality of the products limited in these directions compared with the official data supplied to guest observers. The XIS data were screened with XSELECT as standard (see e.g. Koyama et al 2006). The XIS products were extracted from circular regions of 4.3' radius centred on the source, while background products were extracted from two smaller circular regions offset from the source (and avoiding the chip corners with calibration sources) with a total area equal to that of the source region. The latest response and ancillary response files provided by the instrument teams were used. For the HXD/PIN (Takahashi et al 2006), instrumental background spectra were extracted from time dependent models provided by the HXD instrument team, based upon a database of non X-ray background observations made by the PIN diode to date. Since the background modeling is the key issue for the hard X-ray measurement with the HXD, the HXD team has provided two independent background models, model A and B, which use different algorithms (Kokubun et al. 2006). Spectral analysis of the source spectrum using the two models was found to give statistically indistinguishable results for the three 2006 January MCG-6-30-15 observations over the whole 1445 keV band used here, and we use the background model A in our analysis. The response files appropriate for the XIS nominal position observation were chosen dated as of 2006/08/07. \nWe have extracted products for the three frontilluminated CCD XIS detectors (XIS0, XIS2, and XIS3) and for the back-illuminated CCD (XIS1). The XIS2 and XIS3 detectors are found to produce remarkably similar spectra in the whole band used here. The XIS0 spectrum is slightly flatter than that from the XIS2 and XIS3 detectors so we proceed by co-adding just the XIS2 and XIS3 products in our analysis (see Yaqoob et al 2006 for mention of structures in the XIS0 and XIS1 spectra). In Fig. 1, we show the broad-band 0.5-12 keV background subtracted light curve from the XIS2 and XIS3 detectors during the three pointed observations in 2006. We also \nFig. 1. The light curve from the XIS2/XIS3 detectors in the 0.5-12 keV band. We also show the count rates chosen to select the High Flux (HF) and Low Flux (LF) states. They are chosen so that the HF and LF spectra have approximately the same number of counts (6 . 5 × 10 5 in the whole 0.5-12 keV band). \n<!-- image --> \nshow, as a reference, the count rate levels selected to define the High Flux (HF) and Low Flux (LF) states which will be used to study the spectral variability of MCG6-30-15. As normal for this source (thought to harbour a black hole with a mass of ∼ 3 × 10 6 M /circledot , McHardy et al 2005) the light curve exhibits large amplitude and relatively short timescale variability with variations up to factors 2-3 in a few ks.", '3. The 3-12 keV XIS spectrum': 'We start our analysis of the Suzaku data by considering the 3-12 keV time-averaged co-added spectrum from the XIS2 and XIS3 front-illuminated CCD detectors. (XIS response and ancillary files 20060213.rmf and 20060415.arf, with a 6 mm extract radius, were used.) The most important feature in this energy band is the strong, skewed, relativistic Fe K α line which characterizes the X-ray spectrum of MCG-6-30-15 enabling us to explore the nature and geometry of the accretion flow close to the central black hole with much higher accuracy than in any other object so far. In Fig. 2 we show the ratio of the data to a simple power law model fitted in the 3-4 keV and 7.512 keV band and absorbed by the Galactic column density (4 . 08 × 10 20 cm -2 ). The residuals clearly show the asymmetric and broad Fe K α line profile (top panel). In the bottom panel, we superimpose the Fe K α profile as observed with the XMM-Newton EPIC-pn camera in 2001. The agreement between the two instruments is remarkable and demonstrates the excellent level of the Suzaku XIS calibration even in the early stages of the mission. The 2-10 keV flux is 4 . 1 × 10 -11 erg cm -2 s -1 in the XMMNewton observation and 4 . 0 × 10 -11 erg cm -2 s -1 in the Suzaku one. Notice the absorption/emission structures in the blue wing of the relativistic line which are clearly detected by both the EPIC-pn and the XIS detectors with excellent agreement.', '3.1. The Fe K band and the relativistic line': "As a first attempt to fit the Fe K α line profile we consider the simplest possible spectral model comprising a power law continuum absorbed by a column of neutral matter (fixed at the Galactic value N H =4 . 08 × 10 20 cm -2 ) and a set of Gaussian emission lines. We consider first three Gaussian emission lines: a narrow unresolved ∼ 6.4 keV emission line (the narrow component of the Fe line from distant matter), two narrow unresolved ∼ 6.7 keV and ∼ 6.97 keV absorption lines (the Fe xxv and Fe xxvi resonant absorption line already detected by Chandra , Young et al. 2005), and an additional Gaussian emission line with width free to vary (representing the broad relativistic Fe line). We obtain a fit with χ 2 =2550 for 2235 degrees of freedom (dof). The narrow component of the Fe line is at 6 . 43 +0 . 01 -0 . 02 keV (Uncertainties are quoted throughout the paper at the 90 per cent confidence level) and has an equivalent width (EW) of only 30 ± 5 eV. Absorption lines are detected at 6 . 74 ± 0 . 03 keV and 7 . 04 ± 0 . 05 keV. The broad Fe line is at 5 . 88 ± 0 . 05keV, has a width of 840 ± 40 eV and an EW of 305 ± 20 eV. \nHowever, the best-fitting model leaves clear residuals in the 3-12 keV band. In particular the broad Fe line is not properly modelled and the residuals suggest the introduction of a double Gaussian model with one broad Gaussian around 6.4 keV and an even broader one at lower energies to model the extended red wing. We obtain a significant improvement of the statistics with χ 2 = 2398 for 2232 (Table 1). The energies of the three unresolved lines are 6 . 41 +0 . 03 -0 . 02 keV for the narrow Fe emission line, and 7 . 04 ± 0 . 05 keV, and 6 . 73 ± 0 . 04 keV for the absorption lines. Their equivalent widths are 25 ± 5 eV, -(12 ± 8), and -(15 ± 6) eV respectively. The upper limit on the EW of the narrow Fe K α emission line (only 30 eV) indicates that reflection from distant matter plays a minor and marginal role in MCG-6-30-16, as already demonstrated by high- \nFig. 2. Top: The Suzaku XIS data divided by a power law model fitted in the 3-4 keV and 7.5-10 keV band are shown in the most relevant 3-8 keV band. Bottom: The data from the XMM-Newton EPIC-pn detector (red) are superimposed on the XIS data (black). In both cases the model is a power law absorbed by the Galactic column density and fitted in the 3-4 keV and 7.5-12 keV band. \n<!-- image --> \nion spectroscopy with the HETG Chandra gratings (Lee et al 2002; Young et al 2005). The two absorption line energies are slightly higher than the rest-frame energies of Fe xxv and Fe xxvi resonant absorption and are consistent with an origin in a common outflow with velocity of few thousand km s -1 . In particular, they are both consistent in equivalent width and outflow velocity with the values measured with Chandra (2 . 0 +0 . 7 -0 . 9 × 10 3 km s -1 , Young et al 2005). The photoelectric absorption implied by such a thin highly ionized absorber is small and has a negligible effect on the fits described here. \nAs for the broad Fe line, the parameters of the two broad Gaussian lines describing its profile are reported in Table 1 (first model 'Double Gaussian'). They are both clearly resolved and their cumulative EW is 320 ± 45 eV, although this value should be taken with care since the continuum is a simple absorbed power law and does not include the reflection continuum which must be associated with the broad Fe line. It is interesting to note that the width of the Fe line core at ∼ 6 . 45 keV is σ > 260 eV (i.e. FWHM /similarequal 31950 km s -1 ) suggesting that it originates in \nthe outer disc at < 80 r g from the central black hole, while the red wing has a width > 760 eV indicating that the emitting matter is located within 6 . 5 r g from the centre, already implying, even by means of a very simple and phenomenological model, that the black hole in MCG-630-15 is most likely a spinning Kerr black hole in which the accretion disc extends down within the marginal stable orbit for a non-rotating Schwarzschild black hole (6 r g ).", '3.2. A self-consistent reflection model': "The multiple Gaussian fit described above is merely a phenomenological description of the hard spectrum. The Gaussian emission lines and, most importantly, the broad Fe line are the clear signature of X-ray reflection and the above spectral model did not include any reflection continuum. Here we build a much more self-consistent model in which the broad Fe line is computed together with the associated reflection continuum. We use a grid of models from Ross & Fabian (2005) to obtain the X-ray reflection spectrum. The X-ray reflection model has the spectral slope of the illuminating power law Γ, the ionization parameter ξ , the Fe abundance, and the normalization as free parameters. However, we forced the photon index to be the same as the power law continuum for consistency. Since the reflection model does not include Ni as an element, we also include a Ni K α line with energy fixed at 7.47 keV. \nIn order to reproduce the relativistic broad Fe line profile, the reflection spectrum is convolved with a relativistic kernel derived from the Laor (1991) code. The relativistic blurring parameters are the emissivity indexes q in and q out (where the emissivity is /epsilon1 = r -q in within the innermost 6 r g and /epsilon1 = r -q out outside), the inner disc radius r in , and the observer inclination i . The outer disc radius is fixed at its maximum allowed value of 400 r g . The choice of a broken power law emissivity profile is motivated by the previous long XMM-Newton observation of MCG-6-30-15 (Vaughan & Fabian 2004). \nThe model is applied to the co-added XIS2 and XIS3 time-averaged spectrum in the 3-12 keV band and we keep the three unresolved Gaussian lines as above. The results of the spectral fitting are reported in Table 1 ('Blurred Reflection' model). We obtain a better fit with χ 2 = 2360 for 2230 dof. The better fit with respect to the phenomenological Double Gaussian model described above is due to a better description of the overall relativistic Fe line profile. In addition, the model accounts for some high-energy residuals above about 10 keV which were present in the above best-fitting solution due to the unmodelled reflection component. \nWe measure a relatively standard Γ = 2 . 18 +0 . 07 -0 . 06 continuum slope which compares very well with previous results with XMM-Newton . The reflection component contributes significantly to the hard spectrum of MCG-6-3015 and we measure a reflection fraction of R = 2 . 8 ± 0 . 9 ( R =1 corresponds to the level of reflection expected from 2 π sr), fully consistent with previous results from a simultaneous XMM-Newton and BeppoSAX observation (Vaughan & Fabian 2004). The precise value of the re- \nTable 1. Results of spectral fits to the 3-12 keV XIS2 and XIS3 time-averaged co-added spectrum with the different models used to describe the relativistic Fe line of MCG-6-30-15. We present results for a phenomenological double-Gaussian fit and for a much more self-consistent relativistically blurred reflection model. For the reflection model, we measure an Fe abundance of 2 . 0 +1 . 4 -0 . 6 times solar. We only report results for the relativistic Fe line here. A more detailed fit is presented in Table 2, where high-energy data from the HXD/PIN detector are also included. A subscript p indicates that the parameter reached its min/max allowed value. \n| Double Gaussian | Double Gaussian | Double Gaussian | Double Gaussian | Double Gaussian | Double Gaussian | Double Gaussian | Double Gaussian |\n|---------------------------|-----------------------|-----------------------------|---------------------------|----------------------|--------------------|--------------------|--------------------|\n| Continuum | K α Red Wing | K α Red Wing | K α Red Wing | K α Blue Core | K α Blue Core | K α Blue Core | χ 2 /dof |\n| Γ | E | σ | EW | E +0 | σ | EW | |\n| 1 . 96 ± 0 . 02 | 5 . 38 ± 0 . 10 | 840 +70 - 80 | 130 ± 15 | 6 . 45 . 02 - 0 . 05 | 290 ± 30 | 190 ± 30 | 2398/2232 |\n| Blurred Reflection | Blurred Reflection | Blurred Reflection | Blurred Reflection | Blurred Reflection | Blurred Reflection | Blurred Reflection | Blurred Reflection |\n| Continuum | Relativistic Blurring | Relativistic Blurring | Relativistic Blurring | Reflector | Reflector | Reflector | χ 2 /dof |\n| Γ 2 . 18 +0 . 07 - 0 . 06 | i 38 ± 4 | r in . 6 +0 . 6 - 0 . 365 p | q in 4 . 6 +0 . 6 - 0 . 9 | q out 2 . 6 ± 0 . 3 | R 2 . 8 ± 0 . 9 | ξ 65 ± 45 | 2360/2230 | \nction fraction will be better constrained in a subsequent analysis when the high energy data from the HXD/PIN detector are considered as well. The reflector is only mildly ionized with ξ = 65 ± 45 erg cm s -1 and the inclination is constrained to be 38 · ± 4 · . \nAs for the relevant relativistic blurring parameters, the XIS data are able to constrain the inner disc radius to be smaller than r in < 2 . 2 r g . If, as it is customary, r in is identified with the innermost stable circular orbit around a Kerr black hole, such a small 90 per cent upper limit implies that the black hole spin in MCG-6-30-15 is a > 0 . 917, i.e. the black hole is an almost maximally spinning Kerr black hole. The inner and outer emissivity profiles are not consistent with each other within the errors and q in /similarequal 4 . 8 is steeper than q out /similarequal 2 . 6. However, if the two indices are forced to be one and the same, we obtain q in ≡ q out =3 . 1 +0 . 5 -0 . 3 with only a slight worsening of the fit statistic (∆ χ 2 =6).", '4. Adding the HXD/PIN data; the 3-45 keV spectrum': 'In the above analysis the relativistic Fe line is associated with its own reflection continuum and a self-consistent model to the 3-12 keV spectrum is found. However, one of the main characteristics of the Suzaku mission is the presence of the HXD PIN hard X-ray detector providing high quality data above 12 keV (Fig. 3). The spectral analysis of the XIS data revealed the presence of a strong reflection component associated with the broad relativistic Fe line which would produce a strong Compton hump around 2030 keV. The HXD data above 12 keV are thus crucial to investigate the reflection component further and will enable us to infer the reflection parameters and to measure the direct and reflection continua with high accuracy. \nIn Fig. 4, we show the XIS and HXD/PIN time- \nFig. 3. PIN data displayed as total countrate (blue, upper), backgrounds A and B (red and green, respectively, middle) and source (black lower) spectra. \n<!-- image --> \naveraged spectrum in the 0.5-45 keV band. The Figure shows the data to model ratio for a power law model fitted above 3 keV and ignoring the bands where reflection is expected to dominate. The ratio clearly shows the presence of both the relativistic Fe line in the XIS data and also the presence of a large Compton hump around 20 keV visually confirming that the hard spectrum of MCG-6-30-15 is dominated by a strong reflection component. Absorption is present affecting the soft band below 3 keV. We avoid the complication of modelling the complex warm absorber and we concentrate, in this first paper on the Suzaku observation of MCG-6-30-15, on the data above 3 keV. A detailed model of the warm absorber will be presented elsewhere where the data from the back-illuminated (more sensitive at soft energies) XIS detector will be also considered. \nWe consider our best-fitting spectral model to the 3- \nFig. 5. The best-fit model components to the time-averaged spectrum of MCG-6-30-15. \n<!-- image --> \nFig. 4. The XIS2/XIS3 and PIN data in the 0.5-45 keV band are compared with a simple power law fitted in the 3-45 keV band ignoring the Fe K and Compton hump energy bands (4-7.5 keV and 14-30 keV). The ratio plot shows the main spectral components which are the power law continuum, absorption below about 3 keV, and a strong smeared reflection component comprising the relativistic broad Fe line and Compton hump. Data have been re-binned for visual clarity. \n<!-- image --> \n12 keV XIS data and extend it to the PIN data between 14 keV and 45 keV. We include a cross-normalization constant between the XIS and PIN data and find that it is constrained to be 1 . 10 ± 0 . 05 (where the XIS constant is set to 1). This factor is consistent with the results obtained from Crab observations (Kokubun et al. 2006). Since the available HXD/PIN background model A does not include the contribution from the Xray cosmic background, a spectral model of the form 2 . 06 × 10 -6 ( E/ 100 keV ) -1 . 29 exp( -E/ 41 . 13 keV ) was included in all spectral fits in the HXD/PIN band (the model being valid up to about 70 keV). The model is based on the HEAO -A1 spectrum, re-scaled to account for the HXD field of view. \nThe best-fitting XIS model provides an acceptable description of the 3-45 keV data and we obtain a good quality fit of χ 2 = 2495 for 2312 dof. The model parameters are very much consistent with those obtained from the 3-12 keV parameters. In fact, extending the model up to 45 keV provides a reasonable description of the hard data, reproducing most of the large Compton hump seen in Fig. 4, even before fitting. The results for the TimeAveraged (TA) spectrum in the 3-45 keV band are reported in Table 2 together with similar fits to the High Flux (HF) and Low Flux (LF) spectra which will be discussed below. The reflection component contributes significantly to the total flux and we measure a large reflection fraction of 3 . 8 ± 0 . 7. As a reference, and since it will be useful in a spectral variability analysis below, we report here that its contribution represents the 33 ± 6 per cent of the total flux in the 3-12 keV band and the 51 ± 10 per cent in the 14-45 keV band. The relativistic blurring parameters are consistent within the errors with those already reported in Table 1 and the same is true for the continuum power law slope and for the ionization \nparameter. \nAs a consistency check, we have also determined the level of the reflection component by using the spectrum above 7.5 keV, so avoiding the iron line. For this the pexrav model in XSPEC was used, with results Γ = 2 . 2, reflection fraction R = 2 -4 and iron abundance poorly constrained ( A Fe = 0 . 5 -2 times solar). We note that such reflection predicts a strong iron line with an equivalent width of at least 300 eV (taking conservatively an iron abundance of unity and reflection fraction of 2), using the predictions of George & Fabian (1991). The limit on any narrow component to the iron emission feature is at least ten times smaller than this (this work; Lee et al 2002; Young et al 2005), thereby ruling out any non-smeared origin for the reflection component. The reflection must be due to matter well within the optical broad line region. We also stress that if the evidence for relativistic reflection is taken into account and the pexrav component is blurred accordingly, the shape of the reflection continuum is broader and redshifted requiring a higher Fe abundance ( A Fe =1 . 5 -3 . 5 times solar) and reflection fraction ( R = 3 -5) to fit the data, while the continuum photon index is basically unaffected. The blurred and Fe overabundant reflector described by pexrav is thus fully consistent with our best-fitting results with a much more complex spectral model including the relativistic Fe line (see Table 2).', '5. Flux and spectral variability': 'The HXD/PIN detector has no imaging capabilities and thus there is no simple way to obtain a background subtracted light curve by selecting appropriate regions. However, background subtracted light curves can be obtained by extracting spectra in each time interval, subtracting the background from the available models in the same time interval, and reading out the background subtracted count rate. In background model A, the background of the HXD is calculated from the data base by using the count rate of the upper discriminator, which has \n<!-- image --> \nFig. 6. The background subtracted light curve of MCG-6-30-15 during the three 2006 pointed observations (first in the top panel, second and third in the bottom panel). The dotted black light curve is the XIS2 and XIS3 light curve rescaled to the mean HXD/PIN count rate in each orbit. In red, we show the HXD/PIN light curve. The time bin is about 45 ks. Time elapses from the start of the first observation. \n<!-- image --> \na strong correlation with the flux of the non-X-ray background. The procedure must also include a model for the cosmic X-ray background (not included in the available background models) and we use the same model as for spectral fitting by simulating the cosmic X-ray background HXD/PIN spectrum and adding it to the instrumental background. We have applied the above procedure to the HXD/PIN data by selecting, after some experimentation, a timescale of 45 ks enabling us to obtain about 5000 background subtracted counts per time interval in the 14-45 keV PIN band on average. In Fig. 6 we show the XIS2 and XIS3 and the HXD/PIN light curves for the three 2006 observations in the 0.5-12 keV and 1445 keV respectively. To ease the comparison, the XIS light curves have been re-scaled so that they have the same mean count rate as the PIN ones in each segment (i.e. observation). It is visually clear that the harder band (solid red) has a much lower variability amplitude than the softer one (dotted black). In the following we explore a set of different techniques with the aim of exploring the \n<!-- image --> \nFig. 7. Top: The XIS2 and XIS3 count rate in the hard 3-12 keV band is plotted against that in the 1-2 keV band from the 45 ks light curve. The relationship is linear and reveals a clear hard off-set. Bottom: Flux-flux plot for the 14-45 keV HXD/PIN data against the 1-2 keV reference band. Due to larger spread in the PIN band (see text for details), we have re-binned the original flux-flux plot according to the 1-2 keV flux so that each data point shown here comprises three original data points. The binned flux-flux plot exhibits a linear relationship, although some scatter is present. \n<!-- image --> \nflux and spectral variability of the source.', '5.1. Flux-flux plots': 'As a first model-independent way to characterize the flux/spectral variability of MCG-6-30-15 we analysed the relationship of the fluxes (count rates) in different energy bands following the technique of flux-flux plots introduced by Churazov, Gilfanov & Revnivtsev (2001) in a study of Cygnus X-1 spectral variability, and Taylor, Uttley & McHardy (2003) as a model-independent tool to disentangle the main drivers of the spectral variability in AGN. \nIn Fig. 7 we show the 3-12 keV XIS count rate from the 45 ks light curve against the count rate in the 1-2 keV band which is chosen as reference (see also Vaughan & Fabian 2004). The relationship between the count rates in the two bands is remarkably linear and leaves a clear hard offset on the y-axis, as already noticed by Taylor, \nUttley & McHardy (2003) and Vaughan & Fabian (2004) in previous RXTE and XMM-Newton observations. The linearity of the relationship indicates that the flux variations are strongly dominated by changes in the normalization of a spectral component with constant spectral shape. On the other hand, the hard offset suggests the presence of a component that varies little and that contributes more to the hard than to the reference 1-2 keV band. We measure a hard offset of a = 0 . 48 ± 0 . 02 cts/s which can be used to infer the contribution of the weakly variable component at mean flux level: since the mean count rate in the 3-12 keV band is 1 . 79 ± 0 . 01 cts/s, the hard offset a represents 26 . 8 +1 . 3 -1 . 2 per cent of the 3-12 keV flux at mean flux level. This compares very well with the 33 ± 6 per cent contribution of the reflection component to the 3-12 keV band, as obtained from direct spectral fitting. \nWe have applied the same technique to the HXD/PIN light curve and plotted the 14-45 keV count rate against the 1-2 keV reference one. The relationship is much more noisy than when data from the XIS alone were used. To reduce the scatter and decrease the PIN error bars, we have binned the original flux-flux plot according to the soft reference count rate so that each new point comprises three original data points. In the lower panel of Fig. 7 we show the binned flux-flux plot obtained by plotting the 14-45 keV HXD/PIN count rate against the reference 1-2 keV one. \nA fit with a linear relationship is perfectly acceptable ( χ 2 =5 . 8 for 5 dof), again indicating the presence of a clear hard offset in the PIN band. In this case, the contribution of the weakly variable (reflection) component to the 1445 keV band at mean flux level is estimated to be as high as 55 ± 17 per cent. Again, this is in good agreement with the contribution of the reflection component in the 1445 keV band obtained from direct spectral fitting (51 ± 10 per cent). \nThe flux-flux plot analysis detailed above indicates that the spectral variability of MCG-6-30-15 can be decomposed into two main components: a highly variable component which varies in normalization only but not in spectral shape, and a weakly variable one which has a much harder spectral shape. The excellent agreement between the contribution of the weakly variable component in the different energy bands as inferred from the flux-flux plot analysis and that of the reflection component as obtained by direct spectral fitting strongly suggest that the weakly variable component is the relativistically smeared X-ray reflection from the accretion disc. Obviously, the highly variable component can be identified with the other component required by spectral fitting, i.e. the power law continuum. The flux-flux plots suggest that as the source varies, the power law maintains the same spectral slope and only varies in normalization.', '5.2. The Flux RMS spectrum': 'A further approach to studying the spectral variability of a source is the RMS spectrum, which, for a chosen timescale, is the fractional variability seen at each energy \nFig. 8. The RMS spectrum of MCG-6-30-15 on a 45 ks timescale is shown in the 2-45 keV band. The solid red line represents the theoretical RMS that would be obtained from the best-fitting spectral model of the time-averaged spectrum if all the variability could be explained with a two-component model in which the power law has a constant photon index and varies in normalization only, while the disc reflection component is perfectly constant. \n<!-- image --> \nexpressed as a flux normalized root mean square variability function. The RMS spectrum allows us to quantify the fractional variability as a function of energy in a model-independent manner and is therefore an important piece of information, complementary to the flux-flux plot analysis described above. Such RMS spectra have previously been determined for MCG-6-30-15 (Matsumoto et al 2003; Fabian et al 2002; Ponti et al 2004). However, as far as data below 10 keV are available, the RMS spectrum has proved to be ambiguous to model. In particular, the typical trend that is observed in AGN is that the fractional variability decreases with energy, a behaviour that can be explained either in terms of a power law softening at high flux levels (i.e. a pivoting power law) or in the framework of the two-component model discussed above. The new ingredient here is to include the HXD/PIN data above 10 keV with the goal of confirming/rejecting the interpretation that comes out from the flux-flux plots analysis. \nOur result for a 45 ks timescale is shown in Fig. 8. The general trend is that the RMS decreases with energy with a marked drop at ∼ 6 . 4 keV reassuringly confirming that the Fe line is less variable than the continuum, as already noticed in previous works (e.g. Vaughan & Fabian 2004; Ponti et al 2004). The novelty of the Suzaku observation is represented by the HXD/PIN data. They confirm the trend of lower fractional variability at higher energies but they lie above and not on the extrapolation of the 2-6 keV trend. \nWe have theoretically modelled the RMS spectrum by considering our best-fitting spectral model to the timeaveraged spectrum and reproducing the 45 ks data by allowing only the power law normalization to vary, i.e. enforcing a perfect two-component model. The result of this exercise is shown as a solid red line in Fig. 8 and shows that the RMS spectrum is fully consistent with a two- \nFig. 9. Ratio plot for the HF (black) and LF (red) spectra with a power law model fitted ignoring the 4-7.5 keV and 14-25 keV energy bands. The ratio shows that the Compton hump contributes by about 80 per cent in the LF state and by about 40-50 per cent in the HF one, while its contribution in the time-averaged spectrum is about 60 per cent (see Fig. 4). This shows that the reflection component is less variable than the power law and thus, the reflection fraction is expected to increase as the overall flux decreases. \n<!-- image --> \ncomponent model in which the only variable component is a constant-slope power law. Some variability of the reflection component cannot be excluded, but its variability amplitude is certainly not comparable to the variability amplitude of the power law. Indeed, small mismatches between the model and the data could be due to some intrinsic variability of the reflection component which deserves more detailed study to be performed in future work.', '5.3. The High and Low Flux states': 'Having analysed the spectral variability of MCG6-30-15 with two model-independent (and calibrationindependent) techniques we now consider a more direct approach through spectral fitting. We extracted High Flux (HF) and Low Flux (LF) from both the XIS and the PIN spectra according to the selection criterion shown in Fig. 1. We first consider a simple power law fit in the 3-45 keV band ignoring, as done before, the Fe K and Compton hump bands. The result for the HF and LF spectra is shown in Fig. 9 as a ratio plot. There is a clear evidence, both in the XIS and in the HXD/PIN, that the reflection component contributes more to the LF than to the HF spectrum, which again indicates an almost constant reflection. \nWe have then applied the same spectral model as for the 3-45 keV time-averaged (TA) spectrum to the HF and LF spectra and our results are reported in Table 2. Almost all parameters are consistent with being the same in the HF and LF spectra. In particular, the power law slope is consistent with being constant and equal to ∼ 2 . 25 in the TA, HF, and LF spectra. The same is true for the Fe abundance, disc inclination, inner disc radius, outermost emissivity index q out and reflector ionization state (although there is some indication for a lower ionization \nFig. 10. The spectra, best-fitting model, and residuals for the HF (upper panel) and LF (lower panel) states. \n<!-- image --> \nat low flux levels). The innermost emissivity index q in appears to be steeper in the LF than in the HF spectrum, but the two indexes are still consistent with each other within the 90 per cent errors. A steeper emissivity profile in the LF state would produce a more extended red wing as expected in the framework of the light bending model where low flux states are characterized by a more centrally concentrated illumination of the inner disc (Miniutti & Fabian 2004). \nOn the other hand, the reflection fraction definitely appears to vary and it is higher in the LF spectrum ( R =4 . 8 ± 0 . 8) than in the HF one ( R =2 . 5 ± 0 . 6) as already strongly suggested by the ratio plot in Fig. 9. This result is fully consistent with the indication inferred from the flux-flux plot and rms spectrum analysis discussed above: if the reflection component is only weakly variable, its contribution increases in the LF states because the direct power law decreases while the reflection stays approximately constant. We point out that this is exactly the behaviour predicted by the light bending model (Miniutti & Fabian 2004). The spectra, best-fitting models and residuals for the HF and LF states are shown in Fig. 10. The best-fit spectral components are as in Fig. 5 ( which refers to the best-fit parameters to the time-averaged spectrum) with the only major difference in the HF/LF state being \nFig. 11. The difference spectrum (HF minus LF) fitted with s power law absorbed by the Galactic column. \n<!-- image --> \ndue to a higher/lower power law normalization.', '5.4. The difference spectrum': 'We also produced a difference spectrum obtained by subtracting the LF spectrum from the HF one for the XIS and HXD/PIN. By definition, the difference spectrum only shows the variable components of the spectrum (modified by absorption), while any non variable component is subtracted away. In Fig. 11 we show the residuals to a simple power law fit in the 3-45 keV band where Galactic absorption is included. The power law fit is very good and the power law slope is consistent with that found in our spectral analysis (Γ = 2 . 2 ± 0 . 1) but clear positive residuals are left in a broad hump around 6 keV at the ∼ 2 σ level. This may indicate either that the red wing of the relativistic Fe line has varied in flux or that the shape of the line is slightly different in the HF and LF spectra. \nIf the residuals are due to flux variability of the reflection component, we should also see some positive residuals at the Compton hump energy (20-30 keV) which are not seen in the HXD/PIN difference spectrum. Some reflection variability is in fact allowed by the results of our spectral fits of the HF and LF spectra (see Table 2) although the reflection flux is consistent with being the same at the two flux levels. If the reflection flux varied but the Fe line keeps the same shape, the main residual in the difference spectrum should be at the energy of the blue peak of the line, around 6.4 keV, while the residuals in Fig. 11 indicate that only the red wing of the line has varied. \nThe remaining possibility is that the Fe line has a different shape (rather than a different flux) in the HF and LF spectra. This is already hinted at by the inner emissivity index ( q in , see Table 2) which is marginally steeper in the LF state (although consistent within the errors). If the emissivity is really different at the two flux levels, residuals are produced in the red wing of the line and could potentially explain those seen in the difference spectrum (and at the same time the lack of residual at Compton hump energies). The result is in the sense expected from the light-bending model (see Fig. 4 of Miniutti & Fabian \n2004).', '6. Discussion': "The Suzaku observation of MCG-6-30-15 has, for the first time, enabled the hard X-ray variability of the source to be determined. A wide range of techniques including direct spectral fitting, flux-flux plots and rms spectrum all show that the spectral variability on a 45 ks timescale can be decomposed into just two broad components above 3 keV, namely a variable power-law continuum and a harder constant component. This last component has the broad iron line and reflection hump expected from X-ray reflection in the very innermost regions of an accretion disc around a rapidly spinning black hole. \nJust the strength and shape of the reflection component predicts a high equivalent width iron line, much stronger than the weak narrow emission component seen in the spectrum. The only way for such a strong iron line to be 'hidden' in the spectrum is for it to be smeared into the broad emission structure that we observe. \nThe behaviour of the power-law and reflection components is consistent with the light bending model. In which case much of the rapid (power-law component) variability seen in the source is due to strong gravitational light bending as the site of the emission region changes location within a few gravitational radii of the black hole. We assume that the emission region is associated with some magnetic structure produced by the strong differential motions in the accretion disc. Note that to produce observed variability the emission region does not need to physically move, just that the main emission site changes position. \nThe broad iron line is strong in MCG-6-30-15 because both the iron abundance is high, at around 2 times the solar value, and the large degree of light bending around its Kerr black hole has emphasised the reflection spectrum relative to the power-law continuum.", 'Acknowledgements': 'We are deeply grateful to the whole Suzaku team for building, launching, calibrating and operating the spacecraft and instruments. GM thanks the PPARC, ACF the Royal Society, JL the Isaac Newton Trust, Corpus Christi College and PPARC, JC the PPARC for support. \nTable 2. Results of fits to the XIS2/XIS3 and PIN Time-Averaged (TA), High Flux (HF) and Low Flux (LF) spectra in the 3-45 keV band. The power law normalization ( N PL ) is given in units of 10 -2 ph cm -2 s -1 . As for the reflection, we prefer to give the reflection fraction which is a more interesting measure of its strength. The power law and reflection fluxes are reported in the 3-45 keV band in units of 10 -11 erg cm -2 s -1 for completeness. The reflector ionization parameter ( ξ ) is in erg cm s -1 and the inner disc radius ( r in ) and inclination ( i ) are in units of r g = GM/c 2 and degrees respectively. The energies of the lines are in keV, their EW in eV, and their width is fixed to 10 eV. The narrow Fe K α normalization is in units of 10 -6 ph cm -2 s -1 . A subscript p indicates that the parameter reached its min/max allowed value. \n| Parameter | TA spectrum | HF spectrum | LF spectrum |\n|-----------------------------------|------------------------------|------------------------------------|---------------------------------------------------------|\n| Γ | 2 . 26 ± 0 . 04 | 2 . 25 ± 0 . 05 | 2 . 25 ± 0 . 05 |\n| N PL | 1 . 7 ± 0 . 1 | 2 . 7 ± 0 . 1 | 1 . 2 ± 0 . 1 |\n| R | 3 . 8 ± 0 . 7 | 2 . 5 ± 0 . 6 70 ± 37 | 4 . 8 ± 0 . 8 |\n| ξ | 68 ± 31 1 . 9 +1 . 4 - 0 . 5 | | 48 ± 35 |\n| A Fe | | 1 . 8 +1 . 3 - 0 . 4 | 1 . 9 +1 . 5 - 0 . 6 |\n| r in | 1 . 7 +0 . 4 - 0 . 465 p | 1 . 235 +0 . 7 - 0 . 0 p | 1 . 7 +0 . 3 - 0 . 465 p |\n| i | 38 ± 3 | 37 ± 4 | 41 ± 5 |\n| q in | 4 . 4 +0 . 5 - 0 . 8 | 3 . 6 +0 . 8 - 0 . 7 | 4 . 5 ± 0 . 6 |\n| q out | 2 . 5 ± 0 . 3 | 2 . 8 ± 0 . 4 | 2 . 6 ± 0 . 3 |\n| | 6 . 41 +0 . 03 | ± | ± |\n| E FeK α EW FeK α | - 0 . 02 35 ± 5 | 6 . 40 0 . 03 19 ± 4 | 6 . 41 0 . 03 44 ± 6 |\n| N FeK α | 8 . 5 ± 1 . 3 | ± | 8 . 5 ± 1 . 3 |\n| E abs 1 EW abs 1 E abs 2 EW abs 2 | 6 . 73 ± 0 . 04 | 8 . 3 1 . 2 ± | |\n| | - (22 ± 8) - ± | 6 . 72 0 . 05 - (20 +23 - 14 ) - ± | 6 . 75 ± 0 . 04 - (45 ± 30) 7 . 03 ± 0 . 04 - (28 ± 18) |\n| | 7 . 07 ± 0 . 04 (13 8) | 7 . 08 ± 0 . 05 (15 10) | |\n| F PL 3 - 45 keV F REF 3 - 45 keV | 4 . 0 ± 0 . 2 4 . 1 ± 0 . 8 | 6 . 4 ± 0 . 2 4 . 0 ± 0 . 9 | 2 . 9 ± 0 . 3 3 . 8 ± 0 . 6 |\n| χ 2 /dof | 2495/2312 | 1910/1848 | 1900/1925 |', 'References': "| Churazov E., Gilfanov M., Revnivtsev M. 2001, MNRAS, 321, 759 | |\n|---------------------------------------------------------------------------------------------------------------------------------------------------|---------------------------------------------------------------|\n| Dabrowski Y., Fabian A.C., Iwasawa K., Reynolds C.S., 1997, MNRAS, 288, L11 | Lasenby A.N., |\n| Fabian A.C., Rees M.J., Stella L., White N.E., 1989, MNRAS, 238, 729 | |\n| Fabian A.C., Iwasawa K., Reynold C.S., Young A.J., 2000, PASP, 112, 1145 | |\n| Fabian A.C. et al, 2002, MNRAS, 335, L1 Fabian A.C., Vaughan S., 2003, MNRAS, 340, L28 Fabian A.C., Miniutti G., Gallo L., Boller Th., Tanaka Y., | |\n| Vaughan S., Ross R.R., 2004, MNRAS, 353, 1071 | |\n| Fabian A.C., Miniutti G., Iwasawa K., Ross R.R., 2005, MNRAS, 361, 795 Fabian A.C., Miniutti G., 2006, to appear in 'Kerr Spacetime: | |\n| Rotating Black Holes in General Relativity' eds. 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2009PhRvD..79l4028B
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Effective-one-body waveforms calibrated to numerical relativity simulations: Coalescence of nonspinning, equal-mass black holes
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2009-01-01
| 18 | 0.45 | 162 |
['-', '-', '-', '-', 'methods numerical', 'methods numerical', '-', 'perturbation theory', '-', '-', '-']
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[]
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We calibrate the effective-one-body (EOB) model to an accurate numerical simulation of an equal-mass, nonspinning binary black-hole coalescence produced by the Caltech-Cornell Collaboration. Aligning the EOB and numerical waveforms at low frequency over a time interval of ∼1000M, and taking into account the uncertainties in the numerical simulation, we investigate the significance and degeneracy of the EOB-adjustable parameters during inspiral, plunge, and merger, and determine the minimum number of EOB-adjustable parameters that achieves phase and amplitude agreements on the order of the numerical error. We find that phase and fractional amplitude differences between the numerical and EOB values of the dominant gravitational-wave mode h<SUB>22</SUB> can be reduced to 0.02 radians and 2%, respectively, until a time 20M before merger, and to 0.04 radians and 7%, respectively, at a time 20M after merger (during ringdown). Using LIGO, Enhanced LIGO, and Advanced LIGO noise curves, we find that the overlap between the EOB and the numerical h<SUB>22</SUB>, maximized only over the initial phase and time of arrival, is larger than 0.999 for equal-mass binary black holes with total mass 30-150M<SUB>⊙</SUB>. In addition to the leading gravitational mode (2, 2), we compare the dominant subleading modes (4, 4) and (3, 2) for the inspiral and find phase and amplitude differences on the order of the numerical error. We also determine the mass-ratio dependence of one of the EOB-adjustable parameters by calibrating to numerical inspiral waveforms for black-hole binaries with mass ratios 2∶1 and 3∶1. The results presented in this paper improve and extend recent successful attempts aimed at providing gravitational-wave data analysts the best analytical EOB model capable of interpolating accurate numerical simulations.
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https://arxiv.org/pdf/0902.0790.pdf
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{'Effective-one-body waveforms calibrated to numerical relativity simulations: coalescence of non-spinning, equal-mass black holes': 'Alessandra Buonanno, 1 Yi Pan, 1 Harald P. Pfeiffer, 2 Mark A. Scheel, 2 Luisa T. Buchman, 2 and Lawrence E. Kidder 3 1 Maryland Center for Fundamental Physics, Department of Physics, University of Maryland, College Park, MD 20742 2 Theoretical Astrophysics 130-33, California Institute of Technology, Pasadena, CA 91125 3 Center for Radiophysics and Space Research, Cornell University, Ithaca, New York, 14853 (Dated: October 29, 2018) \nWe calibrate the effective-one-body (EOB) model to an accurate numerical simulation of an equalmass, non-spinning binary black-hole coalescence produced by the Caltech-Cornell collaboration. Aligning the EOB and numerical waveforms at low frequency over a time interval of ∼ 1000 M , and taking into account the uncertainties in the numerical simulation, we investigate the significance and degeneracy of the EOB adjustable parameters during inspiral, plunge and merger, and determine the minimum number of EOB adjustable parameters that achieves phase and amplitude agreements on the order of the numerical error. We find that phase and fractional amplitude differences between the numerical and EOB values of the dominant gravitational wave mode h 22 can be reduced to 0 . 02 radians and 2%, respectively, until a time 20 M before merger, and to 0 . 04 radians and 7%, respectively, at a time 20 M after merger (during ringdown). Using LIGO, Enhanced LIGO and Advanced LIGO noise curves, we find that the overlap between the EOB and the numerical h 22 , maximized only over the initial phase and time of arrival, is larger than 0 . 999 for equal-mass binary black holes with total mass 30-150 M /circledot . In addition to the leading gravitational mode (2 , 2), we compare the dominant subleading modes (4 , 4) and (3 , 2) for the inspiral and find phase and amplitude differences on the order of the numerical error. We also determine the mass-ratio dependence of one of the EOB adjustable parameters by calibrating to numerical inspiral waveforms for black-hole binaries with mass ratios 2:1 and 3:1. The results presented in this paper improve and extend recent successful attempts aimed at providing gravitational-wave data analysts the best analytical EOB model capable of interpolating accurate numerical simulations. \nPACS numbers: 04.25.D-, 04.25.dg, 04.25.Nx, 04.30.-w', 'I. INTRODUCTION': "The first-generation gravitational-wave detectors the Laser Interferometer Gravitational Wave Observatory (LIGO) [1, 2], GEO [3] and Virgo [4] - have operated at design sensitivity for a few years, providing new upper limits for several astrophysical sources. They are now undergoing an upgrade to Enhanced LIGO and Virgo + ; this will improve their sensitivity by a factor of ∼ 2. The second-generation interferometers, Advanced LIGO [5] and Advanced Virgo, will start operating in 2013-2015 with an overall improvement in sensitivity by a factor of ∼ 10, thus increasing the event rates for many astrophysical sources by a factor of one thousand. \nOne of the most promising sources for these detectors is the inspiral and merger of compact binary systems of black holes. The search for gravitational waves (GWs) from coalescing binaries and the extraction of parameters are based on the matched-filtering technique [6, 7], which requires a rather accurate knowledge of the waveform of the incoming signal [8]. In particular, the detection and subsequent data analysis of GW signals are made by using a bank of templates modeling the GWs emitted by the source. \nThe effective-one-body (EOB) formalism was introduced [9, 10] as a promising approach to describe analytically the inspiral, merger, and ringdown waveforms emitted during a binary merger. Necessary inputs for \nthe EOB approach include high-order post-Newtonian (PN) results [11] for two-body conservative dynamics, radiation-reaction force, and gravitational waveforms. For compact bodies, the PN approximation is essentially an expansion in the characteristic orbital velocity v/c or, equivalently, in the gravitational potential, GM/ ( rc 2 ), with r the typical separation and M the total binary mass. The EOB approach, however, does not use the PN results in their original Taylor-expanded forms (i.e., as polynomials in v/c ), but instead in some resummed forms [9, 10, 12, 13, 14, 15, 16, 17]. The latter are designed to incorporate some of the expected nonperturbative features of the exact results. \nAs it is now possible to produce very accurate numerical simulations of comparable mass binary black-hole coalescences (see e.g. [18, 19, 20, 21, 22, 23, 24, 25]), we can compare in detail the EOB predictions with numerical results, and when necessary, introduce new features into the EOB model in order to improve its agreement with the numerical results. This is an important avenue to LIGO, GEO and Virgo template construction, as eventually thousands of waveform templates may be needed to detect the GW signal within the detector noise, and to extract astrophysical information from the observed waveform. Given the high computational cost of running the numerical simulations, template construction is currently an impossible demand for numerical relativity alone. \nThis paper builds upon a rather successful recent ef- \nfort [22, 26, 27, 28, 29, 30, 31] aimed at producing the best analytical EOB model able to interpolate accurate numerical simulations. Other approaches based on phenomenological waveforms have also been proposed [32, 33]. Here we calibrate the EOB model to the most accurate numerical simulation to date of an equal-mass, non-spinning binary black-hole merger, that has been produced with a pseudospectral code by the Caltech-Cornell collaboration [21, 23]. Taking into account the uncertainties in the numerical simulation, we investigate the significance and degeneracy of the EOB adjustable parameters and determine the minimal number of adjustable parameters that achieves as good agreement as possible between the numerical and EOB GW's phase and amplitude. In addition to the leading GW mode ( /lscript, m ) = (2 , 2), we also compare the leading subdominant modes (4 , 4) and (3 , 2). By reducing the phase difference between the EOB and numerical inspiral waveforms of black-hole binaries with mass ratios q = m 1 : m 2 of 2:1 and 3:1, we explore the dependence of one of the adjustable parameters on the symmetric mass ratio ν = m 1 m 2 / ( m 1 + m 2 ) 2 . \nThe paper is organized as follows. In Sec. II, we briefly review the EOB dynamics and waveforms. In Sec. III, we calibrate the EOB model to the numerical simulation of an equal-mass non-spinning binary black-hole coalescence and determine the region of the parameter space of the EOB adjustable parameters that leads to the best agreement with the numerical results. We also discuss the impact of our results on data analysis, and calibrate the EOB model with inspiral waveforms from accurate numerical simulations of non-spinning black hole binaries with mass ratios 2:1 and 3:1. Sec. IV summarizes our main conclusions. Finally, the Appendix compares the numerical h /lscriptm extracted with the Regge-WheelerZerilli (RWZ) formalism with the h /lscriptm obtained by two time integrals of the Newman-Penrose (NP) scalar Ψ /lscriptm 4 .", 'II. EFFECTIVE-ONE-BODY MODEL': 'In this section we briefly review the EOB dynamics and waveforms, focusing mainly on the adjustable parameters. More details can be found in Refs. [9, 10, 13, 15, 22, 26, 28, 29, 30, 31]. Here we follow Refs. [22, 28].', 'A. Effective-one-body dynamics': "We set M = m 1 + m 2 , µ = m 1 m 2 /M = ν M , and use natural units G = c = 1. In absence of spins, the motion is constrained to a plane. Introducing polar coordinates ( r, Φ) and their conjugate momenta ( p r , p Φ ), the EOB effective metric takes the form [9] \nds 2 eff = -A ( r ) dt 2 + D ( r ) A ( r ) dr 2 + r 2 ( dθ 2 +sin 2 θ d Φ 2 ) . (1) \nFollowing Ref. [16, 34], we replace the radial momentum p r with p r ∗ , the conjugate momentum to the EOB tortoise radial coordinate r ∗ : \ndr ∗ dr = √ D ( r ) A ( r ) . (2) \nIn terms of p r ∗ the non-spinning EOB Hamiltonian is [9] \nH real ( r, p r ∗ , p Φ ) ≡ µ ˆ H real = M √ 1 + 2 ν ( H eff -µ µ ) -M, (3) \nwith the effective Hamiltonian [9, 13, 34] \nH eff ( r, p r ∗ , p Φ ) ≡ µ ̂ H eff = µ √ p 2 r ∗ + A ( r ) [ 1 + p 2 Φ r 2 +2(4 -3 ν ) ν p 4 r ∗ r 2 ] . (4) \nThe Taylor-approximants to the coefficients A ( r ) and D ( r ) can be written as [9, 13] \nA k ( r ) = k +1 ∑ i =0 a i ( ν ) r i , (5) \nD k ( r ) = k ∑ i =0 d i ( ν ) r i . (6) \nThe functions A ( r ), D ( r ), A k ( r ) and D k ( r ) all depend on the symmetric mass ratio ν through the ν -dependent coefficients a i ( ν ) and d i ( ν ). These coefficients are currently known through 3PN order (i.e. up to k = 4) and can be read from Eqs. (47) and (48) in Ref. [22]. Previous investigations [15, 22, 28, 29, 30, 31] have demonstrated that, during the last stages of inspiral and plunge, the EOB dynamics can be adjusted closer to the numerical simulations by including in the radial potential A ( r ) a pseudo 4PN (p4PN) coefficient a 5 ( ν ). This coefficient has so far been treated as a linear function in ν , i.e. a 5 ( ν ) = λ 0 ν , with λ 0 a constant 1 . In this paper, however, we shall also explore the possibility of going beyond this linear dependence, such that \na 5 ( ν ) = ν ( λ 0 + λ 1 ν ) , (7) \nwhere λ 0 and λ 1 are constants. In order to assure the presence of a horizon in the effective metric (1), a zero needs to be factored out from A ( r ). This is obtained by applying a Pad'e resummation [13]. The Pad'e coefficients for the expansion of A ( r ) and D ( r ) at p4PN order are denoted A 1 4 ( r ) and D 0 4 ( r ), and their explicit form can be read from Eqs. (54) and (59) in Ref. [22]. \nThe EOB Hamilton equations are written in terms of the reduced (i.e., dimensionless) quantities ̂ H real [defined in Eq. (3)], t = t/M , and ̂ Ω = Ω M [10]: \n̂ ̂ \n̂ \n̂ dp r ∗ d ̂ t = A ( r ) √ D ( r ) [ -∂ ̂ H real ∂r ( r, p r ∗ , p Φ ) + ̂ F r ( r, p r ∗ , p Φ ) ] , (10) \n̂ dr d t = A ( r ) √ D ( r ) ∂ ̂ H real ∂p r ∗ ( r, p r ∗ , p Φ ) , (8) \n√ d Φ d t = ∂ ̂ H real ∂p Φ ( r, p r ∗ , p Φ ) , (9) \ndp Φ d t = ̂ F Φ ( r, p r ∗ , p Φ ) , (11) \n̂ with the definition ̂ Ω ≡ d Φ /d ̂ t . Furthermore, for the Φ component of the radiation-reaction force we use the nonKeplerian Pad'e-approximant to the energy flux [12, 35] \n̂ \nF Φ = nK ̂ F 4 4 ≡ -v 3 Ω νV 6 Φ F 4 4 ( V Φ ; ν, v pole ) , (12) \n̂ \n̂ where v Ω ≡ ̂ Ω 1 / 3 , V Φ ≡ ̂ Ω r Ω , and r Ω ≡ r [ ψ ( r, p Φ )] 1 / 3 . Here ψ is defined by Eqs. (66)-(68) of Ref. [22]. As the EOB Hamiltonian is a deformation of the Schwarzschild Hamiltonian, the exact Keplerian relation ̂ Ω 2 r 3 Ω = 1 holds. The quantity F 4 4 in Eq. (12) is given by Eqs. (39) and (40) in Ref. [22] 2 and it uses the Taylor-expanded energy flux (as given by Eq. (19) in Ref. [22]) in the form \nF 8 ( ν ) = -323105549467 3178375200 + 232597 4410 γ E -1369 126 π 2 + 39931 294 log 2 -47385 1568 log 3 + 232597 4410 log v Ω + νA 8 , (13) \nwhere we combine the known test-mass-limit terms [36] with a p4PN adjustable parameter A 8 [22]. 3 \nThe radial component of the radiation-reaction force ̂ F r ( r, p r ∗ , p Φ ) in Eq. (10) was neglected in previous studies [22, 26, 28, 29, 30, 31] because Ref. [10] showed that for quasi-circular motion, in some gauges, it can be set to zero. Furthermore, it was shown in Ref. [10] that if the motion remains quasi-circular even during the plunge, ̂ F r ( r, p r ∗ , p Φ ) does not affect the dynamics considerably. However, since we are trying to capture effects in the numerical simulations which go beyond the quasicircular motion assumption, we find it interesting to add \n̂ \n̂ F r ( r, p r ∗ , p Φ ) [see Eq. (3.18) of Ref. [10] and the discussion around it]. We set \nF r ( r, p r ∗ , p Φ ) = a F r RR ( ν ) ˙ r r 2 Ω ̂ F Φ ( r, p r ∗ , p Φ ) , (14) \nFinally, the tangential force described by Eq. (12) applies only to quasi-circular motion. This tangential force could also in principle contain terms describing the departure from quasi-circular motion during the last stages of inspiral and plunge. There are several ways to include such non-quasi-circular (NQC) terms [15, 22, 31]; here we do so by replacing the quantity ̂ F Φ on the right-hand side of Eq. (11) [but not the ̂ F Φ on the right-hand side of Eq. (14)] with NQC ̂ F Φ , where \n̂ where a F r RR ( ν ) is an adjustable parameter. \n̂ NQC F Φ ≡ ̂ F Φ ( 1 + a F Φ RR ( ν ) ˙ r 2 ( r Ω) 2 ) , (15) \n̂ \n̂ and a F Φ RR ( ν ) is an additional adjustable parameter. The form of this NQC correction will be discussed further in Sec. III B. Note that alternative NQC terms have been proposed in the literature-for example, in Ref. [15] the authors used p 2 r / ( p Φ /r 2 ) while Ref. [31] employed p 2 r ∗ / ( r Ω) 2 . In summary, in the notation of Ref. [22], the EOB model used here is nK F 4 4 /H 4 with adjustable parameters { a 5 ( ν ) , v pole ( ν ) , a F Φ RR ( ν ) , a F r RR ( ν ) , A 8 } .", 'B. EOB waveform: Inspiral & Plunge': 'Having the inspiral dynamics in hand, we need to compute the gravitational waveform h /lscriptm . Reference [21] compared the numerically extracted gravitational waveform h 22 to the PN result with amplitude expressed as a Taylor -expansion [37, 38]; even when expanded to 3PN order, the amplitude disagreed by about one percent at times several hundred M before merger. As previous investigations [29, 30, 31] have shown, more accurate agreement with the numerical h 22 amplitude can be obtained by applying several resummations to the Taylorexpanded h 22 amplitude. These resummations have recently been improved using results in the quasi-circular test-particle limit [17]. We follow Ref. [17] and write the EOB modes h /lscriptm as \nh 22 ( t ) = -8 M R √ π 5 ν e -2 i Φ V 2 Φ F 22 , (16a) \n̂ h 32 ( t ) = -8 M 3 R √ π 7 ν (1 -3 ν ) e -2 i Φ V 4 Φ F 32 , (16c) \n̂ \n̂ \n̂ h 44 ( t ) = -64 M 9 R √ π 7 ν (1 -3 ν ) e -4 i Φ V 4 Φ F 44 , (16b) \nwhere R is the luminosity distance from the binary, and with \nF lm = { ˆ H eff T /lscriptm ( ρ /lscriptm ) /lscript e i δ /lscriptm ( /lscript + m even) ˆ J eff T /lscriptm ( ρ J /lscriptm ) /lscript e i δ /lscriptm ( /lscript + m odd) (17) \nwhere ˆ H eff and ˆ J eff are effective sources that in the testparticle, circular-motion limit contain a pole at the EOB light ring (photon orbit); here ˆ H eff is given in Eq. (4), and ˆ J eff = p Φ v Ω is equal to the orbital angular momentum p Φ normalized to the circular-orbit Newtonian angular momentum v -1 Ω . The quantities T /lscriptm , δ /lscriptm , ρ /lscriptm , ρ J /lscriptm can be read from Eqs. (19), (20), (23), (25), (C1), (C4) and (C6) in Ref. [17], respectively. More specifically, T /lscriptm is a resummed version [16] of an infinite number of leading logarithms entering the tail effects; δ /lscriptm is a supplementary phase [16] which corrects the phase effects not included in the complex tail factor; ρ /lscriptm and ρ J /lscriptm are the resummed expressions of higher-order PN effects as recently proposed in Ref. [17] in the test-particle circular-orbit limit. The latter resummation was proposed to cure, among other effects, the linear growth with /lscript of the 1PN corrections in the Taylor-expanded amplitude. \nFurthermore, motivated by the PN expansion for generic orbits, to include NQC effects in h /lscriptm we write \nh insp -plunge /lscriptm ≡ NQC h /lscriptm = ̂ h /lscriptm [ 1 + a h /lscriptm 1 ˙ r 2 ( r Ω) 2 +˙ r 2 ( a h /lscriptm 2 ˙ r 2 ( r Ω) 2 + a h /lscriptm 3 M r 1 ( r Ω) 2 ) +˙ r 4 a h /lscriptm 4 M r 1 ( r Ω) 2 ] . (18) \nAs we shall discuss in detail below, for the (2,2) mode, one of the four adjustable parameters a h 22 i in Eq. (18) will be fixed by requiring that the peak of the EOB h 22 occurs at the same time as the peak of the EOB orbital frequency [31] (i.e., at the EOB light-ring); this requires no matching to a numerical waveform. Another of the a h 22 i will be fixed by requiring that the peak amplitude of the EOB and numerical waveforms agree. The final three a h 22 i parameters will be determined by minimizing the overall amplitude difference with respect to the numerical waveform. We note that an alternative NQC factor has been proposed in Ref. [31], notably 1 + ap 2 r /star / (Ω 2 r 2 + /epsilon1 ). We shall compare those different choices below.', 'C. EOB waveform: Merger & Ringdown': 'The merger-ringdown waveform in the EOB approach is built as follows [10, 26, 28, 30, 31, 35]. For each mode ( /lscript, m ) we write \nh merger -RD /lscriptm ( t ) = N -1 ∑ n =0 A /lscriptmn e -iσ /lscriptmn ( t -t /lscriptm match ) , (19) \nwhere n is the overtone number of the Kerr quasi-normal mode (QNM), N is the number of overtones included in our model, and A /lscriptmn are complex amplitudes to be determined by a matching procedure described below. The quantity σ /lscriptmn = ω /lscriptmn -iα /lscriptmn , where the oscillation frequencies ω /lscriptmn > 0 and the inverse decay-times α /lscriptmn > 0, \nare numbers associated with each QNM. The complex frequencies are known functions of the final black-hole mass and spin and can be found in Ref. [39]. The final black-hole masses and spins can be obtained from several fitting formulae to numerical results [28, 34, 40, 41]. Here we use the more accurate final black-hole mass and spin computed in Ref. [23]: M BH /M = 0 . 95162 ± 0 . 00002, a/M BH = 0 . 68646 ± 0 . 00004. While these numbers differ from the predictions of the fitting formulae in Ref. [28] by only 0 . 3%, such disagreement would be noticeable in our comparison. The matching time t 22 match ( ν ) is an adjustable parameter that will be chosen to be very close to the EOB light-ring [10] when matching the mode h 22 . \nThe complex amplitudes A /lscriptmn in Eq. (19) are determined by matching the EOB merger-ringdown waveform with the EOB inspiral-plunge waveform. In order to do this, N independent complex equations are needed. In Refs. [10, 26, 28, 35, 42], the N equations were obtained at the matching time by imposing continuity of the waveform and its time derivatives \nd k dt k h insp -plunge /lscriptm ( t /lscriptm match ) = d k dt k h merger -RD /lscriptm ( t /lscriptm match ) , ( k = 0 , 1 , 2 , · · · , N -1) , (20) \nand we denote this approach point matching . In Refs. [30, 31], the comb matching approach was introduced. In this approach, N equations are obtained at N points evenly sampled in a small time interval ∆ t /lscriptm match centered at t /lscriptm match \nh insp -plunge /lscriptm ( t /lscriptm match + 2 k -N +1 2 N -2 ∆ t /lscriptm match ) = h merger -RD /lscriptm ( t /lscriptm match + 2 k -N +1 2 N -2 ∆ t /lscriptm match ) , ( k = 0 , 1 , 2 , · · · , N -1) . (21) \nFinally, the full (inspiral-plunge-merger-ringdown) EOB waveform reads \nh /lscriptm = h insp -plunge /lscriptm θ ( t /lscriptm match -t )+ h merger -RD /lscriptm θ ( t -t /lscriptm match ) . (22) \nThe point matching approach gives better smoothness around the matching time, but it is not very stable numerically when N is large and higher order numerical derivatives are needed. As we include eight QNMs in our ringdown waveforms, we find that the comb matching approach is more stable. To improve the smoothness of the comb matching we use here a hybrid comb matching : We choose a time interval ∆ t /lscriptm match ending at t /lscriptm match , we impose the continuity of the waveform at N -4 points evenly sampled from t /lscriptm match -∆ t /lscriptm match to t /lscriptm match , but we also require continuity of the first and second order time derivatives of the waveform at t /lscriptm match -∆ t /lscriptm match and t /lscriptm match , thus guaranteeing the continuity of h /lscriptm . Furthermore, we fix t /lscriptm match to be the time when the EOB orbital frequency reaches its maximum, and tune ∆ t 22 match in the range 2 . 5 M -3 . 5 M depending on the EOB dynamics. \nIt is worth noticing that the lowest frequency among the eight QNMs included in our merger-ringdown waveform is Mω 227 ∼ 0 . 44, which is larger than the EOB inspiral-plunge waveform frequency Mω ( t 22 match ) ∼ 0 . 36. Therefore, generically the EOB GW frequency will grow very rapidly from Mω ∼ 0 . 36 to Mω ∼ 0 . 44 immediately after the matching time, and this growth can be much more rapid than what is seen in the numerical simulation. We find that we can avoid this rapid growth by carefully fine-tuning the matching interval ∆ t 22 match , and this is what we do for the comparisons presented here. Quite interestingly, we find that the h 22 matching can be made much less sensitive to ∆ t 22 match if we include a pseudo QNM that has a frequency Mω ( t 22 match ) ∼ 0 . 36 and a decay time comparable to that of the highest overtone τ 227 ∼ 0 . 7 M . We refer to this QNM as pseudo because its frequency and decay time do not coincide with any of the QNMs of our final Kerr BH [39, 43]. Although we do not use this pseudo QNM in the present analysis, we expect that its inclusion can help when matching higher modes of equal and unequal mass binaries and we shall consider it in the future.', 'III. CALIBRATING THE EFFECTIVE-ONE-BODY WAVEFORMS TO NUMERICAL RELATIVITY SIMULATIONS': "We shall now calibrate the EOB model against a numerical simulation of an equal-mass non-spinning binary black hole. This simulation was presented as run '30c1/N6' in Scheel et al. [23], and the inspiral part of the waveform was used in previous comparisons with PN models [21, 22]. In addition to the NP scalars Ψ /lscriptm 4 extracted from this simulation, we will be using gravitational waveforms h /lscriptm extracted with the RWZ formalism [44, 45, 46, 47]. The Appendix discusses details of the numerical implementation used to obtain h /lscriptm from the RWZ scalars, and presents a comprehensive comparison of the numerical Ψ /lscriptm 4 and RWZ h /lscriptm waveforms. Consistency between the two wave-extraction schemes is good, with phase differences less than 0 . 02 radians for the (2,2)mode until about a time 20 M after the peak of h 22 | . \n| \n| Because we have more experience with the NP scalars during the inspiral, and because Ψ 22 4 appears to behave better than RWZ h 22 during ringdown (see Fig. 14 in the Appendix), we prefer to use the numerical Ψ 4 data. Therefore, during the inspiral phase, we will calibrate the EOB adjustable parameters by comparing the second time derivative of EOB h 22 against the numerical Ψ 22 4 . During the plunge-merger phase, when the time derivatives of the waveform vary most rapidly, it is more difficult to calibrate the EOB h 22 since the resummation techniques in the EOB model were aimed at providing us with the best h 22 . Therefore, around time of merger, we shall calibrate the EOB h 22 to the RWZ h 22 . Note also that data analysis is based on h /lscriptm , further motivating our choice to build the best EOB model for h /lscriptm . Neverthe-le \nafter calibration, in Sec. III C, we show comparisons of the EOB waveforms with both the numerical RWZ h 22 and Ψ 22 4 . The ringdown part of the numerical waveform is not used in the calibration of the EOB parameters; the QNMs are determined solely from the mass and spin of the final hole.", 'A. Waveform alignment and uncertainties in numerical waveforms': "As previous investigations [21, 26, 31, 48, 49] have shown, the phase error between two waveforms depends crucially on the procedure used to align them in time and phase. For the inspiral phase , we shall adopt here the alignment procedure introduced in Ref. [22] (see also Ref. [33]) that consists of minimizing the quantity \nΞ(∆ t, ∆ φ ) = ∫ t 2 t 1 [ φ 1 ( t ) -φ 2 ( t -∆ t ) -∆ φ ] 2 dt , (23) \nover a time shift ∆ t and a phase shift ∆ φ , where φ 1 ( t ) and φ 2 ( t ) are the phases of the two waveforms. This alignment procedure has the advantage of averaging over the numerical noise and residual eccentricity when aligning numerical and EOB waveforms. The range of integration ( t 1 , t 2 ) is chosen to be as early as possible, where we expect the PN-based EOB waveform to be most valid, but late enough so that it is not contaminated by the junk radiation present in the numerical initial data. Moreover, the range of integration should be large enough for the integral to average over noise and residual eccentricity. Here we fix t 1 = 1040 M and t 2 = 2260 M (measured from the start of the numerical waveform), so that we include three full cycles of phase oscillations due to eccentricity. \nUsing this alignment procedure, we estimate the errors on the numerical Ψ 22 4 by comparing Ψ 22 4 computed at different numerical resolutions and/or using different extrapolation procedures. In particular, Fig. 1 summarizes the phase errors for a set of numerical Ψ 22 4 computed in Ref. [23]. The numerical waveform labeled 'N6, n=3' (identical to the run '30c1/N6, n=3' from [23]) is the reference numerical waveform used throughout this paper unless otherwise noted. This waveform is the most accurate waveform from Ref. [23], extracted at various radii and then extrapolated to infinity. The waveforms with different values of n vary the order of the extrapolation and are used to quantify the uncertainty in the phase due to extrapolation, while those labeled by N5 (as opposed to N6) are from a simulation with a lower numerical resolution and are used to quantify the uncertainty due to numerical truncation errors. Figure 1 also includes a comparison between waveforms extracted at finite coordinate radius r ex = 225 M . \nExtrapolation with n = 2 leads to systematic errors in the extrapolated waveform (see, Fig. 10 of Ref. [21]), which in turn results in a systematic error in ∆ t . Therefore, the blue dashed line in Fig. 1 represents a possibly \nFIG. 1: Numerical error estimates. Phase difference between numerical Ψ 22 4 waveforms, when aligned using the same procedure as employed for the EOB-NR alignment [see Eq. (23)]. 'N6' and 'N5' denote the highest- and next-tohighest numerical resolution, n denotes the order of extrapolation to infinite extraction radius, and ' r = 225 M ' denotes waves extracted at finite radius r = 225 M . The data are smoothed with a rectangular window of width 10 M ; the light grey dots represent the unsmoothed data for the N5-N6 comparison at r ex = 225 M . \n<!-- image --> \noverly conservative error estimate. The feature of the solid brown curve around t ≈ 3700 M is due to an issue with data processing of the lower resolution 'N5' run. \nThe primary use of Fig. 1 is to assess numerical errors relevant for the calibration of the EOB inspiral phase. By construction of the alignment procedure, this figure shows the numerical errors for waveforms that are aligned in the interval [ t 1 , t 2 ], several orbits before merger. Calibrating the EOB inspiral phase in this manner is appropriate, because it ensures that early in the inspiral, the EOB-model and the numerical simulation agree well, i.e. that we expect little de-phasing at lower frequencies. This is important for waveform templates of low mass binaries, where the early inspiral waveform lies in LIGO's sensitive frequency band. \nFigure 1 shows that the numerical Ψ 22 4 waveforms are accurate to a few hundredths of a radian until very close to merger, when compared with our alignment procedure. Furthermore, Fig. 15 in the Appendix demonstrates that NP and RWZ waveforms differ by only 0 . 02 radians through inspiral and merger. Therefore, we shall adopt a deviation of 0 . 02 radians between EOB- and NR inspiral-waveforms as our goal for the EOB inspiral calibration. The horizontal line in Fig. 1 indicates this phase difference of 0 . 02 radians and it will be our requirement when calibrating the EOB values of Ψ 22 4 . The numerical phase errors exceed 0 . 02 radians at times t = 3660 M , 3850 M , 3900 M and 3933 M , respectively, and so our goal will be for EOB to agree to 0 . 02 radians at least up to \nTABLE I: Summary of all possible adjustable parameters of the EOB model considered in this paper. As we shall discuss in the main text, we will not need all of these parameters. In particular, we find that for the black-hole binary simulations investigated here, the choices a F r RR ( ν ) = 0 = a F Φ RR ( ν ) , A 8 = 0, t /lscriptm match ( ν ) at the peak of the EOB orbital frequency, allow the numerical and EOB values of the GW phase and amplitude to agree within numerical error. Furthermore, we find that for an equal-mass black-hole binary coalescence it is sufficient to set a 5 ( ν ) = νλ 0 [see Eq. (7) with λ 1 = 0 ] and calibrate λ 0 , v pole (1 / 4), ∆ t 22 match (1 / 4) and a h 22 i (1 / 4). For an equal-mass black-hole binary coalescence it is even possible to calibrate only one EOB-dynamics adjustable parameter, λ 0 [see Eq. (7)] and let v pole → ∞ . Finally, for an unequalmass binary inspiral it is sufficient either to set λ 1 = 0, use the value of λ 0 from the equal-mass binary case, and calibrate v pole ( ν ); or alternatively to let v pole →∞ and calibrate both λ 0 and λ 1 in a 5 ( ν ) [see Eq. (7)]. \n| EOB-dynamics | EOB-waveform adjustable parameters adjustable parameters |\n|-----------------------------------------------------------|-----------------------------------------------------------------------------------------|\n| a 5 ( ν ) v pole ( ν ) a F r RR ( ν ) or a F Φ RR ( ν A 8 | t /lscriptm match ( ν ) ∆ t /lscriptm match ( ν ) ) a h /lscriptm i ( ν ) i = 1 , ... 4 | \nt ≈ 3900 M . The choice of 0 . 02 radians is motivated by the goal of bringing the disagreement between the EOB and numerical phases at least to the level of the numerical error (see Fig. 15).", 'B. Tuning the adjustable parameters of the equal-mass effective-one-body dynamics': "We divide the adjustable parameters into two groups and tune them separately in two steps. The first group of EOBdynamics parameters includes { a 5 ( ν ) , v pole ( ν ) , a F Φ RR ( ν ) , a F r RR ( ν ) , A 8 } . These parameters determine the inspiral and plunge dynamics of the EOB model and affect the merger-ringdown waveform only indirectly through the waveform's phase and frequency around the matching point. [We note that the inspiral phase is independent of the parameters a h /lscriptm i , see Eq. (18).] These parameters are calibrated to the numerical NP Ψ 22 4 . The second group of EOBwaveform parameters includes { a h /lscriptm i , t /lscriptm match , and ∆ t /lscriptm match } , and affect only the plunge-merger-ringdown but not the inspiral EOB waveform. These parameters are calibrated to the numerical RWZ h 22 . All the possible adjustable parameters of the EOB model employed in this paper are summarized in Table I. In the first step of our calibration procedure, we reduce the phase difference before merger by tuning the EOB-dynamics parameters. In the second step, we use these fixed values of the EOB-dynamics parameters, and tune the EOB-waveform parameters. \nAmong the EOB-dynamics parameters, a 5 ( ν ) and v pole ( ν ) are the most important as they affect the entire quasi-circular evolution of the inspiral. The two \nradiation-reaction parameters a F Φ RR and a F r RR are introduced to adjust the dynamics of late inspiral when we expect that the quasi-circular assumption is no longer valid. The p4PN parameters in the energy flux, A 8 , also influences the entire evolution, but we find that A 8 is strongly degenerate with a 5 (1 / 4) throughout the inspiral until a time ∼ 100 M before merger. Based on these considerations, we shall tune a 5 (1 / 4) and v pole first and consider a F Φ RR (1 / 4), a F r RR (1 / 4) and A 8 only when exploring how to further improve the late evolution. \nTherefore, in our first step, we set a F Φ RR (1 / 4) = a F r RR (1 / 4) = A 8 = 0 and vary a 5 (1 / 4) and v pole (1 / 4). Applying the alignment procedure presented at the beginning of Sec. III A, we shift each EOB Ψ 22 4 in time and phase to agree with the reference numerical waveform at low frequency, and determine the time when the phase difference between the numerical and EOB Ψ 22 4 waveforms becomes larger than 0 . 02 radians. We denote this reference time as t ref . \nFigure 2 is a contour plot of the time t ref in the a 5 (1 / 4)v pole (1 / 4) parameter space. For all points inside the largest contours (blue curves), the associated EOB Ψ 22 4 phase evolutions agree with the numerical ones up to t = 3660 M , which is the earliest reference time considered in Sec. III A. In order to get EOB models that have phase differences less than 0 . 02 radians until t = 3900 M , a 5 (1 / 4) and v pole (1 / 4) have to be inside the innermost two separate thin contours (red curves). One might view these contours as encompassing all values of a 5 (1 / 4) and v pole (1 / 4) that are consistent with the numerical inspiral waveform, given the fixed choices of the various other EOB parameters. There are a 5 (1 / 4) and v pole (1 / 4) values that make the EOB phase differences less than 0 . 03 radians until t = 3933 M , but not less than 0 . 02 radians until t = 3933 M (the latest reference time). We find that phase errors of the EOB Ψ 22 4 corresponding to the upper left contours in Fig. 2 grow rapidly after t = 3900 M , whereas phase errors of EOB Ψ 22 4 corresponding to the lower right contours grow only mildly until around t = 3940 M . For this reason, we shall restrict the tuning of the other adjustable parameters to the lower right region of Fig. 2 inside the innermost contour. As a reference set, we choose a 5 (1 / 4) = 6 . 344 and v pole (1 / 4) = 0 . 85. 4 We note that the latter value is rather different from the value obtained in Ref. [31] when a 5 (1 / 4) = 6 . 25 is used. This is due to differences \nFIG. 2: In the parameter space of the EOB-dynamics adjustable parameters a 5 (1 / 4) and v pole (1 / 4) we show the contours of the time t ref at which the phase difference between the numerical '30c1/N6, n=3' and EOB Ψ 22 4 becomes larger than 0 . 02 radians. Note that the innermost red contours cover two disjoint regions. The inset shows the effect of numerical uncertainty: The filled contours are the t ref = 3850 M and 3900 M contours from the main panel. The open contours are identical, except computed using the '30c1/N6, n=2' numerical Ψ 22 4 . The reference model is shown as a black dot. \n<!-- image --> \nbetween the EOB models - for example Ref. [31] employs the Pad'e-resummed GW energy flux with constant logarithms, whereas we use the Pad'e-resummed GW energy flux with factorized logarithms. \nQuite interestingly, looking more closely at the red lines in the right corner of Fig. 2, as v pole increases, we find another possible reference set a 5 (1 / 4) = 4 . 19 and v pole → ∞ . With this choice, the pole in the Pad'e flux of Eq. (12) disappears. \nIn order to understand whether further tunings of radiation-reaction effects by adjusting the parameters ( v pole , a F Φ RR , a F r RR , A 8 ) can modify the phasing during plunge, we compute how sensitive the phasing is to radiation-reaction effects once the binary has passed the last stable orbit (LSO) defined as ( ∂H eff /∂r ) LSO = 0 = ( ∂ 2 H eff /∂r 2 ) LSO . Reference [50] pointed out that the phasing during the plunge is not affected much by radiation reaction, but driven mostly by the conservative dynamics. We want to quantify the latter statement more fully. \nIn order to do this, we need to define when the plunge starts. In the absence of radiation reaction, the plunge starts beyond the LSO where r = r LSO , ω = ω LSO and p Φ = p LSO Φ . But in the presence of radiation reaction, Ref. [10] observed that there is not a unique t LSO at which the conditions r = r LSO , ω = ω LSO and p Φ = p LSO Φ are satisfied. In fact, the above conditions may happen at different times (see Fig. 12 in Ref. [10]). Indeed, for \nFIG. 3: For the case a 5 (1 / 4) = 6 . 344 and v pole (1 / 4) = 0 . 85 ( A 8 = 0, a F Φ RR = 0 and a F r RR = 0), we show the phase difference between the numerical and EOB mode h 22 versus the numerical GW frequency Mω 22 for EOB models in which the GW energy flux is shut down at several EOB orbital frequencies. The vertical line marks the maximum EOB orbital frequency. \n<!-- image --> \nthe case a 5 (1 / 4) = 6 . 344 and v pole (1 / 4) = 0 . 85, we find that with radiation reaction, r ( t r LSO ) = r LSO , ω ( t ω LSO ) = ω LSO and p Φ ( t p Φ LSO ) = p LSO Φ where t r LSO = 3914 . 50 M , t ω LSO = 3919 . 83 M and t p Φ LSO = 3885 . 53 M , and where the orbital frequencies, corresponding to the three different t LSO values are M Ω = 0 . 975 , 0 . 106, and 0 . 074, respectively. Following Ref. [10], we will say that the plunge starts during the time interval spanned by the values of t LSO which in this case is t LSO 34 M before merger. \nIn Fig. 3, we show the phase difference between the numerical and EOB h 22 as a function of the numerical GW frequency Mω 22 for EOB models in which the GW energy flux is suddenly shut down at several EOB orbital frequencies. The cyan curve in Fig. 3 is obtained when the GW energy flux is not shut down. Note that in this case the phase difference increases fast close to the EOB matching point, which is marked by the vertical line in Fig. 3. The phase difference can change considerably its shape (including the sign of the slope close to the EOB matching point) when the energy flux is shut down before M Ω = 0 . 12-0 . 13, but it does not change much, especially the fast increase close to the matching point, when the energy flux is shut down after M Ω = 0 . 12-0 . 13, immediately after the LSO defined by the condition ω ( t ω LSO ) = ω LSO above. \n∼ \nThis study suggests that it is difficult to modify the behaviour of the EOB phasing during plunge by tuning only the adjustable parameters entering the radiationreaction terms or the GW energy flux, a F r RR ( ν ), a F Φ RR ( ν ) and A 8 , v pole ( ν ). The behaviour of the EOB phasing during plunge is more sensitive to adjustable parameters in the EOB conservative dynamics, e.g., a 5 ( ν ) at 4PN \nFIG. 4: Effect of a F r RR on contours of acceptable EOB parameters. The solid contours are the t ref = 3850 M and 3900 M contours from Fig. 2. The open contours shifted to the lowerright \n<!-- image --> \nare the same, but computed with a F r RR = 0 . 5 instead of a F Φ RR = 0. The reference model is shown as a black dot. \norder or a 6 ( ν ) at 5PN order, etc. However, the parameters a i ( ν ) also affect the phasing during the very long inspiral, and a careful tuning is needed to reach excellent agreement both during inspiral and plunge. \nNevertheless, it is possible to modify the behaviour of the EOB phasing during the late inspiral by tuning A 8 , a F r RR (1 / 4) and a F Φ RR (1 / 4) together with a 5 (1 / 4) and v pole (1 / 4). As an example of this, we redo the contour plot shown in Fig. 2, but with a F r RR (1 / 4) = 0 . 5 instead of zero. The result is shown as dashed curves in Fig. 4. We still find EOB models that have phase differences less than 0 . 02 radians until t = 3900 M . In particular, with the reference value v pole (1 / 4) = 0 . 85 and choosing a 5 (1 / 4) = 6 . 013, we find that the behaviour of the EOB phasing is substantially modified only for the last 40 M of evolution before merger. In this case, the change in phase difference is in the range of 0 . 01-0 . 1 radians, and the slope of phase difference at the matching point can change sign. Similar results are obtained when repeating this analysis with a F Φ RR (1 / 4) or A 8 different from zero. We also observe that the effect on the dynamics of the adjustable parameter a F Φ RR (1 / 4) is almost equivalent to the effect of the adjustable parameter a F r RR (1 / 4), except for a minus sign and a different scaling. So it is not necessary to consider both of these radiation-reaction adjustable parameters. \nAlthough time consuming, in principle it is possible to perform a comprehensive search over the complete set of the EOB-dynamics parameters a 5 ( ν ), v pole ( ν ), A 8 , a F r RR ( ν ) or a F Φ RR ( ν ). However, at this point there is no need \nFIG. 5: We compare the numerical and EOB h 22 amplitudes when the EOB model with reference values a 5 (1 / 4) = 6 . 344 and v pole (1 / 4) = 0 . 85 are used. We show the EOB amplitudes without the NQC corrections and the EOB amplitude with the NQC terms suggested in Ref. [31], where the NQC parameters take the values a = 0 . 75 and /epsilon1 = 0 . 09. When the NQC corrections are not included, we show the EOB amplitude of Eq. (16a) which uses the resummation procedure of Ref. [17], and also the EOB amplitudes of Eq. (16a) when the Pad'e-resummations P 1 4 and P 2 3 of ρ 22 suggested in Ref. [17] are applied. Note that in this plot, the EOB amplitudes do not contain the merger-ringdown contribution. \n<!-- image --> \nto further improve the EOB evolution close to merger, and achieve better agreement with the equal-mass, nonspinning numerical data, since the agreement is already at the level of the numerical error. Thus, in the following, we shall use the values of a 5 (1 / 4) and v pole (1 / 4) based on Fig. 2, obtained by setting to zero all the other EOBdynamics adjustable parameters in Table I. We will leave a comprehensive study of the other EOB-dynamics adjustable parameters to future work when highly accurate numerical merger waveforms of unequal-mass black-hole binaries become available. \nWe shall now discuss the EOB model with reference values a 5 (1 / 4) = 6 . 344 and v pole (1 / 4) = 0 . 85, and tune the EOB-waveform adjustable parameters. We shall comment at the end of this section on the results when the other reference values a 5 (1 / 4) = 4 . 19 and v pole (1 / 4) → ∞ are used. In Fig. 5 we compare the numerical and EOB h 22 amplitudes with and without including NQC terms. The agreement of the numerical amplitude with the EOB amplitude of Eq. (16a) without NQC terms, which uses the resummation procedure of Ref. [17], is rather remarkable. The relative difference at the peak is only ∼ 1 . 5%, and the EOB peak amplitude occurs only ∼ 6 M before the numerical peak amplitude. \nWe notice that this excellent agreement is due to the presence in ρ 22 of test-particle corrections through 5PN order. Were the test-particle corrections through 4PN or 5PN orders not included, the disagreement at the peak would become 4 . 9% and 11 . 3%, respectively. 5 \nFigure 5 also shows the EOB amplitudes of Eq. (16a) when the Pad'e-resummations P 1 4 and P 2 3 of ρ 22 suggested in Ref. [17] are applied. In these cases, the EOB peak amplitude almost coincides in time with the numerical peak amplitude, but the relative difference in the value of the peak amplitude is rather large. However, those large differences may be resolved if the resummed version of the GW energy flux [17] consistent with the resummed h /lscriptm were used. Figure 5 also contains the EOB h 22 amplitude with NQC terms as suggested in Refs. [30, 31] [see Eq. (12) in Ref. [31]]. The relative difference with the numerical amplitude is ∼ 20% at the peak. It is rather interesting to observe, as pointed out in Ref. [31], that by aligning the numerical and EOB waveforms at low frequency, we find that the peak of the numerical h 22 coincides with the peak of the EOB orbital frequency. Here, to improve the amplitude agreement during plunge and merger, we include the NQC corrections of Eq. (18). We fix two of the adjustable parameters, a h 22 1 and a h 22 2 , by requiring that a local extremum of the EOB h 22 amplitude occurs at the same time as the peak of the EOB orbital frequency (i.e., the EOB light-ring), and that the EOB amplitude at the peak coincides with the numerical amplitude at the peak. In fact, we expect that in the near future, the peak of the numerical h 22 will be able to be predicted by numerical relativity with high accuracy for several mass ratios. Thus, the peak can be fit with a polynomial in ν . (Preliminary studies which use results from Ref. [28] confirm this expectation.) The other two adjustable parameters, i.e., a h 22 3 and a h 22 4 , are calibrated to the numerical results to further reduce the disagreement. Specifically, we do a two-parameter least-square-fit of the ratio of the numerical RWZ and EOB h /lscriptm on Eq. (18) in which a h 22 1 and a h 2 2 are fixed as functions of a h 22 3 and a h 2 4 by the requirements described above. We notice that the strategy of improving the amplitude agreement followed in this paper might change in the future, when accurate numerical unequal-mass black-hole binary inspiralmerger-ringdown waveforms become available. A smaller number of adjustable parameters might suffice if more requirements on the EOB model itself can be imposed or if a different matching procedure, such as the one suggested in Ref. [51], is employed. \nFIG. 6: Comparison of numerical waveform to EOB waveform with a 5 (1 / 4) = 6 . 344 and v pole (1 / 4) = 0 . 85, i.e. the same model used in Fig. 5. The top panels show the real part of numerical and EOB h 22 , the bottom panels show amplitude and phase differences between them. The left panels show times t = 0 to 3900 M , and the right panels show times t = 3900 to t = 4070 M on a different vertical scale. \n<!-- image -->", 'C. Comparing the gravitational-wave modes h /lscript m of equal-mass coalescing black-hole binaries': 'In this section, we focus on the model whose EOBdynamics and EOB-waveform adjustable parameters were calibrated to numerical RWZ h 22 and NP Ψ 22 4 in Sec. III B. Using this EOB model, we generate the GW modes h 22 , h 32 and h 44 and compare them to the corresponding numerical modes. We choose these three modes because they are the most dominant ones for an equalmass, non-spinning black-hole binary. \nIn Fig. 6, we show the numerical and EOB mode h 22 aligned with the procedure of Sec. III A. Using the reference values a 5 (1 / 4) = 6 . 344 and v pole (1 / 4) = 0 . 85, we find that the best phase and amplitude agreement is obtained when the matching occurs at an interval of ∆ t 22 match = 3 . 0 M ended at t 22 match = 3942 . 5 M , i.e., at the peak of M Ω, with a h 22 1 (1 / 4) = -2 . 23 and a h 22 2 (1 / 4) = 31 . 93, a h 22 3 (1 / 4) = 3 . 66 and a h 22 4 (1 / 4) = -10 . 85. The phase difference is strictly within ± 0 . 02 radians until the merger, i.e., the peak of h 22 , which happens at t = 3942 . 5 M (early numerical data contaminated by junk radiation was discarded until t = 200 M ). The relative amplitude difference is also within ± 0 . 02 in this range. The phase difference becomes 0 . 04 radians at t = 3962 M , before a rather large error starts contaminating the numerical h 22 . A more careful tuning on the EOB-waveform adjustable parameters could further improve the phase agreement. However, we do not think it is worthwhile to improve the agreement at this point since we are only examining the equal-mass case. Note that the relative amplitude difference becomes ∼ 7% at t = 3962 M , and \nFIG. 7: Comparison between EOB h 22 and the numerical Ψ 22 4 . The top four panels show the real part of the waveform, on a linear and logarithmic y -axis. The bottom two panels show the phase difference (in radians) and the fractional amplitude difference between the two waveforms. The left panels show times t = 0 to 3900 M , and the right panels show times t = 3900 to t = 4070 M with different vertical scales. (The quantities in the lower left panel have been smoothed; the grey data in the background of that panel presents the raw data.) This figure uses the same EOB model as Figs. 5 and 6, namely a 5 (1 / 4) = 6 . 344 and v pole (1 / 4) = 0 . 85. \n<!-- image --> \nincreases during the ringdown. \nThe numerical GW strain h 22 plotted in Fig. 6 is computed using RWZ wave extraction. During the ringdown, this waveform is noisier than the extracted NP scalar Ψ 22 4 (see the Appendix). Therefore, in Fig. 7, we compare the numerically extracted Ψ 22 4 with the second time derivative h 22 of the EOB waveform. Overall, the agreement is much better than for the comparison of h 22 in Fig. 6. Phase and relative amplitude differences are smaller than 0 . 002 during most of the inspiral, and remain smaller than 0 . 01 up to t = 3920 M . In the interval around merger, t = 3930 M to 3960 M , the agreement is slightly worse than in Fig. 6; the disagreement in this region is caused by the differences between the inspiral EOB h 22 and numerical NP Ψ 22 4 frequencies, as discussed at the beginning of Sec. III. \nIn the ringdown region, t > 3960 M , Fig. 7 shows excellent agreement, and this agreement persists until late times. In contrast to the h comparison shown in Fig. 6, in Fig. 7 both phase and amplitude differences remain bounded ; during the ringdown, the phase difference between EOB h 22 and Ψ 22 4 oscillates around 0 . 08 radians, \nFIG. 8: We show the amplitude and frequency of the numerical and EOB mode h 22 , the EOB orbital frequency and the frequency of the numerical mode Ψ 22 4 . The vertical line marks the peak of the EOB amplitude and orbital frequency. \n<!-- image --> \nand the amplitude differs by about 8%. Apart from small oscillations likely caused by gauge effects (see the Appendix), ∆ φ remains constant to an excellent degree during about 9 ringdown oscillations, i.e. during an accumulated phase of about 56 radians. If the quasi-normal mode frequency used in the EOB ringdown waveform were different from the numerical ringdown frequency by as little as 0.1%, a linearly accumulating phase-difference of ∼ 0 . 056 radians would accumulate, which would be clearly noticeable in the lower right panel of Fig. 7. Thus, we find agreement at the 0.1% level between the numerical quasi-normal mode frequency and the prediction based on final mass and spin of the numerical simulation. \nIn Fig. 8, we compare the amplitude and frequency of numerical and EOB h 22 waveforms together with the orbital frequency of the EOB model. The peak of the latter is close to the EOB light ring, and is aligned with both the EOB and numerical h 22 amplitudes (as required by our choice of a h 22 1 and a h 22 2 ). During the ringdown, the frequency computed from the numerical h 22 shows increasingly large oscillations. We also plot the frequency computed from the numerical Ψ 22 4 mode. This frequency shows much smaller, and bounded, oscillations deep into the ringdown regime. \nHaving constructed our EOB waveform purely by considering the (2,2) mode, we now discuss agreement between higher modes of the EOB model and the numerical simulation. Figure 9 shows phase and amplitude differences for the two next largest modes, the (4,4) and the (3,2) mode. The EOB model is identical to the one that has been calibrated to agree with the (2,2) mode, and the parameters a h 32 i and a h 44 i , which appear in Eq. (18) to correct the amplitude of the higher-order modes for non- \nFIG. 9: Upper panel: Amplitude and phase differences of numerical and EOB mode h 32 over the inspiral range. Lower panel: Amplitude and phase differences of numerical and EOB mode h 44 over the inspiral range. \n<!-- image --> \nquasi-circular motion, are set to zero. But although the EOB model has not been calibrated in any way to match the higher-order modes, the agreement between numerical and EOB waveforms shown in Figure 9 is rather good for t /lessorsimilar 3700 M . In fact, the differences between EOB and NR modes are comparable to the estimated numerical errors in these modes (as estimated by convergence tests between different numerical resolutions, and the comparison between the numerical h and Ψ 4 waveforms which are presented in the Appendix. Around t ≈ 3700 M , the numerical (3,2) and (4,4) modes begin to show additional features, which we believe are unphysical, and are described in more detail in the Appendix. These features prevent a meaningful comparison of the (3,2) and (4,4) modes at later times. \nFigure 10 shows amplitude and frequency of the (4,4) and (3,2) modes for both the EOB model and the numerical simulation for the last few hundred M of inspiral. This figure begins approximately where the NR-EOB differences in Fig. 9 exceed the vertical scale of that figure. The EOB amplitude and phase follow roughly the average of numerical results, which show oscillations resulting from numerical errors. At earlier times, the EOB and NR amplitudes, phase and frequencies track each other very closely, as can be seen from Fig. 9. Please compare also with Fig. 8 which plots the frequencies for the (2,2) mode. \nFinally, in Fig. 11, we show the numerical and EOB mode h 22 using the reference values a 5 (1 / 4) = 4 . 19 and v pole → ∞ . In this case we find that the best phase and amplitude agreement is obtained when the matching occurs over a range of ∆ t 22 match = 2 . 2 M ended at the peak \nFIG. 10: Upper panel: Amplitude of the numerical and EOB modes h 32 and h 44 . Lower panel: Frequency of the numerical and EOB modes h 32 and h 44 . The EOB orbital frequency 2 M Ω (4 M Ω) is indistinguishable from the frequency of the h 32 ( h 44 ) mode on the scale of this plot. \n<!-- image --> \nof M Ω, with a h 22 1 (1 / 4) = -2 . 50 and a h 22 2 (1 / 4) = 35 . 43, a h 22 3 (1 / 4) = 4 . 91 and a h 22 4 (1 / 4) = -32 . 40. Comparing the result with that of Fig. 6, we notice that the phase and amplitude differences are only slightly worse than the reference model of Fig. 6, but still within numerical error.', 'D. Impact on data analysis': "Using the EOB model with reference values a 5 (1 / 4) = 6 . 344 and v pole (1 / 4) = 0 . 85, we now quantify the disagreement between numerical and EOB waveforms by calculating their maximized overlaps which are important for analysis of data [52] from GW detectors. Here we restrict ourselves to the dominant mode h 22 . Given two time-domain waveforms h 1 ( t ) and h 2 ( t ; t 0 , φ 0 ) generated with the same binary parameters, the maximized overlap, otherwise known as a fitting factor (FF), is given explicitly by [27] \nwhere \nFF ≡ max t 0 ,φ 0 〈 h 1 , h 2 ( t 0 , φ 0 ) 〉 √ 〈 h 1 , h 1 〉〈 h 2 ( t 0 , φ 0 ) , h 2 ( t 0 , φ 0 ) 〉 , (24) \n〈 h 1 , h 2 〉 ≡ 4 Re ∫ ∞ 0 ˜ h 1 ( f ) ˜ h ∗ 2 ( f ) S h ( f ) df . (25) \nHere ˜ h i ( f ) is the Fourier transform of h i ( t ), and S h ( f ) is the detector's power spectral density. We compute the \nFIG. 11: Comparison of the numerical data to an EOB model with v pole = ∞ . This figure is analogous to Fig. 6, but uses an EOB-model that was calibrated with the restriction v pole = ∞ (parameters are given in the main text). Even without v pole , the inspiral can be matched equally well as in Fig. 6; during the ringdown, the phase differences are somewhat larger, but it is possible that refined tuning will reduce them further. \n<!-- image --> \nFFs for binary black holes with total mass 30-150 M /circledot , using LIGO, Enhanced LIGO and Advanced LIGO noise curves 6 , and find in all cases FFs larger than 0 . 999. Note that the FFs are computed maximizing over time of arrival and initial phase, but not over the binary parameters. We note that FF ≥ 0 . 999 gives a mismatch /epsilon1 ≡ 1 -FF between the numerical and the analytical h 22 of /epsilon1 NR -EOB ≤ 0 . 001. For the noise curves of LIGO, Enhanced LIGO and Advanced LIGO, we find that the mismatch between all extrapolated numerical waveforms h is less than 0 . 0001 for black-hole binaries with a total mass of 30-150 M /circledot . If we take this mismatch as an estimate of the difference between the numerical and the exact physical waveforms, we have /epsilon1 e -NR ≤ 0 . 0001. The mismatch between the exact and the analytical h 22 is therefore /epsilon1 e -EOB ≤ 0 . 0017. This mismatch is smaller than the bound 0 . 005 presented in Ref. [8], and therefore our EOB model is sufficiently accurate for GW detection in LIGO, Enhanced LIGO, and Advanced LIGO. \nAs a check of the robustness of our EOB model calibrated to numerical waveforms of equal-mass blackhole binaries, we extend the model to a set of unequalmass black-hole binaries by comparing numerical and EOB Ψ 22 4 inspiraling waveforms for mass ratios 2:1 and 3:1. These simulations were performed with the CaltechCornell SpEC code, last about eight orbits and have phase errors similar to the equal mass simulation discussed so far. Details of simulations will be published separately. We explore here the possibility of setting a 5 ( ν ) = νλ 0 with λ 0 constant and let v pole depend on the mass ratio. Indeed, in the test-particle limit we expect 7 from 1 / √ 3. v pole (0) = 1 / √ 3 = 0 . 57735, whereas in the equal-mass case we find v pole (1 / 4) = 0 . 85. As a preliminary study, we do not perform a comprehensive search over the λ 0 -v pole parameter space for unequal-mass binaries, as we did for equal-mass binaries in Sec. III B. We fix λ 0 to our reference value 25 . 375 and tune v pole ( ν ) to require phase differences on the order of the numerical error. \nIn Figs. 12 and 13, we compare the numerical and EOB Ψ 22 4 waveforms and their amplitude and phase differences for binaries with mass ratios q = m 1 : m 2 of 2:1 and 3:1. The alignment procedure of Sec. III A was used with t 1 = 310 M and t 2 = 930 M . The figures also show the numerical phase error obtained from runs with two different numerical resolutions. The specifics of these numerical runs will be published separately. In the case of mass ratios q = 2:1 and 3:1, we find that by tuning v pole ( ν ), the difference between numerical and EOB waveforms can be reduced to values smaller than the numerical error. The best values of v pole we find are v pole = 0 . 76 ± 0 . 01 for mass ratio 2:1, and v pole = 0 . 70 ± 0 . 01 for mass ratio 3:1. Choosing parameters outside this range results in differences between numerical and EOB waveforms that are at least twice the numerical error. Combining v pole values for mass ratios 1:1, 1:2, 1:3, and the test-particle limit, we find a least-square fitting formula v pole ( ν ) = 0 . 57 -0 . 65( ± 0 . 35) ν +7 . 0( ± 1 . 5) ν 2 . \nFinally, we observe that the phase and amplitude differences between numerical and EOB waveforms can be reduced to values smaller than the numerical error, if we choose the EOB reference model of Sec. III B where we set v pole → ∞ and let a 5 ( ν ) = ν ( λ 0 + λ 1 ν ). In particular, calibrating the mass ratio 2:1 and 3:1, we find a 5 ( ν ) = ν [ -7 . 3( ± 0 . 1) + 95 . 6( ± 0 . 3) ν ]. These EOB models agree with the numerical data as well as the EOB models shown in Figs. 12 and 13. \nFIG. 12: EOB-NR comparison for a BH binary with mass ratio 2 : 1. The upper panel shows the numerical and EOB mode Ψ 22 4 , and the lower panel shows phase and amplitude differences between EOB and numerical run. The dashed brown line is the estimated phase-error of the numerical simulation, obtained as the difference between simulations at high resolution 'N6' and lower resolution 'N5'. \n<!-- image -->", 'IV. CONCLUSIONS': "In this paper, building upon recent, successful results [17, 22, 26, 27, 28, 29, 30, 31] of the EOB formalism [9, 10, 13, 14, 15], we have concentrated on the EOB model denoted nK F 4 4 /H 4 in Ref. [22], with adjustable EOB-dynamics and EOB-waveform parameters defined in Table I. We have calibrated this EOB model to a very accurate numerical simulation of an equal-mass nonspinning binary black-hole coalescence [23]. \nWhen comparing EOB and numerical waveforms, or when comparing numerical waveforms with each other, we determine the arbitrary time offset and phase offset between the waveforms by minimizing the phase differences between the waveforms over a time interval of ∼ 1000 M at low frequency, where the PN-based EOB waveforms are expected to be most accurate [22]. Compared to aligning waveforms at a particular time or frequency, this procedure is less sensitive to numerical noise and residual eccentricity. \nAmong the EOB-dynamics adjustable parameters { a 5 (1 / 4) , v pole (1 / 4) , a F Φ RR (1 / 4) , a F r RR (1 / 4) , A 8 } , the parameters a 5 (1 / 4) and v pole (1 / 4) have the largest effect upon the long inspiral phase. Thus, we set { a F Φ RR (1 / 4) = 0 , a F r RR (1 / 4) = 0 , A 8 = 0 } in our EOB model, and we considered the phase difference between the numerical and \nFIG. 13: EOB-NR comparison for a BH binary with mass ratio 3 : 1. The upper panel shows the numerical and EOB mode Ψ 22 4 , and the lower panel shows phase and amplitude differences between EOB and numerical run. The dashed brown line is the estimated phase-error of the numerical simulation, obtained as the difference between simulations at high resolution 'N6' and lower resolution 'N5'. \n<!-- image --> \nEOB Ψ 22 4 as a function of the parameters a 5 (1 / 4) and v pole (1 / 4). This phase difference increases with time, so we have sought parameters for which this phase difference remains small for as long a time as possible. We found regions of the ( a 5 (1 / 4) , v pole (1 / 4)) parameter space where this phase difference is less than 0 . 02 radians either until t = 282 M or until t = 42 M (red curves) before the time when the numerical h 22 reaches its maximum amplitude (see blue and red curves in Fig. 2), respectively. \nMoreover, building on Refs. [10, 50], we have found that the EOB-dynamics adjustable parameters entering the GW energy flux cannot modify the phase of the EOB Ψ 22 4 during the plunge and close to merger. This is because any modification of the GW energy flux beyond the LSO has negligible effect on the phasing, as the evolution is driven mostly by the conservative part of the dynamics. We also found that A 8 is strongly degenerate with a 5 (1 / 4) until almost 100 M before merger, and that a F r RR (1 / 4) and a F Φ RR (1 / 4) have an almost equivalent effect on the phasing, except for a minus sign and a different scaling. Overall, for the equal-mass non-spinning case, we have found that the EOB-adjustable parameters { a F Φ RR (1 / 4) , a F r RR (1 / 4) , A 8 } have a minor effect in reducing the phase and amplitude differences between the EOB model and the numerical simulation (see also Fig. 4). To achieve differences on the order of the numerical error, we can restrict ourselves to the EOB parameter space \nwith a F Φ RR (1 / 4) = 0 , a F r RR (1 / 4) = 0 , A 8 = 0 } . \n{ \n} Furthermore, using our alignment procedure, we have found that the peak of the numerical h 22 coincides with the peak of the EOB orbital frequency, confirming what was pointed out in Ref. [31]. As in Ref. [31], we require that the EOB dominant mode h 22 peaks at the maximum of the EOB orbital frequency (i.e., the EOB light-ring). We also require that the EOB amplitude at the peak coincides with the numerical amplitude at the peak. In fact, we expect that in the near future, the peak of the numerical h 22 will be able to be predicted by numerical relativity with high accuracy for several mass ratios. Thus, the peak can be fit with a polynomial in ν . (Preliminary studies which use results from Ref. [28] confirm this expectation.) These requirements determine the EOB-waveform parameters a h 22 1 (1 / 4) and a h 22 2 (1 / 4). To further improved the agreement close to merger, we then tune a h 22 3 (1 / 4), a h 22 4 (1 / 4), and ∆ t 22 match (1 / 4), so that the phase and amplitude differences between the EOB and numerical h 22 are minimized. In particular, we found that this happens if ∆ t 22 match (1 / 4) is chosen to be around 3 M (while t 22 match (1 / 4) is fixed at the maximum of the EOB orbital frequency M Ω). For the EOB reference model with a 5 (1 / 4) = 6 . 344 and v pole = 0 . 85, we have found that the phase and amplitude differences between EOB and numerical h 22 waveforms are 0 . 02 radians and 2%, respectively, until 20 M before merger, and 0 . 04 radians and 7%, respectively, during merger and early ringdown, until the numerical h 22 starts to be affected by numerical oscillations (see Fig. 6). These agreements were obtained by comparing EOB and numerical values of h 22 , the latter having been extracted from the RWZ scalars. We also compared the EOB and numerical Ψ 22 4 . In this case, the agreement is even better during the long inspiral and through the late ringdown, with phase and amplitude disagreements of 0 . 02 radians and 2% until 20 M before merger, and 0 . 08 radians and 8%, respectively, during merger and ringdown (see Fig. 7). However, around the transition between plunge and ringdown, the EOB h 22 has some oscillations because the EOB resummation provides us with h 22 , whereas when taking time derivatives of h 22 non-resummed higher order PN terms are generated, spoiling in part the agreement of h 22 . \nQuite interestingly, we have found that phase and amplitude differences between EOB and numerical waveforms can also be reduced to numerical errors, at least during the inspiral, if we let v pole →∞ and calibrate the coefficients λ 0 and λ 1 in a 5 ( ν ) = ν ( λ 0 + λ 1 ν ), see Fig. 11. \nFor data analysis purposes, we have also computed the maximized overlaps or fitting factors (FFs) between the EOB reference model with a 5 (1 / 4) = 6 . 344 and v pole = 0 . 85 and numerical h 22 . We maximized only over the initial phase and time of arrival. We have found that for black-hole binaries with total mass 30-150 M /circledot , using LIGO, Enhanced LIGO and Advanced LIGO noise curves, the FFs are larger than 0.999. We have also computed the FFs between values of numerical h 22 that were computed in slightly different ways (e.g. different nu- \nmerical resolutions, different extraction procedures), and have estimated the mismatch between the exact and EOB h 22 . We have concluded, in the spirit of Ref. [8], that our analytical h 22 satisfies the requirements of detection with LIGO, Enhanced LIGO and Advanced LIGO. \nFinally, to test the robustness of the EOB model, we have also compared it to a few equal-mass subdominant modes ( /lscript, m ), notably (4 , 4) and (3 , 2), and to the dominant mode (2 , 2) of a set of unequal-mass inspiraling binaries. Without changing the EOB-dynamics adjustable parameters, we have found that, in the equal-mass case, the phase and amplitude differences of EOB and numerical h 44 and h 32 are within the numerical errors throughout the inspiral (see Figs. 9 and 10). Furthermore, in the unequal-mass case, we have found that we can reduce the phase difference of the EOB and numerical Ψ 22 4 of inspiraling binaries of mass ratios 2:1 and 3:1 on the order of the numerical error (see Figs. 12 and 13). This can be obtained either (i) by setting a 5 ( ν ) = νλ 0 with λ 0 fixed by the equal-mass case, and calibrating v pole ( ν ), or (ii) by letting v pole →∞ and calibrating a 5 ( ν ). \nIn the near future, we plan to compare the nonspinning EOB model defined in this paper to a larger set of accurate numerical simulations of black-hole binary coalescences (for both equal and unequal-mass binaries), and complete the tuning of all the EOBdynamics and -waveform adjustable parameters. In particular, we expect to improve the EOB plunge-mergerringdown matching either by reducing the number of EOB-waveform adjustable parameters or by employing different matching procedures or GW energy fluxes. \nWhile polishing this manuscript for publication, an independent calibration of the EOB model which uses the equal-mass binary black-hole data of the Caltech-Cornell collaboration employed in this paper and made public on January 20, 2009 appeared on the archives [54].", 'Acknowledgments': 'We thank Oliver Rinne for his work on implementing Regge-Wheeler-Zerilli wave extraction, and Fan Zang for extrapolating waveforms to infinite extraction radius. We also thank Emanuele Berti and Evan Ochsner for useful discussions, and Emanuele Berti for providing us with the quasi-normal mode frequencies and decay times used in this paper. A.B. and Y.P. acknowledge support from NSF Grant No. PHY-0603762. L.B., L.K., H.P., and M.S. are supported in part by grants from the Sherman Fairchild Foundation to Caltech and Cornell, and from the Brinson Foundation to Caltech; by NSF Grants No. PHY-0601459, No. PHY-0652995, and No. DMS0553302 at Caltech; by NSF Grants No. PHY-0652952, No. DMS-0553677, and No. PHY-0652929 at Cornell.', 'APPENDIX: COMPARING DIFFERENT METHODS OF COMPUTING h /lscript m': "The analysis in Sec. III relies to some extent on the GW strain h extracted from the numerical simulation. Earlier papers describing generation of the numerical data [21, 22, 23] focused on the behavior of the NP scalar Ψ 4 , and performed comparisons to PN theory based on the numerical Ψ 4 . \nWe have two means of computing a GW strain h from the numerical simulations. The first is a double time integration of Ψ 4 , exploiting the relation \nΨ 4 = h. (A.1) \n[Note that throughout this Appendix, we suppress indices /lscriptm denoting the components of the decomposition into spin-weighted spherical harmonics. Thus, Eq. (A.1) is meant to apply to each complex component ( /lscript, m )]. For each time integration [and each mode ( /lscript, m )], a complex integration constant needs to be determined. These constants are fixed with the procedure described in Sec. II of Ref. [22], in which a certain functional of temporal variations of the amplitude of the integrated data is minimized. The minimization is performed over 25 separate integration intervals [ t 1 , t 2 ] with t 1 /M = 1000 , 1100 , . . . , 1400 and t 2 /M = 2600 , 2700 , . . . , 3000. We then compute the time average of these 25 integrated waveforms, and we use this time average, which we denote as ∫∫ Ψ 4 , as the GW strain. Note that we perform the above operations on the numerical Ψ 4 data after it has been extrapolated to infinite extraction radius. \nOur second means of extracting a GW strain is using the RWZ equations [44, 45] generalized to arbitrary spherically symmetric coordinates, as formulated by Sarbach & Tiglio [46]. An advantage to the Sarbach & Tiglio formalism in contrast to the more widely-used ZerilliMoncrief formalism ( [55] and references therein) is that in the former case, the GW strain is obtained directly from the gauge-invariant RWZ scalars (at leading order in the inverse radius), without any time integration. With Oliver Rinne, we have implemented the Sarbach & Tiglio formalism for a Minkowsi background in standard coordinates in the Caltech-Cornell spectral code [47]. From the RWZ scalars (extracted at finite radii), we compute the GW strain and then extrapolate to infinite extraction radius in order to obtain the final waveform h RWZ . \nIn order to gain insight into the accuracy and reliability of the computed GW strain, we explore the differences between waveforms extracted with either technique (see also [56] for a similar comparison). Figure 14 shows the real part of the numerical (2 , 2) mode. On the scale of the full waveform, no disagreement between h RWZ and ∫∫ Ψ 4 is visible. However, the lower two panels of Fig. 14 show differences between h RWZ and Ψ 4 deep in the ringdown phase: While Ψ 4 continues to decay exponentially through many orders of magnitude, h RWZ exhibits noticeable deviation from a pure exponential decay at about a \nFIG. 14: The ( l, m ) = (2 , 2) mode of the numerical waveform. \n<!-- image --> \ntenth of peak amplitude. Decay of h RWZ stops completely at about one per cent of peak amplitude. \nWe suspect that this unexpected behavior is caused by gauge effects: All simulations in the numerical relativity community are performed using gauges in which the coordinates dynamically respond to the changing geometry, so as to avoid pathologies such as coordinate singularities. Ideally, the procedures used to extract gravitational radiation from the simulations should be gauge invariant, so that the choice of gauge used in the simulation is irrelevant. In practice, however, wave extraction techniques are not perfect. For example, the RWZ technique is gauge invariant only to first order in perturbation theory about fixed background coordinates. Likewise, the NP technique is strictly gauge-invariant only if applied at future null infinity, rather than at a finite distance from the source. Gauge effects are expected to manifest themselves differently in NP and RWZ wave extraction techniques, so by comparing the results of these two extraction techniques, we can get a handle on the size of our uncertainties that arise from gauge effects. \nTherefore, we will examine the differences between the numerical h RWZ and Ψ 4 . Using (A.1), we can compute a meaningful difference in two ways. The first way is to \nFIG. 15: Phase and relative amplitude difference between the ( l, m ) = (2 , 2) modes of the RWZ waveform h RWZ and NP scalar Ψ 4 [see Eqs. (A.2)-(A.5)]. The right panel shows an enlargement of merger and ringdown, with the dotted vertical lines indicating time of maximum of | Ψ 4 | , and where | Ψ 4 | has decayed to 10% and 1% of the maximal value. (The blue lines are smoothed; the grey data in the background represents the unsmoothed data.) \n<!-- image --> \ndifferentiate h RWZ twice and compute \n∆ φ NP = arg(Ψ 4 ) -arg( h RWZ ) (A.2) \n∆ A NP A = | Ψ 4 | - | h RWZ | ( | h RWZ | + | Ψ 4 | ) / 2 . (A.3) \nThe subscript 'NP' indicates that the comparison is made on the level of the NP scalars, i.e. Ψ 4 appears undifferentiated on the right-hand-sides. The second way is to time-integrate Ψ 4 and to calculate \n∆ φ RWZ = arg( Ψ 4 ) -arg( h RWZ ) (A.4) \n∫∫ The results of these comparisons are presented in Fig. 15. An examination of this figure reveals several properties of the extracted Ψ 4 and h RWZ waveforms. First, we note that during the inspiral and merger (up to t /lessorsimilar 3960 M , that is 18 M after the peak of h RWZ ), the RWZ and NP waveforms agree to better than 0.02 radians. ∆ φ NP contains more noise because noise is amplified by the double time differentiation to compute h , and because Ψ 4 is contaminated by junk-radiation from the initial data up to time t ≈ 1000 M . The blue lines in this plot have been smoothed (by convolution with a Gaussian of width 5 M ) to reduce the effect of noise due to junk radiation. (The grey data in the background of Fig. 15 shows the unsmoothed ∆Φ NP ). In contrast, ∆ φ RWZ does not show similar high frequency noise (the red dashed curves in Fig. 15 are not smoothed). Integration naturally smooths noise and apparently, the RWZ wave extraction is less \n∫∫ \n∆ A RWZ A = | ∫∫ Ψ 4 | - | h RWZ | ( | h RWZ | + | ∫∫ Ψ 4 | ) / 2 . (A.5) \nFIG. 16: The ( l, m )=(4 , 4) mode of the numerical waveform. \n<!-- image --> \nsusceptible to the noise introduced by junk radiation. Unfortunately, because of an imperfect choice of integration constants for the time integration, ∫∫ Ψ 4 does not precisely oscillate around zero at all times. This results in oscillations of ∆ φ RWZ and ∆ A RWZ /A during the inspiral; the frequency of these oscillations coincides with the GW frequency. The choice of integration constants, however, is good enough to confine these oscillations to less than about 0.02 radians in phase and 0.5 per cent in amplitude during the inspiral. \nAround merger, differences of the wave strain, i.e. ∆ φ RWZ and ∆ A RWZ /A , begin to grow, and during ringdown this growth accelerates. This large disagreement is caused by two effects. The first effect is the contamination of h RWZ in the ringdown phase, presumably by gauge effects, as shown in Fig. 14. The second effect is related to the time integration used to obtain ∫∫ Ψ 4 . During the inspiral phase, with an appropriate choice of integration constants the average value of ∫∫ Ψ 4 is very nearly zero (see top left panel of Fig. 15). Thus, the inspiral phase fixes all integration constants. When we now extend the integration through merger and ringdown, we find that ∫∫ Ψ 4 during ringdown has a contribution that grows linearly in time. Because the desired oscillatory part of ∫∫ Ψ 4 decays exponentially, this linearly growing contribution contaminates arg ∫∫ Ψ 4 to an increasing degree as time increases. The linearly growing contribution \nFIG. 17: Phase and relative amplitude difference between the ( l, m ) = (4 , 4) modes of the RWZ waveform h RWZ and NP scalar Ψ 4 , cf. Eqs. (A.2)-(A.5) The right panels shows an enlargement of merger and ringdown, with the dotted vertical line indicating the position of the maximum of | Ψ 4 | . \n<!-- image --> \nto ∫∫ Ψ 4 is just barely visible in the top panel of Fig. 14; for the (3,2) and (4,4) modes discussed below, it will be much more obvious. \nWhen matching an analytical model waveform to numerical results, one must choose whether to match to Ψ 4 , ∫∫ Ψ 4 , or h RWZ , and we have just seen that these three numerical waveforms differ by systematic effects that arise from properties of the numerical simulation. Given Figs. 14 and 15, it appears that Ψ 4 is preferable over ∫∫ Ψ 4 because Ψ 4 lacks the low-frequency oscillations during inspiral that are introduced in ∫∫ Ψ 4 by time integration, and furthermore Ψ 4 lacks the linear drift during the ringdown. Similarly, Ψ 4 has an advantage over h RWZ because it has much cleaner behavior during ringdown (see Fig. 14). \nWe now turn our attention to the next largest mode, ( l, m ) = (4 , 4), which is shown in Figure 16. Concentrating on the top panel first, we see that ∫∫ Ψ 4 agrees with h RWZ very well for a large fraction of the inspiral. However, for t /lessorsimilar 1000 M and t /greaterorsimilar 3900 M , ∫∫ Ψ 4 contains contributions that grow linearly in time. Note that these contributions cannot be removed by a different choice of integration constants, because integration constants result in addition of a linear term a + bt uniformly at all times. Hence, if the integration constants were changed to yield agreement for t /lessorsimilar 1000 M , the linearly growing discrepancy would appear at t /greaterorsimilar 1000 M . The reason that the transition is around t ∼ 1000 M may be related to the so-called junk radiation that is present in numerical simulations, and arises because the initial data do not correspond precisely to a snapshot of an evolution. A small fraction of the outgoing junk radiation is reflected when passing through the outer boundary. The reflected waves pass through the computational domain \nFIG. 18: The ( l, m )=(3 , 2) mode of the numerical waveform. \n<!-- image --> \nat retarded time t ≈ 1000 M . While the reflected junk radiation is small, apparently it is sufficient to contaminate Ψ 4 , as seen in the top left panel of Fig. 16. \n∫∫ \nAround merger, t ≈ 3950 M , ∫∫ Ψ 4 picks up another linearly growing contribution which renders ∫∫ Ψ 4 basically useless during merger and ringdown. This contamination might be related to oscillations in Ψ 4 and h RWZ that become visible at t /greaterorsimilar 3750 M (see middle and lower panel of Fig. 16). It is presently unclear what causes these effects, but we conjecture that they are related to gauge effects that influence either our current wave-extraction procedure, or our current wave-extrapolation procedure. It is quite possible that a refined understanding of gauge effects will reduce these features in the future. \nBecause of the apparent contamination of the wave- \n- [1] B. C. Barish and R. Weiss, Phys. Today 52 , 44 (1999).\n- [2] S. J. Waldman (LIGO Scientific Collaboration), Class. Quantum Grav. 23 , S653 (2006).\n- [3] S. Hild (LIGO Scientific Collaboration), Class. Quantum Grav. 23 , S643 (2006).\n- [4] F. Acernese, P. Amico, M. Alshourbagy, F. Antonucci, S. Aoudia, S. Avino, D. Babusci, G. Ballardin, F. Barone, L. Barsotti, et al., Class. Quantum Grav. 23 , S635 (2006). \nforms for early and late times, we will restrict the EOBNR comparison of the higher order modes to the time interval 1000 /lessorsimilar t/M /lessorsimilar 3600. Figure 17 shows that within this interval,Ψ 4 and h RWZ agree to better than 0 . 02 radians in phase and 1% in amplitude. \nFinally, Figures 18 and 19 present an analogous comparison for the ( l, m ) = (3 , 2) mode. Qualitatively, these figures are similar to Figs. 16 and 17. Agreement between Ψ 4 and h RWZ is very good for the time interval 1000 /lessorsimilar t/M /lessorsimilar 3700 M , with the phases differing by less than 0 . 1 radians and the amplitudes by less than about 1%. The larger disagreement might be due to the smaller RWZ vs. Ψ -- (l,m)=(3,2) \nFIG. 19: Phase difference between the ( l, m ) = (3 , 2) modes of the RWZ waveform h and NP scalar Ψ 4 , cf. Eqs. (A.2)(A.5). The right panels shows an enlargement of merger and ringdown, with the dotted vertical line indicating the position of the maximum of | Ψ 4 | . \n<!-- image --> \namplitude of the (3,2) mode of Ψ 4 during the inspiral phase relative to the (4,4) mode. One potentially interesting difference between the (4,4) and (3,2) modes lies in the relative size of the variations in | h RWZ | and | Ψ 4 | in the time range 3700 /lessorsimilar t/M /lessorsimilar 3900 M : For the (4,4) mode, variations in | Ψ 4 | are clearly smaller than variations in | h RWZ | (see Fig. 16. For the (3,2) mode this is reversed, with | h RWZ | showing somewhat smaller variations than | Ψ 4 | . \n- [5] P. Fritschel, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series , edited by M. Cruise and P. Saulson (2003), vol. 4856 of Society of PhotoOptical Instrumentation Engineers (SPIE) Conference Series , pp. 282-291, gr-qc/0308090v1.\n- [6] L. S. Finn, Phys. Rev. D 46 , 5236 (1992).\n- [7] L. S. Finn and D. F. Chernoff, Phys. Rev. D 47 , 2198 (1993).\n- [8] L. Lindblom, B. J. Owen, and D. A. Brown, Phys. Rev.\n- D78 , 124020 (2008).\n- [9] A. Buonanno and T. Damour, Phys. Rev. D 59 , 084006 (1999).\n- [10] A. Buonanno and T. Damour, Phys. Rev. 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2014SSRv..183..189C
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Massive Binary Black Holes in Galactic Nuclei and Their Path to Coalescence
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2014-01-01
| 20 | 0.51 | 162 |
['black hole physics', 'dynamical systems', '-', 'black hole physics', '-']
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Massive binary black holes form at the centre of galaxies that experience a merger episode. They are expected to coalesce into a larger black hole, following the emission of gravitational waves. Coalescing massive binary black holes are among the loudest sources of gravitational waves in the Universe, and the detection of these events is at the frontier of contemporary astrophysics. Understanding the black hole binary formation path and dynamics in galaxy mergers is therefore mandatory. A key question poses: during a merger, will the black holes descend over time on closer orbits, form a Keplerian binary and coalesce shortly after? Here we review progress on the fate of black holes in both major and minor mergers of galaxies, either gas-free or gas-rich, in smooth and clumpy circum-nuclear discs after a galactic merger, and in circum-binary discs present on the smallest scales inside the relic nucleus.
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https://arxiv.org/pdf/1407.3102.pdf
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{'No Header': 'Noname manuscript No. (will be inserted by the editor)', 'Massive binary black holes in galactic nuclei and their path to coalescence': "Monica Colpi \n25 June 2014, to appear in Space Science Reviews - DOI: 10.1007/s11214-014-0067-1 \nAbstract Massive binary black holes (10 5 M glyph[circledot] -10 9 M glyph[circledot] ) form at the centre of galaxies that experience a merger episode. They are expected to coalesce into a larger black hole, following the emission of gravitational waves. Coalescing massive binary black holes are among the loudest sources of gravitational waves in the Universe, and the detection of these events is at the frontier of contemporary astrophysics. Understanding the black hole binary formation path and dynamics in galaxy's mergers is therefore mandatory. A key question poses: during a merger, will the black holes descend over time on closer orbits, form a Keplerian binary and coalesce shortly after? Here we review progress on the fate of black holes in both major and minor mergers of galaxies, either gas-free or gas-rich, in smooth and clumpy circum-nuclear discs after a galactic merger, and in circum-binary discs present on the smallest scales inside the relic nucleus. \nKeywords black hole physics · dynamics · galaxy mergers · black hole binaries", '1 Massive binary black holes as tracers of black hole seed formation and galaxy assembly, along cosmic history': "In the universe, black holes come in two flavours : the 'stellar black holes' relic of the most massive stars, weighing ∼ 5 -30 M glyph[circledot] ( Ozel et al. 2010), and the 'supermassive black holes' residing in the nuclear regions of galaxies which carry large masses, typically in excess of 10 8 M glyph[circledot] (Vestergaard et al. 2008). The black holes of stellar origin are observed in X-ray binaries as accreting objects, while the super-massive ones power the bright QSOs and the less luminous active galactic nuclei (AGN) (Merloni and Heinz 2013). Super-massive black holes are also observed in their quiescent state as massive dark objects in nearby galaxy spheroids (Gultekin et al. 2009; Kormendy and Ho 2013), and a compelling case is that of the Milky Way housing an (almost) inactive black hole of 4 × 10 6 M glyph[circledot] (Ghez et al. 2008; Gillessen et al. 2009). \nMonica Colpi Department of Physics G. Occhialini University of Milano Bicocca, Piazza della Scienza 3, I20123, Milano, Italy Istituto Nazionale di Fisica Nucleare (INFN) - Milano Bicocca, Piazza della Scienza 3, I20123, Milano, Italy email: [email protected] \nA black hole desert exists between ∼ 30 M glyph[circledot] and ∼ 10 6 M glyph[circledot] . These black holes are often called intermediate mass or middleweight black holes with boundaries of the desert zone that are not physically constrained, due to the lack of observations. The maximum mass of a black hole of stellar origin can be as large as a few × 10 2 M glyph[circledot] , according to theoretical studies, depending on the metallicity of the collapsing progenitor stars and on the role of radiative feed-back in limiting the final mass (Omukai and Palla 2001; Heger et al. 2003). The minimum mass of super-massive black holes is not constrained as unknown is the process of formation of these black holes (Volonteri 2010; Schleicher et al. 2013). \nLimits on the density of the X-ray background light (resulting from accretion of an unresolved population of massive black holes) and on the local black hole mass density (Marconi et al. 2004; Merloni and Heinz 2013), suggest that super-massive black holes have grown their mass across cosmic ages through repeated episodes of accretion and via coalescences driven by galaxy mergers. This has led to the concept of black hole seed and black hole growth from seeds in concordance with the rise of cosmic structure. The characteristic mass or mass interval of the seed population is unknown and weakly constrained theoretically. Thus, aim of contemporary astrophysics is to disclose the mechanism of black hole seed formation through the detection of middleweight black holes in galaxies (Reines et al. 2013). \nThe discovery of tight correlations between the black hole mass M · and stellar velocity dispersion σ ∗ , and between M · and the stellar mass of the spheroid M ∗ ( M · /M ∗ ∼ 10 -3 ) highlighted the existence of a process of symbiotic evolution between black holes and galaxies (Marconi and Hunt 2003; Haring and Rix 2004; Ferrarese and Ford 2005; Gultekin et al. 2009; Kormendy and Ho 2013). The current interpretation is that the huge power emitted by the central black hole when active may have affected the rate of star formation in the host, turing the galaxy into a red and dead elliptical (Mihos and Hernquist 1996; Di Matteo et al. 2005; Hopkins et al. 2006). At present, there is a live debate on whether the correlation extends to bulge-less disc galaxies or in general to lower mass galaxies (Kormendy and Ho 2013). Lower mass galaxies are expected to house lighter super-massive black holes, according to the above relations. Thus low-mass galaxies are the preferred sites for the search of the middleweight black holes in the desert zone. Many galaxies (up to 75%) host at their centre a Nuclear Star Cluster, a compact sub-system of stars with mass M NSC typically < ∼ 10 7 M glyph[circledot] , and halfmass radius of < ∼ 10 pc. In a number of Nuclear Star Clusters a central middleweight black hole has been discovered which co-habit the cluster (Ferrarese et al. 2006). Less tight correlations have been found between M NSC and the mass M ∗ of the host galaxy (Scott and Graham 2013), indicating that while in bright spheroids the presence of a central black hole appears to be compulsory 1 , in less bright (disc) galaxies a Nuclear Star Cluster, with or without a central black hole, is preferred. \nThe search for middleweight black holes in less massive disc galaxies will be central for understanding the process of formation and co-evolution of black holes and galaxies (Reines et al. 2013; Kormendy and Ho 2013). But, this is not the only strategy. A powerful and promising new route exists to unveil infant black holes, forming at high redshift z < ∼ 15: this is the route proposed by The Gravitational Universe , the science theme selected by ESA for the next large mission L3, for the search and detection of \nlow frequency gravitational waves from coalescing binary black holes in merging galactic halos (eLISA Consortium 2013). \nAccording to the current paradigm of the Λ -CDM cosmology, galaxies form following the baryonic infall of gas into collapsing dark matter halos and assemble hierarchically through mergers of (sub-)galactic units (White and Rees 1978). Black hole seeds growing in these pristine merging halos inescapably undergo coalescences (Volonteri et al. 2003). This is illustrated in Figure 1, where we show characteristic tracks of black holes along cosmic history computed from semi-analytical models of galaxy formation (Volonteri and Natarajan 2009). The tracks are reported in the mass versus redshift plane, from 10 2 M glyph[circledot] up to 10 9 M glyph[circledot] and for 0 < ∼ z < ∼ 20: black holes form from seeds of different masses (according to different seed formation models) and grow via accretion and mergers, the last denoted as circles in the diagram. The gravitational wave signal from these mergers will be detected with a high signal-to-noise ratio from the forthcoming science mission of The Gravitational Universe , eLISA (eLISA Consortium 2013). This will allow to explore black hole seed formation and evolution as early as z > ∼ 10, just at the end of the dark ages but well before the epoch of cosmic re-ionisation of the intergalactic hydrogen. \nBlack holes come in binaries when two galaxies merge, and the gravitational wave signal emitted at coalescence offer a unique environment to measure, with exquisite precision, the black hole masses and spins, every time there is a merger (Amaro-Seoane et al. 2013). Thus, it has become mandatory to study how and when binary black holes form and evolve inside galactic halos, during the formation of cosmic structures. This is a multi-face problem crossing the boundaries between astrophysics and cosmology. \nThis review aims at describing one aspect of the astrophysics of binary black holes: that of their dynamics in merging galaxies. This is a central problem if we want to consider binary black holes as powerful sources of gravitational waves and as unique tracers of the cosmic assembly of galaxies, as proposed in The Gravitational Universe (eLISA Consortium 2013). This problem carries some resemblance to the problem of formation, in stellar dynamics, of close binary neutron stars fated to coalesce: rapid shrinking of the star's orbits occurs through phases of unstable mass transfer and common envelope evolution to avoid the risk of supernova disruption, before the two compact stars reach the phase of inspiral by gravitational waves. Likewise, binary black holes experience a phase of rapid sinking by dynamical friction when the two galaxies merge, but only after experiencing close encounters with stars and strong coupling with gas, they enter the phase of gravitational wave driven inspiral. In the next sections, we will describe the fate of massive black holes in merging galaxies, indicating whether they form close binaries ready to coalesce or wide pairs fated to wander in the host galaxy, and the conditions for this to happen. \nSection 2 starts with a brief historical recollection of the problem of black hole dynamics in stellar environments, and expands these early findings to account for recent advances in this field. Section 3 addresses the problem of black hole dynamics in gas-rich galaxy major and minor mergers, while Section 4 describes the fate of black holes in gaseous circum-nuclear and circum-binary gas discs. Section 5 presents a short summary of the timescales along the path to coalescence, and Section 6 contains the main conclusion and the directions in which the field may evolve into. \nFig. 1 Paths of black holes forming at high redshift from light (10 2 -3 M glyph[circledot] ) and heavy (10 5 -6 M glyph[circledot] ) seeds. The black holes evolve along tracks, in the mass versus redshift diagram, as they experience accretion episodes and coalescences with other black holes. Circles mark the loci of black hole coalescences. Four paths are selected: two ending with a black hole powering a z ∼ 6 QSO (starting from a massive seed, blue curve, and from a seed resulting from the collapse of a massive metal-free star, yellow curve); a third ending with a typical 10 9 M glyph[circledot] black hole in a giant elliptical galaxy (red curve); and finally the forth ending with the formation of a Milky Way-like black hole (green curve). The tracks are obtained using state-of-the-art semianalytical merger tree models. The grey transparent area in the bottom right corner roughly identifies the parameter space accessible by future electromagnetic probes which will observe black holes powered by accretion. Over-lied are contour levels of constant sky and polarisation angle-averaged Signal-to-Noise-Ratios (SNRs) for eLISA, for equal mass non-spinning binaries as a function of their total rest frame mass (eLISA Consortium 2013). It is remarkable that black hole mergers can be detected by eLISA with a very high SNR across all cosmic ages. Courtesy of eLISA Consortium (2013). \n<!-- image -->", '2.1 Super-massive black hole binaries in galactic nuclei': "In a pioneering Letter to Nature , Begelman, Blandford and Rees (1980) write: 'There are straightforward reasons for surmising that super-massive black hole binaries exist: mergers between galaxies appear to be frequent; cD galaxies in clusters or groups quite probably formed in this manner; and there is direct evidence that the near-by active galaxy Centaurus A is a merger product.' In particular, they correlate the bending and apparent precession of radio jets, observed in a number of active nuclei, with the presence of two black holes in a binary, exploring the dynamics of its formation inside the violently relaxed stellar core of the newly formed galaxy. \nBegelman et al. depicted the existence of three main phases along the path to coalescence: an early phase of pairing (phase I) under dynamical friction in the stellar bulge of the post-merger galaxy, ending with the formation of a close Keplerian binary ; a phase of hardening (phase II) during which the binary separation decreases due to \nenergy loss by close encounters with single stars plunging on nearly radial orbits on the binary; a phase of gravitational wave inspiral (phase III), ending with the coalescence of the two black holes due to the emission of gravitational waves. In all the phases, gravitational torques act to decrease the orbital angular momentum and energy of the black holes, to promote their pairing and sinking toward more bound states. \nThe black hole that forms has a new mass, new spin according to mass-energy conservation (Rezzolla et al. 2008) and because gravitational waves carry away linear momentum, the new black hole receives a gravitational recoil that can be as large as < ∼ 5000 km s -1 depending on the orientation and magnitude of the black hole spins and orbital angular momentum at the time of binary coalescence (Lousto and Zlochower 2013). Thus, an additional phase IV subsequent to merging should be considered corresponding to a recoiling black hole moving-inside or escaping-from it host galaxy (Gualandris and Merritt 2008; Merritt et al. 2009; Devecchi et al. 2009). This phase and the relation between spin, mass ratio and recoil are not considered here (Bogdanovi'c et al. 2007; Dotti et al. 2010; Centrella et al. 2010) nor the observability of active black holes along their path to coalescence [we defer the reader to e.g. Komossa (2006); Schnittman (2011); Bode et al. (2010); Eracleous et al. (2011); Liu et al. (2011); Komossa (2012); Koss et al. (2012); Liu et al. (2013); Decarli et al. (2013); Comerford and Greene (2014); Lusso et al. (2014); Liu et al. (2014)]. \nReturning to phase I, it is known that dynamical friction against the stars acts on each black hole individually to cause their progressive sinking (Chandrasekhar 1943; Begelman et al. 1980; Colpi et al. 1999; Yu 2002), until they come close enough to form a Keplerian binary . As dynamical friction is proportional to the background density of stars and to the square of the black hole mass, more massive black holes in denser environments sink more rapidly. In a stellar background of N stars described by a singular isothermal sphere, with density profile ρ ∗ = σ 2 ∗ / (2 πGr 2 ) and one-dimensional (1D) velocity dispersion σ ∗ , a black hole of mass m · at distance r sinks by dynamical friction on a timescale \nτ df ∼ 2 × 10 8 ln -1 N ( 10 6 M glyph[circledot] m · )( r 100 pc ) 2 ( σ ∗ 100 km s -1 ) yr . (1) \nThis timescale decreases with decreasing distance from the galaxy's nucleus, so that dynamical friction becomes more and more rapid with orbital decay. Eventually, the black holes end forming a Keplerian system. Binary formation occurs approximately when the mass in stars enclosed in their orbit drops below twice the total mass of the binary m · , t = m · , 1 + m · , 2 ; hereon m · , 1 ( m · , 2 ) is the mass of the primary (secondary) black hole, and q = m · , 2 /m · , 1 ≤ 1 the mass ratio. In a singular isothermal sphere, a Keplerian binary forms when a ∗ binary glyph[similarequal] Gm · , t /σ 2 ∗ , i.e. at a separation comparable to the gravitational sphere of influence of the black holes viewed as a single point mass m · , t . Dynamical friction guides the inspiral, with no significant amplification of the eccentricity (Colpi et al. 1999), approximately down to a ∗ binary . The weakening of dynamical friction is due to the high velocity that the black holes acquire when the form a binary, as the drag is inversely proportional to the square of the orbital velocity. \nPhase I ends when the binary separation a has decayed below \na ∗ hard = a ∗ binary µ 3 m · , t ∼ Gµ 3 σ 2 ∗ ∼ 0 . 1 q (1 + q ) 2 ( m · , t 10 6 M glyph[circledot] )( 100 km s -1 σ ∗ ) 2 pc , (2) \nwhere µ = m · , t q/ (1 + q ) 2 is the reduced mass of the binary (Quinlan 1996; Yu 2002; Merritt and Milosavljevi'c 2005). The hardening radius a ∗ binary is defined as the binary separation at which the kinetic energy per unit mass of the binary equals the kinetic energy per unit mass of the stars in the galactic potential. During the hardening phase II, the black hole orbital energy and angular momentum are extracted via scattering of single stars off the binary, in close three-body encounters. As a single star impinging on the binary causes a fractional energy change of the order of ∼ ξm ∗ /m · , t (where ξ ∼ 0 . 2 -1 is a coefficient calculated after averaging over many star-binary scattering experiments), a large number of stars, or the order of ∼ m · , t /m ∗ , is necessary for a sizeable change of the binary binding energy E · = Gm · , 1 m · , 2 / 2 a. The binary offers a cross section A ∼ πaGm · , t /σ 2 ∗ to the incoming flow of stars and this leads to a hardening rate s ≡ d (1 /a ) /dt ∼ ξπGρ ∗ /σ ∗ for the semi-major axis a, and a corresponding hardening time (independent on the number N of stars in the galaxy) \nτ ∗ hard ∼ σ ∗ πGρ ∗ a ∼ 70 ( σ ∗ 100 km s -1 ) ( 10 4 M glyph[circledot] pc -3 ρ ∗ )( 10 -3 pc a ) Myr . (3) \nOpposite to τ df , the hardening time τ ∗ hard increases with decreasing a , as the binary cross section decreases with a . Thus, a potential stalling of the binary can occur at the smallest binary separations, during phase II. \nPhase III starts when the coalescence time driven by gravitational wave emission \nτ gw ∼ 5 . 4 × 10 8 f ( e ) -1 (1 + q ) 2 q a 4 m 3 · , t ( 1 0 . 001 pc ) 4 ( 10 6 M glyph[circledot] m · , t ) 3 yr (4) \ndrops below τ ∗ hard , where f ( e ) = [1 + (73 / 24) e 2 +(37 / 96) e 4 ](1 -e 2 ) -7 / 2 . The crossing condition, τ ∗ hard = τ gw thus provides the binary separation at which the black holes transits from phase II into III: \na ∗ II → III = ( G 2 c 5 256 5 π ) 1 / 5 ( σ ∗ ρ ∗ ) 1 / 5 f 1 / 5 ( e ) ( q (1 + q ) 2 ) 1 / 5 m 3 / 5 · , t . (5) \nIf τ gw evaluated at a ∗ II → III exceeds the age of the universe, than the binary stalls and does not reach coalescence. From equation [4], we can define as a gw the distance at which the coalescence time τ gw equals the Hubble time τ Hubble : \na gw = 2 × 10 -3 f ( e ) 1 / 4 q 1 / 4 (1 + q ) 1 / 2 ( m · , t 10 6 M glyph[circledot] ) 3 / 4 ( τ Hubble 13 . 6 Gyr ) 1 / 4 pc . (6) \nExpressed in units of the Schwartzschild radius r S = 2 Gm · , t /c 2 associated to m · , t , a gw = 1 . 4 × 10 4 ( m · , t / 10 6 M glyph[circledot] ) -1 / 4 r S for the case of an equal mass circular binary. Coalescence occurs as long as a ∗ II → III < a gw . \nAccording to equation [3], for a wide interval of stellar densities and velocity dispersions, the coalescence time τ gw , evaluated at a ∗ II → III , is less than the Hubble time τ Hubble , so that the binary is excepted to enter the gravitational wave driven regime shortly after it has become hard. However, the estimate of τ ∗ hard , in equation [3], severely underestimates the true hardening time since a large number of stars in 'loss cone' orbits is necessary to drive the binary down to phase III. The loss cone in the black hole binary system is identified as the domain, in phase-space, populated by stars with sufficiently low angular momentum, J 2 < ∼ J 2 lc < ∼ 2 Gm · , t a , to interact with the binary. If \nhardening occurs at a constant rate s , the number of stars necessary to complete the hardening phase is as large as N lc ∼ ( µ/m ∗ ) ln( a ∗ hard /a gw ) , comparable to the mass of the binary. In the case of massive black holes ( m · , t > 10 8 M glyph[circledot] ) in elliptical galaxies and spheroids, such a large reservoir of stars may not be available (Merritt 2013b). \nAt the end of phase I, when stellar encounters begin to control the contraction of the newly formed binary, the black holes start ejecting stars from the loss cone at a high clearing rate. Refilling of stars in the phase-space requires a lapse time comparable to the two-body relaxation timescale τ rel ∝ N which in galactic nuclei, viewed as spherical systems, is often longer than the Hubble time (Yu 2002). Thus, the lack of stars in phase-space causes the binary to stall , at a separation a ∗ stall typically of ∼ 0 . 1 -1 pc, much larger than a gw (eq. [6]). Thus the binary can not reach coalescence in a Hubble time, and this is referred to as the last parsec problem . This represents an obstacle to the path to coalescence during the transit across phases II and III, for a large range of black hole masses, and mass ratios q (Yu 2002). We are therefore left with a major uncertainty on the estimate of the true hardening time τ Hard which is expected to be closer to τ rel , in the case of empty loss cone, and to τ ∗ hard as given by equation [3], in the case of full loss cone: thus, τ ∗ hard < τ Hard < τ rel . \nThe binary is a source of kinetic energy as it deposits in the stellar bath an energy \n∆E · ∼ E · ( a gw ) ∼ 2 × 10 55 f ( e ) -1 / 4 q 3 / 4 (1 + q ) 3 / 2 ( m · , t 10 6 M glyph[circledot] ) 5 / 4 ( τ Hubble 13 . 6 Gyr ) -1 / 4 erg , (7) \nin order to enter phase III. Compared to the binding energy of a stellar bulge of mass M ∗ , (3 / 2) M ∗ σ 2 ∗ , energy deposition accounts ∼ 10% of the total energy of the system, if one assumes m · , t ∼ 10 -3 M ∗ , a value of σ ∗ ∼ 100 km s -1 , and an equal mass binary of 10 6 M glyph[circledot] . Binary energy deposition via encounters with single stars can create a stellar core in an otherwise steep density profile, due to star's ejection. Stellar scouring has been observed in a number of core, missing-light elliptical galaxies that are at present indirect candidates for black hole mergers (Milosavljevi'c and Merritt 2001; Kormendy and Ho 2013; Merritt 2013a). The binary carries a larger angular momentum at a ∗ hard compared to the angular momentum at the onset of gravitational wave inspiral, \nJ · ( a ∗ hard ) J · ( a gw ) ∼ 10 (1 -e 2 ) 7 / 16 q 3 / 8 (1 + q ) 3 / 4 ( m · , t 10 6 M glyph[circledot] ) 1 / 8 100 km s -1 σ ∗ ( τ Hubble 13 . 6 Gyr ) -1 / 8 . (8) \nFrom the equation it is clear that the binary can reduce J · when transiting from phase II to III, increasing the eccentricity during orbital decay. \nAfter Begelman et al., the last parsec problem has been considered a major bottleneck to the path of binary coalescence, and has motivated many studies (Milosavljevi'c and Merritt 2001; Yu 2002; Merritt and Milosavljevi'c 2005). Direct N -Body simulations of binary inspiral in isotropic, spherical galaxy models confirmed, on solid grounds, the stalling of the binary: the binary hardening rate s was found to be proportional to the rate of repopulation of loss cone orbits which in turn depends on N . Simulations with a lower number of particles N (corresponding to shorter two-body relaxation times τ rel ) show rapid binary decay. By contrast, more realistic simulations with larger N (longer τ rel ) display a much lower hardening rate s for the binary (Preto et al. 2011). The extrapolation of the result to the limit of very large N , as in elliptical galaxies or bulges of spirals, leads to stalling of the massive binary over a Hubble time. \nYu (2002) noticed that if one drops the assumption of sphericity, the hardening time τ Hard is lower and can be less than the Hubble time in the case of less massive (power-law) galaxies which have a shorter τ rel . Spherical galaxies have all stars on centrophobic orbits, whereas galaxies with a high degree of axisymmetry and triaxialily host a significant fraction of stars on centrophilic orbits, such box orbits, which pass arbitrarily close to the binary and have low angular momentum. Furthermore, chaotic orbits in steep triaxial potentials can enhances the mass flux into the loss cone region (Merritt and Poon 2004). Some non axisymmetric potential can also excite bar instabilities causing a flow of stars toward the binary (Berczik et al. 2006). \nBinary stalling has been recently challenged in models of galaxy's mergers. A number of direct N -Body simulations indicate that the end-product of a merger is not a spherical galaxy (Berczik et al. 2006; Khan et al. 2011; Preto et al. 2011; Khan et al. 2013; Wang et al. 2014). The new galaxy retains substantial amount of rotation or/and a large degree of asphericity or triaxiality such that the binary is seen to harden at a rate independent of N , as if the loss cone were fully refilled, or as if an N -independent mechanism (collisionless relaxation) provides a supply of stars in loss cone orbits. In light of these findings the last parsec problem appears today as an artefact of the oversimplifying assumption of sphericity of the relic galaxy, and that more realistic models, simulated starting from ab initio conditions, point in the direction of hardening times ranging between 0.1 and a few Gyrs, for the models explored. 2 An interesting corollary of these investigations is that in non-spherical models the binary eccentricity e is seen to increase to values very close to 1 (Preto et al. 2011; Khan et al. 2011), indicating rapid transfer of angular momentum to stars from the cumulative action of many scatterings (Sesana 2010; Dotti et al. 2012). \nAs final remark, alternative mechanisms exist that can cause the contraction of the binary orbit, such as recycling of stars ejected by the binary on returning eccentric orbits (Milosavljevi'c and Merritt 2003), massive perturbers scattering stars into loss cone orbits (Perets et al. 2007), or a third black hole in a trio encounter. In the latter case, the third closely interacting black hole can harden the binary due to eccentricity oscillations (Blaes et al. 2002), or energy exchange in the three-body scattering, or can repopulate the loss cone having perturbed the underlying gravitational potential of the host galaxy (Hoffman and Loeb 2007; Kulkarni and Loeb 2012). \nSo far, we considered the hardening of massive black hole binaries in massive galaxies. But, a question to investigate is related to the evolution of middleweight black holes of ∼ 10 3 -4 M glyph[circledot] which tend to inhabit smaller mass halos with shorter relaxation timescales, and that form at high redshift when the universe was younger. This narrows down the interval of time accessible for hardening. Is there a last parsec problem? Extrapolating the results to middleweight black hole masses may not be straightforward, and in the next subsection we shortly explore the hardening in this regime. \n2.2 Middleweight binary black holes in stellar environments: hardening or stalling in high redshift nuclei? \nStudying the hardening and coalescence of middleweight black hole binaries with m · , t > ∼ 10 4 M glyph[circledot] is of importance as black holes of this mass are primary sources for eLISA, as illustrated in Figure 1. In the figure, black hole coalescences occur at a rate equal to the rate of merging of their parent dark matter halos controlled by dynamical friction only. The underlying assumption is there is no or negligible delay between the merger of the halo and that of the nested black holes, caused by the potential stalling of the binary. At present, whether black hole binaries at very high redshift are able to reach coalescence in the short cosmological time lapse between black hole seed formation and halo mergers is unclear (paper in preparation). \nAs an exercise and for illustrative purposes, one can compute limits upon the density ρ ∗ and velocity dispersion σ ∗ that a massive stellar cluster should have to allow rapid hardening of the binary during phase II. In star clusters the density and velocity dispersion are functions of distance. Thus, one should consider ρ ∗ and σ ∗ are characteristic values of the central region of a massive star cluster in an hypothetical galactic nucleus. \nFocus on the case of a black hole forming at z form (e.g. ∼ 20) and coalescing with another black hole, following a halo-halo merger at z coal ( ∼ 15), or the case of two adjacent mergers between redshift z 1 and z 2 over a short interval of cosmic time. The time lapse ∆τ lapse can be as short as < ∼ 0 . 1 Gyr, or more conservatively as short as 1 Gyr. In Figure 2 we plot, in the σ ∗ -ρ ∗ plane, the lines of constant τ rel = 0 . 34 σ 3 ∗ / ( G 2 m ∗ ρ ∗ ln Λ ) (with m ∗ = 1M glyph[circledot] and ln Λ ∼ 10) corresponding to a cosmic time lapse ∆τ lapse equal to 0.1 Gyr and 1Gyr, respectively. The solid lines in Figure 2 refer to the loci where τ rel = ∆τ lapse , so that characteristic densities higher than \n( ρ ∗ 1 . 6 × 10 7 M glyph[circledot] pc -3 ) rel > ∼ ( σ ∗ 100 km s -1 ) 3 ( 0 . 1 Gyr ∆τ lapse ) (9) \nare requested, at a fixed σ ∗ , to allow for binary hardening on the relaxation timescale (corresponding to the empty loss cone regime). The dashed lines in Figure 2 refer instead to the loci where τ ∗ hard ( a gw ) = ∆τ lapse , as given by equation [3] (corresponding to the full loss cone regime), for a black hole binary of m · , t = 10 4 M glyph[circledot] (upper dashed curve for ∆τ lapse equal to 0.1 Gyr, lower dashed curve for 1 Gyr). This condition implies \n( ρ ∗ 6 × 10 6 M glyph[circledot] pc -3 ) hard > ∼ ( σ ∗ 100 km s -1 ) ( 10 4 M glyph[circledot] m · , t ) 3 / 4 ( 0 . 1 Gyr ∆τ lapse ) 5 / 4 . (10) \nFigure 2 shows that true hardening, at early cosmic epochs, requires densities in excess of > ∼ 10 6 -8 M glyph[circledot] pc -3 and comparatively low dispersion velocities < ∼ 70 km s -1 to meet the conditions for coalescence in the short time lapse ∆τ lapse of 0.1 or 1 Gyr. We do not know if such dense stellar environment were present at the centre of unstable pregalactic discs. Today Nuclear Star Clusters, plausible candidates to harbour a central middleweight black hole, have all densities and velocity dispersions that do not meet this condition. Non-equilibrium conditions and/or the presence of gas, abundant in pre-galactic discs, may be instrumental in taxing the black holes to small separations, in this interval of masses, and at earlier cosmic epochs. Thus, a key question to pose is whether gas can fasten the transition along the three phases of pairing, hardening and \nFig. 2 Stellar mass density (in units of M glyph[circledot] pc -3 ) versus velocity dispersion (in units of kms -1 ) of hypothetical nuclear star clusters hosting middleweight black hole binaries on their path to coalescence which harden via single-binary encounters with solar mass stars: solid lines refer to the loci in the ( σ ∗ , ρ ∗ ) plane given by eq. 9 where the central relaxation time τ rel equals ∆τ lapse . The upper-red (lower blue) solid line refers to τ rel = ∆τ lapse = 0 . 1 Gyr (1 Gyr). Dashed lines refer to an equal-mass black hole binary of 10 4 M glyph[circledot] and refer to the loci where the hardening time τ ∗ hard , given by eq. [3], equals ∆τ lapse . The hardening time is computed at a black hole binary separation a gw given by eq. [6]. The upper (lower) dashed line refers to τ ∗ hard = ∆τ lapse = 0 . 1 Gyr (1 Gyr). Empty circles refer to mean stellar densities and velocity dispersion (calculated using the virial theorem) for five Nuclear Star Clusters of known mass and half-mass radii (Seth et al. 2008; Merritt 2013b). \n<!-- image --> \ngravitational wave driven inspiral and this question will be addressed in the incoming sections.", '3 Black holes dynamics in gas-rich mergers': 'Merging galaxies which are the sites of formation of binary black holes are expected to contain large concentrations of cold gas (unless one considers mergers between today elliptical galaxies only). This inevitable abundance of gas, in particular in high redshift disc galaxies, motivated us to inquiry into the role of gas dynamics as an alternative in the process of black hole hardening and coalescence. \nIn this section, we review the pairing of black holes in gas-rich merging galaxies following their dynamics ab initio to highlight the key role played by gas in affecting the black hole inspiral and the remarkable difference between major and minor mergers. \nMajor mergers refer to interactions between galaxies of comparable mass, while minor mergers refer to interactions between a primary massive galaxy and a less massive galaxy, typically with mass ratio 1:10, and below. Boundaries among major or minor mergers are not sharp, as in many cases, the various outcomes depend also on the internal structure and gas content of the interacting galaxies.', '3.1 Major mergers and the formation of a Keplerian binary': "The study of black hole dynamics in gas-rich mergers dates back to (Mayer et al. 2007), yet it is still in its infancy. The rich physics involved and the high computational demand require state-of-the-art simulations, and the body of data is still inhomogeneous, fragmented and incomplete. While black hole dynamics in collisionless mergers of spherical galaxies has been explored (with direct N -Body codes) starting from galaxies on close elliptical bound orbits and followed mainly during the hardening phase (Khan et al. 2011), black hole dynamics in mergers between gas-rich disc galaxies has been studied starting from cosmologically motivated (parabolic) orbits, during the pairing phase over separations > ∼ 10 kpc, down to the scale ( < ∼ 10 pc) when the black holes form a Keplerian binary (Mayer et al. 2007; Colpi et al. 2009; Colpi and Dotti 2011; Chapon et al. 2013; Mayer 2013). Further hardening has been later explored in dedicated simulations (Escala et al. 2005; Dotti et al. 2006, 2007, 2009; Fiacconi et al. 2013). \nDisc galaxies, as observed at low redshifts, are multi-component systems comprising a collisionless dark matter halo, a stellar disc which coexists with a multi-phase gaseous disc, and a central bulge housing (when present) a massive black hole. Simulating a collision between two disc galaxies with central black holes thus requires simulating the dynamics of the collisionless components (dark matter and stars) jointly with that of gas which is dissipative, and thus subject to cooling, star formation, shock heating and stellar feed-back. \nThere are many simulations of disc galaxy mergers in the literature (e.g. (Hopkins et al. 2013)), but there exists only a limited number in which the black hole dynamics is followed self-consistently from the > ∼ kpc scale down to scales < ∼ 10 pc. When two galaxies merge, the two black holes are customarily assumed to merge promptly and form a single black hole. A recent set of N -Body/Smooth Particle Hydrodynamic simulations exists which follow the dynamics of black holes from the 100 kpc scale, typical of a merger, down to a scale < ∼ 10 pc, and which include star formation and feed-back (Van Wassenhove et al. 2012, 2014). There exists a further class of SPH or/and Adaptive Mesh Refinment (AMR) simulations which have enough resolution to witness the formation of a Keplerian binary on the ∼ 1 pc scale, but which treat the gas thermodynamics via a phenomenological energy equation, in the form of a polytrope (Mayer et al. 2007; Chapon et al. 2013). \nEqual mass mergers are disruptive for both progenitor galaxies. The galaxies first experience a few close fly-by during which tidal forces start to tear the galactic discs apart, generating tidal tails and plumes. The discs sink by dynamical friction against the dark matter background, and the massive black holes follow passively the dynamics of the bulge and disc they inhabit. Prior to merging, during the second pericentre passage, strong spiral patterns appear in both the stellar and gaseous discs: non axisymmetric torques redistribute angular momentum so that as much as 60% of the gas originally present in each disc of the parent galaxy is funnelled inside the inner few \nhundred parsecs of the individual galaxy centres. The black holes, still in the pairing phase, are found to be surrounded by a rotating stellar and gaseous disc. \nLater, the gasoues discs eventually merge in a single massive rotationally supported nuclear disc of < ∼ 100 pc in size, now weighing ∼ 10 9 M glyph[circledot] . The disc develops gravoturbulence (with velocities ∼ 60 -100 km s -1 ) that guarantees a Toomre parameter Q > ∼ 2. This prevents fragmentation of gas into stars on the timescale necessary for the black holes to form a Keplerian binary (a few Myr). This short sinking timescale comes from the combination of two facts: that gas densities are higher than stellar densities due to the dissipative nature of the interaction, and that the black holes move relative to the background with mild supersonic velocities. Under these conditions, the hydrodynamical drag is the highest (Ostriker 1999). The subsequent evolution is described in Section 4. \nDuring final revision of this review, a new dedicated N -Body/SPH simulation by (Roˇskar et al. 2014) of two Milky-Way-like galaxy discs with moderate gas fractions, has been carried out at parsec-scale resolution, including a new model for radiative cooling and heating in a multi-phase medium, star formation and feedback from supernovae. The massive black holes weighing ∼ 10 6 M glyph[circledot] are form a pair at a separation of ∼ 100 pc which gradually spirals inward. However, due to the strong starburst triggered by the merger, the gas in the centre most region is evacuated, requiring ∼ 10 Myr for the nuclear disc to rebuild. The clumpy nature of the interstellar medium has a major impact on the dynamical evolution of the pair now subjected to stochastic torquing by both clouds and spiral modes in the disc. These effects combine to delay the orbital decay of the two black holes, just in phase I of gas-dominated dynamical friction. An inspiral timescale of ∼ 100 Myr is found in this simulation which is smaller compared to that estimated in collisionless mergers, but longer of a factor at least 10 compared to the case of mergers with a single-phase gas. The result is in line with what found in Fiacconi et al. (2013) (see Section 4.1.2) who describes black hole dynamics in clumpy nuclear discs. We notice however that a single run may not suffice to pin down the characteristic gas-dynamical friction timescale in dissipative mergers, and that the perturbations induced by a population of massive clumps in the stellar component may alter the star's dynamics, prompting rapid refilling of the loss cone region around the two black holes, an effect that these simulations can not capture. \n3.2 Black hole paring in unequal-mass mergers", '3.2.1 Collisionless unequal-mass mergers': 'Early works on collisionless mergers of unequal-mass spherical dark matter halos (Governato et al. 1994; Taffoni et al. 2003; Boylan-Kolchin et al. 2008) indicated that additional mechanisms are present, besides dynamical friction, that influence the structure and orbital evolution of the interacting halos (primarily the less massive, secondary): (i) progressive mass loss, or tidal stripping , induced by the tidal field of the main halo which reduces the mass of the secondary delaying the sinking by dynamical friction (the force scaling as the square of the satellite mass), and (ii) tidal heating , i.e. the effect of short impulses imparted to bound particles within the secondary satellite galaxy by the rapidly varying tidal force of the primary which heats the system causing its (partial) dissolution (Taffoni et al. 2003). Depending on the energy E of the orbit and its degree of circularity ε , on the relative mass concentration c s /c h between satellite and main \n<!-- image --> \n<!-- image --> \nFig. 3 Outcome of simulations of unequal mass mergers between spherical (dark matter) halos described with a NFW profile (Navarro et al. 1996), and described in the plane circularity ε versus relative concentration c s /c h (measuring the ratio between the scale radii of the two halos, defined as in (Navarro et al. 1996; Taffoni et al. 2003)). The figure depicts the life diagram of a satellite halo with mass M s /M h = 0 . 01. Each plot is labelled by the value x ( E ), the radius of the circular orbit (in units of the half mass of the main halo) at the onset of dynamical evolution. We identify the regions corresponding to merger (M) of the satellite into the main host halo, disruption against the background (D), and survival (S) of part of the satellite in the periphery of the main halo. Satellite halos with low concentration on less circular orbits are fragile to disruption, while satellite halos on wide orbits and high concentration preserve their identity. Mergers are preferred in correspondence of high concentration and close circular orbits. Courtesy of Taffoni et al. (2003). \n<!-- image --> \nhalo, and on the initial mass ratio of the primary to the satellite halos M h /M s , the encounter can lead either to rapid merging toward the centre of the primary halo (M), disruption (D), or survival (S) (when a residual mass remains bound and maintains its identity, orbiting in the main halo for a time longer than the Hubble time). Figure 3 illustrates the various outcomes of these experiments. In this context, merging times can be described by an empirical equation which accounts for the progressive mass loss of the secondary by tidal stripping and the progressive delay in the halo merging process \nτ df , tidal t dyn ≈ Θ ( E,ε, c s /c h ) ln(1 + M h /M s ( t )) M h M s ( t ) (11) \nwhere Θ , function of the initial parameters, and satellite mass M s ( t ) are computed from the numerical simulation. Figure 3 shows the fragility of less concentrated satellites to dispersal and disruption. These findings anticipate the possibility that unequal-mass mergers may release black holes on peripheral orbit inside the primary, due to tidal stripping of the less massive galaxy prior completion of the merger.', '3.2.2 Gas-rich unequal-mass mergers': "Recent suites of N -Body/SPH simulations of unequal-mass galaxy mergers have highlighted the occurrence of new key features in the dynamics of the discs and their embedded black holes that can be ascribed to differences in the central concentration of the interacting galaxies, and to the geometry of the encounter, but that go beyond the results inferred in Section 3.2.1. These new simulations illustrate the pivotal role played by gas which acts, through its cooling, to enhance the central mass concentration of the satellite and favours the sinking of the secondary black hole in the otherwise disrupted galaxy (Kazantzidis et al. 2005; Callegari et al. 2009, 2011; Van Wassenhove et al. 2012, 2014). \nFig. 4 Upper panel: black hole separation as a function of time for a 1:4 merger. The thin and thick lines refer to the dry (gas free) and wet (with gas fraction of 10%) cases, respectively. The inset shows the color-coded density of stars (left) and gas (right) for the wet case at t = 5 . 75 Gyr (marked with a red dot on the curve); each image is 12 kpc on a side, and colors code the range 10 -2 -1 M glyph[circledot] pc -3 for stars, and 10 -3 -10 -1 M glyph[circledot] pc -3 for the gas. Lower panel: black hole separation as a function of time for a 1:10 merger (upper panel). The thin and thick line refer to the dry and wet (with gas fraction of 30%) cases, respectively. The inset shows density maps at t = 1 . 35 Gyr for the wet merger: images are 4 kpc on a side (color coding as in upper panel). Courtesy of Callegari et al. (2009). \n<!-- image --> \nIn unequal-mass mergers, the secondary, less massive galaxy undergoes major transformations. In particular, if the merger is wet, i.e. if the gas fraction in the disc of the secondary is relatively high ( > ∼ 10%), tidal torques during the last peri-centre passage prior merging, trigger inflows which give rise to a nuclear starburst in the vicinity of the secondary black hole. This enhances the resilience of the galaxy's nucleus against tidal stripping due to the increased stellar density and degree of compactness of the nuclear bulge, at the time the secondary starts interacting with the disc of the primary. The denser stellar cusp surrounding the secondary black hole thus sinks rapidly toward the primary, dragging the black hole that reaches a separation of ∼ 100 pc, close to the resolution limit of the simulation. This is illustrated in Figure 4 for 1:4 and 1:10 wet mergers, where the relative separation of the black holes is plotted against time (heavy solid line). In Figure 4 we also plot the stellar and gas distribution at the end of the nuclear starburst that created a denser stellar nucleus in the secondary. The disc of the secondary, rather turbulent and clumpy due to star formation, is later disrupted by ram pressure stripping by the gas of the primary. The secondary black hole continues its sinking toward the centre of the primary, being surrounded by the compact and massive star cluster. In Figure 4, we also contrast the results from dry, i.e. gas free mergers. In the absence of the central starburst, dry mergers leave the secondary black hole wandering on a peripheral orbit at ∼ 1 kpc away from the central, primary black hole. The naked black hole will then sink by dynamical friction on a longer timescale (Callegari et al. 2009, 2011; Khan et al. 2012a). \nHigher-resolution simulations of disc galaxies have recently revealed the occurrence of additional features, indicating how rich is the outcome of mergers under different initial conditions. Van Wassenhove et al. (2014) have shown that, as the gas-rich merger \nFigure 6 summarises these findings, i.e. the correlation between the ability of pairing (measured evaluating the black hole relative separation) and the mass ratio q, evaluated at the end of the simulation. Coplanar prograde mergers with higher fractions of gas lead to higher q and smaller black hole separations. Inclined mergers with large gas fractions can instead fail in bringing the black holes to a small separation. Torques acting on the satellite during the early phases of the merger are weaker for higher \n<!-- image --> \n<!-- image --> \nFig. 5 Time sequence of stellar density snapshots in the 1:4 coplanar, prograde-prograde merger after the second peri-centre passage, at times 1.2, 1.43, and 1.48 Gyr, respectively. The scale of the left and central snapshots is 8 kpc, and 2 kpc for the right snapshot. A black dot marks the black hole in the primary galaxy nucleus which is dissolved during the interaction, while the secondary is at the centre of the highest density region of the secondary galaxy. In the last panel the secondary nucleus and black hole are near the centre of the mass distribution. Courtesy of Van Wassenhove et al. (2014). \n<!-- image --> \nprogresses, the newly formed stellar nucleus of the less massive galaxy, denser on small scales, is able to dissolve the less concentrated nucleus of the primary, via impulsive tidal heating. This is illustrated in Figure 5 which shows how the denser nucleus of the secondary, at the end of the merger, finds itself in the midst of the mass distribution, having dissolved the nucleus of the main galaxy.", '3.3 Black hole pairing in minor mergers: the role of mass accretion': 'Minor mergers among galaxies with mass ratios 1:10 or less show behaviours that are extremes, along the sequence of unequal-mass mergers, and may lead to wandering black holes, even in presence of a sizeable fraction of cold gas (Callegari et al. 2009, 2011). The fate of black holes in minor mergers depends not only on the gas content but on the orbital parameters, such as the degree of co-planarity, and in addition, on new input physics (neglected for seek of simplicity), i.e. accretion. During the encounter the secondary black hole as well as the primary can accrete from the surrounding gas and increase their mass. A mass increase can influence the dynamics of the secondary black hole, as a larger mass implies a more rapid sinking by dynamical friction. This correlation has been found in a number of simulations by Callegari et al. (2011) who showed that the black hole mass ratio q is not conserved during the merger. The secondary black hole is subjected to episodes of accretion which enhance the mass by an order of magnitude when interacting with the gas of the primary galaxy. Thus, the black hole mass ratio does not mirror that of the galaxies, and can be much higher than the initial value indicating that black holes in unequal-mass mergers may carry comparable masses at the time they form a close pair. \nFig. 6 Black hole mass ratio q versus relative separation, at the end of the simulation (when either a pair forms on the scale of the force resolution (10 pc), or the secondary black hole wanders at the periphery of the main galaxy), for the 1:10 mergers explored in (Callegari et al. 2011), labelled according to their initial gas fractions f g , orbital inclination θ and initial peri-centre R p . Courtesy of (Callegari et al. 2011). \n<!-- image --> \ninclinations, and for this reason the increase in mass ratio q during the first three orbits is milder than in the coplanar case with the same gas fraction. Moreover, a higher inclination corresponds to a slower orbital decay so that the satellite galaxy undergoes a larger number of tidal shocks before being disrupted, preventing further episodes of substantial accretion onto the secondary black hole. Finally, gas-rich mergers on closer orbits (i.e. with smaller peri-centre) are less effective in pairing contrary to what expected. Because of the smaller distance of approach and higher relative velocities between the satellite and the surroundings, ram pressure strips gas effectively, reducing the importance of the starburst that made the satellite less susceptible to stripping, and the accretion process onto the black hole. The joint action of these effects is therefore conducive to weak pairing, irrespective of the large amount of gas present initially. \nIn summary, minor mergers appear to fail in forming close black hole pairs in a number of cases, as the less massive galaxy is disrupted by tidal and ram pressure stripping at earlier times during the encounter so that dynamical friction is unable to deliver the secondary black hole to the centre of the main galaxy, within a Hubble time. The boundary between failure and success, i.e. between coalescence and wandering, is still poorly determined as it depends on the geometry, gas content and internal structure of the galaxies, and on the follow-up black hold dynamics on smaller scales (Khan et al. 2012a). Direct N -body simulations of gas-free minor mergers have shown that black hole coalescences can occur on timescales of one to a few Gyrs, regardless the mass ratio provided that its value q > ∼ q crit ∼ 0 . 05 -0 . 1 (Khan et al. 2012b). The rather abrupt transition at q crit appears to result from the monotonic decrease of merger-induced triaxiality in the main galaxy with decreasing mass ratio. The secondary galaxy is too small and light to significantly perturb the massive primary, slowing down the rate of binary single star interactions and hardening. Judging from the results of simulations of galaxy minor mergers from a limited sample comprising gas-poor and gas-rich cases, a very rough boundary between coalescence and wandering appears to be at q crit < ∼ 0 . 1. \nAlong parallel lines, N -Body/SPH cosmological simulations of massive disc galaxies, inclusive of black hole seed formation and growth, have shown that satellite galaxies containing black hole seeds are often tidally stripped as they merge with the primary while building the main galaxy disc. This creates naturally a population of wandering middleweight black holes in the massive spiral (from 5-10 wanderers), remnants of satellite cores (Bellovary et al. 2010).', '4 Black hole dynamics in gaseous nuclear discs': "SPH simulations of black hole dynamics in massive, rotationally supported nuclear discs represent a benchmark for studying the process of binary formation and coalescence, in gas-rich environments (Escala et al. 2005; Dotti et al. 2006, 2007, 2009; Fiacconi et al. 2013). These are not ab initio simulations, since the disc, in rotational equilibrium, is already in place as part of the remnant galaxy, or of the main galaxy, in case a minor merger has delivered the secondary black hole inside the disc of the massive host. \nAt present, there is no analytical model nor simulation that can trace the black hole dynamics in nuclear discs from the disc's periphery (at < ∼ 100 pc) down to the < ∼ 10 -3 pc scale (i.e. close and below a gw ) where gravitational waves drive the inspiral, due to the susceptibility of the gas to undergo gravitational instabilities conducive to star formation episodes and to the complexity of the gas thermodynamics in neutral and ionised media present on the smallest scales. In this context, key elements are the rotation of the underlying background, its self-gravity , the degree of gas dissipation , and the nature of turbulence and viscosity . Large scale gas discs can cool down, develop turbulence and inhomogeneities in the form of massive clumps which become sites of star formation. Gas can also dissipate the kinetic energy of the moving black holes via radiative cooling in a disc on smaller scales when the disc around the binary is nearly Keplerian. Thus, black hole inspiral in gaseous discs is mainly governed by processes of angular momentum exchange and radiative cooling. A compelling question to pose is whether angular momentum transport resulting from the gravitational interaction of the black holes with the gas is faster than that from the slingshot of stars. \nIn gaseous discs, we are led to distinguish three phases. There exists an early phase I-g of nuclear-disc-driven migration during which non axisymmetric perturbations in the density field excited by the gravitational field of the black hole(s) cause the braking of the orbit in regions where the disc dominates the gravitational potential (Escala et al. 2005; Dotti et al. 2006, 2007, 2009). The typical scale covered by I-g is between 100 pc down to ∼ 0 . 1 pc. \nWith time, the gas mass enclosed in the orbit decreases below m · , t and the black hole dynamics is dominated by their own gravitational potential. The binary then forms a Keplerian system. This corresponds to the onset of phase II-g of binary-discdriven migration. In phase II-g, the tidal torques exerted by the binary on the disc are sufficiently intense to repel gas away from the binary clearing a cavity, called gap (Farris et al. 2014; Rafikov 2013; Hayasaki et al. 2013; D'Orazio et al. 2013; Roedig et al. 2012; Kocsis et al. 2012; Shi et al. 2012; Noble et al. 2012; Roedig et al. 2011; Cuadra et al. 2009; Hayasaki 2009; MacFadyen and Milosavljevi'c 2008; Hayasaki et al. 2008, 2007; Ivanov et al. 1999; Gould and Rix 2000). The binary is then surrounded by a circum-binary disc. Rotation in the circum-binary disc is nearly Keplerian but the disc's structure is affected by the binary, acting as a source of angular momentum. In phase II-g, black holes migrate under the combined action of viscous and gravitational \ntorques which ultimately drive the binary into the third phase III of gravitational-driven inspiral where loss of orbital energy and angular momentum is due to the emission of gravitational waves. Not for all black hole masses this gas-assisted inspiral leads to coalescence in a Hubble time and more work is necessary along these lines (Cuadra et al. 2009). \nBelow, we explore phase I-g considering first a smooth nuclear disc and later a clumpy nuclear disc to study black hole migration on pc-scales. In a second step we will explore phase II-g when a circum-binary disc forms which controls the evolution of the binary on smaller scales.", '4.1.1 Smooth circum-nuclear discs': "In a number of targeted studies, the massive nuclear disc is described by a Mestel model: the disc, self-gravitating and axisymmetric, has rotation velocity V rot independent of radius R . With constant V rot , fluid elements in the disc are in differential rotation with Ω = V rot /R , and are distributed following a surface density profile Σ ( R ) = Σ d [ R d /R ], where R d is a scale radius. The disc mass within a radius R is then given by M Mestel ( R ) = M d [ R/R d ], with M d = 2 πR 2 d Σ d , and the circular velocity V 2 rot = GM d /R d . The disc is pressure supported vertically, with aspect ratio h/R d of ∼ 0 . 1 -0 . 05, and isothermal sound speed c s such that the Toomre parameter Q is > ∼ 3 everywhere, to prevent the development of gravitational instabilities. The disc is embedded in a more massive Plummer stellar sphere, representing the innermost region of the galactic bulge: hereon we refer to this configuration as circum-nuclear disc (Escala et al. 2005; Dotti et al. 2006, 2007, 2009). \nIn this smooth background (guaranteed by the large Q ) one can trace the black hole dynamics, assuming a primary black hole of mass m · , 1 at rest in the centre of the circum-nuclear disc, and a secondary black hole of mass m · , 2 initially moving on a wide eccentric co-planar orbit inside the disc. The mass ratio q , the initial binary orbital elements, and the disc mass M Mestel enclosed in the binary orbit (in excess of the binary mass m · , t , in the simulated volume), are free parameters to mimic different encounter geometries and mergers of galaxies with different stellar/gas mass contents. \nAssisted by a series of N -Body/SPH simulations, these studies have highlighted key differences in the black hole dynamics compared to that in spherical collisionless backgrounds, the most remarkable being the dragging of the moving black hole into a co-planar co-rotating orbit with null eccentricity before the black holes form a binary (Dotti et al. 2006, 2007, 2009). \nThe simulations show that any orbit with large initial eccentricity is forced into circular rotation in the disc. In the different panels of Figure 7, we show the overdensity excited by the black hole along the orbital phase, for an initial value of the eccentricity e 0 equal to 0 . 9. The wake, that in a uniform medium, trails the motion of the perturber maintaing its shape, here changes both orientation and shape, due to the differential rotation of the underlying disc. The wake is trailing behind when the black hole is at pericentre, but is leading ahead when at apocentre, since there, the black hole is moving more slowly than the background, and this causes a temporary acceleration on m · , 2 . It is important to remark that this is a rapid change of e occurring on few orbital times. Circularisation is a fast process, and it is faster the cooler is the disc, \nFig. 7 Colour-coded gas surface density of a smooth disc of mass M d = 5 × 10 8 M glyph[circledot] , for a black hole binary with q = 0 . 1 and a primary black hole of 10 7 M glyph[circledot] . The initial eccentricity is e 0 = 0 . 7. Snapshots refer to four different times, covering the process of circularisation lasting a few Myrs. The gas and the secondary black hole rotate counter-clockwise. The position of the black holes is marked by black dots. In the top and bottom left (right) panels the density wake excited by the secondary black hole is leading (trailing) the orbit, resulting in the circularisation of the relative orbit. Courtesy of D. Fiacconi. \n<!-- image --> \ni.e. the denser is the disc. Sinking times are found to depend on the equation of state adopted, i.e on the polytropic index γ used to model the thermodynamic behaviour of the gas. \nA further signature of a rotating background is the angular momentum flip of an initially counter-rotating orbit, if it exists (Dotti et al. 2009). 3 Initially, the gas opposes to the motion of the black hole as the density perturbation is always in the form of a trailing wake which causes an effective brake. The change of the orbital angular momentum from negative to nearly null values is further facilitated by the fact that, while the orbit decays, the black hole interacts with progressively denser regions of the disc. The orbit is nearly radial when the orbital angular momentum changes sign but the change is so rapid, relative to the orbital time, that the black hole is forced to corotate. When co-rotation establishes along an eccentric orbit, the orbital momentum increases under the circularising action of dynamical friction (non axisymmetric wake) in its co-rotating mode. Thus, a further prediction of black hole inspiral in rotating discs is that gas-dynamical friction conspires to turn counter-rotating orbits into corotatating ones, even before the formation of a Keplerian binary. \nAfter circularisation, the secondary black hole continues to spiral in as it experiences a net negative torque, despite having reduced its relative velocity with respect to neighbouring fluid elements. 4 Dynamical friction is a non-local process and in a disc there is a residual velocity difference between the black hole and the more distant rotating fluid elements. One can view the migration process described in the text again as a manifestation of the larger scale gravitational perturbation excited by the black hole, but this time the drag is inside a rotating inhomogeneous background. The net torque results from the sum of positive (inside the black hole orbit) and negative (outside) contributions as the perturbation is highly non axisymmetric due to differential rotation. The black hole m · , 2 is able to excite a non axisymmetric perturbation in the disc structure which produces a net negative torque on m · , 2 . This process is reminiscent to Type I planet migration, but with key differences. While in planet migration, the central star dominates the gravitational potential and the disc's self-gravity is negligible, in disc-driven black hole migration (phase I-g) the disc is dominant, while the gravity of the central black hole is negligible. In Type I migration, the net torque on the planet is the sum of the Lindblad and co-rotating resonances, computed in the linear perturbation theory under the hypothesis that the planet migrates on a timescale much longer than the orbital time, so that the small-amplitude perturbation is periodic in the disc frame. By contrast, during phase I-g of black hole migration, the torque on the secondary black hole comes from the non-linear density perturbations that m · , 2 excites in the disc. Figure 8 shows the fast orbital decay that the secondary black hole experiences after circularisation (Fiacconi et al. 2013). \nTo gain some insight into nuclear-driven-disc migration in a Mestel disc, we estimate the migration time following a simple argument to capture key dependences of τ mig on the disc properties and black hole mass (Armitage 2013). In the two-body approximation, a fluid particle, approaching m · , 2 along a straight-line with impact parameter b and relative velocity v rel ∼ aδΩ ∼ bΩ experiences a velocity change parallel to v rel of the order of δv || ∼ 2 G 2 m 2 · , 2 / ( b 2 v 3 rel ) , where a denotes the black hole distance from the centre of the disc. As gas exterior to the secondary black hole moves more slowly than the black hole in the disc, it gains velocity parallel to v rel increasing its angular momentum. As angular momentum is conserved, this implies a decrease in the angular momentum per unit gas mass for the black hole equal to ∼ -aδv || . Note that gas interior to the black hole orbit exerts a torque of opposite sign, so that the net torque depends on a delicate balance. As simulations show inward migration and larger torques in the black hole vicinity, we compute the rate of change of angular momentum considering only neighbouring gas particles in the trailing side of the spiral density perturbation, contained in a cylinder of scale height b ∼ h . Accordingly, the mass flux on m · , 2 is δm/δt ∼∼ 2 πhΣ v rel ∼ 2 πh 2 ΣΩ , where Σ and Ω are evaluated at a . The resulting torque on the black hole can thus be written as T Mestel I ∼ -4 πζ [ m · , 2 /M Mestel ( a )] 2 Σa 4 Ω 2 , where ζ = ζ ' ( a/h ) 3 brackets uncertainties in the normalisation of the torque and its dependence on the aspect ra- \nFig. 8 The evolution of the angular momentum of the secondary black hole orbiting inside a smooth circum-nuclear disc of mass M d = 5 × 10 8 M glyph[circledot] . The initial eccentricity is e 0 = 0 . 9 . Red and black colours refer to q = 0 . 1 and q = 0 . 2, respectively ( m · , 1 = 10 7 sun in the runs). The dot marks the time at which the circularisation of the orbit by dynamical friction is completed. The inset shows the black hole separation (in pc) versus time (in Myr). At circularisation, M Mestel ( a ) /m · , 2 ∼ 150 and 75 for the two cases, respectively. Courtesy of D. Fiacconi. \n<!-- image --> \npendent of radius a , the black hole sinks from an initial radius a i to a much smaller radius a f on a migration time scale τ I mig , Mestel ∼ CΩ -1 i [ M Mestel ( a i ) /m · , 2 ], where C = ( h/a ) 3 / (4 ζ ' ) . 5 We remark that the scaling of τ I mig , Mestel with the disc and black hole mass holds true provided M Mestel ( a ) > m · , 1 > m · , 2 . \nNuclear discs, stable against fragmentation, are nevertheless ideal. The gas can not be treated as a simple one-phase fluid. Galactic discs are sites of local gravitational instabilities conducive to star formation episodes: massive stars inject energy in the form of winds and supernova blast waves, feeding back energy into the disc and the gas is multi-phase, and clumpy. Thus, it is of importance to understand how the black hole sinking is affected by the inhomogeneous substructure of star forming discs. To this aim, in the next subsection, we explore black hole dynamics in clumpy discs.", '4.1.2 Clumpy circum-nuclear discs and stochastic orbital decay': 'In real astrophysical discs, massive gas clouds coexist with warmer phases and a polytropic equation of state, often used in SPH simulations, provides only an averaged representation of the real thermodynamical state. Cold self-gravitating discs are unstable to fragmentation and attain stability when stars, resulting from the collapse and/or collision of clouds, inject energy in the form of winds and supernova blast waves, feeding back energy into the disc now composed of stars and a multiphase gas. \nFig. 9 Colour-coded face-on view of the gas surface density of a clumpy disc model with M d = 5 × 10 8 M glyph[circledot] , m · , 1 = 10 7 M glyph[circledot] , q = 0 . 1 and initial eccentricity e = 0 . 7, plotted at four different times. The position of the black holes is marked by white dots. Courtesy of D. Fiacconi. \n<!-- image --> \nDue to the complexity of implementing this rich physics at the required level of accuracy, a first step ahead is to insert a phenomenological cooling prescription to allow the formation of clumps in a controlled way (Fiacconi et al. 2013). Clumps of size ∼ 5 pc in the mass interval between 10 5 M glyph[circledot] and 10 7 M glyph[circledot] develop in the disc, and evolve as they mass segregate, collide with each other and interact with the secondary black hole. Acting as massive perturbers, they disturb the otherwise smooth black hole orbital decay due to the stochastic behaviour of their torques that are not coherent in time. Several close encounters between m · , 2 and the massive clumps act as gravitational slingshots, causing an impulsive exchange of orbital energy and angular momentum. Thus, the black hole deviates from its original trajectory either outwards, or inwards or out of the disc plane (above the typical scale height of the disc). When moving on an inclined orbit the black hole experiences the weaker dynamical friction of the stellar background, resulting in a longer orbital decay timescale. The secondary black hole can also be captured by a massive clump forming a pair which segregates rapidly toward the centre. Figure 9 shows the evolution of the gas surface density of a selected run, and the position of the two black holes (marked as white dots) at four different times. \nThe stochastic behaviour of the black hole orbit, resulting from the incoherence of torques, emerges mainly when the clump to black hole mass ratio is M clump /m · , 2 > ∼ 1. This enlarges the values of the decay time which now range from less than < ∼ 1 up to > ∼ 50 Myr. This suggests that describing the cold clumpy phase of the interstellar medium in nuclear discs, albeit so far neglected, is important to predict the black hole \ndynamics. Ongoing simulations in a multi-phase star forming nuclear disc resulting from the collision of two discs following a merger produce results that are intermediate between the smooth and clumpy case (Lupi et al. in preparation).', '4.2 Binary-disc-driven migration': "In Section 4.1.1 we followed the black hole inspiral during phase I-g, in presence of a self-gravitating, rotationally supported disc much heavier than the binary, a condition leading to migration as described in (Escala et al. 2005; Dotti et al. 2006, 2007, 2009). However, with binary decay, the disc mass M Mestel enclosed in the black hole orbit a decreases with time falling below m · , t . The black holes then form a Keplerian binary surrounded by a less massive disc, called circum-binary disc, dominated by the gravity of the binary and its quadrupolar field. \nIf we impose continuity in the physical processes, there might exist an intermediate phase whereby migration of the secondary black hole is controlled by resonant torques. In close resemblance to Type I planet migration, and for very small binary mass ratios q glyph[lessmuch] 1, the resulting torque on m · , 2 is T mig , Kep I = -ζ K [ m · , 2 /m · , 1 ] 2 Σa 4 Ω 2 K , where Ω K is the Keplerian rotational velocity in the gravitational field of m · , 1 evaluated in a , Σ the disc surface density in the immediate vicinity of m · , 2 , ζ K = (1 . 36 + 0 . 54 α )( a/h ) 2 , and α the slope of the surface density profile Σ ∝ a -α of the underlying Keplerian disc (Tanaka et al. 2002; Armitage 2013). Notice that because of the natural scaling present in the problem, the torque T Kep I differs from the expression of T Mestel I having m · , 1 in place of M Mestel ( a ) as reference mass for the gravitational potential. In this case, the migration time reads as τ I mig , Kep ∝ [ m · , 1 /m · , 2 ][ m · , 1 /M disc ( a )] Ω -1 K , under the condition that the disc mass enclosed in the black hole orbit M disc ( a ) < m · , 1 . \nAs described in Section 3, black hole binaries form preferentially in the aftermath of major mergers so that the binary mass ratio q < ∼ 1. Furthermore, accretion drives q to larger and larger values, in the case of minor mergers under specific circumstances (Callegari et al. 2011). Thus, it is quite likely that migration under the action of torques excited by resonances is a missing step, in the dynamical evolution of black hole binaries. At this time the binary is expected to alter profoundly the structure of the underlying disc. \nThus, a key question poses: will the black holes sink down to the domain of gravitational wave inspiral transferring their angular momentum to the disc, or would the binary stall? Is there a phase II-g of migration and under which conditions? This phase is in fact somewhat controversial, as the black hole fate depends on whether the disc is a one-time, short-lived excretion disc (Lodato et al. 2009; Pringle 1991), or an extended long-lived disc (Rafikov 2013). These are conditions that are not recoverable from realistic larger-scale simulations as it is difficult to model the transition from a disc dominated by self-gravity and gravito-turbulence to a disc dominated by magneto-hydrodynamical turbulence stirred by magneto-rotational instabilities in the conducting fluid (Shi et al. 2012). \nUnless the binary is surrounded by a geometrically thick disc or envelope and decays promptly (del Valle and Escala 2012, 2014), the tidal force exerted by the binary on the circum-binary disc is expected to eventually clear a cavity. The picture is that the binary transfers orbital angular momentum to the disc by exciting non-axisymmetric density perturbations in the disc body, causing the formation of a low-density, hollow region, called gap (Farris et al. 2014; Rafikov 2013; Hayasaki et al. 2013; Roedig et al. \n2012; Kocsis et al. 2012; Shi et al. 2012; Roedig et al. 2011; Cuadra et al. 2009; Hayasaki 2009; Haiman et al. 2009; MacFadyen and Milosavljevi'c 2008; Hayasaki et al. 2008, 2007; Ivanov et al. 1999; Gould and Rix 2000; Pringle 1991). Viscous torques in the disc oppose gas clearing by the tidal field of the binary and ensure strong binary-disc coupling. Under these conditions, the binary enters a regime of slow orbital decay [referred to as Type II migration in the case of planets (Artymowicz and Lubow 1994, 1996; Gould and Rix 2000; Armitage and Natarajan 2002; Armitage 2013)] during which the inner edge of the circum-binary disc compresses in coordination with the hardening of the binary, so that the size δ ( t ) of the gap decays remaining close to twice the binary semi-major axis, δ ∼ 2 a ( t ) . \nDue to the tidal barrier offered by the binary, gas piles up at the inner rim of the disc. One can view the binary as acting as a dam, halting the gas inflow. Accordingly, the accretion rate in the circum-binary disc is not constant in radius and no strict steady state is ever attained in the disc body. In 1D modelling of circum-binary discs, the disc is seen to evolve into a state of constant angular momentum flux (Rafikov 2013), and binary decay leads to a secular and self-similar evolution of the disc as first suggested in (Ivanov et al. 1999). This happens when the system loses memory of the natural scale a , set by the size of the cavity and binary orbit, soon after the angular momentum injected by the binary has been transmitted to the larger scale extended disc (Rafikov 2013). Gap opening implies longer hardening time scales compared to nuclear-disc-driven migration, now controlled by the viscous time at the inner disc edge. \nThe migration time can be estimated as τ II mig ∼ τ ν [( m · , 2 + M edge d ) /M edge d ] where M edge d ∼ Σ ( R ) R 2 is the local mass near the inner edge of the disc, at R ∼ 2 a -3 a where the surface density has a peak, and τ ν the disc viscous time there: τ ν ∼ (2 / 3) R 2 /ν ∼ 2 πR 2 Σ/ ˙ M where ˙ M ∼ 3 πνΣ is the mass accretion rate in an unperturbed reference disc (Shakura and Sunyaev 1973). When M edge d > m · , 2 , the secondary black hole behaves as a parcel in the viscous disc and migrates on the viscous timescale, whereas when M edge d < m · , 2 migration slows down and occurs of a timescale longer than τ ν . Condition M edge d < m · , 2 is often referred to as secondary-dominated Type II migration (Haiman et al. 2009; Syer and Clarke 1995). The opposite regime is referred to as discdominated Type II migration. \nThe timescale τ II mig can be recovered if one assumes a torque on the binary of the form T mig II ∼ -ξ j o ˙ M ∼ ξ ' j o M edge d Ω K where j o = ( µ/m · , t )( Gm · , t a ) 1 / 2 is the binary angular momentum per unit mass, and ξ or ξ ' determined by numerical simulations, e.g. (MacFadyen and Milosavljevi'c 2008; Roedig et al. 2012; Shi et al. 2012). The above expression for the torque relates the rate of binary orbital decay to the local disc mass M edge d near the inner edge of the disc. Therefore, depending on how M edge d varies over the relevant timescales, i.e. whether the disc is continuously re-filled of gas to keep M edge d nearly stationary, or the disc mass is consumed before the binary has evolved substantially, orbital decay accelerates or decelerates, returning the problem to the old, outstanding issue on whether black holes are continuously fed in galactic nuclei or not, and on which timescale. \nSemi-analytical expressions of the migration time have been derived in (Haiman et al. 2009) considering orbital decay within a Shakura & Sunyaev accretion disc (Shakura and Sunyaev 1973). This enabled the authors to evaluate the disc surface density, opacity, viscosity and ultimately M edge d as the binary transits through the outer/middle and inner zones of the disc. Under these simplifying assumptions (of a \nFig. 10 Residence time | a/ ˙ a | of equal-mass black hole binaries, embedded in a steady circumbinary disc, as a function of the black hole separation (in units of 2 Gm · , t /c 2 ), as computed in (Haiman et al. 2009) for a reference disc model. The top x -axis label refers to the Keplerian relative orbital velocity of the black holes in the binaries. The four curves correspond to binaries with total masses of m · , t = 10 3 , 10 5 , 10 7 and 10 9 M glyph[circledot] as labeled. The large dots denote the critical radius beyond which the assumed circum-binary Keplerian disc is unstable to fragmentation. Similarly, triangles denote radii beyond which the disc may be susceptible to ionisation instabilities (the gas temperature falls below 10 4 K). In each case, blue/red colors indicate whether the disc mass enclosed within the binarys orbit is larger/smaller than the black hole mass m · , 2 . The dotted/dashed/solid portion of each curve indicates the outer/middle/inner disc region. Note that in the disc-dominated regime (blu segments) the binary residence time is ∼ 10 9 yrs, while it decreases below ∼ 10 7 yrs for all binaries, i.e. independent of their mass, at the entrance in the stable region of a circum-binary disc (red dots). Courtesy of Haiman et al. (2009). \n<!-- image --> \nsteady 1D disc), Haiman et al. (2009) have shown that the sinking time of the binary is a monotonic decreasing function of the binary orbital period (or separation). The residence time t res ∼ | a/ ˙ a | for equal-mass binaries (which is in this context close to τ II mig ) is plotted in Figure 10 from Haiman et al. (2009) for a disc with α viscosity parameter equal to 0.3, a radiative efficiency of 0.1 and an accretion rate equal to 0.1 of the Eddington value. In the disc-dominated regime when M edge d > m · , 2 the migration timescale is of the order of ∼ Gyr and when M edge d < m · , 2 it drops below 10 7 yrs, showing weak dependence of the binary mass. Similar timescales have also been found in Rafikov (2013) when considering 1D disc models undergoing self-similar evolution [see also figure 6 of Haiman et al. (2009)]. Despite these studies, we are nonetheless \nFigure 11 shows the distribution of gas around the black hole binary after gap formation, from two SPH-3D simulations of Newtonian, massive circum-binary discs \n<!-- image --> \nFig. 11 Color-coded gas surface density of two Newtonian, self-gravitating circum-binary discs, showing the presence of a binary region with the two black holes and their mini-discs, a porous cavity filled with streams, the inner rim or edge working as a dam, and the body disc. Left (right) panel refers to a run with gas in the cavity treated with an isothermal (adiabatic) equation of state. Courtesy of Roedig et al. (2012). \n<!-- image --> \nfar from having a reliable estimate of the migration timescale in circum-binary discs under a variety of conditions, given the rich physics involved. 6 \nGap opening and/or maintenence of the inner cavity around massive black holes have been seen in numerous numerical simulations of both Keplerian and self-gravitating circum-binary discs (MacFadyen and Milosavljevi'c 2008; Shi et al. 2012; Cuadra et al. 2009; Roedig et al. 2011; del Valle and Escala 2012). But interestingly, recent 2D and 3D simulations have demonstrated that the binary+disc system contains as many as three discs and that these discs may persist being constantly fed by gas flowing through the gap. The three discs comprise the circum-binary disc plus two mini-discs around each member of the binary (Farris et al. 2014; Shi et al. 2012; Roedig et al. 2012, 2011). This is due to the fact that the disc inner edge is porous (for sufficiently high disc aspect ratios): high velocity, narrow streams of gas leak periodically through the dam into the inner cavity, modulated by the binary orbit (Roedig et al. 2012; Shi et al. 2012; Noble et al. 2012; D'Orazio et al. 2013; Farris et al. 2014). \nFig. 12 Differential torques dT/dR and integrated torque T (averaged over the time span of the simulation, and in code units) exerted by the disc on the binary with mass ratio q = 0 . 1 as a function of the radial distance, in units of the binary separation a , for the adiabatic (left) and isothermal (right) run from (Roedig et al. 2012). In each panel, the differential torque acting on the primary is plotted in green, on the secondary in red, and the sum of the two in blue. Notice that the torque density dT/dR shows different signs and starts oscillating around the zero point at distances far from the binary where the binary-disc coupling decreases sharply. The black line refers to the integrated torque T up to a distance R : T is positive inside a , and negative outside giving a total negative contribution. Courtesy of Roedig et al. (2012). \n<!-- image --> \n(Roedig et al. 2012) which differ from one anther due to a different thermodynamic modelling of the gaseous streams in the cavity: isothermal (on the right side) and adiabatic (on the left side). The figure highlights the occurrence of different domains in the disc (from outside in): the disc body R > 2 . 5 a where spiral patterns develop; the cavity edge , at radii 2 < R < 2 . 5 a, which is porous and leaky; the cavity region or gap, between a < R < 2 a, which is almost devoid of gas except for the presence of tenuous streams; the binary region, at 0 < R < a , with the two black holes and their mini-discs, fed by gas from the disc body flowing through the cavity across the porous dam. The mini-discs and the cavity are sharper in the isothermal case compared to the adiabatic case where the amount of gas impacting the gap is larger. Only a fraction of this gas is captured by the black holes to form the mini-discs, the remaining being swiftly ejected away. The different regions highlighted in Figure 11 contribute to the differential torque dT/dR on the binary with different signs as illustrated in Figure 12, for the case of a Newtonian self-gravitating disc (in the adiabatic and isothermal model, respectively). One can notice that the differential torque shows an oscillatory behaviour with a sharp maximum at the location of the secondary black hole ( R ∼ 0 . 75 a ), and \na deep minimum in the cavity region. Positive and negative peaks alternate in the disc body that almost cancel out, giving a negligible contribution to the total torque. Torques on the secondary black hole are always larger than on the primary, due to its proximity to the inner rim of the disc, resulting in a stronger interaction. \nIn summary, simulations now indicate that clearing a cavity in the disc does not prevent the inflow of gas through streams across the cavity's edge. Thus accretion of a fraction of this gas on the black holes, and preferentially onto the secondary (nearer to the disc's edge) may be a persistent feature (Farris et al. 2014). Thus binary evolution becomes more complex than outlined in the first part of this section. All binary elements evolve over time and in some cases inward migration can turn into outward migration (Hayasaki 2009; Roedig et al. 2012). The evolution equation for the semi-major axis of a binary depends on changes in the eccentricity, mass, reduced mass and on the exchange of angular momentum between the binary and the three discs through a generalised T . All these contribute to ˙ a/a = 2 T/J · -˙ m · , t /m · , t -2 ˙ µ/µ +2 e ˙ e/ (1 -e 2 ) where J · is the binary angular momentum. The sign of this derivative thus depends on different effects. \nThe binary eccentricity tends to increase during the binary-disc coupling (Armitage and Natarajan 2005; MacFadyen and Milosavljevi'c 2008), and the growth of e has also been seen in 3D numerical simulations (Roedig et al. 2011). Progress in the analysis of this process has revealed that such excitation can not grow indefinitely, as saturation occurs due to the interaction of the secondary black hole with gas near the inner rim of the disc body, and to the accumulation of gas around the black holes in the minidiscs (Roedig et al. 2011). The initial rise of e can be understood, as in Section 4.1.1, using dynamical friction in a differentially rotating background as leading argument. The secondary black hole, closer to the circum-binary disc, induces a trailing density wave near the inner rim which reduce its tangential velocity, causing a loss of orbital angular momentum. The eccentricity e grows and continues to grow as long as the gas at the inner edge of the circum-binary disc moves with a lower angular velocity. However the progressive decay of the black hole tangential velocity with increasing e leads eventually to a reversal of the sign of the relative velocities, the gas moving faster than the black holes, thus developing a wake heading in front which leads to an acceleration of the black hole. The process reaches saturation, and this is found to occur about e ∼ 0 . 6 -0 . 8. Furthermore, when the binary becomes very eccentric, the secondary, less massive black hole passes through the mini-disc of the primary suffering a deceleration at peri-centre which in turn decreases e , which then attains a saturation value. \nThe mass of the two black holes tend to increase as well, and the increase of the mass of the secondary black hole is even higher, being closer to the disc, thus driving q toward unity (Farris et al. 2014). The mass accretion rate is not severely limited (compared to the case of a single isolated black hole) and ˙ M is found to be modulated at the binary orbital period and higher harmonics (Farris et al. 2014; Roedig et al. 2011). Interestingly, modulated accretion suggests a promising avenue for producing a modulated electromagnetic signal permitting the identification of binaries during migration in circum-binary discs at different orbital phases along the path to coalescence (Eracleous et al. 2011; Decarli et al. 2013; Montuori et al. 2012).", '5 Timescales: an overview': "Galaxy interactions and mergers are the sites of formation of dual, binary, coalescing and recoiling black holes. Associated to these different dynamical phases there is a zoo of sources: the dual, binary and recoiling AGN if the black holes are active. A residence time is associated to each phase: in phase I, the dynamical friction timescale τ df or /and the dynamical friction timescale which accounts for tidal mass loss τ df , tidal ; in phase II, the hardening time in a stellar background τ Hard (which falls in the interval between τ ∗ hard and τ rel ), or/and (in gas-rich mergers) the timescale of nuclear-discdriven migration τ I mig , Mestel and binary-disc-driven migration τ II mig ; ultimately in phase III, the gravitational wave timescale τ gw . \nThere is no simple recipe to calculate the residence times in terms of fundamental parameters such as the black hole mass and mass ratio since these timescales depend on the morphology of the interacting galaxies, the geometry of the encounter, the gas fraction, and most importantly on the complex input physics of difficult implementation even in current state-of-the-art simulations. \nThe characteristic coalescence time τ coal would be the sum of the timescales associated to the different phases (I, II or I-g,II-g, and III), calculated along each individual pathway. Their value depends, even in the minimal model, on whether the merger is gas-poor (dry) or gas-rich (wet), and major or minor. As no unique pathway exists for a pair, τ coal can be estimated simply considering the maximum of all residence times. This timescale should then be compared with the Hubble time or better with the running age of the universe at the time of coalescence, given that eLISA sources are typically at high redshifts (Amaro-Seoane et al. 2013). \nHere is a tentative summary of the timescales inferred from the whole body of works, in the black hole mass range < ∼ 10 7 M glyph[circledot] and for values of the initial black hole mass ratio q ( which indicates the mass ratio between the two interacting galaxies). If the 'zero' time is calculated when the merger of the baryonic components (bulge and disc) is completed, i.e. when the black holes behave as individual objects moving in the relic galaxy, the relevant timescales in different environments and conditions are expected to cluster approximately around these values: \n- · In dry major mergers ( q > q crit ∼ 0 . 1): (i) dynamical friction time τ df < ∼ [10 Myr 100 Myr](Yu 2002) - (ii) hardening timescale τ Hard ∼ 1 Gyr to a few Gyr (Khan et al. 2011).\n- · In wet major mergers ( q > q crit ∼ 0 . 1): (i) gas-dynamical friction time τ df < ∼ [10 Myr 100 Myr](Mayer et al. 2007; Chapon et al. 2013; Roˇskar et al. 2014) - (ii) nuclear-discdriven migration time τ I mig , Mestel ∼ (5 Myr - 50 Myr) (Escala et al. 2005; Dotti et al. 2006, 2007, 2009; Fiacconi et al. 2013) - (iii) binary-disc-driven migration τ II mig ∼ 10 Myr (Haiman et al. 2009).\n- · In dry minor mergers ( q < q crit ∼ 0 . 1): (i) dynamical friction time τ df , tidal < ∼ [10 Myr - 100 Myr](Yu 2002) - (ii) hardening timescale τ Hard ∼ 1 Gyr up to a few Gyr (Yu 2002; Khan et al. 2012b).\n- · In wet minor mergers ( q < q crit ∼ 0 . 1): (i) dynamical friction time with corrections due to tidal stripping τ df , tidal < ∼ 100 Myr or wandering (Callegari et al. 2009, 2011). The fate is uncertain. Depending on the geometry of the encounter and gas fraction, the secondary black hole may wander in the primary galaxy.", '6 Summary and future prospects': "The study of the dynamics of black holes, with masses from 10 4 M glyph[circledot] up to 10 9 M glyph[circledot] , inside galaxies displaying a large variety of morphologies and masses, is not a side problem: it is central if we want to search for or recognise signs of their duality and/or coalescence at electromagnetic level, and if we want to detect the gravitational waves emitted at the time of black hole coalescence. Observationally the search of dual AGN (accreting black holes in merging galaxies at separations of ∼ kpc), and of binary (pc scales) and recoiling AGN have received attention in recent years (Eracleous et al. 2011; Liu et al. 2011; Komossa 2012; Koss et al. 2012; Liu et al. 2013; Decarli et al. 2013; Comerford and Greene 2014; Lusso et al. 2014; Liu et al. 2014). There has been some major advances in the study of the dynamics of black holes in merging galaxies, over the last years, and the points to remember and to take away for future reference are: \n- 1. Black holes in binaries can reach coalescence under the emission of gravitational waves. But, for this to happen, the black holes have to be driven to separations as small as ∼ 10 -3 pc or less, as gravity is a weak force and gravitational waves are a manifestation of the strong field regime (Sathyaprakash and Schutz 2009). This is a minuscule distance, compared to galaxy's sizes, and merging galaxies are the sites where these events can occur. Nature has thus to provide a series of mechanisms able to extract energy and angular momentum, from the large scale of the merger (at least a few kpc) to the micro-parsec scale, i.e. the scale at which the black hole horizons touch. The path to coalescence is long and complex, and stalling of the binary at some scale is a possibility.\n- 2. Three phases accompany the path to coalescence: the pairing, hardening, and gravitational-wave driven inspiral phases. Stars or gas, or stars and gas drive the black hole inspiral, depending on whether galaxies are gas-rich or gas-poor. Bottlenecks can appear at various scales and a major effort is to identify possible obstacles. The last parsec problem, i.e. the stalling of a massive black hole binary at the centre of a large spherical collisionless galaxy was highlighted as a critical step.\n- 3. Thanks to recent advances in numerical computing, the last parsec problem appears to be an artefact of oversimplifying assumptions. Galaxies, relic of mergers, are not spherical systems and can retain a high degree of triaxiality or asymmetry. Under these circumstances the hardening of the binary via binary-single stellar encounters appears to have no halt, at least in the cases explored and coalescence timescales are close to 1 to a few Gyr. The issue is not settled yet.\n- 4. It is now possible to track the black hole dynamics during galaxy collisions using state-of-the-art simulations. The dynamics of the interacting galaxies is followed ab initio , from the large scale (several kpc) down to the central few parsecs, considering all components - the dark halo, the stellar and gaseous disc, and bulge. This enables us to trace the rise of asymmetries and instabilities in both the stellar and gas components which play a pivotal role in determining whether there is stalling or rapid sinking of the binary.\n- 5. Major gas-rich (wet) mergers are conducive to the formation of close Keplerian binaries. The gas, thanks to its high degree of dissipation, controls the black hole inspiral inside the massive circum-nuclear disc that forms in the end-galaxy. When described as a single phase medium, the gas promotes rapid inspiral before the \ndisc fragments into stars, and before stellar dynamical friction becomes effective. When the gas is multi-phase and clumpy on various mass scales, the black hole orbit shows a stochastic behaviour. The black holes in this case form a binary on timescales typically between 1 Myr and 100 Myr. \n- 6. Coalescences of middleweight black holes of ∼ 10 4 M glyph[circledot] at high redshifts z ∼ 10 -15 require either very dense, low velocity dispersion stellar environments, or large not yet quantified amounts of gas in unstable forming galaxies.\n- 7. Minor mergers can release the less massive black hole on peripheral orbits in the main galaxy, due to the disruptive action of tidal torques on the less massive galaxy, rising a new problem: the last kpc problem. Gas plays a key role in the process of pairing in minor merger, as it makes the satellite galaxy more resilient against tidal stripping because central gas inflows, triggered during the interaction, steepen the stellar cusp. Due to the fragility of the satellite galaxy, the fate of black holes in minor mergers is uncertain: encounter geometry, gas fraction, degree of gas dissipation are key elements for establishing whether the black hole is a sinking or a wandering black hole inside the primary galaxy. The prediction is that there is a large scatter in the outcomes.\n- 8. At sub-parsec scales, the Keplerian binary is likely surrounded by a circum-binary disc. When present, gas-assisted inspiral takes place which can be faster than stardriven inspiral. Both processes likely co-exist but have never been treated jointly. The black holes are expected to migrate on a timescale controlled by the interplay between the binary tidal torque which tends to clear a cavity (repelling the disc's gas in the immediate vicinity of the binary) and viscous torques in the disc which tend to fill and even overfill the cavity. Gas leaks through the gap and the black holes are surrounded by mini accretion discs. All these processes modify the orbital elements in a complex way. Theoretical models indicate that if there is a sufficiently longlived inflow of gas at the inner edge of the circum-binary disc, the binary hardens on timescales < ∼ 10 9 Gyr or even much less (depending on the previous history).\n- 9. The path to coalescence still remains a complex problem to solve and there is no clear-cut answer. To be conservative, coalescence times range between several Myrs to about several Gyrs. \nThe field needs to evolve farther along different lines and directions. Here are a few hints: \n- 1. There is need to continue to study not only the growth of black holes along cosmic history, but their dynamics as they are inter-connected. Attempts to follow the dynamics of black holes during the cosmic assembly of galactic halos have been carried on, albeit at much lower revolution than required to track their accretion history and fate (Bellovary et al. 2010). Any effort along this line is central in order to understand the coevolution of black holes with galaxies.\n- 2. Intriguingly enough, the fate of black hole binaries in galaxies takes us back to the unsolved problem of the feeding of black holes in galactic nuclei over cosmic ages, i.e. of the angular momentum barrier present on parsec scales. Dynamical decay, star formation, accretion and their back-reactions are coupled. Star formation makes the ISM multi-phase and turbulent. Supernova and AGN feed-back may heat/remove gas and the consequences of these effects on black hole migration have not been quantified yet, during orbital evolution.\n- 3. Dual, binary and recoiling AGN and also triple AGN are important observational targets. There is the need to improve upon observational strategies for identifying \nbinary and recoiling AGN in large surveys, assisted by tailored and coordinated hydro-dynamical simulations. \n- 4. Black hole migration in circum-binary disc is a challenging problem which deserves constant attention. Binary eccentricity growth, accretion and outflows, are processes that affect the dynamics and stability of the system as a whole. Understanding the nature of torques in multi-phase models of circum-nuclear discs will become central in order to extrapolate the black hole migration timescale from the parsec scale down to the domain controlled by gravitational wave inspiral, and to asses the observability of sub-parsec binaries during the phases which anticipate the merging. \nAcknowledgements I would like to thank my collaborators Simone Callegari, Massimo Dotti, Davide Fiacconi, Alessandro Lupi, Lucio Mayer, Constanze Roedig, Alberto Sesana and Marta Volonteri for many useful and illuminating discussions over the years. I would like also to thank the International Space Science Institute for kind hospitality.", 'References': "- P. Amaro-Seoane, S. Aoudia, S. Babak, P. Bin'etruy, E. 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2006CQGra..23.5643E
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Black holes in Einstein-aether theory
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2006-01-01
| 21 | 0.45 | 162 |
['-', 'astrophysics', '-']
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We study black hole solutions in general relativity coupled to a unit timelike vector field dubbed the 'aether'. To be causally isolated, a black hole interior must trap matter fields as well as all aether and metric modes. The theory possesses spin-0, spin-1 and spin-2 modes whose speeds depend on four coupling coefficients. We find that the full three-parameter family of local spherically symmetric static solutions is always regular at a metric horizon, but only a two-parameter subset is regular at a spin-0 horizon. Asymptotic flatness imposes another condition, leaving a one-parameter family of regular black holes. These solutions are compared to the Schwarzschild solution using numerical integration for a special class of coupling coefficients. They are very close to Schwarzschild outside the horizon for a wide range of couplings, and have a spacelike singularity inside, but differ inside quantitatively. Some quantities constructed from the metric and aether oscillate in the interior as the singularity is approached. The aether is at rest at spatial infinity and flows into the black hole, but differs significantly from the 4-velocity of freely falling geodesics.
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https://arxiv.org/pdf/gr-qc/0604088.pdf
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{'Christopher Eling and Ted Jacobson': 'Department of Physics, University of Maryland College Park, MD 20742-4111 USA \nE-mail: \[email protected] \nE-mail: \[email protected]', 'Abstract.': "We study black hole solutions in general relativity coupled to a unit timelike vector field dubbed the 'aether'. To be causally isolated a black hole interior must trap matter fields as well as all aether and metric modes. The theory possesses spin-0, spin1, and spin-2 modes whose speeds depend on four coupling coefficients. We find that the full three-parameter family of local spherically symmetric static solutions is always regular at a metric horizon, but only a two-parameter subset is regular at a spin0 horizon. Asymptotic flatness imposes another condition, leaving a one-parameter family of regular black holes. These solutions are compared to the Schwarzschild solution using numerical integration for a special class of coupling coefficients. They are very close to Schwarzschild outside the horizon for a wide range of couplings, and have a spacelike singularity inside, but differ inside quantitatively. Some quantities constructed from the metric and aether oscillate in the interior as the singularity is approached. The aether is at rest at spatial infinity and flows into the black hole, but differs significantly from the the 4-velocity of freely-falling geodesics.", '1. Introduction': "Einstein-Aether theory, nicknamed 'ae-theory', consists of general relativity (GR) coupled to a dynamical, unit timelike vector field u a called the 'aether'. The aether is somewhat like a nonlinear sigma model field with a hyperbolic target space. But, unlike for a sigma model, the hyperboloid is dynamically determined at each point by the spacetime metric and the aether couples to the metric via covariant derivatives. The aether dynamics thus modifies the metric dynamics, and the coupled system differs significantly from the familiar examples of GR coupled to matter fields. \nProperties of ae-theory have been extensively studied for the past few years, and many observational signatures have been worked out (see [1] for a recent review). The primary motivation for studying ae-theory is to explore the gravitational consequences of local Lorentz symmetry violation. The aether itself breaks local boost invariance at each point, while preserving rotational symmetry in a preferred frame. There are four free coupling coefficients in the general ae-theory Lagrangian (besides the gravitational constant), thus providing a somewhat general class of boost violating models ‡ . There is a common range of the four coupling coefficients for which simultaneously all postNewtonian parameters agree with those of GR [4, 5, 6], where all perturbations are stable [7, 8] and have positive energy [8, 9, 10], gravitational and aether vacuum Cerenkov radiation [11] is absent [6], the gravitational constant in the Friedman equation agrees with Newton's constant [12], and gravitational radiation damping of binary pulsar orbits agrees with the rate in GR for weak fields at lowest post-Newtonian order [10]. \nIn spherical symmetry the aether has one degree of freedom, a radial tilting. Restricting to the time-independent case, we found in [13] that there is a three-parameter family of local solutions, of which a two-parameter sub-family is asymptotically flat. In [13] we focused on the one-parameter sub-family of these in which the aether is aligned with the static Killing vector. These 'static aether' solutions were found analytically, up to the inversion of a transcendental function. The solutions are characterized by their total mass, and it was shown in [13] that the solution outside a spherical star is in this family. \nIn this paper we focus on the time-independent, spherically symmetric black hole solutions of ae-theory. This is the first step in comparing ae-theory with astrophysical observations of black holes. It is also of mathematical interest as an example of unusual black hole behavior with non-linear self-gravitating fields. \nThe Lagrangian for ae-theory depends on four dimensionless coupling coefficients c 1 , 2 , 3 , 4 . The static aether solutions referred to above depend only on the single combination c 1 + c 4 , but this is not the case for the black hole solutions. Moreover, unlike those solutions, the black hole solutions cannot (as far as we know) be obtained analytically, so all of the results in this paper are numerically obtained. We do not make an exhaustive study here for all values of the c i , but rather just attempt to determine the generic behavior of the black hole solutions. \nWe begin this paper in section 2 with a discussion of the definition of a black hole in ae-theory, and the conditions for the existence of regular black hole solutions. The qualitative reasons for the existence of a one parameter family of such solutions are explained. In preparation for the subsequent detailed analysis the action, field equations, and field redefinition properties of the theory are reviewed in section 3. This is followed in section 4 by a demonstration using the power series solution of the field equations about a metric horizon that such horizons are generically regular, i.e. there is a three parameter family of solutions in the neighborhood of such a horizon, just as in the neighborhood of any generic point. We then show that spin-0 horizons are generically singular, but a two-parameter family of solutions is regular and the additional condition of asymptotic flatness further reduces this to a one-parameter family. In section 5 we study the properties of typical examples of this one parameter family of black holes, imposing regularity by a power series expansion about the spin-0 horizon and numerically integrating out to infinity, determining the asymptotically flat solutions by tuning the data at the horizon. These black holes are rather similar to Schwarzschild outside the horizon, and like Schwarzschild they have a spacelike curvature singularity inside at (or very near) zero radius. Unlike the static aether solutions of [13], u a is not aligned with the static Killing field in these black hole solutions. The aether flows into the black hole, but differs significantly from the 4-velocity of freely-falling geodesics at rest at infinity. We also note that some functions constructed from the metric and aether exhibit oscillatory behavior in the interior as they approach the singularity. Section 6 concludes the paper with a brief discussion of questions for further work.", '2. General properties of black holes in ae-theory': "The relevant notion of a black hole in ae-theory is not immediately clear. To trap matter influences a black hole must have a horizon with respect to the causal structure of g ab , the metric to which matter couples universally (or almost universally, according to observations). We call this a 'metric horizon'. This is not the only relevant notion of causality however. For general values of the coupling coefficients c i , ae-theory has multiple characteristic hypersurfaces. In particular, perturbing around flat spacetime it was found in [7] that there are spin-2, spin-1, and spin-0 wave modes, with squared speeds relative to the aether given by \nspin-2 1 / (1 -c 13 ) spin-1 ( c 1 -1 2 c 2 1 + 1 2 c 2 3 ) /c 14 (1 -c 13 ) spin-0 c 123 (2 -c 14 ) /c 14 (1 -c 13 )(2 + c 13 +3 c 2 ) (1) \nwhere c 13 = c 1 + c 3 , etc. The speeds s i are generally different from each other and from the metric speed of light 1. Only in the special case where c 4 = 0, c 3 = -c 1 , and c 2 = c 1 / (1 -2 c 1 ) do all the modes propagate at the same speed. In general, the characteristic surfaces for a mode of speed s i are null with respect to the effective metric η ab +( s 2 i -1) u a u b , where η ab is the flat metric and u a the constant background aether. \nIn the nonlinear case characteristics can be defined as hypersurfaces across which the field equations admit a discontinuity in first derivatives [14]. For this paper we presume that these characteristics define the relevant notion of causal domain of dependence for the ae-theory field equations. This seems quite plausible, although no rigorous study has been attempted. The characteristic hypersurfaces are determined by the highest derivative terms in the field equations, so can also be identified by examination of high frequency solutions to the linearized equations about a given background [14]. For such solutions the gradients in the background are irrelevant, so we can infer that the nonlinear characteristics are null surfaces of the effective metrics, \ng ( i ) ab = g ab +( s 2 i -1) u a u b . (2) \nWe refer to the horizons associated with these metrics as the spin-0, spin-1, and spin-2 horizons. If a black hole is to be a region that traps all possible causal influences, it must be bounded by a horizon corresponding to the fastest speed. The coupling coefficients c i determine which speed is the fastest. \nA horizon is potentially a location where a solution to the field equation can become singular. This is because at a characteristic surface the coefficient of a second derivative term usually present in the equation vanishes. As such a surface is approached, the smallness of that coefficient may generically produce a solution in which some second derivative grows without bound, leading to singular behavior. This does not occur at spin-1 and spin-2 horizons in spherically symmetric solutions to ae-theory, presumably since there are no spherically symmetric spin-1 or spin-2 modes. However there is a spherical spin-0 mode, and we find that spin-0 horizons are generically singular. \nThe requirement that the spin-0 horizon be regular reduces the three parameter family of local static, spherically symmetric solutions to a two parameter family, which reduces to a one parameter family when asymptotic flatness is imposed. Hence there is just a one parameter family of regular static, spherically symmetric black hole solutions in ae-theory, just as in GR. Unlike outside a star, the aether in these solutions is not aligned with the Killing vector but rather flows into the black hole. We conjecture that the collapse of a spherical star to form a black hole is nonsingular, and therefore must be accompanied by a burst of spin-0 aether radiation as the aether (and metric) exterior adjusts. \nThe fact that the aether is not aligned with the Killing vector in a static black hole solution is no accident. It cannot be so aligned, since it is everywhere a timelike unit vector and the Killing vector is null on the horizon. Thus at a regular horizon the aether must be 'infalling', although it is nevertheless invariant under the Killing flow. The static aether solution found in [13] has aether aligned with the Killing vector, and can be thought of as an extremal black hole with a singular horizon on which the aether becomes infinitely stretched. \nWhile the aether can be regular at a generic point on the horizon, it cannot smoothly extend to the bifurcation sphere B , i.e the fixed point set of the Killing flow at the intersection of the past and future horizons [1]. The Killing flow acts as a Lorentz \nboost in the tangent space of any point on B , so it is impossible for the aether to be invariant under the flow there. This implies that the aether must blow up, becoming an infinite null vector as B is approached. This in turn raises the concern that there may be no regular metric horizon, since regularity on a future horizon is typically linked to regularity at B . Indeed Racz and Wald [15] have established, independent of any field equations, conditions under which a stationary spacetime with regular Killing horizon can be extended to a spacetime with a regular bifurcation surface, and conditions under which matter fields invariant under the Killing symmetry can also be extended. In spherical symmetry these conditions are satisfied for the metric, but the aether vector field breaks the required time reflection symmetry so it need not be regular at the bifurcation surface (although all scalar invariants must be, as must the aether stress tensor if the field equations hold). \nAnother potential obstruction to the existence of regular ae-theory black hole horizons arises from the form of the aether stress tensor. At a regular stationary metric horizon the Raychaudhuri equation for the horizon congruence with with null generator k a implies that R ab k a k b must vanish, hence the Einstein equation implies that the matter stress tensor component T ab k a k b must also vanish. With common matter fields, e.g., scalar fields, Maxwell or Yang-Mills fields, and nonlinear sigma model fields, it is easy to show from examination of the form of the stress tensors that this condition is automatically satisfied locally for any field invariant under the Killing flow, independent of field equations. This property does not seem to hold kinematically for the aether stress tensor, but since we find a full three-parameter family of regular metric horizons, it is evidently imposed by the field equations. \nThe fact that T ab k a k b does not vanish kinematically might appear to contradict the following general argument. For a Killing horizon with non-zero surface gravity it is not necessary to examine the form of the stress tensor to arrive at the inference that the horizon component of a matter stress tensor vanishes. If χ a is the horizongenerating Killing vector, then the vanishing of the scalar T ab χ a χ b on the horizon is guaranteed by the facts that (i) it is invariant along the flow, (ii) χ a vanishes at the bifurcation surface, and (iii) T ab is regular at the bifurcation surface (as guaranteed by the Racz-Wald extension theorem). But this argument too seems to fail for the aether stress tensor, because (as noted above) there is no purely kinematic way to argue that it (and therefore its stress-tensor) is regular at the bifurcation surface. So, again, the field equations seem to play an essential role in ensuring the existence of regular metric horizons.", '3. Einstein-Aether theory': 'The action for Einstein-Aether theory is the most general diffeomorphism invariant functional of the spacetime metric g ab and aether field u a involving no more than two \nderivatives, \nS = 1 16 πG ∫ √ -g L d 4 x (3) \nwhere \nL = -R -K ab mn ∇ a u m ∇ b u n -λ ( g ab u a u b -1) . (4) \nHere R is the Ricci scalar, K ab mn is defined as \nK ab mn = c 1 g ab g mn + c 2 δ a m δ b n + c 3 δ a n δ b m + c 4 u a u b g mn (5) \nwhere the c i are dimensionless constants, and λ is a Lagrange multiplier enforcing the unit timelike constraint. This constraint restricts variations of the aether to be spacelike, hence ghosts need not arise. A term of the form R ab u a u b is not explicitly included as it is proportional to the difference of the c 2 and c 3 terms in (3) via integration by parts. The metric signature is (+ ---) and the units are chosen so that the speed of light defined by the metric g ab is unity. When u a is a unit vector, the square of the twist ω a = /epsilon1 abcd u b ∇ c u d is a combination of the c 1 , c 3 , and c 4 terms in the action (3), \nω a ω a = -( ∇ a u b )( ∇ a u b ) + ( ∇ a u b )( ∇ b u a ) + ( u b ∇ b u a )( u c ∇ c u a ) . (6) \nIn spherical symmetry the aether is hypersurface orthogonal, hence it has vanishing twist, so the c 4 term can be absorbed by making the replacements \nc 1 → c 1 + c 4 , c 3 → c 3 -c 4 , c 4 → 0 . (7) \nThe field equations from varying (3) plus a matter action (coupled only to the metric) with respect to g ab , u a and λ are given by \nG ab = T ( u ) ab +8 πGT M ab (8) \n∇ a J a m -c 4 ˙ u a ∇ m u a = λu m , (9) \ng ab u a u b = 1 , (10) \nwhere \nJ a m = K ab mn ∇ b u n . (11) \nThe aether stress tensor is given by \nT ( u ) ab = ∇ m ( J ( a m u b ) -J m ( a u b ) -J ( ab ) u m ) + c 1 [( ∇ m u a )( ∇ m u b ) -( ∇ a u m )( ∇ b u m )] + c 4 ˙ u a ˙ u b + [ u n ( ∇ m J mn ) -c 4 ˙ u 2 ] u a u b -1 2 L u g ab , (12) \nwhere L u = -K ab mn ∇ a u m ∇ b u n . The Lagrange multiplier λ has been eliminated from (12) by solving for it via the contraction of the aether field equation (9) with u a .', '3.1. Metric redefinitions': "It is sometimes convenient to re-express the theory in terms of a new metric and aether field, related to the original fields by a field redefinition of the form \ng ' ab = g ab +( σ -1) u a u b (13) \nu ' a = 1 √ σ u a . (14) \nThe constant σ is restricted to be positive so the new metric remains Lorentzian. In effect, the field redefinition 'stretches' the metric tensor in the aether direction by a factor σ . The action (3) for ( g ' ab , u ' a ) takes the same form as that for ( g ab , u a ) up to the values of the coefficients c i . The c i coefficients for the action expressed in terms of the new fields (13)-(14) are [16] \nc ' 1 = σ 2 ( (1 + σ -2 ) c 1 +(1 -σ -2 ) c 3 -(1 -σ -1 ) 2 ) (15) \nc ' 2 = σ ( c 2 +1 σ -1 ) (16) \n- \nc ' 4 = c 4 -σ 2 ( (1 -σ -1 ) 2 c 1 +(1 -σ -2 ) c 3 -(1 -σ -1 ) 2 ) . (18) \nc ' 3 = σ 2 ( (1 -σ -2 ) c 1 +(1 + σ -2 ) c 3 -(1 -σ -2 ) ) (17) \nCertain combinations of the coefficients change by a simple scaling under the field redefinitions: \nc ' 14 = c 14 (19) \nc ' 123 = σc 123 (20) \nc ' 13 -1 = σ ( c 13 1) (21) \n-c ' 1 -c ' 3 -1 = σ -1 ( c 1 -c 3 -1) . (22) \n- \nThe field redefinition relates solutions to the field equations coming from the two actions. \nUsing a field redefinition the general form of the action can be simplified [16] by eliminating one of the c i or a combination of the c i . If we choose σ = ( s 2 ) 2 = 1 / (1 -c 13 ), then c ' 13 vanishes, i.e. c ' 3 = -c ' 1 . In this case the two corresponding terms in the Lagrangian combine to make a Maxwell-like lagrangian, for which the Levi-Civita connection drops out. (This choice of σ is positive and therefore preserves Lorentzian signature provided the original coefficients satisfy c 13 < 1.) In the context of spherical symmetry, we may also exploit the vanishing to the twist of the aether to absorb the c 4 term in (3) by the replacements (7) as explained above. After these two changes, the Lagrangian takes a much simpler reduced form characterized by only two c i coefficients, \nL ae = -R -c ' 1 2 F ab F ab -c ' 2 ( ∇ a u a ) 2 -λ ( g ab u a u b -1) , (23) \nwhere F ab = 2 ∇ [ a u b ] . In the next section we will use this reduced form of the action to investigate the general behavior of stationary spherically symmetric solutions, addressing the existence of solutions around a metric horizon, asymptotic flatness, and the regularity of the spin-0 horizon. \nAnother useful choice of field redefinition is to arrange for the new metric in (13) to coincide with the effective metric for one of the wave modes (2) by choosing σ = s 2 i . In this way we transform to a 'frame' where one of the spin-2, spin-1, or spin-0 horizons coincides with the metric horizon. Under the field redefinition all the squared speeds become s ' i 2 = s 2 i /σ , since the metric tensor in the aether direction is stretched by the factor σ . In section 5 we use this method to make the spin-0 and metric horizons coincide, which simplifies the expansion of the field equations around the spin-0 horizon.", '4. Generic behavior of horizons and spatial infinity': 'In this section we demonstrate that, for generic values of the c i coefficients (at least for the reduced theory (23)), (i) metric horizons are generically regular in the threeparameter family of local solutions, (ii) asymptotic flatness imposes one condition on this family, and (iii) regularity of the spin-0 horizon imposes another condition, leaving a one-parameter family of black hole solutions with regular spin-0 horizons.', '4.1. Metric horizon expansion': "One way to determine the number of independent solutions with a regular metric horizon is to expand the field equations in a power series about a candidate horizon and solve algebraically order by order. We did this using the Maple computer application to carry out the algebra. The computation was prohibitively complicated using our methods with general values of the c i , so we restricted attention to the reduced theory (23). The determination of the solution space can also be done more 'experimentally,' by numerical integration of the field equations with varying initial data. With the latter method no special restriction on the c i is required. \nFor the study of spherical black holes we adopt Eddington-Finkelstein (EF) type coordinates ( v, r, θ, ϕ ) with line element \nds 2 = N ( r ) dv 2 -2 B ( r ) dvdr -r 2 d Ω 2 (24) \nwith advanced (null) time coordinate v and radial 'area coordinate' r . The timetranslation Killing vector is ∂ v , and a metric horizon corresponds to N ( r ) = 0. These coordinates are regular at metric and other horizons so are useful for studying black holes and their interiors. Using these coordinates a stationary spherical aether field takes the form \nu a = a ( r ) ∂ v + b ( r ) ∂ r (25) \nand the unit constraint (10) becomes \nNa 2 -2 Bab = 1 . (26) \nThe field equations (8) and (9) become a set of coupled, second order ordinary differential equations (ODE's) involving the functions N,B,a,b . We use the constraint (26) to solve for b in terms of the other three functions. Even for the reduced case c 13 = c 4 = 0 the equations are sufficiently complicated that it does not seem useful to display them here. \nRegularity at the metric horizon can be imposed by making a power series expansion about the radius r h where N ( r h ) = 0, \nN ( r ) = N ' ( r h )( r -r h ) + 1 2 N '' ( r h )( r -r h ) 2 + · · · (27) \na ( r ) = a ( r h ) + a ' ( r h )( r -r h ) + 1 2 a '' ( r h )( r -r h ) 2 + · · · (29) \nB ( r ) = B ( r h ) + B ' ( r h )( r -r h ) + 1 2 B '' ( r h )( r -r h ) 2 + · · · (28) \nInserting these expansions in the field equations, one can solve order by order for the power series coefficients. This allows the set of free parameters in the initial data at the horizon to be identified. At zeroth order in ( r -r h ) the field equations imply that a ' ( r h ) is a function of N ' ( r h ), B ( r h ), a ( r h ), and r h . The specific result is sufficiently complicated that it too does not seem useful to display here. Solving to higher orders we find that all remaining coefficients in the series expansion are determined by these four initial data parameters. Using the scaling freedom in the v coordinate ( v → λv where λ is a constant) one of the initial values at the horizon can be fixed arbitrarily. Thus, there is a three-parameter family of local solutions with a regular metric horizon. As discussed in [13], we also found a three-parameter family of local solutions expanding about an arbitrary radius (i.e. not imposing a horizon), hence we conclude that regularity of the metric horizon generically imposes no restriction on the solutions.", '4.2. Asymptotic expansion': "In addition to regularity at all the horizons, we require the black hole solutions to be asymptotically flat. To determine the form of such solutions we can change to the inverse radius variable x = 1 /r and expand around x = 0, as was done in [4] (where isotropic coordinates were employed). In the reduced theory (23) this yields the solutions \nN ( x ) = 1 + N 1 x + 1 48 c ' 1 N 3 1 x 3 + · · · (30) \nB ( x ) = 1 + 1 16 c ' 1 N 2 1 x 2 -1 12 c ' 1 N 3 1 x 3 + · · · (31) \na ( x ) = 1 -1 2 N 1 x + a 2 x 2 +( -1 96 c ' 1 N 3 1 + 1 16 N 3 1 -N 1 a 2 ) x 3 + · · · (32) \nwhere N 1 = N ' ( x = 0) and a 2 = a '' ( x = 0), and the freedom to rescale v has been exploited to set N ( x = 0) = 1. No more free parameters appear at higher orders, so the asymptotically flat solutions are determined by the two free parameters N 1 and a 2 . \nAn asymptotically flat solution can be determined by using a simple shooting method, numerically integrating outward from an interior radius where there are three free initial data parameters. As in [13], we find that to match the asymptotic form (30)-(32) requires tuning just one of the three initial parameters, as expected since there are two free parameters in the asymptotic form. We conclude that in particular there is a two-parameter family of asymptotically flat 'black hole' solutions with metric horizon fixed to lie at a given radius r h . In practice, to integrate outward from a metric horizon we found it necessary to first use the perturbative solution about the horizon \nto generate from the horizon data an initial data set some small radial distance away. This is because the ODE's have a singular point at the horizon. \nA more direct way to generate such asymptotically flat black hole solutions is to start the numerical integration near infinity and integrate inward using the inverse radius coordinate x . Since x = 0 is a singular point of the ODE, it is necessary to start the integration at some small non-zero x value. The expansions (30)-(32) can be used to generate initial data as a function of N 1 and a 2 . We find again this way that regularity at the metric horizon does not impose any conditions on N 1 and a 2 . The functions N , B , and a evolve smoothly through a point where N goes to zero.", '4.3. Asymptotically flat solutions and spin-0 horizon regularity': "So far we have shown that there is a two-parameter family of asymptotically flat black hole solutions with a regular metric horizon. Normally we expect just one black hole parameter, the total mass, unless there are conserved charges that can be additional parameters. In ae-theory there seems to be no such conserved charge, so the situation is puzzling. Another puzzling aspect of these black hole solutions not yet discussed here is that some have internal singularities at nonzero radius, rather than just at r = 0 like most known black holes. Moreover, in some solutions we found that such singularities can occur externally, i.e. not inside a metric horizon. All these puzzles are resolved by the recognition that the singularities in question occur precisely at the location of the spin-0 horizon. Imposing regularity at the spin-0 horizon eliminates one free parameter, leaving us with a conventional one-parameter family of asymptotically flat black holes. \nIn the rest of this subsection the full range of behavior of asymptotically flat solutions is discussed. In particular it is demonstrated that when a spin-0 horizon occurs it is singular for generic values of the two initial data parameters N 1 and a 2 at infinity. Evidence is given that by tuning one of these parameters to a special value a regular spin-0 horizon can be obtained. In the next section we show by a power series expansion around the spin-0 horizon that such regular solutions do indeed exist. \nFor the reduced theory (23), the spin-2 and spin-1 speeds are both unity, but the spin-0 mode has squared speed s 2 0 = ( c ' 2 /c ' 1 )(2 -c ' 1 ) / (2 + 3 c ' 2 ) (relative to the aether), which is generically different from unity. At the spin-0 horizon the mode propagates at fixed radius, hence the surface of constant r at that location is null with respect to the effective metric g (0) ab defined in (2). This implies the condition g (0) vv = 0 in EF coordinates, which we find occurs where \nNa 2 = 1 -s 0 1 + s 0 or 1 + s 0 1 -s 0 . (33) \nThe first root is less than 1 so according to (26) occurs when b < 0, i.e. when the aether tips inward. For the second root the aether tips outward. The combination f = Na 2 is independent of the arbitrary scale for the v coordinate and equal to one at infinity. Inserting the expansions (30) and (32) we find the asymptotic form \nf ( x ) = N ( x ) a ( x ) 2 = 1 + 2( a 2 -3 8 N 2 1 ) x 2 -N 1 ( a 2 -3 8 N 2 1 ) x 3 + O ( x 4 ) (34) \nCuriously, this expansion is independent of both c ' 1 and c ' 2 through order x 3 (but not beyond) and depends linearly on a 2 -3 8 N 2 1 through order x 5 . This pattern suggests an analytic solution may be possible, but we will not pursue this here. The static aether solution studied in [13] corresponds to the case where f ( x ) = 1 for all x , which occurs when a 2 /N 2 1 = 3 / 8. \nAs in the previous subsection, we study solutions obtained by integration inwards starting from an asymptotically flat spatial infinity. Since the theory has no length scale, the solution with data ( N 1 , a 2 ) is trivially related to that with data ( λN 1 , λ 2 a 2 ) (as are Schwarzschild solutions with different mass trivially related). If we think of the line element ds 2 as giving a numerical value specified with respect to a given length unit, then to go from one solution to another we need only change the unit of length. Thus without loss of generality we can fix units with N 1 = ± 1. The solutions then depend on the choice of theory through c ' 1 , c ' 2 , on the parameter a 2 , and on the sign of N 1 . A systematic study of these solutions is beyond the scope of this paper; here we just indicate the various behaviors we have encountered, and then focus on the regular positive mass black holes. \nLet us first consider positive mass solutions, i.e. N 1 = -1. As the radial coordinate decreases, N decreases from 1, while the combination Na 2 increases or decreases according as a 2 is greater or less than 3/8. There are solutions where f ( r ) does not reach 0 or (1 ∓ s 0 ) / (1 ± s 0 ) and therefore neither a metric nor spin-0 horizon is attained. In some of these N ( r ) re-curves out to positive infinity, a ( r ) approaches zero and the solution reaches a curvature singularity near r = 0. There are also solutions similar to the static aether of [13], where B ( r ) goes to infinity as N ( r ) approaches a finite value, indicating a minimal area two-sphere. In some cases the larger root for a spin zero horizon in (33) is reached by f ( r ). In other solutions f ( r ) may reach a metric horizon, but does not attain the value corresponding to a spin-0 horizon. This can only happen when s 0 > 1. N ( r ) again re-curves out to infinity and a ( r ) approaches zero near r = 0 and there are outer and inner metric horizons. In still other solutions, the functions and their derivatives are regular up to the point where f ( r ) reaches the spin-0 horizon, but generically the spacetime is singular at that point. If s 0 > 1, f ( r h ) is negative and the singularity is located inside a metric horizon. If s 0 < 1 the singularity occurs without any metric horizon. \nFor a specific example of this last type we choose parameters c ' 1 = 0 . 051 and c ' 2 = 0 . 116, for which the spin-0 speed is 1.37. (These arise from starting with coefficients that satisfy all the observational constraints described in [6] and performing the field redefinition to the reduced action (23).) Figure 1 shows the behavior of N , B , and a for N 1 = -1 and a 2 = -0 . 1. In this case there are 'outer' and 'inner' metric horizons where N = 0, but f ( r ) does not reach as low as -0 . 158, which is required by (33) in this case for a spin-0 horizon. For values of a 2 < -0 . 1 the minimum value of N shifts upward and eventually N never reaches zero, i.e. the metric horizon disappears. On the other hand, for a 2 > -0 . 1 the minimum value of f decreases until the spin-0 horizon is reached. At this point a ' goes to negative infinity, N ' blows up to positive infinity, and \nFigure 1. Solution for reduced theory (23) with c ' 1 = 0 . 051 and c ' 2 = 0 . 116, determined by data N 1 = -1 and a 2 = -0 . 1 at spatial infinity. There is both an outer and inner metric horizon where N vanishes, but f does not decrease enough to reach a spin-0 horizon. The functions N and f go to zero slightly inside r = 1 which would be the horizon radius of the corresponding Schwarzschild solution. As a 2 increases the minimum of f ( r ) decreases until the solution acquires a spin-0 horizon, where is it generically singular. \n<!-- image --> \nthere is a curvature singularity. In contrast, note that in Figure 1 a ' ( r ) has a maximum value while N ' ( r ) goes to negative infinity. This suggests that at some special value of a 2 there is a transition were the concavity of a ' ( r ) and N ' ( r ) changes and the derivatives are finite at a spin-0 horizon. Regularity at the spin-0 horizon seems thus to impose one condition on the asymptotic values N 1 and a 2 . \nIn the negative mass case one might expect only solutions analogous to negative mass Schwarzschild, with N increasing from 1 at spatial infinity and no spin-0 or metric horizon. While the solution does take this form for all a 2 in the theory with c ' 1 = 0 . 051, c ' 2 = 0 . 116, for other values of c ' 1 , 2 there are ranges of a 2 where N increases from 1 at infinity, but then decreases to a metric horizon at finite r , and all the functions and their derivatives are regular until f ( r ) reaches the value less than zero required for a spin-0 horizon. This peculiar behavior of a negative mass solution with metric and spin-0 horizons remains to be studied more closely. In particular, it is not clear whether a negative mass solution with a regular spin-0 horizon could exist.", '5. Black holes with regular spin-0 horizons': 'In this section we discuss the behavior of black hole solutions possessing regular spin-0 horizons. Rather than imposing regularity at the spin-0 horizon by the shooting method \nintegrating in from infinity, we instead expand the field equations in a power series about a non-singular spin-0 horizon.', '5.1. Horizon expansion': "Due to the complexity of the field equations and their singular nature at the horizon, we were unable to implement the power series solution about a spin-0 horizon in the generic reduced theory (23) (even with computer aided algebra). It might be possible to obtain the perturbative solution by a more well-adapted method, but instead we simplified the computation by making a field redefinition to a new metric for which the spin-0 and metric horizons coincide. Starting from an arbitrary set of coefficients c i , this is implemented by the choice σ = s 2 0 in (14), after which we have s 0 = 1 without loss of generality in the theory. As before we can also then exploit spherical symmetry to absorb c 4 by making the replacements (7), which do not disrupt the coincidence of the spin-0 and metric horizons since this is just a re-expression of the same Lagrangian without changing the field variables Σ. This reduces the distinct parameter space to just ( c 1 , c 3 ) (omitting the prime in the notation for c 1 , 3 ). After this field redefinition the coefficient c 2 is given by \nc 2 = -2 c 3 -c 3 1 -2 c 3 c 2 1 -c 1 c 2 3 2 -4 c 1 +3 c 2 1 +3 c 3 c 1 . (35) \nA further simplification of the equations is achieved by trading the metric function N for the combination of metric and aether functions f = Na 2 . We have no insight into why this simplifies the expansion of the field equations about the common metric and spin-0 horizon at N = 0 = f , although as stated above the combination Na 2 is invariant under a rescaling of the v coordinate. The field equations in this set of field variables involve a , a ' , a '' , f , f ' , f '' , B , and B ' . At the horizon f ( r ) vanishes linearly, f ( r ) = f ' ( r 0 )( r -r 0 ) + · · · By a constant rescaling of v we can furthermore set B ( r h ) equal to 1. Using this along with (28) and (29) the field equations can be expanded and solved order by order for the coefficients of the power series. \nSolving the field equations for this theory as algebraic equations for the expansion coefficients we find that at zeroth order in ( r -r h ) the quantities a ( r h ), a ' ( r h ), a '' ( r h ), and f '' ( r h ) are determined by free parameters r h , f ' ( r h ), B ' ( r h ). We succeeded in solving the equations to the next order in ( r -r h ) only in the special cases c 3 = 0, c 3 = c 1 , and c 3 = -c 1 . In these cases we find that B ' ( r h ) is determined by r h and f ' ( r h ). Hence, consistent with the expectation of the previous section, there is a two-parameter family of local solutions around the regular spin-0 horizon. These solutions are generically not asymptotically flat. \nΣ The spin-0 speed is invariant under (7), as guaranteed by this argument. The spin-1 speed is not invariant, but this does not contradict the argument since there is no spherically symmetric spin-1 mode. Note however that in diagnosing whether spin-1 perturbations are trapped in a given black hole it is important to use the value of the spin-1 speed written in (1) before the c 4 coefficient has been absorbed. \nFigure 2. Plots of f , N , B , a , and S (the Schwarzschild version of N ) vs. z = r -r h for c 1 = 0 . 3, in units with r 0 = 2. The horizon radius is r h ≈ 2 . 07 for the ae-theory black hole, so S = 0 at z ≈ -0 . 07. The solutions agree closely outside the horizon. Deviations are noticeable near the horizon and become significant in the interior, where N blows up more rapidly. Near the singularity f begins to oscillate rapidly. \n<!-- image -->", '5.2. Asymptotically flat black holes': "To produce asymptotically flat solutions we numerically integrate outward, starting with the horizon data and matching onto (30), (31), and (32) by tuning f ' ( r h ) until f ( r ) is constant and equal to 1 at very large r values. The asymptotic flatness boundary condition at infinity thus reduces the number of free parameters to one, namely the horizon radius itself. Solutions with different horizon radii are trivially related. Since r h is a singular point of the ODE's, it is necessary to start the integration with initial data at some small positive value of r -r h . We used the series solution determined by a given r h and f ' ( r h ) to generate this initial data. To examine the solution inside the horizon, we numerically integrated inward, starting at a small negative value of r -r h with data generated by the same series solution. \nHere we will discuss the properties of the solutions for the c 3 = 0 theory only, whose behavior is typical of the three special cases c 3 = 0 , ± c 1 . Figure 2 displays the solution for c 1 = 0 . 3, c 2 = -. 025, c 3 = c 4 = 0, together with S ( r ) = 1 -2 /r , the Schwarzschild version of N ( r ) with the same mass. For this plot we use the scaling freedom of v to convert the numerical solution to a 'gauge' where the metric functions and v component of the aether are all equal to 1 at infinity. The two metric functions B ( r ) and N ( r ) in GR and ae-theory are in very close agreement outside the horizon, while inside they differ noticeably. \n5.2.1. Black hole mass The ADM mass M ADM of an asymptotically flat spacetime whose asymptotic metric takes the Schwarzschild form at O (1 /r ) is directly determined by the coefficient r 0 = 2 GM ADM of the O (1 /r ) part of g tt . In ae-theory the relation between M ADM and the total energy E of the spacetime is GM ADM = G N E [17, 9], where G N = G/ (1 -c 14 / 2) is the Newton constant appearing in the force law between two weakly gravitating masses [12]. We shall refer to the quantity r 0 / 2 with dimensions of length as the 'mass' in what follows, and denote it by M . For a Schwarzschild black hole in GR, r 0 is equal to the horizon radius r h . In ae-theory the ratio r 0 /r h is a constant (since there is only one length scale) determined by the coupling coefficients c i . \nThe EF line element (24) transforms to Schwarzschild form \nds 2 = N dt 2 -( B 2 /N ) dr 2 -r 2 d Ω 2 (36) \nwith time coordinate t defined by dt = dv -( B/N ) dr . The asymptotic form of B (31) shows that B = 1 + O (1 /r 2 ), so up through O (1 /r ) the line element (36) has the standard asymptotic form if N and B are normalized to 1 at infinity. In generating an asymptotically flat numerical solution we fixed the scale freedom of the v coordinate by imposing B ( r h ) = 1 at the horizon however, so the asymptotic form of N is N ∞ + N 1 /r + O (1 /r 2 ). The mass is given by M = r 0 / 2 = N 1 / 2 N ∞ , which can be extracted from the numerical solution at large r . \n5.2.2. Horizons The solution displayed in Figure 2 has metric and spin-0 horizons at z = 0, but how about spin-1 and spin-2 horizons? Is the fastest speed actually trapped? The condition for a horizon corresponding to a speed s 0 is given in (33). As the speed approaches infinity the horizon value of f approaches -1 from above. In Figure 2 (and for all values of c 1 that we studied up to 0 . 7), the minimum value of f ( r ) is less than -1, which is sufficient to trap any wave mode. The fact that f ( r ) curves back to being greater than -1 indicates that an inner horizon might exist for some wave modes in certain parameter ranges of c i . \nIn the theory under discussion we have c 3 = c 4 = 0 and c 2 = -c 3 1 / (2 -4 c 1 +3 c 2 1 ), so the squared mode speeds in (1) are given by 1 / (1 -c 1 ) for spin-2 and (1 -c 1 / 2) / (1 -c 1 ) for spin-1. With 0 < c 1 < 1 both of these are greater than 1, and the spin-2 speed is the highest. In the particular case shown in the figure, the spin-1 speed is 1 . 10 and the spin-2 speed is 1 . 20, which correspond to horizons at f = -0 . 049 and f = -0 . 089 respectively, which do not seem to be reached a second time. \n5.2.3. Oscillations A notable aspect of the black hole interior displayed in Figure 2 is the oscillation in f . The function h = Ba also oscillates in a similar manner, but is 180 degrees out of phase. In addition, there are related oscillations in the curvature scalar and aether congruence behavior discussed below. These oscillations are reminiscent of the interior behavior found in Einstein-Yang-Mills black holes [18], where the metric functions and derivative of the Yang-Mills potential oscillate an infinite number of times before the singularity. \nFigure 3. Oscillations of f = Na 2 near the singularity inside the black hole solution shown in figure 2, plotted vs. z = r -r h in units with M = 1. \n<!-- image --> \nWhile N decreases monotonically and B increases monotonically, a goes to zero, so the oscillations of Na 2 and Ba arise because of variations in the magnitude of their derivatives. Since the oscillations inside z = -2 are not clearly visible in Figure 2 a zoomed in graph of f ( z ) is provided in Figure 3. From this graph it is clear that f smoothly turns over at least once more before the singularity. Although the number of oscillations before the singularity appears finite, it is possible that the numerical integration employed is not capable of resolving additional or even infinitely more oscillations. More information may be obtained in the future by improved numeric methods or analytic methods around r = 0. \n5.2.4. Curvature singularity There appears to be a spacelike curvature singularity at or near r = 0, as in the Schwarzschild solution of GR. In Figure 2, the approach of N to negative infinity near r = 0 suggests a singularity. In Figure 4 the logarithm of the Kretschmann scalar K = R abcd R abcd is plotted vs. ln r for the ae-theory solution together with its value in the corresponding Schwarzschild solution with the same mass. In the latter case K = 48 /r 6 in units with M = 1, so log K = -6 ln r +ln48. The rate d ln K/d ln r for the ae-theory solution seems to alternate between roughly -6 and -4 . 5. The location of the transitions may be correlated with the oscillations discussed above. The location of the singularity seems to be at r = 0 for all the values of c 1 , although our numerical solutions do not permit a determination of the exact location. \nFigure 4. Plot of ln R abcd R abcd vs. ln r for c 1 = 0 . 3 ae-theory black hole (wiggly curve) and Schwarzschild black hole (stright line) of the same mass, in units with M = 1. r is plotted on a logarithmic scale. The slope for the GR case is -6, while for the ae-theory case it alternates between roughly -6 and -4 . 5. \n<!-- image --> \n5.2.5. Aether congruence The aether field defines a congruence of radial timelike curves at rest at infinity and flowing into the black hole. It is interesting to compare this with the static frame and with the congruence of freely falling radial geodesics with 4-velocity v a that are also at rest at infinity. Being unit vector fields, at each point u a and v a can be fully characterized by their Killing energy, i.e. their inner product with the Killing vector. The free-fall congruence has a conserved energy that is equal to one if the Killing vector is normalized to one at infinity. The aether does not fall as quickly outside the black hole. In fact it remains rather aligned with the Killing vector up until quite close to the horizon. \nTo characterize and contrast the free-fall and aether congruences we plot in Figure 5 the derivative dr/dτ of radius with respect to proper time along each congruence. Let us call this the quantity the 'proper velocity'. ‖ The aether and free-fall are both at rest at infinity, but only as the horizon is approached is the aether finally pulled away from the Killing direction. As close as r = 3 r h ( z ≈ 4), the proper velocity of the aether \nFigure 5. Radial proper velocity dr/dτ of free-fall (lower, solid curve) and aether (upper, dashed curve) vs. z = r -r h , in units with M = 1, for c 1 = 0 . 3. In contrast to the free-falling geodesics, the aether does not begin to fall significantly inward until close to the horizon. \n<!-- image --> \nz \nis still about fifteen times smaller than that of free-fall. Inside the horizon the aether proper velocity is equal to the free-fall one around z = -1 . 3, but the 4-velocities do not agree there. The aether is still going inward faster, but its proper time is 'running slower' so it can have the same proper velocity. \nTo compare the aether and free-fall motions inside the horizon we plot in Figure 6 the inward 3-velocity of the aether with respect to the free-falling frame. The relative velocity is initially negative, meaning that the aether is not falling in as fast as the the free-fall frame. It is clear from this plot at around z = -1 . 3 the aether still lags well behind free-fall. However, around z = -1 . 9 the relative velocity is zero, and after that it oscillates a couple of times (at least) before reaching the singularity. \n5.2.6. Surface gravity and the first law of black hole mechanics The laws of black hole mechanics have been shown to apply to a wide class of generally covariant metric theories of gravity coupled to matter [19]. There appears to be no straightforward extension of the first law and the concept of black hole entropy to ae-theory however [17], a difficulty that is tied to the fact that there is no smooth extension of the aether to the bifurcation surface of the Killing horizon. Moreover, it is not clear to which horizon the law should apply, in a theory with multiple characteristic surfaces. For example, in the solutions considered in this section, the spin-2 horizon is inside the spin-1 horizon which is inside the joint spin-0 and metric horizon. One might imagine that the relevant horizon is \nFigure 6. Inward 3-velocity of the aether relative to free-fall inside the horizon for c 1 = 0 . 3. The velocity is initially negative and the aether lags behind the free-fall. Near the singularity the velocity oscillates between between faster and slower than free-fall. \n<!-- image --> \nalways the Killing horizon, but recall that by a field redefinition we can make any one of these horizons be the Killing horizon. We shall not try to shed any light on these puzzling issues here. Rather, we just briefly examine the variational relation between mass, surface gravity and area of the spin-0 horizon, for possible future use. \nThe first law of black hole mechanics for spherically symmetric neutral black holes in GR takes the form \nδM = α κδA 8 πG , (37) \nwhere A = 4 πr 2 h is the horizon area, κ is the surface gravity, and α = 1. By dimensional analysis such a variational relation must also hold in ae-theory, with some value for the dimensionless constant α that depends on the dimensionless coupling coefficients c i . Presumably for M we should put the total energy E of the spacetime, and for ' G ' we should put the Newton constant G N governing the attractive force between distant bodies. Alternatively one might use the ADM mass M ADM and the constant G appearing in the ae-theory action (3). As discussed in section 5.2.1, GM ADM = G N E = r 0 / 2, so these two choices actually yield identical 'first laws'. If we express the mass and area in terms of r 0 and r h respectively, (37) thus becomes \nδr 0 = 2 ακr h δr h . (38) \nAs pointed out in section 5.2.1, r 0 and r h are proportional, so we infer that α is determined by the dimensionless combination \nα = r 0 2 κr 2 h , (39) \nTable 1. Properties of black hole solutions for several c 1 values, in units with r h = 1. \n| c 1 | f ' ( r h ) | γ = u a v a | r 0 | κ | α |\n|-------|---------------|---------------|-------|-------|-------|\n| 0.1 | 2.096 | 1.619 | 0.99 | 0.507 | 0.976 |\n| 0.2 | 2.072 | 1.608 | 0.979 | 0.517 | 0.947 |\n| 0.3 | 2.039 | 1.592 | 0.966 | 0.528 | 0.914 |\n| 0.4 | 1.997 | 1.568 | 0.951 | 0.543 | 0.876 |\n| 0.5 | 1.941 | 1.535 | 0.933 | 0.562 | 0.83 |\n| 0.6 | 1.867 | 1.49 | 0.911 | 0.588 | 0.787 |\n| 0.7 | 1.767 | 1.429 | 0.881 | 0.625 | 0.704 | \nwhich depends on the coefficients c i defining the theory. \n5.2.7. Black hole properties for different values of c 1 Various properties of the black hole solutions for different values of c 1 are displayed in Table 1. The other coupling coefficients have the values c 3 = c 4 = 0 and c 2 is given by (35). For each c 1 there is a one-parameter family of black hole solutions with regular spin-0 horizon, labeled by horizon radius. For the values in the table we compare black holes with the same horizon radius, and adopt units with r h = 1. The Killing vector which enters the definition of κ and α is normalized to unity at spatial infinity. \nThe values of f ' ( r h ) that yield asymptotically flat solutions for different choices of c 1 are displayed in the 2nd column. These values decrease as c 1 grows. For c 1 = 0 . 8 and larger we could not find a f ' ( r h ) that yielded an asymptotically flat solution. The third column shows the gamma factor between the aether and free-fall velocity at the horizon. The fourth column shows r 0 = 2 GM ADM . This is equal to r h for c 1 = 0 (a Schwarzschild black hole), and decreases by 12% as c i increases up to 0.7. Conversely, for a given mass the black hole horizon is larger for larger c 1 . The fifth column shows the surface gravity, which for c 1 = 0 is 1 / 2 r h and increases by 25% as c 1 increases up to 0 . 7. The last column is the dimensionless ratio (39) appearing in the first law (37), which is unity for c 1 = 0 and decreases by 30% as c 1 increases to 0.7.", '6. Discussion': 'In this paper we considered the meaning of a black hole in Einstein-Aether theory, arguing that the fastest wave mode must be trapped if the configuration is to qualify as a causal black hole. Regularity at the spin-0 horizon was identified as a key property of black holes in ae-theory. It was found that, for generic values of the coupling constants c i , regularity at a metric horizon imposes no restrictions on spherically symmetric, static local solutions but regularity at a spin-0 horizon imposes one condition. At least for a class of coupling constants, there is a one parameter family of asymptotically flat black hole solutions with all horizons (metric and spin-0,1,2) regular. \nFrom an astrophysical point of view an essential question is what happens when \nmatter collapses. It is a plausible conjecture that nonsingular spherically symmetric initial data will evolve to one of the regular black holes whose existence has been demonstrated here, but this has certainly not been shown. It would be very interesting to answer this question by numerical evolution of the time-dependent field equations. To do so, one could add scalar matter to form a collapsing pulse, but this is likely not necessary since the aether itself has a spherically symmetric radial tilt mode that can serve the purpose. This would correspond to formation of a black hole by an imploding spherical aether wave. \nWe examined several properties of regular black hole solutions for a special class of coupling coefficients defining the ae-theory. A complete classification of the solutions for different coefficients remains an open research problem, as does the study of non-black hole solutions and negative mass solutions, which strangely enough could include black holes. Oscillating behavior approaching the internal singularity has been identified, but not studied in detail. Finally, only the static, spherically symmetric case was examined; the question of rotating black hole solutions remains untouched.', 'Acknowledgments': 'We are grateful to B.Z. Foster for helpful suggestions on the presentation. This research was supported in part by the NSF under grant PHY-0300710.', 'References': '- [1] Eling C, Jacobson T and Mattingly D 2006 Einstein-aether theory Deserfest ed J Liu, M J Duff, K Stelle, and R P Woodard (Singapore: World Scientific) ( Preprint gr-qc/0410001).\n- [2] Kostelecky V A 2004 Gravity, Lorentz violation, and the standard model Phys. Rev. D 69 105009 ( Preprint hep-th/0312310).\n- [3] Bailey Q G and Kostelecky V A 2006 Signals for Lorentz violation in post-Newtonian gravity Phys. Rev. D, to appear ( Preprint gr-qc/0603030).\n- [4] Eling C and Jacobson T 2004 Static post-Newtonian equivalence of GR and gravity with a dynamical preferred frame Phys. Rev. D 69 064005 ( Preprint gr-qc/0310044).\n- [5] Graesser M L, Jenkins A and Wise M B 2005 Spontaneous Lorentz violation and the long-range gravitational preferred-frame effect Phys. Lett. B 613 5 ( Preprint hep-th/0501223).\n- [6] Foster B Z and Jacobson T 2006 Post-Newtonian parameters and constraints on Einstein-aether theory Phys. Rev. D 73 , 064015 ( Preprint gr-qc/0509083).\n- [7] Jacobson T and Mattingly D 2004 Einstein-aether waves Phys. Rev. D 70 024003 ( Preprint gr-qc/0402005).\n- [8] Lim E A 2005 Can we see Lorentz-violating vector fields in the CMB? Phys. Rev. D 71 063504 ( Preprint astro-ph/0407437)\n- [9] Eling C 2006 Energy in the Einstein-aether theory Phys. Rev. D 73 084026 ( Preprint gr-qc/0507059).\n- [10] Foster B Z 2006 Radiation damping in Einstein-aether theory Phys. Rev. D 73 104012 ( Preprint gr-qc/0602004).\n- [11] Elliott J W, Moore G D and Stoica H 2005 Constraining the new aether: Gravitational Cherenkov radiation JHEP 0508 , 066 ( Preprint hep-ph/0505211).\n- [12] Carroll S M and Lim E A Lorentz-violating vector fields slow the universe down 2004 Phys. Rev. D 70 123525 ( Preprint hep-th/0407149). \n- [13] Eling C and Jacobson T 2006 Spherical Solutions in Einstein-Aether Theory: Static Aether and Stars Class. Quantum Grav. , to appear ( Preprint gr-qc/0603058).\n- [14] Courant R and Hilbert D 1962 Methods of Mathematical Physics , (New York: Interscience Publishers).\n- [15] Racz I and Wald R M 1996 Global extensions of space-times describing asymptotic final states of black holes Class. Quantum Grav. 13 539 ( Preprint gr-qc/9507055).\n- [16] Foster B Z 2005 Metric redefinitions in Einstein-aether theory Phys. Rev. D 72 044017 ( Preprint gr-qc/0502066).\n- [17] Foster B Z 2006 Noether charges and black hole mechanics in Einstein-aether theory Phys. Rev. D 73 024005 ( Preprint gr-qc/0509121).\n- [18] Donets E E, Galtsov D V and Zotov M Y 1997 Internal structure of Einstein Yang-Mills black holes Phys. Rev. D 56 3459 ( Preprint gr-qc/9612067).\n- [19] Wald R M 2001 The thermodynamics of black holes Living Rev. Rel. 4 6 ( Preprint gr-qc/9912119).'}
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2015MNRAS.448.1504S
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A refined sub-grid model for black hole accretion and AGN feedback in large cosmological simulations
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2015-01-01
| 37 | 0.51 | 162 |
['black hole physics', 'methods numerical', 'galaxies active', 'galaxies evolution', 'galaxies nuclei', 'galaxies quasars', '-']
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In large-scale cosmological hydrodynamic simulations simplified sub-grid models for gas accretion on to black holes and AGN feedback are commonly used. Such models typically depend on various free parameters, which are not well constrained. We present a new advanced model containing a more detailed description of AGN feedback, where those parameters reflect the results of recent observations. The model takes the dependence of these parameters on the black hole properties into account and describes a continuous transition between the feedback processes acting in the so-called radio-mode and quasar-mode. In addition, we implement a more detailed description of the accretion of gas on to black holes by distinguishing between hot and cold gas accretion. Our new implementations prevent black holes from gaining too much mass, particularly at low redshifts, so that our simulations are successful in reproducing the observed present-day black hole mass function. Our new model also suppresses star formation in massive galaxies slightly more efficiently than many state-of-the-art models. Therefore, the simulations that include our new implementations produce a more realistic population of quiescent and star-forming galaxies compared to recent observations, even if some discrepancies remain. In addition, the baryon conversion efficiencies in our simulation are - except for the high-mass end - consistent with observations presented in the literature over the mass range resolved by our simulations. Finally, we discuss the significant impact of the feedback model on the low-luminous end of the AGN luminosity function.
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https://arxiv.org/pdf/1409.3221.pdf
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{'A refined sub-grid model for black hole accretion and AGN feedback in large cosmological simulations': "Lisa K. Steinborn 1 glyph[star] , Klaus Dolag 1 , 2 , Michaela Hirschmann 3 , M. Almudena Prieto 4 , 5 , Rhea-Silvia Remus 1 \n- 1 Universitats-Sternwarte Munchen, Scheinerstr.1, D-81679 Munchen, Germany\n- 2 Max-Plank-Institut fur Astrophysik, Karl-Schwarzschild Strasse 1, D-85740 Garching, Germany\n- 3 UPMC-CNRS, UMR7095, Institut d'Astrophysique de Paris, Boulevard Arago, F-75014 Paris, France\n- 4 Instituto de Astrof'ısica de Canarias (IAC), V'ıa L'actea s/n, La Laguna, E-38200, Spain\n- 5 Departamento de Astrof'ısica, Facultad de F'ısica, Universidad de La Laguna, Astrof'ısico Fco. S'anchez s/n, La Laguna, E-38207, Spain \nAccepted 2015 January 13. Received 2014 December 16; in original form 2014 September 10", 'ABSTRACT': 'In large scale cosmological hydrodynamic simulations simplified sub-grid models for gas accretion onto black holes and AGN feedback are commonly used. Such models typically depend on various free parameters, which are not well constrained. We present a new advanced model containing a more detailed description of AGN feedback, where those parameters reflect the results of recent observations. The model takes the dependency of these parameters on the black hole properties into account and describes a continuous transition between the feedback processes acting in the so-called radio-mode and quasar-mode. In addition, we implement a more detailed description of the accretion of gas onto black holes by distinguishing between hot and cold gas accretion. Our new implementations prevent black holes from gaining too much mass, particularly at low redshifts, so that our simulations are successful in reproducing the observed present-day black hole mass function. Our new model also suppresses star formation in massive galaxies slightly more efficiently than many state-of-the-art models. Therefore, the simulations that include our new implementations produce a more realistic population of quiescent and star-forming galaxies compared to recent observations, even if some discrepancies remain. In addition, the baryon conversion efficiencies in our simulation are - except for the high mass end consistent with observations presented in literature over the mass range resolved by our simulations. Finally, we discuss the significant impact of the feedback model on the low-luminous end of the AGN luminosity function. \nKey words: black hole physics, methods: numerical, galaxies: active, galaxies: evolution, galaxies: nuclei, quasars: supermassive black holes', '1 INTRODUCTION': "Black holes play an essential role in the formation and evolution of galaxies. They can even influence galaxy clusters and the intra cluster medium (ICM). However, observations of active galactic nuclei (AGN) indicate that gas accretion onto black holes and AGN feedback are complex processes, which are not yet fully understood (e.g. Merloni & Heinz 2007, McNamara et al. 2011, Ma et al. 2013). There is evidence for two distinct phases of AGN activity and feedback: the radio-mode and the quasar-mode. The radio-mode is characterized by large radio jets generating hot X-ray cavi- \nties (Russell et al. 2013, Mezcua & Prieto 2014), whereas in the quasar-mode the emission is dominated by the accretion disc, which is visible as the so-called blue bump in the spectrum of quasars and Seyfert galaxies (e.g. Elvis et al. 1994, Prieto et al. 2010). \nChurazov et al. (2005) characterized this distinction in a theoretical model by describing AGN feedback with two components: radiation and mechanical outflow. In their model the amount of energy associated with each component depends on the Eddington ratio f Edd = ˙ M · / ˙ M Edd . When a black hole accretes with the Eddington accretion rate ˙ M Edd , gas cooling and AGN feedback are in equilibrium. Churazov et al. (2005) also took advection-dominated accretion flows (ADAFs) into account, although a jet con- \ntion can successfully replace an ADAF (Falcke et al. 2004, Fern'andez-Ontiveros et al. 2011). \nTo constrain this model and to really understand the origin of different types of AGN and how they influence their environment, large cosmological simulations play a key role. They have two major advantages: firstly, they provide a statistically large sample of black holes. This allows to compare the simulations to the newest and currently most complete observations of the M · -M ∗ relation (e.g. McConnell & Ma 2013) or black hole mass functions (e.g. Marconi et al. 2004, Shankar et al. 2004, Shankar et al. 2009) and stellar mass functions (e.g. Muzzin et al. 2013, Bernardi et al. 2013), in particular the very massive end. Secondly, having large enough cosmological boxes where also massive galaxy clusters form, allows to probe the influence of black holes across all scales of cosmic environment. \nThere already exist a number of studies discussing large cosmological simulations that include black holes (e.g. Di Matteo et al. 2005, Di Matteo et al. 2008, Robertson et al. 2006, Teyssier et al. 2011, Degraf et al. 2011, Booth & Schaye 2009, Khandai et al. 2014, Rosas-Guevara et al. 2013, Hirschmann et al. 2014, Vogelsberger et al. 2014, Schaye et al. 2015). Those simulations mostly use the black hole model implemented by Springel et al. (2005) or are based on it. In these models - in contrast to some more simplified black hole models (e.g. Battaglia et al. 2010) black holes are typically described as sink particles which have fundamental properties like mass and accretion rate, which can be linked directly to observables. Hence, we can study black hole growth and the co-evolution between black holes and their host galaxies to constrain and improve the parametrization of the underlying model. In the model from Springel et al. (2005) the gas accretion onto black holes is calculated according to the Bondi formula (Hoyle & Lyttleton 1939, Bondi 1952, Bondi & Hoyle 1944), multiplied by a so-called boost factor α . This factor was introduced to account for the limited resolution in simulations leading to smaller densities and larger temperatures near the black hole (Booth & Schaye 2009). To estimate the AGN feedback, a constant value for the radiative efficiency is typically used (Shakura & Sunyaev 1973). \nFor low resolutions this model works reasonably well. However, to study not only the origin of the observed fundamental relations between black holes and their host galaxies (Haring & Rix 2004, Tremaine et al. 2002, McConnell & Ma 2013), but also the impact of gas accretion and AGN feedback on the morphology of the galaxy, simulations with higher resolution are needed. Until now, this was only studied in simulations of isolated galaxies and mergers of galaxies (e.g. by Hopkins et al. 2008, Debuhr et al. 2011, Van Wassenhove et al. 2014, Capelo et al. 2015) as well as in cosmological zoom simulations (e.g. by Angl'es-Alc'azar et al. 2013, Marinacci et al. 2014, Dubois et al. 2013, Choi et al. 2014). To reproduce both statistical black hole and galaxy properties within a fully cosmological context and across various environments in a statistically relevant sample size, large cosmological boxes with high resolution are needed. This is still a challenge, but thanks to increasing computational power it now becomes feasible. However, despite of this success, new challenges arise as simulations typically over-estimate the high-mass end of the black hole and stellar mass function (e.g. Sijacki et al. 2014, Khandai et al. 2014, \nVogelsberger et al. 2014, Genel et al. 2014, Hirschmann et al. 2014). Therefore, a more detailed black hole model is necessary. \nIn this work we extend the model by Springel et al. (2005) by improving the treatment of the two modes of AGN feedback: radiation and mechanical outflows. Following theoretical predictions (Churazov et al. 2005, White & Frenk 1991, Narayan & Yi 1995) as well as recent observational results (Davis & Laor 2011, Chelouche 2013, Russell et al. 2013) gives us estimates for the corresponding two efficiencies depending on the black hole mass and the accretion rate, which outreaches the simplified black hole model commonly used in simulations. \nFollowing Sijacki et al. (2007), a steep transition between radio-mode and quasar-mode is often used in current simulations (e.g. Fabjan et al. 2010, Hirschmann et al. 2014). This is only a rough approximation to the smooth transition which is observed and also theoretically expected. Adopting the model by Churazov et al. (2005) - which was already constrained by observations, e.g. Russell et al. 2013 - allows us to get a smooth transition between the two modes. This was used by Hirschmann et al. (2014) to calculate AGN luminosities, but it was never implemented into simulations. Such modifications were also suggested by a recent paper of Sijacki et al. (2014), who studied the AGN luminosity function within a cosmological simulation using a constant radiative efficiency. They concluded that in the radio-mode radiative efficiencies might depend on the accretion rate and on average should be lower than the value 0.1 used in the original black hole model from Springel et al. (2005). Furthermore, Davis & Laor (2011) and Chelouche (2013) found that the radiative efficiency not only correlates with the accretion rate, but also with the black hole mass. \nAnother deficiency in current implementations of black holes in cosmological simulations is that the (original) Bondi model predicts far too low accretion rates during the quasarmode so that black holes do not reach the observed masses for a given bulge mass. Therefore, a so-called boost factor is commonly used to artificially raise the accretion rates. This results in realistic accretion rates for the accretion of cold gas. However, it has the disadvantage that it also raises the accretion rate when the hot gas content is large enough to fulfil the assumptions of the Bondi model, namely when the gas is distributed in an isotropic sphere. This typically is the case in old quiescent galaxies. Consequently, black holes become too massive at low redshifts. Hence, accretion rates have to be lower in the radio-mode (Li et al. 2013). \nIndeed, several studies adapt the black hole model for higher resolution simulations by using a boost factor which depends on the resolution (Choi et al. 2012, Choi et al. 2014), density (Booth & Schaye 2009), pressure (Vogelsberger et al. 2013) or angular momentum (Rosas-Guevara et al. 2013), although none of them contains a direct distinction between the accretion of cold and hot gas, even if the existence of such two distinct accretion modes has been shown by observations (e.g. Hlavacek-Larrondo et al. 2013) and predicted by high-resolution simulations of black hole accretion on sub-kpc scales (Gaspari et al. 2013, Bourne et al. 2014) as well as semi-analytical models (e.g. Somerville et al. 2008, Hirschmann et al. 2012, Fanidakis et al. 2011, Fanidakis et al. 2013). A distinction between accretion of cold and hot gas based on the multi-phase model from Springel & \nHernquist (2003) was implemented in the simulations from Pelupessy et al. (2007). In their study, the molecular gas of the star forming particles was evaluated from a multi-phase model, in which the accretion of this cold gas was evaluated separately without any boost factor, assuming the corresponding temperature as fixed in the underlying multi-phase model. \nA black hole mainly grows in the quasar-mode, where cold gas forms an accretion disc around the black hole which leads to higher accretion rates. During that period, black holes grow until the AGN feedback and gas cooling are in equilibrium. At that point, they reach the M · -σ relation (Churazov et al. 2005) and thus, the M · -M ∗ relation. Consequently, the accretion rate drops until the black hole crosses the threshold towards the radio-mode. As reviewed by several authors (e.g. Yuan & Narayan 2014, Heckman & Best 2014), the accretion in the radio-mode, sometimes also called jet-mode, can be described with ADAFs containing hot gas (Yuan et al. 2009). Alternatively, the accretion of hot adiabatic gas can be described with the Bondi model (Gaspari et al. 2013). Therefore, we distinguish between hot and cold gas and estimate the accretion rate separately for both gas phases. This allows us to use different boost factors for hot and cold gas and thus, to account for both observed accretion modes. \nThe outline of this paper is as follows: in section 2 we describe our black hole model. The set-up of the cosmological simulations is presented in section 3. In section 4, adopting different models for black hole accretion and AGN feedback, we show the results for our simulations, in particular the evolution of the black hole mass, the stellar mass and the star formation rate. In section 5 we discuss the radiative efficiency in the radio-mode and its influence onto the AGN luminosity functions. Furthermore, we compare our results with other cosmological simulations. Finally, in section 6, we summarize our main results.", '2.1 Black hole accretion': 'The Bondi model is commonly used in simulations to estimate the black hole accretion rate. The Bondi accretion rate ˙ M B (Bondi 1952, Shima et al. 1985) is given by \n˙ M B = 4 πG 2 M 2 · ρ ∞ ( v 2 + c 2 s ) 3 / 2 , (1) \nwhere M · is the black hole mass, ρ is the density, c s is the sound speed of the accreted gas and v is the velocity of the gas relative to that of the black hole. Since Bondi (1952) assumed an isotropic and isothermal sphere of gas for his estimation, it is not straight forward to adopt this Bondi accretion model for hydrodynamic, cosmological simulations aiming to follow a self consistent accretion history of black holes. For the implementations based on Springel et al. (2005), the accretion rate of the black hole is estimated by \n˙ M B = 4 παG 2 M 2 · 〈 ρ 〉 ( 〈 c s 〉 2 + 〈 v 〉 2 ) 3 / 2 , (2) \nwhere 〈 ρ 〉 , 〈 v 〉 and 〈 c s 〉 are computed using kernel weighted SPH estimations. Due to limited numerical resolution in \nsuch simulations, the original equation (1) is multiplied by a boost factor α , which in Springel et al. (2005) is set to a value of α = 100. Note that the SPH estimates also depend on the type of SPH kernel and the number of neighbours. To make this estimation less sensitive to the actual structure of the multi phase media in the vicinity of the black hole and therefore the algorithm less dependent on resolution and on the actual choice of numerical parameters for the kernel weighted interpolation, Choi et al. (2012) suggested to use a different way of building the averages: \n˙ M B = 〈 4 παG 2 M 2 · ρ ( c 2 s + v 2 ) 3 / 2 〉 . (3) \nStill, choosing the correct value for the boost factor α is not trivial. Since due to the limited resolution the density in the not resolved vicinity of black holes is large, it will be underestimated and - in turn - the temperature (and thus the sound speed) will be overestimated. Following this argument, Booth & Schaye (2009) parametrize α , which is chosen to be α = 1 as long as the density is below the critical value where one can assume the gas to be in the hot phase. For larger densities, when gas is accreted mainly in a cold phase, α increases with density. Alternatively, Vogelsberger et al. (2013) have presented a recipe for modelling α based on the equilibrium between cooling losses and AGN feedback. However, both models do not directly account for the different accretion modes of hot and cold gas phase, where cold gas usually is accreted in turbulent streams, whereas hot gas indeed can be assumed to be isotropic and isothermal. \nIn our model, we use a sixth-order Wendland kernel (Dehnen & Aly 2012) with 295 neighbours, building the mean values according to equation (2) and directly distinguishing between the accretion of hot and cold gas. In this way, we can safely use the original estimate of building the averages, which has the advantage to be more sensitive to density structures close to the black hole. In general, we assume hot gas has temperatures above T ≈ 10 6 K , whereas cold gas has temperatures below T ≈ 10 5 K (Gaspari et al. 2013). Since we do not account for a third warm phase, we choose T = 5 · 10 5 K as threshold between hot and cold gas. In contrast to Pelupessy et al. (2007), who use the molecular fraction of the gas for star-forming particles from the multiphase model (Springel & Hernquist 2003) to account for cold gas accretion, we also assign gas with a temperature below our threshold in addition to the star forming gas to the cold phase. For both gas phases the accretion rate is calculated separately according to equation 2, but with different values for α according to the result by Gaspari et al. (2013), who argue that due to turbulence the assumptions of the Bondi model are not fulfilled for the cold gas. When they include cooling and turbulence in their simulation, they find an accretion rate which is around 100 times larger than the Bondi accretion rate. Interestingly, this is the same value which is used as boost factor α in the original model from Springel et al. (2005). But for adiabatic accretion, the difference, Gaspari et al. (2013) find, is about one order of magnitude smaller. Hence, we use α = 10 for hot gas and α = 100 for cold gas. \nFurthermore, the black hole accretion rate ˙ M · is limited to the Eddington accretion rate \n˙ M Edd = 4 πGM · m p η Edd σ T c , (4) \nwhere m p is the proton mass, σ T the Thompson scattering cross section and η Edd the feedback efficiency if the black hole would accrete with ˙ M Edd . Then the accretion rate is given by: \n˙ M · = min( ˙ M B , hot + ˙ M B , cold , ˙ M Edd ) . (5) \nThe distinction between hot and cold gas accretion leads to a faster black hole growth in the quasar-mode, because when calculating the mean value of the sound speed 〈 c s 〉 and the gas velocity 〈 v 〉 only for cold gas, the accretion rate estimated with equation (2) is higher than calculating the mean values of both cold and hot gas together. This solves the well known problem of too low gas accretion, which was addressed in other simulations by increasing the maximum accretion rate to a few times ˙ M Edd (e.g. Di Matteo et al. 2012), which is not needed in our simulations.', '2.2 AGN feedback': "In the commonly used black hole model by Springel et al. (2005), the feedback energy per unit time is calculated as \n˙ E = glyph[epsilon1] f glyph[epsilon1] r ˙ M · c 2 , (6) \nwhere glyph[epsilon1] f is the efficiency with which the energy radiated from the black hole is coupled to the ISM (Springel et al. 2005, Booth & Schaye 2009) and glyph[epsilon1] r is the radiative efficiency. \nThe original model as used in Hirschmann et al. (2014) is simplified, since it uses a constant radiative efficiency and thus does not allow for a smooth transition between quasarand radio-mode. Furthermore, it neglects mechanical feedback, which was already implemented in other simulations as AGN driven winds (i.e. Choi et al. 2014). To account for both mechanical and radiative feedback, we adopt a new feedback scheme based on Churazov et al. (2005). In this study, they propose that AGN feedback can be split up into two components: \n(i) Outflow: The outflow component is a mechanical feedback which dominates at accretion rates below ∼ 0 . 01 ˙ M Edd and diminishes at accretion rates above ∼ 0 . 1 ˙ M Edd . The corresponding gas heating power is given by: \nP o = glyph[epsilon1] o ˙ M · c 2 , (7) \nwhere glyph[epsilon1] o is the outflow efficiency. \n(ii) Radiation: The radiative component dominates near the Eddington limit ( f Edd > 0 . 1) and has the luminosity \nL = glyph[epsilon1] r ˙ M · c 2 . (8) \nWe implement both radiative and mechanical AGN feedback as thermal feedback due to the inability to resolve the sub-kpc scales, where the jets provide the mechanical feedback. The feedback energy per unit time in this model is then the sum of P o and the fraction glyph[epsilon1] f of the luminosity: \n˙ E = ( glyph[epsilon1] o + glyph[epsilon1] f glyph[epsilon1] r ) ˙ M · c 2 . (9) \nThe effect of accreted matter can be split into outflow and radiation components: \n˙ M · ˙ M Edd = P o L Edd + L L Edd , (10) \nFigure 1. The lines show the predictions by Churazov et al. 2005 (C05) for the power of the radiation (red line), the mechanical outflow (blue line) and the sum of both (black dashed line). Observations of jet powers (blue errorbars and edges) and luminosities (red errorbars and edges) constrain the difference between both components. This figure includes two different observations: The big stars and squares show recent observations by Mezcua & Prieto 2014 (MP14) and the data with blue and black errorbars are observations by Russell et al. 2013 (R13). Black triangles mark upper limits. Furthermore, the black hole masses are indicated by the colors of the symbols. Since the masses used by R13 are based on K-band magnitudes, which are known to be inaccurate, we used the dynamical masses by McConnell & Ma (2013) for the sources included in both samples. \n<!-- image --> \nwhere the Eddington accretion rate \n˙ M Edd = L Edd η Edd c 2 (11) \ndepends on the total efficiency \nη := glyph[epsilon1] o + glyph[epsilon1] r . (12) \nThis model is shown as solid lines (blue corresponds to mechanical outflow and red to radiation) in Fig. 1, which were adopted from Churazov et al. (2005). For the outflowdominated regime they assume \nL L Edd = 10 · ( ˙ M · ˙ M Edd ) 2 (13) \nas a lower limit for the radiation, which is a consequence of advection-dominated accretion flows (Narayan & Yi 1995). In the radiation-dominated regime the outflow decreases with the Eddington ratio: \nP o L Edd = 10 -4 · ( ˙ M · ˙ M Edd ) -1 . 8431 . (14) \nThis guarantees that the minimum value for the outflow efficiency is glyph[epsilon1] o = 10 -5 , which was calculated by Churazov et al. (2005) assuming that gas cooling and AGN feedback balance each other at the Eddington limit. We choose ˙ M · ˙ M Edd = 0 . 05 as the threshold between radio and quasar mode. The value \nfor the outflow at ˙ M · ˙ M Edd = 1 follows the calculations of Churazov et al. (2005), who find glyph[epsilon1] o ≈ 10 -5 for black holes accreting with the Eddington accretion rate. \nThe feedback model of Churazov et al. (2005) was recently confirmed by observations (see also Russell et al. 2013) measuring luminosities and cavity powers of a large sample of unresolved nuclear X-ray sources. Most of the selected brightest cluster galaxies (BCGs) have large X-ray cavities. The data from Russell et al. (2013) show a large scattering of the luminosities in the radio regime illustrated by round filled circles with black errorbars in Fig. 1, implying that a secondary quantity influences the luminosity. A few data points are below the theoretical lower limit, albeit the uncertainties in the observations are relatively high. Uncertainties can occur, for example, when measuring the cavity volume due to projection effects. In Fig. 1, the black hole masses are color-coded as indicated by the colorbar. The masses from Russell et al. (2013) are based on K-band magnitudes, which is known to be problematic. Therefore, we use the dynamical masses from McConnell & Ma (2013) for the sources included in both samples. Nearly all black holes that lie below the prediction are very massive ( > 10 9 M glyph[circledot] ). For lower masses, the observations are in better agreement with the predictions. We will discuss the uncertainties in section 5.2 in more detail. \nRecently, Mezcua & Prieto (2014) presented measurements of luminosities of a much smaller sample of AGN, but with sufficiently larger angular resolution and sensitivity. Their estimations for L bol are more reliable than those presented in Russell et al. (2013), because they measure L bol after integrating the radio to X-ray Spectral Energy Distribution (SED). Furthermore they explicitly provide values for X-ray cavity powers. For CenA, M87 and NGC1052, they used X-ray cavities of maser emission from the literature (Prieto et al. 2010, Russell et al. 2013, Fern'andez-Ontiveros et al. 2012). All other values were estimated using the correlation between core radio luminosity at 5 GHz and P o of Merloni & Heinz (2007). The data from Mezcua & Prieto (2014) is also included in Fig. 1, where the filled stars represent the luminosities and the squares the cavity powers. Since equation (13) is a lower limit, their luminosities are in very good agreement with the predictions. The cavity powers do not always match the blue line, but as described by Mezcua & Prieto (2014), they are expected to be lower limits, because the estimations of P o do not take into account the energy which is used to compress the gas when the jet advances the ISM/ICM. \nIn simulations, the theoretical and observational results shown in Fig. 1 can be used to calculate the efficiencies glyph[epsilon1] o and glyph[epsilon1] r . To estimate the radiative and outflow efficiencies, we first have to assume a value for the total efficiency η and then use the predictions from Churazov et al. (2005) to separate the AGN feedback into radiation and mechanical outflow. In theoretical studies, the total efficiency is often assumed to be 0 . 1 (e.g. Churazov et al. 2005), however, observations of Davis & Laor (2011) and Chelouche (2013) suggest a mass dependence of this parameter. In the model from Churazov et al. (2005), both glyph[epsilon1] o and glyph[epsilon1] r depend on the accretion rate and the total efficiency. For ˙ M · / ˙ M Edd < 0 . 05 the lower limit \nfor glyph[epsilon1] r can be calculated with equation (8) and (13), i.e. \nglyph[epsilon1] r , min = 10 η ˙ M · ˙ M Edd (15) \nSince this is only a lower limit, all solutions between glyph[epsilon1] r , min and glyph[epsilon1] r , max = η are possible. Therefore, we introduce the slope β , which is in the range between 0 and 1, to get a general expression for glyph[epsilon1] r : \nglyph[epsilon1] r = A · η ( ˙ M · ˙ M Edd ) β , (16) \nwhere A = 10 -4 · 0 . 05 -2 . 8431 -β . The outflow efficiency is calculated with equation (16) and (12). \nFor ˙ M/ ˙ M Edd > 0 . 05 the radiation dominates. The origin of the blue line in Fig. 1 in this regime is the analytical calculation by Churazov et al. (2005), which is based on the equilibrium between gas cooling and heating of gas due to AGN feedback. Hence, it is not only a lower limit and it is not necessary to introduce a slope as in the radio regime. In that respect from equation (7) and (14) follows \nglyph[epsilon1] o = 10 -4 η ( ˙ M · ˙ M Edd ) -2 . 8431 (17) \nand thus glyph[epsilon1] r = η -glyph[epsilon1] o . This is shown in Fig. 2 for different black hole masses. The filled circles and diamonds in Fig. 2 are the observations from Davis & Laor (2011) and Chelouche (2013) illustrating that they are not consistent with the model for η = 0 . 1 (green lines). Therefore, we account for the observed spin of black holes by following the observations of Davis & Laor (2011) for quasars and of Chelouche (2013) for Seyfert 1 AGN, who both find a correlation between the radiative efficiency and the black hole mass. Hence, we use the relation found by Davis & Laor (2011) to estimate the total efficiency at the Eddington limit, which is approximately the same as the radiative efficiency at the Eddington limit: \nη Edd ( M · ) ≈ glyph[epsilon1] r , Edd ( M · ) = 0 . 089 ( M · 10 8 M glyph[circledot] ) 0 . 52 . (18) \nWe limit η Edd ( M · ) by the value 0.42, which is the theoretical maximum efficiency of a rotating black hole. To calculate the outflow efficiency, the constant value of η = 0 . 1 is used as it is currently difficult to estimate outflow efficiencies with observations (see section 5.2 for further discussion). Equation (12), (16) and (17) then lead to the following set of equations: \nglyph[epsilon1] r = Aη Edd ( M · ) ( ˙ M · ˙ M Edd ) β , if ˙ M · ˙ M Edd < 0 . 05 , η Edd ( M · ) -10 -4 η Edd ( M · ) ( ˙ M · ˙ M Edd ) -2 . 8431 , otherwise (19) \nand \nglyph[epsilon1] o = 0 . 1 -A · 0 . 1 ( ˙ M · ˙ M Edd ) β , if ˙ M · ˙ M Edd < 0 . 05 , 10 -5 ( ˙ M · ˙ M Edd ) -2 . 8431 , otherwise. (20) \nIn our simulations both radiative and mechanical feedback are implemented as thermal feedback, since we do not resolve jets. \nFigure 2. Our new feedback model includes both outflow (dotted line) and radiation (dashed lines) as described by Churazov et al. (2005) as well as a mass dependent radiative efficiency following Davis & Laor (2011). The solid lines show the sum of glyph[epsilon1] o and glyph[epsilon1] r . The small dots and diamonds are observations by Davis & Laor 2011 (D11) and Chelouche 2013 (Ch13), who both estimated radiative efficiencies. In the radio regime we assume η = 0 . 1. The large stars and squares correspond to recent observations by Mezcua & Prieto 2014 (MP14) of the outflow and radiation. From left to right the observed galaxies are M87, NGC 4594, NGC 1097, NGC 3169, NGC 1386, NGC 2911, NGC 1052 and Cen A. Small stars and squares correspond to observations by Russell et al. 2013 (R13). The black hole masses are color-coded as indicated by the colorbar. \n<!-- image --> \nThe three coloured lines in Fig. 2 show the model from Churazov et al. (2005) for β = 0 . 5 (thick dashed lines) and β = 1 (thin dashed lines) and different black hole masses. The red lines correspond to M · = 10 10 M glyph[circledot] , the green ones to M · = 10 8 M glyph[circledot] and the blue ones to M · = 10 6 M glyph[circledot] . This is in much better agreement with the observations than choosing a constant total efficiency. In the radio regime, we included observations by Russell et al. (2013) and Mezcua & Prieto (2014), who measured the power of the radiation and outflow as well as L Edd . With equation (10) they calculated ˙ M/ ˙ M Edd . Using the equations (7), (8) and (11) we can derive the efficiencies \nglyph[epsilon1] o = η · P 0 /L Edd ˙ M · / ˙ M Edd (21) \nand \nglyph[epsilon1] r = η · L/L Edd ˙ M · / ˙ M Edd . (22) \nIn the radio regime, it is justified to use η = 0 . 1. As can be seen in Fig. 2, the data points for the radiative efficiency do not show the simple trend as assumed in Churazov et al. (2005). In fact, they seem to be consistent with randomly scattering between 10 -1 and 10 -5 . There also seems to be no mass dependency in the radio regime 1 . \nFor NGC 1097 and NGC 1386, the radiation dominates. The observations by Mezcua & Prieto (2014) show that these sources have small jets, whereas the other sources have larger jets. Interestingly both NGC 1097 and NGC 1386 have a bar at large scales, but they show no evidence of a bar on small scales. They both also have a ring of star-forming regions. This indicates that the morphology of the galaxies will play a key role for future studies. For simulations this implies that the resolution has to be high enough to resolve the morphology of galaxies. Note that this is not the case for the simulations performed in this work, but will be the aim for forthcoming studies.", '3 THE SIMULATIONS': "The present work is based on a set of cosmological simulations called the Magneticum Pathfinder Simulations 2 (Dolag et al. in prep.). The simulations are performed with an updated version of the TreePM-SPH code P-GADGET3 (Springel 2005). \nWe adopt a ΛCDM-cosmology with parameters according to the seven year results of the Wilkinson Microwave Anisotropy Probe with Ω m = 0 . 272, Ω Λ = 0 . 728, Ω b = 0 . 0456 and h = 0 . 704 (Komatsu et al. 2011). We follow the hydrodynamics of the gas using the smoothed particle hydrodynamics method (see Price 2012 for a recent review on the SPH method). We use an entropy conserving formulation (Springel & Hernquist 2002), where star formation is based on a multi-phase sub-resolution model by Springel & Hernquist (2003). Additionally, we include complex treatment for a wide range of physical processes such as isotropic thermal conduction (Dolag et al. 2004) with an efficiency of κ = 1 / 20 of the classical Spitzer value, stellar evolution, metal enrichment and supernova feedback (Tornatore et al. 2003, Tornatore et al. 2007), a cooling function which depends on the individual metal species following Wiersma et al. (2009) as well as the treatment of black holes and their associated feedback based on the model implemented by Springel et al. (2005). We improve the accuracy, stability and reliability of our hydrodynamical method with several state-of-the-art improvements of the SPH method. This includes the higher-order Wendland kernel functions (Dehnen & Aly 2012) as well as time dependent artificial viscosity to properly track turbulence within galaxy clusters (Dolag et al. 2005, Donnert et al. 2013). \nRegarding the black hole physics we use the modifications as described by Fabjan et al. (2010), in contrast to the original model implemented by Springel et al. (2005), and made changes to the seeding and further treatment of black holes as described in detail by Hirschmann et al. (2014). The most important one of these changes is that we do not pin the black holes to the most bound particles anymore. This 'pinning' is used in other simulations to keep the black holes in the centre of their host galaxy, but it also has the side effect that black holes 'jump' from the less massive galaxy to the more massive one during merger events. To avoid that \nfrom McConnell & Ma (2013) were taken if available. If not, the same masses were taken which Russell et al. (2013) used to calculate L Edd . \nthe black hole particles are wandering away from the centre of galaxies by numerical effects, we firstly implemented the conservation of momentum and centre of mass when two black hole particles are merging. Secondly, we enforce momentum conservation for the smooth accretion of gas and therefore do not model any momentum transfer when swallowing gas. Without pinning, we have black holes not only in central galaxies, but also keep them in satellite systems until they fully merge. Thus, we are able to track black hole growth much better, in particular in massive galaxy clusters (following all the black holes in satellite galaxies). \nHirschmann et al. (2014) already presented a detailed analysis of black hole growth in the Magneticum Pathfinder Simulations particularly focusing on the origin of the antihierarchical growth of black holes within a hierarchical structure formation scenario. Various observational trends can be already explained using the simplified black hole model described by Springel et al. (2005). However, implementing the more detailed description of AGN feedback and black hole accretion as described in section 2 leads to further improvements in predicting a more realistic population of black holes and AGN in our hydrodynamic simulations. \nWe performed six simulation runs with the same resolution as in the large (500Mpc) 3 box with an initial particle number of 2 · 1564 3 analysed by Hirschmann et al. (2014). In the context of the set of Magneticum Pathfinder Simulations from Dolag et al. (in prep.) we refer to this resolution as hr ('high resolution'). The particle masses are M dm = 6 . 9 · 10 8 M glyph[circledot] /h , M gas = 1 . 4 · 10 8 M glyph[circledot] /h and M stars = 3 . 5 · 10 7 M glyph[circledot] /h and the softening length is 3.75 kpc/h for dark matter and gas and 2.0 kpc/h for stars. Black holes are represented as collisionless sink particles. They are seeded in galaxies with stellar masses above 2 . 3 · 10 10 M glyph[circledot] with an initial mass of 4 . 6 · 10 5 M glyph[circledot] . \nFour of our simulations are 'test' runs with a smaller box size of (68Mpc) 3 , which were performed to be able to test the effect of the new black hole accretion and AGN feedback model separately. The first run adopts the 'original' black hole model as described in Hirschmann et al. (2014) to which we refer as the fiducial model. The second run adopts only the new accretion model (NAM), the third run only adopts the new feedback model (NFM), and finally, our fourth run combines both new implementations (NFAM). \nThe other two simulations have the same resolution but a larger box size of (182Mpc) 3 to achieve a larger statistical sample of galaxies and black holes. The first box uses the original implementation of black hole growth and the second box adopts the NFAM model, enabling us to statistically see the effects of the new model, in particular on the more massive galaxy and black hole population. \nAs described in section 2 in detail, the NAM, NFM and NAFM models contain improvements of the black hole model regarding the calculation of the accretion rate and/or the feedback energy of black holes: \n- (i) NAM: For the estimation of the black hole accretion rate we use different boost factors for cold ( α = 100) and hot ( α = 10) gas. For this run we use the fiducial feedback model.\n- (ii) NFM: For the calculation of the energy of the AGN feedback we consider not only radiative, but also mechanical feedback. The two different feedback mechanisms have \nTable 2. Best-fit parameters and standard deviation for our runs in comparison to the observations by McConnell & Ma (2013). All black holes with masses smaller than 5 · 10 7 M glyph[circledot] have been excluded for the fit. For the 182Mpc/hr runs we took only stellar masses below 10 12 M glyph[circledot] into account to exclude clusters. \n| | a | b | σ |\n|--------------------------|-----------------|-----------------|------|\n| McConnell & Ma (2013) | 8 . 46 ± 0 . 08 | 1 . 05 ± 0 . 11 | 0.45 |\n| 68Mpc/hr fiducial model | 8.53 | 1.28 | 0.17 |\n| 68Mpc/hr NFM | 8.52 | 1.03 | 0.16 |\n| 68Mpc/hr NAM | 8.44 | 1.24 | 0.19 |\n| 68Mpc/hr NFAM | 8.51 | 1.00 | 0.16 |\n| 182Mpc/hr fiducial model | 8.46 | 0.93 | 0.15 |\n| 182Mpc/hr NFAM | 8.40 | 1.09 | 0.14 | \ndifferent efficiencies. The radiative efficiency glyph[epsilon1] r depends on the black hole mass and the Eddington ratio, whereas the outflow efficiency glyph[epsilon1] o depends only on the Eddington ratio. Like in the fiducial model only a fraction glyph[epsilon1] f of the radiation couples to the surrounding medium. Both kinds of feedback are implemented as thermal feedback. Hence, the total feedback energy is computed with equation (9). We use the old accretion model for this simulation. \n(iii) NFAM: Our final run contains both the new feedback and the new accretion model. \nThe new feedback model as shown in Fig. 2 was implemented into the code using equation (19) and (20). In reality the slope β can be between 0 and 1. However, the choice of β does not play a significant role for the simulations, as the mechanical outflow dominates over the radiation in the radio regime. Furthermore, the AGN luminosities are not calculated during the simulation but only for the analysis afterwards. Thus, we choose the fixed value of β = 0 . 5 for all simulations. \nFor the NAM run and the two fiducial runs we use the standard feedback model with glyph[epsilon1] f = 0 . 15 and a constant radiative efficiency glyph[epsilon1] r = 0 . 2 (Hirschmann et al. 2014). In the other runs we use glyph[epsilon1] f = 0 . 2. The parameters of the simulations used in this work are summarized in Table 1. \nNote that we identify the dark matter haloes and the corresponding galaxies in the simulation using the friendsof-friends and then the SUBFIND algorithm (Dolag et al. 2009, Springel et al. 2001).", '4.1.1 Black hole-galaxy mass scaling relations at z = 0': "The upper panel in Fig. 3 shows the predictions for the present-day M · -M ∗ relation for the 68Mpc/hr NFAM simulation. In our simulations M ∗ is the total stellar mass of a galaxy and not only the stellar mass of the bulge, because our resolution is not high enough to resolve the internal structures of the individual galaxies. Hence, all galaxies consist mainly of a spheroidal component. The solid black lines in Fig. 3 indicate the observations of McConnell & Ma (2013) and the dashed line is the fit for all black holes in our simulations with M · > 5 · 10 7 . This threshold is necessary to exclude newly seeded black holes, as they are \nTable 1. General settings of the simulations performed in this study. Variable values of glyph[epsilon1] r and glyph[epsilon1] o are calculated with equations (19) and (20). \n| | Box size [(Mpc/h) 3 ] | initial particle number | glyph[epsilon1] f | glyph[epsilon1] r | glyph[epsilon1] o |\n|--------------------------|-------------------------|---------------------------|---------------------|---------------------|---------------------|\n| 68Mpc/hr fiducial model | 48 3 | 2 · 216 3 | 0.15 | 0.2 | - |\n| 68Mpc/hr NFM | 48 3 | 2 · 216 3 | 0.2 | variable | variable |\n| 68Mpc/hr NAM | 48 3 | 2 · 216 3 | 0.15 | 0.2 | - |\n| 68Mpc/hr NFAM | 48 3 | 2 · 216 3 | 0.2 | variable | variable |\n| 182Mpc/hr fiducial model | 128 3 | 2 · 576 3 | 0.15 | 0.2 | - |\n| 182Mpc/hr NFAM | 128 3 | 2 · 576 3 | 0.2 | variable | variable | \nseeded far below the relation and need time to grow onto the relation. Black holes with masses above M · > 5 · 10 7 are close enough to the M · -M ∗ relation to exclude seeding effects. The figure shows the excellent agreement of our NFAM model with observations, in particular in comparison to other simulations, i.e. the Illustris simulation (Sijacki et al. 2014) and the MassiveBlack-II simulation (Khandai et al. 2014). The dark grey shaded area marks the 1 σ -scatter of the observations and the light grey shaded area the 1 σ -scatter for our simulation. For a quantitative comparison with the observations, Table 2 shows the bestfitting parameters a and b corresponding to the fit function log( M · /M glyph[circledot] ) = a + b · log( M ∗ / 10 11 M glyph[circledot] ) for all six runs. It also contains the 1 σ scatter of McConnell & Ma (2013) and our simulations. For the 182Mpc/hr runs, we consider only stellar masses below 10 12 M glyph[circledot] to exclude the central galaxies of very massive clusters (see discussion in section 4.2). \nWhile the slope of the M · -M ∗ relation turns out to be relatively insensitive to the values of glyph[epsilon1] r and glyph[epsilon1] f , the normalization depends strongly on these parameters as already shown by Di Matteo et al. (2005), because the final black hole mass follows the proportionality M · ∝ ( glyph[epsilon1] f glyph[epsilon1] r ) -1 . Hence, many recent simulations which include black holes (e.g. Di Matteo et al. 2005, Robertson et al. 2006, Degraf et al. 2011, Hirschmann et al. 2014) tuned these parameters in order to reproduce the normalization of the observed M · -M ∗ relation. In addition, the normalization depends on the cooling function (Churazov et al. 2005), i.e. the values of glyph[epsilon1] r and glyph[epsilon1] f must be larger to get the same normalization if the cooling is more effective. Since glyph[epsilon1] r is not a constant parameter in our new AGN feedback model, the slope of the M · -M ∗ relation changes. This is shown in the lower panel of Fig. 3. Here we show the ratio of the simulated to the observed black hole mass (from McConnell & Ma 2013) versus the galaxy stellar mass for all different models, i.e. the Fiducial, NFM, NAM and NFAM runs (colored dashed lines), as well as for the results from Sijacki et al. (2014) and Khandai et al. (2014) (black dotted and dotted-dashed lines, respectively). Since they use a constant radiative efficiency, their slopes are similar to our fiducial simulation. In our new feedback model, however, glyph[epsilon1] r is not a free parameter anymore. Therefore, it is encouraging that both the slope and the normalization of the M · -M ∗ relation are self-consistently predicted with less free parameters than in the standard model. \nHowever, even in our new model one free parameter remains, i.e. the fraction of radiation coupling to the surrounding medium glyph[epsilon1] f , for which we choose a value of glyph[epsilon1] f = 0 . 2 (to be \nconsistent with the observed relation) 3 . For lower efficiencies the feedback would be higher and the black holes would grow too much. We would like to remark that the normalization of the M · -M ∗ relation in simulations always depends on the observations used for the calibration of glyph[epsilon1] f . However, there are discrepancies in observational estimations of the M · -M ∗ relation. For example, Scott et al. (2013) find a slightly higher normalization, but a similar slope as McConnell & Ma (2013), which would change the calibration of glyph[epsilon1] f . \nIn our simulations, the NFAM model reproduces the observed slope better than the Fiducial model, in which the black holes accrete slightly too much gas, resulting in too large masses, particularly at low redshifts and in the most massive galaxies. The new AGN feedback model is more efficient in preventing gas accretion onto massive black holes. Thus, the gas in the vicinity of the black hole has a higher thermal and kinetic energy, which results in lower accretion rates. Consequently, as can be seen in Fig. 3, the massive end of the M · -M ∗ relation is now in excellent agreement with the observations from McConnell & Ma (2013). \nOur second implementation is the separation of hot and cold gas (NAM). For an increasing amount of hot gas in the vicinity of the black hole, this results in slightly lower accretion rates due to the smaller boost factor. Even if the new accretion model by itself cannot prevent the most massive black holes from growing too much, it can decrease the black hole masses slightly. Consequently, a combination of both modifications results in the best match with the observed M · -M ∗ relation. \nThe best-fitting parameters in Table 2 summarize the excellent agreement of the NFM-run and the NFAM-run with the observations. Particularly, the slope b is in better agreement with the observations than in the other runs and also in the analysis of the Illustris simulation shown by Sijacki et al. (2014). Note that in the simulations, the 1σ scatter is significantly smaller than in the observations. As the typical measurement errors in the observations are still substantial, future observations are needed to distinguish, whether this relation indeed has such a small scatter as seen in the simulations, or if there are still additional processes missing in the simulations which influence the growth and evolution of the black holes. \nFurthermore, the scatter in the black hole mass in the \nFigure 3. Upper panel: present-day relation between the black hole mass and the host galaxy stellar mass for 68Mpc/hr NFAM run. The dots represent the black holes in the simulations at z = 0. The solid black line shows the fit to the observations by McConnell & Ma 2013 (M&M13) and the dark shaded area the corresponding 1 σ -error. The dashed lines illustrate the fit to our simulation for M · > 5 · 10 7 M glyph[circledot] (to exclude seeding effects) and the light shaded area the corresponding 1 σ -error. For comparison with other simulations we also show the results from Sijacki et al. 2014 (S14) and Khandai et al. 2014 (K14) as dotted and dotted-dashed lines. Lower panel: Ratio of the simulated black hole mass in all different models (Fiducial: dark blue, NFM: light blue, NAM: green, NFAM: red) to the observed black hole mass M · obs (McConnell & Ma (2013), black solid line and grey shaded area) versus the galaxy stellar mass. \n<!-- image --> \nsimulations decreases with increasing black hole mass. This is most likely a consequence of statistical merging (Peng 2007, Hirschmann et al. 2010, Jahnke & Macci'o 2011) and is also visible in the Illustis simulation (Sijacki et al. 2014). Nevertheless, the relative role of AGN feedback and stasticial merging in establishing the M · -M ∗ relation and producing the observed slope still remains a matter of debate. \nTo explore black hole growth in our simulations in more detail, Fig. 4 shows the cosmic evolution of four black holes selected due to their different present-day mass (different colors) on the M · -M ∗ relation 4 . When black holes are merging, the most massive progenitor is followed back in time. As can be seen in this figure we can distinguish between two different phases of black hole growth: during the first \nFigure 4. Evolution of the total black hole mass and the corresponding host galaxy stellar mass of four haloes (diamonds in different colors) in the 68Mpc/hr NFAM simulation. The black line shows the fit from McConnell & Ma (2013) \n<!-- image --> \nphase, they grow rapidly until they reach the M · -M ∗ relation and thus the Eddington limit. In this phase black hole accretion is primarily triggered by smooth accretion of cold gas, because below the Eddington limit AGN feedback is not strong enough to suppress gas cooling. Hence, the cold gas reservoir is large enough to trigger black hole growth. In our simulations, this phase is a consequence of the small black hole seeding mass. However, recent observations seem to indicate that the slope of the M · -M ∗ relation is steeper for black holes with masses below 10 8 M glyph[circledot] (Graham & Scott 2013, Scott et al. 2013). Therefore, we can speculate that the phase of rapid black hole growth is actually present and that simulations in which black holes are seeded on or above the M · -M ∗ relation might miss the first phase of black hole growth. \nIn the second phase black holes grow along the M · -M ∗ relation. In this phase, gas cooling and AGN feedback are in equilibrium and hence both star formation and black hole growth are suppressed. Only the in-fall of cold gas either in the form of streams or clumps as well as merger events can trigger star formation and black hole growth during this period. \nTo demonstrate that at low redshifts black holes grow faster compared to the growth of the stellar mass than at high redshifts, we show exemplarily the results for four typical objects, where we verified that they reflect the typical growth of BHs with the chosen final mass. For example, the stellar mass of the host galaxy corresponding to the red diamonds grows very little, whereas the black hole mass increases by more than two orders of magnitude. This galaxy reaches the M · -M ∗ relation already 1.08 Gyr after the seeding. In contrast, the stellar mass of the host galaxy corresponding to the black and blue diamonds grows much more during the first phase of black hole growth. Here, the ob- \nreaches 5 the M · -M ∗ relation after 2.29 Gyr. This trend is also visible in Fig. 6, which shows the M · -M ∗ relation at different redshifts, in particular when looking at the data points corresponding to the lowest stellar masses. The figure will be discussed later in more detail. Hence, we suspect that the black hole mass at the threshold between the two phases - namely when the M · -M ∗ relation is reached - depends on the seeding redshift. We suggest, that these differences might be a consequence of the star formation rate, which decreases with time (see section 4.3). \nFurthermore, since black holes are seeded upon a certain galaxy mass, they are seeded earlier in a dense environment and can thus become more massive. We plan to study the evolution of black holes and their host galaxies in a forthcoming study in more detail, performing a simulation with resolution high enough to resolve the internal structure of galaxies. In particular, we are interested in the effect of merger events on black hole growth and star formation, because the black hole and stellar masses in Fig. 4 seem to grow mainly in steps after reaching the M · -M ∗ relation. These steps also explain the scatter around the M · -M ∗ relation in our simulations. It furthermore indicates, that black hole growth and star formation are both triggered by merger events. However, for this study it is more important to increase the box size instead of the resolution, in particular to extend our simulation results towards more massive galaxies and black holes.", '4.1.2 Evolution of the black hole mass function': 'Fig. 5 shows the black hole mass function of both the fiducial and the NFAM 182Mpc/hr run. We compare our simulations to observed black hole mass functions of the local universe by Marconi et al. (2004), Shankar et al. (2004), Shankar et al. (2009) and Shankar (2013). We would like to remark that the uncertainties in these relations are large, in particular because the black hole masses are estimated using different scaling relations as recently discussed by Shankar (2013) and therefore, we also show the black hole mass functions derived from the best fit velocity dispersion function and stellar mass function from Bernardi et al. (2010) using different scaling relations, i.e. from McConnell & Ma 2013 (dotted grey lines) and Kormendy & Ho 2013 (dashed grey lines). Since the high mass end of all of these curves is lower than in Shankar (2013), we take - following their discussion - the two data points at the high mass end of Shankar (2013) as upper limits. One should also keep in mind that as discussed in Tundo et al. (2007), the different black hole scaling relations are not necessarily consistent with each other or with the M · -M ∗ relation from McConnell & Ma (2013), which we use in this work to calibrate the value of the free parameter glyph[epsilon1] f . The uncertainties in the scaling relations are also reviewed and discussed in Kormendy & Ho 2013. \nThe high mass end of the fiducial simulation is just in agreement with the upper limits of Shankar (2013), but the NFAM simulation matches previously published black hole mass functions much better, because the new accretion and feedback models suppress the growth of massive black \nFigure 5. Black hole mass function of the fiducial (dashed coloured lines) and the NFAM (solid coloured lines) 182Mpc/hr simulation at different redshifts. For comparison we show observations from Marconi et al. 2004 (black solid line), Shankar et al. 2004 (black diamonds and lines with grey shaded areas), Shankar et al. 2009 (dark grey shaded area) and Shankar 2013 (black dots). To show the uncertainties in deriving black hole mass functions from observations, we show as dotted and dashed grey curves the black hole mass functions derived from the best fit velocity dispersion function and stellar mass function from Bernardi et al. (2010) using different scaling relations, i.e. from McConnell & Ma 2013 (MM) and Kormendy & Ho 2013 (KK). \n<!-- image --> \nholes more efficiently. As already shown in Fig. 3, the smaller masses of the most massive black holes are mainly caused by the new feedback scheme, where the mass dependency of the radiative efficiency for the model is taken from Davis & Laor (2011), which is quite similar to the results presented in Trakhtenbrot (2014). From a theoretical point of view, this relation is motivated by the fact that the spin of the black hole should increase with mass. However, the slope of this relation might actually be flatter than in Davis & Laor (2011) due to selection effects (see discussion in Raimundo et al. 2012 and Laor & Davis 2011). Thus, the massive end of the black hole mass function of the NFAM simulation could be a lower limit. Furthermore, we already mentioned that it is uncertain whether in general the normalization of the M · -M ∗ relation could be larger than in McConnell & Ma (2013). \nFor less massive galaxies, the effects of the seeding become dominant which cause the deviation from the observed black hole mass function at small masses. However, especially at low masses, observations are uncertain and only give an upper limit (Shankar 2013), in particular because pseudo-bulges do probably not follow the observed scaling relations like the M · -σ relation or the M · -M ∗ relation as reviewed by Kormendy & Ho (2013).', '4.1.3 Evolution of the black hole-galaxy mass scaling relations': 'Fig. 6 shows the relation between the black hole mass and the stellar mass of the host galaxy for our NFAM 182Mpc/hr \nFigure 7. Eddington ratio distributions for the two 182Mpc/hr simulations at different redshifts. The black dotted vertical line marks the threshold between radio-mode and quasar-mode. The vertical lines in the top show the mean values. \n<!-- image --> \nFigure 6. Evolution of the relation between the black hole mass and the host galaxy stellar mass for the NFAM 182Mpc/hr run (red dots). The dashed lines are fits for both 182Mpc/hr runs including all black holes with masses larger than 5 · 10 7 M glyph[circledot] and stellar masses with masses smaller than 10 12 M glyph[circledot] to exclude clusters. The light grey shaded area marks the corresponding 1 σ -error of the NFAM run. The black line with the dark grey shaded area represents the fit through the observations from McConnell & Ma (2013) with the 1 σ -error. The dotted and dotted-dashed lines show the results from other simulations, i.e. from Sijacki et al. (2014) and Khandai et al. (2014). \n<!-- image --> \nrun at different redshifts, again in comparison to the observations by McConnell & Ma (2013) and the simulations from Sijacki et al. (2014) and Khandai et al. (2014). Again, we only show black holes with masses above 5 · 10 7 M glyph[circledot] . Below this limit black holes generally grow fast, while M ∗ stays relatively constant until they reach the M · -M ∗ relation. The reason is the equilibrium between AGN feedback and gas cooling, when black holes accrete with ˙ M Edd as described by Churazov et al. (2005). Afterwards black holes can only grow along the M · -M ∗ relation together with their host galaxy through smooth accretion or merging. \nIn the NFAM run, the M · -M ∗ relation is much earlier in place than in the original run, namely already at z = 3. Furthermore, the panels at z = 2 and z = 1 show that in the fiducial simulation the slope of the M · -M ∗ relation is larger than at z = 0, where it is in agreement with the observed M · -M ∗ relation. \nIn our very massive galaxies ( M ∗ ≈ 10 13 M glyph[circledot] ), i.e. the central galaxies of galaxy clusters, most black holes are lying slightly below the M · -M ∗ relation. This is most likely caused by a still too large stellar mass in these very massive galaxies, also visible in the high mass excess of the stellar mass function and the still too large baryon conversion efficiency for large haloes as discussed later on. The reason for the overestimation of stellar masses of cluster galaxies might be the purely thermal feedback in our model, which fails to reproduce the mechanical feedback in such massive systems, visible as large X-ray cavities in observed clusters. Hence, an implementation of mechanical jets (e.g. Ostriker et al. 2010, \nDubois et al. 2013, Choi et al. 2014) might play an important role for future simulations, in which both the resolution and the size of the cosmological boxes will get larger and larger. Furthermore, in our analysis we do not distinguish between the stars belonging to the central galaxy and the ones which would be related to the intra cluster light (ICL), which can be substantial for such massive systems. It is also possible that some merging systems are identified as one galaxy. Thus, the predicted stellar mass for cluster galaxies might actually be slightly larger than in observations. \nFor comparison, Fig. 6 also includes the fit to the data points of the fiducial model, where black holes in galaxy clusters are substantially more massive compared to the stellar mass, especially at redshifts around z = 1. Although the fit at z = 0 is in agreement with the fit from McConnell & Ma (2013), it is evident from the black hole mass function that the black hole masses are too large at the high-mass end implying that the galaxy stellar masses must be too large (compensating for the large black hole masses) which will be investigated in more detail in section 4.2.', '4.1.4 Eddington ratio distribution': 'The modifications in our NFAM simulations are also expected to significantly affect the Eddington ratios of the black holes. Therefore, in Fig. 7 we present the Eddington ratio distributions of both 182Mpc/hr simulations at different redshifts. The black dotted vertical line shows the threshold between radio-mode and quasar-mode and the vertical lines in the top mark the mean values. For redshifts below z = 3 the Eddington ratios are clearly smaller in the NFAM run than in the fiducial simulation. For higher redshifts the Eddington ratios in the NFAM run are larger than in the fiducial simulation. We suggest that the wide range of values for the feedback efficiency leads to broader distributions. Es- \npecially the range of very low accretion rates is represented much better in the NFAM simulation than in the fiducial run. \nIn contrast to the recent study from Sijacki et al. (2014) our simulations - in particular the NFAM run - show two peaks in the Eddington ratio distribution for z < 4, one in the radio-mode and a second peak either in the radiomode or in the quasar-mode. This indicates that we have a clear separation between two accretion modes. In the fiducial model, where a step function was used to distinguish between radio-mode and quasar-mode (Hirschmann et al. 2014), the two peaks are only visible at z = 1. In the NFAM simulation, the second peak appears at z = 3 in the quasarmode. For smaller redshifts it is much more distinct. Interestingly, at z = 1 and z = 2, which is the redshift range where most quasars are observed, a very clear second peak is visible in the quasar-mode. For z = 4 the Eddington ratios are even higher, because here the first phase of black hole growth is dominant. At z = 0 both peaks are in the radiomode and even a third peak is visible at very low Eddington ratios.', '4.2 Evolution of the stellar mass function': "Fig. 8 shows the evolution of the stellar mass function in the simulations (blue: fiducial model, red: NFAM model) and observations (black symbols from Panter et al. 2004, Cole et al. 2001, Bell et al. 2003, P'erez-Gonz'alez et al. 2008, Borch et al. 2006, Bundy et al. 2005, Drory et al. 2004, Fontana et al. 2006 and Marchesini et al. 2007 and black lines from Muzzin et al. 2013 and Bernardi et al. 2013). The figure illustrates that the new feedback scheme can slightly suppress late star formation at the high-mass end, mainly because the radiative efficiency now depends on the black hole mass. Hence, compared to the fiducial model, the modifications in the NFAM model lower the amount of massive galaxies resulting in an overall better match with the massive end of the observed SMF, at least down to z = 0 . 2. \nFor the entire redshift range, a small peak in the SMFs is visible at stellar masses of about 2 · 10 10 M glyph[circledot] . The origin of this peak is caused by a subtle effect of our black hole seeding. Since black holes are seeded below the M · -M ∗ relation, the AGN feedback is efficient during the first phase of black hole growth and hence suppresses star formation until the equilibrium between cooling and AGN feedback is reached. During that phase, the stellar mass stops growing and consequently, there are more galaxies with a certain stellar mass. The peak moves towards higher stellar masses at higher redshifts because of the effect seen in Fig. 4, namely that black holes which are seeded earlier have larger stellar masses when they reach the M · -M ∗ relation. \nThe overestimation of the low-mass end of the stellar mass function at high redshifts happens most likely due to the chosen wind model (constant winds as in Springel & Hernquist 2003) as described by Hirschmann et al. (2014) in more detail. Apart from that, our simulations - especially the NFAM run - are in good agreement with observations at high redshifts. \nFor z < 0 . 2, the high-mass end is still overestimated. However, we have to keep in mind that observations in this mass range contain also relatively large uncertainties. Bernardi et al. (2013) showed that different measurements of \nstellar masses differ from each other significantly, especially at the high-mass end. They demonstrate that the stellar masses are higher using a Sersic model instead of standard models. Their fits using a single Sersic and a Sersic-bulge + exponential-disc model are shown as black dashed and dotted dashed line in the upper left panel of Fig. 8. In comparison to other observational estimates this is in better agreement with our simulations. Nevertheless, the high-mass end still appears to be slightly overestimated in our simulations as also indicated by the massive end of the M · -M ∗ relation (see lower right panel of Fig. 6). \nTo study the effect of our new accretion and feedback models on the stellar masses in more detail, Fig. 9 shows the stellar mass functions separately for quiescent and starforming galaxies in our simulations - again in comparison to the observations from Muzzin et al. (2013). Following Franx et al. (2008) we use a specific star formation rate of 0 . 3 /t Hubble as threshold to distinguish between quiescent and star-forming galaxies. We would like to mention that this is a different selection criterion than in the observations, where a threshold in the UVJ diagram is used (Muzzin et al. 2013). Hence, this criterion might lead to discrepancies with the observations, which may e.g. falsely identify metal-rich, star-forming galaxies to be red and thus quiescent. \nFig. 9 illustrates that our new implementations increase the amount of quiescent galaxies at z > 1 . 5. Consequently, for this redshift range, the discrepancies between simulated and observed SMFs are much smaller for the NFAM simulation than for the Fiducial run. Star formation is suppressed, when cooling and AGN feedback are in equilibrium (Churazov et al. 2005) and the gas in the vicinity of the AGN cannot cool enough to form stars. Hence, the increase of the amount of quiescent galaxies can be explained with the upper left panel in Fig. 6, which shows that the M · -M ∗ relation - and thus the phase of equilibrium - is earlier in place for the NFAM run. This is due to higher black hole accretion rates during the phase of rapid black hole growth as a consequence of both new implementations: firstly, the new accretion model leads to higher accretion rates when cold gas dominates. Secondly, the new feedback model results in less AGN feedback for low black hole masses and thus to lower gas temperatures. \nIn contrast to the equilibrium phase, which can be associated with the radio-mode, the phase of star formation and rapid black hole growth is not much affected by our new implementations. We conclude that the overestimation of the high-mass end is mainly due to star-forming galaxies. At z < 1 the amount of star-forming galaxies is too low for 2 · 10 10 M glyph[circledot] < M ∗ < 2 · 10 11 M glyph[circledot] . Firstly, this is an effect of the low seeding mass of black holes, which also leads to the overproduction of quiescent galaxies. Secondly, it is a consequence of the overestimation of the high-mass end. \nFor both runs, Fig. 9 shows an artefact at low redshifts, namely that the amount of star-forming galaxies decreases rapidly after the seeding of black holes. We speculate that this decrease might be due to our very low black hole seeding mass, which leads to artificially high accretion rates. This also explains why the number of star-forming galaxies is reduced in the NFAM model compared to the fiducial one. Fig. 4 illustrates why this artefact becomes even larger with decreasing redshift: for black holes that are seeded later, the evolutionary track during the first phase of black hole growth \nFigure 8. Stellar mass functions in different redshift ranges for the fiducial (blue lines) and the NFAM (red lines) 182Mpc/hr runs. The solid black lines with the shaded areas show the observed stellar mass functions presented by Muzzin et al. 2013 (M13) and their Poisson errors. The black diamonds are observations from Panter et al. (2004), Cole et al. (2001), Bell et al. (2003), P'erez-Gonz'alez et al. (2008), Borch et al. (2006), Bundy et al. (2005), Drory et al. (2004), Fontana et al. (2006) and Marchesini et al. (2007). The black dashed and dotted-dashed lines show the result from Bernardi et al. 2013 (B13) using a Sersic model and a Sersic-bulge + exponential-disc model. \n<!-- image --> \nFigure 9. Stellar mass functions of quiescent (dashed lines) and star-forming (solid lines) galaxies in different redshift ranges for the fiducial (blue lines) and the NFAM (red lines) 182Mpc/hr runs. For the threshold between quiescent and star-forming galaxies we use the specific star formation rate of 0 . 3 /t Hubble following Franx et al. (2008). The black lines with the shaded areas (light grey for star forming and dark grey for quiescent galaxies) show the observations from Muzzin et al. 2013 (M13) and their Poisson errors. \n<!-- image --> \nis steeper then for early black hole seeds. All in all Fig. 9 shows that our new implementations cannot significantly improve the stellar mass functions at low redshifts, but at high redshifts they predict a larger amount of quiescent galaxies, which is in better agreement with observations. \nTo quantify how efficient baryons are converted into \nstars for a given halo mass, we calculate the mean baryon conversion efficiencies, which are defined as M ∗ / ( f bar M halo ), where f bar = 0 . 17 is the baryon fraction of the universe, for different redshifts. To be comparable to other studies we do not use M vir for the halo mass, but M 200c , which is the mass inside the radius where the density is 200 times larger than \nFigure 10. Mean baryon conversion efficiencies versus halo mass at different redshifts for the two 182Mpc/hr runs. The grey shaded area shows the 1 σ -error of the NFAM run. The dashed and solid red vertical lines mark the minimum and mean value of M 200c in the NFAM simulation corresponding to the minimum stellar mass for black hole seeds. Below the mean seeding limit our resolution does not allow reliable predictions (dashed lines). The black vertical line shows the resolution limit for the baryon content as estimated by Vazza et al. (2011), which is given by 500 dark matter particles. We compare our simulation with abundance matching models (Moster et al. 2013, Behroozi et al. 2013) and with observations estimating the halo mass with weak lensing (Mandelbaum et al. 2006, Hudson et al. 2013, Reyes et al. 2012) or X-ray temperatures (Kravtsov et al. 2014). \n<!-- image --> \nthe critical density of the universe. Fig. 10 shows the conversion efficiencies versus halo mass for our two 182Mpc/hr runs (different panels illustrate z = 0 , 1 , 2). The black vertical line shows the resolution limit for the baryon content as estimated by Vazza et al. (2011), which is given by 500 dark matter particles. Furthermore, the dashed and solid red vertical lines mark the minimum and mean value of M 200c , respectively, in the NFAM simulation corresponding to the minimum stellar mass for black hole seeds. Below the mean seeding limit our resolution does not allow reliable predictions (dashed lines). The figure clearly shows, that the new implementations lower the stellar content in a halo for a given mass above this limit, which is also reflected by the reduced high-mass end of the stellar mass functions (see Fig. 8). At z = 2 and z = 1, this effect is even stronger than at z = 0. The dotted and dotted-dashed black lines show the predictions of the abundance matching models by Moster et al. (2013) and Behroozi et al. (2013). The peak at M halo ≈ 10 12 M glyph[circledot] is in agreement with these models, which also find a maximum baryon conversion efficiency of around 20 per cent. At larger halo masses, the stellar content decreases due to AGN feedback and because the gas is consumed by star formation. Although the baryon conversion efficiencies in the NFAM simulation are smaller than in the fiducial run, they are still higher than in the abundance matching models of Moster et al. (2013) and Behroozi et al. (2013) for M 200c > 10 13 M glyph[circledot] galaxies. \nFor the NFAM simulation, at low redshifts a slight 'upturn' of the baryon conversion efficiencies occurs for stellar masses above 10 14 M glyph[circledot] corresponding to galaxy clusters due to too inefficient AGN feedback. This might indicate that other AGN feedback processes like mechanical jets should be included in future simulations. Since the most massive black holes accrete less in the NFAM model we suspect that there is more cold gas left to form stars than in the fiducial run. Therefore, the upturn is only visible in the NFAM simulation. However, except for the high-mass end, our simulations - in particular the NFAM run - are in agreement with observations using weak lensing (Mandelbaum et al. (2006), Reyes et al. (2012) and Hudson et al. (2013)) or X-ray temperatures Kravtsov et al. (2014) to estimate the total halo mass. 6", '4.3 Evolution of the star formation rate': "Fig. 11 shows the SFR-stellar mass plane (number density is color-coded) for our two 182Mpc/hr runs at different redshifts. The panels illustrate all galaxies classified as subhaloes using the SUBFIND algorithm (Dolag et al. 2009, Springel et al. 2001). For comparison with observations, we also show the main sequence for star-forming galaxies estimated by Steinhardt et al. (2014) for 4 < z < 6 (red line), by Daddi et al. (2007) for z = 2 (orange line) and by Elbaz et al. (2007) for z = 1 and z = 0 (yellow line). At z = 2 and z = 1, the simulated SFRs at a given stellar mass are slightly below the observations. This trend is also visible in the recently published analysis of the Illustris simulation by Sparre et al. (2014). At z = 0 and at redshifts above z = 4 \nFigure 11. Comparison of the star formation rates of all galaxies in the two 182Mpc/hr runs at different redshifts. The solid lines represent the observed main sequence of galaxies derived by Steinhardt et al. 2014 (S14), Daddi et al. 2007 (D07) and Elbaz et al. 2007 (E07). \n<!-- image --> \nlog(M ∗ ) [M /circledot ] \nour simulation results are in very good agreement with the observed main sequence, independent of the adopted black hole model. The redshift evolution of the SFR-stellar mass plane nicely demonstrates that the most massive galaxies become more and more quiescent with cosmic time. Furthermore, in the NFAM simulation star formation is suppressed earlier than in the fiducial one. This is consistent with Fig. 9, where we demonstrated that in the NFAM run the amount of quiescent galaxies is larger at earlier times. In the NFAM simulation, the SFRs of the most massive galaxies decrease already at redshifts above z = 4 . 8 such that they lie below the observed main sequence of star forming galaxies. In the fiducial simulation, this decrease starts at redshifts below z = 4. This may be unrealistic, because - as shown in Fig. 9 - Muzzin et al. (2013) observe much more quiescent galaxies at high redshifts ( z > 3) than in our fiducial simulation. Looking at the star formation main sequence of the Illustris simulation (Sparre et al. 2014) shows that this is not only a problem in our fiducial run, but seems to be a general issue. Therefore, it is encouraging that in the NFAM run galaxies become quiescent much earlier due to both of our new implementations, even if there are still discrepancies between the observed and simulated SMFs for star-forming and quiescent galaxies. The new feedback model leads to a lower feedback energy for low black hole masses, whereas for large black hole masses the AGN feedback is stronger as long as the black holes are accreting in the quasar-mode and star formation is suppressed. \nThe new accretion model leads to lower accretion rates when the hot gas phase dominates. Hence, black holes grow less strongly and the SFR decreases already in less massive galaxies as can be seen in the panels corresponding to z = 1. From the earlier and more rapid decrease of the SFR follows that at z = 1 star-forming galaxies with stellar masses above 2 · 10 10 M glyph[circledot] are more concentrated along the observed main sequence in the NFAM simulation than in the fiducial one. At z = 0 there are only very few star-forming galaxies above log( M ∗ /M glyph[circledot] ) = 10 . 5, which is the mass at which AGN feedback becomes important. At that redshift both runs predict galaxies with similar SFRs at a given stellar mass. Hence, our modifications mainly affect the evolution of high redshift galaxies. \nFig. 12 depicts the redshift evolution of the mean specific SFR for our two 182Mpc/hr runs. As in Biffi & Maio (2013) - who studied early proto-galaxies at z > 9 - we compare our simulations with other theoretical models (i.e. Biffi & Maio 2013, Dayal et al. 2013, Dav'e et al. 2011) and observations (i.e. Noeske et al. 2007, Daddi et al. 2007, Dunne et al. 2009, Pannella et al. 2009, Stark et al. 2009, Yabe et al. 2009, Michaglyph[suppress]lowski et al. 2010, Schiminovich et al. 2010, Reddy et al. 2012, Bouwens et al. 2014, Gonz'alez et al. 2012, Zheng et al. 2012, Stark et al. 2013 and Coe et al. 2013). Irrespectively of the assumed accretion and feedback models, our simulations are both in better agreement with observations than many other theoretical models, especially at low redshifts (where the observational constraints are tighter). Fig. 12 also demonstrates that our new implementations have no effect on the specific SFR. Hence, the changes in the SFR and in the stellar mass are the same. \nHowever, star formation is certainly not only regulated by AGN feedback. Recent studies (e.g. Hopkins et al. 2014, Hirschmann et al. 2013, Aumer et al. 2013, Kannan et al. \nFigure 12. History of the specific star formation rate in our 182Mpc/hr runs in comparison to different observations and other theoretical predictions. \n<!-- image --> \nFigure 13. History of the star formation (orange lines) and black hole accretion rate (red lines) density in both 182Mpc/hr runs (fiducial model: dashed lines, NFAM: solid lines) in comparison to observations from Hopkins & Beacom 2006 (squares). \n<!-- image --> \n2014) showed that stellar feedback also plays an important role, particularly for low mass galaxies. Fig. 13 provides further evidence that our model is still not sufficient for reproducing galaxies with realistic SFRs. It illustrates the history of the star formation and the black hole accretion rate densities as shown by Hirschmann et al. (2014) for our two 182Mpc/hr runs compared to observations of the SFR density (squares) by Hopkins & Beacom (2006). In comparison to the fiducial model, the star formation rate density in the NFAM model is slightly lower above z ≈ 1 . 5, although it is still too high in comparison to the observations except for very high redshifts, which are, however, affected by resolution. \nAs expected due to the lower black hole masses in the NFAM model, the black hole accretion rate density is significantly lower at z < 4 . 5 than in the fiducial model. For higher redshifts, it is larger than in the fiducial model, which leads \nto a much shallower increase up to the maximum. Fig. 13 demonstrates that in the NFAM simulation the SFR and the black hole accretion rate evolve very similar with redshift. The reason is that both depend on the amount of cold gas. With our new accretion model the analogy between SFR and black hole accretion is even stronger, because the accretion factor for hot gas is smaller than for cold gas. Thus, in the NFAM simulation, hot gas results not only in less star formation, but also in smaller black hole accretion rates. This shows that the gas temperature plays a key role in both galaxy formation and black hole growth. A similar accordance between the history of the star formation and black hole accretion rate density was also found by Zheng et al. (2009), who adopted the luminosity functions from Hopkins et al. (2007) to estimate the black hole accretion rate densities.", '5.1 The effect of the feedback model onto the luminosity functions': 'As already mentioned before, the choice of the slope β of the feedback model should not have a significant influence on the resulting galaxy and black hole properties in the simulations since glyph[epsilon1] r is much smaller than glyph[epsilon1] o . However, it has an influence on the AGN luminosity functions, which are calculated during post-processing using the accretion rates calculated by the simulation and the radiative efficiencies, which can be varied. \nIn that way we can test the effect of the parameter β on the AGN luminosity function. We calculate the bolometric AGN luminosities of the NFAM simulation for different values of β using equation (8) and (19). Fig. 14 shows the resulting luminosity functions in comparison to the observational compilation of Hopkins et al. (2007). For a comparison of moderately luminous AGN, particularly at high redshifts, one has to keep in mind that simulations are affected by resolution (see discussion of Hirschmann et al. 2014). In addition, dust obscuration effects in observational data typically result in an underestimation of their number density (e.g. Hasinger 2008, Merloni et al. 2014) which complicates a comparison between simulations and observations. Even if luminosity-dependent obscuration effects on a torus level are already considered in Hopkins et al. (2007), an additional redshift-dependence (of X-ray luminosities, as suggested by e.g. Hasinger 2008 and Merloni et al. 2014) may change the low luminous end at high redshifts. \nFig. 14 shows that the effect of the choice of β on the AGN luminosity functions is not significant, especially at high redshifts, because β changes only the efficiencies in the radio-mode and not in the quasar-mode. For lower redshifts, when more black holes accrete with low Eddington ratios, it has an influence on the amount of AGN with luminosities smaller than 10 45 erg/s in the sense that with decreasing β the radiative efficiency and thus the amount of moderately luminous AGN is increasing and thus the result is in better agreement with the observational constraints. However, due to the fact that observations constrain very low values of glyph[epsilon1] r we suspect that the accretion rates in the quasar-mode are slightly underestimated in our simulations. \nFigure 14. AGN luminosity function of our 182Mpc/hr NFAM run at different redshifts for different values for the slope β in comparison to the observational compilation by Hopkins et al. (2007). \n<!-- image --> \nFigure 15. AGN luminosity function of our 182Mpc/hr NFAM run at different redshifts for different values of glyph[epsilon1] r in the radio regime in comparison to the observational compilation by Hopkins et al. (2007). The green and blue curves show the result for two constant values of glyph[epsilon1] r . For the purple and red curve we took random values in two different intervals. \n<!-- image --> \nAs shown in Fig. 2, the actual value of glyph[epsilon1] r is entirely unconstrained in the radio regime. It might depend on many properties like the morphology of the host galaxy or the merger history of an individual black hole. For that reason, calculating a more realistic value of glyph[epsilon1] r is beyond the current feasibility. \nNevertheless, according to the observations by Russell et al. (2013), one should consider different models to estimate glyph[epsilon1] r in the radio-mode. Fig. 15 shows the AGN luminosity functions in comparison to observational compilation by Hopkins et al. (2007) for four models adopting different values for glyph[epsilon1] r in the radio regime: \n- (i) The commonly used value glyph[epsilon1] r = 0 . 1 (green lines) seems \nto match the observations reasonably well, although such a value is unlikely according to the results from Russell et al. (2013) and Mezcua & Prieto (2014). \n(ii) glyph[epsilon1] r = 10 -3 is the mean value of the data points from Russell et al. (2013). Because we change only values in the radio regime, the high luminosity end is not affected. At lower luminosities, the AGN number densities are significantly underestimated as AGN become way too faint 7 (blue lines). \n(iii) We choose random values in log space in the range 10 -5 < glyph[epsilon1] r < 0 . 4. This is approximately the range of the data points from Russell et al. (2013) with a maximum value equal to the theoretical maximum efficiency of a rotating black hole (since we assumed η = 0 . 1). It leads to a reasonably good match (magenta line) with the observational constraints, even if the low luminous end is slightly lower than when adopting the commonly used value (green lines). Since we may speculate that the curve will probably be shifted upwards when choosing a higher resolution (Hirschmann et al. 2014), the concordance with the observations might be even better. \n(iv) Now we exclude very low values for glyph[epsilon1] r and hence choose random values in the range 10 -3 < glyph[epsilon1] r < 0 . 4. This leads to a slightly, but not significantly larger number density of moderately luminous AGN (red lines) and hence to a better agreement with observations. \nIn comparison to the AGN luminosity functions from the Illustris simulation (Sijacki et al. 2014), we have less luminous AGN for redshifts below z = 1, although our cosmological box is larger. Nevertheless, to investigate the high-mass end in more detail larger cosmological boxes are needed. Hirschmann et al. (2014) already presented luminosity functions of a larger box from the set of Magneticum Pathfinder Simulations, which are in good agreement with the observations from Hopkins et al. (2007). Furthermore, our simulation matches better with the observed amount of AGN with luminosities below L ≈ 10 45 erg/s than in Sijacki et al. (2014). This confirms the conclusion from Sijacki et al. (2014) that the radiative efficiency is not constant and might actually be very low in the radio regime. \nThis analysis shows that the efficiency of the radiative component in the radio regime is indeed not yet understood because the theoretical lower limit is not captured by observations. Interestingly, choosing random values for the radiative efficiency in the range of the observed values leads to a good agreement with observed AGN luminosity functions. This may indicate that in the radio regime the radiative efficiency depends neither on the mass of the black hole, nor on its accretion rate. It also implies that - as we are matching the observed luminosity function by randomly choosing the radiative efficiency within the observed values - the distribution of the accretion rates as predicted by the simulations are similar to the observed ones. We conclude, that it is theoretically not fully understood how efficient AGN radiate and we suspect that the morphology of the galaxy, but also turbulence or even magnetic fields might play an im- \nportant role. Since jets dominate in the radio-mode, they can also prevent efficient accretion. The similar morphologies of the two radiation dominated sources from Mezcua & Prieto (2014), i.e. NGC 1097 and NGC 1386, give a first evidence for these speculations, because they both have a ring of star forming regions and a bar on large scales, but no bar on small scales. However, a better understanding of black hole accretion and AGN feedback processes is a great challenge for the future, because more accurate observations are needed to learn in which cases ADAF/Bondi models are a good estimate and in which cases we have to include additional physical processes.', '5.2 The unconstrained total efficiency in the radio regime': 'Besides the radiative efficiency, the total efficiency η in the radio regime is also unconstrained. Throughout this study, we always assumed η = 0 . 1 to calculate glyph[epsilon1] r and glyph[epsilon1] o , making, thus, our conclusions for the radio regime rather uncertain. The reason for this assumption are missing or unconstrained estimations of ˙ M · . According to equation (11), η is given by \nη = L Edd ˙ M Edd c 2 = L bol ˙ M · ˙ M Edd L bol L Edd ˙ M · c 2 . (23) \nIn observations, however, usually only the AGN luminosity, the jet power and the black hole mass are measured. Using the black hole mass, one can calculate L Edd . Equation (10) is then used to calculate ˙ M · / ˙ M Edd . Hence ˙ M · is the parameter which is typically missing. Nevertheless, for some of the sources from Russell et al. (2013) and Mezcua & Prieto (2014), ˙ M · has been estimated. We use these estimations to calculate the corresponding total efficiencies with equation (23). With these values and equations (21) and (22) we then compute glyph[epsilon1] o and glyph[epsilon1] r . \nBefore we calculate the efficiencies for the selected sources, we want to focus on the nearest SMBH, namely Sagittarius A* (Sgr A*). For the luminosity we adopt L bol = 2 . 1 · 10 36 erg / s (Narayan et al. 1998) and for the power of the mechanical outflow we assume P o = 1 . 2 · 10 41 erg / s (Yusef-Zadeh et al. 2012). With these values and the mass M SgrA ∗ = 4 · 10 6 M glyph[circledot] we calculate the Eddington ratio using equation (10). Although Sgr A* is the nearest SMBH, there are different estimates for the accretion rate. Quataert et al. (1999) estimated a Bondi accretion rate of ∼ 3 · 10 -5 M glyph[circledot] / yr. However, there are other models suggesting the actual accretion rate might be much lower than the Bondi accretion rate (e.g. Quataert & Gruzinov 2000). Cuadra et al. (2006) derived ˙ M ≈ 3 · 10 -6 M glyph[circledot] / yr from stellar winds. We calculated the efficiencies corresponding to both values using equation (21) and (22). They are shown in Fig. 16. The upper data points belong to ˙ M ≈ 3 · 10 -6 M glyph[circledot] / yr and the lower ones to ˙ M ≈ 3 · 10 -5 M glyph[circledot] / yr. Assuming that the ADAF model really provides a lower limit, this illustrates that ˙ M ≈ 3 · 10 -6 M glyph[circledot] / yr is in good agreement with our model for the radiative efficiency. It also indicates that it is necessary to choose different lower limits for different black hole masses, because the dashed green line - which corresponds to η ≈ 0 . 1 - is far above the data point. However, the corresponding value for glyph[epsilon1] o is larger then the commonly used value 0.1. This indicates, that the outflow efficiencies \nFigure 16. Same as in Fig. 2, but with efficiencies calculated using values for ˙ M · from Russell et al. 2013 (R13) and from other authors, i.e. Evans et al. 2004, Allen et al. 2006 and Li et al. 2011 (R13 ∗ , MP14). The three data points from Mezcua & Prieto (2014), for which we know estimations of ˙ M · are from left to right M87, NGC 4594 and CenA. We also included values for Sgr A*, which have been calculated using different estimations of ˙ M · , i.e. ˙ M ≈ 3 · 10 -6 M glyph[circledot] / yr from Cuadra et al. 2006 (upper symbols) and ˙ M ≈ 3 · 10 -5 M glyph[circledot] / yr from Quataert & Gruzinov 2000 (lower symbols). \n<!-- image --> \nmight differ significantly from this value, which is not well constrained. For the second estimation of the accretion rate, i.e. ˙ M ≈ 3 · 10 -5 M glyph[circledot] / yr, the radiative efficiency is clearly below the prediction, although glyph[epsilon1] o is near 0.1. This implies that Bondi estimations of the accretion rate indeed tend to be too high. \nNow, we consider the sources from Russell et al. (2013) and Mezcua & Prieto (2014), for which ˙ M · has been estimated using the Bondi model. Russell et al. (2013) investigated a subsample of 13 objects for which they estimated ˙ M · . The efficiencies corresponding to these sources are plotted in Fig. 16 (R13). Other authors also estimated ˙ M · : for Centaurus A and NGC 4216 we use the result from Evans et al. (2004) and for the Sombrero galaxy (NGC 4594) we take ˙ M · from Li et al. (2011). For M87, M84, M89, NGC 4636, NGC 4472, NGC407 and NGC5846 we take values from Allen et al. (2006). The efficiencies calculated with these values and the data from Russell et al. (2013) are marked with grey symbols (R13 ∗ ). Most of these sources are also in the selected sample from Russell et al. (2013). We can, thus, directly compare the results of two independent measurements. This shows a clear discrepancy between different estimations of ˙ M · . Overall, the efficiencies are larger using the ˙ M · from Russell et al. (2013). In contrast to Fig. 2, the lowest values of the radiative efficiency now tend to increase with increasing Eddington ratio as predicted by theory. Nevertheless, the observations are in better agreement with theory using only the 13 objects of the selected subsample. Furthermore, Fig. 16 indicates that the value glyph[epsilon1] o = 0 . 1 is indeed a reasonable assumption for the mean value of the observed values, although the observations can be nearly two dex lower. \nHowever, all these estimations are highly uncertain and very speculative. On the one hand, all data points are upper limits due to the approximation of using the Bondi model. On the other hand, there are studies showing that accretion rates can also be much smaller than ˙ M B (i.e. Li et al. 2013, Baganoff et al. 2003, Quataert & Gruzinov 2000). Moreover, values for L bol might be underestimated when the jet is emitting in the plane of the sky. In that case, the measured flux is smaller than if the jet were located close to the line of sight. This would lead to higher radiative efficiencies and to an even better agreement with our model. Furthermore, uncertainties in the determination of black hole masses make it almost impossible to investigate whether the lower limit for the radiative efficiency splits up for different black hole masses as seen in the quasar regime (Davis & Laor 2011,Chelouche 2013). \nNevertheless, the data shown in Fig. 16 is one of the best constrained samples. The comparison between Fig. 16 and Fig. 2 shows that we need more accurate measurements to learn more about the feedback of radio jets and the corresponding efficiencies. Due to the fact that knowing the efficiencies is (at least with the currently available computational power) essential for performing large-scale cosmological simulations, it is worth and necessary spending more effort on observational estimates of black hole accretion rates.', '5.3 Comparison with other simulations': "During the last couple of years, several other groups have also been working on large cosmological simulations including baryons and black holes. As our simulations, some of these simulations, for example the MassiveBlack-II simulation (Khandai et al. 2014), earlier simulations from Di Matteo et al. (2008) and the new EAGLE simulation (e.g. Schaye et al. 2015), are based on the SPH code GADGET-3, but differ in their physical sub-resolution models, including the model for black hole growth. In contrast, the recent Illustris simulation (e.g. Vogelsberger et al. 2014, Genel et al. 2014) has been performed with a different hydrodynamic scheme, the moving mesh code AREPO (Springel 2010), and also slightly different sub-resolution models. A comparison between these models can help to understand which effects the different sub-resolution models for black hole growth and AGN feedback may have on basic galaxy and black hole properties. \nFig. 17 shows the stellar mass function in the NFAM model below z = 0 . 2 in comparison to other simulations. As for the black hole mass function, the number density of massive galaxies in the Illustris simulation (Genel et al. 2014, green lines) is by half an order of magnitude larger than the one in the Magneticum simulation. For stellar masses below 4 · 10 11 M glyph[circledot] the galaxy number densities in the Illustris simulation are in reasonably good agreement with the observations, while our simulations produce slightly too few low mass galaxies. Since the difference between the SMFs of the fiducial model and the NFAM model are very small at z = 0 we suggest that other physical processes (e.g. stellar feedback or cooling) or the lower resolution might be the reason for the lower stellar masses. The prediction from the MassiveBlack-II simulation (Khandai et al. 2014, orange line) has no pronounced exponential cut-off with the consequence that they over-estimate the low and the high \nFigure 17. Comparison of the SMF in the NFAM model (red line) at z = 0 with the Illustris simulation (Genel et al. 2014/G14, green line), the MassiveBlack-II simulation (Khandai et al. 2014/K14, orange line) and the EAGLE simulation (Schaye et al. 2015/Sch14, blue line). The observations shown are the same as in Fig. 8. \n<!-- image --> \nmass end, but slightly under-estimate the number density of galaxies around the exponential cut-off. In contrast, the stellar mass function obtained by the EAGLE simulation (Schaye et al. (2015), blue line), where the feedback is especially calibrated to match the stellar mass functions, is in good agreement with observations for the entire stellar mass range. \nCompared to our results - the black holes in the Illustris simulation are much more massive than in the Magneticum simulation (as shown in Fig. 18). This discrepancy might have several reasons, for example the different implementations of radiative AGN feedback. Furthermore, given that there may still be resolution dependent details of the black hole feedback model (e.g. the estimation of the Bondi accretion rate or the distribution of the feedback) the higher resolution of the Illustris simulation could contribute to these differences. In addition, there could be differences due to the different numerical techniques, namely SPH and moving mesh, especially in the way the feedback gets transported away from the centre of the galaxies. In addition, a more efficient gas cooling in AREPO (Nelson et al. 2013) might lead to higher black hole accretion rates. Furthermore, the underlying physics referring to the energy transport might influence how much gas is driven outward and which fraction of this gas is recycled as for instance discussed by Nelson et al. (2014). \nDue to the large uncertainties in different observational estimates it is not clear which simulation matches the observations of the local Universe best. At z = 1 we also compare the black hole mass function of our NFAM model to the predictions of Di Matteo et al. (2008). This simulation produces slightly more massive black holes than the Magneticum simulation, which might be due to a more inefficient AGN feedback of massive black holes in Di Matteo et al. (2008). \nObviously, the other simulations shown here capture \nFigure 18. Comparison of the black hole mass function in the 182Mpc/hr NFAM run with that in the Illustris simulation (Sijacki et al. 2014) at z = 0 and with that in the MassiveBlack-II simulation (Di Matteo et al. 2008) at z = 1. The observations shown are the same as in Fig. 5. \n<!-- image --> \nblack holes down to smaller black hole masses. Firstly, this is due to the higher resolutions. Secondly, they use the socalled 'pinning' to keep the black holes at the potential minimum and therefore in the centre of the galaxies. Hence, they can seed the black holes in less massive galaxies. In our simulations this is not possible, because the black holes in less well defined galaxies would not be able to stay in the centre of their host galaxy due to numerical effects. However, not using the so-called 'pinning' avoids other drawbacks of this method as discussed in Hirschmann et al. (2014). As discussed by Shankar (2013), also the low mass end of the black hole mass function is relatively uncertain and depends on the black hole scaling relations. For example, the low mass end could be significantly smaller when excluding galaxies with pseudo-bulges. Therefore, it will be quite challenging to compare observed black hole mass functions to any simulation at the low mass end. \nFig. 19 shows a comparison of the AGN luminosity function in our NFAM run (purple line) with the predictions from the Illustris simulation (Sijacki et al. 2014, green solid line) and from the MassiveBlack-II simulation (Khandai et al. 2014, orange solid line). The luminosity function of the Illustris simulation matches both the observations and our simulation, whereas the MassiveBlack-II simulation widely fails to reproduce the observed shape of the observed luminosity functions of Hopkins et al. (2007). Since the latter simulation contains the original model from Springel et al. (2005) with only one mode of AGN feedback, we can speculate that this might be one possible reason for the discrepancies. The Illustris simulation uses a so-called 'radiative' efficiency, which is implemented as a change in the net cooling rate and is most efficient in the quasar-mode (Sijacki et al. 2014). This seems to have a similar effect as our variable radiative efficiency, which increases for large black hole masses in the quasar mode. \nNevertheless, we want to emphasize that despite of the general importance for understanding the (physical or nu- \nFigure 19. Comparison of the AGN luminosity function in the NFAM model (using random radiative efficiencies in the radio regime in the range 10 -5 < glyph[epsilon1] r < 0 . 4) with the predictions of the Illustris simulation (Sijacki et al. 2014/S14, green solid lines) and of the MassiveBlack-II simulation (Khandai et al. 2014/K14, orange solid lines). \n<!-- image --> \nmerical) origin of different simulation predictions, such a comparison must remain speculative: besides different models for black hole growth and AGN feedback, many other physical details (e.g. models for star formation, stellar feedback) or different hydrodynamic schemes may cause more fundamental changes in basic galaxy properties. Such an investigation is, however, clearly beyond the scope of this work.", '6 SUMMARY AND CONCLUSIONS': 'In this paper, we presented an improved implementation of the black hole model originally introduced by Springel et al. (2005). We combined theoretical predictions of Churazov et al. 2005, Narayan & Yi 1995 and Gaspari et al. (2013) with observations from Russell et al. 2013, Mezcua & Prieto 2014, Davis & Laor 2011 and Chelouche 2013 in order to model the underlying sub-grid processes more realistically. \nThe new model includes a combination of mechanical outflow and radiation, which we both implemented as thermal feedback due to the inability of resolving sub-kpc scales, where jets provide the mechanical feedback. Both feedback processes are modelled as a function of the actual accretion rate with respect to the Eddington rate, which leads to a smooth transition between the outflow-dominated radiomode and the radiation-dominated quasar-mode. In addition, our model includes a mass dependent radiative efficiency to account for the observed spin of the black holes. \nFurthermore, we distinguish between the hot and the cold gas component within the environment of the black holes and calculate the accretion rate for these two components separately. This allows us to model the Bondi accretion differently for the two phases, where we use two different boost factors ( α = 10 for the hot and α = 100 for the cold gas) according to the results of small-scale simulations of Gaspari et al. (2013). \nBesides that, free parameters of the model (like the various efficiencies) are now more strictly linked to values inferred from observations. Compared to the fiducial model, our new implementations predict a more realistic population of black holes and their host galaxies, when compared to fundamental observational constraints, in several aspects: \n- (i) The slope and normalization of the produced M · -M ∗ relation are in much better agreement with observations over a larger range of galaxy masses and redshifts than in the fiducial model. In particular, these improvements are due to the faster black hole growth at large redshifts and the lower black hole masses at the massive end for redshifts below z ≈ 2.\n- (ii) Our new feedback scheme is also able to efficiently suppress the late growth of massive black holes. Hence, the resulting present-day black hole mass function provides an excellent match to the observed one.\n- (iii) In the NFAM simulations, the equilibrium between gas cooling and AGN feedback within the galaxies is reached earlier. Consequently, star formation starts to be suppressed at earlier times. This leads to a better agreement with observed stellar mass functions than before. In particular, in the NFAM simulation there are much more quiescent galaxies at high redshifts than in the fiducial simulation, in which galaxies become quiescent far too late. However, some inconsistencies between observed and simulated SMFs for quiescent and star-forming galaxies remain.\n- (iv) The baryon conversion efficiencies are more consistent with observations and abundance matching predictions than before, although they are still too high by a factor of 2-3 at very high stellar masses. \nA comparison with other large cosmological simulations (e.g. Illustris, MassiveBlack-II) illustrates that the original black hole model from Springel et al. (2005) needs to be extended to be able to reproduce observations. In particular, we find that \n- (i) our NFAM simulation successfully matches the observed M · -M ∗ relation. As our fiducial model, the simulations from Sijacki et al. (2014) and Khandai et al. (2014) do not manage to entirely reproduce the observed slope. This may be due to the constant values adopted for their radiative efficiencies.\n- (ii) In contrast to the MassiveBlack-II simulation, both our NFAM simulation and the Illustris simulation are able to reproduce the observed luminosity functions. We suggest that this might be due to the distinction between quasarmode and radio-mode.\n- (iii) our model predicts a lower high mass end of the black hole mass functions than other simulations (i.e. Di Matteo et al. 2008, Sijacki et al. 2014), because the new AGN feedback model is more efficient in limiting black hole growth at higher masses. Although all simulations are compatible with the upper limits of the black hole mass function estimated from observations by Shankar (2013), our model is in excellent agreement with the observational data from Marconi et al. (2004), Shankar et al. (2004) and Shankar et al. (2009).\n- (iv) We predict lower stellar masses than Genel et al. (2014) and Khandai et al. (2014). Since our new implementations do not change the SMFs at z = 0 significantly, we \nsuggest that other physical processes like stellar feedback or cooling might be the reason for the differences. In addition, we find that improvements in the model for star formation and stellar feedback like in Schaye et al. (2015) might be necessary to better reproduce the observed shape of the SMFs. \nDespite of the overall success of the NFAM model, open questions regarding the actual values of the feedback efficiencies remain. In contrast to the quasar-mode, the radiative efficiency in the radio-mode does not show clear trends in observations, which generally have large uncertainties, especially due to the difficulties in accurately determining the accretion rate. At high redshifts, the quasar luminosity function predicted by the simulations is quite insensitive to the choice of the radiative efficiency in the radio-mode. However, the best match between simulated and observed quasar luminosity functions - especially at low redshifts - is obtained when applying a random radiative efficiency to the simulated AGN in the radio-mode with no dependency on black hole mass or actual accretion rate. \nStudying the growth of black holes in more detail (i.e. for individual objects) provides evidence for a two phase process controlling the evolution of the accretion onto the black hole and the associated feedback: \n- (i) As long as black holes have masses below the M · -M ∗ relation, they grow mainly due to continuous gas accretion. This phase is primarily driven by cold gas accretion with an accretion rate that increases up to the Eddington limit. In this phase, AGN are observed as luminous X-ray sources. This means that the most luminous AGN are not necessarily driven by merger events as long as they are below the M · -M ∗ relation.\n- (ii) When the M · -M ∗ relation is reached, gas cooling and AGN feedback are in equilibrium. Consequently, hot gas accretion begins to dominate. This means that the accretion rate, compared to the original implementation, is lowered since we correctly reduce the boost factor for the hot phase. In this phase, AGN feedback is mostly visible as radio jets. This low accretion phase can be disturbed by mergers or other processes driving cold gas into the centre of the galaxy. In a forthcoming study of the most luminous AGN in a simulation with higher resolution we will investigate in more detail whether those objects are mainly triggered by major mergers. \nRegarding the latter point, more detailed studies are needed to better differentiate the AGN triggering mechanisms (as galaxy major and minor mergers) and their correlation with the black hole accretion processes within a cosmological context. The next generation of simulations will also allow to distinguish between morphological types of galaxies in more detail and thus, to investigate the connection between AGN luminosities and the host galaxy morphologies, hopefully shedding more light on the main trigger mechanisms for AGN activity in different redshift and luminosity regimes. Such future simulations will also help to understand the dependency of the AGN driving mechanisms on the large-scale environment. \nIn addition, we plan to further improve the current implementations by taking the angular momentum of the accreted material into account, which in turn would allow to better model the direction of the feedback. This would es- \npecially have an important effect on the spatial distribution of the feedback energy in the surroundings of the AGN. Indeed, current black hole accretion and feedback models are purely empirically motivated and have the major drawback that they do not capture the underlying small-scale physical processes, which is, within the framework of large-scale cosmological simulation, currently not feasible due to limited computational power. Nevertheless, despite of the rather crude approximations, the black hole model, in particular with our new modifications, seems to capture the essence of how black holes grow and how feedback affects the host galaxies in reality. \nFuture observations will improve our understanding of the different accretion modes and their relation to the multi phase nature of the ICM/IGM. In particular, studies of Seyfert galaxies (Mezcua et al. in prep.) will allow an investigation of the role of warm H2 gas (with temperatures of ∼ 10 3 K). In combination with X-ray observations, this will shed more light on the complicated interplay between the various accretion modes of AGN.', 'ACKNOWLEDGMENTS': "We thank the referee for a careful and constructive reading of our paper. We also thank Alexander Beck, Veronica Biffi, Andreas Burkert, Massimo Gaspari, Mar Mezcua, David Schlachtberger, Francesco Shankar and Adelheid Teklu for many fruitful discussions. Particularly, we thank Madhura Killedar for carefully reading the text and editing. Furthermore we would like to thank Shane W. Davis and Helen Russell for providing us with observational data and Veronica Biffi and Umberto Maio for providing us with the data for Fig. 12. \nWe are especially grateful for the support by M. Petkova through the Computational Center for Particle and Astrophysics (C2PAP). Computations have been performed at the 'Leibniz-Rechenzentrum' with CPU time assigned to the Project 'pr86re'. \nThis research was supported by the DFG Cluster of Excellence 'Origin and structure of the universe' and the SFB-Tansregio TR33 'The Dark Universe'. \nMichaela Hirschmann acknowledges financial support from the European Research Council under the European Communitys Seventh Framework Programme (FP7/20072013)/ERC grant agreement n. 202781 and support from the European Research Council via an Advanced Grant under grant agreement no. 321323NEOGAL. \n- M. 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2020JHEP...06..085H
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Islands in Schwarzschild black holes
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2020-01-01
| 25 | 0.45 | 162 |
['black hole physics', 'black hole physics', '-', '-']
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[]
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We study the Page curve for asymptotically flat eternal Schwarzschild black holes in four and higher spacetime dimensions. Before the Page time, the entanglement entropy grows linearly in time. After the Page time, the entanglement entropy of a given region outside the black hole is largely modified by the emergence of an island, which extends to the outer vicinity of the event horizon. As a result, it remains a constant value which reproduces the Bekenstein-Hawking entropy, consistent with the finiteness of the von Neumann entropy for an eternal black hole.
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[]
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https://arxiv.org/pdf/2004.05863.pdf
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{'Koji Hashimoto, Norihiro Iizuka, and Yoshinori Matsuo': 'Department of Physics, Osaka University, \n1-1 Machikaneyama, Toyonaka, Osaka 560-0043, Japan \nE-mail: \[email protected] , [email protected] , \[email protected] \nAbstract: We study the Page curve for asymptotically flat eternal Schwarzschild black holes in four and higher spacetime dimensions. Before the Page time, the entanglement entropy grows linearly in time. After the Page time, the entanglement entropy of a given region outside the black hole is largely modified by the emergence of an island, which extends to the outer vicinity of the event horizon. As a result, it remains a constant value which reproduces the Bekenstein-Hawking entropy, consistent with the finiteness of the von Neumann entropy for an eternal black hole.', 'Contents': '| 1 | Introduction and our strategy | 1 |\n|-----|-----------------------------------------|-----|\n| 2 | No island, no entropy bound | 6 |\n| 3 | Island saves the entropy bound | 7 |\n| | 3.1 Close look at the black hole | 7 |\n| | 3.2 View from a distance | 8 |\n| 4 | Higher dimensions | 10 |\n| 5 | Page time and scrambling time | 12 |\n| | A Early time growth of the entropy | 15 |\n| | B Geodesic distance and extremal volume | 16 |', '1 Introduction and our strategy': "The information paradox [1] is the most fundamental problem in quantum gravity. The Hawking radiation behaves as thermal radiation [2], which implies that the entanglement entropy outside the black hole is monotonically increasing. On the other hand, quantum mechanics requires that the entanglement entropy goes to zero at the end of the evaporation since it must be the pure state. The time evolution of the entanglement entropy is described by the so-called Page curve [3, 4]. Thus, the information loss paradox is translated to the problem how the Page curve is reproduced for the entanglement entropy of the Hawking radiation. \nRecently it was proposed that the Page curve emerges from the effect of islands [5-8]. Regarding the state of the Hawking radiation as that in a region R outside the black hole, the density matrix of R is normally defined by taking the partial trace over the states in R , which is the complementary region of R . According to the prescription of the minimal quantum extremal surface [9-11], states in some regions in R , which are called islands I ( ⊂ R ) , should be excluded from the states to be traced out. Thus, the entanglement entropy of the Hawking radiation R is effectively given by that of states in R ∪ I . Explicitly, the entanglement entropy of the Hawking radiation is give by \nS ( R ) = min { ext [ Area( ∂I ) 4 G N + S matter ( R ∪ I ) ]} , (1.1) \nby using the prescription of the quantum extremal surface. \nThe island rule was first proposed as a result of the conjectured quantum extremal surface prescription, and recently the island rule was derived by using the replica method for the gravitational path integral. When one applies the replica trick [14-16] to gravitational theories, one can fix only the boundary conditions of the replica geometries, and new saddles, where bulk wormholes are connecting different copies of spacetime, need to be taken into account. These new saddles, called replica wormholes , lead to islands [12, 13]. In the semi-classical limit of gravity, the partition function of the geometry with replicas is dominated by that giving the minimum entanglement entropy. In this way, the replica trick for gravitational theories leads to the same formula (1.1) as the quantum extremal surface prescription. \nSince the replica wormhole is merely a consequence of the replica trick in models with gravitation, the island conjecture is expected to be applicable to any black hole. So far, among recent works [5-8, 12, 13, 17-27], the island rule has been studied mainly in two spacetime dimensions, 1 which offer a tractable treatment of the entanglement entropy of the Hawking radiation. In this paper, we make one more step for general black holes. We study the effect of islands in the Schwarzschild black holes, in asymptotically flat spacetime in generic dimensions. Needless to say, the asymptotically flat four-dimensional Schwarzschild black hole is the first solution historically [29] and the simplest and the most interesting black hole. We start with the four-dimensional case. \nThe gravitational part of the action is given by the Einstein-Hilbert action with the Gibbons-Hawking term, \nI = I gravity + I matter , (1.2) \nI gravity = 1 16 πG N ∫ M d 4 x √ -g R + 1 8 πG N ∫ ∂ M d 3 x √ -hK , (1.3) \nwhere G N is the Newton constant. 2 Our goal in this paper is to show that the entanglement entropy of the Hawking radiation of the Schwarzschild black hole follows a Page curve once islands are taken into account. The Schwarzschild black hole metric we consider is \nds 2 = -r -r h r dt 2 + r r -r h dr 2 + r 2 d Ω 2 (1.4) \nwith the horizon radius r h . Its temperature is \nT H = 1 β = 1 4 πr h . (1.5) \nIn the following we summarize our analyses and necessary ingredients for them. First, we will apply the quantum extremal surface (or equivalently, the replica wormhole) prescription to gravity theory with matter fields. We do not resort to holographic correspondences, nor to embedding into higher-dimensional (AdS) spacetime, nor to coupling with an auxiliary system to absorb the radiation. We will use the global two-sided geometry. \nBefore we proceed, we comment on the formula (1.1). The formula (1.1) consists of two terms; the gravitational part 3 of the generalized entropy, S gravity , which is proportional to the total area (or volume for D > 4 ) of the boundaries of an island, ∂I , as [30-33] \nS gravity = Area( ∂I ) 4 G N , (1.6) \nand the matter entanglement entropy S matter on the region I and R in the curved background. Note that the formula given by eq. (1.6) is consistent with our action (1.3). Note also that without islands, the gravitational entropy of an island, I , vanishes. \nUnlike two-dimensions, in four-dimensions, it is well-known that the matter entanglement entropy has area-like divergences, which depend on the short distance cut-off [36, 37]. Therefore, this yields the following divergence for the matter entropy \nS matter ( R ∪ I ) = Area( ∂I ) glyph[epsilon1] 2 + S (finite) matter ( R ∪ I ) , (1.7) \nwhere glyph[epsilon1] is the short distance cut-off scale. 4 This divergence can be absorbed by the renormalization of the Newton constant as [38] \n1 4 G ( r ) N = 1 4 G N + 1 glyph[epsilon1] 2 , (1.8) \nwhere G N is bare Newton constant and G ( r ) N is renormalized Newton constant . In this respect, if we regard G N in the formula (1.1) as the renormalized Newton constant, then the leading cut-off dependent divergence of S matter ( R ∪ I ) in eq. (1.7) is already taken into account, and therefore S matter ( R ∪ I ) yields only a finite contribution, i.e., S (finite) matter ( R ∪ I ) ; Therefore our proposal formula in higher dimensions is \nS ( R ) = min { ext [ Area( ∂I ) 4 G ( r ) N + S (finite) matter ( R ∪ I ) ]} . (1.9) \nBy evaluating the formula (1.9) as the prescription of the minimal quantum extremal surface in higher dimensions, in this paper, we will derive the Page curve. For the finite matter entropy contribution, S (finite) matter ( R ∪ I ) , we will use eq. (1.11) and eq. (1.12). This can be understood as follows. \nThe region for the Hawking radiation R in the Schwarzschild spacetime is the union of two regions R + and R -which are located in the right and left wedges in the Penrose diagram, respectively (see Fig. 1). The distance between R + and R -becomes very large at late times (see appendix B), therefore at late times, the entanglement entropy of the Hawking radiation without islands is expected to be very large and the configuration with islands is expected to give the dominant contribution. \nFirst, we consider the configuration without islands. The matter entanglement entropy we will evaluate is that of separated two regions R + and R -(see Fig. 1 Left). In this case, the finite contributions of the matter entanglement entropy is, S ( R + ∪ R -) minus S ( R + )+ S ( R -) , which is essentially the minus of the mutual information I ( R + ; R -) . 5 This is because the leading contributions of the entanglement entropy of each region, S ( R + ) and S ( R -) , are the divergences of the form (1.7), proportional to the area of the boundary surface. Hence these cut-off dependent leading boundary-area divergences are already taken into account by the renormalized Newton constant and do not contribute to S (finite) matter ( R ∪ I ) . 6 Thus, in the case of no island, the finite part of the matter entanglement entropy is given in terms of the mutual entropy as \nS (finite) matter ( R ∪ I ) = S (finite) matter ( R ) = -I ( R + ; R -) (without island) (1.11) \nNext, we consider the configuration with an island I . At late times, each of two boundaries of I is much closer to the boundary of R in the same wedge, than to the boundaries of R and I in the other wedge (see Fig. 1 Right). 7 The correlation between the left and right wedges is negligible since the neighboring boundary (hyper)surfaces of different regions behaves like charges with opposite sign. Thus the total entanglement entropy is well dominated by that in each side separately. In fact, in the case with an island, a symmetric configuration will give the minimal entropy. We may consider only the right wedge, since the contributions from the right and left wedges will be equal to each other, so in total it is twice of the contribution of only the right wedge. The finite contributions of the matter entanglement entropy from only the right wedge is S ( R + ∪ I ) minus S ( R + ) + S ( I ) , which is essentially the minus of the mutual information I ( R + ; I ) . Therefore, in the case with an island, the finite part of the matter entanglement entropy is given as \nS (finite) matter ( R ∪ I ) = -2 I ( R + , I ) (with an island I ) (1.12) \nIn this paper, for the finite matter entropy contribution, S (finite) matter ( R ∪ I ) for eq. (1.9), we use eq. (1.11) and eq. (1.12). \nThe generic expression for the mutual information I ( A ; B ) in curved spacetimes is not known, so we need to make some assumptions and take some limits. In this paper, we \nI ( A ; B ) ≡ -S ( A ∪ B ) + S ( A ) + S ( B ) . (1.10) \n6 There are higher order corrections to the area terms of the entanglement entropy, which are also renormalized into the higher order gravitational constants. \n7 The distance between the right and the left boundaries is characterized by the volume of an extremal surface connecting the boundaries. The calculation of that extremal surface resembles that of a holographic complexity ('complexity = volume' conjecture [34, 35]). As shown in appendix B, at late times, the volume of the three-dimensional extremal surface in the analytically continued Schwarzschild geometry grows linearly in time. The section of this extremal surface is at most 4 πb 2 where b is the value of the radial coordinate of the Schwarzschild geometry, thus the extremal three-dimensional surface is a long cylinder at late times. This means that the four-dimensional matter free fields can be treated as a two-dimensional massless fields which are the lowest mode in the KK towers. \n<!-- image --> \nFigure 1 . Penrose diagram of the static Schwarzschild spacetime without island (left) and that with an island I (right). The region R whose states are identified with the Hawking radiation has two parts R + and R -, which are located in the right and the left wedge, respectively. The boundary surfaces of R + and R -are indicated as b + and b -, respectively. The island extends between the right wedge and the left wedge. The boundaries of I are located at a + and a -. At late times, the distance between the right wedge and the left wedge is very large. \n<!-- image --> \nconsider only free massless matter fields. We use the following two limits: the distance between the boundary surfaces of A and B is (i) large or (ii) small, compared to the scale of the size of the boundary surfaces. \n- (i) When the distance is much larger than the correlation length of the massive modes in the KK tower of the spherical part, only the s-waves can contribute to I ( A ; B ) . The mutual information I ( A ; B ) is approximated by that of the two-dimensional massless fields, \nI ( A ; B ) = -c 3 log d ( x, y ) (1.13) \nwhere c is the central charge and d ( x, y ) is the distance between x and y which are the boundaries of A and B , respectively. \n- (ii) When the distance L between the parallelly placed boundary (hyper)surfaces of A and B is sufficiently small, the mutual information I ( A ; B ) is given by [39, 40] \nI ( A ; B ) = κc Area L 2 (1.14) \nfor c free massless matter fields, where κ is a constant. 8 Although the formula above is for the flat spacetime, we expect that it can be used when the length scale of the curvature is large compared to L . \nIn this paper, we evaluate the entanglement entropy of the Hawking radiation in the asymptotically flat eternal Schwarzschild black hole, and investigate the effect of the islands, by using the formulae eqs. (1.9), (1.11) and (1.12) with eqs. (1.13) and (1.14). Sec. 2 shows \nthat the entropy without the island grows linearly in time at late times. In Sec. 3 we include the island and the extremization about the location of it results in the time-independent behavior of the entropy. We treat two cases: Sec. 3.1 for the entanglement region R close to the horizon, and Sec. 3.2 for R far away from the horizon. Both cases show that the emergent island extends to the outer vicinity of the horizon, and the total entropy at late times is almost twice the Bekenstein-Hawking entropy formula. For simplicity the calculations are given for D = 4 spacetime dimensions, and in Sec. 4 we show that all results are qualitatively the same in generic higher dimensions. Finally in Sec. 5 we draw the Page curve and discuss the Page time and the scrambling time. From now on, we write the renormalized Newton constant simply as G N .", '2 No island, no entropy bound': 'In this section we evaluate the entanglement entropy at late times, in the case of the absence of the island. For the two-dimensional s-wave approximation to make sense, all the points in eq. (1.13) need to be well-separated, compared to the scale of the radius of the sphere. In the absence of the island, we have only two points which are the boundaries of the entanglement regions at the right R and the left R (see Fig. 1 Left). So, at late time, we can use the formulas (1.11) and (1.13) as \nS matter = c 3 log d ( b + , b -) (2.1) \nwhere b + and b -stand for the boundaries of the entanglement regions in the right and the left wedges of the Schwarzschild geometry. Here ( t, r ) = ( t b , b ) for b + and ( t, r ) = ( -t b + iβ/ 2 , b ) for b -, respectively. 9 Following a conformal map, we find that the matter part of the entanglement entropy in the Schwarzschild geometry is \nS matter = c 6 log ( U ( b -) -U ( b + )) ( V ( b + ) -V ( b -)) W ( b + ) W ( b -) . (2.2) \nHere the Schwarzschild metric in the Kruskal coordinates is given by \nds 2 = -dUdV W 2 + r 2 d Ω 2 , (2.3) \nwhere we have defined the coordinates as \nr ∗ = r -r h + r h log r -r h r h , (2.4) \nU ≡ -e -t -r ∗ 2 r h = -√ r -r h r h e -t -( r -r h ) 2 r h , V ≡ e t + r ∗ 2 r h = √ r -r h r h e t +( r -r h ) 2 r h . (2.5) \nThe conformal factor W of the Schwarzschild black hole geometry is \nW = √ r 4 r h UV r -r h = √ r 4 r 3 h e r -r h 2 r h . (2.6) \nThen, the total entanglement entropy is calculated as \nS = c 6 log [ 16 r 2 h ( b -r h ) b cosh 2 t b 2 r h ] . (2.7) \nAs we mentioned the two-dimensional approximation is valid for late time, t b glyph[greatermuch] b ( > r h ) , so the above result is approximated as \nS glyph[similarequal] c 6 t b r h , (2.8) \nwhich grows linearly in time. \nAt the late times where \nc t r h glyph[greatermuch] r 2 h G N , (2.9) \nthis entropy becomes much larger than the black hole entropy. This contradicts with the finiteness of the von Neumann entropy for a finite-dimensional black hole system. In such a case, an island is expected to emerge. In the next section, we calculate the entanglement entropy with a single island and show that in fact the Page curve is reproduced once we take into account the effects of an island.', '3 Island saves the entropy bound': 'In this section we calculate the entanglement entropy with a single island. The configuration is shown in Fig. 1 Right. To capture the entropy of the full degrees of freedom of the radiation, the entanglement region R had better be close to the event horizon. So first we consider such a case of looking at the black hole closely, b -r h glyph[lessmuch] r h , in Sec. 3.1. Then later in Sec. 3.2 we consider the other case when the boundary of the region R is far away from the horizon, that is, a view from a distance.', '3.1 Close look at the black hole': 'Let us consider the case b -r h glyph[lessmuch] r h , the close look at the black hole from the region R . The boundaries of the island I , a ± , are located at ( t, r ) = ( t a , a ) for a + and ( t, r ) = ( -t a + iβ/ 2 , a ) for a -. It is plausible that t a = t b would extremize the entropy, and in this subsection we assume it. We also assume that at late times, due to the fact that the left wedge and the right wedge are separated by the volume growing linear in time (see appendix B), we just need to focus on the right-hand side of the Penrose diagram for the calculation of the entropy (and the final result is twice of it). Then the total entropy is \nS glyph[similarequal] 2 πa 2 G N -2 κc 4 πb 2 L 2 , (3.1) \nwhere the distance between the end point of the island I and that of the entanglement region R is the geodesic distance, \nL = ∫ b a dr √ 1 -r h r . (3.2) \nWeneed to extremize the entropy with respect to a which is the location of the boundary of the island. Physically, when we regard the entropy S as a potential energy for a particle located at r = a , this extremization is due to the harmonic (gravitational) potential 2 πa 2 G N and the attractive potential -2 κc 4 πb 2 L 2 which pushes the particle closer to r = b . The entropy formula (1.14) is valid only if L glyph[lessmuch] a , which is fine because we here consider the case b -r h glyph[lessmuch] r h (and resultantly a -r h glyph[lessmuch] r h ). \nIn that case, the geodesic distance (3.2) is \nL glyph[similarequal] 2 √ r h ( √ b -r h -√ a -r h ) . (3.3) \nTo minimize the entropy (3.1) with respect to a , we change the variable to x ≡ √ a -r h r h , and consider the equation ∂S ∂x = 0 which is equivalent to \nx ( √ b -r h r h -x ) 3 = κc G N 2 r 2 h , (3.4) \nwhere approximations x glyph[lessmuch] 1 and b ≈ r h are taken into account. This equation has at most two solutions for x . The minimization occurs at a smaller x solution, satisfying x glyph[lessmuch] √ b -r h r h , and with the fact that the right-hand side of eq. (3.4) is very small, we find the location of the island as \na = r h + ( κc G N ) 2 4( b -r h ) 3 . (3.5) \nSo the boundary of the island is located very close to, and slightly outside of, the black hole horizon. \nSubstituting this expression to the total entropy, we find \nS = 2 πr 2 h G N -2 πκc r h b -r h . (3.6) \nWe find a very natural interpretation of this result. First of all, this is constant, as opposed to the late time result without the island, eq. (2.8). Therefore, the configuration with the island is preferred, and the entropy stops growing at late times. The first term in eq. (3.6) is exactly (twice of) the Bekenstein-Hawking entropy formula [2, 41]. The second term is the effect of the quantum matter.', '3.2 View from a distance': 'Next, let us consider the case when the boundary r = b of the entanglement region R is far away from the horizon, b glyph[greatermuch] r h . In this case, we assume that the s-wave approximation is valid, 10 and use the matter entropy formula (1.13) for calculating the total entropy. \nThe entanglement entropy for the conformal matter is given by \nS matter = c 3 log d ( a + , a -) d ( b + , b -) d ( a + , b + ) d ( a -, b -) d ( a + , b -) d ( a -, b + ) . (3.7) \nUsing the Kruskal coordinates given in Sec. 2, the total entanglement entropy is calculated as \nS = 2 πa 2 G N + c 6 log [ 2 8 r 4 h ( a -r h )( b -r h ) ab cosh 2 t a 2 r h cosh 2 t b 2 r h ] + c 3 log cosh ( r ∗ ( a ) -r ∗ ( b ) 2 r h ) -cosh ( t a -t b 2 r h ) cosh ( r ∗ ( a ) -r ∗ ( b ) 2 r h ) +cosh ( t a + t b 2 r h ) , (3.8) \nwhere \ncosh r ∗ ( a ) -r ∗ ( b ) 2 r h = 1 2 [ √ a -r h b -r h e a -b 2 r h + √ b -r h a -r h e b -a 2 r h ] . (3.9) \nThe island is expected to show up near the black hole horizon, so we assume a ∼ r h and check if this approximation is correct or not, later. For a ∼ r h , the second term in the right-hand side of eq. (3.9) dominates, so we ignore the first term. \nLet us consider the late time behavior. We take the late time approximation 11 \n1 2 √ b -r h a -r h e b -a 2 r h glyph[lessmuch] cosh t a + t b 2 r h . (3.10) \nWe also consider the approximation \ncosh t a -t b 2 r h glyph[lessmuch] 1 2 √ b -r h a -r h e b -a 2 r h (3.11) \nwhich will be checked to be satisfied later. Then the entanglement entropy (3.8) is approximated as 12 \nS = 2 πa 2 G N + c 6 log [ 2 8 r 4 h ( a -r h )( b -r h ) ab cosh 2 t a 2 r h cosh 2 t b 2 r h ] -c 3 log [ 1 2 √ a -r h b -r h e a -b 2 r h cosh t a + t b 2 r h ] -2 c 3 √ a -r h b -r h e a -b 2 r h cosh t a -t b 2 r h = 2 πa 2 G N + c 6 log [ 16 r 4 h ( b -r h ) 2 ab e b -a r h ] -2 c 3 √ a -r h b -r h e a -b 2 r h cosh t a -t b 2 r h . (3.14) \nd ( b + , b -) glyph[similarequal] d ( a + , a -) glyph[similarequal] d ( b ± , a ∓ ) glyph[greatermuch] d ( b ± , a ± ) , (3.12) \nand the entanglement entropy of the matter is approximated as \nS matter = c 3 log [ d ( a + , b + ) d ( a -, b -)] . (3.13) \nThis simplified expression indeed results in the expression same as eq. (3.14). \nThis allows a local minimum at \na glyph[similarequal] r h + ( cG N ) 2 144 π 2 r 2 h ( b -r h ) e r h -b r h cosh 2 t a -t b 2 r h , (3.15) \nand with that value of a the total entropy (3.14) is calculated as \nS = 2 πr 2 h G N + c 6 log [ 16 r 3 h ( b -r h ) 2 b e b -r h r h ] -c 2 G N 36 πr h ( b -r h ) e r h -b r h cosh 2 t a -t b 2 r h . (3.16) \nWe vary this expression for t a and find that t a = t b extremizes it. For t a = t b , the value of a given in eq. (3.15) in fact satisfies eq. (3.11). The late time condition (3.10) is rewritten as \nr h log r h ( b -r h ) c G N glyph[lessmuch] t b . (3.17) \nThen in eq. (3.16) we put t a = t b and ignore higher order terms in G N , to obtain the final expression of the entanglement entropy \nS = 2 πr 2 h G N + c 6 [ log ( 16 r 3 h ( b -r h ) 2 b ) + b -r h r h ] . (3.18) \nThis does not grow in time. The interpretation of this result is the same as in the result of the close look, eq. (3.6). The first term, which emerged as a result of the island, provides the Bekenstein-Hawking entropy formula [2, 41] for the four-dimensional Schwarzschild black hole. \nSo summarizing the close look result (3.6) and the distant look result (3.18), we have confirmed that the island shows up at late times and the entropy growth disappears. The boundary of the island is located very close to the event horizon, and in fact the island provides the renowned Bekenstein-Hawking entropy of the Schwarzschild black hole.', '4 Higher dimensions': "The arguments presented in this paper can go through for the case of Schwarzschild black holes in higher spacetime dimensions, D ≥ 4 . In this section, we provide results in generic D dimensions, and find that the results obtained in Sec. 2 and Sec. 3 are universal. \nThe Schwarzschild metric in D dimensions is \nds 2 = -f ( r ) dt 2 + dr 2 f ( r ) + r 2 d Ω 2 D -2 , f ( r ) ≡ 1 -r D -3 h r D -3 . (4.1) \nThe area of the ( D -2) -sphere at radius r is r D -2 Ω D -2 , where Ω D -2 is the volume of the unit ( D -2) -sphere. \nFirst, we look at the case with no island. Similarly to Sec. 2, the Kruskal coordinates are given 13 just by generalizing the factor f ( r ) . We arrive at the expression for the total \nU ≡ -exp [ -( D -3) t -r ∗ 2 r h ] , V ≡ exp [ ( D -3) t + r ∗ 2 r h ] , (4.2) \nentropy at late times, \nS glyph[similarequal] c 6 ( D -3) t b r h . (4.4) \nFor D = 4 this reproduces eq. (2.8). There is no physical difference; the entropy grows linearly in time. \nNext, we consider the entanglement entropy with the island. For the close look at the black hole as in Sec. 3.1, instead of the four-dimensional formula (1.14), we use the D -dimensional formula \nI ( A ; B ) = κ D c Area L D -2 . (4.5) \nHere the constant κ D also depends on D . The total entropy with the island contribution is \nS glyph[similarequal] Ω D -2 2 G N a D -2 -2 κ D c Ω D -2 b D -2 L D -2 (4.6) \nwith the distance in the short distance approximation b, a glyph[similarequal] r h , \nL glyph[similarequal] 2 √ D -3 √ r h ( √ b -r h -√ a -r h ) . (4.7) \nWe minimize S by varying the location a of the boundary of the island, and find \na = r h + ( κ D c G N ) 2 r 2 D -5 h 2 6 -2 D ( D -3) D -2 ( r h b -r h ) D -1 . (4.8) \nSo the island is located very close to the black hole horizon. The total entanglement entropy is found as \nS = Ω D -2 2 G N r D -2 h -Ω D -2 κ D c 2 3 -D ( D -3) ( D -2) / 2 ( r h b -r h ) ( D -2) / 2 . (4.9) \nThis reproduces eq. (3.6) for D = 4 . Again, the contribution of the Bekenstein-Hawking entropy emerges as the island contribution, and there exists a small contribution from the matter field (the second term). \nWe can also work out the higher dimensional case for the view at a distance given in Sec. 3.2. The total entanglement entropy with the island at late times is \nS glyph[similarequal] Ω D -2 2 G N a D -2 + c 6 [ log [ 2 4 r 4 h f ( b ) f ( a ) ( D -3) 4 ] +2( D -3) r ∗ ( b ) -r ∗ ( a ) 2 r h ] -2 c 3 exp [ ( D -3) r ∗ ( a ) -r ∗ ( b ) 2 r h ] cosh [ ( D -3) t a -t b 2 r h ] . (4.10) \nwith r ∗ ≡ ∫ r dr/f ( r ) , giving the metric of the form (2.3) with \nW ≡ ( D -3) 2 r h √ f ( r ) exp [ ( D -3) r ∗ 2 r h ] . (4.3) \nThe location of the boundary of the island is again found to be very close to the horizon, \na = r h + ( c G N ) 2 r 2 D -5 h ( 2 3( D -2)Ω D -2 ) 2 exp [ (3 -D ) ( b r h -1+ g ( b r h ))] cosh 2 [ ( D -3) t a -t b 2 r h ] , (4.11) \nwhere we have defined 14 \ng ( x ) ≡ -x 4 -D D -4 2 F 1 ( 1 , D -4 D -3 , 2 D -7 D -3 ; x 3 -D ) -1 D -3 ( γ +log( D -3)+ Γ ' (( D -4) / ( D -3)) Γ(( D -4) / ( D -3)) ) . \nSubstituting eq. (4.11) to the total entropy, we find again t a = t b extremizes it, and the final expression for the entanglement entropy is \nS glyph[similarequal] Ω D -2 2 G N r D -2 h + c 6 [ log [ 2 4 r 4 h f ( b ) ( D -3) 3 ] +( D -3) ( b r h -1+ g ( b r h ))] . (4.12) \nThe first term of this late-time expression of the entanglement entropy is the BekensteinHawking entropy which emerged from the island. This eq. (4.12) shares the same structure as eq. (4.9).", '5 Page time and scrambling time': 'In this paper, we have calculated the entanglement entropy of the Hawking radiation of the asymptotically flat eternal Schwarzschild black hole in D ( ≥ 4 ) spacetime dimensions, for the configuration without islands and that with an island. We can summarize the findings, eqs. (2.8), (4.4), (3.6), (3.18), (4.9), and (4.12), as follows. The entanglement entropy of a given region R outside of the horizon linearly grows with time t for the configuration without islands; \nS = c 6 ( D -3) t r h , (5.1) \nwhere r h is the Schwarzschild radius and c is the number of massless matter fields. For the case with an island at late times, the saddle point analysis for the boundary of the island a shows that it emerges at the outer vicinity of the horizon, \na = r h + O ( ( c G N ) 2 r 2 D -5 h ) . (5.2) \nThe resultant entanglement entropy for the region R is 15 \nS = 2 S BH + O ( c ) , (5.3) \nwhere S BH is the Bekenstein-Hawking entropy of the Schwarzschild black hole, S BH ≡ Area ( r = r h ) / 4 G N , which is time-independent at late times. O ( c ) effects arise from the quantum effects by the matters. \nFigure 2 . The Page curve for the eternal Schwarzschild black hole. In this plot we ignore terms of higher order in c G N /r D -2 h , which are small compared to t Page or S BH . \n<!-- image --> \nGenerally, the dominant contribution to the entanglement entropy comes from the configuration with a minimum entropy. Thus, for the eternal Schwarzschild black holes, at early times the entanglement entropy is given by that of the configuration without the island, then at late times it is by the one with the island. So the dominant configurations switch at the time identified with Page time t Page , at which the time evolution of the entanglement entropy drastically changes: the linear growth is replaced by a time-independent constant, see Fig. 2. Equating the asymptotic constant value of the entropy (5.3) with the entropy without the island (4.4), we find the Page time for the eternal Schwarzschild black hole, \nt Page = 3Ω D -2 ( D -3) r D -1 h c G N + O ( r h ) . (5.4) \nAlthough the higher order corrections would depend on b , which is the boundary location of the entanglement region R , the leading term is universal. Using the Hawking temperature T H = ( D -3) 4 πr h , the universal term is written as \nt Page = 3 π S BH c T H . (5.5) \nAfter the Page time, the entropy is given by 2 S BH for the Hawking radiation in the both sides of the Penrose diagram. Thus the entanglement entropy for the Hawking radiation observed only in a single side approximately agrees with S BH , as expected. \nLet us compare the Page time with a semiclassical estimate of the lifetime of the black hole [42]. In four dimensions, the radiation power reduces the mass M of the black hole as \ndM dt = -cα G 2 N M 2 (5.6) \nwhere α is a constant dependent on the spin of the radiating particle. Solving this gives a time-dependent Schwarzschild radius as \nr h ( t ) = r h ( t = 0) [ 1 -24 cα G N t ( r h ( t = 0)) 3 ] 1 / 3 , (5.7) \nso the semiclassical estimate of the black hole lifetime is \nt evaporate = ( r h ( t = 0)) 3 24 c αG N = 1 96 π 2 α S BH ( t = 0) c T H ( t = 0) . (5.8) \nOn the other hand, the Page time (5.5) can be modified once we include this semiclassical reduction of the black hole mass. Since the contribution of the change of the geometry to the matter entropy is at higher order, we can just substitute the time-dependent Schwarzschild radius (5.7) to our entropy formulas eqs. (5.1) and (5.3), to evaluate the Page time with the effect of the black hole evaporation. At the early stage, the time evolution of the entropy slightly deviates from the linear growth. And at late times, the entropy is not constant but decreases in time. The intersection of the two curves gives the Page time. We find, in four dimensions, the Page time is obtained by solving \nc 6 t r h ( t ) = 2 4 π ( r h ( t )) 2 4 G N , (5.9) \nwhich yields \nt Page = 3 π S BH ( t = 0) c T H ( t = 0) 1 1 + 2 5 3 2 πα . (5.10) \nLooking at the original eq. (5.5), we find that the last factor in the expression above is due to the backreaction of the black hole evaporation. Comparing this Page time (5.10) with the black hole lifetime (5.8), we see that both are proportional to S BH ( t = 0) / ( c T H ( t = 0)) , so they are at the same order. 16 \nWith the concrete location of the emergent island, we can also discuss the time scale for scrambling. According to the island prescription, the density matrix of the Hawking radiation in R is effectively given by that of R and I . This implies that the information thrown into the island I would be able to be collected from the Hawking radiation. If we send a message from the point r = b toward the island at time t = t 0 , it reaches the island r = a , at time \nt a = t 0 + b -a + r h log b -r h a -r h , (5.11) \nat the earliest. 17 Supposing that the message would be reconstructable from the Hawking radiation once they are in the island, I at t b = t a , we can identify the scrambling time t scr = t a -t 0 , since the information is no longer contained in the black hole but in the Hawking radiation R ∪ I . Using eq. (5.2), this yields the scrambling time estimation as \nt scr glyph[similarequal] 2 r h log r 2 h G N glyph[similarequal] 1 2 πT H log S BH . (5.12) \nSince the leading contribution comes from log G N , the most dominant part of the scrambling time t scr is universal and expressed in terms of the Hawking temperature T H and the Bekenstein-Hawking entropy S BH . This expression is valid in general dimensions, D ≥ 4 . The scrambling time obtained is proportional to 1 /T H log S BH , as predicted first in ref. [43]. \nSeveral comments are in order. In this paper, we have studied only the configuration without islands and that with an island. Configurations with more islands also might contribute to the entanglement entropy. As the configuration with a single island agrees with the entropy of the black hole, those with more islands would not have dominant contributions at late times. They would contribute around the Page time, so that the sharp change of the time evolution of the entanglement entropy may be smoothed away. \nOne remaining problem, which is important in the viewpoint of information, is how the information in the island is transported to the Hawking radiation. We have found that the expected Page curve is reproduced by the effect of the island, and the entanglement entropy of the Hawking radiation agrees with that of the black hole. However these do not tell how the information is restored concretely. Further study of islands will reveal the mystery of the black hole information paradox.', 'Acknowledgments': 'K. H. would like to thank Sotaro Sugishita for discussions on ref. [44]. N. I. would like to thank Takanori Anegawa for various helpful discussions on related project [27]. This work is supported in part by JSPS KAKENHI Grant Number JP17H06462 (K.H.), JP18K03619 (N.I.).', 'A Early time growth of the entropy': 'In this appendix we study the early time growth of the entanglement entropy. At the very early stage of the time evolution, the geodesic distance between the two points is short compared to the scale of the area, so we can use the short distance formula (1.14). \nIn the formula (1.14) the separation L between the two regions R is approximated by the geodesic distance in two-dimensional part of the Schwarzschild spacetime. At the short distance expansion it should coincide 18 with d in Sec. 2, \nL glyph[similarequal] 4 r h √ b -r h r h cosh t 2 r h . (A.1) \nFor this L to be smaller than the scale of the area ∼ r 2 h , we need t glyph[lessmuch] r h log r h b -r h and thus b -r h glyph[lessmuch] r h . \nSubstituting eq. (A.1) to eq. (1.14), we find the total entropy, \nS = -π 2 κc 4 ( b r h -1 ) cosh 2 t 2 r h . (A.2) \nThis grows as ∼ t 2 at early time 0 ≤ t glyph[lessmuch] r h log r h b -r h . \nSo, together with the result given in Sec. 2, we find that the total entanglement entropy grows in time, when we do not include the contribution from the island. At early times it grows as t 2 , and at late times it grows linearly in time. \nNow, let us consider if we need to include the contribution from the island, at early times. We look at the early time behavior of eq. (3.8). 19 Suppose \n1 2 √ b -r h a -r h e b -a 2 r h glyph[greatermuch] cosh t a + t b 2 r h , cosh t a -t b 2 r h (A.3) \nwhich is valid at early times, t a , t b glyph[lessmuch] r h . The entanglement entropy (3.8) is approximated as \nS = 2 πa 2 G N + c 6 log [ 2 8 r 4 h ( a -r h )( b -r h ) ab cosh 2 t a 2 r h cosh 2 t b 2 r h ] -4 c 3 √ a -r h b -r h e a -b 2 r h cosh t a 2 r h cosh t b 2 r h . (A.4) \nThe location of the boundary of the island, r = a , is determined by minimization of eq. (A.4). However, we find no saddle point, as seen in the following way. Writing a -dependent terms in eq. (A.4) by x ≡ √ ( a -r h ) /r h , eq. (A.4) is roughly written as \nS ∼ r 2 h G N x 2 + c log x -cx . (A.5) \nSo, the saddle point equation has the structure \n0 = ∂S ∂x = r 2 h G N x + c x -c . (A.6) \nFor G N c/r 2 h glyph[lessmuch] 1 , this does not allow a solution for x . Since there is no minimum for the entropy as we vary a , we conclude that at early times the island is not generated.', 'B Geodesic distance and extremal volume': "In appendix A, we have argued that the length scale d used in Sec. 2 is equal to the separation L between the two regions R , given by the geodesic distance in the two-dimensional part of the Schwarzschild spacetime, \nL = ∫ dt √ -( 1 -r h r ) + ( 1 -r h r ) -1 ˙ r 2 . (B.1) \nHere we show that it is indeed the case. \nBy integrating the equation of motion of eq. (B.1), or equivalently, since the Hamiltonian must be constant, we obtain a conservation law \nλ = (1 -r h r ) √ -( 1 -r h r ) + ( 1 -r h r ) -1 ˙ r 2 , (B.2) \nwhere λ is a constant. By solving eq. (B.2), we find dt/dr as \ndt dr = 1 ( 1 -r h r ) √ 1 + λ -2 ( 1 -r h r ) . (B.3) \nWe refer the 'reversing' point ˙ r = 0 as (0 , r 0 ) , then we have \nt = ∫ b r 0 dr ( 1 -r h r ) √ √ √ √ ( 1 -r 0 r h ) ( 1 -r 0 r ) , (B.4) \nand the integration constant is determined as λ 2 = r h r 0 -1 . Now, early time corresponds to r 0 glyph[similarequal] r h , so we define r 0 = r h (1 + α ) and b = r h (1 + β ) , and consider the parameter regions α glyph[lessmuch] 1 and β glyph[lessmuch] 1 . The latter is natural as we need a shorter distance. In this approximation, we find that eq. (B.4) is solved for a relation between α and t as \nβ -α = β cosh 2 t 2 r h . (B.5) \nThe geodesic distance (B.1) to which the expression for ˙ r is substituted, is \nL = 2 ∫ b r 0 dr √ r 0 r h 1 √ 1 -r 0 r . (B.6) \nSince 0 < r 0 < r h < b , the integrands above are divergent at r = r h , but the divergence can be regularized by cutting out the region r h -glyph[epsilon1] < r < r h + glyph[epsilon1] with glyph[epsilon1] glyph[lessmuch] r h . In the approximation β glyph[lessmuch] 1 , we find \nL glyph[similarequal] 4 r h √ β -α = 4 r h √ β cosh t 2 r h . (B.7) \nIn the second equality we substituted eq. (B.5). This indeed coincide with eq. (A.1). \nAnother issue which we would like to settle here is the assumption we have used in Sec. 3.1: the growth of the volume V ( t ) of the extremal surface between the boundary at b + and that at b -. The calculation of V ( t ) is quite similar to that of the holographic complexity [44], except that we are now working in the Schwarzschild spacetime. Following the method developed in ref. [44], after a straightforward calculations similar to the above, we find \nV = ∫ b r 0 dr r 4 √ r 4 f ( r ) -r 4 0 f ( r 0 ) , t = ∫ b r 0 dr -√ -r 4 0 f ( r 0 ) f ( r ) √ r 4 f ( r ) -r 4 0 f ( r 0 ) . (B.8) \nwith f ( r ) ≡ 1 -r h /r . If we drop r 4 and r 4 0 from the expression above, they reduce to eq. (B.6) and eq. (B.4). These equations give V ( r 0 ) and t ( r 0 ) where r 0 is the value of r at the 'reversing' point. So, eliminating r 0 , we obtain V ( t ) . Numerically, we can easily find that V ( t ) grows linearly in time at late times, as in the case of the holographic complexity [44].", 'References': "| [1] S. Hawking, 'Breakdown of Predictability in Gravitational Collapse,' Phys. Rev. D 14 (1976), 2460-2473. |\n|-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|\n| [2] S. W. 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2017ApJ...846...82Z
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Constraining Formation Models of Binary Black Holes with Gravitational-wave Observations
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2017-01-01
| 40 | 0.48 | 162 |
['-', 'gravitational waves', 'methods statistical', 'stars black holes', 'stars novae;cataclysmic variables', '-', '-', '-', '-']
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[]
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Gravitational waves (GWs) from binary black hole (BBH) mergers provide a new probe of massive-star evolution and the formation channels of binary compact objects. By coupling the growing sample of BBH systems with population synthesis models, we can begin to constrain the parameters of such models and glean unprecedented knowledge about the inherent physical processes that underpin binary stellar evolution. In this study, we apply a hierarchical Bayesian model to mass measurements from a synthetic GW sample to constrain the physical prescriptions in population models and the relative fraction of systems generated from various channels. We employ population models of two canonical formation scenarios in our analysis—isolated binary evolution involving a common-envelope phase and dynamical formation within globular clusters—with model variations for different black hole natal kick prescriptions. We show that solely with chirp mass measurements, it is possible to constrain natal kick prescriptions and the relative fraction of systems originating from each formation channel with { O }(100) of confident detections. This framework can be extended to include additional formation scenarios, model parameters, and measured properties of the compact binary.
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https://arxiv.org/pdf/1704.07379.pdf
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{'CONSTRAINING FORMATION MODELS OF BINARY BLACK HOLES WITH GRAVITATIONAL-WAVE OBSERVATIONS': 'Michael Zevin 1 , Chris Pankow 1 , Carl L. Rodriguez 2 , Laura Sampson 1 , Eve Chase 1 , 3 , Vassiliki Kalogera 1 , and Frederic A. Rasio 1 \n- 1 Center for Interdisciplinary Exploration and Research in Astrophysics (CIERA) and Dept. of Physics and Astronomy, Northwestern University, 2145 Sheridan Road, Evanston, IL 60208, USA\n- 2 MIT-Kavli Institute for Astrophysics and Space Research, 77 Massachusetts Avenue, 37-664H, Cambridge, MA 02139, USA and 3 LSSTC Data Science Fellow \nDraft version September 11, 2017', 'ABSTRACT': 'Gravitational waves (GWs) from binary black hole (BBH) mergers provide a new probe of massivestar evolution and the formation channels of binary compact objects. By coupling the growing sample of BBH systems with population synthesis models, we can begin to constrain the parameters of such models and glean unprecedented knowledge about the inherent physical processes that underpin binary stellar evolution. In this study, we apply a hierarchical Bayesian model to mass measurements from a synthetic GW sample to constrain the physical prescriptions in population models and the relative fraction of systems generated from various channels. We employ population models of two canonical formation scenarios in our analysis - isolated binary evolution involving a common-envelope phase and dynamical formation within globular clusters - with model variations for different black hole natal kick prescriptions. We show that solely with chirp mass measurements, it is possible to constrain natal kick prescriptions and the relative fraction of systems originating from each formation channel with O (100) of confident detections. This framework can be extended to include additional formation scenarios, model parameters, and measured properties of the compact binary. \nKeywords: gravitational waves, black holes, data analysis, hierarchical Bayesian modeling, stellar evolution', '1. INTRODUCTION': "Recent observations of gravitational waves (GWs) have launched a new branch of observational astronomy. The confident detections of binary black hole (BBH) mergers GW150914, GW151226, and GW170104, as well as BBH candidate LVT151012 by the advanced LIGO (aLIGO) detectors marked the discovery of BBH systems in our universe, and enticed deeper exploration of massive-star evolution (Aasi et al. 2013; Abbott et al. 2016a,c,b, 2017). The final stages of BBH evolution enable the measurement of their physical properties, which connects us to their preceding history and can potentially constrain the environments responsible for facilitating BBH formation, the relative fraction of systems produced through various formation channels, and the physical processes underlying binary stellar evolution. With expected BBH merger rates ranging from 12 to 213 Gpc -3 yr -1 (Abbott et al. 2016e, 2017), it is an opportune time to develop methods for utilizing all existing and future observations to constrain and inform astrophysical models. \nTwo canonical formation channels are generally considered for contributing to the full population of BBHs: isolated binary evolution (i.e., 'the field') and dynamical formation (i.e., 'clusters'). In the isolated evolution scenario, binaries are predicted to evolve and tighten through a common-envelope phase (e.g., Voss & Tauris 2003; Dominik et al. 2013; Belczynski et al. 2014), or through chemically homogeneous evolution of close binaries that attain rapid rotation (de Mink & Mandel 2016; Marchant et al. 2016; Mandel & de Mink 2016). Alternatively, the dynamical channel predicts BBHs that become bound and tighten through three-body encoun- \nters in dense star clusters such as globular clusters (e.g., Portegies Zwart & McMillan 2000; Downing et al. 2010, 2011; Rodriguez et al. 2016a, 2015), galactic nuclei and AGN disks (Antonini & Rasio 2016; Bartos et al. 2017; Stone et al. 2017), or young stellar clusters (Ziosi et al. 2014; Chatterjee et al. 2017). In addition to these canonical scenarios, more exotic formation channels have been suggested for facilitating BBH mergers, such as field triples tightened by Lidov-Kozai cycles (Antonini et al. 2017; Silsbee & Tremaine 2017), primordially formed black holes (Bird et al. 2016), or remnants of population III stars (Inayoshi et al. 2016). While these models maintain the ability to predict heavy BBHs such as GW150914 (e.g., Belczynski et al. 2016; Rodriguez et al. 2016b; Stevenson et al. 2017b), the rates and property distributions predicted from population synthesis simulations are highly sensitive to the prescriptions chosen for uncertain physical processes such as black hole natal kick prescriptions, wind mass-loss, and common envelope physics (Dominik et al. 2012; Stevenson et al. 2015). \nThe analysis of compact binary populations through GW observations provides a unique and powerful mechanism for determining the models that describe the true underlying BBH population. By pairing measured BBH properties from the growing sample of GW observations with population synthesis models that account for various formation scenarios and physical prescriptions, constraints can be placed on the relative fraction of systems produced by each formation channel (i.e., 'branching ratio') and the inherent physical processes that underpin binary stellar evolution. As the merger rates predicted by various channels are highly uncertain and overlap- \nping, this approach to astrophysical model selection in the context of GWs primarily explores the distributions of the BBH source parameters. Neglecting eccentricity and finite size effects, the masses and spins of component black holes primarily determine the GW signal during the evolution of the BBH through inspiral, merger, and ringdown (Abbott et al. 2016d), allowing for parameter estimation of these quantities by comparing a measured GW signal with template waveforms generated from a sample of the physical parameters (Cutler & Flanagan 1994; Poisson & Will 1995; Veitch et al. 2015). \nOne confounding aspect of this model selection problem is the uncertain physical mechanisms underlying population modeling of binary compact objects. To circumvent the intricacies and uncertainties of binary evolution modeling, studies such as Mandel et al. (2017) have taken an agnostic, model-independent approach toward model selection by developing methods for distinguishing populations through clustering of source parameters, such as black hole masses. Though such an approach can help to identify multiple populations, it lacks the ability to directly identify the physical processes inherent to stellar evolution models. To this end, Stevenson et al. (2015) assessed the potential of using GW observations for differentiating population synthesis models that have various prescriptions for common envelope binding energy, maximum neutron star mass, black hole natal kick prescriptions, and stellar winds, finding that certain models could be ruled out in the near-future given expected merger rates. However, the inclusion of alternative formation channels would complicate this process. \nSeveral studies, such as Vitale et al. (2017) and Stevenson et al. (2017a), have performed model selection to infer branching ratios using BBH spin distributions, finding that one may converge upon the branching ratio between field and cluster formation channels with dozens to hundreds of detections. Though spin distributions for differing population models can be constructed purely by geometrical arguments, isolated binary evolution and dynamical formation rely on vastly different procedures for compact binary formation and evolution, and physically motivated modeling is therefore required for accurate and comparable distributions in mass and redshift. Using such models, Rodriguez et al. (2016c) found that certain combinations of masses and spins can be produced exclusively by dynamical formation channels, and the detection of such an outlier could be a clear indication of this formation process. Furthermore, Farr et al. (2017) demonstrated that the clustering of effective spin measurements in the current catalog of GW observations hints at an isotropic spin angle distribution rather than an aligned one. However, there is still much work to be done in utilizing population synthesis and catalogs of GW observations to infer properties of the true compact binary populations. \nIn this paper, we present an approach of hierarchical model selection that utilizes physically motivated models of BBH populations from multiple environments to infer underlying physical prescriptions and branching ratios between formation channels. In particular, we focus on the utility of chirp mass measurements for inferring black hole natal kick prescriptions and the branching ratio between isolated binary evolution and dynamical formation models. However, this approach can easily scale to in- \nclude more measured BBH properties, additional population models, and submodels accounting for different uncertain physical prescriptions. \nThe outline for the paper is as follows. In Section 2, we discuss the BBH population models used in this analysis, which model field binaries (Section 2.1) and cluster binaries (Section 2.2), accounting for selection biases (Section 2.3). Section 3 outlines the algorithm for hierarchical model selection (Section 3.1), the mock observations and analytical approximations for measurement uncertainty (Section 3.2), and sampling procedure (Section 3.3). In Section 4, we discuss our inference on branching ratio (Section 4.1) and kick prescription (Section 4.2) through this methodology. We conclude in Section 5 with a discussion of the analysis and future prospects.", '2. BBH POPULATION MODELS': 'In this section, we describe the population models used in our analysis. These models are identical to those used in Rodriguez et al. (2016c), except that the various natal kick prescriptions used as submodels of the field population are also incorporated into the globular cluster models. Our inference relies on output parameter distributions of BBHs that merge as potential LIGO sources, such as component masses, spin-tilts, and redshifts.', '2.1. Isolated Binary Populations': "Populations of field binaries are generated using an upgraded version of the binary evolution code BSE (Hurley et al. 2002). This code rapidly models stellar populations using metallicity-dependent fits for single-star evolution, while also modeling binary interactions such as mass transfer and changes in orbital angular momentum from black hole natal kicks. Furthermore, BSE now uses a radius-dependent fitting formula for the common envelope binding energy parameter λ (Claeys et al. 2014). Our modifications to this code implement physical prescriptions from more recent work related to stellar winds and supernova prescriptions. These include massloss prescriptions for O and B stars known as the 'Vink prescription' (Vink et al. 2001), metallicity dependence to the evolution of Wolf-Rayet stars (Vink & de Koter 2005), and prescriptions for the supernova mechanism developed in Fryer et al. (2012). Despite these upgrades, many of the physical mechanisms governing binary stellar evolution are still poorly constrained, and incorrect physical prescriptions may propagate inaccuracies to the physical parameter distributions of our populations. We evolve 10 5 binaries in 11 metallicity bins ranging from 0 . 005 to 1 . 5 Z glyph[circledot] , with masses from 18 M glyph[circledot] -150 M glyph[circledot] drawn from an initial mass function of p ( m ) dm ∝ m -2 . 3 dm (see, e.g. Kroupa 2001). Mass ratios are drawn from a uniform distribution on the interval [0 , 1], and initial semi-major axes are drawn from a distribution flat in log space on the interval 10 R glyph[circledot] -10 5 R glyph[circledot] . All binaries that evolve into BBHs are maintained as GW candidates, scaled appropriately by their merger time. \nWhile the following scenarios may not cover all the physical uncertainties in population models, they do provide a representative sample of possibilities. As such, we consider 12 permutations of physical assumptions that all affect the final population parameters: \n1. Three different natal kick prescription models, im- \nFigure 1. Chirp mass distributions for the field and cluster population models. Each panel shows the independently normalized distributions of sources generated (filled histogram) and sources weighted by detectability (unfilled histogram). For reference, the chirp masses of the four likely gravitational-wave events (GW150914, GW151226, GW170104, and LVT151012) are plotted, with the outer lines representing the 90% credible region. The top, middle, and bottom panels show the distributions for fallback, proportional, and full natal kick prescriptions, respectively. We construct each model using one kick magnitude prescription, comprised of equal abundances from the four submodels described in items 2 and 3 of Section 2.1. \n<!-- image --> \nparting different amounts of linear velocity to the newly formed black holes in the binary. One model, the fallback kick prescription, assumes that some fraction of the mass ejected during core collapse will 'fallback' on the black hole: \nV BH kick = (1 -f fallback ) V NS kick . (1) \nThe fraction of material that falls back is proportional to the core mass of the black hole progenitor. The second model, the proportional kick prescription, assumes that the kick imparted to the black hole is reduced by the ratio of the neutron star mass to the black hole mass: \nV BH kick = m NS m BH V NS kick (2) \nwhere we assume m NS = 2 . 5 M glyph[circledot] for all systems, as this value represents the hypothetical 'boundary' between neutron stars and black holes in most population synthesis codes. The final kick prescription, called the full kick prescription, assumes that the black hole kick is equal to the full kick velocity imparted on the neutron star: \nV BH kick = V NS kick . (3) \n- 2. Two differing kick directions. In one model we assume kicks are isotropically distributed in solid angle around the exploding star, which is the common assumption in population models. However, observations of pulsars have suggested a correlation between the kick direction and spin axis (Kaplan et al. 2008), motivating the inclusion of a polar kick prescription where the kicks are confined to 10 · cones about the rotational axis of the progenitor star.\n- 3. Two different methods of accounting for uncertainties in the realignment of the component spin axes after the first supernova. One model allows for realignment of the binary after the first kick, whereas the other model does not realign. Though this does not have an effect on the mass distributions of the field population models, it has a substantial effect on the spin distributions of the resultant BBHs. \nAll these variations in model assumptions largely affect the resultant spin-tilt distributions of the binaries. However, only kick magnitudes play a substantial role in the final distribution of BBH chirp masses. As seen in Figure 1, stronger kick prescriptions flatten out the relative abundance of low-mass binaries in field models; these systems acquire larger linear velocities from the \nkicks, allowing the kinetic energy of the binary component to more easily overcome the gravitational potential and become unbound. As this paper focuses on chirp mass measurements, we construct each model using one kick magnitude prescription, and equal abundances from the four submodels that are described in items 2 and 3 of Section 2.1. Furthermore, we expect only one kick magnitude to be true, whereas kick direction and binary realignment prescriptions may be dependent on processes such as stellar rotation. Future work will incorporate spin measurements in the inference and address these submodels with more detail.", '2.2. Cluster Binary Populations': "In this study, we consider the 'classical' channel of dynamical formation in old, metal-poor globular clusters. Cluster binaries are drawn from a few dozen globular cluster models generated using the Cluster Monte Carlo (CMC) code (see, e.g. Chatterjee et al. 2010). Black holes sink to the centers of globular clusters due to dynamical friction, fostering mass segregation through energy equipartition and turning the globular cluster cores into dynamical factories for forming heavy stellar-mass BBHs. Though these models are sensitive to initial conditions, dynamically formed binaries rely on single-star evolution and N -body dynamics rather than binary stellar evolution and therefore maintain fewer uncertainties in the physical processes (e.g., common envelope evolution, mass transfer) involved in generating BBHs (Rodriguez et al. 2016a). However, as the choice of kick magnitude prescription may alter the distribution of black holes within a cluster or eject black holes entirely, we include cluster submodels with the same variations in kick prescription presented for the field models in Section 2.1. This ensures a putative population mixing has a consistent kick prescription. As seen in Figure 1, the stronger kick magnitude prescriptions tend to flatten out peaks in the normalized chirp mass distributions for the cluster population as well as the field population. Note that in the cluster case the natal kicks do not act to disrupt individual binary systems, as the final partners are usually found long after the components have evolved into compact objects. \nThe globular cluster models generated in Rodriguez et al. (2016a) used the fallback prescription for black hole natal kicks described above. To create equivalent populations using the proportional and full kick prescriptions, we implement the following approximate procedure: for each of the 48 globular cluster models from Rodriguez et al. (2016a), we take the initial ( t = 0) snapshot of the cluster, and evolve the massive stars (above 18 M glyph[circledot] ) forward with BSE until the stars have completed their evolution and the initial population of black holes has formed. We then record the velocity of the natal kick, and only retain those black holes for which V kick < √ -2Φ( r ), where Φ is the gravitational potential of the cluster, and r is the initial radial position of the star in the cluster. 1 \n1 Because the black hole-formation timescale for massive stars ( ∼ 5Myr) is significantly smaller than the mass-segregation timescale ( ∼ 100 Myr, see O'Leary et al. 2006), we can safely ignore the change in position of the star between birth and black holeformation. \nOnce we have an initial population of black holes for each cluster model, we proceed to eject black holes and BBHs from our synthetic population, assuming that the rate at which black holes are ejected is identical to that found with CMC. 2 We select black hole masses without replacement from a list of N black holes, according to \np ( i ) di ∝ i 2 di for 1 < i < N / 3 (4) \nwhere i is the index of the list of black holes, sorted in order of decreasing mass. Equation (4) is physically motivated by the fact that globular clusters preferentially eject the most massive black holes first, continuing to eject black holes until depletion (Morscher et al. 2015). The functional form of p ( i ) was found through trial-anderror to reproduce the masses and mass ratios of ejected BBHs from the CMC simulations. For every BBH, we also remove four single black holes from the list (see, e.g. Heggie & Hut 1993). The reasoning for this is that once a binary is nearing the hardness necessary to eject it from the cluster, the scatterings it undergoes will eject single objects with the same strong three-body encounters responsible for hardening and ejecting the BBH. However, since the binary is about twice as massive as the single objects that are scattered, there are a few scatterings where the velocity of the single object surpasses the cluster escape speed, while the velocity of the binary does not. Numerical tests indicate that on average 3-4 single black holes are ejected for each binary that is ejected. Finally, at the time of ejection, we set the eccentricity and semi-major axis of the binary using the half-mass radius and mass of the cluster at the time of ejection, according to: \nP ( e ) de = 2 e de (5) \nP ( a | M GC , R h , µ bin ) da = 1 aσ √ 2 π × (6) exp -( log µ bin R h aM GC -a ∗ ) 2 2 σ 2 da \nwhere µ bin = ( m 1 m 2 ) / ( m 1 + m 2 ) is the reduced mass of the binary, and a ∗ and σ are the parameters of a lognormal distribution with mean a ∗ = 3 . 98 and σ = 0 . 59 (see Rodriguez et al. 2016a, Equations (7) and (8)). The merger time of each binary is computed by adding the time each binary is ejected (assuming all globular clusters to be 12 Gyr old) to the GW merger time from Peters (1964). The redshift assigned to each merger is the redshift at that cosmological lookback time, assuming a flat ΛCDM cosmology with Ω M = 0 . 306 and H 0 = 67 . 9kms -1 Mpc -1 (Planck Collaboration et al. 2016). \nOnce we have a population of ejected BBHs from clusters with different kick prescriptions, we resample the \n2 This assumption is well justified, as it is the total energy flux of the cluster, not of the black hole sub-system, that determines the ejection rate of black holes from a globular cluster (Breen & Heggie 2013). However, for clusters where there are not sufficiently many black holes to meet the energy requirements of the cluster (e.g., the full kick prescription), this assumption will overestimate the early ejection rate of black holes. \npopulation of BBHs to better represent what we expect to see from globular clusters in the local universe. First, we draw a population of binaries from our effective globular cluster models by preferentially selecting binaries from globular clusters closer to the peak of the observed globular cluster mass function (i.e., massive globular cluster models that more-closely resemble the population of observed globular clusters in the local universe; see Rodriguez et al. 2016a). We then take this population of BBHs from globular clusters, where we have assumed a universal globular cluster mass function and constant spatial density of globular clusters, and create a 3D kernel-density estimate (KDE) of the binary mergers in m 1 , m 2 , and redshift. We then draw as many binaries as we want from this distribution using an MCMC (Foreman-Mackey et al. 2013), with the KDE as our likelihood and a prior on the redshift, which is uniform in comoving volume. \nThe stronger kick prescriptions (i.e., full kicks) retain more low-mass binaries relative to their high-mass counterparts. This is because in the full kick case the natal kick velocity does not decrease with increasing black hole mass, contrary to fallback and proportional kicks (see equations 1-3. Therefore, the full kick prescription will kick out all black holes from the cluster with equal likelihood regardless of the black hole mass, whereas the velocity of the kick is stifled for higher-mass objects in the fallback and proportional cases, allowing more of these objects to be retained relative to their lower-mass analogs. However, as the stronger kick prescriptions will cause more newly formed black holes to be ejected from the cluster, it also results in a decrease to the overall merger rate. The above procedure is necessary to generate new BBH populations without having to generate new, computationally expensive models of massive globular clusters. We use this approximate method for all three populations, including the fallback prescription (for which we do have complete CMC models from Rodriguez et al. 2016a). This was done to avoid any systematic differences that our approximate technique may have introduced, and it was found that this method matched well with the true fallback population generated from CMC. Finally, we assume that the isotropic and polar kick models for clusters should be identical, since BH retention in clusters should be independent of kick angle.", '2.3. Incorporation of Selection Biases': 'The distributions described above represent all BBH systems that are generated by these populations in the local universe. As the detectability of a given source is dependent on both its physical and orientation parameters (e.g., masses, spins, redshift, frequency content, detector network antenna pattern, inclination), the distribution of observed parameters will be different from the true source distribution. Therefore, we translate the raw source distributions into distributions of detectable sources by the expected design-sensitivity power spectrum and antenna pattern of a single detector assuming isotropic sky location and inclination distributions. A signal-to-noise (S/N) threshold of 8 is applied, defined by \nρ 2 = 4 glyph[Rfractur] ∫ ˜ h glyph[star] ( f ) ˜ h ( f ) S n ( f ) df (7) \nwhere ˜ h ( f ) is the gravitational waveform in the frequency domain and S n ( f ) is the one-sided power spectral density of the noise. 3 We then calculate the probability of a system with a given mass and redshift passing this threshold and then weight the distributions accordingly. 4 As seen in the right panel of Figure 1, this tends to flatten out the low-mass peaks and amplify the number of higher-mass systems.', '3. MODEL SELECTION': 'With population models in hand, we leverage BBH mass measurements to infer properties of the underlying distribution. The two questions we aim to address in this paper are as follows: \n- 1. Given a catalog of N BBH chirp mass measurements from GW observations that come from a population made up of field and cluster binaries with a particular branching ratio ( β ), how well can one discern the inherent black hole natal kick prescription?\n- 2. Assuming one prescription is correct, how many observations are required to confidently converge on the true value of this branching ratio? \nWe now describe the machinery behind this inference. \nFigure 2. Example realization of the sampling, where 5000 samples are drawn from the RJMCMC chain. This particular realization is for 100 observations from the fallback model, with a cluster branching ratio of β = 0 . 4. β C and β F are the fraction of systems that are drawn from the cluster and field populations, respectively. The left panels show the value of β inferred for each step in the sampling, with colors indicating the model chosen. The right panels show the total binned histograms. \n<!-- image --> \n3 For S/N calculations, we assume non-spinning component black holes and utilize the IMRPhenomPv2 waveform approximate (Khan et al. 2016). \n4 Though the cluster models maintain redshift information, our field populations do not. We assume for simplicity that field binaries are distributed uniformly in comoving volume in the local universe and sample redshifts accordingly.', '3.1. Hierarchical Modeling': "As described by Mandel (2010) and Hogg et al. (2010), among many others, the objective of hierarchical modeling is to infer a set of model parameters glyph[vector] λ given N observations { x i } , which are characterized by a set of physical parameters { glyph[vector] θ i } and constrained by prior assumptions { glyph[vector] α i } . The astrophysical model described by parameters glyph[vector] λ gives a probability distribution for physical parameters, in our case chirp masses. By Bayes' theorem, the posterior on glyph[vector] λ is \np ( glyph[vector] λ | glyph[vector] θ ) = p ( glyph[vector] θ | glyph[vector] λ ) p ( glyph[vector] λ ) p ( glyph[vector] θ ) (8) \nwhere p ( glyph[vector] θ | glyph[vector] λ ) is the likelihood of observing a particular set of physical parameters, p ( glyph[vector] λ ) is the prior on the model parameters, and p ( glyph[vector] θ ) is a normalization constant. \nHowever, as we observe N independent GW signals rather than the physical parameters directly, we rewrite the likelihood as \np ( { x i }| glyph[vector] λ ) = N ∏ i =1 p ( x i | glyph[vector] λ ) = N ∏ i =1 ∫ d glyph[vector] θp ( x i | glyph[vector] θ ) p ( glyph[vector] θ | glyph[vector] λ ) (9) \nAgain applying Bayes' Theorem, we write p ( x i | glyph[vector] θ ) as p ( glyph[vector] θ | x i ) p ( x i ) /p ( glyph[vector] θ ) to get \np ( { x i }| glyph[vector] λ ) = N ∏ i =1 p ( x i ) ∫ d glyph[vector] θ p ( glyph[vector] θ | x i ) p ( glyph[vector] θ | glyph[vector] λ ) p ( glyph[vector] θ ) (10) \nFigure 3. Mock observations drawn from a population specified by a kick prescription and branching ratio. This particular realization draws five observations from the fallback kick population model with β = 0 . 4. Each panel shows the distributions for a given kick prescription, where the blue and orange lines represent the field and cluster models, respectively, and the dashed black line shows the (normalized) combined population given the value of β . The box-and-whisker plots at the bottom of each panel show the median value of the posterior samples for an observation with an orange line, the upper and lower quartile of these samples as the edges of the box, and the maximum and minimum value of the posterior samples as whiskers. In this case, all observations are drawn from the fallback population model (dotted line in the middle panel). \n<!-- image --> \nwhere p ( glyph[vector] θ ) is the prior on the physical parameters that are used to generate the posterior samples. \nWe approximate the integral as a discrete sum over posterior samples \n∫ d glyph[vector] θp ( glyph[vector] θ | x i ) f ( glyph[vector] θ ) ≈ 1 S S ∑ k =1 f ( glyph[vector] θ k ) (11) \nand ignore the multiplicative constant p ( x i ) as it is not a function of glyph[vector] θ or glyph[vector] λ and will not affect the sampling of the posterior. Therefore, the full expression for the likelihood that we wish to sample is \np ( { x i }| glyph[vector] λ ) = N ∏ i =1 1 S S ∑ k =1 p ( glyph[vector] θ k | glyph[vector] λ ) p ( glyph[vector] θ k ) (12) \nwhere, again, N is the number of observed events, S is the number of posterior samples, glyph[vector] θ k are the astrophysical parameters, and glyph[vector] λ are the model parameters. \nWe aim to do model selection between different kick prescriptions while simultaneously performing inference on branching ratios between field and cluster formation channels. The parameters of our astrophysical model are thus glyph[vector] λ = ( ι, β ), where ι is an indexing parameter that indicates the kick prescription ( ι ∈ [0 , 1 , 2] where 0, 1, and 2 designate the proportional, fallback, and full kick prescriptions, respectively) and β is the branching ratio parameter, defined as the fraction of observations that are drawn from cluster models (0 ≤ β ≤ 1). \nAs the branching ratios between the various formation channels are highly uncertain, we maintain minimal assumptions on our prior knowledge of the model parameters. For the prior on β , we use a Dirichlet distribution, which is a multivariate generalization of the beta distribution. This allows for minimal prior assumptions while ensuring the values of β for all channels sum to unity. Though X-ray binary observations (Repetto et al. 2017) and the current catalog of BBH observations (Belczynski et al. 2016) provide moderate evidence for certain black hole natal kick prescriptions, we use a uniform prior on the kick prescription, which for this discrete parameter puts equal weight on each prescription.", '3.2. Mock observations': "We represent the chirp mass distributions for our populations with a Gaussian KDE and draw 'observations' from this model. In practice, the observations themselves are manifested as a set of samples drawn from a posterior computed for each candidate event. Instead of employing very accurate, but computationally expensive, Markovian methods to estimate the parameter posteriors, we instead use the Fisher matrix as a proxy for the inverse covariance of a simpler Gaussian parameter distribution (see Cutler & Flanagan 1994). We are justified in this procedure in the case of chirp mass ( M c ), because it determines the leading-order evolution of GWs from compact binary coalescence and is therefore the best-measured physical property from a GW signal (see Abbott et al. 2016d, and references therein). However, this methodology is less accurate for parameters that are more correlated and less constrained, such as effective \nFigure 4. Convergence on the true value of β as a function of number of observations. The dark and light shadings represent the 68% and 90% credible intervals, respectively . The black vertical lines show the point in our discrete samplings of N obs at which the 68% credible interval for β is constrained to less than 20% of the full range of β ; due to our discrete sampling this threshold is in fact reached before this point. The left, center, and right panels show the convergence on β when the injected kick prescription is fallback, proportional, and full, respectively. Note that the convergence rate varies depending both on the injected value of β and the injected kick prescription. \n<!-- image --> \nspin and symmetric mass ratio (see, e.g. Vallisneri 2008; Rodriguez et al. 2013; O'Shaughnessy et al. 2014). \nAs the Fisher matrix tends to overestimate the distributional width in M c , this also provides a conservative estimate for the true measurement uncertainty. However, as chirp mass measurements are highly constrained, we find that the inclusion of measurement uncertainty does not drastically affect our results. Nonetheless, we draw 100 mock posterior samples from a Gaussian distribution with a mean centered on the true value and standard deviation σ M c as each 'observation'. \nN smeared observations are drawn from a 'true' distribution, described by a particular natal kick prescription and a value of the branching ratio β (such that N cluster = βN and N field = (1 -β ) N ). Figure 3 shows one realization of this procedure. We then use these mock observations as the basis behind our statistical inference through hierarchical modeling.", '3.3. Sampling': "The technique we use for sampling the posterior on the model parameters is Reverse Jump Markov Chain Monte Carlo (RJMCMC; see e.g. Green 1995). In this method, the calculation of the Bayes factor between two models does not require the explicit calculation of the evidence integral. Rather, the model itself becomes a parameter of the chain. Depending on which value an indexing parameter takes, the likelihood and prior are evaluated using one of a set of models, which may or may not be of the same dimensionality. Said another way, the sampler first 'jumps' in model index space, and then estimates the value for β within the particular model it lands. We assume that all models contain the same set of model parameters with the same meaning - in other words, the branching ratio β is the fraction between \nfield and cluster in all models. The samples can then be sorted by model index at the end in order to generate posteriors for the individual kick prescriptions. Since our models have the same number of parameters with the same priors, the Bayes factor is then simply the ratio of the number of iterations that the chain spends in each model: \nB ij = # of iterations in model i # of iterations in model j (13) \nWe use the open source MCMC python library emcee (Foreman-Mackey et al. 2013) for the implementation of this algorithm.", '4. ASTROPHYSICAL INFERENCE WITH CHIRP MASS MEASUREMENTS': "We now seek to utilize this inference for constraining branching ratios and properties of stellar evolution, given a catalog of BBH observations. Because the current number of BBH observations is likely too few to make any substantial claim about formation channels or physical prescriptions, we demonstrate the method using a mock catalog of BBH observations. We then apply one kick prescription model and formation channel branching ratio value to be the 'correct' description of nature, and gauge how well one can converge on these injected values over an increasing number of observed systems. For the purposes of this study, we consider 100 realizations for each combination of the branching ratio β , kick prescription ι , and number of observations N obs .", '4.1. Branching Ratios': "Convergence to the true branching ratio is a strong function of the number of observations drawn from the \nFigure 5. Convergence on the branching ratio β for various number of observations and injected values of β . Colors represent the number of observations, and are identical to the colors used in Figure 4. All observations are drawn from a population in which the 'fallback' model is deemed true. In each panel, the 10 realizations of the inference on β are shown as faded lines, and the dark line is the median value of 100 realizations at each bin of the histogram. The dashed black vertical line marks the injected value of β . \n<!-- image --> \ntrue population, and is also sensitive to the injected branching ratio itself. To summarize the convergence as a function of observations, we plot the marginalized posterior on β for different values of β in Figure 4. To demonstrate this convergence in another way, Figure 5 shows the injected value of β for different combinations of β and N obs , (i.e., many realizations of the sampling visualized in Figure 2), as well as the median value of the 100 realizations for each combination. As expected, there is rapid convergence on the true value of β as the number of observations increases, as well as smaller variance in the individual realizations. From Figure 5, it is also noted that the inference on β is unbiased. \nOther model parameters aside, with only chirp mass measurements we converge on the true branching ratio to an accuracy of ± 10% with O (100) observations of BBH systems. This convergence on branching ratio is similar to that found in Vitale et al. (2017) and Stevenson et al. (2017a) using population models with varied spin distributions; with dozens to a hundred observations, we will begin to see strong convergence on the true value of the branching ratio if BBH observations are dominated by the two canonical channels. Cumulative distribution functions of the searched intervals for β are shown in Figure 6. When only one mode in the distribution is present, the searched interval refers to the distance between the mode of the distribution and its true value. Quantitatively, when one channel dominates the overall event rate, convergence on the true value is noticeably better.", '4.2. Natal Kick Model Selection': "Our methodology also provides inference on the model index - that is, the underlying physical prescriptions assumed in the models. For the purposes of this study, the only physical prescription altered between models was the black hole natal kick magnitude. Since all models have the same parameters and prior ranges, the Bayes factor for one model compared to another can be simply \nFigure 6. Cumulative distribution function on searched area for various injected values of β and number of observations (colored lines). The searched area represents the distance between the mode of the distribution and its true value. The searched area for significant N obs is notably smaller when one channel dominates, e.g. β ∼ 1 or β ∼ 0. \n<!-- image --> \nFigure 7. Bayes factors between different models as a function of the number of observations. The dark lines show the median value, and shaded regions a 68% credible interval derived from the 100 realizations. For this figure, observations were drawn from the 'fallback' kick prescription, and various injected values of β are shown with different colors. The top and bottom panels show the Bayes factor between fallback kicks and proportional kicks, and the Bayes factor between fallback kicks and full neutron star kicks, respectively. The upper limit of the plot is the maximum Bayes factor calculable given our number of samples, and leads to the apparent flattening of the function near this maximal value. The Bayes factor between fallback and full kick models increases much more rapidly than the Bayes factor between fallback and proportional kick models as a function of N obs , since the chirp mass distributions produced by fallback and full kick models are morphologically much more distinct. \n<!-- image --> \ncomputed as the ratio of the number of iterations the chain spends in each model. \nAs the kick prescription has a noticeably different effect on the distribution of chirp masses in detectable cluster models relative to detectable field models (see Figure 1), the confidence for one prescription relative to another is a strong function of the branching ratio, as well as the number of observations. For example, branching ratios closer to β = 1 draw more observations from the cluster models, which have more distinctive features in the physical parameter distributions of detectable binaries compared to field models and allow for easier discrimination between populations. Furthermore, the growth of Bayes factors as a function of N obs is expedited when comparing two kick prescriptions with dramatically different effects on the physical parameter distributions. This can also be seen in Figure 7: the Bayes factor between fallback kick and full kick increases much more rapidly than the Bayes factor between fallback kick and proportional kick. \nWe achieve Bayes factors between natal kick prescriptions of ∼ 20 as the number of observations reaches O (100), though, as can be seen in Figure 7, the rate of increase of Bayes factors is extremely sensitive to the injected value of the branching ratio. Given the predicted discovery rates and projected interferometer sensitivity \nincreases in the next few years, this indicates that we can begin to confidently infer the natal kick from supernovae within the lifetime of a design-sensitivity interferometer network. Even if the detection rate remains low, ruling out one physical model compared to others is within reach, especially given the conservative assumptions we have made. As population models are parameterized by many other discrete and continuous variables, we can expand this analysis to constrain other uncertain physical prescriptions of population synthesis using upcoming GW observations. Furthermore, this hierarchical approach does not require the models to be parameterized in the exact same way.", '5. DISCUSSION AND CONCLUSIONS': "With the detection rates predicted for the advanced network of GW observatories, we can look forward to dozens to hundreds of BBH observations in the next decade. These systems provide a unique tool for studying massive-star evolution and the environments in which BBH systems arise, and by pairing a catalog of detections with detailed population models we can begin to constrain many of the uncertain processes driving binary stellar evolution. \nThis work investigates how hierarchical modeling can infer the parameters of binary stellar evolution from multiple formation channels using solely chirp mass measure- \nFigure 8. Bayes factors between different models as a function of N obs , with observations drawn from models with different injected kick prescriptions. Dotted lines, dashed lines, and solid lines represent full kicks, proportional kicks, and fallback kicks as the correct distribution from which observations are drawn, respectively. All models in this plot have an injected value of β = 0 . 8. The median values and credible regions are indicated as in Figure 7. Interestingly, we find that when the proportional kick model is injected as the 'true' model, our inference does not necessarily prefer proportional kicks as a function of N obs . This is particularly apparent when comparing the Bayes factor between proportional kicks and fallback kicks, when observations are drawn from the proportional kick model. Though the populations are strikingly similar, we believe this issue arises from our conservative approximation of measurement uncertainty. Our approximations, which rely on Fisher matrix formalism for determining the spread of the posterior distribution, provide symmetric widths in our sampled posterior to both lower and higher values for M c . This may be unrealistically bolstering posterior samples in the low-mass peak of the fallback models (see Figure 3). Furthermore, as we limit to δ -function observations, this effect disappears. \n<!-- image --> \nments. We find that with O (100) observations, we will see convergence on the value for the branching ratio and the preferred natal kick prescription, provided the two channels considered dominate BBH rates. Furthermore, 'extreme' values of β (i.e., domination by a single formation channel) facilitate quicker convergence on both the branching ratio and natal kick prescription. \nNotably, we find a trade-off between inference on the branching ratio and inference on the natal kick prescription. This effect is dependent on the injected kick prescription itself; models with the fallback prescription shows the largest disparity between field and cluster models, thereby allowing quicker convergence on branching ratio, whereas models using the full kick prescription are less distinguishable and require more observations to converge on the branching ratio, as seen in Figure 4. However, as full kicks predict drastically different combined distributions of field and cluster populations relative to fallback and proportional kicks, comparisons with this model allow for accelerated inference on kick prescription, as we demonstrate in Figure 8. \nUsing our methodology, the current number of GW observations from BBH systems is far fewer than the number of observations needed to make any meaningful statement about kick prescription and branching ratio. Nonetheless, we inject the chirp masses of the three current GW events and one GW candidate from the first and second aLIGO observing runs of as our observations, using the 90% credible intervals for chirp mass measurements cited in Abbott et al. (2016a) and Abbott et al. (2017) to generate mock posterior samples. As expected, our analysis recovers the priors for both quantities and provides no discernment on branching ratio and true natal kick prescription. \nThe methodologies in this paper provide a framework for many extensions and refinements, both in the context of inferring additional parameters of population models and including more measured properties of BBH systems. One very simple extension, for example, is to measure the actual physical event rates, and compare with the rates derived from GW observations. Another possibility would be to constrain parameters that define a mass gap, either at low masses, between the maximum observed neutron star mass and minimum observed black hole mass, or at high masses due to pair instability supernovae. \nMultiple studies, such as Stevenson et al. (2017a) and Vitale et al. (2017), have also found that inference using spin distributions can converge on branching ratios with a similar number of GW observations. Though subject to higher measurement uncertainty and highly influenced by an unknown spin magnitude distribution, the inclusion of spin parameters in our analysis could help distinguish models, as dynamical formation predicts an isotropic distribution in spin-tilts whereas isolated field binaries are believed to preserve the memory of their initial spin alignment and can further align their spins through mass transfer, common envelope evolution, and tidal torquing. Furthermore, kick prescriptions have a large effect on spin distributions and the inclusion of spin parameters could act to bolster the confidence of one kick model compared to another. Notably, the detection of a single outlier event, such as those described in Rodriguez et al. (2016c), could go a long way toward discriminating \nbetween models. \nThe inclusion of spins would also allow inference on other poorly constrained model parameters of population synthesis; though they have minimal impact on the mass distribution, the direction of the natal kick or partial realignment of the binary after the first supernova, for example, strongly affect the resultant spin distributions in population synthesis models. As the models used in this study are equipped with spin-tilts for the BBH systems, we plan to include spin measurements for the purposes of model selection in future work. However, we note that other mechanisms unaccounted for in our models likely affect the spin-tilts of the binary components as well, and may act to contaminate the information we extract about the physical processes that are accounted for. \nBesides taking a hierarchical approach to model selection using chirp mass distributions, this study provides the framework for determining Bayes factors between a discrete set of population models. In future work, we plan to expand this methodology to include other population synthesis model parameters, such as common envelope efficiency, natal kick direction, and the rate of binary coalescences as a function of redshift. Furthermore, the framework is extensible, allowing the inclusion of other proposed formation channel models, such as young stellar clusters, galactic nuclei, and chemically homogeneous evolution. As more BBHs are observed by the advanced network of GW detectors, this type of inference will evolve into a powerful tool for constraining the correct physical prescriptions in population synthesis models, thereby improving our understanding of the physical processes governing binary stellar evolution. \nWe would like to thank Simon Stevenson, Steven Reyes, and our anonymous referee for helpful suggestions on this manuscript, as well as Scott Coughlin for assistance in debugging code. This work was supported by NSF Grant AST-1312945, NSF Grant PHY-1607709, and NASA Grant NNX14AP92G at Northwestern University. M.Z. greatly appreciates financial support from the IDEAS Fellowship, a research traineeship program supported by the National Science Foundation under grant DGE-1450006. C.R. is grateful for the hospitality of the Kavli Institute for Theoretical Physics, supported by NSF Grant PHY11-25915, and is supported at MIT by a Pappalardo Fellowship in Physics. L.S. acknowledges support from the L'Oreal FWIS Fellowship program. E.C. thanks the LSSTC Data Science Fellowship Program; her time as a fellow has benefited this work. V.K. and F.A.R. also acknowledge support from NSF Grant PHY-1066293 at the Aspen Center for Physics. The majority of our analysis was performed using the computational resources of the Quest high performance computing facility at Northwestern University, which is jointly supported by the Office of the Provost, the Office for Research, and Northwestern University Information Technology. This paper has been assigned LIGO document number ligo-p1700064.", 'REFERENCES': "Aasi, J., Abadie, J., Abbott, B. P., et al. 2013, Phys. Rev. D, 88, 062001 \n- Abbott, B. P., Abbott, R., Abbott, T. 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2014MNRAS.437.2744T
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Swift J1644+57 gone MAD: the case for dynamically important magnetic flux threading the black hole in a jetted tidal disruption event
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2014-01-01
| 44 | 0.53 | 162 |
['accretion', 'accretion disks', 'black hole physics', 'mhd', 'gamma rays', 'astronomy x rays', '-', '-', '-']
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[]
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The unusual transient Swift J1644+57 likely resulted from a collimated relativistic jet, powered by the sudden onset of accretion on to a massive black hole (BH) following the tidal disruption (TD) of a star. However, several mysteries cloud the interpretation of this event, including (1) the extreme flaring and `plateau' shape of the X-ray/γ-ray light curve during the first t - t<SUB>trig</SUB> ∼ 10 d after the γ-ray trigger; (2) unexpected rebrightening of the forward shock radio emission at t - t<SUB>trig</SUB> ∼ months; (3) lack of obvious evidence for jet precession, despite the misalignment typically expected between the angular momentum of the accretion disc and BH; (4) recent abrupt shut-off in the jet X-ray emission at t - t<SUB>trig</SUB> ∼ 1.5 yr. Here, we show that all of these seemingly disparate mysteries are naturally resolved by one assumption: the presence of strong magnetic flux Φ<SUB>•</SUB> threading the BH. Just after the TD event, Φ<SUB>•</SUB> is dynamically weak relative to the high rate of fall-back accretion dot{M}, such that the accretion disc (jet) freely precesses about the BH axis = our line of sight. As dot{M} decreases, however, Φ<SUB>•</SUB> becomes dynamically important, leading to a state of `magnetically arrested disk' (MAD). MAD naturally aligns the jet with the BH spin, but only after an extended phase of violent rearrangement (jet wobbling), which in Swift J1644+57 starts a few days before the γ-ray trigger and explains the erratic early light curve. Indeed, the entire X-ray light curve can be fitted to the predicted power-law decay dot{M} ∝ t^{-α } (α ≃ 5/3 - 2.2) if the TD occurred a few weeks prior to the γ-ray trigger. Jet energy directed away from the line of sight, either prior to the trigger or during the jet alignment process, eventually manifests as the observed radio rebrightening, similar to an off-axis (orphan) γ-ray burst afterglow. As suggested recently, the late X-ray shut-off occurs when the disc transitions to a geometrically thin (jetless) state once dot{M} drops below ∼the Eddington rate. We predict that, in several years, a transition to a low/hard state will mark a revival of the jet and its associated X-ray emission. We use our model for Swift J1644+57 to constrain the properties of the BH and disrupted star, finding that a solar mass main-sequence star disrupted by a relatively low-mass M<SUB>•</SUB> ∼ 10<SUP>5</SUP>-10<SUP>6</SUP> M<SUB>⊙</SUB> BH is consistent with the data, while a white dwarf disruption (though still possible) is disfavoured. The magnetic flux required to power Swift J1644+57 is much too large to be supplied by the star itself, but it could be collected from a quiescent `fossil' accretion disc that was present in the galactic nucleus prior to the TD. The presence (lack of) of such a fossil disc could be a deciding factor in what TD events are accompanied by powerful jets.
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https://arxiv.org/pdf/1301.1982.pdf
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{'Swift J1644 + 57 gone MAD: the case for dynamically-important magnetic flux threading the black hole in a jetted tidal disruption event': 'Alexander Tchekhovskoy 1 ? , Brian D. Metzger 2 , Dimitrios Giannios 3 , and Luke Z. Kelley 4 \n- 1 Center for Theoretical Science, Jadwin Hall, Princeton University, Princeton, NJ 08544, USA; Center for Theoretical Science Fellow\n- 2 Department of Physics, Columbia University, 538 West 120th Street, 704 Pupin Hall, New York, NY 10027, USA\n- 3 Department of Physics, Purdue University, 525 Northwestern Avenue, West Lafayette, IN 47907, USA\n- 4 Astronomy Department, Harvard University, 60 Garden Street MS 10, Cambridge, MA 02138, USA \nAccepted . Received ; in original form', 'ABSTRACT': "The unusual transient Swift J1644 + 57 likely resulted from a collimated relativistic jet, powered by the sudden onset of accretion onto a massive black hole (BH) following the tidal disruption (TD) of a star. However, several mysteries cloud the interpretation of this event, including (1) the extreme flaring and 'plateau' shape of the X-ray / GLYPH<13> -ray light curve during the first t GLYPH<0> t trig GLYPH<24> 10 days after the GLYPH<13> GLYPH<0> ray trigger; (2) unexpected rebrightening of the forward shock radio emission at t GLYPH<0> t trig GLYPH<24> months; (3) lack of obvious evidence for jet precession, despite the misalignment typically expected between the angular momentum of the accretion disk and BH; (4) recent abrupt shut-o GLYPH<11> in the jet X-ray emission at t GLYPH<0> t trig GLYPH<24> 1 : 5 years. Here we show that all of these seemingly disparate mysteries are naturally resolved by one assumption: the presence of strong magnetic flux GLYPH<8> GLYPH<15> threading the BH. Just after the TD event, GLYPH<8> GLYPH<15> is dynamically weak relative to the high rate of fall-back accretion ˙ M , such that the accretion disk (jet) freely precesses about the BH axis = our line of site. As ˙ M decreases, however, GLYPH<8> GLYPH<15> becomes dynamically important, leading to a state of 'magnetically-arrested' accretion (MAD). MAD naturally aligns the jet with the BH spin, but only after an extended phase of violent rearrangement (jet wobbling), which in Swift J1644 + 57 starts a few days before the GLYPH<13> -ray trigger and explains the erratic early light curve. Indeed, the entire X-ray light curve can be fit to the predicted power-law decay ˙ M / t GLYPH<0> GLYPH<11> ( GLYPH<11> ' 5 = 3 GLYPH<0> 2 : 2) if the TD occurred a few weeks prior to the GLYPH<13> -ray trigger. Jet energy directed away from the line of site, either prior to the trigger or during the jet alignment process, eventually manifests as the observed radio rebrightening, similar to an o GLYPH<11> -axis (orphan) gamma-ray burst afterglow. As suggested recently, the late X-ray shut-o GLYPH<11> occurs when the disk transitions to a geometrically-thin (jet-less) state once ˙ M drops below GLYPH<24> the Eddington rate. We predict that, in several years, a transition to a low / hard state will mark a revival of the jet and its associated X-ray emission. We use our model for Swift J1644 + 57 to constrain the properties of the BH and disrupted star, finding that a solar-mass main sequence star disrupted by a relatively low mass M GLYPH<15> GLYPH<24> 10 5 GLYPH<0> 10 6 M GLYPH<12> BH is consistent with the data, while a WD disruption (though still possible) is disfavored. The magnetic flux required to power Swift J1644 + 57 is much too large to be supplied by the star itself, but it could be collected from a quiescent 'fossil' accretion disk that was present in the galactic nucleus prior to the TD. The presence (lack of) of such a fossil disk could be a deciding factor in what TD events are accompanied by powerful jets. \nKey words: MHD - black hole physics - gamma-rays: galaxies - X-rays: galaxies - accretion, accretion discs", '1 INTRODUCTION': "The unusual soft GLYPH<13> -ray / X-ray / radio transient Swift J164449.3 + 573451 (hereafter Sw J1644 + 57) has been broadly interpreted as resulting from a relativistic outflow, powered by accretion following the tidal disruption (TD) of a star by a massive \nblack hole (BH) (Bloom et al. 2011; Burrows et al. 2011; Levan et al. 2011; Zauderer et al. 2011). Evidence supporting this model includes the rapid onset of Sw J1644 + 57 and its location at the center of a compact galaxy at redshift z ' 0 : 353 (Levan et al. 2011). At least until recently, the SED showed two distinct components, which led Bloom et al. (2011) and Burrows et al. (2011) to suggest di GLYPH<11> erent sources for the X-ray and radio emission (see also Liu et al. 2012). The X-ray / GLYPH<13> -ray emission is highly variable, which indicates an origin from relatively small radii, likely from a location internal to the jet itself (although see Socrates 2012). Figure 1 shows that the emission was particularly variable for the first GLYPH<24> 10 days after the Swift / BAT trigger, t trig (though roughly constant in a time-averaged sense), after which point undergoing a power-law decline, \nLX / ( t GLYPH<0> t trig) GLYPH<0> GLYPH<11> ; (1) \nwith GLYPH<11> GLYPH<24> 5 = 3, consistent with the rate of fall-back accretion in simple TD models (e.g. Rees 1988; Lodato et al. 2009; Guillochon & Ramirez-Ruiz 2012; Stone et al. 2012; although see Cannizzo et al. 2011). The X-ray flux has recently abruptly dropped by more than two orders of magnitude, indicating that the jet has apparently 'shut o GLYPH<11> ' approximately 500 days after the initial trigger (Zauderer et al. 2012). The total isotropic X-ray energy radiated to date is GLYPH<24> 5 GLYPH<2> 10 53 ergs. \nIn contrast to the X-ray emission, brightness temperature constraints place the radio emission from Sw J1644 + 57 at much larger radii, suggesting that it instead results from synchrotron emission powered by the shock interaction between the relativistic jet and the surrounding circumnuclear medium (Giannios & Metzger 2011; Zauderer et al. 2011; Metzger et al. 2012; Berger et al. 2012; Wiersema et al. 2012; Zauderer et al. 2012). By modeling the observed radio emission based on the first several weeks of data, Metzger, Giannios & Mimica (2012) (hereafter MGM12) derived values for the bulk Lorentz Factor GLYPH<0> j GLYPH<25> 10, opening angle GLYPH<18> j GLYPH<24> 1 = GLYPH<0> j GLYPH<24> 0 : 1, and beaming fraction f b GLYPH<25> 3 GLYPH<2> 10 GLYPH<0> 3 of the jet which are remarkably similar to those of AGN jets. Berger et al. (2012) (hereafter B12) presented updated radio light curves of Sw J1644 + 57, which showed a distinct rebrightening starting at t GLYPH<0> t trig GLYPH<24> 1 month and peaking on a timescale of several months. This behavior is surprising since the emission is significantly brighter than expected if the blast wave were evolving with a relatively constant energy, as would be expected if the instantaneous jet power tracked the X-ray light curve. B12 proposed that this large additional energy results from slower material catching up to the forward shock at late times. Regardless of its interpretation, however, the radio rebrightening clearly indicates the jet structure (angular or temporal) is more complex than those commonly and successfully applied to normal gamma-ray burst afterglows (Panaitescu & Kumar 2002; Cao & Wang 2012; Liu et al. 2012). \nThe discovery of a jetted TD event presents several theoretical mysteries. Relativistic jets from AGN are thought to result from magnetic, rather than hydrodynamic, collimation and acceleration (e.g. Rees et al. 1982). 1 If the jet energy derives from the Penrose-Blandford-Znajek process, then the total jet power is given \nby (Tchekhovskoy et al. 2010): \nPj = GLYPH<20> c 16 GLYPH<25> r 2 g GLYPH<8> 2 GLYPH<15> ! 2 H f ( ! H) \n= 1 : 2 GLYPH<2> 10 47 GLYPH<8> 2 GLYPH<15> ; 30 M GLYPH<0> 2 GLYPH<15> ; 5 ! 2 H f ( ! H) erg s GLYPH<0> 1 ; (2) \n; ; = 0 : 5 GLYPH<2> 10 47 GLYPH<8> 2 GLYPH<15> ; 30 M GLYPH<0> 2 GLYPH<15> ; 5 erg s GLYPH<0> 1 ; (3) \nwhere GLYPH<20> GLYPH<25> 0 : 045, rg = GM GLYPH<15> = c 2 is BH gravitational radius; GLYPH<8> GLYPH<15> = 10 30 GLYPH<8> GLYPH<15> ; 30 cgs is the magnetic flux threading the hole; M GLYPH<15> = M GLYPH<15> ; 510 5 M GLYPH<12> is the BH mass; ! H = a = [1 + (1 GLYPH<0> a 2 ) 1 = 2 ] is dimensionless angular frequency of BH horizon (equals unity for a maximally spinning BH); and f ( ! H) = 1 + 0 : 35 ! 2 H GLYPH<0> 0 : 58 ! 4 H is a high-spin correction, while the normalization in the third line has been calculated for a = 0 : 9. \n: Sw J1644 + 57 radiated GLYPH<24> 2 GLYPH<2> 10 53 ergs over the first GLYPH<24> 10 days after the trigger, corresponding to an average isotropic X-ray luminosity L trig X GLYPH<24> 2 GLYPH<2> 10 47 erg s GLYPH<0> 1 . The total (true) jet power during this interval was thus P trig j = 2( fb GLYPH<15> bol GLYPH<15> GLYPH<0> 1 X ) L X GLYPH<25> 10 46 ( f b GLYPH<15> bol GLYPH<15> GLYPH<0> 1 X = 0 : 03) erg s GLYPH<0> 1 , where GLYPH<15> b GLYPH<24> few is a bolometric correction; GLYPH<15> X < 1 is the jet radiative e GLYPH<14> ciency; and the factor of 2 accounts for the other jet beamed away from Earth. Equation (3) shows that in order to explain P trig j , the required magnetic flux GLYPH<8> GLYPH<15> ; 30 GLYPH<24> M GLYPH<15> ; 5 for a = 0 : 9, is several orders of magnitude larger than that through a typical main sequence star (Bloom et al. 2011). In x 5 : 2 we discuss possible alternative sources of magnetic flux, such as could be supplied by a pre-existing quiescent accretion disk. \nRegardless of its origin, the high luminosity of Sw J1644 + 57 requires a large magnetic flux. In fact, at least two independent arguments suggest that such magnetic flux was actually present. First, note that a significant fraction of the mass of the disrupted star, and hence of its magnetic flux or that of a quiescent disk, is accreted on the characteristic fall-back time t fb [eq. (12)] of the most tightly bound tidal debris (e.g. Ulmer 1999; Strubbe & Quataert 2009). Equation (3) would thus naively imply that the average jet power should be constant, or rising, at times t GLYPH<29> t fb, in contradiction to the observed power-law decline in the X-ray luminosity. Using 3D GRMHD simulations, Tchekhovskoy et al. (2011) have shown that if the magnetic flux GLYPH<8> GLYPH<15> is su GLYPH<14> ciently high, then magnetic forces impede accretion onto the BH, causing the flow to enter a 'magnetically-arrested' (MAD; e.g. Narayan et al. 2003; Igumenshchev 2008). MAD flows achieve a quasi-steady state as the result of 3D instabilities, which allow matter to slip past the field lines towards the BH. This process regulates the jet power [eq. (3)] to be proportional to the BH feeding rate, Pj / ˙ M [eq. (17)]. Hence, the fact that the late-time jet power in Sw J1644 + 57 faithfully tracks the expected rate of fall-back accretion ˙ M / t GLYPH<0> GLYPH<11> is only naturally understood if the flow is in a magnetically-arrested state . \nAdditional evidence for a strong magnetic flux is related to the mystery raised by Stone & Loeb (2012), who noted that in general a TD jet should precess if it is pointed along the angular momentum axis of the accretion disk. 2 Lack of clear evidence 3 for large-scale jet precession in Sw J1644 + 57 thus requires either a set of highly unlikely circumstances, such as an unphysically low BH spin or \nnear-perfect alignment between the angular momentum of the BH and the original orbit of the disrupted star (Stone & Loeb 2012), or some mechanism for aligning the angular momentum of the disk with the BH spin. In fact, recent numerical simulations by McKinney et al. (2012a) (hereafter MTB12a) show that such an alignment between the disk and BH spin axis can occur due to MHD forces, but only if the strength of the magnetic field threading the BH is similar to that required for MAD accretion. If such an alignment process occurred in Sw J1644 + 57, then the lack of observed precession also provides indirect evidence for a strong magnetic flux. \nIn this paper, we present a new physical scenario for Sw J1644 + 57 which addresses the seemingly disparate mysteries raised above, including the shape of the X-ray / GLYPH<13> -ray light curve; lack of jet precession; and late radio rebrightening. We show that all of these features are naturally expected given a single assumption: the presence of a strong magnetic flux threading the BH. We begin in x 2 with some basic phenomenological considerations. Then in x 3 we overview the timeline of our proposed scenario for Sw J1644 + 57. In x 4 we use our model to constrain the properties of the BH and stellar progenitor. In x 5 we discuss our results, including the nature of the disrupted star ( x 5 : 1); the origin of the magnetic flux ( x 5 : 2); the nature and duration of the flaring state ( x 5 : 3); the origin of the radio rebrightening ( x 5 : 4); and future predictions ( x 5 : 5). We present our conclusions in x 6. Throughout the paper we use Gaussian-cgs units and set the zero time to the point of disruption, t disr = 0.", '2 PHENOMENOLOGICAL CONSIDERATIONS': 'We begin by discussing the origin of several features in the X-ray and radio light curves of Sw J1644 + 57 from a phenomenological perspective. Then in x 3 we present a more systematic overview of our model.', '2.1 Relativistic Jet as the Origin of X-ray Emission': "Figure 1 shows the soft X-ray (0.3-10 keV) light curve of Sw J1644 + 57. Upper limits on the mass of the host galaxy and the observed variability timescale place a rough upper limit of M GLYPH<15> . 10 7 M GLYPH<12> on the mass of the central BH (e.g. Bloom et al. 2011). The jet luminosity was thus highly super-Eddington over at least the first several days of activity, even after correcting for jet beaming (MGM12). At t & 10 days, the time-averaged emission follows a power-law decline with temporal index GLYPH<11> ' 5 = 3 (eq. [1]), similar to the rate of mass fall-back in standard TD scenarios (Rees 1988). This decay rate has been used as evidence that Sw J1644 + 57 was in fact a TD event, but as we discuss below, it is not clear a priori why the jet power should so faithfully track the accretion rate.", "2.2 Early Time Light Curve 'Plateau'": "The first GLYPH<24> 10 days of Sw J1644 + 57 was characterized by particularly intense flaring (Fig. 1). The luminosity during this period, though variable by several orders of magnitude, was approximately constant on average. Such a 'plateau' has no obvious explanation in TD scenarios, but, as we now discuss, it naturally results if the GLYPH<13> GLYPH<0> ray trigger was delayed with respect to the time of disruption. \nFigure 1 shows that a plateau consistent with the data is reproduced simply by shifting the zero-point of time, even for a purely power-law decay in the assumed flux. In order to reproduce the duration of the plateau and match the predicted accretion rate to \nFigure 1. X-ray light curve of Sw J1644 + 57, as measured by the Swift X-ray Telescope (XRT) and Chandra . When plotted as a function of time since the GLYPH<13> GLYPH<0> ray trigger t GLYPH<0> t trig the average emission shows a 'plateau-like' phase lasting for GLYPH<24> 10 days, which naively appears inconsistent with the predicted power-law decay ˙ M fb / ( t GLYPH<0> t disr) GLYPH<0> GLYPH<11> in the rate of fall-back accretion following a tidal disruption event ( t = t disr). However, if the trigger time is delayed with respect to the disruption, then a 'plateau'-like shape in Fx ( t GLYPH<0> t trig) is naturally produced ( x 2 : 2). The two solid curves show ˙ M fb / t GLYPH<0> GLYPH<11> (arbitrary normalization) for complete ( red , GLYPH<11> = 5 = 3 for trigger time delay t trig GLYPH<0> t disr = 15 days) and partial stellar disruption ( blue , GLYPH<11> = 2 : 2, t trig GLYPH<0> t disr = 30 days), while the dotted curves show the conventional versions of power-law fits that neglect the trigger time delay. The trigger time delay for the solid lines are chosen to match ˙ M fb with the the average luminosity of the early plateau phase ( t GLYPH<0> t trig . 10 days). If we instead match ˙ M fb to the 'envelope' created in the light curve by the brightest flares, then adopting a shorter trigger time delay is also consistent with the data, e.g., t trig GLYPH<0> t disr = 5 days for a complete disruption ( red dashed line , GLYPH<11> = 5 = 3). This timescale is consistent with the first evidence for activity from Sw J1644 + 57 (Burrows et al. 2011; Krimm & Barthelmy 2011; Zauderer et al. 2011) 4 days prior to the first BAT trigger. \n<!-- image --> \nthe average X-ray flux, we find that a trigger delay t trig GLYPH<0> t disr GLYPH<24> weeks-month is required, depending on whether the TD event was a complete ( GLYPH<11> ' 5 = 3) or partial disruption ( GLYPH<11> = 2 : 2; we discuss this distinction in x 3). If we instead fit the 'envelope' of X-ray emission set by the brightest flares (a factor GLYPH<24> 10 times higher flux than the plateau average), then a shorter trigger delay t trig GLYPH<0> t disr GLYPH<24> 5 days is also consistent with the data (justification of such a possibility is discussed in x 4.2.1). There in fact was evidence for jet activity GLYPH<24> 4 days prior to the Swift trigger (on March 28, 2011) as seen in both Swift / BAT data (Krimm & Barthelmy 2011; Burrows et al. 2011), as well as inferred by the rise time of the radio emission (Zauderer et al. 2011; MGM12). However, our fits in Figure 1 show that the TD event could have occured much earlier than this time.", '2.3 Jet Activity Prior to Trigger and Radio Rebrightening': "What could cause such a long delay before the onset of jet emission? One possibility is that the process of jet formation requires special conditions which only became satisfied at late times after the disruption, such as the accumulation of a critical quantity of magnetic flux ( x 5 : 3). It is also possible that the jet was active soon after disruption, but that it was initially pointed away from our line of site, as could be expected if the black hole spin and the angular momentum of the fall-back disk were initially misaligned. Even \nStage 1 Precessing disk-aligned jet » few weeks \n<!-- image --> \nStage 2 Wobbling jet » few weeks \n<!-- image --> \nno jet \n~a \nthin disk \nStage 4 No jet » 5 ¡ 10 years \n<!-- image --> \nStage 5 \nJet revival \nUnlimited \nif the jet eventually fully aligns towards our line of site, it might not be pointed towards us at all times during initial stages in the alignment process (see below). \nAlthough X-ray emission from a misaligned jet is unobservable, it still imparts kinetic energy into the ambient medium surrounding the black hole. Synchrotron radio emission from this o GLYPH<11> -axis blast-wave would not be visible initially due to relativistic beaming, but could become visible once the ejecta slows to mildly relativistic Lorentz factor GLYPH<13> GLYPH<24> 2(0 : 5 =GLYPH<18> ma) ; where GLYPH<18> ma is the angle between the jet and line of site, normalized to a typical value GLYPH<24> 30 GLYPH<14> . A blast wave of [isotropic] energy E iso = 10 54 E 54 erg that interacts with an external medium of density n = 10 n 1 cm GLYPH<0> 3 (see MGM12, B12 for motivation for these characteristic values) will decelerate to GLYPH<13> = 2 at a distance R dec ' 2 : 4 GLYPH<2> 10 18 ( E 54 = n 1) 1 = 3 cm from the BH. The observer time corresponding to emission from this radius t obs ' R dec(1 GLYPH<0> GLYPH<12> cos GLYPH<18> ma) =GLYPH<12> c . 1yr, for GLYPH<18> ma . 0 : 5. Delayed emission from such an o GLYPH<11> -axis jet thus provides a possible explanation for the otherwise mysterious radio rebrightening observed to peak GLYPH<24> 4 months after the onset of Sw J1644 + 57 (B12; see x 5.4 for further discussion).", '2.4 Intense Early-Time Flaring Followed by a Steady Decline': "Large-amplitude flares in the early GLYPH<13> -ray / X-ray lightcurve of Sw J1644 + 57 (Fig. 1) could result from geometric e GLYPH<11> ects, such as changes in jet orientation which cause emission to periodically beam in and out of our line of sight. This erratic 'wobbling' of the \njets is a natural consequence of the alignment process. Eventually, once the jet direction becomes stable, one of the jets continuously points towards us. This produces relatively steady X-ray emission at late times ( & 10 days), which decays as a power-law in time, reflecting the rate at which stellar debris returns to the BH.", '2.5 Jet Shuto GLYPH<11>': "The X-ray emission from Sw J1644 + 57 recently declined abruptly at t GLYPH<24> 500 days, after which time the XRT is able to place only upper limits on the flux (Fig. 1). Sw J1644 + 57 was subsequently detected by Chandra at at flux nearly 2 orders of magnitude lower than that just prior to the decline (Sbarufatti et al. 2012; Zauderer et al. 2012); however, this residual flux is consistent with it being the high energy extension of the same forward shock synchrotron emission observed at radio frequencies (Zauderer et al. 2012). Such a jet 'shut o GLYPH<11> ' may occur once the accretion rate drops below a fraction of the Eddington accretion rate (Zauderer et al. 2012), since after this time the disk becomes geometrically-thin and enters a thermally-dominant accretion state, which are not observed to produce powerful jets (Russell et al. 2011).", '3 TIMELINE OF PROPOSED SCENARIO': "We now describe the timeline for our proposed scenario for Sw J1644 + 57, the stages of which are illustrated in Figure 2. \nFigure 2. Stages in the proposed model for Sw J1644 + 57. [Panel (a)]: Stage 1. Shortly after the stellar disruption, the magnetic flux threading the BH GLYPH<8> GLYPH<15> increases, as it is dragged inward by the accreting material. Stellar debris returning to the BH forms a tilted accretion disk, with a rotation axis that is misaligned with the BH spin. The latter points towards our line of site (shown as vertical in the figure). At early times, when the mass accretion rate ˙ M is highest, the magnetic flux GLYPH<8> GLYPH<15> is not dynamically important. Under these conditions, the disk undergoes precession due to Lense-Thirring torques (Fragile et al. 2007) and the jets point along the disk axis. Their high-energy emission is beamed away from Earth and is not detectable. [Panel (b)]: Stage 2. As ˙ M decreases with time, the magnetic flux eventually becomes dynamically important, leading to a state of magnetically-arrested disk (MAD) (Narayan et al. 2003; Tchekhovskoy et al. 2011). In the MAD state, the magnetic field is su GLYPH<14> ciently strong to o GLYPH<11> set gravitational forces acting on the inner disk and to align the axis of the disk axis (and hence the jets) with the BH spin (so-called 'magneto-alignment e GLYPH<11> ect'; MTB12a). However, it takes time for the entire disk and jet to align with the BH spin. As ˙ M decreases, the BH magnetic flux and jet power Pj / ˙ M also decrease. As excess magnetic flux leaks out of the BH into the disk, MAD encompasses a larger and larger fraction of the disk. During this process, the jet pushes against the disk and wobbles erratically (see movies in MTB12a). As the jet comes in and out of our line of sight, this produces large amplitude variations ('flares') in lightcurve of Sw J1644 + 57 during the first GLYPH<24> 10 6 seconds after the trigger (see Fig. 1). [Panel (c)]: Stage 3. Once MAD encompasses the entire disk, the disk / jet alignment with the BH spin completes. Since the jet direction points steadily towards Earth, its X-ray emission becomes less variable as it continues to track the rate of fall-back accretion. [Panel (d)]: Stage 4. Once ˙ M deceases below GLYPH<24> 30% of the Eddington accretion rate ˙ M Edd, the disk transitions to a geometrically-thin state and the jet shuts o GLYPH<11> , producing an abrupt decline in X-ray emission at t & 500 days. [Panel (e)]: Stage 5. At very late times, once the accretion rate drops below a few percent of ˙ M Edd, the disk will again enter a geometrically-thick regime, and the jet may turn back on, again analogous to state transitions in X-ray binaries. For Sw J1644 + 57 this 'jet revival' is estimated to occur between the years GLYPH<24> 2016-2022 ( x 5 : 5). \n<!-- image --> \n(d) \nFigure 3. Various quantities as a function of time since disruption, assuming a partial disruption of a MS star and a delay of t trig GLYPH<0> t disr = 60 days between the disruption and the GLYPH<13> -ray trigger ( x 2 : 2). Di GLYPH<11> erent stages in our model for Sw J1644 + 57 are indicated with color coding and numbers ( x 3; see Fig. 2 for a detailed explanation). [Panel (a)]: Mass accretion rate GLYPH<21> = ˙ M = ˙ M Edd as a fraction of the Eddington accretion rate ˙ M Edd ( blue solid line ; left axis), which peaks at t ' 1 : 5 t fb ' 30 days and subsequently decays as GLYPH<21> / t GLYPH<0> 2 : 2 . The same curve (right axis) gives the maximum value of the magnetic flux threading the BH, GLYPH<8> MAD GLYPH<15> ; 30 / GLYPH<21> 1 = 2 , in units of 10 30 G cm 2 ( red dashed line ). As stellar debris returns to the vicinity of the BH, it drags in magnetic flux from a pre-existing, 'fossil' accretion disk ( x 5 : 2). This accumulated flux due to the swept-up fossil field increases with time as GLYPH<8> fb ; 30 / t 2 = 3 ( green dotted line ; eq. 36). At early times, most of this flux ends up on the BH, so GLYPH<8> GLYPH<15> ; 30 = GLYPH<8> fb ; 30 ( red dashed line ). For our choice of parameters (see below), BH receives a substantial amount of magnetic flux early on, enough to overcome the ram pressure of the infalling gas and produce the jets, GLYPH<8> GLYPH<15> ; 30 > GLYPH<8> on ; 30 ( red dashed line lies above blue dotted line ). Eventually, the central magnetic flux becomes dynamically-important, i.e., GLYPH<8> GLYPH<15> ; 30 ' GLYPH<8> MAD GLYPH<15> ; 30 ( red dashed line crosses blue solid line ). Since the inner disk can only hold GLYPH<8> MAD GLYPH<15> ; 30 worth of flux on the hole (Stage 2), the rest leaks out and instead contributes to the disk flux ( cyan long-dashed line ), i.e. GLYPH<8> D ; 30 = GLYPH<8> fb ; 30 GLYPH<0> GLYPH<8> MAD GLYPH<15> ; 30 . However, the disk flux eventually also saturates (depending on its radial extent, eq. 43), after which it also tracks the BH magnetic flux (Stage 3). Once the Eddington ratio, GLYPH<21> , falls below a critical value ( GLYPH<21> cr = 0 : 2 in this figure) the accretion disk becomes geometricallythin; the central BH and disk lose their magnetic flux; and the jets shut o GLYPH<11> (Stage 4). At a much later time, when GLYPH<21> . 0 : 02, the accretion disk becomes geometrically-thick again and produces powerful jets (Stage 5). [Panel (b)]: Dimensionless values (eq. 9) of the magnetic flux shown in panel (a). In this figure we have assumed following parameters: BH spin a = 0 : 7; BH mass M GLYPH<15> = 3 GLYPH<2> 10 5 M GLYPH<12> ; stellar mass M ? = 0 : 44 M GLYPH<12> ; fraction of star accreted by BH f 0 : 4 f acc = 0 : 75; Eddington fraction of the fossil disk GLYPH<21> fossil = 1 : 8 GLYPH<2> 10 GLYPH<0> 3 ; and Pj = LX = f b GLYPH<15> bol GLYPH<15> GLYPH<0> 1 X = 0 : 05. For computing flux accumulation timescale at early time (eq. 37), we assume disk thickness h = r = 1 as expected for a highly super-Eddington flow. Since at later times h = r decreases, for computing jet power we take a single representative value, h = r = 0 : 3. \n<!-- image --> \nFigure 4. Similar to Figure 3, but shown for the case of a complete disruption of a WD, assuming a disruption-trigger delay of t trig GLYPH<0> t disr = 30 days. Note that unlike the MS star scenario (Fig. 3), the mass accretion rate is highly super-Eddington near the peak ( GLYPH<21> GLYPH<29> 100), and initially jet production is suppressed (see discussion in x 3 : 2). However, as the mass accretion rate decreases, GLYPH<21> / t GLYPH<0> 5 = 3 , eventually the magnetic flux becomes su GLYPH<14> ciently large to overcome the ram pressure of the accretion flow, GLYPH<30> GLYPH<15> > GLYPH<30> on ' 20 ( red dashed line overtakes blue dotted line ), and the jets emerge. After this point, the jet evolution is similar to that described in Figure 3. In this figure, we have assumed the following parameters: a = 0 : 87; M = 10 4 M GLYPH<12> ; M ? = 0 : 6 M GLYPH<12> ; f acc = 0 : 54; GLYPH<21> fossil = 10 GLYPH<0> 8 ; GLYPH<21> cr = 0 : 4; and f b GLYPH<15> bol GLYPH<15> GLYPH<0> 1 X = 0 : 003. \n<!-- image -->", '3.1 Stage 0: Tidal Disruption and Flux Accumulation': "A star of mass M ? = m ? M GLYPH<12> and radius R ? = r ? R GLYPH<12> is tidally disrupted by a BH of mass M GLYPH<15> = 10 5 M GLYPH<15> ; 5 M GLYPH<12> if its pericenter radius R p lies within the tidal radius R t ' R ? ( M GLYPH<15> = M ? ) 1 = 3 . After disruption, approximately half of the star is immediately unbound, while the other half remains marginally bound and is placed on highly elliptic orbits, with the most tightly bound material falling back on a characteristic timescale (Stone et al. 2012) \nt fb ' 17 : 3 d M 1 = 2 GLYPH<15> ; 5 m GLYPH<0> 1 ? r 3 = 2 ? : (4) \nThe fall-back accretion rate ˙ M fb peaks (at t peak GLYPH<25> 1 : 5 t fb; Ulmer 1999) at a characteristic value \n˙ M fb ; peak ' GLYPH<11> GLYPH<0> 1 2 = 3 M ? 3 t fb GLYPH<25> 2 GLYPH<2> 10 26 g s GLYPH<0> 1 GLYPH<11> GLYPH<0> 1 2 = 3 M GLYPH<0> 1 = 2 GLYPH<15> ; 5 m 2 ? r GLYPH<0> 3 = 2 ? ; (5) \nor in terms of the Eddington accretion rate ˙ M Edd GLYPH<17> L Edd = 0 : 1 c 2 , \n˙ M fb ; peak ˙ M Edd ' 4 GLYPH<2> 10 3 GLYPH<11> GLYPH<0> 1 2 = 3 M GLYPH<0> 3 = 2 GLYPH<15> ; 5 m 2 ? r GLYPH<0> 3 = 2 ? ; (6) \nand subsequently decays as a power law \n˙ M fb ' ˙ M fb ; peak t t fb ! GLYPH<0> GLYPH<11> (7) \nwhere GLYPH<11> = 5 = 3 for a complete disruption and GLYPH<11> = 2 : 2 for a partial disruption (Guillochon & Ramirez-Ruiz 2012; Figs. 3, 4). Until recently (e.g. Ulmer 1999; Strubbe & Quataert 2009), TD models \npredicted that t fb and ˙ M fb ; peak should depend also on the impact parameter GLYPH<12> GLYPH<17> R t = Rp in addition to the stellar and BH properties; however, recent numerical (Guillochon & Ramirez-Ruiz 2012) and analytic (Stone et al. 2012) work has shown that these estimates were made on the faulty assumption that the energy distribution of the disrupted stellar material is frozen-in at pericenter instead of the tidal radius. \nMatter falls back and circularizes to form an accretion disk at the radius \nR circ ' 2 R p = 2 R t GLYPH<12> GLYPH<0> 1 ' 430 r ? m GLYPH<0> 1 = 3 ? M GLYPH<0> 2 = 3 GLYPH<15> ; 5 GLYPH<12> GLYPH<0> 1 rg ; (8) \naccreting soon thereafter. When the accretion rate is superEddington at early times, the disk may be prone to outflows driven by radiation pressure (e.g. Ohsuga et al. 2005; Strubbe & Quataert 2009), in which case the accretion rate reaching the BH is less than ˙ M fb. \nIf the magnetic flux responsible for powering the jet in Sw J1644 + 57 originates from the star itself, then a substantial fraction of the total flux is accumulated on the relatively short fall-back time t . t fb (eq. [4]). However, flux accumulation can last significantly longer if the field is swept up from a quiescent fossil disk by the infalling debris (in which case GLYPH<8> GLYPH<15> / ( t = t fb) 2 = 3 ; x 5 : 2). On timescales t & t fb the jet power Pj / GLYPH<8> 2 GLYPH<15> [eq. (3)] thus either saturates to a constant value [stellar flux], or increases as Pj / ( t = t fb) 4 = 3 [fossil disk]. In what follows, we express the flux in a dimensionless form, \nGLYPH<30> GLYPH<15> GLYPH<17> GLYPH<8> GLYPH<15> ( ˙ Mr 2 g c ) 1 = 2 GLYPH<25> 30 GLYPH<8> GLYPH<15> ; 30 ˙ M ˙ M fb ; peak ! GLYPH<0> 1 = 2 M GLYPH<0> 3 = 4 GLYPH<15> ; 5 m GLYPH<0> 1 ? r 3 = 4 ? ; (9) \nthat quantifies its dynamical importance in relation to the accreting gas, where rg GLYPH<17> GM GLYPH<15> = c 2 . Figures 3 and 4 show the time evolution of the GLYPH<8> GLYPH<15> and GLYPH<30> GLYPH<15> in our model for Sw J1644 + 57, based on two scenarios for the disrupted star ( x 4 : 1).", '3.2 Stage 1: Precessing Disk-Aligned Jet (Fig. 2a).': "In general the angular momentum of the initial stellar orbit and fall-back accretion disk will not be aligned with the spin of the BH. Such tilted, geometrically-thick accretion disks undergo precession due to Lense-Thirring torques, and their jets are also expected to precess (Fig. 2a). Though expected, evidence for precession is not obviously present in the light curve of Sw J1644 + 57 (Stone & Loeb 2012; although see Saxton et al. 2012; Lei et al. 2012). This mystery is resolved if we postulate that our line of site is aligned with the BH spin axis ( x 2), such that the emission from the tilted jet is beamed away from us and hence is not initially detectable. \nBecause the magnetic field is dynamically weakest relative to the accretion flow ( GLYPH<30> GLYPH<15> is smallest) when the accretion rate is highest ˙ M GLYPH<24> ˙ M fb ; peak (eq. [9]), the magnetic flux does not appreciably influence the disk inclination at these early times. This allows for a phase of jet precession as described above. \nAt even earlier times, however, when GLYPH<30> GLYPH<15> . GLYPH<30> on GLYPH<25> 20, the ram pressure of the quasi-spherical accretion flow (as expected for super-Eddington accretion) is so high that it can stifle jet formation altogether (Komissarov & Barkov 2009). Since GLYPH<30> GLYPH<15> / t GLYPH<11>= 2 if GLYPH<8> GLYPH<15> = constant, (eq. 9; assuming ˙ M fb / t GLYPH<0> GLYPH<11> ) this limits the duration of Stage 1 to GLYPH<24> 4 GLYPH<0> 2 =GLYPH<11> t MAD GLYPH<24> (1 = few) t MAD, where t MAD is the onset of MAD accretion (start of Stage 2), which occurs when GLYPH<30> MAD GLYPH<15> GLYPH<24> 50. The possibly substantial duration of this early jet smothering phase is illustrated by Figure 4, which shows the time evolution of GLYPH<30> GLYPH<15> in the case of a tidally-disrupted WD. In this case, the disruption itself happens on a timescale of GLYPH<24> tens of minutes, but Stage 1 begins \nonly at t GLYPH<24> 10 days. Also note that depending on how fast magnetic flux is accumulated, Stage 1 could be even briefer. If GLYPH<30> GLYPH<15> & GLYPH<30> MAD GLYPH<15> at peak accretion, then the jet will enter the 'wobbling' stage described next essentially from its onset.", '3.3 Stage 2: MAD Onset, Erratic Wobbling Jet (Fig. 2b).': "As ˙ M decreases from its peak value, GLYPH<30> GLYPH<15> / ˙ M GLYPH<0> 1 = 2 (eq. [9]) increases and the magnetic field becomes increasingly important dynamically. Once GLYPH<30> GLYPH<15> exceeds a critical value GLYPH<30> MAD GLYPH<15> GLYPH<24> 50 (depending weakly on BH spin; eq. [13]), the field is su GLYPH<14> ciently strong to feed-back on the accreting gas, leading to a state of 'magnetically-arrested disk' accretion, or MAD (Narayan et al. 2003; Tchekhovskoy et al. 2011; McKinney et al. 2012b). \nThe strong magnetic flux also acts to orient the rotational axis of the inner accretion disk / jet with the BH spin axis (MTB12a). This realignment does not, however, happen instantaneously, nor is it clean. The jets undergo a period of vigorous rearrangement during which they ram against the accretion disk, partially obliterate it, and intermittently punch holes through the disk as they work to reorient the disk's angular momentum along BH spin axis (see Fig. 2b). Recent numerical simulations show that during this stage the jets wobble intensely between the orientation of the outer disk axis and the orientation of the BH axis (MTB12a), so that jet emission transiently comes in and out of our line of sight. This jet wobbling state may explain the epoch of intense flaring comprising the first GLYPH<24> 10 days of the Sw J1644 + 57 light curve ( x 2 : 4). \nAlignment between the disk / jet and BH spin completes once su GLYPH<14> cient magnetic flux is able to leak out of the immediate vicinity of the BH and new flux is brought in by the infalling debris, such that the entire disk (at least out to the circularization radius R circ) becomes magnetically arrested. Depending on R circ, this process requires GLYPH<30> GLYPH<15> to increase by a factor of a few, and hence ˙ M to decrease by a factor of several. In x 5 : 3 we show that this timescale for the entire disk to 'go MAD' is broadly consistent with the observed duration of the early flaring state in Sw J1644 + 57. In Figures 3 and 4 this process is shown by the rising value of the disk flux GLYPH<8> D during Stage 2, until it eventually reaches a constant fraction of the BH flux GLYPH<8> GLYPH<15> at the onset of Stage 3.", '3.4 Stage 3: Steady Spin-Aligned Jet (Fig. 2c).': 'Once the entire disk is magnetically-arrested, the system enters a scale-free MAD solution that depends on just one parameter: the mass accretion rate (Fig. 2c). 4 The strong centrally-accumulated magnetic field not only aligns the jets along the direction of BH spin axis ( x 3.3), it also explains why the jet luminosity faithfully tracks mass fallback rate. When GLYPH<30> GLYPH<15> is small at early times (Stage 1), the BH flux GLYPH<8> GLYPH<15> and jet power Pj (eq. [3]) are approximately constant or increase in time (Stage 1), which is inconsistent with the power-law decrease in the X-ray lightcurve of Sw J1644 + 57 between GLYPH<24> 10 and GLYPH<24> 500 days since the trigger (Fig. 1). In contrast, when the central magnetic field is su GLYPH<14> ciently strong ( GLYPH<30> GLYPH<15> & GLYPH<30> MAD GLYPH<15> GLYPH<24> 50; eq. [13]) to be dynamically-important (MAD state), then the magnetic flux GLYPH<8> GLYPH<15> threading the BH is not determined by initial conditions, but instead by the ram pressure of the accretion flow. Its value regulates such that the jet power P j GLYPH<24> GLYPH<17> j ˙ M c 2 is a constant fraction GLYPH<17> j GLYPH<25> 1 : 3 a 2 of the accretion power (eq. [17]). \nThus, as the fall-back rate decreases as a power-law in time, so do the jet power and luminosity, as is observed. \nAnother expected feature of MAD is a stable QPO at a frequency that is 1 = 4 of BH horizon angular frequency (McKinney et al. 2012b). Reis et al. (2012) detected a potential QPO with period GLYPH<28> QPO GLYPH<24> 200 s in the power-law decay portion of the Sw J1644 + 57 lightcurve, which could also be evidence for MAD. We discuss the constraints implied by this period ( x 4 : 2 : 6) as a part of our analysis in x 4.', '3.5 Stage 4: No Jet (Fig. 2d).': 'Once the accretion rate drops below a few tens of per cent of ˙ M edd, the flow transitions to a geometrically thin disk state, or thermal state, which is not observed (nor theoretically expected) to power a jet (e.g., Fender et al. 2004; Russell et al. 2011). The abrupt decrease in the X-ray flux at t GLYPH<24> 500 days (Fig. 1; by more than two orders of magnitude) can thus plausibly be associated with the point at which ˙ M GLYPH<24> 0 : 3 ˙ M Edd (Steiner et al. 2009, 2010; Abramowicz et al. 2010). The jet luminosity and timescale of this transition also constrain the properties of the disrupted object and central BH ( x 4 : 2 : 3).', '3.6 Stage 5: Jet Revival (Fig. 2e).': "As ˙ M fb continues to decrease as a power-law in time, eventually it will reach a few percent of ˙ M Edd. After this point the disk will again transition to a geometrically-thick disk, analogous to the 'low / hard' state observed in X-ray binaries. Since this state is conducive to jet formation (Narayan & Yi 1995; Fender et al. 2004), the jet in Sw J1644 + 57 and its associated X-ray emission may suddenly turn back on (this is estimated to occur sometime around 2016-2022; x 5 : 5).", '4.1 Stellar Progenitor Scenarios': "We begin by overviewing the possible classes of stellar progenitors which could be responsible for Sw J1644 + 57. TD scenarios usually consider the disruption of a lower main sequence star. However, in principle the star could have been a giant (MacLeod et al. 2012), white dwarf (e.g. Krolik & Piran 2011; Haas et al. 2012; Shcherbakov et al. 2012), or even a planet. Low mass planets seem to be ruled out because the [beaming-corrected] energy budget of Sw J1644 + 57 of & few GLYPH<2> 10 51 erg already exceeds the rest mass energy of Jupiter. A giant star is also unlikely, because the fall-back time of the stellar debris would greatly exceed the observed trigger time ( x 4 : 2 : 1), incompatible with the observed X-ray light curve. However, a white dwarf companion cannot be ruled out a priori , especially considering their potential for harboring a large magnetic flux. \nIn what follows, we thus consider constraints based on two progenitor scenarios: a lower main sequence star (MS) and a white dwarf (WD). We employ approximate mass-radius relations for each given by \nr ? = m ?; [MS] r ? = 0 : 013( m ?= 0 : 6) GLYPH<0> 1 = 3 ; [WD] (10) \nwhere m ? is in solar masses and r ? is in solar radii. Since the stellar radius must exceed the tidal radius, only low mass black holes with \nFigure 5. Sw J1644 + 57's soft X-ray lightcurve accounting for the best-fit time of the disruption, t trig GLYPH<0> t disr = 15 days before the trigger, for a complete disruption (see also Fig. 1). Note that just this simple shift in the zero point in time causes the early-time 'plateau' in the light curve (see Fig. 1) to disappear and the entire light curve to become consistent with a single power-law dependence in time, / t GLYPH<0> 5 = 3 . Various stages of the disruption process are indicated with color coding and text. These stages are explained in Fig. 2 and its caption. \n<!-- image --> \nFigure 6. Sw J1644 + 57's soft X-ray lightcurve accounting for the best-fit time of the disruption, t trig GLYPH<0> t disr = 30 days before the trigger, for a partial disruption (see also Fig. 1). The steeper time-dependence of mass-fallback rate than in a complete disruption, / t GLYPH<0> 2 : 2 , allows for a longer delay between the time of the disruption and the time of the GLYPH<13> -ray trigger. \n<!-- image --> \nM GLYPH<15> . 10 5 GLYPH<0> 10 6 M GLYPH<12> are capable of disrupting a WD, depending on WDmassand BH spin (Kesden 2012). For a centrally-concentrated MSstar, the disruption process can be either 'full' or'partial' (Guillochon & Ramirez-Ruiz 2012), resulting in di GLYPH<11> erent predictions for the rate of fall-back accretion ( x 4 : 2 : 1). \nIn what follows we thus consider three scenarios: (i) a complete or (ii) partial tidal disruption of a lower mass main-sequence star by a supermassive BH; or (iii) a complete disruption of a white dwarf by an intermediate-mass BH.", '4.2 Observational Constraints': 'For each progenitor scenario our goal is to determine the following 4 unknowns: BH mass, M GLYPH<15> ; 5, BH spin, a , stellar mass, m ? , and magnetic flux threading the hole, GLYPH<8> GLYPH<15> . We now discuss what observational data constrain these properties within the framework of our model for Sw J1644 + 57.', '4.2.1 Shape of the X-Ray Light Curve': "First consider what constraints can be placed on Sw J1644 + 57 based on the shape of the X-ray light curve. Recall that although the time-averaged light curve does not follow a single power-law decay if plotted against the time since the GLYPH<13> GLYPH<0> ray trigger t GLYPH<0> t trig, a much better fit is achieved by moving the TD 'zero point' prior to the trigger (Fig. 1). We thus first consider what range of trigger delay time t trig GLYPH<0> t disr produces a light curve shape consistent with the predicted power-law decline in the X-ray luminosity LX / ˙ M fb / ( t GLYPH<0> t disr) GLYPH<0> GLYPH<11> given the expected range in GLYPH<11> . \nAlthough the canonical value of GLYPH<11> = 5 = 3 is often quoted, GLYPH<11> actually depends on the fraction of mass lost by the star during TD, GLYPH<1> M ?= M ? = 0 : 4 f 0 : 4, which di GLYPH<11> ers between complete and partial disruptions (Guillochon & Ramirez-Ruiz 2012). If GLYPH<1> M ?= M ? GLYPH<25> 10% GLYPH<0> 50% then GLYPH<11> = 2 : 2, but if GLYPH<1> M ?= M ? & 50% then the value of GLYPH<11> approaches that for a complete disruption, GLYPH<11> = 5 = 3 (Guillochon & Ramirez-Ruiz 2012). By considering the two limiting cases of complete disruptions with GLYPH<11> = 5 = 3 and partial disruptions with GLYPH<11> = 2 : 2, we bracket the allowed range of possibilities. \nFor complete disruptions ( GLYPH<11> = 5 = 3) we find an allowed range of t trig GLYPH<0> t disr GLYPH<24> 15 + 15 GLYPH<0> 7 days when fitting to the shape of the time-averaged X-ray light curve. Figure 5 shows the trigger delay-corrected light curve for a fiducial value t trig GLYPH<0> t disr = 15 days. For partial disruptions ( GLYPH<11> = 2 : 2), we find required delay times that are somewhat longer t trig GLYPH<0> t disr GLYPH<24> 30 + 30 GLYPH<0> 15 days. Figure 6 shows the corrected light curve in this case for a fiducial delay t trig GLYPH<0> t disr = 30 days. \nOur fits quoted above (Figs. 5, 6) were made by matching the fall-back accretion rate / jet power to the time-averaged X-ray flux. However, if the early phase of high variability indeeds results from a 'wobbling' jet ( x 3 : 3), then bright flares may arise from transient episodes when the jets points towards our line of site. In this case, it is more physical to match the fall-back rate / jet power to the 'envelope' in the X-ray light curve comprised by the flare peaks, since these better characterize the true jet power. Since the observed duty cycle of the flaring state is GLYPH<24> 10%, this implies that the jet points towards us only about 1 = 10 of the time if the wobbling scenario is correct. The total jet power [accounting for both on- and o GLYPH<11> -axis jet emission] could thus exceed the observed [on-axis] emission by an order of magnitude (see also x 5.4). Figure 1 shows that such fits allows for a trigger delay t trig GLYPH<0> t disr as short as a few days, consistent with the first evidence for a jet in Sw J1644 + 57 GLYPH<24> 4 days prior to the GLYPH<13> GLYPH<0> ray trigger (Krimm & Barthelmy 2011). \nGiven the allowed range in t trig, we require that the fallback accretion rate must be past its peak at the trigger: \nt peak = 1 : 5 t fb < t trig = (1 + z ) ; (11) \nwhere the fall-back time (eq. [4]) specialized to WD and MS scenarios is given by \nt fb = 0 : 02 d M 1 = 2 GLYPH<15> ; 5 m GLYPH<0> 3 = 2 ? ; [WD] (12a) \nt fb = 17 : 3 d M 1 = 2 GLYPH<15> ; 5 m 1 = 2 ? ; [MS] ; (12b) \nsince otherwise the light curve would still be rising and hence \nwould not match the LX / ˙ M fb / ( t = t fb) GLYPH<0> GLYPH<11> decay predicted for t GLYPH<29> t fb.", '4.2.2 Magnetic Flux': 'Although it does not represent an independent constraint, the magnetic flux GLYPH<8> GLYPH<15> is determined once the spin of the BH is known, if one assumes that the jet is in a MAD state after the trigger, up until the point when the jet shuts o GLYPH<11> . During MAD accretion, the BH magnetic flux is regulated by the mass accretion rate to a dimensionless value (Tchekhovskoy et al. 2012; Tchekhovskoy & McKinney 2012a), which is well approximated as \nIn a MAD: GLYPH<30> MAD GLYPH<15> GLYPH<17> GLYPH<8> MAD GLYPH<15> ( ˙ Mr 2 g c ) 1 = 2 GLYPH<25> 70(1 GLYPH<0> 0 : 38 ! H) h 1 = 2 0 : 3 ; (13) \ncorresponding to absolute flux \nIn a MAD: GLYPH<8> MAD GLYPH<15> ; 30 = 0 : 067 M 3 = 2 GLYPH<15> ; 5 GLYPH<21> 1 = 2 (1 GLYPH<0> 0 : 38 ! H) h 1 = 2 0 : 3 ; (14) \nwhere GLYPH<21> = ˙ M = ˙ M Edd is the Eddington ratio and h = r = 0 : 3 h 0 : 3 is the thickness of the accretion flow. Thus, if the dimensionless BH spin and the jet luminosity at a given Eddington ratio are known (such as at the point of jet shut-o GLYPH<11> ; see eq. [15] below), then GLYPH<8> GLYPH<15> can be determined.', '4.2.3 Jet Shut-O GLYPH<11> Power': "Another constraint is that the time at which the jet was observed to shut-o GLYPH<11> t o GLYPH<11> GLYPH<0> t trig GLYPH<25> 500 days (see Fig. 1) happens simultaneously with the expected state transition occurring at a fraction of the Eddington accretion rate ( x 2.5), i.e. \n˙ M cr = 0 : 3 GLYPH<21> cr 0 : 3 ˙ M Edd ; (15) \nwhere theoretical and observational uncertainties place the threshold value in the range 0 : 1 . GLYPH<21> cr . 0 : 5. \nJust prior to when the jet shut o GLYPH<11> , its X-ray luminosity / jet power P o GLYPH<11> j was GLYPH<24> 200 times smaller than its value at the end of the GLYPH<24> 10 day plateau phase after the trigger, \nP o GLYPH<11> j ' (1 = 200) P trig j GLYPH<25> 5 GLYPH<2> 10 43 f b GLYPH<15> bol GLYPH<15> GLYPH<0> 1 X 0 : 03 erg s GLYPH<0> 1 : (16) \nNow, by combining equations (3) and (13), the jet power can be written: \nIn a MAD: P j GLYPH<25> F ( ! H) h 0 : 3 ˙ Mc 2 GLYPH<25> 1 : 3 h 0 : 3 a 2 ˙ Mc 2 GLYPH<25> 1 : 6 GLYPH<2> 10 44 h 0 : 3 GLYPH<21> M GLYPH<15> ; 5 a 2 erg s GLYPH<0> 1 ; (17) \nwhere GLYPH<21> GLYPH<17> ˙ M = ˙ M Edd and we used the fact that the spin-dependent factor entering jet power, F ( ! H) = 4 : 4 ! 2 H (1 GLYPH<0> 0 : 38 ! H) 2 f ( ! H), can be approximated as F GLYPH<25> 1 : 3 a 2 to 10% accuracy for 0 : 3 GLYPH<20> a GLYPH<20> 1 (Tchekhovskoy & McKinney 2012a). \nMatching the jet power with the observed power (eq. [16]) at ˙ M = ˙ M cr (eq. [15]) thus requires \na 2 M GLYPH<15> ; 5 = f b GLYPH<15> bol GLYPH<15> GLYPH<0> 1 X 0 : 03 0 : 3 GLYPH<21> cr 0 : 3 h = r : (18) \nThis constraint is independent of the nature of the disrupted object.", '4.2.4 Jet Shut-O GLYPH<11> Accretion Rate': "Another constraint is that the BH mass accretion rate must equal the critical accretion rate (eq. [15]) at the observed jet shuto GLYPH<11> time, t o GLYPH<11> GLYPH<0> t trig ' 500 days, viz. \n˙ M ( t o GLYPH<11> ) = ˙ M cr ; (19) \nwhere ˙ M = f acc ˙ M fb and the predicted fall-back accretion rate (eqs. [4]-[6]) specialized to the MS and WD scenarios: \n˙ M fb = 4 : 2 GLYPH<2> 10 28 M 1 = 3 GLYPH<15> ; 5 m 4 = 3 ? t GLYPH<0> 5 = 3 1 g s GLYPH<0> 1 ; [MS ; complete] (20) \n˙ M fb = 1 : 2 GLYPH<2> 10 29 f 0 : 4 M 3 = 5 GLYPH<15> ; 5 m 8 = 5 ? t GLYPH<0> 2 : 2 1 g s GLYPH<0> 1 ; [MS ; partial] (21) \n˙ M fb = 4 : 6 GLYPH<2> 10 26 M 1 = 3 GLYPH<15> ; 5 t GLYPH<0> 5 = 3 1 g s GLYPH<0> 1 ; [WD ; complete] (22) \nwhere t 1 GLYPH<17> t = day. \nThe factor f acc < 1 accounts for the possibility that only a fraction of the fall-back material actually reaches the BH horizon, with the rest either expelled in the form of the accretion disk winds and or lost during the circularization of the tidal streams. SuperEddington accretion is susceptible to outflows driven by radiation pressure (e.g. Ohsuga et al. 2005, Strubbe & Quataert 2009), but the magnitude of this e GLYPH<11> ect is uncertain. Fortunately, we find that the lower limit on f acc is not constraining ( x 4.3), so without the loss of generality we allow the full range 0 < f acc GLYPH<20> 1 in our calculations.", '4.2.5 Spin Su GLYPH<14> cient for Alignment': 'We require that the spin of the BH to be su GLYPH<14> ciently high, \na & 0 : 5 ; (23) \nsuch that the BH is able to magnetically align the disk and the jets (MTB12a).', '4.2.6 MAD QPO': 'A final possible constraint is to match the QPO period measured in the power-law X-ray light curve of Sw J1644 + 57 (Reis et al. 2012) \nGLYPH<28> QPO = 210 GLYPH<6> 30 s : (24) \nwith the predicted MAD QPO, which occurs at a frequency that is 1 = 4 of BH horizon angular frequency (McKinney et al. 2012b), GLYPH<10> H = ac = (2 r H) = ! H c = (2 r g), resulting in a predicted period \nGLYPH<28> MAD = 2 GLYPH<25> 0 : 25 GLYPH<10> H = 16 GLYPH<25> ! H r g c = 24 : 8 M GLYPH<15> ; 5 ! GLYPH<0> 1 H s : (25) \nEquating (25), multiplied by (1 + z ) GLYPH<25> 1 : 353, with (24) gives a final constraint, \nM GLYPH<15> ; 5 = (6 : 3 GLYPH<6> 0 : 9) ! H : (26) \nWe show below that this requirement is constraining only in the WDscenario.', '4.3 Results': 'Equations (11), (18), (19), (23) and (26) provide 4 or 5 constraints on the unknown parameter space ( M ?; M GLYPH<15> ; a ; ), depending on whether one adopts the (speculative) QPO constraint described in x 4 : 2 : 6. Once M GLYPH<15> and a are determined, the magnetic flux GLYPH<8> GLYPH<15> follows from equation (14). We now apply these constraints individually to each of our proposed scenarios ( x 4 : 1): Complete Disruption \nFigure 7. Constraints on BH mass, M GLYPH<15> , and spin, a , in the scenario of a complete disruption of a lower-mass main-sequence star. The constraint on jet power at jet shut o GLYPH<11> [eq. (18)] gives the blue-shaded regions, with the darker shaded region more likely. The constraint on disruption timescale [eq. (11)] gives the dark (light) shaded red regions for m ? > 1 ( m ? > 0 : 1, respectively). The QPO period constraint [eq. (26)] gives the green curve. The common region seems to favor a lower-mass star, m ? GLYPH<24> few GLYPH<2> 0 : 1, and a light BH, M GLYPH<15> ; 5 GLYPH<24> few, with a & 0 : 5. \n<!-- image --> \nof a Main Sequence Star ( x 4 : 3 : 1); Partial Disruption of a Main Sequence Star ( x 4 : 3 : 2); and Complete Disruption of a White Dwarf ( x 4 : 3 : 3).', '4.3.1 Complete Disruption of a Main-Sequence Star': 'Figure 7 summarizes the constraints on the BH mass and spin for the complete tidal disruption of a low mass MS star. The first constraint, based on the shape of the X-ray light curve (eq. [11]), can be written \nM GLYPH<15> ; 5 < 1 : 3 m GLYPH<0> 1 ? ; (27) \nwhere we have adopted the highest value for t trig = 30 days that allows us to reproduce the observed shape of the X-ray lightcurve (Figs. 1, 5). This constraint is shown in Figure 7 as the dark (light) red shaded regions for m ? GLYPH<21> 1 ( m ? GLYPH<21> 0 : 1). \n? ? For a solar mass star, equation (27) places a tight upper limit on the BH mass, M GLYPH<15> < 1 : 3 GLYPH<2> 10 5 M GLYPH<12> . Such a low mass BH would be novel since they are quite rare (either intrinsically, or due to the observational challenges in detecting them; e.g. Greene 2012) and might be unexpected given the host galaxy of Sw J1644 + 57. A higher mass M GLYPH<15> & few GLYPH<2> 10 5 M GLYPH<12> is allowed if the mass of the star is lower, m ? . 0 : 5, below the peak of the standard IMF (Chabrier 2003). \nThe second constraint (eq. [18]) cuts out an stripe in the M GLYPH<15> GLYPH<0> a plane, as shown as a blue-colored region in Figure 7. The width of the dark (light) blue region reflect an optimistic (conservative) factor of 20 (1000) uncertainty in the value of the right-hand side of equation (18). The chief e GLYPH<11> ect of this constraint is to place a lower limit on the BH mass and spin, the latter consistent with the fourth constraint ( x 4 : 2 : 5). \nThe third constraint (eq. [19]) does not place an interesting limits on BH mass or spin due to the uncertainty in the fraction \nFigure 8. Constraints on BH mass, M GLYPH<15> , and spin, a , in the scenario of a partial disruption of a lower-mass main-sequence star. The constraint on jet power at jet shut o GLYPH<11> [eq. (18)] gives the blue-shaded regions, with the darker shaded region more likely. The constraint on disruption timescale [eq. (11)] gives the dark (light) shaded red regions for m ? > 1 ( m ? > 0 : 1, respectively). The QPO period constraint [eq. (26)] gives the green curve. The common region (including the QPO constraint) allows a Sun-like star, m ? . 1, and a super-massive BH, M GLYPH<15> ; 5 . 5, with a & 0 : 5. \n<!-- image --> \nof the stellar material accreted f acc. However, it does pick out a preferred range in the value of f acc given the other parameters: \nf acc = 0 : 02 GLYPH<21> cr 0 : 3 M 2 = 3 GLYPH<15> ; 5 m GLYPH<0> 4 = 3 ? < 0 : 024 GLYPH<21> cr 0 : 3 m GLYPH<0> 2 ? ; (28) \nwhere in the inequality we have used equation (27). Equation (28) shows that if the tidally disrupted star was solar-like ( m ? GLYPH<24> 1), then large mass loss is required, as could be the result of outflows from the disk or at the impact point of tidal streams. Alternatively, a higher value f acc GLYPH<24> 1 is allowed if the progenitor is instead a low mass M star m ? GLYPH<24> 0 : 1 GLYPH<0> 0 : 2 M GLYPH<12> near the hydrogen-burning limit. \n? The fifth constraint, on the QPO frequency, produces an allowed region shown with green in Fig. 7, which is consistent with (and does not appreciably alter) our conclusions above. If taken seriously, this constraint places an upper limit on the BH mass M GLYPH<15> < 6 GLYPH<2> 10 5 M GLYPH<12> , but otherwise a wide range of BH spin, a & 0 : 5, is allowed.', '4.3.2 Partial Disruption of a Main-sequence Star': "Figure 8 summarizes the constraints on the BH mass and spin based on our second scenario, the partial tidal disruption of a lower-mass MS star. As in equation (27), the first constraint can be written \nM GLYPH<15> ; 5 < 5 : 3 m GLYPH<0> 1 ? ; (29) \nwhere here we have taken t trig = 60 days (again as the maximum allowed by fitting the observed shape of the light curve; x 4.3.1). Constraint (29) is a factor of several less restrictive than that for a complete disruption (eq. 27). For example, the data now allow a relatively massive BH with M GLYPH<15> & few GLYPH<2> 10 5 M GLYPH<12> , even for a Sun-like star, m ? ' 1. \n? The region allowed by the second constraint (eq. [18]; again shown in blue) is exactly the same as for the full disruption (Fig. 7). \nFigure 9. Constraints on BH mass, M GLYPH<15> , and spin, a , in the scenario of a complete disruption of a white dwarf. The constraint on jet power at jet shut o GLYPH<11> [eq. (18)] gives the blue-shaded regions, with the darker shaded region more likely. This darker shaded region appears marginally within the bounds of this plot and requires extreme BH spin: this is due to limited mass supply of the WD that has hard time to power the jets as strong as they are observed. The constraint on disruption timescale [eq. (11)] allows all of the parameter space. We conclude that WD disruption scenario is possible for intermediate mass BHs, with M GLYPH<15> ; 5 = 0 : 1 GLYPH<0> 2 and a & 0 : 5. In addition to requiring an intermediate-mass BH, the WD scenario also requires very small jet opening angles and small bolometric correction, f b GLYPH<15> bol < 5 GLYPH<2> 10 GLYPH<0> 4 . Thus, we suggest that the WD disruption scenario is less likely than a MS star disruption. Accounting for the QPO period constraint [eq. (26)], which is shown with the green curve, constrains the BH spin to a GLYPH<24> 0 : 5. \n<!-- image --> \nThe QPO condition is again only moderately constraining, placing an upper limit M GLYPH<15> ; 5 < 6 GLYPH<2> 10 5 M GLYPH<12> on the BH mass. \n; Similar to the full disruption, the third constraint does not place any interesting limits on the BH or stellar parameters, but it does tell us the fraction, \nf acc = 0 : 2 f 0 : 4 GLYPH<21> cr 0 : 3 M 2 = 5 GLYPH<15> ; 5 m GLYPH<0> 8 = 5 ? < 0 : 39 f 0 : 4 GLYPH<21> cr 0 : 3 m GLYPH<0> 2 ? : (30) \nof the fallback material reaching the BH, where equation (29) is used in the last inequality. For low mass stars the required value of f acc may even exceed unity, potentially ruling out such progenitors depending on the value of GLYPH<21> cr GLYPH<24> 0 : 1 GLYPH<0> 0 : 5. This behavior is opposite to that in the complete MS disruption scenario, where we were forced to conclude that f acc GLYPH<28> 1 [eq. (28)]. As we discussed in x 4.2.1, the two extreme possibilities-of a complete and partial disruption-bracket a continuous family of allowed solutions that, by continuity, are consistent with the data.", '4.3.3 Complete Disruption of a White Dwarf': "Figure 9 summarizes the constraints on the BH properties based on our third scenario, the disruption of a WD. Since a WD is much denser than a MS star, the fallback time in the WD scenario is much shorter than in the MS scenario. In fact, t fb GLYPH<28> t trig, where t trig & 5 days is required to explain the shape of the X-ray lightcurve to within a factor of few (see x 4.2.1). For these reasons, the first constraint, (11), does not place interesting limits on the system parameters. \nThe most important constraint is that on the accretion rate, eq. (19), which gives \nf acc = 2 GLYPH<21> cr 0 : 3 M 2 = 3 GLYPH<15> ; 5 : (31) \nThe physical requirement that f acc < 1 places an upper limit on the BH mass, M GLYPH<15> ; 5 . 2 (for GLYPH<21> cr > 0 : 1), thus requiring an intermediatemass BH . \nSince the mass fallback rate is independent of the WD mass (eq. [22]), this makes the second constraint, eq. (18), on jet power particularly constraining on the BH spin, \na = 1 : 6 f b GLYPH<15> bol GLYPH<15> GLYPH<0> 1 X 0 : 03 ! 1 = 2 GLYPH<18> GLYPH<21> cr 0 : 3 GLYPH<19> 1 = 4 f GLYPH<0> 3 = 4 acc h GLYPH<0> 1 = 2 0 : 3 : (32) \nThe constraints resulting from eqs. (31) and (32) are illustrated in Figure 9 with the dark (light) blue area, whose width reflects optimistic (conservative) modeling and observational uncertainties of a factor 20 (100) in the right-hand side of eq. (18). The most likely (dark blue region) favors a high BH spin, a & 0 : 9. \nThe addition of the less certain QPO constraint (eq. 26) in combination with the spin constrain (eq. 23) favors a BH mass in the range M GLYPH<15> ; 5 GLYPH<24> 0 : 6 GLYPH<0> 2 with an intermediate spin a ' 0 : 5. However, we caution against over-interpreting the QPO constraint, as the observed QPO could be caused by other processes than those resulting from MAD accretion (McKinney et al. 2012b). \nFinally, we note that partial disruption of a WD is less plausible than a complete disruption due to the lower mass fall-back rate and the steeper time-dependence of ˙ M fb (eq. [22]). This makes constraint (32) much more severe, causing tension with the observations. For this reason, we focus exclusively on complete disruption in the WD case.", '5.1 Nature of the Disrupted Star': 'In x 4 : 1 we introduced three plausible scenarios for the nature of the disrupted object: a MS star (full or partial disruption) and a WD(full disruption). We find that a MS stellar disruption comfortably satisfies all of the observational constraints ( x 4.2; Figs. 7, 8) for a BH with mass M GLYPH<15> GLYPH<24> (few GLYPH<0> 10) GLYPH<2> 10 5 M GLYPH<12> which is consistent with upper limits based on X-ray variability and the host galaxy of Sw J1644 + 57(Levan et al. 2011; Bloom et al. 2011; Burrows et al. 2011). Both complete and partial disruption are allowed. A complete disruption favors lower-mass stars, M ? . 0 : 5 M GLYPH<12> , while a partial disruption (in which a star loses 10 to 50% of its mass while its core survives) allows a wide range of stellar masses, M ? = (0 : 1 GLYPH<0> 1) M GLYPH<12> . Complete disruptions of more massive stars ( M ? GLYPH<24> M GLYPH<12> ) are also allowed but require (potentially rare) smallermass BHs ( M GLYPH<15> . 10 5 M GLYPH<12> ). \nUnlike a MS star disruption, WD disruptions require the central object to be an intermediate-mass BH of mass M GLYPH<15> = (1 GLYPH<0> 10) GLYPH<2> 10 4 M GLYPH<12> ( x 4.3.3; Fig. 9). In addition, the mass fallback rate in the WDcase is just barely su GLYPH<14> cient to power the observed jets, which thus greatly restricts the allowed parameter space ( x 4.3.3). \nAlthough the WD scenario is formally allowed, there are several reasons to favor the MS scenario. In addition to the narrow allowed parameter space in the WD case, the probability of WD disruption is much lower than in the MS since its smaller tidal radius limits the allowed range of impact parameters capable of resulting in disruption. Furthermore, the intermediate-mass BH required in the WD case is an exotic, and probably rare, object (Greene \n2012); it is also unclear how frequently such an object would be expected to reside near the nucleus of a galaxy. Given the convergence of several rare events in the WD scenario, we conclude that a MS disruption is more likely. The existence of plausible and constrained solutions provides a consistency check on our model for Sw J1644 + 57. \nIn addition to jetted emission, TD events are accompanied by thermal optical / UV / soft X-ray emission from the accretion disk (e.g. Ulmer 1999) or super-Eddington outflows (Strubbe & Quataert 2009, 2011). No such optical / UV emission was detected from Sw J1644 + 57, possibly due to substantial dust extinction (Bloom et al. 2011). However, if an otherwise similar event with less extinction were to occur in the future, a measurement of the disk flux just after the jet shuts o GLYPH<11> would directly determine GLYPH<21> cr, the critical Eddington ratio at which the disk transitions from being thick to thin. Combined with the jet-related constraints in x 4 : 2, such a measurement would also better determine the properties of the BH and disrupted star.', '5.2 Origin of the Magnetic Flux': "Regardless of the nature of the disrupted star, the magnetic flux threading the BH must be su GLYPH<14> cient to power jet responsible for Sw J1644 + 57. Equation (2) can be written as: \nGLYPH<8> trig GLYPH<15> ; 30 GLYPH<25> 0 : 4 M GLYPH<15> ; 5 0 B B B B B B @ P trig j 10 46 erg s GLYPH<0> 1 1 C C C C C C A 1 = 2 ! H f 1 = 2 ( ! H) 0 : 64 ! GLYPH<0> 1 (33) \nwhere ! H f 1 = 2 ( ! H) is normalized to its value for BH spin a = 0 : 9. If the onset of MAD indeed occurred near the time of the GLYPH<13> GLYPH<0> ray trigger (at Eddington ratio GLYPH<21> trig GLYPH<25> 200 GLYPH<21> cr GLYPH<25> 60( GLYPH<21> cr = 0 : 3); see x 4.2.3), then we require GLYPH<8> MAD GLYPH<15> ; 30 GLYPH<24> 0 : 1 GLYPH<0> 10 for BH masses M GLYPH<15> ; 5 GLYPH<24> 0 : 1 GLYPH<0> 10 consistent with our modeling of Sw J1644 + 57 (Figs. 7-9). This flux is GLYPH<24> 4 GLYPH<0> 6 orders of magnitude greater than that through a solar-type star GLYPH<8> ? GLYPH<24> GLYPH<25> B ? R 2 ? = 10 25 ( B ?= kG)( R ?= R GLYPH<12> ) 2 G cm 2 , even for an optimistically large B ? GLYPH<24> kG average stellar magnetic field. Likewise, a WD ( R ? GLYPH<24> 0 : 01 R GLYPH<12> ) would require a field B ? & 10 11 G for M GLYPH<15> ; 5 GLYPH<24> 1 (Fig. 9) which exceeds the largest measured [surface] fields by two orders of magnitude (e.g. Kepler et al. 2012). \nIf the star itself cannot explain the flux, what could be its source? It has been suggested (Krolik & Piran 2011, 2012) that the requisite flux is generated by a turbulent dynamo in the accretion flow. However, since the net vertical magnetic flux is a conserved quantity in ideal MHD, then the vertical component of the field must undergo a random walk about zero (if not, then what determines a preferred direction?), such that the BH periodically receives patches of random polarity. The characteristic timescale between such flips in the mean field would presumably be set by the accretion timescale near the outer radius of the disk R circ, which is t visc GLYPH<24> 10 3 s and GLYPH<24> 10 6 s in the case of WD and MS stars, respectively (see x 5.3). Without a large-scale magnetic flux to produce a sustained jet, such polarity flips would cause the jet power to transiently switch o GLYPH<11> each time the flux changes sign (Beckwith et al. 2008; McKinney & Blandford 2009; McKinney et al. 2012b) at characteristic intervals GLYPH<24> t visc. Such large-scale variability is not obviously seen in Sw J1644 + 57 lightcurve after the first GLYPH<24> 10 days, once it has settled into a magnetically-arrested state. Instead, the fact that Sw J1644 + 57 was continuously active over nearly 1 : 5 years may suggest that the BH must be threaded by large-scale magnetic flux of the same sign. Although it is di GLYPH<14> cult to rule out a dynamo process completely, since the physics of largescale magnetic field generation in accretion disks is at best poorly \nunderstood, we instead focus on the possibility that a reservoir of large-scale magnetic flux is required. \nApossible source of large-scale flux is that contained in a preexisting ('fossil') accretion disk, which was present at the time of TD but was not detectable in pre-imaging of the host of Sw J1644 + 57 since its accretion rate was relatively low (Eddington ratio GLYPH<21> fossil . 10 GLYPH<0> 2 ). During the disruption process, stellar debris is flung outwards onto a series of highly eccentric orbits with an apocenter radius (e.g. Strubbe & Quataert 2009) \nr a rg GLYPH<25> 2 ct 2 GLYPH<25> rg ! 2 = 3 GLYPH<25> 1 : 3 GLYPH<2> 10 4 M GLYPH<0> 1 = 3 GLYPH<15> ; 5 m GLYPH<0> 2 = 3 ? r ? t t fb ! 2 = 3 : (34) \nthat increases for material with fall-back times longer than that of the most bound stellar debris t fb, where the prefactor in equation (34) is calculated for a solar mass star. Thus, as debris returns to the BH, it sweeps up a significant fraction 5 of the magnetic flux threading the fossil disk at radii r < ra ( t ). \nWe now estimate how luminous the jet from such a fossil disk would have to be in order to supply the required flux, assuming that the fossil disk is itself in a MAD state. In a MAD with a vertical thickness h = r GLYPH<25> 0 : 3 GLYPH<0> 0 : 6 (as characterizes both super-Eddington and highly sub-Eddington accretion), every GLYPH<1> r GLYPH<24> 30 h 0 : 3 rg of the accretion disk contains as much magnetic flux as the BH itself (Tchekhovskoy & McKinney 2012b; McKinney et al. 2012a), where h = r = 0 : 3 h 0 : 3. Thus, the magnetic flux contained by a magnetically-arrested fossil disk out to a distance r is given by (Tchekhovskoy & McKinney 2012b; McKinney et al. 2012b), \nGLYPH<8> fossil D ( r ) GLYPH<25> r 30 rg GLYPH<21> fossil GLYPH<21> MAD ! 1 = 2 h GLYPH<0> 1 0 : 3 GLYPH<8> MAD GLYPH<15> ; (35) \nwhere we used eq. (14) to relate the flux threading the BH by the quiescent disk to that in the MAD state of Sw J1644 + 57, viz. GLYPH<8> fossil GLYPH<15> = ( GLYPH<21> fossil =GLYPH<21> MAD) 1 = 2 GLYPH<8> MAD GLYPH<15> . Since the apocenter distance of the infalling debris increases as ra / t 2 = 3 (eq. 34), the cumulative amount of 'fallback' magnetic flux, which is brought to the BH by the infalling tidal streams, is given by \nGLYPH<8> fb 30 ( t ) ' 0 : 43 GLYPH<21> fossil =GLYPH<21> MAD 10 GLYPH<0> 6 ! 1 = 2 GLYPH<8> MAD GLYPH<15> ; 30 M GLYPH<0> 1 = 3 GLYPH<15> ; 5 m GLYPH<0> 2 = 3 ? r ? h GLYPH<0> 1 0 : 3 t t fb ! 2 = 3 (36) \n' 0 : 38 GLYPH<21> fossil =GLYPH<21> MAD 10 GLYPH<0> 6 ! 1 = 2 M 2 = 3 GLYPH<15> ; 5 m GLYPH<0> 2 = 3 ? r GLYPH<3> GLYPH<2> P j 10 46 erg s GLYPH<0> 1 ! 1 = 2 h GLYPH<0> 1 0 : 3 t t fb ! 2 = 3 ; (37) \nwhere in the last line we have used equation (33) and have assumed a = 0 : 9. Figures 3 and 4 show the time-evolution of GLYPH<8> fb in fiducial MS partial disruption and WD scenarios, respectively. \nBy demanding in equation (36) that at t = t peak ' 1 : 5 t fb the accreted flux GLYPH<8> fb equal GLYPH<8> MAD GLYPH<15> , i.e. su GLYPH<14> cient to magnetically-arrest the disk at t = t MAD, this places a lower limit on the Eddington ratio of the fossil disk (using GLYPH<21> MAD GLYPH<25> 60): \nGLYPH<21> fossil > 2 GLYPH<2> 10 GLYPH<0> 4 M 2 = 3 GLYPH<15> ; 5 m 4 = 3 ? r GLYPH<0> 2 GLYPH<3> ( t peak = t MAD) 4 = 3 ; (38) \ncorresponding to a jet power of the fossil disk given by (eq. 17) \nP fossil j & 3 GLYPH<2> 10 40 M 2 = 3 GLYPH<15> ; 5 m 4 = 3 ? r GLYPH<0> 2 GLYPH<3> ( t peak = t MAD) 4 = 3 erg s GLYPH<0> 1 : (39) \n5 Although the tidal debris traverses only a small fraction of the azimuthal extent of the disk, the time it spends at pericenter is comparable to the local orbital time. Thus, a large fraction of the disk will have su GLYPH<14> cient time to rotate into, and be collected by, the debris before it falls back. \nIf we adopt bolometric and beaming corrections similar to that applied in Sw J1644 + 57, then the resulting X-ray luminosity L X ; obs & 5 GLYPH<2> 10 41 M 2 = 3 GLYPH<15> ; 5 m 4 = 3 ? r GLYPH<0> 2 GLYPH<3> erg s GLYPH<0> 1 to t MAD GLYPH<24> t peak is comfortably consistent with ROSAT upper limits L X . 2 GLYPH<2> 10 44 erg s GLYPH<0> 1 on prior activity from the host galaxy of Sw J1644 + 57 (Bloom et al. 2011) for 1 . M GLYPH<15> ; 5 . 10. 6 Thus, a fossil disk su GLYPH<14> ciently dim to go undetected prior to Sw J1644 + 57 nevertheless could supply su GLYPH<14> cient magnetic flux to power the observed jet.", '5.3 Nature and Duration of the Flaring State': "In our model for Sw J1644 + 57, the jets initially point along the rotation axis of the accretion disk, which is misaligned relative to the BH spin and our line of site (Stage 1 in Fig. 2). As ˙ M decreases, however, magnetic flux accumulated on the BH becomes increasingly dynamically important and the disk enters a magneticallyarrested state, MAD (Narayan et al. 2003; Tchekhovskoy et al. 2011). Once in a MAD state, strong magnetic fields cause the inner disk and jet to align with the BH spin axis (Stage 2). However, this alignment process is erratic, with the jets periodically ramming against the accretion flow (MTB12a). Occasionally a strong flare is produced when the jet transiently aligns with our line of site, but at most times we observe emission originating from large angles o GLYPH<11> the core of the jet. This violent transition to a fully-aligned jet accounts for the extreme variability observed in Sw J1644 + 57 over the first 10 days after the GLYPH<13> -ray trigger (Bloom et al. 2011). Since the first evidence for a jet GLYPH<24> 4 days prior to the trigger (Krimm & Barthelmy 2011; Burrows et al. 2011), the total duration of this flaring phase was at least two weeks. \nBecause the jet alignment is controlled by the magnetic field of the BH, the characteristic interval between flares is set by the timescale for magnetic flux accumulation. Recent simulations show that large fluctuations in the jet power ('flares') are produced in MAD flows due to periodic accumulation and expulsion of flux by the BH on semi-regular intervals of (0 : 5 GLYPH<0> 2)10 3 M GLYPH<15> ; 5 seconds (Tchekhovskoy et al. 2011; Tchekhovskoy & McKinney 2012b). To understand this result, note that the inner GLYPH<24> 30 h 0 : 3 rg of a MAD accretion disk contain as much flux as the BH itself (Tchekhovskoy & McKinney 2012b; McKinney et al. 2012b). Therefore, when the BH expels an order-unity fraction of its flux, this flux is replenished on a characteristic timescale set by the accretion time at r ' 15 h 0 : 3 rg , \nGLYPH<1> t flare GLYPH<24> t acc = GLYPH<11> GLYPH<0> 1 ss h r ! GLYPH<0> 2 GLYPH<10> GLYPH<0> 1 K ' 3 GLYPH<2> 10 3 s GLYPH<18> GLYPH<11> ss 0 : 1 GLYPH<19> GLYPH<0> 1 h GLYPH<0> 1 = 2 0 : 3 M GLYPH<15> ; 5 r 15 rg ! 3 = 2 ; (40) \nwhere GLYPH<11> ss parameterizes the disk viscosity and GLYPH<10> K = ( GM GLYPH<15> = r 3 ) 1 = 2 is the Keplerian orbital velocity. Equation (40) shows that GLYPH<1> t flare is consistent with the observed interval few GLYPH<2> 10 4 seconds between the large-amplitude flares in Sw J1644 + 57 if the BH is moderately massive, M GLYPH<15> ; 5 & few (depending on GLYPH<11> ss). \n; In contrast to the interval between flares, the total duration of the flaring state must be at least as long as the accretion time scale \n6 That said, it is not at all clear whether the jet from the fossil disk would indeed be pointed along our line of site ( = spin axis of the BH) since although the disk is in a MAD state at small radii, the jet direction could be set by the plane of the disk on larger scales, where it is not magnetically-arrested and in general is misaligned with our line of site. \nof the entire disk near its outer radius GLYPH<24> the circularization radius. Substituting R circ (eq. [8]) into equation (40), we find a flaring duration \nGLYPH<28> acc GLYPH<17> t acc( r = R t) \n' 5 GLYPH<2> 10 5 s GLYPH<18> GLYPH<11> ss 0 : 1 GLYPH<19> GLYPH<0> 1 h GLYPH<0> 1 = 2 0 : 3 m ?GLYPH<12> GLYPH<0> 3 = 2 [MS] (41) \n' 10 3 s GLYPH<18> GLYPH<11> ss 0 : 1 GLYPH<19> GLYPH<0> 1 h GLYPH<0> 1 = 2 0 : 3 ( m ?= 0 : 6) GLYPH<0> 1 GLYPH<12> GLYPH<0> 3 = 2 : [WD] (42) \nAgain, GLYPH<28> acc is consistent with the duration of the flaring state in Sw J1644 + 57 ( GLYPH<24> 10 6 s) for a solar-type star with impact parameter GLYPH<12> GLYPH<24> 1, independent of the BH mass. However, for a WD GLYPH<28> acc appears to be too short. \nAlthough GLYPH<28> acc [eq. (42)] sets the minimum duration of the flaring state, the state can last longer if the BH requires more time to fully align the accretion disk with the BH spin. Alignment completes only once the entire disk is magnetically-arrested. At the onset of a MAD, most of the [large-scale] magnetic flux in the system is concentrated near the BH. As time goes on, two processes take place. Firstly, the mass accretion rate drops, causing the centrallyconcentrated magnetic flux to be redistributed to the outer regions of the accretion disk. Secondly, new flux is added at the outer edge of the disk by the infalling stellar debris that has picked up magnetic flux from the fossil accretion flow ( x 5.2). Since in a MAD every GLYPH<24> 30 h 0 : 3 rg in radius holds roughly the same amount magnetic flux as the BH itself (Tchekhovskoy & McKinney 2012b), the whole disk goes MAD once its flux reaches a value \nGLYPH<8> max D GLYPH<25> R circ 30 rg h GLYPH<0> 1 0 : 3 GLYPH<8> GLYPH<15> GLYPH<25> 15 r GLYPH<3> m GLYPH<0> 1 = 3 GLYPH<3> M GLYPH<0> 2 = 3 GLYPH<15> ; 5 GLYPH<12> GLYPH<0> 1 h GLYPH<0> 1 0 : 3 GLYPH<8> GLYPH<15> ; (43) \nwhere GLYPH<8> D is the total flux through the midplane of the disk and the last equality makes use of equation (8). \nNow, the flux through the BH evolves as (see eq. 14): \nGLYPH<8> GLYPH<15> = GLYPH<8> MAD GLYPH<15> ( t = t MAD) GLYPH<0> GLYPH<11>= 2 ; (44) \nwhere GLYPH<8> MAD GLYPH<15> is BH magnetic flux and t MAD is the time at the onset of MAD near the hole. The fallback magnetic flux, brought in by the infalling debris, evolves as (see eq. 37): \nGLYPH<8> fb = GLYPH<8> MAD GLYPH<15> t t MAD ! 2 = 3 : (45) \nThe flux through the disk evolves as \nGLYPH<8> D = ( GLYPH<8> MAD GLYPH<15> GLYPH<0> GLYPH<8> GLYPH<15> ) + ( GLYPH<8> fb GLYPH<0> GLYPH<8> MAD GLYPH<15> ) ; (46) \nwhere the disk flux increases as the result of both flux leaving the BH (first term in parentheses) and new flux brought in by the infalling debris (second term in parentheses). \nCondition (43) can now be written: \nGLYPH<28> align t MAD GLYPH<25> 1 + GLYPH<8> max D GLYPH<8> GLYPH<15> ! GLYPH<13> GLYPH<25> GLYPH<16> 1 + 15 r GLYPH<3> m GLYPH<0> 1 = 3 GLYPH<3> M GLYPH<0> 2 = 3 GLYPH<15> ; 5 GLYPH<12> GLYPH<0> 1 h GLYPH<0> 1 0 : 3 GLYPH<17> GLYPH<13> (47) \nGLYPH<25> 2 6 6 6 6 4 1 + 0 : 8 GLYPH<18> m ? 0 : 5 GLYPH<19> 2 = 3 M GLYPH<15> ; 5 3 ! GLYPH<0> 2 = 3 GLYPH<18> GLYPH<12> 2 GLYPH<19> GLYPH<0> 1 h 0 : 3 3 ! GLYPH<0> 1 3 7 7 7 7 5 2 = 3 ; [MS ; complete] \n(48) \nGLYPH<25> 2 6 6 6 6 4 1 + 1 : 9 GLYPH<18> m ? 0 : 5 GLYPH<19> 2 = 3 M GLYPH<15> ; 5 3 ! GLYPH<0> 2 = 3 GLYPH<18> GLYPH<12> 0 : 8 GLYPH<19> GLYPH<0> 1 h 0 : 3 3 ! GLYPH<0> 1 3 7 7 7 7 5 0 : 57 ; [MS ; partial] \n(49) \nGLYPH<25> 2 6 6 6 6 4 1 + 0 : 4 GLYPH<18> m GLYPH<3> 0 : 6 GLYPH<19> GLYPH<0> 2 = 3 M GLYPH<15> ; 5 0 : 1 ! GLYPH<0> 2 = 3 GLYPH<12> GLYPH<0> 1 h 0 : 3 3 ! GLYPH<0> 1 3 7 7 7 7 5 2 = 3 ; [WD ; complete] \n(50) \nwhere GLYPH<13> = 6 = (4 + 3 GLYPH<11> ), and the last three lines are evaluated for our three progenitor scenarios ( x 4 : 1). Equation (47) shows that the jet alignment (flaring) phase can last for a timescale comparable (or somewhat longer than) than that required for the MAD to form. For a MS star, both GLYPH<28> acc [eq. (42)] and GLYPH<28> align [eq. (47)] are su GLYPH<14> -ciently long to account for the observed duration of the flaring state in Sw J1644 + 57. However, for a WD, although GLYPH<28> acc is very short, the duration of the flaring state is set by GLYPH<28> align GLYPH<24> t MAD and hence (since t MAD GLYPH<24> t trig GLYPH<24> days) is also consistent with observations. \nMany of the points above are illustrated explicitly in Figures 3 and 4, which shows the time-evolution of GLYPH<8> D and GLYPH<8> GLYPH<15> in fiducial MS partial disruption and WD scenarios, respectively. In both cases, GLYPH<8> GLYPH<15> initially rises with the accumulated flux GLYPH<8> fb, until the inner disk near the BH becomes magnetically arrested. After this point, flux leaks out of the BH into the surrounding disk and new flux is added by fallback material at the outer edge of the disk ( GLYPH<8> D rises). However, eventually ˙ M drops su GLYPH<14> ciently for the entire disk to become magnetically arrested (magnetic field marginally dynamically-important everywhere). At this point, even the disk itself cannot hold the accumulated flux, which begins to leak out, causing GLYPH<8> D to fall. The jet alignment / flaring phase described above (Stage 2) occurs during the interval when GLYPH<8> D is still rising.", '5.4 Origin of Radio Rebrightening': "Our fits to the X-ray light curve of Sw J1644 + 57 (Figs. 1, 5, 6) show that the jet could have been active weeks-month prior to the first detection. However, since the jet was pointed away from GLYPH<0> and possibly precessing about GLYPH<0> our line of site ( = spin axis of the BH), its emission was not initially observable due to geometric or relativistic beaming. Nevertheless, since the jet still injects relativistic kinetic energy into the surrounding ISM during this phase, this gives rise to delayed radio afterglow emission on a timescale of months GLYPH<0> year for a typical misalignment angle ( x 2 : 3), consistent with observed radio re-brightening several months after the trigger (B12). \nTo produce radio rebrightening of the magnitude observed in Sw J1644 + 57, the additional energy released by the early o GLYPH<11> -axis jet must exceed that released later during the on-axis phase (i.e. after the initial GLYPH<13> -ray detection). Since the mass accretion rate decreases as ˙ M / t GLYPH<0> GLYPH<11> and the jet power P j / ˙ M [eq. (17)] during MAD accretion, then the maximum 7 energy released by the jet prior to time t is given by \nE ( t ) / Z t t peak t 0GLYPH<0> GLYPH<11> dt 0 = t 1 GLYPH<0> GLYPH<11> peak GLYPH<0> t 1 GLYPH<0> GLYPH<11> GLYPH<11> GLYPH<0> 1 ; (51) \nwhere GLYPH<11> GLYPH<24> 5 = 3 GLYPH<0> 2 : 2. If one demands that the jet energy released before the trigger exceeds the energy released after the trigger, \n1 < E ( t trig) E ( 1 ) GLYPH<0> E ( t trig) = t trig t peak ! GLYPH<11> GLYPH<0> 1 GLYPH<0> 1 ; (52) \nthen this requires \nt trig > 2 1 = ( GLYPH<11> GLYPH<0> 1) t peak : (53) \n7 This is a maximum since we assume that accretion is in a MAD state at all times, such that Pj GLYPH<24> ˙ M c 2 (eq. [17]) : Prior to the onset of MAD (Stage 1 in Fig. 2) the jet e GLYPH<14> ciency is instead limited by the amount of accumulated magnetic flux (eq. [3]). Also, the jet can be stifled at early times when the magnetic field is dynamically weak ( GLYPH<30> GLYPH<15> < GLYPH<30> on GLYPH<25> 20) by the high ram pressure of the thick misaligned inflow (Komissarov & Barkov 2009). \nAside from a di GLYPH<11> erent prefactor, this constraint is identical to that based on the shape of the X-ray light curve (eq. [11]). Selfconsistency of our model thus requires that the o GLYPH<11> -axis power be similar to that required to explain the observed radio rebrightening. \nAlthough we focus above on energy injected o GLYPH<11> -axis prior to the GLYPH<13> -ray trigger, this is not the only means by which the jets could inject 'invisible' energy into the ambient medium. The onset of MAD accretion after the GLYPH<13> -ray trigger may cause the jet to wobble in and out of our line of sight, giving rise to high amplitude variability ( x 3 : 3; Stage 2 in Fig. 2). Since during this process the jets are pointed towards our line of site only a fraction of the time, most of their energy is released during a mis-aligned state. Indeed, the GLYPH<24> 10 per cent duty cycle of the observed flaring (Fig. 1) suggests that the energy injected during misaligned phases could exceed that injected along our line of site by an order of magnitude. Since GLYPH<24> 1 = 2 of the total X-ray fluence occurred during the first t GLYPH<0> t trig . 10 days (wobbling jet phase), misaligned jets could produce o GLYPH<11> axis relativistic ejecta with GLYPH<24> 5 times more energy than that directly probed by the observed GLYPH<13> -ray / X-ray emission. This alone might be su GLYPH<14> cient to power the observed rebrightening, without the need for significant energy injection prior to the trigger.", '5.5.1 X-ray Transients': "We postulate that Sw J1644 + 57 resulted from a somewhat special geometric configuration, in which the BH spin axis was pointed along our line of site. This favourable geometry resulted in several bright flares over the first GLYPH<24> 10 days, which are produced as the jet settles down into its fully aligned configuration (Stage 2 in Fig. 2; x 4.2.1, 5.3). If, however, we had instead been positioned along a more 'typical' line of site not aligned with the BH, then we might only have observed a small number of flares, possibly just one in the majority of cases. Such a single flare would appear as an Xray / soft GLYPH<13> -ray transient of duration GLYPH<24> 10 3 s, yet unaccompanied by an extended luminous X-ray tail as characterized Sw J1644 + 57. \nCould such a population of o GLYPH<11> -axis jetted TDE flares be contributing to the known population of high energy transients? Given their long duration and low luminosity, such flares could be mistaken for a long-soft gamma-ray burst (GRB) or an X-ray Flash. Perhaps the closest known analog is the class of 'low-luminosity GRBs' (LLGRBs) (e.g. Soderberg et al. 2006; Cobb et al. 2006), although most of these are probably not standard TD events due to their locations o GLYPH<11> the nucleus of their host galaxies and their observed association with core-collapse supernovae (e.g. Chornock et al. 2010; but see Shcherbakov et al. 2012). The volumetric rate of LLGRBs actually exceeds that of classical GRBs (Soderberg et al. 2006), but only a handful are known since they are more challenging to detect than luminous high redshift GRBs (see Nakar & Sari 2012 for a recent compilation). Thus, even if the rate of single-flare o GLYPH<11> -axis TD jets is a factor of GLYPH<24> 10 times higher than the rate of Sw J1644 + 57-like events, it is unclear how many should have been detected yet. If such an event is eventually detected, perhaps by the next generation of wide-field X-ray telescopes, it may be distinguished from normal LLGRBs by its [1] nuclear position; [2] associated optical / UV / soft X-ray emission produced by isotropic thermal emission from the accretion disk or ionized stellar ejecta (Strubbe & Quataert 2011,Clausen et al. 2012), rather than a corecollapse SNe; [3] delayed radio emission from the o GLYPH<11> -axis jet (Giannios & Metzger 2011; x 5.5.3).", '5.5.2 Jet Revival': "Although the jet in Sw J1644 + 57 is presently o GLYPH<11> , our model predicts that the BH continues to accrete through a geometrically-thin accretion disk in a jet-less 'thermal state'. However, eventually the mass accretion rate will decrease below GLYPH<25> 2% of ˙ M Edd (Maccarone 2003), after which point the flow may transition to a 'hard' state, as characterized by a radiatively-ine GLYPH<14> cient geometrically-thick accretion flow. Once this transition occurs, magnetic flux can once again accumulate near the black hole on a short timescale ( GLYPH<24> t acc; eq. 42), resulting in a new MAD accretion phase (Stage 5 in Fig. 2). A jet aligned with the Earth and BH spin axis will thus again form, with a power that again tracks the accretion rate ˙ M . From Figures 5 and 6 we estimate that the X-ray flux will be GLYPH<24> 2 GLYPH<2> 10 GLYPH<0> 14 GLYPH<0> 10 GLYPH<0> 13 erg cm GLYPH<0> 2 s GLYPH<0> 1 when the jet turns back on, well within the detection limits of current X-ray observatories. \nIn order to test this idea, we strongly encourage regular Xray follow-up of Sw J1644 + 57 over the next decade. The timescale and flux of the observed revival would inform [1] the rate at which the accretion rate is decreasing, and therefore help to distinguish between partial and complete disruption scenarios, or whether the disk has transitioned to a spreading evolution (e.g. Cannizzo et al. 2011) [2] the ratio of accretion rates which characterize the thick ! thin disk and thin ! thick disk transitions, respectively; [3] and whether the jet beaming correction (related to the opening angle and bulk Lorentz factor) during the low-hard state are similar to those during the super-Eddington state.", '5.5.3 Ubiquity of Radio Transients from TD Events': 'Due to the enormous energy released in relativistic ejecta, Sw J1644 + 57 remains a bright radio source ( F GLYPH<23> GLYPH<24> 1 GLYPH<0> 10 mJy at GLYPH<23> GLYPH<24> 1 GLYPH<0> 50 GHz) even now, almost two years after the TD event. Since most of the jet activity occurred GLYPH<24> months around the trigger time, by now the ejecta has slowed down appreciably due to interaction with the circumnuclear medium. The current expansion velocity of the ejecta is at most mildly relativistic GLYPH<13> GLYPH<24> 2 (B12), in which case the radio emission should be relatively isotropic (i.e. flux varying by less than an order of magnitude from front to side). \nThe fact that Sw J1644 + 57 is a bright isotropic radio source has two implications: First, given the relatively close proximity ( z GLYPH<24> 0 : 1) of most candidate events detected over the past several years, any jet from these events as remotely as powerful as that in Sw J1644 + 57 should be easily detectable by now, even if the jet remains always pointed away from our line of site. Bower et al. (2012) and van Velzen et al. (2012) have conducted radio followup observations of previous thermal TD flares; since most of these observations produced only deep upper limits (with a few interesting exceptions), this already constrains the fraction of TD events accompanied by powerful jets to be . 10 per cent. In hindsight it is perhaps unsurprising that Sw J1644 + 57 is unique, given the enormous magnetic flux required to power its jet, which could require special conditions not satisfied by most TDs ( x 5 : 2; De Colle et al. 2012). A second consequence of the radio luminosity of Sw J1644 + 57 is that o GLYPH<11> -axis emission from other jetted TD events (albeit rare) is isotropic and hence should be detectable out to much higher redshifts ( z GLYPH<24> 1). Such events are one of the most promising sources for future wide-field radio surveys (Giannios & Metzger 2011; van Velzen et al. 2011; Frail et al. 2012; see Cenko et al. 2012 for a potential high redshift analog to Sw J1644 + 57), which will help constrain the rate and diversity of jetted TD events.', '6 CONCLUSIONS': "We have presented a self-consistent model that explains many of the previously disparate puzzles associated with the jetted TD event Sw J1644 + 57 (Fig. 2; x 3). This model relies on just one major assumption: the accumulation of a large, dynamically-important magnetic flux near the central BH, such that the accretion flow from the returning stellar debris becomes magnetically-arrested (MAD; x 3.3) on a timescale of GLYPH<24> week-month after the TD event (Figs. 56). The onset of MAD accretion in Sw J1644 + 57 naturally accounts for (i) the period of intense flaring that lasted for the first GLYPH<24> 10 days after the trigger; (ii) the approximate constancy (in a timeaverage sense) of the luminosity during this period; (iii) the subsequent transition of X-ray luminosity to a steady (non-precessing) jet with a power that tracks the predicted power-law decline in the accretion rate; (iv) the sudden shut o GLYPH<11> of the jet emission at GLYPH<24> 500 days after the trigger; and (v) the potential origin of the mysterious late-time radio rebrightening that started about a month after the trigger. Our model also naturally predicts a QPO (at a frequency closely tied to that of a BH spin) that is consistent with that seen in the lightcurve of Sw J1644 + 57. \nWe emphasize that the strong magnetic flux required by our model is not an entirely independent assumption: a flux of at least this magnitude is necessary to explain the observed jet power in the first place (eq. 3; x 5 : 2; given plausible values of the BH mass and spin), while a much weaker flux would be unable to produce a jet at all (the jet would be 'stifled') against the powerful ram pressure of the misaligned disk (Komissarov & Barkov 2009).", 'ACKNOWLEDGEMENTS': "We thank James Guillochon, Michael Kesden, Serguei Komissarov, Julian Krolik, Morgan MacLeod, Jonathan C. McKinney, Petar Mimica, Ramesh Narayan, Ryan O'Leary, Asaf Pe'er, Tsvi Piran, Eliot Quataert, Roman Shcherbakov, Nicholas Stone, and Sjoert van Velzen for insightful discussions. 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2018PhRvL.121p1103F
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Black Hole Mergers from an Evolving Population of Globular Clusters
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2018-01-01
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[]
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The high rate of black hole (BH) mergers detected by LIGO/Virgo opened questions on their astrophysical origin. One possibility is the dynamical channel, in which binary formation and hardening is catalyzed by dynamical encounters in globular clusters (GCs). Previous studies have shown that the BH merger rate from the present day GC density in the Universe is lower than the observed rate. In this Letter, we study the BH merger rate by accounting for the first time for the evolution of GCs within their host galaxies. The mass in GCs was initially ∼8 ×higher , which decreased to its present value due to evaporation and tidal disruption. Many BH binaries that were ejected long before their merger originated in GCs that no longer exist. We find that the comoving merger rate in the dynamical channel from GCs varies between 18 to 35 Gpc<SUP>-3</SUP> yr<SUP>-1</SUP> between redshift z =0.5 to 2, and the total rate is 1, 5, 24 events per day within z =0.5 , 1, and 2, respectively. The cosmic evolution and disruption of GCs systematically increases the present-day merger rate by a factor ∼2 relative to isolated clusters. Gravitational wave detector networks offer an unique observational probe of the initial number of GC populations and their subsequent evolution across cosmic time.
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https://arxiv.org/pdf/1806.02351.pdf
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{'Black hole mergers from an evolving population of globular clusters': "Giacomo Fragione 1 and Bence Kocsis 2 \n1 Racah Institute for Physics, The Hebrew University, Jerusalem 91904, Israel 2 Institute of Physics, Eotvos University, P'azm'any P. s. 1/A, Budapest, 1117, Hungary (Dated: October 10, 2018) \nThe high rate of black hole (BH) mergers detected by LIGO/Virgo opened questions on their astrophysical origin. One possibility is the dynamical channel, in which binary formation and hardening is catalyzed by dynamical encounters in globular clusters (GCs). Previous studies have shown that the BH merger rate from the present day GC density in the Universe is lower than the observed rate. In this Letter , we study the BH merger rate by accounting for the first time for the evolution of GCs within their host galaxies. The mass in GCs was initially ∼ 8 × higher, which decreased to its present value due to evaporation and tidal disruption. Many BH binaries that were ejected long before their merger, originated in GCs that no longer exist. We find that the comoving merger rate in the dynamical channel from GCs varies between 18 to 35 Gpc -3 yr -1 between redshift z = 0 . 5 to 2, and the total rate is 1, 5, 24 events per day within z = 0 . 5, 1, and 2, respectively. The cosmic evolution and disruption of GCs systematically increases the present-day merger rate by a factor ∼ 2 relative to isolated clusters. Gravitational wave detector networks offer an unique observational probe of the initial number of GC populations and their subsequent evolution across cosmic time. \nIntroduction.The LIGO-Virgo Collaboration 1 has recently detected gravitational waves (GWs) from five binary black hole (BH) mergers, opening an entirely new window into high-energy physics [1, 2]. The astrophysical origin of these mergers is among the most puzzling open questions of our time. Possibilities include isolated binary evolution through a common envelope phase [3] or through chemically homogeneous evolution in shortperiod stellar binaries [4, 5], triple systems [6-9], gasassisted mergers [10-12], and dynamically assembled binaries in dense stellar systems such as globular clusters (GCs) [13-16] or galactic nuclei [17-21]. \nIn contrast to most other channels, the dynamical formation channel is theoretically well-understood, as it is determined by N -body gravitational interactions. In dense systems, chance multibody close encounters lead inexorably to the formation of BH binaries [22]. Further scattering encounters decrease the BH binary separation. In at least 50% of merging systems, the binary is ejected from the GC [23, 24], and the binary merges due to GW emission after several Gyr of its ejection [15, 25]. \nPrevious studies of dynamically formed mergers in GCs have shown that the expected BH merger rate is ∼ 5 Gpc -3 yr -1 [15, 25, 26], which is lower than the observed rate of R = 40-240Gpc -3 yr -1 reported by LIGO/Virgo, corresponding to a power-law mass function prior, and R = 12-65Gpc -3 yr -1 for a log-uniform mass function [27]. The expected rates are sufficiently high that this contribution may be measured and distinguished from other channels statistically using the mass, spin, eccentricity, and redshift distribution [23, 24, 2830]. In these studies the rates were estimated using the \nobserved present-day GC density in the Universe and the GCs were evolved in isolation. \nIn this Letter , we point out the importance of including GC evolution within their host galaxies for studying the dynamically formed BH mergers. The initial mass in GCs is expected to have been a factor ∼ 8 × higher than today [31, 32], since many GCs have evaporated and were tidally disrupted during interactions with the host galaxy. This expectation is confirmed by the observed radial and mass distribution of GCs in the Galaxy [31], and the observed high-energy emission from the Galactic bulge. The so-called Fermi-excess may have been produced by a population of millisecond pulsars that were formed in long-disrupted GCs [32-35]. Does the increased initial GC mass increase the BH merger rate significantly? We determine the BH merger rate by accounting for the evolution and disruption of GCs in their host galaxies. We predict the redshift evolution of the merger rate, which may be measured with upcoming GW detectors. This may offer an observational probe to distinguish mergers of the dynamical GC channel from other astrophysical channels. If so, future GW measurements of the merger rate distribution offers an observational probe of the initial number of GC populations and their evolution. \nGC evolution.We follow Ref. [31] to evolve the initial GC population in their host galaxies using a semianalytical method. The GC formation rate is assumed to be a fixed fraction f GC ,i = 0 . 011 of the total galactic star formation rate assuming that clusters formed at z = 3 [31, 32]. We draw the initial mass of the clusters from a power-law distribution dN/dM ∝ M -2 from 10 4 M glyph[circledot] ≤ M ≤ 10 7 M glyph[circledot] . \nAfter their formation, GCs lose mass via three mechanisms, i.e. dynamical ejection of stars through two-body relaxation, removing stars by the galactic tidal field and \nstellar winds [36-38]. We evolve the cluster mass loss due to isolated evaporation due to two-body relaxation and stripping by the galactic tidal field according to \ndM dt = -M min( t iso , t tid ) , (1) \nwhere [39] \nt iso ( M ) ≈ 8 . 5 M 5 Gyr , (2) \nt tid ( r, M ) ≈ 2 . 07 M 2 / 3 5 r kpc ( V c ( r ) 200km s -1 ) -1 Gyr (3) \nand where M 5 = M/ 10 5 M glyph[circledot] and t tid takes into account the strength of the local galactic field through the circular velocity V c ( r ) at a distance r from the galactic center. While in case of strong tidal field ( t tid < t iso ) the stars loss is dominated by the galactic tidal stripping, in the limit of a weak tidal field ( t tid > t iso ), the evaporation of stars is mostly controlled by internal dynamics. \nIn proximity to the galactic center, the clusters may be torn apart due to the strong tidal forces. We assume that the cluster is disrupted when the average stellar density at half-mass radius falls below the mean galactic density \nρ h < ρ ∗ ( r ) = V 2 c ( r ) 2 πGr 2 . (4) \nWe adopt the average density at the half-mass radius \nρ h = 10 3 M glyph[circledot] pc -3 min [ 10 2 , max(1 , 0 . 25 M 2 5 ) ] . (5) \nThis equation limits ρ h to 10 5 M glyph[circledot] pc -3 in the most massive clusters, consistent with observations [31]. We note that ρ ∗ ( r ) in Eq. (4) takes into account both the adopted field stellar mass, as well as the growing mass of the galactic bulge, that begins to build up as clusters are disrupted in the innermost galactic regions. \nFor what concerns the cluster orbit, following Ref. [31] for simplicity we evaluate the cluster at an instantaneous radial distance r from the host galaxy center which represents the time-averaged radius 2 of the true (probably eccentric) cluster orbit [31]. We evolve the cluster orbits by evolving r according to dynamical friction as[40] \ndr 2 dt = -r 2 t df , (6) \nwhere \nt df ( r, M ) ≈ 90 M -1 5 ( r kpc ) 2 ( V c ( r ) 200km s -1 ) Gyr . (7) \nWe describe the Milky Way's potential with a central 4 × 10 6 M glyph[circledot] black hole, a Sersic profile with total mass 5 × 10 10 M glyph[circledot] and effective radius 4 kpc, and a dark matter halo (10 12 M glyph[circledot] with r s = 20 kpc). Throughout the simulation, we constantly update the galactic mass distribution to include the stellar and gaseous debris from the disrupted clusters [31]. \nThese initial conditions lead to the observed spatial and mass distribution of GCs surviving until z = 0 in our Galaxy [31, 32]. \nRate of black hole mergers.We calculate the rate of mergers from the convolution of the P ej ,M ejection probability at comoving time t ej and the conditional merger probability among ejected binaries P merg | ej ,M within inspiral time ∆ t following ejection in a cluster of mass M as \nP M ( t ) = ∫ t 0 P ej ,M ( t ej ) P merg | ej ,M ( t -t ej ) dt ej . (8) \nWe fit the cumulative probability of BH-BH ejections from Morscher et al. [42] as \nC ej ,M ( t ) = f BH , ej ,M f bin , ej ,M ( t t H ) 0 . 4 , (9) \nwhere f BH , ej ,M ≈ 0 . 5 is the fraction of ejected BHs among all BHs in the cluster, and f bin , ej ,M ≈ 0 . 25 is the fraction of BHs in ejected binaries relative to the number of all ejected BHs in a Hubble time t H 3 . We adopt the results of Monte-Carlo simulations by Rodriguez et al. [25] for the cumulative conditional probability distribution of the inspiral times of BH mergers C merg | ej ,M (∆ t ) = ∫ ∆ t 0 P merg | ej ,M ( t ' ) dt ' (see Figure 1 therein). We find that this function is fitted by \nC M ( t ) = c 1 erf[(ln x ) /c 2 )] + c 0 , (10) \nwhere c 1 = 0 . 497235, c 0 = 0 . 517967, c 2 = 5 . 69292 and \nx = t 7 . 6 Gyr ( M 10 6 M glyph[circledot] ) 4 . (11) \nThis functional dependence follows from the conditions of binary ejection and the subsequent GW-driven evolution [25]. Note that the timescale for a fixed fraction of BHs to merge within a GC is proportional to M -4 . \nWe compute the comoving merger rate density from the evolving population of globular clusters as a function of redshift by summing over the merger rate Γ BH ( z ) over all GCs in a simulation in a Milky Way type galaxy, \nR ( z ) = ρ GC N GC (0) N GC ( z ) ∑ i =1 Γ BH ( M i , z ) , (12) \n<!-- image --> \nFIG. 1: The comoving BH merger rate density as a function of redshift z (left) and the total number of sources that merge per unit observer time up to a maximum redshift z (right). The black solid and red-dashed lines represent upper and lower limits on the expected rate from evolving GCs assuming respectively that merging binaries are all ejected before the cluster may be disrupted or that they merge within the cluster (see text). Blue dashdotted line represents the result of Ref. [25] for isolated clusters. The merger rates are higher for evolving clusters that lost mass due to evaporation and tidal stripping, since they were initially more massive and more numerous to match the present day observed GC distribution. The shaded regions represent the model uncertainty by assuming ρ GC in the range 0 . 32-2 . 31 Mpc -3 [14, 25, 41], the lines represent ρ GC = 0 . 77 Mpc -3 . The lower boundary of our model (red-shadowed region) is computed by scaling ρ GC for the red-dashed curve, while the upper boundary is calculated similarly using the black-solid curve. The shaded blue region coresponds to Ref. [25]. \n<!-- image --> \nwhere M i is the initial mass of the i -th cluster, N GC ( z ) is the number of GCs at redshift z per Milky Way type galaxy taken from the simulation [31] 4 and ρ GC = 0 . 77 Mpc -3 is the comoving number density of globular clusters [14]. 5 In equation (12), the comoving rate of BH-BH mergers per GC of mass M is obtained from the probability of BH-BH mergers as \nΓ BH ( M,t ) = 1 2 f BH MP M ( t ) , (13) \nwhere f BH is the number of BHs per unit mass in a GC \nf BH = ∫ m max m crit f IMF ( m ) dm ∫ m max m min mf IMF ( m ) dm (14) \nand f IMF ( m ) is the stellar initial mass function [38] \nf IMF ( m ) = k { ( m/ 0 . 5 M glyph[circledot] ) -1 . 3 m min ≤ m ≤ 0 . 5 M glyph[circledot] , ( m/ 0 . 5 M glyph[circledot] ) -2 . 3 0 . 5 M glyph[circledot] ≤ m ≤ m max , (15) \nwhere m min = 0 . 08 M glyph[circledot] , m max = 150M glyph[circledot] , and m crit = 20 M glyph[circledot] is the critical mass above which BHs form. In \nreality there may be a transitional mass range allowing either NSs or BHs to form below 20 M glyph[circledot] [43] and the effective m crit may also depend on metallicity [44] or the host cluster [45]. A lower effective value, e.g. m crit = 17 M glyph[circledot] , would produce 25% more BHs and imply a higher merger rate, and vice versa for a higher m crit . Further, the details of the BH ejections due to kick-velocities at birth [46], and the relative retention fraction, may affect the compactness of the host GC [47]. These factors, as well as primordial binaries, may have some effect on the results presented in this paper and deserve further study. We convert time from the initial redshift z in = 3 to z to use in Eqs. (12) and (13) using the cosmological relation [48]. \nDynamically formed BH binaries merge both within the host GC and far outside of it after ejection. Recently it was pointed out that accounting for GW losses during the dynamical evolution of GCs increases the fraction of mergers within the host GC significantly to 50% [23, 24]. This affects the merger rate given by equation (12). For mergers that happen inside the GCs at redshift z merger , only the existing GCs ( N GC ( z merger )) at that redshift contribute to the rate. However, for BHs that merge after their ejection from the host cluster (Eq. 9), all clusters ( N GC ( z ejection )) must be included at the point of ejection, z ejection . The latter assumption implies a higher N GC , thus producing a higher rate of mergers. For a \nFigure 2 presents the comoving BH merger rate density as a function of redshift for different intervals of initial GC masses (left panel) and position in the galaxy (right panel). Although the most abundant, low-mass clusters (10 4 M glyph[circledot] < M GC < 10 5 M glyph[circledot] ) contribute to a negligible fraction to the total rate as they are inefficient at merging (see Eqs. 10-11) and because they dissolve. The largest contribution near the formation epoch of GCs comes from the most massive population (10 6 M glyph[circledot] < M GC < 10 7 M glyph[circledot] ), whose contribution is ∼ 6-7 times that of 10 5 M glyph[circledot] < M GC < 10 6 M glyph[circledot] . In terms of the radial distribution of mergers (right panel), the rate is dominated by clusters in the inner galaxy ( r < 5 kpc), apart from small redshifts ( z glyph[lessorsimilar] 0 . 3). Roughly 25% of the rate comes from GCs in the outer halo ( r > 10 kpc). \n<!-- image --> \nFIG. 2: The comoving BH merger rate density as a function of redshift z subdivided in the different contributions by GCs of different initial masses (left) and different initial position in the host galaxy (right). \n<!-- image --> \nrobust estimate, we calculate the BH merger rate as a function of redshift from the probability distribution of mergers (Eq. 12) in two limiting cases: either extending the sum over all GCs 6 or up to N GC ( z ), respectively. The true merger rate must be between these bounds. \nApart from the rate density, it is useful to calculate the total merger rate in the Universe within redshift z : \nC ( z ) = ∫ z 0 dV c dz ' R ( z ' ) 1 + z ' dz ' , (16) \nwhere dV c /dz ' is the comoving volume at redshift z ' and the 1 / (1 + z ' ) factor accounts for the observed redshifted time at Earth compared to the source's comoving time. \nResults.Figure 1 shows the BH merger rate from the evolving population of GCs in the Universe as a function of redshift. The left panel shows the comoving BH merger rate density R . The red dashed and black solid lines provide the lower and upper limits assuming mergers inside surviving clusters at redshift z and ejected mergers from all clusters as specified above. The black line represents the case where most ejections leading to mergers happen early, well before possible GC disruptions. The blue curve shows the previous result [25] for isolated GCs for comparison. The difference between our upper and lower bounds are relatively small (factor glyph[lessorsimilar] 2 at z ∼ 0 and smaller at higher z ), since the rate is dominated by the initially more massive clusters that survive until today. The merger rate is a factor of ∼ 2 larger for the evolving GC population than for the case of isolated clusters computed in Ref. [25] at z glyph[lessorsimilar] 1. This increased merger rate is a result of the originally more massive clusters in comparison to isolated clusters due to tidal stripping (see Eqs. 10-11). The right panel of Fig. 1 shows \nC ( z ), the total rate of mergers within redshift z . The shaded regions represent model uncertainty by assuming ρ GC in the range 0 . 32-2 . 31 Mpc -3 [14, 25, 41]. \nDiscussionIn this Letter , we have determined the BH merger rate from dynamically formed binaries produced in GCs that coevolve with their host galaxies in the Universe. At redshift z = 0, we have found a rate ∼ 460 Gpc -3 yr -1 , within the uncertainties of our model. We found that the expected merger rate ranges between R ∼ 18 Gpc -3 yr -1 to ∼ 35 Gpc -3 yr -1 for redshift between z = 0 . 5 to 2, and the total rate is 1, 5, 24 events per day within z = 0 . 5, 1, and 2, respectively. This corresponds to a factor ∼ 3 to a ∼ 2 higher rate from z = 0 . 5 to z = 2 with respect to the case neglecting the evolution of GCs in their host galaxies. If a significant fraction of mergers from GCs is from ejected binaries, the rate at low redshift z < 0 . 1 is ∼ 10 Gpc -3 yr -1 , a factor of ∼ 2 higher than previous estimates. For comparison, the current observational limit on the BH merger rate in the local Universe is R = 12-65Gpc -3 yr -1 , as- \nniform mass distribution, and R = 40240 Gpc -3 yr -1 for a power-law BH mass distribution with dN/dm ∝ m -2 . 35 . \nThis result highlights the need for more detailed simulations of GCs tracking their evolution with their host galaxies. Our results were derived by scaling to MilkyWay type hosts. Future work is needed to study the discrepancy between the rates of evolving GC populations and isolated GCs for different host galaxies, whose GC population correlates with the dark matter masses [49]. Furthermore, the possible presence of an intermediatemass black hole (IMBH) may significantly change the evolution of GCs and the distribution of merger rates [50, 51]. \nAt design sensitivity, LIGO-Virgo is expected to observe BH mergers up to z ∼ 1 [52]. Assuming that the LIGO-Virgo observational completeness is 1-10% (i.e. the fraction of all mergers that LIGO-Virgo detects within this volume), our results suggest that one detection per day to ∼ 1 per week is expected from the dynamical channel. Recently, Fishbach et al. [53] suggested that with ∼ 100 -300 LIGO-Virgo detections it will be possible to distinguish among different models of the merger rate evolution within the coming 2-5 years. Indeed, the redshift evolution of merger rate from the dynamical channel shown in Figure 1 is distinct from other formation channels as it increases from z = 0 until the epoch of globular cluster formation with a particular shape as shown (see [3, 53] and references therein, for the redshift evolution for other channels). The cosmic evolution and disruption of GCs increases the present-day merger rates by a factor ∼ 2 in comparison to isolated clusters (see Fig. 1). The redshift evolution of the merger rates carries information on the cosmic history of GCs. Thus, measuring the redshift evolution of the rates will represent an observation probe of GC formation, their initial numbers in the Universe and their evolution across cosmic time. With a sufficiently large sample of mergers the relative contribution of intergalactic GCs [54, 55] may be distinguished from GCs evolving in galaxies. Future instruments such as the Voyager, Einstein Telescope, or Cosmic Explorer will make this endeavor more feasible [52, 56]. \nWe conclude that GW detectors have the potential to provide a view on the evolution of faint GCs which are practically invisible to electromagnetic observatories. 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1995CQGra..12.1699C
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The off-shell black hole
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1995-01-01
| 15 | 0.45 | 161 |
['-', '-']
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[]
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The standard (Euclidean) action principle for the gravitational field implies that for spacetimes with black hole topology, the opening angle at the horizon and the horizon area are canonical conjugates. It is shown that the opening angle bears the same relation to the horizon area that the time separation bears to the mass at infinity. The dependence of the wave function on this new degree of freedom is governed by an extended Wheeler-DeWitt equation. Summing over all horizon areas yields the black hole entropy.
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https://arxiv.org/pdf/gr-qc/9312002.pdf
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{'The Off-Shell Black Hole': 'S teven C arlip ∗ Department of Physics University of California Davis, CA 95616 USA \nand \nC laudio T eitelboim † Centro de Estudios Cientificos de Santiago Casilla 16443, Santiago 9 Chile and Institute for Advanced Study Olden Lane, Princeton, NJ 08540 USA', 'Abstract': "The standard (Euclidean) action principle for the gravitational field implies that for spacetimes with black hole topology, the opening angle at the horizon and the horizon area are canonical conjugates. It is shown that the opening angle bears the same relation to the horizon area that the time separation bears to the mass at infinity. The dependence of the wave function on this new degree of freedom is governed by an extended Wheeler-DeWitt equation. Summing over all horizon areas yields the black hole entropy. \nIASSNS-HEP-93/84 UCD-93-34 gr-qc/9312002 November 1993 \nIt has long been known that when space is not closed, the wave functional of the gravitational field possesses an extra argument in addition to the intrinsic geometry of a spatial section. This additional argument is the time separation at spatial infinity, which is conjugate to the total mass, given by a surface integral. Thus the wave functional ψ = ψ [ T, (3) g ] obeys a Schrodinger equation \n¯ h i ∂ψ ∂T + Mψ = 0 (1) \nH µ ψ = 0 . (2) \nHere M is the ADM mass, a surface integral over the boundary of space at infinity (a two-dimensional surface in four spacetime dimensions). Equations (1) and (2) are not independent, and are more appropriately written in the form [1] \n¯ h i δψ + [∫ δtN µ H µ d 3 x + δT · M ] ψ = 0 (3) \nfor any N µ = ( N,N i ) obeying the asymptotic conditions \nN ( t 2 -t 1 ) → T, N i → 0 (4) \nat r →∞ . (See [2] for a precise statement of these conditions.) If spacetime has the topology IR 4 , equation (3) extends the Wheeler-DeWitt equation. \nThe purpose of this note is to point out that for black hole topologies, the wave function ψ acquires an extra argument Θ, which is a sort of 'dimensionless internal time' associated with the horizon. The dependence of ψ on Θ is governed by an equation analogous to (1), \n¯ h i ∂ψ ∂ Θ -Aψ = 0 , (5) \nwhere A is a surface integral at the horizon, the horizon area. Equation (3) is replaced by \n¯ h i δψ + [ -δ Θ · A + ∫ δξ µ H µ d 3 x + δT · M ] ψ = 0 . (6) \nThe 'off-shell' meaning of the term horizon and the analog of (4) for Θ will be given below. The transition amplitude now depends on an additional argument, \nK = K [ (3) G 2 , (3) G 1 ; A 2 , A 1 ; M 2 , M 1 ] = δ ( A 2 -A 1 ) δ ( M 2 -M 1 ) K [ (3) G 2 , (3) G 1 ; A 2 ; M 2 ] . (7) \nIf one decrees that 'the horizon area A is not observable,' one may introduce a reduced amplitude \nK [ (3) G 2 , (3) G 1 ; M 2 ] = ∫ dAK [ (3) G 2 , (3) G 1 ; A ; M ] , (8) \nwhich may be rewritten as \nK [ (3) G 2 , (3) G 1 ; M ] = ˜ K [ (3) G 2 , (3) G 1 ; Θ E ; M ] ∣ ∣ ∣ Θ E =2 π , (9) \nin addition to the constraints \nwhere ˜ K [ (3) G 2 , (3) G 1 ; Θ E ; M ] is a Laplace transform with respect to A of the amplitude appearing on the right-hand side of (7). (The shift by 2 π in the argument of the Laplace transform will be explained below.) The trace of (9) in (3) G 2 and (3) G 1 yields the exponential of the black hole entropy. \nThe analysis leading to the preceding statements goes as follows. ∗ \nWe start from the Euclidean point of view. The spacetimes admitted in the action principle will have the topology IR 2 × S d -2 ('one black hole sector'). We introduce a system of polar coordinates r , τ in IR 2 , with an arbitrary origin r = r + for the radial coordinate. In the semiclassical approximation, it is useful to take the origin to be the horizon of the black hole and the angle τ to be the Killing time. By abuse of language, we will call r + the horizon even away from the extremum ('off shell'). \nAlthough it is unnecessarily complicated, we will conform to standard practice and use 'Schwarzschild coordinates' near r + . That is, we write the generic Euclidean metric as \nds 2 E = N 2 ( r ) dτ 2 + N -2 ( r ) dr 2 + γ mn ( r, x p ) dx m dx n (10) \nup to terms of order O ( r -r + ), with \n( τ 2 -τ 1 ) N 2 = 2Θ E ( r -r + ) + O ( r -r + ) 2 . (11) \nHere the x m are coordinates on the two-sphere S 2 . The parameter Θ E is the total proper angle (proper length divided by proper radius) of an arc of very small radius and coordinate angular opening τ 2 -τ 1 . For this reason it will be called the 'opening angle.' If one identifies the surfaces τ = τ 1 and τ = τ 2 , thus considering a disk in IR 2 , then the deficit angle 2 π -Θ E is the strength of a conical singularity in IR 2 at r + . For the moment, we assume for simplicity that Θ E is independent of x m ; we shall see below that this restriction may be lifted without changing the conclusions. It is important to emphasize that no a priori relation between Θ E and N ( ) is assumed. \nEquation (11) is the Euclidean analog of (4) for Θ. Note that if we continue the metric (10) to Minkowskian signature by setting τ = it , we find \n∞ \nds 2 = -N 2 dt 2 + N -2 dr 2 + γ mn ( r, x p ) dx m dx n (12) \nwith \nWe therefore learn that the opening angle must also be rotated, \ni ( t 2 -t 1 ) N 2 = 2Θ E ( r -r + ) + O ( r -r + ) 2 . (13) \nΘ E = i Θ . (14) \nNow, it was shown in reference [3] by a geometrical argument based on the Gauss-Bonnet theorem that the covariant Hilbert action \nI H = 1 2 ∫ M √ gRd 4 x -∫ ∂M √ hK d 3 x (15) \nfor IR 2 × S d -2 and the canonical action \nI can = ∫ M ( π ij ∂g ij ∂τ -N µ H µ ) d 4 x (16) \nthat uses the polar angle in IR 2 as time are related by \nI H = 2 πA + I can + B ∞ , (17) \nwhere B ∞ is a local boundary term at spatial infinity. The action (17) is suitable for fixing the intrinsic geometry of the boundary at infinity, for either the full manifold IR 2 × S d -2 or the wedge τ 1 ≤ τ ≤ τ 2 . If we instead wish to implement black hole boundary conditions, fixing N ( ∞ ) with a prescribed rate of fall-off for N -N ( ∞ ) and holding the angular momentum and gauge charges fixed [2], then the term B ∞ (which diverges) must be supplemented by a further surface term, leading to an action \nI = 2 πA + I can -T E M, (18) \nwhere T E is the Euclidean time separation at infinity. The fields held fixed in the action (18) are the following: (i) the three-geometries of the hypersurfaces at τ = τ 1 and τ = τ 2 for r > r + ; (ii) the two-geometry at the horizon; and (iii) the asymptotic time separation T E . \nThe variation of (18) is given by \nδI = (2 π -Θ E ) δA -δT E M + ∫ π ij δg ij d 3 x ∣ ∣ ∣ 2 1 + terms vanishing on shell . (19) \n∣ \nThe contribution -Θ E δA to the variation comes technically from an integration by parts in the variation of H ⊥ in I can . A more geometric way to understand this term is to recall that the action (18) differs from the scalar curvature by a boundary term at infinity, so the contribution to the action of a neighborhood of the horizon is just the integrated scalar curvature. This immediately brings in the deficit angle 2 π -Θ E , a measure of the curvature in IR 2 per unit area in S 2 . \nIt follows from (19) that in addition to the ordinary degrees of freedom (i.e., those present when the topology is IR 4 ) there is one additional dynamical variable, Θ E , conjugate to the area A . The action takes a more symmetric form if one passes to a representation suitable for fixing Θ E instead of A . This can be achieved by subtracting (2 π -Θ E ) A from (18), yielding \nI ' = Θ E A + I can -T E M. (20) \nIn this form the symmetry between Θ and T is manifest, and equations (5) and (6) are established. (Recall that Θ E = i Θ and T E = iT .) \nOne may further allow for a dependence of Θ E on the coordinates x m of the two-sphere at r + . The first term on the right-hand side of (20) must then be replaced by \n∫ Θ E ( x ) γ 1 / 2 ( x ) d 2 x. \nThus Θ E ( x ) is canonically conjugate to the local measure of area on the horizon, γ 1 / 2 ( x ), and equation (5) may be obtained as the integral over the horizon of the 'many-time equation' \n¯ h i δψ δ Θ( x ) -γ 1 / 2 ( x ) ψ = 0 . (21) \nThe reduced amplitude is obtained by summing over all γ 1 / 2 ( x ). This amounts to setting Θ E ( x ) = 2 π for all x , so (9) remains valid. \nIn the semiclassical approximation, the trace of (9) is given by the exponential of the action (20) evaluated on the black hole solution. If one takes τ to be the Killing time, then I can has the value zero, and one finds the standard result \nS = (8 πG ¯ h ) -1 2 πA horizon (22) \nfor the entropy. We have restored the universal constants, in order to make explicit the geometrical nature of the factor multiplying the area in (22) as the opening angle. In this sense, the most natural choice of units is G = (8 π ) -1 rather than G = 1. \nThe above analysis shows that one may regard the black hole entropy as arising from summing over all horizon geometries. We still lack a 'microscopic' explanation for the exponential weight in the integration measure for the surface degrees of freedom, or equivalently for the ¯ h -1 dependence in (22). Note, however, that the factor multiplying the area in (22) comes from the action of a small disk in IR 2 . This would suggest that the entropy per unit area may arise from counting the geometries in such a 'thickened horizon.'", 'Acknowledgements': "We would like to thank M'aximo Ba˜nados and Jorge Zanelli for enlightening discussions. This work was partially supported by grants 0862/91 and 193.1910/93 from FONDECYT (Chile), by institutional support to the Centro de Estudios Cientificos de Santiago provided by SAREC (Sweden) and a group of Chilean private companies (COPEC, CMPC, ENERSIS, CGEI). This research was also sponsored by CAP, IBM and Xerox de Chile. S.C. was supported in part by U.S. Department of Energy grant DE-FG03-91ER40674 and National Science Foundation National Young Investigator award PHY-93-57203.", 'References': "- [1] See, for example, C. Teitelboim, Phys. Rev. D28 (1983) 310.\n- [2] T. Regge and C. Teitelboim, Ann. Phys. 88 (1974) 286.\n- [3] M. Ba˜nados, C. Teitelboim, and J. Zanelli, 'Black Hole Entropy and the Dimensional Continuation of the Gauss-Bonnet Theorem,' submitted to Phys. Rev. Lett. .\n- [4] S. Carlip and C. Teitelboim, 'The Quantum Mechanics and Thermodynamics of the (2+1)-Dimensional Black Hole,' in preparation.\n- [5] See, for example, L. Susskind, 'Some Speculations about Black Hole Entropy in String Theory,' Rutgers preprint RU-93-44."}
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2009JHEP...07..011O
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Three-dimensional black holes, gravitational solitons, kinks and wormholes for BHT massive gravity
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2009-01-01
| 11 | 0.45 | 161 |
['-', '-']
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The theory of massive gravity in three dimensions recently proposed by Bergshoeff, Hohm and Townsend (BHT) is considered. At the special case when the theory admits a unique maximally symmetric solution, a conformally flat solution that contains black holes and gravitational solitons for any value of the cosmological constant is found. For negative cosmological constant, the black hole is characterized in terms of the mass and the ``gravitational hair'' parameter, providing a lower bound for the mass. For negative mass parameter, the black hole acquires an inner horizon, and the entropy vanishes at the extremal case. Gravitational solitons and kinks, being regular everywhere, can be obtained from a double Wick rotation of the black hole. A wormhole solution in vacuum that interpolates between two static universes of negative spatial curvature is obtained as a limiting case of the gravitational soliton with a suitable identification. The black hole and the gravitational soliton fit within a set of relaxed asymptotically AdS conditions as compared with the one of Brown and Henneaux. In the case of positive cosmological constant the black hole possesses an event and a cosmological horizon, whose mass is bounded from above. Remarkably, the temperatures of the event and the cosmological horizons coincide, and at the extremal case one obtains the analogue of the Nariai solution, dS<SUB>2</SUB> × S<SUP>1</SUP>. A gravitational soliton is also obtained through a double Wick rotation of the black hole. The Euclidean continuation of these solutions describes instantons with vanishing Euclidean action. For vanishing cosmological constant the black hole and the gravitational soliton are asymptotically locally flat spacetimes. The rotating solutions can be obtained by boosting the previous ones in the t-phi plane.
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https://arxiv.org/pdf/0905.1545.pdf
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{'Three-dimensional black holes, gravitational solitons, kinks and wormholes for BHT massive gravity': "Julio Oliva 1 , 2 , David Tempo 1 , 3 , 4 and Ricardo Troncoso 1 , 5 \n1 Centro de Estudios Cient'ıficos (CECS), Casilla 1469, Valdivia, Chile. \n2 Instituto de F'ısica, Universidad Austral de Chile, Casilla 567, Valdivia, Chile. \n3 Departamento de F'ısica, Universidad de Concepci'on, Casilla 160-C, Concepci'on, Chile. \n4 Physique th'eorique et math'ematique, Universit'e Libre de Bruxelles, \nULB Campus Plaine C.P.231, B-1050 Bruxelles, Belgium, \nand \n5 Centro de Ingenier'ıa de la Innovaci'on del CECS (CIN), Valdivia, Chile.", 'Abstract': "The theory of massive gravity in three dimensions recently proposed by Bergshoeff, Hohm and Townsend (BHT) is considered. At the special case when the theory admits a unique maximally symmetric solution, a conformally flat solution that contains black holes and gravitational solitons for any value of the cosmological constant is found. For negative cosmological constant, the black hole is characterized in terms of the mass and the 'gravitational hair' parameter, providing a lower bound for the mass. For negative mass parameter, the black hole acquires an inner horizon, and the entropy vanishes at the extremal case. Gravitational solitons and kinks, being regular everywhere, can be obtained from a double Wick rotation of the black hole. A wormhole solution in vacuum that interpolates between two static universes of negative spatial curvature is obtained as a limiting case of the gravitational soliton with a suitable identification. The black hole and the gravitational soliton fit within a set of relaxed asymptotically AdS conditions as compared with the one of Brown and Henneaux. In the case of positive cosmological constant the black hole possesses an event and a cosmological horizon, whose mass is bounded from above. Remarkably, the temperatures of the event and the cosmological horizons coincide, and at the extremal case one obtains the analogue of the Nariai solution, dS 2 × S 1 . A gravitational soliton is also obtained through a double Wick rotation of the black hole. The Euclidean continuation of these solutions describes instantons with vanishing Euclidean action. For vanishing cosmological constant the black hole and the gravitational soliton are asymptotically locally flat spacetimes. The rotating solutions can be obtained by boosting the previous ones in the t -φ plane.", 'Contents': '| I. Introduction | 2 |\n|-----------------------------------------------------------------|-----|\n| II. BHT massive gravity at the special case m 2 = λ | 7 |\n| III. Negative cosmological constant | 8 |\n| A. Black hole | 8 |\n| B. Asymptotically AdS boundary conditions with relaxed behavior | 11 |\n| C. Conserved charges as surface integrals at infinity | 12 |\n| D. Black hole thermodynamics | 14 |\n| 1. Black hole entropy | 14 |\n| E. Gravitational solitons, kinks and wormholes | 15 |\n| 1. Wormhole | 16 |\n| 2. Gravitational soliton | 17 |\n| 3. Gravitational kink | 17 |\n| IV. Positive cosmological constant | 18 |\n| A. Black hole | 18 |\n| B. Gravitational soliton | 20 |\n| C. Euclidean action | 21 |\n| V. Vanishing cosmological constant | 21 |\n| VI. Rotating solutions | 23 |\n| VII. Discussion and final remarks | 24 |\n| References | 26 |', 'I. INTRODUCTION': "As shown by Brown and Henneaux, General Relativity with negative cosmological constant in three dimensions appeared as the first example of a field theory admitting a classical \ncentral charge given by [1] \nc = 3 l 2 G , (1) \nwhere l is the AdS radius, and G is the Newton constant. This is possible due to the enhancement of the asymptotic symmetries from SO (2 , 2) to the infinite dimensional conformal group in two dimensions. Remarkably, the AdS/CFT correspondence [2] was foreseen during the 80's within this context. \nThe asymptotic behavior of the metric is given by [1] \n∆ g rr = f rr r -4 + O ( r -5 ) , ∆ g rm = f rm r -3 + O ( r -4 ) , ∆ g mn = f mn + O ( r -1 ) . (2) \nHere f µν = f µν ( t, φ ), and the indices have been split as µ = ( r, m ), where m includes the time and the angle. The asymptotic metric is written as g µν = ¯ g µν + ∆ g µν , where ∆ g µν corresponds to the deviation from the AdS metric, \nd ¯ s 2 = -(1 + r 2 /l 2 ) dt 2 +(1 + r 2 /l 2 ) -1 dr 2 + r 2 dφ 2 . (3) \nThe asymptotic conditions (2) map into themselves under diffeomorphisms of the form \nη + = T + + l 2 2 r 2 ∂ 2 -T -+ · · · η -= T -+ l 2 2 r 2 ∂ 2 + T + + · · · (4) η r = -r 2 ( ∂ + T + + ∂ -T -) + · · · , \nwhere T ± = T ± ( x ± ), with x ± = t l ± φ , and the dots stand for lower order terms that do not contribute to the surface integrals. Thus, the boundary conditions (2) are invariant under two copies of the Virasoro group, generated by T + ( x + ) and T -( x -). The Poisson brackets of the canonical generators, defined by surface integrals at infinity that depend on the metric and its derivatives, reproduces then two copies of the Virasoro algebra with central charge given by (1). \nIt is known that there are instances where the asymptotic behavior (2) for pure gravity with localized matter fields can be relaxed in order to accommodate solutions of physical interest, without spoiling the asymptotic symmetries (4). \nThis occurs for General Relativity with negative cosmological constant coupled to scalar fields of mass within the range \nm 2 ∗ ≤ m 2 < m 2 ∗ + 1 l 2 , (5) \nwhere \nm 2 ∗ = -( d -1) 2 4 l 2 , (6) \ndefines the Breitenlohner-Freedman bound [3]. In this case, the scalar field possesses a very slow fall-off at infinity that generates a strong back reaction in the metric, so that the standard AdS asymptotic conditions of [1, 4, 5] have to be relaxed. As a consequence, the charges do not only depend on the metric and its derivatives, but acquire an explicit contribution from the matter field. The role of this additional contribution is to cancel the divergences coming from the purely gravitational contribution in order to render the surface integrals defining the charges to be finite. This was investigated at length in [6, 7, 8] for any dimension (see also [9, 10]). \nAs a consequence of the softening of the boundary conditions, the space of admissible solutions is enlarged so as to include hairy black holes [6, 9, 11] 1 , solitons and instantons [14]. \nOne could envisage that a generic effect of relaxing the asymptotic conditions in gravitation is the allowance of hairy solutions, since it is known that this effect extends beyond General Relativity with scalar fields. Indeed, it has been recently shown in [21] that a similar phenomenon occurs for topologically massive gravity [15], where the action of Einstein gravity with negative cosmological constant in three dimensions is supplemented by the Lorentz-Chern-Simons term. For the range 0 < | µl | ≤ 1, where µ is the topological mass parameter, it was found that topologically massive gravity admits a set of relaxed asymptotically AdS boundary conditions allowing the inclusion of the AdS waves solutions discussed in Refs. [16, 17, 18]. In the case of 0 < | µl | < 1, one can see that even though the asymptotic conditions are relaxed with respect to the standard ones (2) the charges acquire the same form as if one had considered the Brown-Henneaux boundary conditions. This \nis because the diverging pieces associated with the slower fall-off cancel out, so that the charges acquire no correction involving the terms associated with the relaxed behavior. As a consequence, the terms with slower fall-off, which cannot be gauged away, can be seen as defining a kind of 'hair' 2 . \nIt is natural then wondering whether these effects could also appear for the theory of massive gravity that has been recently proposed by Bergshoeff, Hohm and Townsend (BHT) [24]. The action for the BHT massive gravity theory is given by \nI BHT = 1 16 πG ∫ d 3 x √ -g [ R -2 λ -1 m 2 K ] , (7) \nwhere K stands for a precise combination of parity-invariant quadratic terms in the curvature: \nK := R µν R µν -3 8 R 2 . (8) \nThe field equations are then of fourth order and read \nG µν + λg µν -1 2 m 2 K µν = 0 , (9) \nwhere \nK µν := 2 ∇ 2 R µν -1 2 ( ∇ µ ∇ ν R + g µν ∇ 2 R ) -8 R µρ R ρ ν + 9 2 RR µν + g µν [ 3 R αβ R αβ -13 8 R 2 ] , (10) \nfulfills 3 K = g µν K µν . Remarkably, the BHT massive gravity theory was shown to be equivalent at the linearized level to the (unitary) Fierz-Pauli action for a massive spin-2 field [24]. The unitarity of the BHT theory has been revisited in [25]. Exact solutions have also been found, including warped AdS black holes [26] and AdS waves [27, 28]. Further aspects of the BHT theory have been explored in [29, 30]. \nAs pointed out in [24], generically the theory admits solutions of constant curvature ( R µν αβ = Λ δ µν αβ ) with two different radii, determined by \nΛ ± = 2 m ( m ± √ m 2 -λ ) . (11) \nThis means that at the special case defined by \nm 2 = λ , (12) \nfor which Λ + = Λ -, the theory possesses a unique maximally symmetric solution of fixed curvature given by \nΛ = 2 λ = 2 m 2 . (13) \nIn this sense, the behavior of the BHT theory is reminiscent to the one of the EinsteinGauss-Bonnet (EGB) theory, which could be regarded as a higher-dimensional cousin of the same degree (but of lower order). This can be seen as follows: \nIn d > 4 dimensions, the action for the EGB theory reads \nI EGB = κ ∫ d d x √ -g R -2 λ + α ( R αβγδ R αβγδ -4 R µν R µν + R 2 , (14) \n( \n( where the quadratic terms appear in a precise combination so that the field equations are of second order [31]. Generically, the EGB theory admits two solutions of constant curvature R µν αβ = Λ δ µν αβ , whose radii are fixed according to \n)) \nΛ ± = 1 2˜ α [ 1 ± √ 1 + 4˜ α ˜ λ ] , (15) \nwith \n˜ α := ( d -3)( d -4) α ; ˜ λ := 2 λ ( d -1)( d -2) . (16) \nHence, at the special case for which Λ + = Λ -, given by \n1 + 4˜ α ˜ λ = 0 , (17) \nthe theory possesses a unique maximally symmetric solution [33]. \nThe static and spherically symmetric solution was found by Boulware and Deser [34], and it is given by \nds 2 = -f 2 ± ( r ) dt 2 + dr 2 f 2 ± ( r ) + r 2 dφ 2 , (18) \nf 2 ± ( r ) = 1 + r 2 2˜ α [ 1 ± √ 1 + 4˜ α ˜ Λ + µ r d -1 ] . (19) \nwith \nThus, for a generic choice of the Gauss-Bonnet coupling α , the solution possesses two branches, each of them approaching to the maximally symmetric solution at infinity according to \nf 2 ± ( r ) = Λ ± r 2 -µ ± r d -3 + · · · , (20) \nNote that for the generic case, the asymptotic behavior has the same fall-off as the one for the Schwarzschild-(A)dS solution of General Relativity (GR) in d dimensions. Nevertheless, for the special case (17), the solution has the following fall-off \nf 2 ( r ) = Λ r 2 -µ r d -5 2 + · · · (21) \nwhich is slower than the one for the Schwarzschild-(A)dS solution ( O ( r 3 -d )). One may then fear that the surface integrals at infinity defining the conserved charges blow up. However, as shown in [33], the conserved charges have to be computed from scratch and they turn out to be finite. It was also shown that the black hole fits within a relaxed set of asymptotic conditions possessing the same asymptotic symmetries as for GR (see also [35]). \nAs it occurs for their three-dimensional counterparts, the consequence of relaxing the asymptotic conditions for the EGB theory is to enlarge the space of admissible solutions so as to include wormholes in vacuum [36, 37, 38, 39], gravitational solitons [40] and rotating spacetimes [41]. \nThus, one may naturally expect that a similar behavior occurs for the BHT massive gravity theory. The purpose of this paper is to show that this is indeed the case.", 'II. BHT MASSIVE GRAVITY AT THE SPECIAL CASE m 2 = λ': "The field equations of the BHT massive gravity theory (9), at the special case m 2 = λ , admit the following exact Euclidean solution: \nds 2 = ( -Λ r 2 + br -µ ) dψ 2 + dr 2 -Λ r 2 + br -µ + r 2 dϕ 2 , (22) \nwhere b and µ are integration constants, and Λ := 2 λ . When the constant b is switched on, as it is apparent from (22), as r → ∞ , the Riemann curvature approaches to a constant ( R µν αβ → Λ δ µν αβ ). The Ricci scalar of this metric is given by \nR = 6Λ -2 b r . (23) \nAs it was shown in [42], the metric (22) is conformally flat, and hence also corresponds to a solution of conformal gravity in three dimensions, as well as for the BHT theory supplemented by the gravitational Lorentz-Chern-Simons form. \nAs it is shown below, this metric describes instantons for a suitable range of the coordinates and of the parameters b and µ . Furthermore, it is possible to perform different Wick rotations so that the corresponding metric of Lorentzian signature describes either asymptotically (A)dS or asymptotically locally flat black holes, as well as gravitational solitons and kinks. Further interesting solutions, including wormholes in vacuum can also be obtained as limiting cases of the black holes and the gravitational solitons. The corresponding rotating solutions can then be obtained by boosting the previous ones in the ' t -φ ' plane. \nThe different cases are examined according to the sign of the cosmological constant.", 'A. Black hole': 'For positive cosmological constant, Λ := 1 l 2 , a solution of Lorentzian signature of the field equations (9), at the special case m 2 = λ , is obtained from the Wick rotation of (22), through ψ → it and ϕ = φ . The metric reads \nds 2 = -( -r 2 l 2 + br -µ ) dt 2 + dr 2 -r 2 l 2 + br -µ + r 2 dφ 2 , (57) \nwhere -∞ < t < + ∞ , 0 ≤ φ < 2 π , and it describes asymptotically dS black holes provided the lapse function g tt admits two positive real roots. In terms of the corresponding roots, r ++ > r + , the metric reads \nds 2 = -1 l 2 ( r -r + )( r ++ -r ) dt 2 + l 2 dr 2 ( r -r + )( r ++ -r ) + r 2 dφ 2 , (58) \nwhere the gravitational hair and mass parameters are given by \nb = 1 l 2 ( r + + r ++ ) > 0 , (59) \nµ = r + r ++ l 2 > 0 . (60) \nNote that dS spacetime is recovered for b = 0, and µ = -1. \nThe black hole (57) exists for the range \n0 < µ ≤ 1 4 b 2 l 2 , (61) \nand possesses a spacelike singularity at the origin ( r = 0) that is enclosed by the event horizon located at r = r + , which is surrounded by the cosmological horizon at r = r ++ . In the case of µ = 0, there is a NUT at the origin, on top of the singularity which becomes null. The upper bound on the mass parameter is saturated at the extremal case, for which both horizons coincide ( r + = r ++ = bl 2 / 2). The causal structure is depicted in Fig. 5. \n<!-- image --> \nFIG. 5: The causal structure of the black holes with positive cosmological constant \n<!-- image --> \nNote that the black hole exists due to the presence of the integration constant b . For a fixed mass parameter within the range (61), if the parameter b decreases, the event horizon radius increases, while the cosmological horizon shrinks. \nRemarkably, since the Hawking temperatures of the event and of the cosmological horizon coincide, i.e., \nT + = T ++ = r ++ -r + 4 πl 2 , (62) \nthe solution can be regarded as a pair of black holes on dS. As a consequence, the Euclidean continuation of the black hole describes a regular instanton whose metric can be written as \nds 2 = l 2 [ sin 2 θdτ 2 + dθ 2 ] + 1 4 (( r ++ + r + ) + ( r + -r ++ ) cos θ ) 2 dφ 2 , (63) \nwhere 0 ≤ τ < 2 π , and 0 ≤ θ < π . The instanton (63) is homeomorphic to S 2 × S 1 and it is obtained from (22) fixing the Euclidean time period according to β = T -1 + = T -1 ++ , and performing the following change of coordinates \nr = 1 2 ( r ++ + r + ) + 1 2 ( r + -r ++ ) cos θ , (64) \nwhich covers the region between the event horizon (located at the north pole, θ = 0) and the cosmological horizon (located at θ = π ). \nThe temperature vanishes for the extremal case ( r + = r ++ ), for which the Euclidean metric (63) reduces exactly to S 2 × S 1 , where the two-sphere is of radius l . Hence, this product space, as well as its Lorentzian continuation, dS 2 × S 1 , solve the field equations (9) for the special case (12). This means that the near horizon geometry of the extremal black hole is given by the three-dimensional analogue of the Nariai solution, where the parameter b , in Eq. (59), can be identified with the period of the circle S 1 . \nNote that a three-dimensional Einstein Universe, R × S 2 , with a two-sphere of radius l is obtained from a double Wick rotation of dS 2 × S 1 once the circle S 1 is unwrapped.', 'B. Asymptotically AdS boundary conditions with relaxed behavior': 'The suitable set of asymptotically AdS conditions that contains the black hole solution (24) possesses a relaxed behavior as compared with the one of Brown and Henneaux, given by (2). The deviation with respect to the AdS metric (3) in our case is given by \n∆ g rr = h rr r -3 + f rr r -4 + ... , ∆ g rm = h rm r -2 + f rm r -3 + ... , (30) ∆ g mn = h mn r + f mn + ... , \nwhere f µν and h µν depend only on the time and the angle, but not on r . Here the f -terms correspond to the deviation from the AdS metric proposed by Brown and Henneaux (for General Relativity), while the h -terms take into account the relaxation of the standard boundary conditions that is required in order to include the black hole solution (24) with slower fall-off. \nThe asymptotic symmetry group associated to (30) contains the standard two copies of the Virasoro group, generated by the diffeomorphisms η in Eq. (4), and it is augmented to a semi-direct product by the additional asymptotic symmetries generated by \nζ = Y ( x + , x -) ∂ r . (31) \nAs it is shown in the next subsection, conserved charges as surface integrals at infinity exist and they turn out to be finite for the relaxed asymptotic conditions proposed in Eq. (30). The corresponding central charge is then found to be twice the value found by Brown and Henneaux for GR, i.e., \nc = 3 l G . (32)', 'C. Conserved charges as surface integrals at infinity': "Despite the fact that the asymptotic conditions (30) are relaxed by additional terms that grow instead of decaying as one approaches to infinity, one can see that they are mild enough in the sense that finite charges as surface integrals can be consistently constructed through standard perturbative methods. Here we follow the approach of Abbot and Deser [45] to construct conserved charges for asymptotically AdS spacetimes, which has been extended to the case of gravity theories with quadratic terms in the curvature by Deser and Tekin [46]. The conserved charges for the BHT theory can be written as \nQ DT ( ξ ) = ( 1 + Λ 2 m 2 ) Q AD ( ξ ) -Λ m 2 Q K ( ξ ) , (33) \nSo that in the limit m 2 → ∞ one recovers the standard expression for GR. Note that the quadratic terms in the action (7) change the factor in front of the Abbott-Deser charges, given by Q AD ( ξ ), and contribute with an additional piece given by Q K ( ξ ). The precise definition of the surface integrals, Q AD ( ξ ) and Q K ( ξ ), can be extracted from Refs. [46], [47]. \nEvaluating the surface integrals appearing in the Deser-Tekin charges (33) on the asymptotic conditions (30), for the asymptotic symmetries generated by ξ = η + ζ , where η and ζ are given by Eqs. (4) and (31), respectively, one obtains \nQ AD ( ξ ) = -1 32 πGl 3 ∫ dφ { T + (4 l 2 ( f + --f ++ ) -f rr ) + T -(4 l 2 ( f + --f --) -f rr ) -r [ ( T + + T -) ( h rr -2 l 2 h + -) +2 l 2 ( T + h ++ + T -h --)]} , (34) Q K ( ξ ) = -1 32 πGl 3 ∫ dφ { T + (4 l 2 f + --f rr ) + T -(4 l 2 f + --f rr ) -r ( T + + T -) ( h rr -2 l 2 h + -) +2 l 2 ( T + h ++ + T -h --)]} , (35) \n[ \nso that the full charge reads \nQ DT ( ξ ) = 1 32 πGl 3 ∫ dφ {( 1 -1 2 m 2 l 2 ) (4 l 2 ( T + f ++ + T -f --) + ( 1 + 1 2 m 2 l 2 ) ( T + + T -)( f rr -4 l 2 f + -) (36) + ( 1 + 1 2 m 2 l 2 ) r [ ( T + + T -) ( h rr -2 l 2 h + -) +2 l 2 ( T + h ++ + T -h --)] } . \nIn the generic case m 2 /negationslash = -1 2 l 2 , for the Brown-Henneaux boundary conditions, where the h -terms are absent, the result of Liu and Sun [47] is recovered. \nNote that the linearly divergent pieces coming from the standard and the purely quadratic pieces, in Eqs. (34) and (35), respectively, combine such that for the special case m 2 = -1 2 l 2 they cancel out. It is worth pointing out that the second line of (36), which is a term of order one, also vanishes in the special case. This goes by hand with the fact that the combination f rr -4 l 2 f + -is generically required to vanish by the field equations, except at the special case. This brings in the freedom to introduce the additional integration constant b in our black hole solution. Therefore, in the special case the charges are given by \nQ DT ( ξ ) = 1 4 πGl ∫ dφ T + f ++ + T -f --) . (37) \n( \n) Note that as this expression does not depend on the h -terms in the asymptotic conditions (30), which cannot be gauged away in general, they could be regarded as a kind of 'gravitational hair'. \nThe central charge can then be obtained from the variation of the charge (37) along an asymptotic symmetry, δ ξ 1 Q DT ( ξ 2 ), evaluated on the AdS background, and it is found to be \nc ± = c = 3 l G . (38) \nFor the special case, this result is in agreement with [47, 48] for BHT massive gravity with Brown-Henneaux boundary conditions. This is natural since according to the general theorems of Ref. [49], as the central charge depends on the parameters of the theory and on the chosen background, its value is not expected to change for a relaxed set of asymptotic conditions that includes the standard asymptotic symmetries. \nThe mass of the black hole (24), measured with respect to an AdS background is given by \nM = Q DT ( ∂ t ) = 1 + µ 4 G , (39) \nwhich does not depend on the integration constant b . Note that the mass of the BTZ black hole ( b = 0) for the BHT theory at the special case is twice the value obtained for GR.", 'D. Black hole thermodynamics': 'It is useful to express the metric of the Euclidean continuation of black hole (24) as \nds 2 = l 2 [ sinh 2 ρ dτ 2 + dρ 2 + 1 4 (( r + + r -) + ( r + -r -) cosh ρ ) 2 dφ 2 ] , (40) \nwhere \nr = l 2 [( r + -r -) cosh ρ + r + + r -] , (41) \nexcludes the region inside the event horizon, so that 0 ≤ ρ < ∞ , and 0 ≤ τ < β . The Hawking temperature is then given by the inverse of the Euclidean time period β , which is found requiring the Euclidean metric to be smooth at the origin \nT = 1 β = 1 4 πl b 2 l 2 +4 µ , (42) \n= r + -r -4 πl 2 . (43) \n√ \nThe temperature vanishes for the extremal case ( b < 0), for which the Euclidean metric (40) reduces to H 2 × S 1 , where H 2 is the two-dimensional hyperbolic space of radius l . Although the coordinate transformation (41) is ill-defined in this case, it is simple to show that H 2 × S 1 , as well as its Lorentzian continuation, AdS 2 × S 1 , solve the field equations (9) for the special case (12). This means that the near horizon geometry of the extremal black hole is given by AdS 2 × S 1 , and from Eq. (40), it is amusing to verify that the parameter b , in Eq. (26), can be identified with the period of the circle S 1 . \nAs it is shown below, a spacetime of the form R × H 2 , which corresponds to a double Wick rotation of AdS 2 × S 1 where the circle S 1 is unwrapped, yields an interesting solution.', '1. Black hole entropy': "The entropy of the black hole (24) can be obtained by Wald's formula [50]. Following, the conventions of [51], the entropy can be obtained from \nS = -2 π ∫ Σ h δL δR µναβ /epsilon1 µν /epsilon1 αβ ¯ /epsilon1 , (44) \nwhere L is the Lagrangian, and ¯ /epsilon1 , /epsilon1 µν , correspond to the volume form and the binormal vector to the space-like bifurcation surface Σ h , respectively. Here /epsilon1 µν is normalized as /epsilon1 µν /epsilon1 µν = -2. \nFor the BHT action one obtains \nδL δR µναβ = g µα [ g νβ -2 m 2 ( R νβ -3 8 g νβ R )] , (45) \nand for the black holes discussed here the binormal vector is given by \n/epsilon1 µν = -2 δ t [ µ δ r ν ] . (46) \nHence, the entropy of the black hole (24) is found to be \nS = πl 2 G √ b 2 l 2 +4 µ , = 1 4 G ( A + -A -) , (47) \nwhere A ± = 2 πr ± corresponds, for b < 0, to the area of the event and the inner horizons. \nNote that for the BHT theory at the special case λ = m 2 , the entropy of the BTZ black hole ( b = 0) is twice the one obtained from GR. \nIt is reassuring to verify that the mass computed form the Deser-Tekin approach and the entropy (47) fulfill the first law dM = TdS .", 'E. Gravitational solitons, kinks and wormholes': 'A different class of exact solutions of the field equations (9) for the special case m 2 = λ , can be obtained from a double Wick rotation of the black hole (24). The solution generically describes a gravitational soliton, and for the extremal case, corresponds to a kink that interpolates between different vacua. Performing a suitable identification, a wormhole solution in vacuum can also be obtained as a limiting case of the gravitational soliton. \nThis can be seen as follows: A smooth Lorentzian solution is obtained from (40), unwrapping the angular coordinate, making φ → it and rescaling the Euclidean time as τ → β 2 π φ . The metric reads: \nds 2 = l 2 [ -1 4 (( r + + r -) + ( r + -r -) cosh ρ ) 2 dt 2 + dρ 2 +sinh 2 ρdφ 2 ] , (48) \nwhere the range of the coordinates is given by -∞ < t < + ∞ , 0 ≤ φ < 2 π , and 0 ≤ ρ < + ∞ . In this case, the integration constants r + and r -are no longer interpreted as horizons.', '1. Wormhole': 'Note that for the case r + = r -the metric reduces to a static universe of negative spatial curvature, i.e., R × H 2 , where H 2 is of radius l . This spacetime can also be obtained from a double Wick rotation of AdS 2 × S 1 , but once the circle S 1 is unwrapped, the link with the gravitational hair given by the period of S 1 is lost. \nA wormhole solution in vacuum can be constructed performing an identification of the hyperbolic space H 2 along a boost of its isometry group, parametrized by a constant. The metric then reads \nds 2 = -dt 2 + l 2 dz 2 + ρ 2 0 cosh 2 zdφ 2 ] , (49) \n[ \n] where -∞ < z < + ∞ . This spacetime then corresponds to the product of the real line with a quotient of the hyperbolic space of the form R × H 2 / Γ, (where Γ is a boost of SO (2 , 1) parametrized by ρ 0 (see, e.g. [52])). The solution describes a wormhole in vacuum whose neck, of radius ρ 0 l , is located at z = 0 and connects two static universes of negative spatial curvature located at z →±∞ . \nFIG. 3: Causal structure of the wormhole. The dotted line corresponds to the location of the neck. \n<!-- image --> \nNote that since the whole spacetime is devoid of matter, no energy conditions can be violated. The causal structure of the wormhole (49) coincides with the one of Minkowski spacetime in two dimensions, as it is depicted in Fig. 3.', '2. Gravitational soliton': 'For the generic case, r + /negationslash = r -, after a suitable rescaling of time, the metric (48) depends on a single integration constant and can be written as \nds 2 = l 2 -( a +cosh ρ ) 2 dt 2 + dρ 2 +sinh 2 ρdφ 2 ] . (50) \n[ \nThis spacetime is regular everywhere, whose Ricci scalar is given by \nR = -2 l 2 a +3cosh ρ a +cosh ρ , (51) \nand describes a gravitational soliton provided a > -1. For a = 0, AdS in global coordinates is recovered. The soliton can then be regarded as a smooth deformation of AdS spacetime, sharing the same causal structure. This is an asymptotically AdS spacetime included within the set of asymptotic conditions given in Eq. (30). This can be explicitly verified changing to Schwarzschild-like coordinates making \nr → l sinh ρ ; t → lt , (52) \nso that the metric reads \nds 2 = -( a + √ r 2 l 2 +1 ) 2 dt 2 + dr 2 r 2 l 2 +1 + r 2 dφ 2 . (53) \nThus, the mass of this solution measured with respect to an AdS background is easily obtained from the surface integral (37), and it is given by \nM = -a 2 4 G . (54)', '3. Gravitational kink': 'A gravitational kink can be obtained from the corresponding double Wick rotation of the black hole (24) for the extremal case. Making \nr -r + = le z ; t → lt , (55) \nthe metric reads \nds 2 = -( a + e z ) 2 dt 2 + l 2 [ dz 2 + e 2 z dφ 2 ] , (56) \nwhere -∞ < z < + ∞ , and a := r + l > 0. For a = 0 one recovers the massless BTZ black hole. The kink interpolates AdS and a static Universe of negative spatial curvature ( R × H 2 ). The causal structure is shown in Fig. 4. \nFIG. 4: Causal structure of the kink, that interpolates between R × H 2 ( z → -∞ ) and AdS ( z → + ∞ ) \n<!-- image -->', 'B. Gravitational soliton': 'A gravitational soliton can be obtained unwrapping and Wick-rotating the angle φ → it and making τ → φ in the Euclidean black hole metric (63). The metric then reads \nds 2 = -(( r ++ + r + ) + ( r + -r ++ ) cos θ ) 2 dt 2 + l 2 dθ 2 +sin 2 θdφ 2 ] , (65) \n] so that the range of the coordinates is given by -∞ < t < + ∞ , 0 ≤ φ < 2 π , and 0 ≤ θ < π . \n[ \nNote that for r + = r ++ , this metric reduces to the static Einstein Universe R × S 2 ; otherwise, after a suitable rescaling of time, the metric (65) depends on a single integration constant and reduces to \nds 2 = -( a +cos θ ) 2 dt 2 + l 2 [ dθ 2 +sin 2 θdφ 2 ] , (66) \ndescribing a gravitational soliton provided | a | > 1.', 'C. Euclidean action': 'In the case of positive cosmological constant, the Euclidean black hole metric (63) coincides with the Euclidean continuation of the soliton in Eq. (66). It is also worth pointing out that neither the Euclidean black hole nor S 2 × S 1 have a boundary. Thus, it is simple to show that the Euclidean continuation of the action (7) evaluated on these solutions vanishes, i.e., \nI ( bh ) = I ( S 2 × S 1 ) = 0 . (67) \nThis is to be compared with the value of the Euclidean action for the three-sphere of radius l (Euclidean dS space), given by \nI ( S 3 ) = l G , (68) \nwhich allows to estimate the pair creation ratio [53].', 'V. VANISHING COSMOLOGICAL CONSTANT': 'In the case of vanishing cosmological constant, the BHT action (7) at the special point λ = m 2 reduces to \nI = ∫ d 3 x √ -gK , (69) \nwhich, as it has been recently shown in [30], enjoys remarkable properties. An asymptotically locally flat black hole solution of this theory can be obtained from the metric (22) making Λ = 0, and ψ → it \nds 2 = -( br -µ ) dt 2 + dr 2 br -µ + r 2 dϕ 2 . (70) \nThis black hole possesses a spacelike singularity at the origin, which is surrounded by an event horizon located at r + = µ/b , provided b > 0 and µ > 0. In the case of µ = 0, there is a NUT at the origin, on top of the null singularity. The causal structure is shown in Fig. 6. \nA gravitational soliton can also be obtained from a double Wick rotation of (70), of the form t → iφ and ϕ → 2 it . After a change of coordinates, given by ρ = √ br -µ , the spacetime possesses a conical singularity at the origin, which is removed for b = 2. The metric is then smooth everywhere and reads \nwith 0 ≤ ρ < ∞ . \nds 2 = -( ρ 2 + µ ) 2 dt 2 + dρ 2 + ρ 2 dφ 2 , (71) \nFIG. 6: Penrose diagrams for the asymptotically locally flat black holes \n<!-- image --> \nAs shown in [30] the degree of freedom associated with the massive graviton, at the linearized level, is captured by the h ti component of the metric deviation. Note that for GR, the graviton degrees of freedom cannot be excited in this way while keeping spherical symmetry. For our solutions (70) and (71) it is apparent that the massive graviton degree of freedom is switched off. It would be interesting to explore the existence of analytic solutions in the full nonlinear theory, for which the degree of freedom could be consistently switched on in the presence of the black hole (70) or the gravitational soliton (71). \nOne may also wonder about whether the asymptotically locally flat black hole (70), and the gravitational soliton (71) can be accommodated within a suitable set of asymptotic conditions at null infinity, along the lines of Ref. [54].', 'VI. ROTATING SOLUTIONS': 'The solutions discussed here can be generalized to the rotating case by means of an (improper) boost in the t -φ plane (For a discussion about this subject in three-dimensional General Relativity see, e.g. [55]). For instance, the rotating extension of the asymptotically AdS black hole (24) is given by \nwith \nand \nHere Ξ := 1 -a 2 /l 2 , and the angular momentum is given by J = Ma , where M is the mass (measured with respect to the zero mass black hole) and -l < a < l is the rotation parameter. This spacetime is then asymptotically AdS and naturally fulfills the relaxed asymptotic conditions in Eq. (30). \nds 2 a = -NFdt 2 + dr 2 F + r 2 ( dφ + N φ dt ) 2 , (72) \n[ \n4 ( \nN = [ 1 -bl 2 4 G ( 1 -Ξ -1 2 ) ] 2 , N φ = -a 2 r 2 ( 4 GM -b Ξ -1 2 G ) , F = G 2 r 2 [ G 2 l 2 + b 2 ( 1 + Ξ -1 2 ) G + b 2 l 2 16 ( 1 -Ξ -1 2 ) 2 -4 GM Ξ 1 2 ] , G = r 2 -2 GMl 2 ( 1 -Ξ 1 2 ) -b 2 l 16 1 -Ξ -1 2 ) 2 ] 1 2 . (73) \nDepending on the range of the parameters M , a and b , the solution possesses an ergosphere and a singularity that can be surrounded by event and inner horizons. As for the nonrotating case, there are different branches of solutions according to the sign of the integration constant b . One can show that the location of the event horizon, the temperature and the entropy are given by \nr a + = γr + , (74) \nT a = γ -1 T , (75) \nS a = γS , (76) \nwhere γ 2 = 1 2 ( 1 + Ξ -1 / 2 ) , and r + , T and S correspond to the horizon radius, the temperature and the entropy for the static case, respectively. For the rotating BTZ black hole ( b = 0) \nthe entropy becomes twice the one obtained in GR, i.e., S = A + 2 G . In the case of b < 0 the mass is bounded from below as \nM ≥ -b 2 l 2 16 G Ξ . \nThis bound is saturated for the extremal case. Further details about the rotating case will be discussed elsewhere.', 'VII. DISCUSSION AND FINAL REMARKS': "The black hole metrics of the form \nds 2 = -( -Λ r 2 + br -µ ) dt 2 + dr 2 -Λ r 2 + br -µ + r 2 dϕ 2 , (77) \nand the wormhole in Eq. (49) were known to be solutions of conformal gravity in three dimensions [42]. Here it was shown that they are nice spacetimes, in the sense that they solve the field equations for theories beyond the one they were intended to. This is also the case for the rest of the solutions discussed here, including the rotating black holes (72), the gravitational solitons in Eqs. (50), (66) and (71), the kink (56), the static universes of constant spatial curvature R × S 2 and R × H 2 , as well as the product spaces AdS 2 × S 1 and dS 2 × S 1 . Therefore, they are all solutions of the BHT field equations for the special case, m 2 = λ , even in presence of the topological mass term \nG µν + λg µν -1 2 m 2 K µν + 1 µ C µν = 0 , (78) \nwhere C µν := /epsilon1 κσµ ∇ κ ( R σν -1 4 g σν R ) is the Cotton tensor. \nIn the case of the black hole with positive cosmological constant (57), since the temperatures of the event and the cosmological horizons coincide, the solution can be regarded as a pair of black holes on dS, whose Euclidean continuation describes an instanton of vanishing Euclidean action. Thus, the pair creation ratio could be estimated from the value of the Euclidean action for S 3 , given by l/G . \nIn the case of negative cosmological constant, the black hole and the gravitational soliton were shown to fit within the set of asymptotically AdS conditions given by (30), having a relaxed behavior as compared with the one of Brown and Henneaux. The asymptotic symmetries contain two copies of the Virasoro group, which is enlarged to a semi-direct \nproduct by the additional asymptotic symmetries associated to local shifts in the radial coordinate generated by (31). Note that this additional asymptotic symmetry can be used to eliminate the function h rr appearing in the asymptotic conditions, so that for h rr = 0 the asymptotic symmetry group reduces to the standard conformal group in two dimensions. Furthermore, this additional symmetry has vanishing charges, as it is apparent from Eq. (37), which suggests that the gauge could always be fixed in this way. Nevertheless, since the remaining charges, as the mass, change nontrivially under shifts of the radial coordinate, this is not the case. This puzzling enhancement of the asymptotic symmetries could be related to the fact that the Deser-Tekin charges are constructed from the linearized theory. Indeed, there are known cases where nonlinear corrections are needed in order to obtain finite charges (as it occurs in the presence of scalar fields [6, 8]). Therefore, it would be desirable to explore this problem within a fully nonlinear approach. It is worth pointing out that this curious behavior may occur even for a nonlinear approach, as it has been observed for different classes of degenerate dynamical systems [58], even in classical mechanics [59], for which the rank of the symplectic form may decrease for certain regions within the space of configurations. Thus, around certain special classes of solutions, additional gauge symmetries arise, and hence the system losses some degrees of freedom. \nIt is also worth pointing out that further asymptotically AdS solutions with a different behavior at infinity exist for the BHT theory at the special point, as it is the case for the AdS waves recently found by Ay'on-Beato, Giribet and Hassa¨ıne [27]. A suitable set of asymptotically AdS conditions, containing the black hole (24) as well as the AdS waves, is such that the deviation with respect to the AdS metric (3) is supplemented by additional terms with logarithmic behavior. Fixing the asymptotic symmetry under local shifts in the radial coordinate, the asymptotic conditions that are invariant under the two copies of the Virasoro algebra, generated by (4), read \n∆ g rr = f rr r -4 + ... , ∆ g rm = j rm r -2 log( r ) + h rm r -2 + f rm r -3 + ... , (79) ∆ g mn = j mn r log( r ) + h mn r + f mn + ... , \nwhere j + -can be consistently switched off, and j µν depends only on the time and the angle, but not on r . One can also verify that the Deser-Tekin charges, once evaluated on the \nasymptotic conditions (79), reduce to the expression given in Eq. (33), which depend only on f ++ , and f --. \nFor the BHT theory in the generic case m 2 /negationslash = λ , black holes which are not of constant curvature, that depend on an additional integration constant that is not related to the mass exist, and they fulfill the Brown-Henneaux boundary conditions [56]. It is amusing to see that this behavior has a similar pattern as the one of the Boulware-Deser solution for the Einstein-Gauss-Bonnet theory. In the generic case, relaxed asymptotic conditions including logarithmic branches have also been studied in Ref. [47]. \nAcknowledgments. We thank D. Anninos, M. Becker, G. Comp'ere, S. Detournay, J. Gegenberg, G. Giribet, M. Guica, M. Henneaux, G. Kunstatter, C. Martinez, W. Song and J. Zanelli for useful discussions and comments. We also thank A. Maloney for let us now about the existence of further new solutions for the generic case. Special thanks to O. Hohm and E. Bergshoeff for the nice discussions about the special case in Vienna, as well as for kindly providing advance access to their manuscript [57], in collaboration with P.K. Townsend. J. Oliva thanks ICTP for the kind hospitality during the Spring School on Superstring Theory and Related Topics. D.T. thanks Conicyt for financial support. R.T. wish to thank D. Grumiller and the organizers of the Workshop on Gravity in Three Dimensions, hosted during April 2009 at the Erwin Schrodinger Institute (ESI), Vienna, for the opportunity of presenting this work. R. T. also thanks the kind hospitality at the Physique th'eorique et math'ematique at the Universit'e Libre de Bruxelles and the International Solvay Institutes. This research is partially funded by Fondecyt grants 1061291, 1071125, 1085322, 1095098, 3085043. 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2012PhRvD..85d4024B
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On black holes in massive gravity
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2012-01-01
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In massive gravity, the black hole solutions found so far on Minkowski space happen to convert horizons into a certain type of singularities. Here, we explore whether these singularities can be avoided if space-time is not asymptotically Minkowskian. We find an exact analytic black hole (BH) solution, which evades the above problem by a transition at large scales to self-induced de Sitter space-time, with the curvature scale set by the graviton mass. This solution is similar to the ones discovered by Koyama, Niz, and Tasinato, and by Nieuwenhuizen, but differs in detail. The solution demonstrates that in massive general relativity, in the Schwarzschild coordinate system, a BH metric has to be accompanied by the Stückelberg fields with nontrivial backgrounds to prevent the horizons to convert into the singularities. We also find an analogous solution for a Reissner-Nordström BH on de Sitter space. A limitation of our approach is that we find the solutions only for specific values of the two free parameters of the theory, for which both the vector and scalar fluctuations lose their kinetic terms; however, we hope our solutions represent a broader class with better-behaved perturbations.
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https://arxiv.org/pdf/1111.3613.pdf
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{'On Black Holes in Massive Gravity': 'L. Berezhiani a , G. Chkareuli a , C. de Rham bc , G. Gabadadze ad and A.J. Tolley c \na Center for Cosmology and Particle Physics, Department of Physics, New York University, New York, NY, 10003 b Départment de Physique Théorique and Center for Astroparticle Physics, Université de Genève, 24 Quai E. Ansermet, CH-1211 Genève \nc Department of Physics, Case Western Reserve University, Euclid Ave, Cleveland, OH, 44106 d Department of Physics, Columbia University, New York, NY, 10027', 'Abstract': 'In massive gravity the so-far-found black hole solutions on Minkowski space happen to convert horizons into a certain type of singularities. Here we explore whether these singularities can be avoided if space-time is not asymptotically Minkowskian. We find an exact analytic black hole (BH) solution which evades the above problem by a transition at large scales to self-induced de Sitter (dS) space-time, with the curvature scale set by the graviton mass. This solution is similar to the ones discovered by Koyama, Niz and Tasinato, and by Nieuwenhuizen, but differs in detail. The solution demonstrates that in massive GR, in the Schwarzschild coordinate system, a BH metric has to be accompanied by the Stückelberg fields with nontrivial backgrounds to prevent the horizons to convert into the singularities. We also find an analogous solution for a Reissner-Nordström BH on dS space. A limitation of our approach, is that we find the solutions only for specific values of the two free parameters of the theory, for which both the vector and scalar fluctuations loose their kinetic terms, however, we hope our solutions represent a broader class with better behaved perturbations.', '1 Introduction and Summary': "According to the representation theory of the Poincaré group in 4D, a massive spin-2 state has to have five degrees of freedom; these can be thought of as the helicity0 , ± 1 , ± 2 states. A good Lagrangian for the massive spin-2 has to be able to describe these states. The Fierz-Pauli mass term [1] is the only ghost- and tachyon-free term at the quadratic order that describes the above 5 states [2]. However, in the zero mass limit it does not recover the linearized Einstein's gravity, since the helicity0 mode couples to the trace of the matter stress-tensor with strength equal to that of the helicity-2; this is called the vDVZ discontinuity [3]. If true, it would rule out massive gravity on the grounds of solar system observations. However, Vainshtein [4] argued that the troublesome longitudinal mode is suppressed at measurable distances by nonlinear effects, making the nonlinear theory compatible with current empirical data [5]. On the other hand, in a broad class of models, the same nonlinear terms that are responsible for the above-mentioned suppression, give rise to an instability known as the Boulware-Deser (BD) ghost [6]. This ghost appears as a 6th degree of freedom in the theory, and even though it is infinitely heavy on the Minkowski background, it becomes sufficiently light on locally nontrivial backgrounds, thus invalidating the theory [7, 8, 9, 10]. \nMore recently, however, using the effective field theory formalism of [7], it has been shown in Ref. [11] that there exists a two parameter family of nonlinear generalization of the linear Fierz-Pauli theory, that is free of the BD ghost order-by order and to all orders, at least in the decoupling limit. \nMost importantly, it was shown in Ref. [12] that the absence of the BD ghost in the decoupling limit is such a powerful requirement that it leads to the resummation of the entire infinite series of the terms in the effective Lagrangian. As a result, a candidate theory of massive General Relativity free of BD ghost, was proposed [12]. \nUsing the Hamiltonian analysis in the unitary gauge it was shown that for a certain choice of the free parameters of the theory, and in the 4th order in nonlinearities, the Hamiltonian constraint that forbids the BD ghost is maintained in the theory of [12]. Note that the quartic order is special, since the lapse necessarily enters nonlinearly in all massive theories precisely in this order [8], and it may appear that the hamiltonian constraint should necessarily be lost then. In spite of this, the constraint is maintained in a subtle way for special theories, as was shown for a toy model in [11], and shown in the 4th order for massive GR in [12]. \nThe existence of the Hamiltonian constraint to all orders in the unitary gauge, and for generic values of the parameters, was shown in Ref. [13], using the method of dealing with the lapse and shift proposed in [11, 12]. Moreover, Ref. [13] has also argued for a secondary constraint, that follows from the conservation of the Hamiltonian constraint 1 . Very recently, the existence of the secondary constraint was explicitly confirmed in Ref. [15]. \nThe absence of the BD ghost among the local fluctuations of the theory of [12] \nin a generic gauge has been shown using the Stückelberg decomposition [16], as well as the helicity decompositions [17] to quartic orders in nonlinearities (in the latter two references, previous misconceptions in the literature claiming the presence of the BD ghost were also clarified). Motivated by the above developments, in the present work we will proceed to study certain subtle properties black holes (BH) in the theory of [12]. \nIn the unitary gauge Lagrangian of the theory the object h µν ≡ g µν -η µν , is the gravitational analog of the Proca field of massive electrodynamics, describing all the five modes of the graviton. The diffeomorphism invariance can be restored by introducing the four scalar fields φ a (the Stückelberg fields) [18, 7, 19], and replacing the Minkowski metric by the covariant tensor ∂ µ φ a ∂ ν φ b η ab \ng µν = ∂ µ φ a ∂ ν φ b η ab + H µν , (1) \nwhere H µν denotes the covariantized metric perturbation, and η ab = diag( -1 , 1 , 1 , 1) . The existence of the 4 Stückelberg scalars φ a in this theory leads to the existence of new invariants in addition to the ones usually encountered in GR (Ricci scalar, Ricci tensor square, Riemann tensor square, etc); one new basic invariant is I ab = g µν ∂ µ φ a ∂ ν φ b . Note that the unitary gauge is set by the condition φ a = x µ δ a µ . In this gauge, I ab = g µν δ a µ δ b ν . Hence, any inverse metric that has divergence (even those which are innocuous in GR) would exhibit a singularity in the invariant I ab . Is this singularity of any significance? The singularity in the above invariant does not necessarily affect the geodesic motion of external observers - the geodesic equation is identical to that of GR, and due to its covariance, one could remove from it what would have been a coordinate singularity in g µν in GR. However, one would expect the singularities in I ab = g µν δ a µ δ b ν to be a problem for fluctuations around classical solutions exhibiting it. Since g µν could change signs on either side of the singularity, this could lead to emergence of ghosts and/or tachyons in the fluctuations around a given classical solution. In what follows, we will take a conservative point of view and will only accept solutions that have non-singular I ab . These arguments, in a somewhat different form, have already been emphasized recently by Deffayet and Jacobson [20]. \nThe above arguments give rise to the following seeming puzzle. On the one hand, according to the Vainshtein mechanism [4], spherically symmetric solutions of massive gravity should approximate those of GR better and better, as we increase the mass of the source and come closer to it. This would imply that the metric of a BH near its horizon should very much be similar to that of GR. On the other hand, the conventional Schwarzschild metric - if it were the solution of massive gravity in unitary gauge - would be singular according to the arguments above. \nWe reiterate this central point in more general terms: In order for a metric to qualify as a valid description of a BH configuration, the physical singularities must be absent at the horizon. Then, in the unitary gauge of massive gravity the \nSchwarzschild-like metric \nds 2 = -(1 -f ) dt 2 + dr 2 1 -f + r 2 d Ω 2 , with e.g. f = r g /r, (2) \ncannot be a legitimate BH solution of the theory. The same applies to the metric of de Sitter (dS) space in the static coordinates for which f = m 2 r 2 . \nRecently, interesting BH solutions of massive gravity have been found in Refs. [21, 22, 23] (for other interesting solutions, which will not be discussed here, see, [24]- [32]). Following Koyama, Niz and Tasinato (KNT) [21], one can start in the unitary gauge, and consider a most general stationary spherically symmetric metric. Then, using the method developed by KNT, very interesting full non-linear solutions for stars and black holes with Minkowskian asymptotics were found by Gruzinov and Mirbabayi in [23]. These solutions do exhibit the Vainshtein mechanism, and therefore are potentially viable classical solutions for stars and other compact objects in massive gravity (although their stability still remains to be studied). Nevertheless, it is not clear, as emphasized in [23], whether these are appropriate solutions for BHs. Even in the best case solution, when all the GR invariants are finite, the invariant g µν ∂ µ φ a ∂ ν φ b η ab diverges, [23]. As noted above, this divergence does not affect the geodesic motion of any external observer, however, we expect it to be a problem for fluctuations. \nCould there be any solution that avoids the above issue? The answer is positive, and the resolution is in the identification of the unitary gauge to the coordinate system in which the black hole has no horizon (the Kruskal-Szekeres, EddingtonFinkelstein, or Gullstrand-Painlevé systems come to our mind). The most convenient one for our purposes is the Gullstrand-Painlevé (GP) system, in which the metric has the following form \nds 2 = -dt 2 +( dr ± fdt ) 2 + r 2 d Ω 2 , (3) \nand is free of horizon singularities. It corresponds to the frame of an in-falling observer and covers half the whole space (for either choice of sign). \n√ \nFurthermore, if one has the metric (3) as a solution in unitary gauge, then the coordinate transformation to the metric (2) will lead us to a background with φ a /negationslash = x a . This means that if the configuration is described by the metric (2), the presence of a halo of helicity ± 1 and/or 0 fields around the BH is unavoidable. We will show in the present work that massive gravity in unitary gauge admits BH solutions precisely of this type. \nInterestingly, the dS-Schwarzschild solution found in [21, 33] do happen to satisfy our conservative criterion of non-singularity. However, the solution that we present here is not among the ones of [21, 33]. \nOne more point worth emphasizing is that the BH solutions of [23] do exhibit the 'helicity-0 hair' (e.g., produce an extra scalar force), while the ones found in [21, 33], and in the present work do not. The status of the 'no-hair' theorems in \nGR with the galileon field (which should capture some properties of the helicity-0 of massive gravity) will be discussed in Ref. [34]. \nA limitation of our work is that we only manage to find these exact analytic solutions for a specific choice of the two free parameters of massive gravity. Such a choice is peculiar since on the obtained background, as we will show, the kinetic terms for both the vector and scalar fluctuations vanish in the decoupling limit. \nThis fact would imply infinitely strong interactions for these modes (unless these modes happen to be nondynamical to all orders, e.g., due to the specific choice of the coefficients of the theory). Because of this issue, we would like to regard the solutions obtained here as just examples demonstrating how non-singular solutions should emerge. We also hope that our solutions are representative of a broader class of solutions which may have better behaved fluctuations. \nIn this regard, there seems to be a few directions in which the studies of massive gravity BH's can be extended. First, one could look at the metric in the unitary gauge which would be some generalization of the Kruskal-Szekeres form. Second, one can extend the massive theory of [12] by adding more degrees of freedom to the existing 5 helicity states of massive graviton. In fact, two consistent extensions have already been discussed so far: (I) adding one real scalar field that makes the graviton mass dynamical [28]; (II) adding one massless tensor field with two degrees of freedom [35] that makes the internal space metric of the Stückelberg field dynamical (bigravity). In the latter case cosmological solutions were found recently in [31] and [32], while BH's were studied in [36]. \nThe work is organized as follows. Section 2 gives a brief review of the theory of massive gravity [12]. In section 3 we find an exact Schwarzschild-de Sitter solution, and in section 4 an exact Reissner-Nordström-de Sitter, solution which have nonsingular I ab . These solution are similar to those discovered by Koyama, Niz and Tasinato, and by Th. Nieuwenhuizen, but differ in detail: our dS solution has no ghost even though the vector field is present (compare to [33]). Moreover, on the obtained solution the singularities in the invariant I ab are absent (compare to [22]). In the Appendix A we give another exact Schwarzschild solution that asymptotes to a conformaly rescaled Minkowski space, and briefly mention its peculiarities. In the Appendix B we discuss fluctuations on the selfaccelerated solution of section 3.", '2 The Theory': "A massive graviton is described by the Lagrangian density of [12] specified below \nL = M 2 pl 2 √ -g ( R + m 2 U ( g, φ a ) ) , (4) \nwhere U is the potential for the graviton that depends on two free parameters α 3 , 4 \nU ( g, φ a ) = ( U 2 + α 3 U 3 + α 4 U 4 ) , (5) \nand the individual terms in the potential are defined as follows: \n2 = [ K ] 2 [ K 2 ] , (6) \nU U \nK K 3 = [ K ] 3 3[ K ][ 2 ] + 2[ K 3 ] , (7) \n-- \nK K K U 4 = [ K ] 4 -6[ K 2 ][ K ] 2 +8[ K 3 ][ K ] + 3[ K 2 ] 2 -6[ K 4 ] , (8) \nK \nwhere K µ ν ( g, φ a ) = δ µ ν -√ g µα ∂ α φ a ∂ ν φ b η ab ; rectangular brackets denote traces, [ K ] ≡ Tr( K ) = K µ µ . The above potential is unique - no further polynomial terms can be added to the action without introducing the BD ghost. \nThe tensor H µν represents the covariantized metric perturbation, as discussed in the introduction, which reduces to the h µν in unitary gauge. While in a gauge unfixed theory we have \nH µν = g µν -∂ µ φ a ∂ ν φ b η ab . (9) \nMoreover, U is constructed in such a way that the theory admits the Minkowski background \ng µν = η µν , φ a = x µ δ a µ . (10) \nHence, it is natural to split φ 's as the background plus the 'pion' contribution φ a = x a -π a , and as it was already mentioned in the introduction, the unitary gauge is defined by the condition π a = 0 . In the non-unitary gauge, on the other hand, it proves to be useful to adopt the following decomposition \nπ a = mA a + ∂ a π Λ 3 , (11) \nwhere A µ describes in the decoupling limit the helicity ± 1 , while π is the longitudinal mode of the graviton (in the decoupling limit [7], M pl → ∞ and m → 0 , while Λ 3 ≡ M pl m 2 is held fixed). This limit captures the approximation in which the energy scale is much greater than the graviton mass scale, E m . \nFor convenience, in what follows, we define the coefficients α and β which are related to those of (5) by α 3 = -( -α + 1) / 3 and α 4 = -β/ 2 + ( -α + 1) / 12 . For generic values of the parameters α and β the theory exhibits the Vainshtein mechanism, as show in the decoupling limit [11], and beyond [21, 26]. As was emphasized in [11], for one special choice, α = β = 0 , the nonlinear interactions vanish in the decoupling limit with fixed Λ , leaving the theory weakly coupled (i.e., no Vainshtein mechanism) in this limit. For this particular choice of the coefficients the action of massive gravity with the potential (6)-(8) (which can be rewritten in terms of just [ K ] and tuned to it cosmological constant [25], referred as a minimal model in Ref. [25]), was shown not to exhibit the Vainshtein mechanism also away from the decoupling limit [21]. \n/greatermuch", '3 A Black Hole on de Sitter': 'In this section we present the Schwarzschild-de Sitter solution in the theory of massive gravity described above. The obtained solution is free of singularities (except from the conventional one appearing in GR). \nFor convenience we choose unitary gauge for the metric. In this gauge the symmetric tensor g µν is an observable describing all the five degrees of freedom of a massive graviton. The equations of motion in empty space read as follows \nG µν + m 2 X µν = 0 , (12) \nwhere X µν is the effective energy-momentum tensor due to the graviton mass, \nX µν = -1 2 [ K g µν -K µν + α ( K 2 µν -KK µν + 1 2 g µν ( [ K ] 2 -[ K 2 ] ) ) (13) +6 β ( K 3 µν -KK 2 µν + 1 2 K µν ( [ K ] 2 -[ K 2 ] ) -1 6 g µν ( [ K ] 3 -3[ K ][ K 2 ] + 2[ K 3 ] ) ) ] . \nUsing the Bianchi identities, from (12) we obtain the following constraint on the metric \nm 2 ∇ µ X µν = 0 , (14) \nwhere µ denotes the covariant derivative. \nIn order to obtain the expression for X µν we make use of the fact that the Lagrangian is written as the trace of the polynomial of the matrix K µ ν . Thus, following the method by Koyama, Niz and Tasinato [21], we choose the basis which diagonalizes the expression appearing under the square root in the definition of K µ ν [One should bear in mind that this is not a coordinate transformation, but rather a trick to simplify the procedure of getting the equations of motion]. As a result, the potential becomes a function of the components of the inverse metric, rather than the combination of square roots of matrices. Having done this, one is free to vary the action with respect to the inverse metric components to obtain explicit expression for (12). Since these expressions are quite cumbersome we will not give them here. \n∇ \nBelow, we concentrate on one particular family of the ghost-free theory of massive gravity in which there is the following relations between the two free coefficients: \nβ = -α 2 6 . (15) \nThat this choice of the coefficients is special was first shown by Th. Nieuwenhuizen [22] (see also [23]). In particular, it was shown in [22] that for this choice the equation (14) is automatically satisfied for a certain diagonal (in spherical coordinates) and time-independent metrics. It is interesting, however, that the above property persists \nfor a more general class of non-diagonal spherically-symmetric metrics written as follows: \nds 2 = -A ( r ) dt 2 +2 B ( r ) dtdr + C ( r ) dr 2 + w 2 r 2 d Ω 2 , (16) \nwhere w is a constant, while A ( r ) , B ( r ) and C ( r ) are arbitrary functions. \nIn subsection 3.1 we find an exact de Sitter solution to (12), and in subsection 3.2 we find an exact BH solution on the obtained dS background. Note that the dS background is entirely due to the graviton mass.', '3.1 The de Sitter Solution': 'We note that we would find an exact dS solution if we required that \nm 2 X µν = λg µν , (17) \nwhere λ is some constant. The solution of the equations (12) that also satisfies (17) with a positive but otherwise arbitrary α is given by \nds 2 = -κ 2 dt 2 + ( α α +1 dr ± κ √ 2 3 α α ( α +1) mrdt ) 2 + α 2 ( α +1) 2 r 2 d Ω 2 . (18) \nHere, κ is a positive integration constant. It is straightforward to check that for (18) we have X µν = (2 /α ) g µν , leading to the expression for the Ricci scalar \nR = 8 α m 2 , (19) \nas expected. Hence, this is a dS space with curvature scale set by the graviton mass and one free parameter α . One could imagine that m ∼ (0 . 1 -1) H 0 , and α ∼ (0 . 01 -1) , in which case the obtained dS solution (if stable) could describe dark energy. \nUp to a rescaling of the coordinates, the expression (18) looks exactly like the de Sitter solution of GR written in the Gullstrand-Painlevé frame. Either ± solution covers half of dS space. One can rotate the obtained solution to the static coordinate system at the expense of nonzero Stückelberg fields. This will be done in the next subsection. In either form, the solution has no additional singularities.', '3.2 Schwarzschild-de Sitter Background': "Having the solution of the previous subsection worked out, it is straightforward to show that the system of equations (12) admits the following exact solution \nds 2 = -κ 2 dt 2 + ( ˜ αdr ± κ √ r g ˜ αr + 2˜ α 2 3 α m 2 r 2 dt ) 2 + ˜ α 2 r 2 d Ω 2 , (20) \nwhere ˜ α ≡ α/ ( α +1) , and as before, κ is an integration constant. In order to bring this solution to a more familiar form let us perform the following rescaling \nr → α +1 α r, dt → 1 κ dt. (21) \nThe resulting metric reads \nds 2 = -dt 2 + ( dr ± √ r g r + 2 3 α m 2 r 2 dt ) 2 + r 2 d Ω 2 . (22) \nThis is the Schwarzschild-de Sitter solution in the GP coordinates. \nHowever, the above rescaling takes us away from the unitary gauge \nφ 0 = t → t -( 1 -1 κ ) t, (23) \nφ r = r → r + 1 α r. (24) \nIn terms of the 'pions', π µ ≡ x µ -φ µ , which can be decomposed as π µ = ( mA µ + ∂ µ π ) / Λ 3 , we have \nπ = Λ 3 2 [ -( 1 -1 κ ) t 2 -1 α r 2 ] , (25) \nA µ = 0 . (26) \nThe fields in (26) correspond to the canonically normalized fields carrying the helicity eigenstates in the decoupling limit. \nLet us now rewrite our solution into a more familiar coordinate system. The metric can be transformed to a static slicing by means of the following coordinate transformation \ndt → dt + f ' ( r ) dr, (27) \nwith f ' ( r ) ≡ -g 01 /g 00 given by \nf ' ( r ) = ± √ r g r + 2 3 α m 2 r 2 1 -r g r -2 3 α m 2 r 2 . (28) \nThe resulting expression for the metric reads as follows: \nds 2 = -( 1 -r g r -2 3 α m 2 r 2 ) dt 2 + dr 2 1 -r g r -2 3 α m 2 r 2 + r 2 d Ω 2 . (29) \nThis is nothing but the metric of the Schwarzschild-de Sitter solution of GR in the static coordinates. However, this metric should be accompanied by a nontrivial backgrounds for the Stückelberg fields. Indeed, it is evident that (27) gives rise to the shift δφ 0 = f ( r ) . In turn, this gives rise to a background for the 'vector mode' \nA 0 = -Λ 3 κm f ( r ) , (30) \nA i = 0 . (31) \nThis particular field assignment has been chosen according to scaling in the decoupling limit. Namely, f ( r ) vanishes in the decoupling limit linearly in m hence it was ascribed to the 'vector mode'. \nWe would like to make two important comments in the remainder of this section. The first one concerns the integration constant κ . Although, all the invariants of GR are independent of κ , the new invariant that is characteristic of massive gravity \nI ab ≡ g µν ∂ µ φ a ∂ ν φ b , (32) \ndoes depend on this integration constant; in the unitary gauge I ab is just the inverse of the GP metric (18) which reads as follows \n -1 κ 2 ± 1 κ α +1 α √ α +1 α r g r + 2 3 α α 2 ( α +1) 2 m 2 r 2 0 ± 1 κ α +1 α √ α +1 α r g r + 2 3 α α 2 ( α +1) 2 m 2 r 2 ( α +1 α ) 2 ( 1 -α +1 α r g r -2 3 α α 2 ( α +1) 2 m 2 r 2 ) 0 0 0 α +1 α ) 2 Ω -1 2 × 2 . \n( \n) Thus, the backgrounds with different values of κ correspond to distinct superselection sectors labeled by the values of I ab . \nThe second comment concerns the issue of small fluctuations on top of this solution. One may worry that the scalar perturbations on this background are infinitely strongly coupled in the light of the results of [24]. In the latter work it was found that, for the parameters chosen as in (15), the de Sitter background has infinitely strongly coupled fluctuations in the decoupling limit. However, we should point out that the self-accelerated background discussed in this section is different from that studied in [24]. This distinction is manifest in (31) by the presence of the background for A 0 , which vanishes in the case of [24]. \nStill, one could argue that it is unnecessary to perform the transformation of variables (27) responsible for this difference, and limit oneself to the rescaling of the coordinates \nr → α +1 α r , t → 1 κ t , (33) \nwhich clearly does not give rise to the vector background. As a result, the 'pion' configuration will become similar to that of [24], while the metric itself will be quite \ndifferent, namely 2 \nds 2 = -(1 -m 2 r 2 ) dt 2 +2 mrdtdr + dr 2 + r 2 d Ω 2 . (34) \nNow, if we were to take this metric as the one in which the decoupling limit should be taken, then we would find that the gauge freedom that is left in this limit \ng µν → g µν + ∂ ( µ ξ ν ) , (35) \nwould not be enough for bringing (34) to the form of de Sitter space in either conformal or static slicing. Furthermore, the canonically normalized (34) diverges in the decoupling limit, in such a way that this divergence can be isolated only in the vector mode. If so, then no conclusion can be drawn about the perturbations around our background based on the results of [24]. This, on the other hand, does not necessarily imply that the fluctuations are fine. As we show in the Appendix B the vector and scalar fluctuations may be infinitely strongly coupled.", '3.3 From Gullstrand-Painlevé to Kruskal-Szekeres': "In this section we ask the question whether the BH solution of the GP form could be analytically continued to cover the other half of the space-time as well. This can be done by going to the Kruskal-Szekeres (KS) coordinates and analyzing the Stückelberg fields. \nLet us start addressing this point by considering the following background \nds 2 = -dt 2 GP + ( dr + √ r g r dt GP ) 2 + r 2 d Ω 2 , φ a = x a . (36) \nFirst we rewrite the metric in static slicing by performing the following change of the time variable \nφ 0 = t GP = t +2 r g √ r r g + r g ln √ r r g -1 √ r r g +1 . (37) \nAs a result the metric takes on the Schwarzschild form. In order to go to KS coordinates we use reparametrizations identical to the one used in GR \n( r r g -1 ) e r/r g = X 2 -T 2 , (38) \nt = r g ln ( X + T X -T ) . (39) \nFor the analysis of the φ 's in KS coordinates we make the 'near the horizon' approximation r/r g → 1 , since this is the the region of our interest. In this limit \nthe above coordinate transformation simplifies to (near the horizons T = ± X , with signs corresponding to the black- and white-hole respectively) \n( r r g -1 ) = 1 e ( X 2 -T 2 ) , (40) \nt = r g ln ( X + T X -T ) . (41) \nAs a result the φ 's take the following form (using the fact that T 2 -X 2 is small) \nφ 0 = 2 r g ln ( X + T ) + r g ( ln (1 / 4) + 1) , (42) \nφ r = r g ( 1 + 1 e ( X 2 -T 2 ) ) . (43) \nNotice that φ 0 is singular on the horizon of the white hole (while being regular on the black hole horizon). The metric in these coordinates is given by \nds 2 = 4 r 3 g e -r/r g r ( -dT 2 + dX 2 ) + r 2 d Ω 2 (44) \nThe invariant I ab = g µν ∂ µ φ a ∂ ν φ b on the above background is singular at X = -T , corresponding to the horizon of the white hole. \nIf one takes the original GP metric (36) to describe the white hole instead of the black hole (this is achieved by flipping the relative sign of the expressions in parentheses) then after analytical continuation to KS coordinates the singularity will appear on the black hole horizon rather than on the one of the white hole. The generalization of this arguments for the case of the dS is straightforward.", '4 Reissner-Nordström solution on de Sitter': "The ghost-free theory of massive gravity with β = -α 2 / 6 , upon its coupling to the Maxwell's theory of electromagnetism, possesses the following Reissner-Nordström solution on dS space \nds 2 = -dt 2 + ˜ αdr ± √ r g ˜ αr + 2˜ α 2 3 α m 2 r 2 -˜ Q 2 ˜ α 4 r 2 dt 2 + ˜ α 2 r 2 d Ω 2 , (45) \nwith ˜ α ≡ α/ ( α +1) and the electromagnetic field given by \nE = ˜ Q r 2 and B = 0 . (46) \nIn order to normalize the radial coordinate appropriately and to rewrite the solution in the static slicing, one needs to perform the rescaling \nr → r ˜ α , (47) \nsupplemented with the following transformation of time \ndt → dt + f ' ( r ) dr, with f ' ( r ) ≡ -g 01 g 00 = ± √ r g r + 2 3 α m 2 r 2 -˜ Q 2 ˜ α 2 r 2 1 -r g r -2 3 α m 2 r 2 + ˜ Q 2 ˜ α 2 r 2 . (48) \nAs a result the metric takes the familiar form \nds 2 = -(1 -r g r -2 3 α m 2 r 2 + ˜ Q 2 ˜ α 2 r 2 ) dt 2 + dr 2 1 -r g r -2 3 α m 2 r 2 + ˜ Q 2 ˜ α 2 r 2 + r 2 d Ω 2 , (49) \nwhile the Stückelberg fields become \nφ 0 = t + f ( r ) , (50) \nφ r = r + 1 α r. (51) \nIn this reference frame the electromagnetic field is \nE = ˜ Q ˜ αr 2 and B = 0 . (52) \nAnd, for obvious reasons the actual charge should be defined by Q ≡ ˜ Q/ ˜ α .", 'Acknowledgments': "We'd like to thank Gia Dvali, Lam Hui, Mehrdad Mirbabayi, Alberto Nicolis, and David Pirtskhalava for useful comments. The work of LB and GC are supported by the NYU James Arthur and MacCraken Fellowships, respectively. CdR is supported by the Swiss National Science Foundation. GG is supported by NSF grant PHY0758032. AJT would like to thank the Université de Genève for hospitality whilst this work was being completed.", 'A Schwarzschild-like Solution': "In this appendix we concentrate on a different choice of the parameters \nβ = -α 2 8 , (I) \nfor which some exact solutions can also be obtained. In particular, we show that there exists an exact non-singular and asymptotically flat BH solution. We choose the unitary gauge and find that X tt and X tr from eq. (12) vanish identically on the ansatz (16), for w = α/ ( α +2) . Suggesting, that there exist fluctuations which are infinitely strongly coupled. \nThen, the solution to the full set of the equations of motion takes the following form \nds 2 = -( α +2) 3 α 2 ( α +2) 5 + α 5 δ ( 1 -r g ( α +2) rα ) dt 2 ± 2 α ( α +2) ( α +2) 5 + α 5 δ √ r 2 g ( α +2) 6 r 2 + r g α 6 δ r dtdr + α 2 ( α +2) 2 ( 1 + r g ( α +2) 6 αr (( α +2) 5 + α 5 δ ) ) dr 2 + α 2 r 2 ( α +2) 2 d Ω 2 , (II) \nwhere δ and r g are positive integration constants. The transformation to the Schwarzschild coordinates is carried out in a way similar to the previous section \nr → α +2 α r, dt → ζ ( dt + f ' ( r ) dr ) , with ζ 2 ≡ ( α +2) 2 α 2 + α 3 ( α +2) 3 δ. (III) \nAs a result the metric takes the conventional form \nds 2 = -( 1 -r g r ) dt 2 + dr 2 1 -r g r + r 2 d Ω 2 , (IV) \nhowever, this should be accompanied by the 'pion' configuration \nπ = Λ 3 2 [ -(1 -ζ ) t 2 -2 α r 2 ] , (V) \nA 0 = -ζ Λ 3 m f ( r ) , (VI) \nA i = 0 . (VII) \nIt is interesting that there exists a choice of parameters for which one gets the background metric identical to that of GR, supplemented with the 'pion' fields listed above. All the invariants are regular on this solution (away from the singularity in the center). It should be pointed out that (II) is the only background among those given in [21] with vanishing cosmological constant. This solution, however, has infinitely strongly coupled fluctuations; this and related issues will be discussed in a forthcoming paper [38].", 'B Fluctuations': "In this section we study the fluctuations on the backgrounds of section 3. The analysis is done in the decoupling limit [7] \nm → 0 , M pl →∞ , with Λ 3 ≡ M pl m 2 -fixed . (VIII) \nIn this limit the Lagrangian can be decomposed into tow pieces. The first describes the dynamics of the helicity0 , ± 2 modes and their interactions with each other. While the second one accounts for the helicity-± 1 modes and their nonlinear couplings to the helicity-0 degree of freedom. \nThe scalar-tensor Lagrangian, with the condition β = -α 2 / 6 , is given by [11] \nL ST = -1 2 h µν E αβ µν h αβ + h µν ( X (1) µν -α Λ 3 X (2) µν -α 2 6Λ 3 X (3) µν ) . (IX) \nHere, the first term represents the linearized Einstein-Hilbert Lagrangian, while X 's are defined in terms of the longitudinal mode as follows \nX (1) µν = -1 2 /epsilon1 µαρσ /epsilon1 νβρσ Π αβ , X (2) µν = 1 2 /epsilon1 µαργ /epsilon1 νβσγ Π αβ Π ρσ , X (3) µν = /epsilon1 µαργ /epsilon1 νβσδ Π αβ Π ρσ Π γδ , \nwith Π µν ≡ ∂ µ ∂ ν π and all the repeated indices contructed by the flat space metric. The scalar-vector Lagrangian, on the other hand, contains an infinite number of terms and schematically is given by \nL SV = -1 4 F 2 µν + ∞ ∑ n =1 ∂A∂A ( ∂∂π Λ 3 ) n , (X) \nwhere F µν denotes the field strength of the helicity-1 mode, A µ , and the whole Lagrangian is invariant under the U (1) gauge transformation δA µ = ∂ µ α . Remarkably, this expression has been recently resummed for the spherically symmetric ansatz [33], making the analysis of the vector fluctuations possible. \nAfter expanding L ST , and the resummed version of L SV (eq. (C.5) of Ref. [33] which is too lengthy to be reproduced here) to the second order in perturbations around the backgrounds of section 3, we find that the kinetic terms for both helicity0 and helicity-± 1 fields vanish identically, on the dS as well as the Schwarzschild-dS space. This does imply that these modes are infinitely strongly coupled in this limit (unless of course they are rendered nondynamical to all orders by some symmetry or constraint). Whether this problem can be remedied by going beyond the decoupling limit, and/or by invoking kinetic terms due to quantum loops (which will be generated as long as they're not prohibited by symmetries), remains to be seen.", 'References': "- [1] M. Fierz and W. Pauli, Proc. Roy. Soc. Lond. A 173 , 211 (1939).\n- [2] P. van Nieuwenhuizen, Nucl. Phys. B 60 (1973) 478. \n- [3] H. van Dam and M. J. G. Veltman, Nucl. Phys. B 22 , 397 (1970);\n- [4] A. I. Vainshtein, Phys. Lett. B 39 , 393 (1972).\n- [5] C. Deffayet, G. R. Dvali, G. Gabadadze and A. I. Vainshtein, Phys. Rev. D 65 , 044026 (2002) [arXiv:hep-th/0106001].\n- [6] D. G. Boulware and S. Deser, Phys. Rev. D 6 , 3368 (1972).\n- [7] N. Arkani-Hamed, H. Georgi, M. D. Schwartz, Annals Phys. 305 , 96-118 (2003). [hep-th/0210184].\n- [8] P. Creminelli, A. Nicolis, M. Papucci and E. Trincherini, JHEP 0509 , 003 (2005).\n- [9] C. Deffayet and J. W. Rombouts, Phys. Rev. D 72 , 044003 (2005) [arXiv:gr-qc/0505134].\n- [10] G. Gabadadze and A. Gruzinov, Phys. Rev. D 72 , 124007 (2005) [arXiv:hep-th/0312074].\n- [11] C. de Rham, G. Gabadadze, Phys. Rev. D82 , 044020 (2010). [arXiv:1007.0443 [hep-th]].\n- [12] C. de Rham, G. Gabadadze, A. J. Tolley, Phys. Rev. Lett. 106 , 231101 (2010). [arXiv:1011.1232 [hep-th]].\n- [13] S. F. Hassan, R. A. Rosen, [arXiv:1106.3344 [hep-th]].\n- [14] J. Kluson, arXiv:1109.3052 [hep-th].\n- [15] S. F. Hassan, R. A. Rosen, [arXiv:1111.2070 [hep-th]].\n- [16] C. de Rham, G. Gabadadze and A. Tolley, arXiv:1107.3820 [hep-th].\n- [17] C. de Rham, G. Gabadadze and A. J. Tolley, arXiv:1108.4521 [hep-th].\n- [18] W. Siegel, Phys. Rev. D49 , 4144-4153 (1994). [hep-th/9312117].\n- [19] S. L. Dubovsky, JHEP 0410 , 076 (2004) [arXiv:hep-th/0409124].\n- [20] C. Deffayet and T. Jacobson, arXiv:1107.4978 [gr-qc].\n- [21] K. Koyama, G. Niz, G. Tasinato, Phys. Rev. Lett. 107 , 131101 (2011). [arXiv:1103.4708 [hep-th]], K. Koyama, G. Niz, G. Tasinato, Phys. Rev. D84 , 064033 (2011). [arXiv:1104.2143 [hep-th]].\n- [22] T. M. Nieuwenhuizen, Phys. Rev. D 84 , 024038 (2011) [arXiv:1103.5912 [gr-qc]]. \n- [23] A. Gruzinov and M. Mirbabayi, arXiv:1106.2551 [hep-th].\n- [24] C. de Rham, G. Gabadadze, L. Heisenberg, D. Pirtskhalava, Phys. Rev. D83 , 103516 (2011). [arXiv:1010.1780 [hep-th]].\n- [25] S. F. Hassan, R. A. Rosen, JHEP 1107 , 009 (2011). [arXiv:1103.6055 [hep-th]].\n- [26] G. Chkareuli, D. Pirtskhalava, [arXiv:1105.1783 [hep-th]].\n- [27] A. H. Chamseddine, M. S. Volkov, Phys. Lett. B704 , 652-654 (2011). [arXiv:1107.5504 [hep-th]].\n- [28] G. D'Amico, C. de Rham, S. Dubovsky, G. Gabadadze, D. Pirtskhalava and A. J. Tolley, arXiv:1108.5231 [hep-th].\n- [29] A. E. Gumrukcuoglu, C. Lin and S. Mukohyama, arXiv:1109.3845 [hep-th].\n- [30] M. Mohseni, Phys. Rev. D 84 , 064026 (2011) [arXiv:1109.4713 [hep-th]].\n- [31] M.S. Volkov, arXiv:1110.6153 [hep-th].\n- [32] D. Comelli, M. Crisostomi, F. Nesti, L. Pilo, [arXiv:1111.1983 [hep-th]].\n- [33] K. Koyama, G. Niz, G. Tasinato, [arXiv:1110.2618 [hep-th]].\n- [34] L. Hui, A. Nicolis, to appear.\n- [35] S. F. Hassan, R. A. Rosen, [arXiv:1109.3515 [hep-th]].\n- [36] D. Comelli, M. Crisostomi, F. Nesti, L. Pilo, [arXiv:1110.4967 [hep-th]].\n- [37] M. Wyman, Phys. Rev. Lett. 106 , 201102 (2011). [arXiv:1101.1295 [astroph.CO]].\n- [38] In preparation."}
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2017PhRvD..95f4024J
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Hierarchical data-driven approach to fitting numerical relativity data for nonprecessing binary black holes with an application to final spin and radiated energy
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2017-01-01
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Numerical relativity is an essential tool in studying the coalescence of binary black holes (BBHs). It is still computationally prohibitive to cover the BBH parameter space exhaustively, making phenomenological fitting formulas for BBH waveforms and final-state properties important for practical applications. We describe a general hierarchical bottom-up fitting methodology to design and calibrate fits to numerical relativity simulations for the three-dimensional parameter space of quasicircular nonprecessing merging BBHs, spanned by mass ratio and by the individual spin components orthogonal to the orbital plane. Particular attention is paid to incorporating the extreme-mass-ratio limit and to the subdominant unequal-spin effects. As an illustration of the method, we provide two applications, to the final spin and final mass (or equivalently: radiated energy) of the remnant black hole. Fitting to 427 numerical relativity simulations, we obtain results broadly consistent with previously published fits, but improving in overall accuracy and particularly in the approach to extremal limits and for unequal-spin configurations. We also discuss the importance of data quality studies when combining simulations from diverse sources, how detailed error budgets will be necessary for further improvements of these already highly accurate fits, and how this first detailed study of unequal-spin effects helps in choosing the most informative parameters for future numerical relativity runs.
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https://arxiv.org/pdf/1611.00332.pdf
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{'Hierarchical data-driven approach to fitting numerical relativity data for nonprecessing binary black holes with an application to final spin and radiated energy': "Xisco Jim'enez-Forteza, 1, GLYPH<3> David Keitel, 1, 2, y Sascha Husa, 1, z Mark Hannam, 3 Sebastian Khan, 3 and Michael Purrer 3, 4 \n1 Universitat de les Illes Balears, IAC3-IEEC, 07122 Palma de Mallorca, Spain 2 School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom 3 School of Physics and Astronomy, Cardi GLYPH<11> University, The Parade, Cardi GLYPH<11> , CF24 3AA, United Kingdom 4 Albert-Einstein-Institut, Am Muhlenberg 1, D-14476 Potsdam-Golm, Germany \n(Dated: 27 March 2017) \n(LIGO document number: LIGO-P1600270-v5) \nNumerical relativity is an essential tool in studying the coalescence of binary black holes (BBHs). It is still computationally prohibitive to cover the BBH parameter space exhaustively, making phenomenological fitting formulas for BBH waveforms and final-state properties important for practical applications. We describe a general hierarchical bottom-up fitting methodology to design and calibrate fits to numerical relativity simulations for the three-dimensional parameter space of quasicircular nonprecessing merging BBHs, spanned by mass ratio and by the individual spin components orthogonal to the orbital plane. Particular attention is paid to incorporating the extreme-mass-ratio limit and to the subdominant unequal-spin e GLYPH<11> ects. As an illustration of the method, we provide two applications, to the final spin and final mass (or equivalently: radiated energy) of the remnant black hole. Fitting to 427 numerical relativity simulations, we obtain results broadly consistent with previously published fits, but improving in overall accuracy and particularly in the approach to extremal limits and for unequal-spin configurations. We also discuss the importance of data quality studies when combining simulations from diverse sources, how detailed error budgets will be necessary for further improvements of these already highly accurate fits, and how this first detailed study of unequal-spin e GLYPH<11> ects helps in choosing the most informative parameters for future numerical relativity runs.", 'I. INTRODUCTION': "According to general relativity, compact binary black holes (BBHs) coalesce through the emission of gravitational waves (GWs), as already observed by LIGO in at least two cases [14]. The merger remnant is a single Kerr BH [5] characterized only by its final spin and mass. An essential and robust tool for predicting BBH evolution, since the 2005 breakthrough [68], is numerical relativity (NR). Due to the large computational cost of each NR simulation, it is still computationally prohibitive to cover the BBH parameter space exhaustively. Hence it is natural to develop simple, yet accurate model fits to the existing set of NR simulations. One then has to study their quality of interpolation between NR simulations, and of extrapolation to regions of parameter space beyond the calibration range. For example, the phenomenological inspiralmerger-ringdown waveform models of [9-13], though used very successfully as one of two waveform families for LIGO O1 data analysis [1-4], still include only a limited set of physical e GLYPH<11> ects and were calibrated to small sets of NR simulations with relatively simple fitting methods. \nIn this paper we develop a more general, hierarchical fitting approach for BBH properties and waveforms. The fit ansatz functions are developed through studying hierarchical structures present in the NR data set itself, and their complexity tailored to the actual predictive power of the data set by the use of information criteria. \nWe first apply this method to the spin and mass of the remnant black hole, which determine the frequencies of the quasinormal-mode ringdown [14-17] to a Kerr black hole. The ringdown is a crucial part in modeling full inspiralmerger-ringdown waveforms [9-13, 18-20]; and from param- \neter estimation with the full waveforms, the final mass and spin can be estimated with accuracy similar to other BBH parameters [2, 4]. Future observations of strong GW signals will allow to test general relativity through consistency tests between inspiral, merger and ringdown [21], significantly improving upon [22]. \nApart from GW observations, the final state of a BBH merger is astrophysically interesting in itself, e.g. for the computation of merger trees [23-30]. The mass and spin of BHs surrounded by matter, e.g. accretion disks, may also be inferred from electromagnetic observations (see [31, 32] for stellar-mass BHs and [33-35] for supermassive BHs). \nFor this paper, we concentrate on nonprecessing quasicircular BBHs, where the black hole spins are parallel or antiparallel to the total orbital angular momentum of the binary. These configurations are fully described in a threedimensional parameter space: given the masses m 1 ; 2 and physical spins S 1 ; 2, we use the two component spins GLYPH<31> 1 = S 1 = m 2 1 and GLYPH<31> 2 = S 2 = m 2 2 and the mass ratio, given either as q = m 1 = m 2 with the convention m 1 > m 2, or as the symmetric mass ratio GLYPH<17> = ( m 1 m 2) = ( m 1 + m 2) 2 = q = (1 + q ) 2 . The total mass is only a scaling factor, and here we work in units of m 1 + m 2 = 1. \nAs suggested by the post-Newtonian (PN) results [36-39] for radiated energy and angular momentum, and confirmed by many previous studies of NR-calibrated models [9, 18, 40, 41], the two dominant parameter dependencies of both final spin and final mass are on the mass ratio and on some appropriately chosen e GLYPH<11> ective spin parameter of the binary, whereas the e GLYPH<11> ects of any di GLYPH<11> erence between the two individual spins are much smaller. However, only by accurately modeling these small unequal-spin e GLYPH<11> ects can the full two-spin information be extracted from GW observations, disentangling any true physical degeneracies from systematic e GLYPH<11> ects due to limitations of the waveform models. \nA similar, but more complex, situation is encountered in the calibration of phenomenological models to precessing binaries, where so far a simple waveform model based on a sin- \nFIG. 1. Flowchart of the hierarchical step-by-step construction leading to a three-dimensional ansatz and fit for the quantity of interest over the ( GLYPH<17>; GLYPH<31> 1 ; GLYPH<31> 2) GLYPH<17> GLYPH<16> GLYPH<17>; b S ; GLYPH<1> GLYPH<31> GLYPH<17> space. \n<!-- image --> \ngle e GLYPH<11> ective precession parameter [11-13] has successfully been employed to analyze the first gravitational wave detections [2, 4]. (A complementary precessing analysis [42] based on the model of [20, 43] has also been published recently.) The current work, while mainly a first step to develop models for the full three-dimensional parameter space of nonprecessing waveforms, can also be considered as a toy model for the numerical calibration of subdominant e GLYPH<11> ects in generic binaries, such as precession and higher modes. \nFor the current three-dimensional aligned-spin parameter space, in addition to the hierarchy of mass ratio, e GLYPH<11> ective spin and spin di GLYPH<11> erence, the sampling by available NR simulations still displays significant bias toward simple subsets: namely the one-dimensional subspaces of nonspinning cases ( GLYPH<17> dependence only) and equal-mass, equal-spin cases ( GLYPH<17> = 0 : 25, q = 1, GLYPH<31> 1 = GLYPH<31> 2, thus e GLYPH<11> ective-spin dependence only) are covered particularly well by existing NR catalogs, while few simulations exist for unequal spins, high spins, and / or very unequal masses. Independently, the extreme-mass-ratio limit ( GLYPH<17> ! 0, q !1 , arbitrary spins) is known analytically [44]. \nHere we exploit and investigate this structure by parametrizing spin e GLYPH<11> ects in terms of an e GLYPH<11> ective spin and a spin-di GLYPH<11> erence parameter. As the e GLYPH<11> ective spin we choose \nb S = S m 2 1 + m 2 2 ; with S = m 2 1 GLYPH<31> 1 + m 2 2 GLYPH<31> 2 ; (1) \nwhich has already been found to work well for final-state quantities in [9]. We discuss other possible choices for the effective spin, for which our method also works robustly, in Appendix C. For spin di GLYPH<11> erence, we use simply GLYPH<1> GLYPH<31> = GLYPH<31> 1 GLYPH<0> GLYPH<31> 2, which makes no assumptions on how spin-di GLYPH<11> erence e GLYPH<11> ects depend on mass ratio. \nWe develop our hierarchical approach, with the aim to ensure an accurate modeling of the subdominant spin-di GLYPH<11> erence e GLYPH<11> ects, along the lines illustrated as a flowchart in Fig. 1: First we consider the one-dimensional subspaces of nonspinning and of equal-mass-equal-spin black holes. We then combine and generalize these subspace fits, adding additional degrees of freedom to cover the entire two-dimensional space of equal-spin black holes, but constrain the generalized ansatz with information from the extreme-mass-ratio limit. In a third step, we investigate the leading subdominant terms, which are \nFIG. 2. The NR data set used in this paper, over mass ratio q = m 1 = m 2 and the two dimensionless spin components GLYPH<31> 1, GLYPH<31> 2, with color indicating the source catalog. Bright green points are cases removed from the analysis for data-quality reasons, as discussed in Appendix A. \n<!-- image --> \nlinear in the di GLYPH<11> erence between spins, and also identify additional nonlinear spin-di GLYPH<11> erence terms. We finally produce a three-dimensional fit to the complete data set with the hierarchically constructed ansatz. This way, we can construct a full ansatz with a relatively low number of free fit coe GLYPH<14> cients and avoid overfitting of spurious e GLYPH<11> ects only due to small sample sizes, while still capturing the essential physical e GLYPH<11> ects that are known from the well-constrained regions. \nAt each step, we evaluate the performance of di GLYPH<11> erent fit choices by several quantitative measures: by the overall residuals, by the Akaike and Bayesian information criteria (AICc, BIC, [45, 46]), and by how well determined the individual fit coe GLYPH<14> cients are. The information criteria are model selection tools to choose between fits with comparable goodness of fit but di GLYPH<11> erent degrees of complexity, i.e. they penalize high numbers of free coe GLYPH<14> cients. See Appendix B for details on these statistical methods. \nPrevious published fits for final spin and / or mass include [9, 27, 47-55], and we will compare our results to the most recent results in the literature, including both fits across the full three-dimensional nonprecessing parameter space, and the e GLYPH<11> ective-spin-based fits from [9], which were used in the aligned-spin IMRPhenomD waveform model [9, 10] and (with in-plane-spin corrections) in the precessing IMRPhenomPv2 model [11-13]. These will be referred to as the 'PhenomD fits' in the following. The plan of the paper is as follows: We first describe our data sets, built from several NR catalogs, in Sec. II, together with the available extreme-massratio limit information. In Sec. III we develop the general fitting recipe for the example of the final spin. We then apply our method also for final mass - or equivalently, radiated energy - in Sec. IV, which illustrates some of the specific choices and adaptations required to apply the general method to each quantity. We summarize our method and results in Sec. V, and give additional details about NR data, fit construction, and fit uncertainty estimates in Appendices A-D. \nOur fits for final spin and radiated energy are also provided in Mathematica and python formats as supplementary material [56], and implemented under the label 'UIB2016' in the nrutils.py package of LALInference [57, 58]. This final version of the paper uses a larger NR calibration set than the initial arXiv submission (1611.00332v1), with fit results fully consistent but better constrained. \nFIG. 3. Input data plotted against symmetric mass ratio GLYPH<17> and e GLYPH<11> ective spin b S . Data consist of the combined set of NR simulations (colored points) and the analytically known [44] extreme-mass-ratio behavior (black line). Left panel: final spin; right panel: radiated energy rescaled by GLYPH<17> . Both GLYPH<31> f and E rad =GLYPH<17> follow a smooth surface in this space, and the well-constrained 1D subspaces together already give a good indication of its curvature. \n<!-- image -->", 'A. Numerical Relativity data sets': 'We combine four data sets of aligned-spin numerical relativity BBH simulations from independent codes and sources: the public catalogs of SXS [59, 60], RIT [52, 61] and GaTech [62, 63]; as well as a set of our own simulations with the BAM code [9, 64, 65], including 27 new cases for which initial configurations and results are listed in Table XIV in Appendix A. After removing 16 cases from the combined data set due to data quality considerations as discussed in Appendix A, we have 161 cases from the SXS catalog, 107 from RIT, 114 from GaTech and 45 from BAM ; for a total of 427 cases. The sampling of our three-dimensional parameter space by the four data sets is shown in Fig. 2. The initial arXiv version of this paper (1611.00332v1) used a smaller data set of 256 NR cases, with the increase coming from an update of the public SXS catalog and new RIT results from [61]. \nTo obtain a qualitative understanding of the hierarchical structure in the two-dimensional parameter space of mass ratio and e GLYPH<11> ective spin, in Fig. 3 we show the NR data set over the ( GLYPH<17>; b S ) plane together with the analytical extrememass-ratio results, discussed below in Sec. II B. For both final spin and radiated energy, we find a reasonably smooth surface spanned by the NR data points. These plots already suggests that - together with the known extreme-mass-ratio results to compensate the sparsity of NR simulations at increasingly unequal masses - good one-dimensional fits in the two best-sampled one-dimensional subsets (equal-massequal-spin and nonspinning BHs) will significantly constrain any two-dimensional fits. \nFurthermore, as a first quantitative check that the assumption of a hierarchical structure in three-dimensional BBH parameter space holds, with unequal-spin e GLYPH<11> ects subdominant to the dependence on GLYPH<17> and b S , we can study the residuals of this data set under the two-dimensional PhenomD fits [9]. For final spin, we find that 90% of relative errors are below 3%. (The only cases over 10% are those with absolute values close to zero, where relative error is not a good measure, and absolute errors (residuals) are limited to GLYPH<1> GLYPH<31> f GLYPH<20> 0 : 025.) Still, this com- \nFIG. 4. Relative errors in final spin of the combined NR data set for this paper under the two-dimensional PhenomD fit [9]. \n<!-- image --> \nparison suggests that unequal-spin e GLYPH<11> ects make a large contribution to these small errors, as shown by four times smaller 90% quantiles when restricting to equal-spin cases only. See also Fig. 4 for histograms of these distributions. For radiated energy, 90% of relative errors are below 2%, with a reduction of that quantile by 1.4 for equal-spin cases only, indicating that spin-di GLYPH<11> erence e GLYPH<11> ects are even smaller for this quantity, which we will also see confirmed in our final results. \nFor details about extraction of final-state quantities, NR data quality and weight assignment, see Appendix A. As explained there, we do not have a full set of NR error estimates available, so we assign heuristic fit weights to each case based on the expected accuracy of the respective NR code in that particular parameter space region. For example, high-massratio cases are down-weighted more for puncture codes.', 'B. Extreme-mass-ratio limit': 'The computational cost of numerical simulations of BH binaries in full general relativity diverges in the extreme-massratio limit GLYPH<17> ! 0. However, this limit is also equivalent to the much simpler case of a test particle orbiting a Kerr black hole. The energy and orbital angular momentum for that configuration have long been known analytically [44]: inserting the radius of the innermost stable circular orbit (ISCO) from Eq. (2.21) of [44] into Eqs. (2.12) and (2.13) of the same reference yields the test-particle energy (equivalent to the radiated \nenergy) and orbital angular momentum at ISCO: \nE ISCO( GLYPH<17>; GLYPH<31> ) = GLYPH<17> 0 B B B B B B @ 1 GLYPH<0> s 1 GLYPH<0> 2 3 GLYPH<26> ISCO( GLYPH<31> f ) 1 C C C C C C A ; (2)a \nL orb ; ISCO( GLYPH<17>; GLYPH<31> ) = 2 GLYPH<17> GLYPH<16> 3 p GLYPH<26> ISCO( GLYPH<31> ) GLYPH<0> 2 GLYPH<31> GLYPH<17> p 3 GLYPH<26> ISCO( GLYPH<31> ) ; (2)b \nwith \nGLYPH<26> ISCO( GLYPH<31> ) = 3 + Z 2 GLYPH<0> sgn( GLYPH<31> ) p (3 GLYPH<0> Z 1)(3 + Z 1 + 2 Z 2) ; (3)a Z 1( GLYPH<31> ) = 1 + (1 GLYPH<0> GLYPH<31> 2 ) 1 = 3 h (1 + GLYPH<31> ) 1 = 3 + (1 GLYPH<0> GLYPH<31> ) 1 = 3 i ; (3)b Z 2( GLYPH<31> ) = q 3 GLYPH<31> 2 + Z 2 1 : (3)c \nNote that both E ISCO and L orb ; ISCO depend linearly on GLYPH<17> . \n; In the test-particle limit, the small BH plunges after reaching the ISCO, and further mass loss scales with GLYPH<17> 2 [66]. Similar to previous work [47, 52, 55, 67], we will exploit this fact to compute the final spin and radiated energy to linear order in GLYPH<17> from the analytical expressions, Eq. (2), holding at the ISCO. To linear order in GLYPH<17> , we thus simply have E rad = E ISCO or M f = 1 GLYPH<0> E ISCO for the final mass, and for the final spin GLYPH<31> f we obtain the implicit equation \nGLYPH<31> f M f ( GLYPH<17>; GLYPH<31> f ) 2 = L orb ; ISCO( GLYPH<17>; GLYPH<31> f ) + S 1 + S 2 ; (4) \nwhere the individual BH spins can be written in terms of our e GLYPH<11> ective spin as \nS 1 + S 2 = (1 GLYPH<0> 2 GLYPH<17> ) b S : (5) \nEquation (4) can then be solved numerically for the final spin GLYPH<31> f as a function of GLYPH<17> and of the e GLYPH<11> ective spin b S . Since this result holds to linear order in GLYPH<17> , and assuming that the final spin and mass are regular functions of GLYPH<17> , we have thus essentially computed the derivatives @ E rad =@GLYPH<17> and @GLYPH<31> f =@GLYPH<17> at GLYPH<17> = 0, in addition to the values at GLYPH<17> = 0, which are E rad(0) = 0 and GLYPH<31> f (0) = S 1 = M 2 . \nAdditionally, assuming that the final state is indeed a Kerr BH, its final spin has to satisfy GLYPH<31> f GLYPH<20> 1. One would also expect the final spin for maximal e GLYPH<11> ective spin, b S = 1, to decrease monotonically with increasing GLYPH<17> . To construct an accurate fit in a neighborhood of b S ! 1 that satisfies these expectations - in particular the Kerr limit - we will constrain our ansatz with the analytically computed value of GLYPH<31> 0 f = @GLYPH<31> f =@GLYPH<17> at ( GLYPH<17> = 0 ; b S = 1). By perturbing Eq. (4) around f GLYPH<17> ! 0 ; GLYPH<31> f ! 1 g to linear order before taking the derivative in GLYPH<17> at the same point, we find \nGLYPH<31> 0 f GLYPH<16> GLYPH<17> ! 0 ; b S ! 1 GLYPH<17> = 0 : (6) \nSeveral variations of this procedure have been used for previous final-spin fits, and di GLYPH<11> erences are due to previous works neglecting the radiated energy in Eq. (4) [47, 52], or not enforcing the derivative for satisfying the Kerr limit [55].', 'III. FINAL SPIN': 'We will now first develop the details of our hierarchical fitting procedure for the example of the final spin of BBH merger remnants, giving more detail here than we will do for the radiated energy in Sec. IV.', 'A. Choice of fit quantity': "We first need to decide which quantity exactly we want to fit. It appears natural to fit a quantity related to the 'final' orbital angular momentum L orb near merger, i.e. separating out the known initial spins Si . This is particularly useful in connection with the extreme-mass-ratio limit, since with Eq. (2)b, L orb is linear in GLYPH<17> to leading order. We can use the relation from Eq. (4) between L orb and the dimensionless Kerr parameter GLYPH<31> f of the remnant BH, M 2 f GLYPH<31> f = L orb + S 1 + S 2 = L orb + S , also outside the extreme-mass-ratio limit. Here M f is the final mass of the remnant BH. \nInstead of the actual angular momentum L orb, we take the liberty of fitting the quantity L 0 orb = M 2 GLYPH<31> f GLYPH<0> S , where (as throughout the paper) M is set to unity. This way, all fit results are easily converted to the final Kerr parameter GLYPH<31> f by adding the total initial spin S , and no correction for radiated energy has to be applied.", 'B. One-dimensional subspace fits': "Motivated by the the unequal sampling of the parameter space by NR simulations, as visualized in Fig. 3, we start our hierarchical fit development with the simplest and bestsampled subspaces of the NR data set, constructing onedimensional fits L 0 orb GLYPH<16> GLYPH<17>; b S = 0 GLYPH<17> and L 0 orb GLYPH<16> GLYPH<17> = 0 : 25 ; b S GLYPH<17> over 92 nonspinning and 37 equal-mass-equal-spin cases. We do not restrict ourselves to polynomial fits, and also include ansatze in the form of rational functions. We have also found good fits for more general functions, but omit these here since we have not explored that option systematically. \nAll fits are performed with Mathematica's NonlinearModelFit function, but also partially crosschecked with the nls package of R. Since rational functions can have singularities, codes such as NonlinearModelFit may not converge to a valid solution without good starting values for the coe GLYPH<14> cients. We solve this problem by first performing a su GLYPH<14> ciently high-order polynomial fit, from which we compute a Pad'e approximant at the desired order, and use the coe GLYPH<14> cients of this approximant as starting values for the rational-function fit. We denote rational functions with a numerator of polynomial order m and denominator of polynomial order k as an ansatz of order ( m ; k ). Before fitting to the NR data, all ansatze are constrained by two facts: For nonspinning BBHs, both GLYPH<31> f and L 0 orb have to vanish for GLYPH<17> ! 0, so that there can be no constant term in the ansatz. Furthermore, from the extreme-mass-ratio prediction Eq. (2), it also follows that the spin-independent coe GLYPH<14> cient linear in GLYPH<17> is 2 p 3 (see also [68]). We will include spin-dependent information linear in GLYPH<17> in Sec. III C. \nThus, we obtain L 0 orb GLYPH<16> GLYPH<17>; b S = 0 GLYPH<17> fits for a large set of polynomial and rational functions. Several of them produce competitive goodness of fit, as measured by the root-mean-squareerror (RMSE) or the full distribution of residuals. However, we do not want to overfit the data, which could induce spurious oscillations in the region of very unequal BH masses that is not covered by NR data. Hence, we rank the fits by information criteria penalizing superfluous free coe GLYPH<14> cients. \nFigure 5 shows the top-ranked fit in terms of Schwarz's Bayesian information criterion (BIC), which is a rational func- \n<!-- image --> \nFIG. 5. Nonspinning NR data and one-dimensional L 0 orb GLYPH<16> GLYPH<17>; b S = 0 GLYPH<17> fit as a function of mass ratio GLYPH<17> . Top panel: best fit in terms of the Bayesian information criterion (BIC), a rational function R(3,1), see Eq. (7). Lower panel: residuals ( GLYPH<1> L 0 orb = data GLYPH<0> fit) of this fit (points) and di GLYPH<11> erences from the three next-best-ranking fits in terms of BIC (lines). See also Fig. 29 in Appendix B for an illustration of BIC ranking for this example. \n<!-- image --> \nTABLE I. Fit coe GLYPH<14> cients for the one-dimensional nonspinning L 0 orb GLYPH<16> GLYPH<17>; b S = 0 GLYPH<17> fit over 92 NR cases, along with their uncertainties (standard errors) and relative errors (Std.err. / estimate). \n| Estimate Standard error Relative error [%] | Estimate Standard error Relative error [%] | Estimate Standard error Relative error [%] | Estimate Standard error Relative error [%] |\n|----------------------------------------------|----------------------------------------------|----------------------------------------------|----------------------------------------------|\n| a 2 | 3 : 833 0 : | 085 | 2 : 2 |\n| a 3 | GLYPH<0> 9 : 49 | 0 : 24 | 2 : 5 |\n| a 5 | 2 : 513 | 0 : 046 | 1 : 8 | \ntion of order (3,1): \nL 0 orb GLYPH<16> GLYPH<17>; b S = 0 GLYPH<17> = 1 : 3 a 3 GLYPH<17> 3 + 5 : 24 a 2 GLYPH<17> 2 + 2 p 3 GLYPH<17> 2 : 88 a 5 GLYPH<17> + 1 : (7) \nThe fit coe GLYPH<14> cients ai along with their uncertainties are given in Table I; all are well determined. The exact ranking of fits can depend on the choice of fit weights (see Appendix A) and on the ranking criterion, but we find that Eq. (7) is topranked by both BIC and AICc. While only ranked 6th by RMSE, none of the considered fits is better than Eq. (7) by more than 6% in that metric either. Additionally, under variations of the weighting scheme, this is robustly the fit among the top-ranked group - by all three criteria - with the lowest number of fitting coe GLYPH<14> cients, indicating it is a robust choice. For comparison, a simple third-order polynomial (two free coe GLYPH<14> cients) is disfavored clearly, by more than a factor of 8 in RMSE and an o GLYPH<11> set of + 452 in BIC, and a fourth-order polynomial (three free coe GLYPH<14> cients, just as Eq. (7)) by almost a factor of 2 and by + 332 respectively. \nThe lower panel of Fig. 5 also compares the preferred fit both to the NR data and to the three next-best ranking fits by BIC. We find that the residuals are centered around zero with \nFIG. 6. Equal-mass-equal-spin NR data and one-dimensional fits of L 0 orb GLYPH<16> GLYPH<17> = 0 : 25 ; b S GLYPH<17> as a function of e GLYPH<11> ective spin b S . Top panel: best fit in terms of BIC, a rational function R(3,1), see Eq. (8). Lower panel: residuals of this fit (points) and di GLYPH<11> erences from the three next-bestranking fits in terms of BIC (lines).TABLE II. Fit coe GLYPH<14> cients for the one-dimensional equal-massequal-spin L 0 orb GLYPH<16> GLYPH<17> = 0 : 25 ; b S GLYPH<17> fit over 37 NR cases. \n| | | Estimate Standard error Relative error [%] | |\n|-----|-----------|----------------------------------------------|--------|\n| b 1 | 1 : 00096 | 0 : 00068 | 0 : 1 |\n| b 2 | 0 : 788 | 0 : 042 | 5 : 3 |\n| b 3 | 0 : 654 | 0 : 074 | 11 : 4 |\n| b 5 | 0 : 840 | 0 : 030 | 3 : 6 | \nno major trends, while the di GLYPH<11> erences among high-ranked fits are much smaller than the scatter of residuals for the wellcovered highGLYPH<17> range, and that the 'systematic uncertainty', as indicated by the di GLYPH<11> erence of high-ranked fits, is still at the same level even in the extrapolatory lowGLYPH<17> region. The BIC ranking for this example is also illustrated in Fig. 29 in Appendix B. \nBefore the second 1D fit in the e GLYPH<11> ective spin parameter b S , we need to decide how we are later going to construct a 2D ansatz combining both 1D fits. We can either subtract or divide the NR data by the nonspinning fit, and find that subtraction exhibits a simpler functional form. Thus we decide to construct a 2D ansatz as the sum of nonspinning and spinning contributions. (We will choose a product ansatz for the radiated energy discussed in the next section.) We thus constrain the constant term of the 1D ansatz in b S to reproduce the GLYPH<17> = 0 : 25 nonspinning result, i.e. L 0 orb ( GLYPH<17> = 0 : 25 ; b S = 0) must be identical for both 1D fits. \nThe top-ranked L 0 orb GLYPH<16> GLYPH<17> = 0 : 25 ; b S GLYPH<17> fits by BIC are shown in Fig. 6, and again we find a unique top-ranked fit by both AICc and BIC, a rational function of order (3,1): \nL 0 orb GLYPH<16> GLYPH<17> = 0 : 25 ; b S GLYPH<17> = 0 : 00954 b 3 b S 3 + 0 : 0851 b 2 b S 2 GLYPH<0> 0 : 194 b 1 b S 1 GLYPH<0> 0 : 579 b 5 b S + 0 : 68637 ; \nFIG. 8. Two-dimensional L 0 orb GLYPH<16> GLYPH<17>; b S GLYPH<17> fit, visualized as L 0 orb GLYPH<16> GLYPH<17>; b S GLYPH<17> =GLYPH<17> . Application of the extreme-mass-ratio limit helps in avoiding extrapolation artifacts which would otherwise appear in lowGLYPH<17> , highj b S j regions that are uncovered by NR simulations. \n<!-- image --> \nFIG. 7. Extreme-mass-ratio comparison of the rescaled final spin: analytical results from solving Eq. (4), the previous PhenomD finalspin fit of [9], and this work. \n<!-- image --> \nTABLE III. Fit coe GLYPH<14> cients for the extreme-mass-ratio limit of the final spin, fitted to discretized analytical results. The fourth coefficient, f 11, is fixed by the derivative constraint in Eq. (13) and its estimate and error computed from the others. \n| | | Estimate Standard error Relative error [%] |\n|------|-------------------|----------------------------------------------|\n| f 21 | 8 : 774 | 0 : 019 0 : 2 |\n| f 31 | 22 : 83 | 0 : 27 1 : 2 |\n| f 50 | 1 : 8805 0 : 0025 | 0 : 1 |\n| f 11 | 4 : 4092 | 0 : 0047 0 : 1 | \nwith four fit coe GLYPH<14> cients bi as listed in Table II. This ansatz is ranked 8th by RMSE, but with only 3% di GLYPH<11> erence from the lowest RMSE, which is attained by a P(5) fit with one more coe GLYPH<14> cients, marginally disfavored by about + 1.7 AICc and + 2.6 in BIC. The best three-coe GLYPH<14> cient fit R(2,1) is significantly worse, with di GLYPH<11> erences of over + 40 in BIC / AICc and 40% in RMSE. Again, the distribution of residuals is well behaved, and di GLYPH<11> erences between the four top-ranked fits by BIC are smaller than the scatter of residuals.", 'C. Two-dimensional fits': 'Next, we want to construct a two-dimensional fit covering the ( GLYPH<17>; b S ) space, as it was illustrated in Fig. 3, by combining both the 1D subspace fits and the extreme-mass-ratio limit. As discussed above, we take the sum of Eq. (7) and the spindependent terms of Eq. (8). We introduce the necessary flexibility to describe 2D curvature and the extreme-mass-ratio limit by generalizing the b S -dependent terms, inserting a polynomial of order J in GLYPH<17> for each bi through the substitution \nbi ! bi j = J X j = 0 fi j GLYPH<17> j : (9) \nThe general 2D ansatz is thus \nL 0 orb GLYPH<16> GLYPH<17>; b S GLYPH<17> = L 0 orb ( GLYPH<17>; 0) + L 0 orb GLYPH<16> 0 : 25 ; b S ; fi j GLYPH<17> GLYPH<0> L 0 orb (0 : 25 ; 0) : (10) \nHere we choose to expand to third order in GLYPH<17> ( J = 3), which is the lowest order leaving enough freedom to incorporate all available constraints from the 1D fits and the extreme-massratio limit, and, as evidenced by the residuals we find below, \nalso high enough to adequately model this data set. Of the resulting 16 coe GLYPH<14> cients, the three fi 0 in the numerator must vanish to preserve the L 0 orb GLYPH<16> GLYPH<17> = 0 ; b S GLYPH<17> = 0 limit, while consistency with the equal-mass fit from Eq. (8) provides four constraints which we use to fix the fi 3 terms: \nfi 3 = 64 GLYPH<0> 64 fi 0 GLYPH<0> 16 fi 1 GLYPH<0> 4 fi 2 : (11) \nFour more coe GLYPH<14> cients are fixed by the extreme-mass-ratio information discussed in Sec. II B: we re-express Eq. (4) in terms of L 0 orb =GLYPH<17> and fit the discretized quantity \nlim GLYPH<17> ! 0 L 0 orb GLYPH<16> GLYPH<17>; b S GLYPH<17> GLYPH<17> GLYPH<0> 2 p 3 = lim GLYPH<17> ! 0 GLYPH<31> f ( GLYPH<17>; S ) GLYPH<0> S GLYPH<17> GLYPH<0> 2 p 3 (12) \nwhere 2 p 3 is the linear contribution from the nonspinning part (cf. Eq. (7)) and the GLYPH<31> f ( GLYPH<17> ! 0 ; S ) values are obtained by solving Eq. (4) numerically for small GLYPH<17> . Before fitting, we apply the derivative constraint from Eq. (6), which for the sum ansatz Eq. (10) implies a coe GLYPH<14> cient constraint \nf 11 ! 0 : 345225 f 21 + 0 : 0321306 f 31 GLYPH<0> 3 : 66556 f 50 + 7 : 5397 : (13) \nWe find this extra physical constraint to be essential in avoiding superextremal GLYPH<31> f results due to fitting artifacts. The extreme-mass-ratio limit fit coe GLYPH<14> cients are listed in Table III, and the improved agreement between analytical results and this new fit, as compared with the previous fit of [9], is illustrated in Fig. 7. \nIn summary, after constraining to the well-covered onedimensional NR data subsets and the analytically known extreme-mass-ratio limit, the 2D ansatz from Eq. (10) has reduced from 16 to 5 free coe GLYPH<14> cients. We fit it to the 60 remaining equal-spin NR cases that were not yet included in the 1D subsets. To remove possible singularities in the GLYPH<16> GLYPH<17>; b S GLYPH<17> plane for this rather general rational ansatz, we set the least-constrained denominator coe GLYPH<14> cient f 52 to zero. Thus we obtain a smooth four-coe GLYPH<14> cient fit as shown in Fig. 8. It has RMSE four times larger than the 1D GLYPH<17> fit and twice as \n<!-- image --> \n<!-- image --> \nFIG. 9. Examples of spin-di GLYPH<11> erence behavior at fixed mass ratios, for residuals GLYPH<1> L 0 orb after subtracting the two-dimensional L 0 orb GLYPH<16> GLYPH<17>; b S GLYPH<17> fit, \n<!-- image --> \nas defined in Eq. (14). Top row: q = 1; lower row: q = 4; left column: surfaces in GLYPH<16> b S ; GLYPH<1> GLYPH<31>; GLYPH<1> L 0 orb GLYPH<17> space; right column: projections onto the GLYPH<1> GLYPH<31> axis with linear and quadratic fits. At equal mass, the surface is parabolic, with the linear term (blue line) and mixture term (not shown) vanishing, but a clear quadratic dependence (orange line). At q = 4 and other intermediate mass ratios, the surface is very close to flat and the linear term dominates. \nhigh as the b S fit, but these residuals are still smaller than expected unequal-spin e GLYPH<11> ects and rather noisily distributed without any clear parameter-dependent trends, thus indicating that the 2D fit su GLYPH<14> ciently captures the dominant GLYPH<16> GLYPH<17>; b S GLYPH<17> dependence to form the basis of studying subdominant spin-di GLYPH<11> erence effects. The least-constrained coe GLYPH<14> cient at this point has a relative error of about 25%, which is good enough to keep it in the ansatz for the next, 3D step where we will refit to a larger data set.', 'D. Unequal-spin contributions and 3D fit': "Nowthe final step in the hierarchical procedure is to explore the subdominant e GLYPH<11> ects of unequal spins, parametrized by the spin di GLYPH<11> erence GLYPH<1> GLYPH<31> = GLYPH<31> 1 GLYPH<0> GLYPH<31> 2. We first study the residuals of the 238 unequal-spin NR cases under the equal-spin 2D fit: \nGLYPH<1> L 0 orb GLYPH<16> GLYPH<17>; b S ; GLYPH<1> GLYPH<31> GLYPH<17> : = L 0 orb ; NR GLYPH<16> GLYPH<17>; b S ; GLYPH<1> GLYPH<31> GLYPH<17> GLYPH<0> L 0 orb j eqSpinFit GLYPH<16> GLYPH<17>; b S GLYPH<17> : (14) \nWe do this at fixed steps in mass ratio, having su GLYPH<14> -cient numbers of NR cases for this analysis at mass ratios q = f 1 ; 1 : 33 ; 1 : 5 ; 1 : 75 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 g . This per-massratio analysis is only used to guide the construction of the full 3D ansatz and as a consistency check, while the final full 3D fit will consist of fitting the constrained 2D ansatz plus spindi GLYPH<11> erence terms directly to the full data set. \nAt each mass ratio, we visually inspect the residuals, which span 2D surfaces in GLYPH<16> GLYPH<31> 1 ; GLYPH<31> 2 ; L 0 orb GLYPH<17> or, equivalently, GLYPH<16> b S ; GLYPH<1> GLYPH<31>; L 0 orb GLYPH<17> \nspace. As illustrated in Fig. 9, we find surfaces close to a plane, indicating a dominant linear dependence on GLYPH<1> GLYPH<31> and possibly a mixture term b S GLYPH<1> GLYPH<31> . The exception is at equal masses, where quadratic curvature in the GLYPH<1> GLYPH<31> dimension dominates. In this case, exchange of GLYPH<31> 1 and GLYPH<31> 2 yields an identical binary configuration, so that terms linear in GLYPH<1> GLYPH<31> indeed have to vanish. We have also exploited this fact in the q = 1 analysis by adding mirror duplicates of each NR data point. Motivated by these empirical findings and symmetry argument, we introduce up to three spin-di GLYPH<11> erence terms, \nGLYPH<1> L 0 orb GLYPH<16> GLYPH<17>; b S ; GLYPH<1> GLYPH<31> GLYPH<17> = A 1( GLYPH<17> ) GLYPH<1> GLYPH<31> + A 2( GLYPH<17> ) GLYPH<1> GLYPH<31> 2 + A 3( GLYPH<17> ) b S GLYPH<1> GLYPH<31> : (15) \nThe full 3D ansatz is then simply the sum of Eqs. (10) and (15): \nL 0 orb GLYPH<16> GLYPH<17>; b S ; GLYPH<1> GLYPH<31> GLYPH<17> = L 0 orb GLYPH<16> GLYPH<17>; b S GLYPH<17> + GLYPH<1> L 0 orb GLYPH<16> GLYPH<17>; b S ; GLYPH<1> GLYPH<31> GLYPH<17> : (16) \nAdding higher orders in the e GLYPH<11> ective spin or spin di GLYPH<11> erence is not supported by visual inspection. At each mass ratio, we now perform four fits in GLYPH<1> GLYPH<31> for the values of the Ai : linear, linear + quadratic, linear + mixed, or the sum of all three terms. Examples are also shown in Fig. 9. \nWe then collect the coe GLYPH<14> cients of each of these fits and use them as data Ai ( GLYPH<17> ) to be fitted as functions of mass ratio (see the 'per-mass-ratio data' in Fig. 10), using as weights the fit uncertainty from each mass ratio rescaled by the average data weight for that mass ratio. We also apply what we know about the extreme-mass-ratio and equal-mass limits: all three Ai ( GLYPH<17> ) \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFIG. 10. Spin-di GLYPH<11> erence behavior of final-spin data after subtraction of the two-dimensional L 0 orb GLYPH<16> GLYPH<17>; b S GLYPH<17> fit, showing the results of fits as in Fig. 9 at GLYPH<17> steps corresponding to q = f 1 ; 1 : 33 ; 1 : 5 ; 1 : 75 ; 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 g and three estimates for the three ansatz functions Ai ( GLYPH<17> ) from Eqs. (15) and (19): (i) unequal-spin part of the final 3D fit from Eq. (16) ('direct 3D fit'), (ii) fit of the unequal-spin terms from Eq. (19) ('fit to residuals') to the residuals of the 2D fit from Eq. (10) over all mass ratios, (iii) fits of Eq. (19) to the per-mass-ratio results. Top-left panel: linear term A 1 only. The remaining panels are for the combined linear + quadratic + mixture fit, in clockwise order: linear term A 1, quadratic term A 2 and mixture term A 3. The A 1 results from the combined fit are very similar to those from the linear-only fit, demonstrating the robustness of extracting leading-order spin-di GLYPH<11> erence e GLYPH<11> ects. For the two lower panels, data points for low GLYPH<17> are outside the displayed range, but the error bars are huge and hence this region does not contribute significantly to the weighted per-mass-ratio fits. In the direct 3D fit to the full data set, however, lowGLYPH<17> information can be better incorporated, leading to the somewhat di GLYPH<11> erent shape of the mixture-term fit. See Sec. III E for more discussion of how well constrained these shapes actually are with the current data set. \n<!-- image --> \nhave to vanish in the limit GLYPH<17> = 0, and the A 1, A 3 linear in GLYPH<1> GLYPH<31> have to vanish for GLYPH<17> = 0 : 25. We thus choose ansatze of the form \nAi = di 0 GLYPH<17> pi GLYPH<16> p 1 GLYPH<0> 4 GLYPH<17> GLYPH<17> qi (1 + di 1 GLYPH<17> ) (17) \nfor Ai = 1 ; 3 linear in GLYPH<1> GLYPH<31> , where the factor GLYPH<16> p 1 GLYPH<0> 4 GLYPH<17> GLYPH<17> qi is motivated from post-Newtonian (PN) results [38, 39], and \nA 2 = d 20 GLYPH<17> p 2 GLYPH<16> 1 + d 21 GLYPH<16> p 1 GLYPH<0> 4 GLYPH<17> GLYPH<17> q 2 GLYPH<17> (18) \nfor the term quadratic in GLYPH<1> GLYPH<31> . We find that the data can be well fit without any higher-order terms and by reducing some of the freedom of these three terms exploratory fits keeping all coe GLYPH<14> cients free give results close to integer numbers for the pi , qi = 1 and d 21 = 0. Hence we choose the three parsimonious ansatze \nA 1( GLYPH<17> ) = d 10(1 GLYPH<0> 4 GLYPH<17> ) 0 : 5 GLYPH<17> 2 ( d 11 GLYPH<17> + 1) (19)a \nA 2( GLYPH<17> ) = d 20 GLYPH<17> 3 (19)b \nA 3( GLYPH<17> ) = d 30(1 GLYPH<0> 4 GLYPH<17> ) 0 : 5 GLYPH<17> 3 ( d 31 GLYPH<17> + 1) : (19)c \nThe blue points and lines in Fig. 10 show these per-massratio results. The shape and numerical results of the dominant linear term A 1 are quite stable under adding one or two of the \nother terms. Fitting two terms, either linear + quadratic or linear + mixture, yields quadratic / mixture e GLYPH<11> ects of very similar magnitude, with the quadratic term following the same basic shape (an intermediate-mass-ratio bulge) as the other two. However, combining all three terms, the results match better with the expectations from symmetry detailed before, with the bulge shape limited to the linear and mixture terms while the quadratic term provides a correction mostly at similar masses. \nUsing again the q = 1, b S = 0 and GLYPH<17> ! 0 constraints on the general ansatz from Eq. (16), we end up with a total of nine free coe GLYPH<14> cients in this final step. We now fit to 298 cases with arbitrary spins not yet used in the 1D fits, with results given in Table IV. Together with the coe GLYPH<14> cients from Tables I-III, these fully determine the fit. To convert back from our fit quantity L 0 orb to the actual dimensionless final spin GLYPH<31> f , just add the total initial spin S = m 2 1 GLYPH<31> 1 + m 2 2 GLYPH<31> 2. \nWe find that the data set is su GLYPH<14> ciently large and clean, and the equal-spin part modeled well enough from the 2D step, to confidently extract the linear spin-di GLYPH<11> erence term and its GLYPH<17> -dependence, which is stable when adding the other terms; and to find some evidence for the combined mixture and quadratic terms, whose shape however is not fully constrained yet. \nTABLE IV. Fit coe GLYPH<14> cients for the final 3D step of the L 0 orb fit to 298 cases not yet used in the 1D fits of Sec. III B. \n| | | Estimate Standard error Relative error [%] |\n|------------------------|----------|----------------------------------------------|\n| d 10 0 : 322 | 0 : 020 | 6 : 2 |\n| d 11 9 : 33 | 0 : 87 | 9 : 3 |\n| d 20 GLYPH<0> 0 : 0598 | 0 : 0021 | 3 : 5 |\n| d 30 2 : 32 | 0 : 28 | 12 : 1 |\n| d 31 GLYPH<0> 3 : 26 | 0 : 20 | 6 : 1 |\n| f 12 0 : 512 | 0 : 085 | 16 : 7 |\n| f 22 GLYPH<0> 32 : 1 | 3 : 6 | 11 : 3 |\n| f 32 GLYPH<0> 154 | 10 | 6 : 5 |\n| f 51 GLYPH<0> 4 : 77 | 0 : 34 | 7 : 1 |", 'E. Fit assessment': "In Fig. 10 we also compare the spin-di GLYPH<11> erence terms from this final 'direct 3D' fit to those obtained from the per-massratio residuals analysis. The linear term is fully consistent, confirming that it is well determined by the data, while for the quadratic and mixture terms both approaches agree on the qualitative shape, but do not match as closely. Under the chosen ansatze, the 3D fit coe GLYPH<14> cients even for those terms are tightly determined (see Table IV). However, we have explicitly chosen the spin-di GLYPH<11> erence terms in Eq. (19) to achieve this goal, while several other ansatz choices (changing the fixed exponents of the multiplicative GLYPH<17> or p 1 GLYPH<0> 4 GLYPH<17> terms, or adding more terms with free coe GLYPH<14> cients in the GLYPH<17> polynomials) can produce fits that are indistinguishable by summary statistics (AICc, BIC, RMSE). Still, most of these have some strongly degenerate and underconstrained coe GLYPH<14> cients, while the reported fit has the desirable property of su GLYPH<14> cient complexity to be within the plateau region of summary statistics while not having any degenerate coe GLYPH<14> cients. \nYet, the shape of the functions A 2( GLYPH<17> ) and A 3( GLYPH<17> ) for the mixture and quadratic terms is not actually as closely constrained from the current data set as the coe GLYPH<14> cient uncertainties alone seem to imply, due to this ambiguity in ansatz selection. This becomes clear from the comparison of direct 3D fit and permass-ratio analysis in Fig. 10. The per-mass-ratio analysis also demonstrates that the data at mass ratios GLYPH<17> < 0 : 16 are not yet constraining enough to help characterize these terms. (The error bars are so large, and hence the weights so low, that they e GLYPH<11> ectively do not contribute to the fit.) It also becomes clear that additional unequal-spin data at intermediate mass ratios would be very useful in constraining the A 2 ; 3( GLYPH<17> ) functions. Meanwhile, it is important to note again that the leading linear spin-di GLYPH<11> erence term is already determined much more narrowly and robustly with the current data set. \nWe can further assess the success of the hierarchical 3D fitting procedure by comparing \n- GLYPH<15> a 2D fit (equal-spin physics only) to equal-spin NR cases only (same as in Fig. 8),\n- GLYPH<15> a 2D fit (equal-spin physics only) to all NR data,\n- GLYPH<15> and the 2D part of the full 3D fit. \nAs shown in Fig. 11, fitting the 2D equal-spin ansatz to the full data set induces strong curvature in the ( GLYPH<17>; b S ) plane, which the full 3D fit is able to correct by the additional degrees of freedom in the spin-di GLYPH<11> erence dimension. This is how it was possible to pull out the subdominant spin-di GLYPH<11> erence e GLYPH<11> ects \nFIG. 11. Green: Di GLYPH<11> erence GLYPH<1> L 0 orb of a 2D fit (equal-spin physics only) to the full data set minus the 2D fit to equal-spin cases only, both including extreme-mass-ratio constraints. The strong curvature at intermediate mass ratios and nonzero spins is due to the equal-spin-physics-only fit trying to compensate for the addition of unequal-spin NR cases. \n<!-- image --> \nOrange: Di GLYPH<11> erence GLYPH<1> L 0 orb of the 2D part of the 3D fit to the full data set minus the 2D-only fit to equal-spin data. The bulk of the parameter space is no longer distorted, and only at high e GLYPH<11> ective-spin magnitudes a small opposite e GLYPH<11> ect to the GLYPH<17> -dependent behavior of the spin-di GLYPH<11> erence terms (cf. Fig. 10) can be seen.TABLE V. Summary statistics for the various steps of the hierarchical final-spin fit. Note that it is not meaningful to compare AICc and BIC between data subsets of di GLYPH<11> erent sizes. There is statistical preference for the 3D fit including all three linear + mixture + quadratic terms, although many di GLYPH<11> erent choices of the Ai ( GLYPH<17> ) ansatz functions yield similar results with just GLYPH<6> a few percent in RMSE and GLYPH<6> a few in AICc / BIC, so that the shape of the mixture and quadratic terms is not yet fully constrained. \n| | N data | N coe GLYPH<11> | RMSE | AICc | BIC |\n|---------------------------------|----------|-------------------|-----------------------------------------------|----------------------------|-------------------|\n| 1D GLYPH<17> | 92 | 3 | 9 : 41 GLYPH<2> 10 GLYPH<0> 5 GLYPH<0> | 1590 : 8 | GLYPH<0> 1580 : 7 |\n| 1D ˆ S | 37 | 4 | 2 : 05 GLYPH<2> 10 GLYPH<0> 4 GLYPH<0> | 563 : 6 | GLYPH<0> 555 : 5 |\n| 2D ( GLYPH<31> 1 = GLYPH<31> 2) | 60 | 4 | 3 : 90 GLYPH<2> 10 GLYPH<0> 4 GLYPH<0> | 880 : 5 GLYPH<0> | 870 : 8 |\n| 2D all | 298 | 4 | 8 : 05 GLYPH<2> 10 GLYPH<0> 3 GLYPH<0> | 2247 : 4 | GLYPH<0> 2229 : 0 |\n| 3D lin | 298 | 6 | 9 : 20 GLYPH<2> 10 GLYPH<0> 4 GLYPH<0> | 3628 : 4 GLYPH<0> 3602 : 9 | |\n| 3D lin + quad | 298 | 7 | 8 : 28 GLYPH<2> 10 GLYPH<0> 4 GLYPH<0> | 3765 : 0 GLYPH<0> 3735 : 8 | |\n| 3D lin + mix | 298 | 8 | 8 : 11 GLYPH<2> 10 GLYPH<0> 4 GLYPH<0> 3693 : | 4 GLYPH<0> 3660 : 6 | |\n| 3D lin + quad + mix | 298 | 9 | 6 : 10 GLYPH<2> 10 4 GLYPH<0> 4087 | GLYPH<0> : 3 GLYPH<0> : 9 | 4050 | \nwith this enlarged data set. The same conclusion is supported by the comparison of summary statistics between the various steps and 2D / 3D fit variants in Table V, showing that the RMSE only increases by 50% from the 2D equal-spin case to the full 3D fit using all data. \nThe distribution of fit residuals for the full data set, projected onto the ( GLYPH<17>; b S ) plane, is shown in Fig. 12, and a comparison of fit residuals with other previously published fits, over the calibration data set of the current work, is shown as histograms in Fig. 13 and summarized in Table VI along with AICc and BIC metrics. The shape of the distributions is consistent, and for all fits the means are much smaller than the standard deviations, showing no evidence for any systematic bias. Our new fit improves significantly over the previous fit [9] used in the calibration of the IMRPhenomD waveform model [10], and also yields some improvement over recent fits from other groups [55, 61], even when those ansatze are refit to our present NR data set. \nRefitting our final hierarchically obtained ansatz directly to the full data set produces slightly better summary statis- \nFIG. 14. Comparison of this work with previously published fits [9, 55, 61] in the limit of extremal aligned spins, GLYPH<31> 1 = GLYPH<31> 2 = 1. The shaded region shows our fit's 90% confidence interval, which is narrow enough to indicate that discrepancies with the referenced fits are significant and due to the di GLYPH<11> erent ansatz constructions, especially in the extreme-mass-ratio limit (cf. Sec. II B), and not just a consequence of insu GLYPH<14> cient data. \n<!-- image --> \nFIG. 12. Residuals of the new 3D final-spin fit, projected to the 2D parameter space of GLYPH<17> and b S . The four NR data sets are distinguished by colors, and unequal-spin points highlighted with stars. \n<!-- image --> \nFIG. 13. Fit residuals of the final spin GLYPH<31> f , for this work and for previously published fits [9, 55, 61], evaluated over the set of 427 NR simulations shown in Fig. 2. Main panel: histograms, with 102 outliers for PhenomD with j NR GLYPH<0> fit j > 0 : 0075 outside of the plot range. Inset: cumulative distributions over the same range. \n<!-- image --> \nTABLE VI. Summary statistics for the new final-spin fit compared with previous fits [9, 52, 55, 61], evaluated over the 427 NR simulations shown in Fig. 2. For Hofmann et al. [55], both the ( nM = 1 ; nJ = 2) fit (6 coe GLYPH<14> cients) and the ( nM = 3 ; nJ = 3) version (16 coe GLYPH<14> cients) are listed. The new fit has a total of 16 coe GLYPH<14> cients calibrated to NR, corresponding to Tables I, II and IV, not counting those constrained from the extreme-mass-ratio limit. We also show results for refitting previous ansatze to the present NR data set, for a refit of our hierarchically obtained ansatz directly using the full data set, and for the same fitting procedure, but using uniform weights. \n| | N coef mean stdev | AICc | BIC |\n|-----------------|--------------------------------------------------------------------------------------------------|---------------------------------------|-------------------|\n| HLZ2014 [52] | 19 GLYPH<0> 4 : 8 GLYPH<2> 10 GLYPH<0> 5 8 : 9 GLYPH<2> 10 GLYPH<0> | 4 GLYPH<0> 5141 : 0 | GLYPH<0> 5061 : 7 |\n| HL2016 [61] | 19 8 : 1 GLYPH<2> 10 GLYPH<0> 7 7 : 9 GLYPH<2> 10 GLYPH<0> | 4 GLYPH<0> 5358 : 1 GLYPH<0> 5278 : 9 | |\n| PhenomD [9] | 11 GLYPH<0> 4 : 7 GLYPH<2> 10 GLYPH<0> 5 7 : 2 GLYPH<2> 10 GLYPH<0> 3 GLYPH<0> 3309 : | 0 GLYPH<0> 3260 : 9 | |\n| (refit) | 11 GLYPH<0> 1 : 7 GLYPH<2> 10 GLYPH<0> 4 7 : 0 GLYPH<2> GLYPH<0> 3 GLYPH<0> | 10 3334 : 5 GLYPH<0> 3286 : 5 | |\n| HBR2016 [55] | 6 GLYPH<0> 1 : 2 GLYPH<2> 10 GLYPH<0> 4 1 : 4 GLYPH<2> 10 GLYPH<0> 3 GLYPH<0> 4717 : | 2 GLYPH<0> 4689 | : 0 |\n| (refit) | 6 GLYPH<0> 1 : 4 GLYPH<2> 10 GLYPH<0> 4 1 : 3 GLYPH<2> 10 GLYPH<0> 3 GLYPH<0> 4791 : 4 | GLYPH<0> : 2 | 4763 |\n| HBR2016 [55] | 16 GLYPH<0> 2 : 8 GLYPH<2> 10 GLYPH<0> 4 1 : 2 GLYPH<2> 10 GLYPH<0> 3 GLYPH<0> 4877 : | 3 GLYPH<0> 4809 : | 7 |\n| (refit) | 16 GLYPH<0> 1 : 4 GLYPH<2> 10 GLYPH<0> 5 1 : 0 GLYPH<2> 10 GLYPH<0> 3 GLYPH<0> 4975 : 8 | GLYPH<0> 4908 : | 2 |\n| This work | 16 GLYPH<0> 2 : 3 GLYPH<2> 10 GLYPH<0> 5 5 : 2 GLYPH<2> 10 GLYPH<0> 4 GLYPH<0> 5991 : 5 GLYPH<0> | 5923 : 9 | |\n| (refit) | 16 GLYPH<0> 2 : 1 GLYPH<2> 10 GLYPH<0> 5 5 : 1 GLYPH<2> 10 GLYPH<0> 4 GLYPH<0> 6011 : 3 GLYPH<0> | 5943 : | 6 |\n| (uniform) | 16 GLYPH<0> 1 : 2 GLYPH<2> 10 GLYPH<0> 5 5 : 0 GLYPH<2> 10 GLYPH<0> 4 GLYPH<0> 5240 : 1 | GLYPH<0> 5172 : | 5 |\n| (uniform refit) | 16 GLYPH<0> 6 : 9 GLYPH<2> 10 GLYPH<0> 6 4 : 9 GLYPH<2> 10 GLYPH<0> 4 GLYPH<0> 5256 : 8 | GLYPH<0> 5189 : | 2 | \ntics, but also allows uncertainties from the less well-controlled unequal-spin set to influence the other parts of the fit, while the stepwise fit gives better control over the extreme-massratio behavior and better-determined coe GLYPH<14> cients for the wellconstrained subspaces. \nAs a further test of robustness, we have repeated the hierarchical fitting procedure with uniform weights instead of the weights used so far and discussed in Appendix A. This yields a fit consistent with our main result, though slightly less well constrained, but still improving over previous fits, thus demonstrating the robustness of the hierarchical fit construction under weighting choice. \nWe have also verified that our new fit does not violate the GLYPH<31> f GLYPH<20> 1 Kerr bound, particularly in the extreme-spin limit ( b S = 1) and at low GLYPH<17> , see Fig. 14.", 'F. Precessing binaries': "While some existing final-spin fits [53-55] also include a calibration to precessing cases, it is also possible to use a simple 'augmentation' procedure [49] (see also [67]) for alignedspin-only calibrations by adding the contribution of in-plane spins in quadrature to the aligned-spin fit result: \nGLYPH<31> aug f = r GLYPH<16> GLYPH<31> aligned f GLYPH<17> 2 + GLYPH<0> S in-plane = M 2 GLYPH<1> 2 : (20) \nThis procedure is known to significantly improve accuracy and reduce bias for precessing binaries. For example, it has been applied to the aligned-spin PhenomD fit [9] for the precessing PhenomPv2 model [12, 13], and to the RIT fit [52] in recent parameter estimation work of the LIGO-Virgo collaboration [4, 42, 69] (including spin evolution according to [41]). \nApplying Eq. (20) to our aligned-spin fit, we find a small overshooting of the j GLYPH<31> f j GLYPH<20> 1 Kerr bound for mass ratios q & 24, when the spin magnitude of the heavier BH is very close to extremal, and for certain orientation angles GLYPH<18> i of the black holes' spins to the angular momentum. The worst cases give an excess in GLYPH<31> f of about 0.12% at q GLYPH<24> 60 and intermediate opening \nangles, comparable to the aligned-spin fit residuals. No overshooting occurs if only the linear-inGLYPH<17> term in the final spin is used. Such a small inaccuracy when extending the alignedspin fit to precessing cases is in principle not surprising, as this parameter-space region is not covered with NR simulations and hence the fit slope in this region is purely determined by extrapolation between the NR data and the extreme-massratio limit, which we have ensured to be smooth with a flat approach to GLYPH<31> f = 1 at ( GLYPH<17> = 0 ; GLYPH<31> 1 = 1) (see Sec. II B and Fig. 14). Very small inaccuracies in the intermediateGLYPH<17> extrapolation region can thus lead to a minimal Kerr violation when adding the in-plane spins according to Eq. (20). A clean solution to this issue would require more calibration NR simulations in the critical region and a study of precessing spin contributions in the extreme-mass-ratio limit. \nHowever, as the overshooting is very small, we have investigated two easy ad hoc solutions: We could take the worst-case point and enforce our 3D fit to be at or below the corresponding L 0 orb value for the aligned-spin projection GLYPH<31> 1 = cos( GLYPH<18> 1) by putting a constraint on one of the fi 2 coe GLYPH<14> -cients. This can remove the overshooting at the worst-case point and nearby, but not over the whole problematic region, as the fit still has enough freedom in other parameters. But the accuracy of the aligned-spin fit already su GLYPH<11> ers from this one extra constraint, and using constraints on more than one coefficient to pull down the augmented GLYPH<31> f over a wider parameter region is fully prohibitive because insu GLYPH<14> cient freedom will remain in the fit to properly calibrate to the actual NR data. \nHence, we opt for an even simpler solution, truncating the augmentation from Eq. (20) at unity: GLYPH<31> f = min GLYPH<16> GLYPH<31> aug f ; 1 : 0 GLYPH<17> . This is justified as the overshooting is very small, on the order of the fit residuals, and limited to an extremal parameter-space region. The need for this ad hoc truncation will reduce or become obsolete when lowGLYPH<17> -high-spin NR simulations and / or precessing extreme-mass-ratio information become available. A detailed comparison of fit accuracies over a representative set of precessing NR runs is left to future work.", 'IV. FINAL MASS AND RADIATED ENERGY': 'To fit the final mass of remnant BHs from BBH mergers, we use the same hierarchical approach as for the final spin, summarized in Fig. 1, with only minor modifications. The quantity we are going to fit here is the dimensionless radiated energy, E rad = M GLYPH<0> M f = 1 GLYPH<0> M f . For GLYPH<17> = 0, it has to vanish even in spinning cases, while the analytical expectation for the leading order in GLYPH<17> , as GLYPH<17> ! 0, is E rad( GLYPH<17> ) =GLYPH<17> GLYPH<24> 1 GLYPH<0> GLYPH<16> 2 p 2 GLYPH<17> = 3. We construct the two-dimensional E rad GLYPH<16> GLYPH<17>; b S GLYPH<17> ansatz as a product of the 1D ansatze, instead of a sum as in the final-spin case. \nIn principle, the fitting procedure is robust enough to use either a sum or product ansatz for either final-state quantity. Actually, by carrying out the full procedure for E rad as described in the following subsections, but using a sum of the 1D contributions, we found a fit statistically at least competitive with that obtained from a product. However, in this case the sum ansatz tends to produce suspicious curvature in the b S = 1, lowGLYPH<17> region, which cannot be suppressed by the extrememass-ratio information. With additional NR data in this region, that problem might be alleviated, but with the current data set we find the product ansatz to be more robust and able to yield a final fit that is both accurate and well determined \nFIG. 15. Nonspinning NR data and one-dimensional E rad GLYPH<16> GLYPH<17>; b S = 0 GLYPH<17> fit as a function of mass ratio GLYPH<17> , Top panel: selected fit, a fourthorder polynomial P(4), see Eq. (21). Lower panel: residuals of this fit (points) and di GLYPH<11> erences from the three next-best-ranking in terms of BIC (lines). \n<!-- image --> \nTABLE VII. Fit coe GLYPH<14> cients for the one-dimensional nonspinning E rad GLYPH<16> GLYPH<17>; b S = 0 GLYPH<17> fit over 92 NR cases. \n| | | Estimate Standard error Relative error [%] |\n|-----|------------------|----------------------------------------------|\n| a 2 | 0 : 5610 | 0 : 0026 0 : 5 |\n| a 3 | GLYPH<0> 0 : 847 | 0 : 027 3 : 2 |\n| a 4 | 3 : 145 | 0 : 069 2 : 2 | \nover the calibration region and without obvious bad behavior in extrapolation.', 'A. One-dimensional fits': 'For the nonspinning 1D fit in symmetric mass ratio GLYPH<17> , a simple fourth-order polynomial \nE rad GLYPH<16> GLYPH<17>; b S = 0 GLYPH<17> = a 4 GLYPH<17> 4 + a 3 GLYPH<17> 3 + a 2 GLYPH<17> 2 + 0 B B B B @ 1 GLYPH<0> 2 p 2 3 1 C C C C A GLYPH<17> (21) \nwith three free coe GLYPH<14> cients, listed in Table VII, is marginally preferred by both AICc and BIC. More complicated rational functions are not able to yield any significant change in residuals (only up to 1% in RMSE), while the di GLYPH<11> erences between Eq. (21) and the next-ranked fits are again much smaller than the remaining residuals, as shown in Fig. 15. \nFor the e GLYPH<11> ective-spin dependence, again the value at GLYPH<16> GLYPH<17> = 0 : 25 ; b S = 0 GLYPH<17> is fixed from the GLYPH<17> fit. A rational function of order (3,1) is top-ranked by AICc, BIC and RMSE and thus unambiguously selected as the preferred ansatz: \nE rad GLYPH<16> GLYPH<17> = 0 : 25 ; b S GLYPH<17> = 0 : 0484161 GLYPH<16> 0 : 128 b 3 b S 3 + 0 : 211 b 2 b S 2 + 0 : 346 b 1 b S + 1 GLYPH<17> 1 GLYPH<0> 0 : 212 b 5 b S \n(22) \nFIG. 16. Equal-mass-equal-spin NR data and one-dimensional fits of E rad GLYPH<16> GLYPH<17> = 0 : 25 ; b S GLYPH<17> as a function of e GLYPH<11> ective spin b S . Top panel: selected fit, a rational function R(3,1), see Eq. (22). Lower panel: residuals of this fit (points) and di GLYPH<11> erences from the three other topranking fits in terms of BIC (lines). \n<!-- image --> \nTABLE VIII. Fit coe GLYPH<14> cients for the one-dimensional equal-massequal-spin E rad GLYPH<16> GLYPH<17> = 0 : 25 ; b S GLYPH<17> fit over 37 NR cases. \n| | Estimate Standard error Relative error [%] | |\n|------------------|----------------------------------------------|--------|\n| b 1 GLYPH<0> 0 : | 209 0 : 016 | 7 : 6 |\n| b 2 | GLYPH<0> 0 : 197 0 : 026 | 13 : 2 |\n| b 3 GLYPH<0> 0 | : 159 0 : 049 | 31 : 1 |\n| b 5 | 2 : 985 0 : 034 | 1 : 1 | \nwith four free coe GLYPH<14> cients listed in Table VIII, and wellbehaved residuals as seen in Fig. 16.', 'B. Two-dimensional fits': 'For the 2D ansatz, we combine the two 1D fits from Eqs. (21) and (22), expanding each b S -dependent term with a polynomial in GLYPH<17> , according to Eq. (9), and removing the GLYPH<16> GLYPH<17> = 0 : 25 ; b S = 0 GLYPH<17> value from the spin ansatz before multiplying with the GLYPH<17> terms: \nE rad GLYPH<16> GLYPH<17>; b S GLYPH<17> = E rad ( GLYPH<17>; 0) E rad GLYPH<16> 0 : 25 ; b S ; fi j GLYPH<17> E rad (0 : 25 ; 0) : (23) \nContrary to the sum ansatz for GLYPH<31> f in Eq. (10), we do not need to set the GLYPH<17> -independent coe GLYPH<14> cients fi 0 of the b S terms to zero, as the E rad GLYPH<16> GLYPH<17>; b S GLYPH<17> = E rad ( GLYPH<17>; 0) (1 + : : : ) form of Eq. (23) already guarantees the correct GLYPH<17> = 0 limit. Hence an expansion up to third order in GLYPH<17> of each b S term, as we chose for the GLYPH<31> f fit, would yield too many free coe GLYPH<14> cients, and instead we only expand up to second order. The four fi 2 coe GLYPH<14> cients are again fixed by the equal-mass boundary conditions: \nfi 2 = 16 GLYPH<0> 16 fi 0 GLYPH<0> 4 fi 1 : (24) \nFIG. 17. Radiated energy: extreme-mass-ratio comparison of analytical results, the previous PhenomD radiated-energy fit of [9], and this work. Note that, constrained to capture the steep rise at b S ! + 1, the new fit actually deviates slightly more from the analytical result at low positive b S . This could be avoided with a more complex 1D b S ansatz, which is however disfavored by the current NR data set. \n<!-- image --> \nTABLE IX. Fit coe GLYPH<14> cients for the extreme-mass-ratio limit of the radiated energy, fitted to discretized analytical results. The fourth coe GLYPH<14> cient, f 10, is fixed by the constraint at b S = 1, cf. Eq. (26), and its estimate and error are computed from the others. \n| | | | Estimate Standard error Relative error [%] |\n|------|-----------|-----------|----------------------------------------------|\n| f 20 | 4 : 27 | 0 : 38 | 8 : 9 |\n| f 30 | 31 : 09 | 0 : 71 | 2 : 3 |\n| f 50 | 1 : 56735 | 0 : 00032 | 0 : 02 |\n| f 10 | 1 : 81 | 0 : 15 | 8 : 2 | \nFIG. 18. Two-dimensional E rad GLYPH<16> GLYPH<17>; b S GLYPH<17> fit, visualized as E rad GLYPH<16> GLYPH<17>; b S GLYPH<17> =GLYPH<17> . Application of the extreme-mass-ratio limit helps in avoiding extrapolation artifacts which would otherwise appear at lowGLYPH<17> , highj b S j regions that are uncovered by NR simulations. \n<!-- image --> \nSimilar to the procedure for GLYPH<31> f , we can use the extrememass-ratio limit to fix the four coe GLYPH<14> cients fi 0 of the linear-inGLYPH<17> terms. Using the analytic result from Eq. (2)a, we force the fit to satisfy the equality \nE rad( GLYPH<17> ! 0 ; b S ) = 1 GLYPH<0> E ISCO( b S ) (25) \nand fit the corresponding leading-order GLYPH<17> dependence of our \nFigure 20 shows that the linear term is again robustly determined and does not change shape much when adding the two additional terms, but already the per-mass-ratio and direct-3D fits for this term do not agree quite as closely as in the GLYPH<31> f fit. The quadratic term is more noisy, and for the mixture term the \n<!-- image --> \n<!-- image --> \nFIG. 19. Examples of spin-di GLYPH<11> erence behavior of the radiated energy at fixed mass ratios, for residuals GLYPH<1> E rad after subtracting the twodimensional E rad GLYPH<16> GLYPH<17>; b S GLYPH<17> fit. Top row: q = 1 (mirror-duplicated data points shown in gray); lower row: q = 4; left column: surfaces in GLYPH<16> b S ; GLYPH<1> GLYPH<31>; GLYPH<1> E rad GLYPH<17> space; right column: projections unto the GLYPH<1> GLYPH<31> axis with linear and quadratic fits. At equal mass, the linear term and mixture term vanish, but the expected quadratic dependence (parabolic surface) is less clearly pulled out from rather noisy residuals than for the final spin (cf. Fig. 9). At q = 4 and other intermediate mass ratios, the surface is not as close to flat as in the final-spin case, and the noisy data still shows some quadratic dependence. \n<!-- image --> \n2D ansatz to discretized values of this quantity. Again we fix one of the four free coe GLYPH<14> cients of E rad GLYPH<16> GLYPH<17> ! 0 ; b S GLYPH<17> by a constraint fixing the value at b S = 1, which is necessary to capture the very steep rise of Eq. (25) as b S ! + 1: \nf 10 !GLYPH<0> 0 : 574752 f 20 GLYPH<0> 0 : 280958 f 30 + 64 : 6408 f 50 GLYPH<0> 88 : 3165 : (26) \nThe agreement between discretized analytical result and fit is shown in Fig. 17, and fit coe GLYPH<14> cients are listed in Table IX. \nWe thus have 12 GLYPH<0> 4 GLYPH<0> 4 = 4 free coe GLYPH<14> cients fi 1, of which f 21 turns out to be extremely poorly constrained, so that we set it to zero before refitting. Results of the 2D fit, calibrated to equal-spin simulations only, are shown in Fig. 18, which shows that the steep shape of the extreme-mass-ratio limit at high b S is smoothly attained by the extrapolated fit. For the curvature at low GLYPH<17> and extremal b S = 1, where there is no NR data, there might be also a contribution from the small remaining fit issues in the extreme-mass-ratio limit (cf. Fig. 17). The residuals again have larger RMSE than the 1D fits in GLYPH<17> and b S , by factors of 6.5 and 1.8 respectively, but show no clear apparent trends, allowing us to use this 2D fit as the basis for an unequal-spin residuals study in the next step.', 'C. Unequal-spin contributions and 3D fit': 'The spin-di GLYPH<11> erence dependence of unequal-spin residuals is less clear here than for the final spin: As seen in the examples of Fig. 19, the general trend is the same with a quadratic dependence on GLYPH<1> GLYPH<31> at equal masses and more dominant linear e GLYPH<11> ects as GLYPH<17> decreases, but the distributions are generally noisier and the second-order terms (quadratic and mixture / b S GLYPH<1> GLYPH<31> ) cannot be as cleanly separated. \nFor both the per-mass-ratio-step analysis and the direct 3D fit, we use the same general functional forms for possible linear, quadratic and mixture terms as in Eqs. (15), (17) and (18). After fixing ill-constrained coe GLYPH<14> cients to integer values, these reduce to \nA 1( GLYPH<17> ) = d 10(1 GLYPH<0> 4 GLYPH<17> ) 0 : 5 GLYPH<17> 2 ( d 11 GLYPH<17> + 1) (27)a \nA 2( GLYPH<17> ) = d 20 GLYPH<17> 3 (27)b \nA 3( GLYPH<17> ) = d 30(1 GLYPH<0> 4 GLYPH<17> ) 0 : 5 GLYPH<17> ( d 31 GLYPH<17> + 1) : (27)c \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFIG. 20. Spin-di GLYPH<11> erence behavior of radiated-energy data after subtraction of the two-dimensional E rad GLYPH<16> GLYPH<17>; b S GLYPH<17> fit, for the three ansatz functions Ai ( GLYPH<17> ) from Eq. (27), with the same mass-ratio steps and fits as in Fig. 10. Top-left panel: linear term A 1 only. The remaining panels are for the combined linear + quadratic + mixture fit, in clockwise order: linear term A 1, quadratic term A 2 and mixture term A 3. The A 1 results from the combined fit are very similar to those from the linear-only fit, demonstrating the robustness of extracting leading-order spin-di GLYPH<11> erence e GLYPH<11> ects. For the two lower panels, results are much more uncertain, and the error bars for low GLYPH<17> go far outside the displayed range, so that this region does not contribute significantly to the weighted per-mass-ratio fits. \n<!-- image --> \nresults are rather uncertain, with an apparent sign change in the e GLYPH<11> ect over GLYPH<17> , but the stepwise cross-checks at least agreeing on the overall shape. Still, we will see below that inclusion of both these e GLYPH<11> ects is statistically justified. \nThe full 3D ansatz for E rad GLYPH<16> GLYPH<17>; b S ; GLYPH<1> GLYPH<31> GLYPH<17> is then built up as \nE rad GLYPH<16> GLYPH<17>; b S ; GLYPH<1> GLYPH<31> GLYPH<17> = E rad GLYPH<16> GLYPH<17>; b S GLYPH<17> + GLYPH<1> E rad GLYPH<16> GLYPH<17>; b S ; GLYPH<1> GLYPH<31> GLYPH<17> ; (28) \nand this time has eight free coe GLYPH<14> cients (three from the 2D ansatz and five from the spin-di GLYPH<11> erence terms). Results for the fit to 298 NR cases not previously used in the 1D fits are listed in Table X. \nTABLE X. E rad fit coe GLYPH<14> cients for the final 3D step, using 298 cases. \n| | | Estimate Standard error Relative error [%] |\n|------------------------|----------|----------------------------------------------|\n| d 10 GLYPH<0> 0 : 098 | 0 : 011 | 11 : 3 |\n| d 11 GLYPH<0> 3 : 23 | 0 : 18 | 5 : 6 |\n| d 20 0 : 0112 | 0 : 0012 | 10 : 5 |\n| d 30 GLYPH<0> 0 : 0198 | 0 : 0036 | 18 : 4 |\n| d 31 GLYPH<0> 4 : 92 | 0 : 19 | 3 : 9 |\n| f 11 15 : 7 | 1 : 2 | 7 : 9 |\n| f 31 GLYPH<0> 243 : 6 | 8 : 0 | 3 : 3 |\n| f 51 GLYPH<0> 0 : 58 | 0 : 13 | 21 : 6 |', 'D. Fit assessment': 'Tbl. XI gives a statistical summary of the various 1D, 2D and 3D fits for E rad as discussed in this section. We find that the full linear + quadratic + mixture fit to all data has better RMSE than the simpler versions, and actually on the same level as the 2D equal-spin fit to equal-spin data only. The additional coe GLYPH<14> cients are also justified by yielding the best AICc and BIC, though not as clearly as for the final-spin case in Table V. Thus, we choose this three-term ansatz, matching the final-spin choice. Also, from Table X we see that all coe GLYPH<14> cients of this ansatz are su GLYPH<14> ciently well constrained, with a worst standard error of 21.6%, to be useful for applications, at least under the assumptions made for the ansatz terms in Eq. (27). Just as was the case for the final-spin fit, and even more so because of the noisier data, we caution that ambiguity over the exact shape of the spin-di GLYPH<11> erence terms remains, because di GLYPH<11> erent choices of the exponents and expansion orders in Eq. (27) can yield statistically indistinguishable fits. \nAs an additional check on the overall functional behavior of the fit, in Fig. 21 we show again comparisons between the intermediate 2D fit calibrated to equal-spin data only, a 2D fit to all data (not working well, as expected) and the 2D part of the final 3D fit. The equal-spin 2D fit and 3D fit agree well, with the strong excess curvature of the all-data 2D fit in the highb S region reduced significantly thanks to the unequal-spin terms. \nTABLE XI. Summary statistics for the various steps of the hierarchical radiated-energy fit. Evidence for spin-di GLYPH<11> erence terms beyond linear order is weaker than for the final-spin fits in Table V. \n| | N data | N coe GLYPH<11> | RMSE | AICc | BIC |\n|---------------------------------|----------|-------------------|-----------------------------------------------|--------------------------------|-------------------|\n| 1D GLYPH<17> | 92 | 3 | 4 : 14 GLYPH<2> 10 GLYPH<0> 5 GLYPH<0> | 1705 : 7 | GLYPH<0> 1695 : 6 |\n| 1D ˆ S | 37 | 4 | 1 : 51 GLYPH<2> 10 GLYPH<0> 4 GLYPH<0> | 577 : 3 | GLYPH<0> 569 : 3 |\n| 2D ( GLYPH<31> 1 = GLYPH<31> 2) | 60 | 3 | 2 : 67 GLYPH<2> 10 GLYPH<0> 4 GLYPH<0> | 875 : 3 | GLYPH<0> 867 : 4 |\n| 2D all | 298 | 3 | 4 : 26 GLYPH<2> 10 GLYPH<0> 4 GLYPH<0> 4070 | : 9 GLYPH<0> | 4056 : 2 |\n| 3D lin | 298 | 5 | 3 : 24 GLYPH<2> 10 GLYPH<0> 4282 | GLYPH<0> 4 : 9 GLYPH<0> 4260 : | 9 |\n| 3D lin + quad | 298 | 6 | 2 : 72 GLYPH<2> 10 GLYPH<0> 4 GLYPH<0> | 4391 : 9 GLYPH<0> 4366 : 3 | |\n| 3D lin + mix | 298 | 7 | 2 : 91 GLYPH<2> 10 GLYPH<0> 4 GLYPH<0> 4339 | : 3 GLYPH<0> | 4310 : 1 |\n| 3D lin + quad + mix | 298 | 8 | 2 : 62 GLYPH<2> 10 GLYPH<0> 4 GLYPH<0> 4417 : | 8 GLYPH<0> | 4385 : 0 | \nFIG. 21. Green: Di GLYPH<11> erence GLYPH<1> E rad of a 2D fit to the full radiated energy data set minus the 2D fit to equal-spin cases only, both including extreme-mass-ratio constraints. Orange: Di GLYPH<11> erence GLYPH<1> E rad of the 2D part of the 3D fit to the full data set minus the 2D-only fit to equal-spin data. \n<!-- image --> \nFIG. 22. Residuals of the radiated-energy fit, projected to the 2D parameter space of GLYPH<17> and b S . Stars: unequal-spin points. \n<!-- image --> \nThe residuals of the new E rad fit are shown over the GLYPH<16> GLYPH<17>; b S GLYPH<17> parameter space in Fig. 22 and as histograms in Fig. 23, compared with several previously published fits [9, 19, 52]. Comparison statistics are also summarized in Table XII. Again we find significant improvement, by a factor of 5 in residual standard deviation over the simple two-coe GLYPH<14> cient fit of [19] and about 40% improvement over the previous PhenomD fit of [9] and the RIT fits of [52, 61]. As the maximum possible emitted energy fraction - for equal-mass BBHs with extremal positive spins - the new fit predicts E rad GLYPH<25> 0 : 1142, consistent with the fit from [70] which was specifically calibrated to equal-mass- \nFIG. 23. Fit residuals of the radiated energy E rad, for this work (cf. Table X) and for previously published fits (SEOBNRv2 2014 [19], Healy & Lousto 2016 [61], PhenomD 2015 [9]), evaluated over the set of 427 NR simulations shown in Fig. 2. Main panel: histograms, with 10 outliers for SEOBNRv2 (a recalibration of the fit from [51]) with j NR GLYPH<0> fit j > 0 : 002 outside of the plot range. Inset: cumulative distributions over the same range. \n<!-- image --> \nTABLEXII. Summary statistics for the new radiated-energy fit compared with previously published fits [9, 19, 52, 61], evaluated over the full set of 427 NR simulations shown in Fig. 2. Also listed are a refit of the PhenomD [9] ansatz to the present NR data set, a refit of our hierarchically obtained ansatz directly to the full data set, and results with the same fitting procedure, but using uniform weights. \n| | N coef | mean stdev | AICc | BIC |\n|-----------------|-------------|-----------------------------------------------------------------------------------------------|-------------------------|-------------------|\n| HLZ2014 [52] | 19 GLYPH<0> | 5 : 4 GLYPH<2> 10 GLYPH<0> 5 3 : 4 GLYPH<2> 10 GLYPH<0> 4 | GLYPH<0> 5802 : 5 | GLYPH<0> 5723 : 2 |\n| HL2016 [61] | 19 GLYPH<0> | 4 : 4 GLYPH<2> 10 GLYPH<0> 5 3 : 0 GLYPH<2> 10 GLYPH<0> 4 GLYPH<0> 5909 : 8 | GLYPH<0> | 5830 : 5 |\n| PhenomD | 10 | 2 : 5 GLYPH<2> 10 GLYPH<0> 5 3 : 4 GLYPH<2> GLYPH<0> 4 GLYPH<0> : | 10 5914 9 GLYPH<0> 5870 | : 8 |\n| (refit) | 10 | 6 : 1 GLYPH<2> 10 GLYPH<0> 5 3 : 3 GLYPH<2> 10 GLYPH<0> 4 GLYPH<0> | 5947 : 7 GLYPH<0> | 5899 : 6 |\n| SEOBNRv2 [19] | 2 | GLYPH<0> 1 : 7 GLYPH<2> 10 GLYPH<0> 4 1 : 0 GLYPH<2> 10 GLYPH<0> 3 GLYPH<0> 5036 : | 1 GLYPH<0> | 5023 : 9 |\n| This work | 15 | 4 : 7 GLYPH<2> 10 GLYPH<0> 5 2 : 2 GLYPH<2> 10 GLYPH<0> 4 GLYPH<0> 6454 : 8 | GLYPH<0> 6391 : | 0 |\n| (refit) | 15 | 6 : 3 GLYPH<2> 10 GLYPH<0> 5 2 : 1 GLYPH<2> 10 GLYPH<0> 4 GLYPH<0> 6482 : 8 GLYPH<0> | | 6419 : 0 |\n| (uniform | 15 | GLYPH<0> 4 : 0 GLYPH<2> 10 GLYPH<0> 6 2 : 1 GLYPH<2> 10 GLYPH<0> 4 GLYPH<0> 5987 : 3 GLYPH<0> | | 5923 : 5 |\n| (uniform refit) | 15 | 1 : 4 GLYPH<2> 10 GLYPH<0> 6 2 : 0 GLYPH<2> 10 GLYPH<0> 4 GLYPH<0> 6034 | : 2 GLYPH<0> | 5970 : 4 | \nequal-spin cases and with [9, 52], but 15% higher than [51], underlining the importance of high-order b S terms at extremal spins.', 'V. CONCLUSIONS': "We have developed a hierarchical step-by-step fitting method for results of numerical-relativity simulations of BBH mergers, assuming nonprecessing spins and negligible eccentricity, so that the parameter space is given by the mass ratio and two spin components: ( GLYPH<17>; GLYPH<31> 1 ; GLYPH<31> 2). We have then applied the method to the spin GLYPH<31> f and mass M f of the remnant Kerr BH, the latter being equivalent to radiated energy E rad. \nAn appropriate fit is constructed in simple steps which reduce the dimensionality of the problem, with the ansatz choice at each step driven by inspecting the data. The full higherdimensional fit is then built in a bottom-up fashion, modeling each parameter's contribution in order of its importance, as illustrated in the flowchart of Fig. 1. \nA key goal of our approach is to avoid overfitting. To achieve this, and to evaluate the quality of the fits, we use in- \nformation criteria (AICc and BIC) described in Appendix B. At least as important, however, is also the hierarchical datadriven nature of our procedure, e.g. when modeling the subdominant dependence on the di GLYPH<11> erence between the spins. Through a reduction to one-dimensional problems, and inspecting the data at di GLYPH<11> erent mass ratios, an appropriate model with a small number of parameters is easily identified. \nWe compare our results with previous fits in the literature, also refitting these previous models to our data set, and we find a clear preference for the new fits in terms of residuals and information criteria. \nOur emphasis on inspecting the underlying data set, and comparing its quality with the statistical analysis of fit errors, highlights that for further improvement of these numerical fits, it will be essential to understand the uncertainties and systematic e GLYPH<11> ects in a data set, and to cleanly combine data from di GLYPH<11> erent codes, rather than simply increasing the number of calibration simulations without controlling the error budget. \nIn addition to the quality of the fit as applied to the available numerical relativity data, we investigate extrapolation properties beyond the calibration region, in particular for extreme spins, where we check that our fit does not overshoot the extreme Kerr limit, and avoids pathological oscillations. The extreme-mass-ratio limit has previously been incorporated into fits for final mass and spin in di GLYPH<11> erent ways, in particular for the final spin, where the influence of radiated energy is not always accounted for (e.g. in [47, 52]). Doing so is however important to avoid oscillatory features due to an unphysical value of the derivative with respect to the symmetric mass ratio GLYPH<17> at GLYPH<17> = 0, and indeed we achieve a more robust behavior for close-to-extremal final spins in comparison to other recent fits [52, 55] - see Fig. 14. In order to avoid numerical errors in this derivative, we use an analytical expansion around the ( GLYPH<17> = 0 ; b S = 1) corner. \nWe have also verified that our fits for final mass and spin are consistent with the Hawking area theorem for black holes [71]: the area of the final horizon is larger than the sum of the individual horizons of the initial black holes. The difference in areas is much larger than the fit errors, thus the area theorem does not provide an additional constraint on the fits. \nFor both GLYPH<31> f and E rad, we can robustly identify the main unequal-spin contribution of the form f ( GLYPH<17> ) ( GLYPH<31> 1 GLYPH<0> GLYPH<31> 2). The shape of the function f ( GLYPH<17> ) is well determined for final spins, but has larger errors for the smaller unequal-spin contribution to radiated energy. We also discuss and find statistical evidence for two further unequal-spin terms, one quadratic in GLYPH<1> GLYPH<31> , and a mixed b S GLYPH<1> GLYPH<31> term, which however are comparable in size to errors in the numerical data, so that their GLYPH<17> -dependent shape cannot be tightly constrained with the current data set. \nThe same hierarchical fitting procedure can be easily applied to other quantities such as peak luminosity [72, 73]. An important goal of our work is the accurate calibration of unequal-spin e GLYPH<11> ects for full inspiral-merger-ringdown waveform models, in particular toward extending the PhenomD model [9, 10]. This will allow investigating under which conditions gravitational-wave observations can reveal the individual spins of a coalescing binary. While no fundamental modifications are necessary to the method itself, a careful analysis of the data sets will be required to make appropriate choices in combining the 1D subspace fits, including extreme-mass-ratio information, and adding spin-di GLYPH<11> erence terms. A more ambitious goal for the future is the extension of our hierarchical \nfit construction to the higher-dimensional problems of generic precessing BBHs.", 'ACKNOWLEDGMENTS': "X.J., D.K. and S.H. were supported by the Spanish Ministry of Economy and Competitiveness grants FPA2016-76821-P, CSD2009-00064 and FPA2013-41042-P, the Spanish Agencia Estatal de Investigaci'on, European Union FEDER funds, Vicepresid'encia i Conselleria d'Innovaci'o, Recerca i Turisme, Conselleria d'Educaci'ı Universitats del Govern de les Illes Balears, and the Fons Social Europeu. M.H. was supported by the Science and Technology Facilities Council grants ST / I001085 / 1 and ST / H008438 / 1, and by European Research Council Consolidator Grant 647839, and M.P. by ST / I001085 / 1. S.K. was supported by Science and Technology Facilities Council. The authors thankfully acknowledge the computer resources at Advanced Research Computing (ARCCA) at Cardi GLYPH<11> , as part of the European PRACE Research Infrastructure on the clusters Hermit, Curie and SuperMUC, on the U.K. DiRAC Datacentric cluster and on the BSC MareNostrum computer under PRACE and RES (Red Espa˜nola de Supercomputaci'on) grants, 2015133131, AECT2016-1-0015, AECT-2016-2-0009, AECT-2017-1-0017. We thank the CBC working group of the LIGO Scientific Collaboration, and especially Nathan Johnson-McDaniel, Ofek Birnholtz, Deirdre Shoemaker and Aaron Zimmerman, for discussions of the fitting method, results and manuscript. This paper has been assigned Document No. LIGO-P1600270-v5.", 'Appendix A: Data sets and NR uncertainties': "For NR calibration of BBH coalescence models, it is useful to combine data sets from di GLYPH<11> erent research groups and numerical codes, both to increase robustness against code inaccuracies and errors in the preparation of data products (such as incorrect metadata), and to benefit from the combined computational resources of di GLYPH<11> erent groups. Computational cost increases significantly with mass ratio and spin, thus the highmass-ratio and high-spin regions are still poorly sampled, and numerical errors are often higher (as we will see below). Very di GLYPH<11> erent spins on the two black holes will typically also increase computational cost due to the need to resolve di GLYPH<11> erent scales in the grids around the two black holes (higher spin leads to smaller black holes for typical coordinate gauges in numerical relativity), which is a challenge for accurately capturing small subdominant e GLYPH<11> ects like the nonlinear spindi GLYPH<11> erence e GLYPH<11> ects we discuss in this work. \nIn this Appendix, we discuss our procedures to eliminate data points of poor quality, to assign fit weights, and to check consistency between assumed error bars and our fit results. We expect two main avenues to significantly improve over the fits we have presented in this paper: (a) providing more data points with high spins and unequal masses, in order to improve the accuracy of the fit near the boundaries of the fitting region and to reduce the need for extrapolation; and (b) determining more accurate and robust error bars for NR data, which would allow one to isolate small subdominant e GLYPH<11> ects. \nFinal spin and final mass are usually computed as surface integrals over the apparent horizon using the isolated-horizon formalism [74] (see [75] for a summary of methods and references for di GLYPH<11> erent codes), from surface integrals over spheres \nFIG. 25. Histograms of the di GLYPH<11> erences between radiated energy computed from either horizon or waveform data (as in Fig. 24), compared with the residuals of the new radiated-energy fit (as in Fig. 23). \n<!-- image --> \nFIG. 24. Di GLYPH<11> erences between radiated energy computed from either horizon or waveform data, across the parameter space. The color scale quantifies the di GLYPH<11> erences between the two computations. Differences are largest for high-mass-ratio and high-spin cases, where high NR accuracy is more demanding. \n<!-- image --> \nat large or infinite radius (as in [64]), or from the energy or angular momentum balance computed from initial and radiated quantities (see Eq. (A1) below). Final spin and final mass can also be obtained from fits to the ringdown [76]. \nFinal mass and spin from the apparent horizon (AH) are generally expected to be more accurate than those based on the evaluation of asymptotic quantities (such as Bondi mass, angular momentum, or radiated energy and angular momentum) at finite radius, where errors may arise due to finite radius truncation, insu GLYPH<14> cient extrapolation to infinity or numerical inaccuracies in propagating the wave content to large distances at su GLYPH<14> cient numerical resolution. On the other hand, AH quantities may su GLYPH<11> er from inaccuracies in finding the apparent horizon and from gauge ambiguities. \nFor SXS, RIT and GaTech results, we take the values provided in the catalogs [52, 59, 60, 62, 63], which in general have been computed at the horizon. For a comparison of horizon vs radiated quantities for the RIT results, see Table V in [52], where error bars for the horizon quantities are significantly smaller. For the BAM code, for some large-mass-ratio cases the AH finder fails due to the unfortunate choice of a shift condition, which results in a coordinate growth in the horizon which is roughly linear in time during the evolution. After several orbits the horizon of the larger BH is then no longer contained within the fine grid of the mesh refinement, which may trigger a failure of the horizon finder code. Due to the high computational cost of the simulations, we have not rerun these cases with improved parameters for the apparent horizon finder code. But rather, we compute the final angular momentum from the angular momentum surface integral at large radius, and the energy from the radiated GWs. For processing GaTech waveforms we have also followed [77]. \nFor all 414 cases where we have both the waveform and AH estimate available, we perform cross-checks between AH quantities and those obtained from integrating radiated energy and angular momentum. In order to compute the final mass from the radiation quantities, we integrate \ndE dt = lim r !1 2 6 6 6 6 6 4 r 2 16 GLYPH<25> Z GLYPH<10> d GLYPH<10> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> Z t GLYPH<0>1 4d ˜ t GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<12> 2 3 7 7 7 7 7 5 (A1) \nover time, starting after the initial burst of 'junk radiation'. \nWe extrapolate the result from di GLYPH<11> erent extraction radii to infinity at linear order in inverse radius. To account for energy radiated at separations larger than the initial separation of the NR simulation, we compute the radiated energy at 3.5 PN order [78-84] from ! 2 [0 ; ! 0], with ! 0 being the initial orbital frequency of the NR simulation (after junk radiation). Then we can consider the di GLYPH<11> erences between radiated energy values from the horizon and from the integrated waveforms, shown in Fig. 24, as an estimate of NR errors. However, this will typically be a pessimistic estimate because horizon quantities are in general more reliable and thus big di GLYPH<11> erences are typically caused by inaccuracies in the integrated emission. In Fig. 25 we show that the distribution of this pessimistic estimate is similar to, but much wider-tailed than, the residuals from our radiated-energy fit. \nA more realistic measure of NR errors is the di GLYPH<11> erence between results from di GLYPH<11> erent codes for equal initial parameters. With a strict tolerance requiring equal initial parameters to within numerical accuracy, \nGLYPH<12> GLYPH<12> GLYPH<12> GLYPH<21> i GLYPH<0> GLYPH<21> j GLYPH<12> GLYPH<12> GLYPH<12> GLYPH<20> GLYPH<15> = 0 : 0002 with GLYPH<21> i = f GLYPH<17> i ; GLYPH<31> 1 i ; GLYPH<31> 2 i g ; (A2) \nwe find 41 such duplicate configurations out of the total of 427 cases. 1 We evaluate di GLYPH<11> erences between these equalparameter cases for final spin and radiated energy. Figure 26 shows that, with strict tolerance, these error estimates (standard deviations of 3 : 1 GLYPH<2> 10 GLYPH<0> 4 for GLYPH<31> f and 1 : 6 GLYPH<2> 10 GLYPH<0> 4 for E rad) 2 are still on the same order but smaller than the respective fit residuals (RMSE of 5 : 2 GLYPH<2> 10 GLYPH<0> 4 for GLYPH<31> f and 2 : 2 GLYPH<2> 10 GLYPH<0> 4 for E rad). However, the set of true duplicates is small and mostly concentrated in equal-spin-similar-mass regions of the parameter space (cf. Fig. 27), preventing us from naively extrapolating this error estimate to the full parameter space. Hence we consider it as a somewhat optimistic estimate of final-state NR errors. \nTABLE XIII. NR cases from the source catalogs not included in the fit calibration, for reasons detailed below. \n| id | q | GLYPH<31> 1 | GLYPH<31> 2 ! 0 | D 0 E rad | GLYPH<1> E rad GLYPH<31> f | GLYPH<1> GLYPH<31> f tag | code |\n|------|----------|---------------|------------------------|-------------------------------|----------------------------------------|----------------------------------------------------------|--------|\n| 1 | | | 1.00 -0.80 -0.80 0.060 | | | 5.88 0.0325 -0.0010 0.4122 -0.0146 D6.2 q1 a-0.8 m100 | GaTech |\n| 2 | | | 1.00 -0.60 -0.60 0.058 | | | 5.93 0.0349 -0.0013 0.4876 -0.0066 D6.2 q1 a-0.6 m100 | GaTech |\n| 3 | 1.00 | | | 0.80 -0.80 0.025 10.92 0.0491 | | 0.0002 0.6839 -0.0000 D11 a0.8 q1.00 m103 As | GaTech |\n| 4 | 1.00 | 0.80 | | | 0.80 0.024 11.07 0.0883 -0.0005 0.9086 | 0.0010 D11 q1.00 a0.8 m200 | GaTech |\n| 5 | 2.50 | 0.60 | 0.60 0.051 | 6.27 0.0528 | 0.0002 0.8255 | 0.0004 Lq D6.2 q2.50 a0.6 th000 m140 | GaTech |\n| 6 | 3.50 | 0.00 | | 0.00 0.015 15.90 0.0258 | 0.0007 0.5046 | 0.0005 BBH CFMS d15.9 q3.50 sA 0 0 0 sB 0 0 0 SXS | |\n| 7 | | 5.00 -0.73 | 0.00 0.030 | 9.53 0.0129 | 0.0004 0.0222 | 0.0460 D10 q5.00 a-0.73 0.00 m240 | GaTech |\n| 8 | | 5.00 -0.72 | 0.00 0.030 | 9.54 0.0129 | 0.0004 0.0164 | 0.0340 D10 q5.00 a-0.72 0.00 m240 | GaTech |\n| 9 | | 5.00 -0.71 | 0.00 0.029 | 9.55 0.0130 | 0.0005 0.0105 | 0.0220 D10 q5.00 a-0.71 0.00 m240 | GaTech |\n| 10 | 5.00 | 0.00 | | | 0.00 0.027 10.07 0.0176 -0.0001 0.4175 | 0.0009 D10 q5.00 a0.0 0.0 m240 | GaTech |\n| 11 | 5.50 | 0.00 | 0.00 0.031 | 9.16 0.0161 | | 0.0001 0.3932 -0.0002 D9 q5.5 a0.0 Q20 | GaTech |\n| 12 | 6.00 | 0.00 | | | 0.00 0.027 10.13 0.0145 -0.0001 0.3732 | 0.0007 D10 q6.00 a0.00 0.00 m280 | GaTech |\n| 13 | 6.00 | 0.40 | | 0.00 0.026 10.35 0.0195 | | 0.0000 0.6257 -0.0000 D10 q6.00 a0.40 0.00 m280 | GaTech |\n| 14 | 8.00 | 0.85 | 0.85 0.048 | | 6.50 0.0248 -0.0027 0.8948 -0.0012 q8 | ++ 0.85 T 80 200 -4pc | BAM |\n| | 15 10.00 | 0.00 | 0.00 0.035 | | | 8.39 0.0082 -0.0000 0.2588 -0.0019 D8.4 q10.00 a0.0 m400 | GaTech |\n| | 16 10.00 | 0.00 | 0.00 0.035 | | 8.39 0.0081 -0.0001 0.2665 | 0.0058 q10c25e T 112 448 | BAM | \nFIG. 26. Di GLYPH<11> erences in final spin GLYPH<31> f (top panel) and radiated energy E rad (lower panel) for equal-parameter configurations but di GLYPH<11> erent NR codes. Solid circles: configurations with parameters equal to within numerical accuracy (narrow tolerance). Open circles: similar configurations but with some deviation in the parameters (wider tolerance, e.g. up to GLYPH<1> GLYPH<17> GLYPH<25> 0 : 001). Pairs of simulations are shown with a small horizontal o GLYPH<11> set for ease of visual identification. \n<!-- image --> \nWe therefore have a rough expectation for the range of possible NR errors bracketed by these pessimistic and optimistic estimates, but no detailed information for each case over the whole parameter space. Instead, we use simple heuristic fit weights. The overall scale of the NR error is not relevant for \nFIG. 27. Di GLYPH<11> erences in the final-state quantities for equal-parameter configurations and di GLYPH<11> erent NR codes. Top panel: final spin GLYPH<31> f , lower panel: radiated energy E rad. Points here correspond to both open and solid circles from Fig. 26 (wider tolerance). \n<!-- image --> \ndetermining fit weights, so we only need to assign relative weights between the cases, emulating the usual quadratic scaling with data errors which can also be deduced from Fig. 24. For SXS data we down-weight cases with GLYPH<17> < 0 : 1 by a factor of 2 2 ; while for the puncture codes ( BAM , GaTech, RIT) we expect larger inaccuracies especially at low GLYPH<17> , and so we \nFIG. 28. Unequal-spin e GLYPH<11> ects for final spin GLYPH<31> f at q = 5, shown as residuals against the 2D equal-spin fit (cf. Fig. 9). The three points highlighted in red are similar configurations from the GaTech catalog, for which it has since been confirmed [85] that the sign of GLYPH<31> f should be negative instead, making L 0 orb fit with the trend - corrected values are shown in green. \n<!-- image --> \ndown-weight by a factor of 2 2 above GLYPH<17> = 0 : 223 and 3 2 below that mass ratio, and a factor of 5 2 below GLYPH<17> = 0 : 05 (including the computationally challenging q = 18 cases). As mentioned before, a more detailed NR error study, leading to better-determined weights, would be a clear avenue to further improve fit results. \nFrom the original set of NR simulations we have removed 16 cases as outliers, which are listed in Table XIII. For this decision, we have considered three main sources of outliers: cases whose NR setup is not appropriate for the purpose of this study, duplicated configurations for which the variations in the final quantities are much larger than the RMSE, and cases that are found to be drastically o GLYPH<11> the trend of otherwise smooth data sets in any of the one-dimensional plots in our hierarchical fitting procedure. Outliers 1,2,5,16 have rather short orbital evolutions, so that they can be used for ringdown-only studies, but not for our purpose of predicting the final state from initial parameters. For outliers 10-12 and 15-16 we have found large variations in the final-state values for di GLYPH<11> erent codes (see Fig. 26). Here we have used only the equivalent SXS configuration, in each case corresponding to longer and presumably more accurate evolutions. The remaining seven outliers have been identified after performing the step-by-step one-dimensional analysis of the data, each deviating so clearly that there must be an underlying systematic problem and not just a statistical fluctuation (in which case they could not be excised from the data set). As an example, we highlight in Fig. 28 three clear GaTech outliers found in the unequal-spin calibration step; however, it was recently confirmed [85] that these three cases should have a negative sign of their final spin, and with this change they are fully consistent with our fits. We note that the overall data quality of the omitted cases may be perfectly adequate for other studies; while for this final-state study, due to good data coverage in the corresponding parameter-space regions and clear global trends in the full data set, the consistency requirements are quite narrow. \nWe also summarize in Table XIV 27 new BAM simulations first used in this paper, including recalculated values for previously published [9] mass ratio q = 18 simulations. \nTABLEXIV. New BAM simulations used in this work, with a focus on high unequal spins; and recalculated values for q = 18 cases from [9]. For each simulation, we list mass ratio q = m 1 = m 2, initial spins GLYPH<31> 1 and GLYPH<31> 2, reference orbital frequency GLYPH<10> 0, initial separation D 0 (after junk radiation), eccentricity e , radiated energy E rad (scaled to unit initial mass) and dimensionless final spin GLYPH<31> f . \n| q | GLYPH<31> 1 | GLYPH<31> 2 ! 0 D 0 e | GLYPH<2> 10 GLYPH<0> 3 i E rad | GLYPH<31> f |\n|-------|---------------|-------------------------|----------------------------------|---------------------|\n| 1.00 | | 0.00 -0.85 0.022 11.97 | 2.25 0.0392 | 0.5514 |\n| 1.00 | | 0.85 -0.85 0.023 11.61 | 2.61 0.0491 | 0.6854 |\n| 1.00 | | 0.50 -0.50 0.023 11.58 | 1.59 0.0485 | 0.6858 |\n| 1.20 | | 0.00 -0.85 0.020 12.79 | 0.74 0.0401 | 0.5747 |\n| 1.20 | | 0.50 -0.50 0.028 10.00 | 1.76 0.0503 | 0.7142 |\n| 1.20 | | 0.85 -0.85 0.028 10.00 | 5.16 0.0527 | 0.7359 |\n| | 1.50 -0.50 | 0.50 0.024 11.00 | 1.80 0.0408 | 0.5865 |\n| 1.75 | 0.00 | 0.85 0.022 11.69 | 2.35 0.0484 | 0.7033 |\n| 1.75 | | 0.00 -0.85 0.021 12.18 | 1.93 0.0369 | 0.5810 |\n| 1.75 | | 0.85 -0.85 0.023 11.59 | | 0.8167 |\n| | 1.75 -0.85 | 0.85 0.021 12.35 | 4.95 0.0567 2.66 0.0343 | 0.4607 |\n| 1.75 | 0.85 | 0.00 0.021 12.35 | 1.00 0.0682 | 0.8724 |\n| | 1.75 -0.85 | 0.00 0.020 12.69 | 0.58 0.0307 | 0.3934 |\n| 2.00 | | 0.50 -0.50 0.024 11.10 | 1.76 0.0464 | 0.7510 |\n| 2.00 | | 0.00 -0.85 0.023 11.47 | 2.85 0.0347 | 0.5693 |\n| 2.00 | 0.00 | 0.85 0.024 11.16 | 2.52 0.0436 | 0.6732 |\n| 2.00 | | 0.85 -0.85 0.024 11.00 | 1.78 0.0556 | 0.8344 |\n| | 2.00 -0.85 | 0.85 0.022 11.73 | 3.07 0.0310 | 0.4002 |\n| | 2.00 -0.50 | 0.50 0.023 11.53 | 2.60 0.0336 | 0.4925 |\n| | 2.00 -0.85 | 0.00 0.022 11.97 | 2.70 0.0292 | 0.3425 |\n| 2.00 | 0.85 | 0.00 0.023 11.36 | 4.02 0.0646 | 0.8782 |\n| | 3.00 -0.50 | 0.50 0.024 11.26 | 1.69 0.0237 | 0.3339 |\n| 3.00 | | 0.50 -0.50 0.025 10.63 | 1.41 0.0373 | 0.7410 |\n| | 3.00 -0.85 | 0.00 0.023 11.74 | 3.25 0.0201 | 0.1562 |\n| 4.00 | 0.00 | 0.85 0.026 10.52 | 14.79 0.0230 | 0.4900 |\n| | 4.00 -0.85 | 0.85 0.023 11.37 | 5.04 0.0158 | 0.0323 |\n| | 4.00 -0.50 | 0.50 0.024 10.99 | 1.68 0.0177 | 0.2152 |\n| 18.00 | 0.80 | 0.00 0.042 9.00 | 5.00 0.0087 | 0.8308 |\n| | 18.00 -0.80 | 0.00 0.027 10.00 | | 2.50 0.0031 -0.5270 |\n| | 18.00 -0.40 | 0.00 0.031 9.00 | | 1.30 0.0035 -0.1813 |\n| 18.00 | 0.00 | 0.00 0.040 7.58 | 1.30 0.0041 | 0.1623 |\n| 18.00 | 0.40 | 0.00 0.040 7.43 | 5.00 0.0057 | 0.5030 |", 'Appendix B: Fit assessment and model selection criteria': "Both while constructing the full ansatz in our hierarchical process, and when selecting the final fit, we rank fits by several standard statistical quantities, which are briefly summarized here for the benefit of the reader. \nA basic figure of merit is the root-mean-square-error, \nRMSE = v u t 1 N data N data X n = 1 GLYPH<2> X NR( GLYPH<17> n ; GLYPH<31> 1 ; n ; GLYPH<31> 2 ; n ) GLYPH<0> fit( GLYPH<17> n ; GLYPH<31> 1 ; n ; GLYPH<31> 2 ; n ) GLYPH<3> 2 ; (B1) \nwhich just checks the overall goodness of fit. One caveat here is that down-weighted NR cases are fully counted in the RMSE, so that a generalized variance estimator using weights can be more useful. \nFurthermore, it is important in model selection to penalize models with too many free coe GLYPH<14> cients, as in principle the RMSE can be made arbitrarily small when the number of coe GLYPH<14> cients approaches the number of data points. A popular \nfigure of merit for model selection considering the number of coe GLYPH<14> cients is the Akaike information criterion [45], \nAIC = GLYPH<0> 2 ln L max + 2 N coe GLYPH<11> s ; (B2) \nwhich intuitively can be understood as weighing up goodness of fit (measured by the maximum log-likelihood L max) against parsimony. Standard implementations, as the one from Wolfram Mathematica, assume Gaussian likelihoods. \nA generalization that corrects the AIC for low numbers of observations and reproduces it for large data sets is the AICc: \nAICc = AIC + 2 N coe GLYPH<11> s( N coe GLYPH<11> s + 1) N data GLYPH<0> N coe GLYPH<11> s GLYPH<0> 1 : (B3) \nIn this work, we always use AICc instead of AIC. \nA related quantity, similar in form but with a completely di GLYPH<11> erent theoretical justification and with subtle di GLYPH<11> erences in practice, is the Bayesian information criterion or Schwarz information criterion [46]: \nBIC = GLYPH<0> 2 ln L max + N coe GLYPH<11> s ln( N data) : (B4) \nThough based on an approximation to full Bayesian model selection (while the AIC is derived from information theory), the BIC in general cannot be interpreted as a direct measure of Bayesian evidence between models. \nThere is much literature on advantages and disadvantages of these two criteria, and several other alternatives exist - see e.g. [86] and references therein. In practice, the BIC tends to impose a slightly stronger penalty on extra parameters than the AIC(c). Both criteria have been criticized [86] for not only penalizing completely extraneous parameters, but also the existence of degeneracies between parameters. However, this is a virtue rather than a problem for our purpose of selecting parsimonious model ansatze. \nFor all of AIC, AICc and BIC, the model with the lowest value is preferred. Higher than unit di GLYPH<11> erences between two models are generally required to count as significant evidence; [86] quotes GLYPH<6> 5 as 'strong' and GLYPH<6> 10 as 'decisive' evidence. \nBy default, we rank the one-dimensional GLYPH<17> and b S ansatze by BIC, and apply the same criterion to judge how many GLYPH<1> GLYPH<31> terms to include in the final 3D model. In general these are not guaranteed to be the best by AICc or RMSE as well, but in practice we check that the three criteria give a very similar ranking of models. \nStill, we find that sometimes the best fit by any of these three criteria (RMSE, AICc, BIC) can su GLYPH<11> er from one or several parameters being not well constrained. In parameter estimation over large measured data sets, such as the CMB example in [86], this is a desirable feature of model selection, as it allows to assess which physics is actually constrained by the data. However, in our case we are well aware that our data set does not constrain all possible functions over the parameter space, and we are more interested in reporting a wellconstrained model that captures the information in the data set than to dig out weak constraints on possible extensions of that model. So we augment our model selection criteria by considering also the well-constrainedness of each individual fit coe GLYPH<14> cient, and allow for picking a fit with slightly worse summary statistics if it has better-constrained coe GLYPH<14> cients; or we drop individual coe GLYPH<14> cients from a high-order ansatz and reassess the quantitative criteria for that reduced model. \nFIG. 29. BIC for the one-dimensional L 0 orb GLYPH<16> GLYPH<17>; b S = 0 GLYPH<17> fits from Sec. III B. The inset panel is a zoom-up of the top-ranked fits. The tested set of ansatze includes all polynomials from second to seventh order in GLYPH<17> and all rational functions of order ( i ; j ), j GLYPH<20> i , up to i + j = 6. The preferred ansatz, a rational function of order (3,1) with three free coe GLYPH<14> cients, is highlighted. \n<!-- image --> \nAs an example of how e.g. the BIC can guide model selection, we show in Fig. 29 the BIC ranking for the onedimensional L 0 orb GLYPH<16> GLYPH<17>; b S = 0 GLYPH<17> fits from Sec. III B. A plateau of almost constant BIC is made up of several fits with N coe GLYPH<11> s GLYPH<21> 3, with the more complex fits yielding no additional improvement, so that we choose the simplest fit among this group. Still, even if it had not come up actually top-ranked, as in this case, choosing a lowN coe GLYPH<11> s fit from within the high-ranked group would be preferable over some slightly higher-ranked, but less-well-constrained fit.", 'Appendix C: Spin parameter selection': "The results of the main text are given in terms of the spin parameter b S . However, there is no unique definition of an 'e GLYPH<11> ective spin', and alternative parametrizations have been used in the literature [9, 55]. We have tested the robustness of our hierarchical approach for two additional spin parameters: \nb S = S m 2 1 + m 2 2 ; S = m 2 1 GLYPH<31> 1 + m 2 2 GLYPH<31> 2 ; GLYPH<31> e GLYPH<11> = m 1 GLYPH<31> 1 + GLYPH<31> 2 m 2 : \nWe redid the hierarchical ansatz construction and fitting for S and GLYPH<31> e GLYPH<11> , making the same ansatz choices for GLYPH<31> e GLYPH<11> as we did for b S in the main text, but changing the 1D spin ansatz to a polynomial P(7) for S (instead of R(3,1) for b S and GLYPH<31> e GLYPH<11> ) because rational functions in S tend to yield singularities. Checking other possible choices, we have not found any ansatz combination that makes these alternatives match or exceed the performance of the b S -based fits presented in the main part of this paper. Results in terms of the RMSE, AICc and BIC are listed in Table XV, and residual histograms shown in Fig. 30. We still obtain better results than most previous fits (see Tables VI and XII) with any parametrization, thus demonstrating the robustness of our method. \nAgain, we have also analyzed the fit in the extrapolation regions to detect any artifacts not reflected by the statistical criteria (which are meaningful only in the calibrated region). In Fig. 31 we check the extrapolation behavior of fits with the alternative parametrizations in the notoriously di GLYPH<14> -cult GLYPH<31> 1 = GLYPH<31> 2 = 1 limit. The approach to this limit is smoother \n| | RMSE | AICc | BIC |\n|-----------------------|-------------------------------|-------------------|-------------------|\n| ˆ S | 5 : 15 GLYPH<2> 10 GLYPH<0> 4 | GLYPH<0> 5991 : 5 | GLYPH<0> 5923 : 9 |\n| S | 5 : 24 GLYPH<2> 10 GLYPH<0> 4 | GLYPH<0> 5930 : 9 | GLYPH<0> 5863 : 3 |\n| GLYPH<31> e GLYPH<11> | 5 : 97 GLYPH<2> 10 GLYPH<0> 4 | GLYPH<0> 5799 : 6 | GLYPH<0> 5731 : 9 | \nTABLE XV. Summary statistics for fits with three di GLYPH<11> erent choices of e GLYPH<11> ective spin parameter and ansatz choices as discussed below, evaluated over the full 427 point NR data set. Top table: Final spin, lower table: radiated energy. \n| RMSE | AICc | BIC |\n|-----------------------|---------------------------------|-------------------|\n| ˆ S | 10 GLYPH<0> 4 GLYPH<0> 6454 : 8 | GLYPH<0> 6391 : 0 |\n| S | 10 GLYPH<0> 4 GLYPH<0> 5526 : 1 | GLYPH<0> 5439 : 1 |\n| GLYPH<31> e GLYPH<11> | 10 GLYPH<0> 4 GLYPH<0> 5962 : 7 | GLYPH<0> 5898 : 9 | \nFIG. 30. Fit residuals for three di GLYPH<11> erent choices of e GLYPH<11> ective spin parameter. Top panel: final spin; lower panel: radiated energy. \n<!-- image --> \nfor the fits using b S and GLYPH<31> e GLYPH<11> than for that using S , which shows some certainly nonphysical oscillations. \nThe conclusion is that the hierarchical fitting method is quite robust under a change of e GLYPH<11> ective-spin parametrization, and indeed we would expect full equivalence in the limit of a huge data set with small, completely known NR errors (using appropriately adapted ansatze for each parametrization). With the current data set, b S and GLYPH<31> e GLYPH<11> perform similarly, while when using S additional high-spin data would be even more important to ensure smooth extrapolation. \n<!-- image --> \nFIG. 31. Final-state quantities in the extremal GLYPH<31> 1 = GLYPH<31> 2 = 1 limit for three di GLYPH<11> erent choices of e GLYPH<11> ective spin parameter. Top panel: final spin, lower panel: radiated energy. \n<!-- image -->", 'Appendix D: Fit uncertainties': "The uncertainty of evaluating a fitted quantity Q at a point ( GLYPH<17>; GLYPH<31> 1 ; GLYPH<31> 2) can be expressed through prediction intervals [87] \nQ ( GLYPH<17>; GLYPH<31> 1 ; GLYPH<31> 2) GLYPH<6> qt ( x ; N data GLYPH<0> N coe GLYPH<11> s) q b GLYPH<27> 2 + GLYPH<27> 2 fit ; (D1) \nwhere qt is the student-t quantile for a confidence level x , b GLYPH<27> 2 is the error variance estimator from the (weighted) meansquare error of the calibration data under the fit, and GLYPH<27> 2 fit is the standard error estimate of the fitted model, which for a singlestage fit is \nGLYPH<27> 2 fit = grad t ( GLYPH<17>; GLYPH<31> 1 ; GLYPH<31> 2) GLYPH<1> C fit GLYPH<1> grad ( GLYPH<17>; GLYPH<31> 1 ; GLYPH<31> 2) (D2) \nwith the gradient vector grad ( GLYPH<17>; GLYPH<31> 1 ; GLYPH<31> 2) of the fit ansatz in the coe GLYPH<14> cients, evaluated at this point, and the covariance matrix C fit of the fit. Note that Eq. (D1) gives the uncertainty for a single additional observation, as opposed to the narrower confidence interval of the mean prediction, which lacks the b GLYPH<27> 2 term. \nIn our hierarchical fitting approach, to propagate the uncertainties from the nonspinning, equal-mass and extreme-massratio limits, we have to assume that the uncertainties in these regimes and that of the final fit are independent, so that we can take the full covariance matrix as a block-diagonal composition of these four contributions. The half width of a prediction \ninterval at confidence x is then \nqt ( x ; N data GLYPH<0> N coe GLYPH<11> s) q b GLYPH<27> 2 + GLYPH<27> 2 final + GLYPH<27> 2 GLYPH<17> + GLYPH<27> 2 b S + GLYPH<27> 2 GLYPH<17> = 0 : (D3) \nAs these three particular regimes are significantly better constrained than the bulk of the parameter space (which is the \n- [1] B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), Phys. Rev. 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2005PhRvD..72j3517B
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New signature of dark matter annihilations: Gamma rays from intermediate-mass black holes
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2005-01-01
| 7 | 0.46 | 161 |
['-', '-', '-', '-', 'cosmology dark matter', 'black hole physics', '-', 'cosmic rays', 'astrophysics', '-']
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[]
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We study the prospects for detecting gamma rays from dark matter (DM) annihilations in enhancements of the DM density (mini-spikes) around intermediate-mass black holes (IMBH) with masses in the range 10<SUP>2</SUP>≲M/M<SUB>⊙</SUB>≲10<SUP>6</SUP>. Focusing on two different IMBH formation scenarios, we show that, for typical values of mass and cross section of common DM candidates, mini-spikes, produced by the adiabatic growth of DM around pregalactic IMBHs, would be bright sources of gamma rays, which could be easily detected with large field-of-view gamma-ray experiments such as GLAST, and further studied with smaller field-of-view, larger-area experiments like Air Cherenkov Telescopes CANGAROO, HESS, MAGIC, and VERITAS. The detection of many gamma-ray sources not associated with a luminous component of the Local Group, and with identical cutoffs in their energy spectra at the mass of the DM particle, would provide a potential smoking-gun signature of DM annihilations and shed new light on the nature of intermediate and supermassive black holes.
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https://arxiv.org/pdf/astro-ph/0509565.pdf
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{'A New Signature of Dark Matter Annihilations: Gamma-Rays from Intermediate-Mass Black Holes': 'Gianfranco Bertone, 1 Andrew R. Zentner, 2 and Joseph Silk 3 \n1 Particle Astrophysics Center, Fermi National Accelerator Laboratory, Batavia, Illinois 60510-0500, USA 2 Kavli Institute for Cosmological Physics and Department of Astronomy and Astrophysics, The University of Chicago, Chicago, IL 60637,USA 3 Astrophysics, Denys Wilkinson Building, Keble Road, Oxford OX1 3RH, United Kingdom (Dated: September 23, 2018) \nWe study the prospects for detecting gamma-rays from Dark Matter (DM) annihilations in enhancements of the DM density (mini-spikes) around intermediate-mass black holes with masses in the range 10 2 ∼ < M/ M /circledot ∼ < 10 6 . Focusing on two different IMBH formation scenarios, we show that, for typical values of mass and cross section of common DM candidates, mini-spikes, produced by the adiabatic growth of DM around pregalactic IMBHs, would be bright sources of gamma-rays, which could be easily detected with large field-of-view gamma-ray experiments such as GLAST, and further studied with smaller field-of-view, larger-area experiments like Air Cherenkov Telescopes CANGAROO, HESS, MAGIC and VERITAS. The detection of many gamma-ray sources not associated with a luminous component of the Local Group, and with identical cut-offs in their energy spectra at the mass of the DM particle, would provide a potential smoking-gun signature of DM annihilations and shed new light on the nature of intermediate and supermassive Black Holes. \nPACS numbers: 95.35.+d,97.60.Lf,98.62.Js,98.70.Sa,98.70.Rz \nFERMILAB-PUB-05-408-A', 'I. INTRODUCTION': "Although many astrophysical and cosmological observations provide convincing evidence for the existence of a 'dark' component in the matter density of the Universe, the nature of this dark matter (DM) remains unkown. It is commonly assumed that DM is made of new, as yet undiscovered, particles, associated with theories beyond the Standard Model of Particle Physics. Among the most widely studied candidates are the supersymmetric neutralino and candidates arising in theories with extradimensions, which appear difficult to constrain with direct searches (i.e. by looking for nuclear recoils due to DMparticles scattering off nuclei) and whose prospects of discovery at future accelerators strongly depend on the details of the particle physics setup (for recent reviews see e.g. Refs. [1, 2]). Indirect searches via the detection of annihilation radiation may provide an interesting alternative, but they are usually affected by large astrophysical and cosmological uncertainties. Furthermore, in many cases, the detection of an annihilation signal may be difficult to distinguish from less exotic astrophysical sources. An example of this is the case of the Galactic center, where such high-energy radiation has been recently observed by several different experiments, without providing any conclusive evidence for or against an interpretation in terms of DM annihilation products (see Refs. [3, 4, 5, 6, 7] and references therein). \nHere we describe a scenario that may provide smokinggun evidence for the annihilation of DM particles. If intermediate-mass black holes (IMBHs), with a mass ranging between 10 2 and 10 6 M /circledot (e.g. [8]), exist in the Galaxy, their adiabatic growth would have modified the DM distribution around them, leading to the formation of 'mini-spikes', that is, large, local enhance- \nments of the DM density [9]. The DM annihilation rate being proportional to the square of the number density of DM particles, these mini-spikes would be bright gamma-ray sources, distributed in a roughly sphericallysymmetric way about the galactic center, and well within the observational reach of the next-generation gammaray experiments. Their brightness and isotropy make them ideal targets of large field-of-view gamma-ray experiments such as GLAST [10]. In case of a positive detection, Air Cherenkov Telescopes such as CANGAROO [11], HESS [12], MAGIC [13] and VERITAS [14] could extend the observations to higher energies and improve the angular resolution. We argue that the observation of numerous (up to ∼ 100) point-like gamma-ray sources with identical cut-offs in their energy spectra, at an energy equal to the mass of the DM particle, would provide smoking-gun evidence for DM particles. \nIn this paper, we make predictions for the number of detectable black holes in two different IMBH formation scenarios. In the first scenario, IMBHs form in rare, overdense regions at high redshift, z ∼ 20, as remnants of Population III stars, and have a characteristic mass-scale of a few 10 2 M /circledot [15] (a similar scenario was investigated in Ref. [9, 16, 17]). In this scenario, these black holes serve as the seeds for the growth supermassive black holes found in galactic spheriods [18]. In the second scenario, IMBHs form directly out of cold gas in early-forming halos, in a sense that will be specified below, and and are typified by a larger mass scale, of order 10 5 M /circledot . We demonstrate that, with respect to Ref. [9], the latter scenario leads to qualitative differences in the mini-spike profiles with dramatic consequences for the detectability of gamma-ray fluxes. For both scenarios, we make detailed estimates of the population of IMBH in the Milky Way DM halo using a complete model of IMBH forma- \non at high redshift, black hole mergers, and halo merger and evolution [19]. This allows us the unique ability to make a detailed study of the detectabilty of mini-spikes as gamma-ray sources. \nThis paper is organised as follows. In Sec. II A, we review the evidence and formation scenarios for IMBHs. In Sec. II B, we describe the model that we employ to estimate the properties of the local IMBH population, and we present the main properties (radial profile, mass function etc.) of IMBH populations in Milky Way-like halos in Sec. II C. Sec. III is devoted to the calculation of the mini-spike profiles and Sec. IV to the DM annihilation fluxes. Sec. V contains our primary results on the observability of gamma-rays from the annihilation of DM around IMBHs. In Sec. VI, we discuss the implications of our results and draw our conclusions. We perform all of our calculations in the context of a standard, flat cosmological constant plut cold DM (ΛCDM) cosmology with Ω M = 0 . 3, Ω Λ = 0 . 7, h = 0 . 7 and a scale-invariant primordial power spectrum with a normalization set by σ 8 = 0 . 9.", 'A. The case for IMBHs': "In the last few years, observational and theoretical evidence has accumulated [8] for the existence of compact objects, heavier than stellar black holes, but lighter than the so-called supermassive black holes (SMBHs) lying at the centers of galactic spheroids. We consider here the mass range 20 ∼ < M IMBH / M /circledot ∼ < 10 6 , where the lower bound of the IMBHs mass corresponds to recent estimates of the maximum mass of the remnant of a massive stellar collapse [20], and the upper limit roughly indicates the minimum mass of SMBHs, assumed to lie in the range 10 6 ∼ < M SMBH / M /circledot ∼ < 10 9 (see e.g. Ref. [18] for a recent review). \nA hint of the existence of IMBHs is provided by the detection of bright, X-ray, point sources, called ultraluminous X-ray sources (ULXs), that are apparently not associated with active galactic nuclei [21, 22, 23]. Although many known X-ray sources are associated with neutron stars and black holes, this intepretation fails in the case of ULXs. ULXs would have to emit radiation far above the Eddington limit, if M ∼ < 20 M /circledot , and their positions in their host galaxies are not compatible with masses M ∼ > 10 6 M /circledot , because dynamical friction would cause these objects to sink to the centers of their hosts on a timescale shorter than a Hubble time (e.g. Ref. [8]). Accretion by IMBHs has been advocated as a possible explanation [17]. Another hint for the existence of IMBHs, although not conclusive, comes from stellar kinematics in globular clusters [24]; the observed relation between the mass and the velocity dispersion in selected globular clusters may fall on the extrapolation of the analogous relation for SMBHs [25, 26, 27]. \nFrom a theoretical point of view, a population of massive seed black holes could help to explain the origin of SMBHs. In fact, observations of quasars at redshift z ≈ 6 in the Sloan Digital survey [28, 29, 30] suggest that SMBHs were already in place when the Universe was only ∼ 1 Gyr old, a circumstance that can be understood in terms of rapid growth starting from massive seeds (see e.g. Ref. [31]). Furthermore, the growth of SMBHs through accretion and merging of heavy seeds may aid in the understanding some of the observed relationships between supermassive black hole masses and the properties of their host galaxies and halos [32, 33, 34, 35, 36, 37]. Scenarios that seek to explain the properties of the observed supermassive black hole population generally result in the prediction of a concomitant population of 'wandering' IMBHs throughout massive DM halos and the intergalactic medium [19, 38, 39]. However, despite their theoretical interest, it is difficult to obtain conclusive evidence for the existence of IMBHs. A viable detection strategy could be the search for gravitational waves produced in the mergers of the IMBH population [19, 40, 41, 42, 43, 44], which may become possible with the advent of space-based interferometers such as LISA.", 'B. IMBHs formation scenarios': "We focus here on two scenarios leading to the formation of black holes at very different mass scales. In the first scenario (which we refer to as scenario A ), black holes are remnants of the collapse of Population III (or 'first') stars [15]. Numerical simulations suggest that the first stars may form when primordial molecular clouds with ≈ 10 5 M /circledot cool by formation and distruction of H 2 into cold pockets at the centers of their DM halos, with typical densities of order 10 4 cm -3 and temperatures of order a few × 10 2 K [45, 46], and become gravitationally unstable. \nNewtonian simulations suggest that the fate of Pop III stars is very different from the case of their metal-enriched, comparably less massive counterparts mentioned above. Zero metallicity Pop III stars with masses in the range M ∼ 60 -140 M /circledot and M ∼ > 260 M /circledot collapse directly to black holes, stars with 140 ∼ < M/ M /circledot ∼ < 260 M /circledot are completely disrupted due to the pulsation-pair-production instability, leaving behind no remnant, and again stars with masses M ∼ > 260 M /circledot collapse directly to black holes [47] (see also Refs. [48, 49, 50, 51]). The evolution timescale of these very massive stars is typically of order t ∗ ∼ 1 -10 Myr. After this timescale, supernovae begin to explode, releasing energy and metals into the surrounding medium. In the standard picture of hierarchical structure formation, the metal-enriched material will be collected at later times at the centers of more massive haloes, where new generations of stars will form. \nInterestingly, if a ∼ 10 2 M /circledot black hole forms halos that \nrepresent ∼ 3 σ peaks of the smoothed density field, the resulting baryonic mass fraction in these objects would be comparable with the mass fraction in SMBHs [15]. Additionally, such a scenario leads to the natural prediction of a population of 'wandering' black holes in the halos of Milky Way-sized galaxies, with masses similar to their initial mass scale M ∼ 10 2 M /circledot , as many of the relatively small halos ( M ∼ 10 7 M /circledot for 3 σ fluctuations at z = 18) that host early-forming black holes do not merge with the central galaxy, but orbit about the periphery of the halo [19, 38, 39]. We stress that black holes in this scenario may not necessarily form at the very centers of their initial host dark matter halos at high redshift, a circumstance that, as we shall see, may have important consequences on the detectability of IMBHs. \nTo represent the predictions of this class of black hole formation scenario where black holes form at ∼ 100 M /circledot from the remnants of the first stars, we use a model similar to that proposed by Madau and Rees [15] and studied in further detail by Islam et al. [38] and Volonteri et al. [39]. Specifically, at z = 18, we populate halos that constitute 3 σ peaks in the smoothed primordial density field with seed black holes of initial mass 100 M /circledot . We evolve these halos using an analytic model of halo growth that is focused on making many statistical realizations of the growth of a Milky Way-sized halo. After populating progenitor halos at high redshift with black holes as described above, these processes of halo growth and evolution are treated as described in detail in [52, 53] and [19, 54]. We refer the reader to these references for details and tests of the halo evolution models. For the purposes of this study, we take the mass of the Milky Way halo to be M MW = 10 12 . 1 h -1 M /circledot and perform 200 statistical realizations in order to ascertain the expected range of observable IMBHs. \nThe second scenario that we consider ( scenario B ) is based on the proposal of Ref. [37] and it is representative of a class of models in which black holes originate from massive objects formed directly during the collapse of primordial gas in early-forming halos [55, 56, 57, 58, 59, 60]. In this class of models, the initial black holes are massive ( ∼ 10 5 M /circledot ) and the growth of SMBH proceeds in such a way that both mergers and accretion play an important role. We use the model of Ref. [37] to represent the predictions of models that start SMBH growth from very massive seeds. The proposal of Ref. [37] is as follows. During the virialization and collapse of the first halos, gas cools, collapses, and forms pressure-supported disks at the centers of halos that are sufficiently massive to contain a relatively large amount of molecular hydrogen (molecular hydrogen is the primary gas coolant in halos in the relevant mass range, see [61] for a review). In halos that are both massive enough that molecular hydrogen cooling is efficient and which do not experience any major mergers over a dynamical time, a protogalactic disk forms and can evolve uninterrupted. An effective viscosity due to local gravitational instabilities in the disk leads to an effective viscosity that transfers mass inward and \nangular momentum outward [62] until supernovae in the first generation of stars heat the disk and terminate this process [37]. By the time the process terminates (of order the lifetimes of Pop III stars, ∼ 1 -10 Myr), a baryonic mass of order ∼ 10 5 M /circledot loses its angular momentum and is transferred to the center of the halo. Such an object may be briefly pressure-supported, but it eventually collapses to form a black hole [47, 63]. \nThe requirements that the early-forming host halo be massive enough to form an unstable disk and that the halo not experience a major merger imprints a typical mass scale for halos within which this process occurs of order ∼ 10 7 M /circledot . In this case the characteristic mass of the black hole forming in a halo of virial mass M v is given by \nM bh = 3 . 8 × 10 4 M /circledot ( κ 0 . 5 ) ( f 0 . 03 ) 3 / 2 ( M v 10 7 M /circledot ) ( 1+ z 18 ) 3 / 2 ( t 10Myr ) , (1) \nwhere we have assumed that a fraction f is the fraction of the total baryonic mass in the halo that has fallen into the disk, z is the redshift of formation, κ is that fraction of the baryonic mass which loses its angular momentum that remains in the remnant black hole, and t is the timescale for the evolution of the first generation of stars [37]. The distribution of black hole masses is a log-normal distribution with a mean given by the characteristic mass above and a standard deviation σ M bh = 0 . 9. The spread is determined by the spread in total angular momentum exhibited by halos of fixed mass in cosmological N -body simulations of DM halo formation [64]. Using the prescriptions of the Koushiappas et al. [37] model, we can again populate halos with black holes at high redshift and evolve them forward to determine the properties of satellite black holes in a statistically large sample of Milky Way-like halos at z = 0. This is precisely what was done in Ref. [19] in order to study the gravity wave background and we refer the reader to this work for further details. As with scenario A, we take the Milky Way halo to have mass M MW = 10 12 . 1 h -1 M /circledot at z = 0 and construct 200 realizations of wandering black hole populations in halos of this mass.", 'C. Intermediate-Mass Black Holes in Milky Way-sized Halos': 'In the previous two sections, we outlined models for the production of IMBHs in the early universe and evolution of IMBHs in their host halos in the context of the hierarchical CDM model of structure formation. Of course, as halos merge to form larger systems that eventually grow to the size of the Milky Way, black holes merge, producing supermassive, central black holes and perhaps a detectable gravity wave signal. These products have been the focus of most previous work regarding these models [19, 37, 39, 65]. Consequently, these studies focused \nFIG. 2: Cumulative radial distribution of unmerged IMBHs in the scenario A (red) and B (black), for a Milky Way Halo at z=0. The mean and error are based on 200 Monte Carlo realizations of IMBH populations in Milky Way-sized halos. Notice that unlike subhalo populations, IMBHs do not exhibit a significant anti-bias with respect to the DM. Rather, they are slightly biased toward being found near the halo center. \n<!-- image --> \nFIG. 1: Mass function of unmerged IMBHs in the scenario B, for a Milky Way Halo at z=0. The distribution is based on an average of 200 Monte Carlo realizations of a halo of virial mass M v = 10 12 . 1 h -1 M /circledot , roughly the size of the halo of the Milky Way. \n<!-- image --> \nmuch attention on the merging of black holes as halos and galaxies merge. On the contrary, we are most interested in those pristine black holes that are orbiting within the Milky Way halo and have not merged with other black holes because these unmerged black holes may still reflect the properties of the dark matter density enhancement in which they formed. \nIn scenario A, the mass spectrum of unmerged black holes is a delta function as described in Section II A. The average number of unmerged black holes per Milky Way halo is N bh , A /similarequal 1027 ± 84, where the errorbar denotes the 1 σ scatter from halo-to-halo. In scenario B, the total number of unmerged black holes per Milky Way halo is N bh , B /similarequal 101 ± 22. We show in Fig. 2 the final mass spectrum (i.e. at redshift z = 0) of black holes in scenario B. As expected the distribution follow closely the initial mass spectrum, with a characteristic mass of order ≈ 10 5 M /circledot . The only deviation is that the overall distribution is slightly broadened by the fact that not all black holes form at the same redshift in halos of the same mass (see Eq. [1] and Refs. [19, 37]). The radial distribution of unmerged black holes is less trivial, and it would be more difficult to derive directly from the models of IMBH formation at high redshift. The distribution is essentially set by the energy and angular momentum distributions of merging objects in a ΛCDM cosmology and dynamical friction (e.g. [53]). Unlike dark matter substructures, which are generally absent from the inner parts of the host halos, because they tend to lose mass via tidal mass loss and heating, black holes and the sur- \nng dark matter distribution in the vicinity of the IMBHs can survive tidal disruption to very small galactocentric distances. The final, cumulative radial distributions of unmerged IMBHs are shown in Fig. 2. They are very similar for scenarios A and B (though the normalization is different), and shows a behavior that scales as dN/dr ∼ r -3 at large scales and tends toward a shallower slope on scales smaller than the scale radii of typical MW-size halos (see the following section).', 'III. THE DENSITY ENHANCEMENT OF DARK MATTER AROUND IMBHS': "In each early-forming halo that hosts a seed black hole, when the black hole forms the DM distribution about the black hole inevitably reacts, adjusting to the new gravitational potential. This process has been studied extensively, particularly in the context of stellar cusps around massive black holes in clusters of stars or at the centers of galaxies (see e.g. [66, 67, 68, 69]). Gondolo and Silk have applied this argument to the distribution of DM at the center of the Galaxy [71] and introduced the term 'spike' for the consequent enhancement in the DM density around the central SMBH, in order to avoid confusion with DM 'cusps' at the centers of halos in the cold dark matter model of structure formation. It was subsequently shown that dynamical processes like off-center formation \nof the seed black hole, or major merger events, may lead to destruction or reduction of the spike [72, 73]. However, steeply rising stellar cusps in the innermost regions of galaxies suggest that such processes were not effective, at least in the case of the Milky Way, or that the stellar cusps were re-generated via star formation [74] or energy exchange between stars [75]. \nRecently, Bertone and Merritt studied the evolution of DM spikes including gravitational scattering off stars and the self-annihilation of DM particles [6, 7], showing that the DM density in spikes is, indeed, substantially reduced by these effects, but the enhancement of the annihilation signal is still significant with respect to ordinary DM cusps. \nIn the present study, we are interested in 'mini-spikes' surrounding IMBHs. Because we track the merger history of each individual black hole, we can select precisely those black holes which never experienced mergers, to ensure that major mergers have not destroyed any cusp that existed around the original black hole. \nFurthermore, the models we explore predict from between a few hundred to a few thousand black holes scattered throughout the Milky Way halo, and as the Milky Way has only 11 luminous companions within ∼ 300 kpc [76], we expect that the majority of the wandering black holes in our models reside in satellite halos with no significant stellar component. \nThis implies that the effects of scattering off of stars should not significantly alter the DM distributions around the wandering IMBHs. The mini-spikes around unmerged, wandering, IMBHs are thus less sensitive to all of the dynamical processes that may have affected the spike at the Galactic center. \nWe proceed now to evaluate the DM enhancements around IMBHs. As a first step, we need to specify the 'initial' DM profile, that is, the DM distribution prior to black hole formation. Let the subscript ' f ' denote quantities at the time when the IMBH formed. The initial DM profile of the mini-halo, before adiabatic growth, can be well approximated with a Navarro, Frenk, and White (NFW) profile [77] \nρ ( r ) = ρ 0 ( r r s ) -1 ( 1 + r r s ) -2 (2) \nThe normalization constant ρ 0 , and the scale radius r s , can be expressed in terms of the virial mass of the halo at the time when the IMBH formed M vir ,f , and the virial concentration parameter c vir ,f \nr s = r vir ,f c vir ,f , ρ 0 = M vir ,f 4 πr 3 s f ( c vir , f ) . (3) \nWe recall that the virial mass is related to the virial radius r vir ,f by \nM vir ,f = 4 π 3 [∆ vir ( z f ) ρ m ( z f )] r 3 vir ,f , (4) \nwhile the function f ( x ) is, apart from constants, simply the volume integral of the NFW profile f ( x ) ≡ ln(1 + x ) -[ x/ (1 + x )]. \nIn Eq. (4), ρ m ( z f ) is the mean DM density at the redshift of formation z f , while ∆ vir ( z f ) is the virial overdensity, for which we have adopted here the fitting form of Bryan and Norman [78]. At the redshifts of interest ( z ∼ > 12) the universe is DM-dominated and the expansion rate and growth of perturbations are described by the standard relations for an Ω M = 1, 'standard' CDM cosmology. In this case, ∆ vir ( z f ) /similarequal 18 π 2 /similarequal 178. For each black hole at redshift z = 0 we extract from its merger tree the parameters M vir ,f , c vir , f , z f and use Eq. (2) to calculate the initial DM profile before the formation of the black hole. Alternatively, we could have chosen the more recent parametrization proposed by Navarro et al. [79] (see also Refs. [80, 81]). However, this profile implies modifications at scales smaller than those we are interested in, where the profile is anyway modified by the presence of the IMBH. \nWe assume that the black holes form over a timescale long enough to guarantee adiabaticity, but short compared to the cosmological evolution of the host halo (in scenario B, both of these assumptions are built into the black hole formation model, see Section II B as well as our discussion below). \nAdiabaticity requires that the formation time of the black hole is much larger than the dynamical timescale at a distance r h from the black hole, where r h is the radius of the sphere of gravitational influence of the black hole, r h /similarequal GM bh /σ 2 , and σ is the velocity dispersion of DM particles at r h . In practice, we estimate r h by solving the implicit equation \nM ( < r h ) ≡ ∫ r h 0 ρ ( r ) r 2 d r = 2 M bh . (5) \nFor a representative case in scenario B, with M bh = 10 8 M /circledot and M vir ,f = 10 8 M /circledot , this gives r h /r s ≈ 0 . 04. In scenario B, the black hole formation time is set by the timescale for viscous angular momentum loss and is limited by the evolutionary timescale of the first stars and the gravitational infall time across the gaseous disk, which is of order Myr (see Ref. [19] for a detailed discussion of timescales). The relevant timescale for the mass build up of the IMBH is then t ev ∼ 1 -20 Myr. In scenario A, we follow Ref. [9] where the characteristic timescale for the growth of the black hole by accretion is taken to be of order 1 -20 Myr for a plausible range of accretion efficiencies. \nThe basics of adiabatic growth can be easily understood (e.g. Ref. [2]), and in most cases the details can be worked out by taking into account the approximate conservation of adiabatic invariants under a certain set of assumptions. If one starts from an initially uniform DM distribution, the final profile will be a mild mini-spike with density ρ sp ∝ ( r/r h ) 3 / 2 (e.g. see [69] and references therein). If one starts from a cuspy profile, such as the NFW profile of Eq. (2), the new profile is essentially a \npower-law, \nρ sp ( r ) = ρ ( r sp ) ( r r sp ) -γ sp (6) \nwhere the radius of the spike is r sp ≈ 0 . 2 r h [70], and γ sp is related to the initial power-law index γ by [71] \nγ sp = 9 -2 γ 4 -γ . (7) \nIn the case of the profile of Eq. (2), this reduces to γ sp = 7 / 3. \nThe DM annihilation flux in this case diverges at small radii. However, the very annihilations that we study here provide an upper limit to the DM number (and thus mass) density. In absence of other processes affecting the distribution of DM, the DM density obeys the equation \n˙ n χ ( r, t ) = -σv n 2 χ ( r, t ) (8) \nwhere σv is the annihilation cross-section times relative velocity (in the non-relativistic limit) and mχ is the DM particle mass. The solution to the evolution equation is \nn χ ( r, t ) = n χ ( r, t f ) 1 + n χ ( r, t f ) σv ( t -t f ) (9) \nwhich shows that efficent annihilations set an upper limit to the matter density of order m χ /σv ( t -t f ). We define r lim as the radius where \nρ sp ( r lim ) = m χ /σv ( t -t f ) ≡ ρ lim . (10) \nWe therefore define an inner cut-off at a radius \nr cut = Max[4 R Schw , r lim ] (11) \nwhere R Schw is the Schwarzschild radius of the IMBH R Schw = 2 . 95 km M bh / M /circledot . For common values of the mass and cross section of the DM particle, r lim ∼ 10 -3 pc so that r cut = r lim . \nIs the adiabatic growth of a central mass a good approximation in our IMBH formation scenarios? We have already discussed the timescales involved, but the derivation of the inner radius r lim provides us with the possibility of checking whether the size of the region where matter accretes, leading to the formation of a black hole, is actually smaller than the characteristic size of the DM spike, r lim . In scenario A, this is not a problem, because in this case the black holes from from Pop. III stars and the spike is produced by the growth of the black hole, thus by processes occuring on scales of order R Schw << r lim . In scenario B, the situation is different, because the minispike is produced by the flow of protogalactic material that lost its angular momentum by viscosity. \nSuch an object may collapse directly to a black hole or it may form a short-lived, pressure-supported object [47, 63]. However, in either case, the characteristic size of the massive object that forms is likely to be much smaller \nthan r lim . We can make an order-of-magnitude estimate of the relative sizes as follows. Massive stars are believed to have a polytropic equation of state with n=3. In other words, the equation of state is described by \nP ( r ) = Kρ ( r ) Γ , Γ = 1 + 1 /n, (12) \nand in this case n = 3 implies Γ = 4 / 3, as appropriate for a star supported by radiation pressure. It is possible to evaluate numerically the properties of a polytropic star in hydrostatic equilibrium, for n = 3 the approximate relation ρ c = 54 . 2¯ ρ between the central and the average density of the star holds (see e.g. [82]). \nWe infer that the typical scale for the radial extent of such an object should be \nR ∗ = ( 54 . 2 3 M ∗ 4 πρ c ) 1 3 ≈ 10 -5 pc ( M 10 5 M /circledot ) 1 3 ( ρ c 10 -2 gcm -3 ) -1 3 . (13) \nThis scale is clearly much smaller than the typical size of the spike r cut .", 'IV. DARK MATTER ANNIHILATIONS IN MINI-SPIKES': "Dark matter particles are expected to have a nonnegligible annihilation cross-section into Standard Model particles, in order to be kept in chemical equilibrium in the early Universe. It should be a weak or weakerthan-weak interaction in order to provide a relic density which satisfies cosmological constraints (for recent reviews of DM candidates and detection techniques see e.g. Refs. [1, 2]). Although it is difficult to make definitive statements on the nature of the DM particles, it is commonly believed that a mass in the range m χ ∼ 100 -1000 GeV would be a reasonable expectation in the most widely-discussed DM scenarios (e.g. minimal supersymmetry or scenarios with unified extradimensions). A na¨ıve estimate of the annihilation cross section, based on the observed relic abundance of DM, suggests that σv ∼ 10 -26 cm 3 s -1 . This value can be more appropriately used as an upper limit to the annihilation cross-section, rather than an actual estimate, since processes like co-annihilations may significantly affect relic density yields (for more details see Refs. [1, 2] and references therein). \nInstead of undertaking a detailed scan of the parameter space for different DM candidates, we limit ourselves here to estimates of the annihilation fluxes for two benchmark models: an optimistic model, with m χ = 100 GeV and σv = 3 × 10 -26 cm 3 s -1 , leading to large annihilation fluxes; and a model with m χ = 1 TeV and σv = 10 -29 cm 3 s -1 , leading to more pessimistic predictions. We note that in both cases, the mini-spike profiles reach their maximum values at a radii r lim >> 4 R Schw , thus r lim provides an estimate of the size of the region \nwhere most of the annihilation radiation originates from. The case of annihilations from the DM spike at the center of the Galaxy has been extensively studied in the literature in terms of neutrino, gamma-ray, and synchrotron emission [6, 7, 71, 83, 84, 85, 86]. \nThe flux of gamma-rays from a mini-spike around an IMBH can be expressed as \nΦ( E,D ) = 1 2 σv m 2 χ 1 D 2 d N d E ∫ r sp r cut ρ 2 sp ( r ) r 2 dr = d N d E ρ 2 sp 4 γ sp -6 σv m 2 χ r 3 sp D 2 ( r cut r sp ) -2 γ sp +3 (14) \nwhere we assumed r sp >> r cut . Inserting typical values of DM and spike parameters we get, for the case γ = 1 ( γ sp = 7 / 3), \nΦ( E,D ) = Φ 0 d N d E ( σv 10 -26 cm 3 / s ) ( m χ 100GeV ) -2 × ( D kpc ) -2 ( ρ ( r sp ) 10 2 GeVcm -3 ) 2 × ( r sp pc ) 14 3 ( r cut 10 -3 pc ) -5 3 , (15) \nwith Φ 0 = 9 × 10 -10 cm -2 s -1 . It is useful here to emphasize the relative luminosities of IMBHs in the MW halo. In particular, consider the case of the relatively more luminous objects of scenario B. Using the fiducial values adopted in eq. (15), which are typical of scenario B, one can easily verify that the 'luminosity' of a mini-spike (proportional to the volume integral of ρ 2 sp ) is of the order of the gamma-ray luminosity of the entire Milky Way halo , a circumstance that has dramatic consequences for the prospects of indirect detection, as we describe in the following section. \nTo estimate the flux, we need now to specify the gamma-ray spectrum per annihilation d N/ d E , which depends on the nature of the DM particle. In most scenarios, direct annihilation in two photons is severely suppressed, but a continuum spectrum is expected from the decay of secondary neutral pions. \nIn Fig. 3 we show the predicted gamma-ray spectra for different annihilation channels. For the b ¯ b channel we show two different curves, corresponding to different parametrization of the process of quark fragmentation and subsequent decay of neutral pions, for 2 different mass scales of the DM particle. The first (FPS 04) corresponds to the parametrization in Ref. [87], while the second (BSS 03) refers to the spectra presented in Ref. [88]. The differences for different parametrizations and mass scales appear to be small, The third curve (BUB 97) corresponds to an analytic fit for the WW and ZZ channels, as discussed in Ref. [89] , a channel leading to harder spectra with respect to the quarkantiquark channel. These channels often represent the most important annihilation channels for neutralino Dark \nFIG. 3: Energy spectra of photons per annihilation for different annihilation channels. The solid and dotted lines both correspond to the b ¯ b annihilation channel, the differences are due to different parametrizations of quark fragmentation and different DM particle mass scales. The solid line shows the parameterization of Ref. [87] with m χ = 1 TeV, while the dotted line shows that of Ref. [88] with m χ = 100 GeV. The short-dashed line corresponds to the spectra for annihilation through the WW and ZZ channels. In particular, we show the fit from Ref. [89]. Lastly, the long-dashed shows the spectrum, summed over contributing channels, for annihilation of Kaluza-Klein DM from Ref. [4]. \n<!-- image --> \nMatter. In the case of Kaluza-Klein DM, other channels become important. Following Ref. [4] we show in Fig. 3 the total spectrum obtained by adding the contribution of different channels, weighted with the appropriate branching ratios. It is evident from the figure that in this case the spectrum is harder than the the quark or gauge bosons channels, due to contributions from internal bremsstrahlung as well to decays of quarks and tau leptons. Internal bremsstrahlung is a general feature of scenarios where DM particles annihilate into pairs of charged fermions, which produces a sharp edge feature in the spectrum, dropping abruptly at a photon energy equal to the WIMP mass [4, 90, 91]. \nIn the next section, we will present our predictions using the BSS 03 spectrum, with the caveat that different annihilation channels may lead to slightly different results. The predictions for different annihilation channels can be easily obtained by plugging the appropriate spectrum per annihilation into Eq. 15.", 'V. RESULTS': 'In this section, we present our results on the detectability of annihilation radiation from the density enhancements about IMBHs. The number and properties of the IMBHs population are slightly different from one realization of Milky Way-sized halos to another, as described in Sec. II. To estimate the prospects of detection of IMBHs in the Milky Way, we thus need to average the results over all realizations. \nIn Fig. 4, we show the (average) integrated luminosity function of IMBHs in scenario B. We define the integrated luminosity function as the number of black holes producing a gamma-ray flux larger than Φ, as a function of Φ. The upper (lower) line corresponds to m χ = 100 GeV, σv = 3 × 10 -26 cm 3 s -1 ( m χ = 1 TeV, σv = 10 -29 cm 3 s -1 ). In a practical sense, the plot shows the number of IMBHs that can be detected with experiments with point source sensitivity Φ above 1 GeV. We show for comparison the point source sensitivity above 1 GeV for EGRET and GLAST, corresponding roughly to the flux for a 5 σ detection of a high-latitude pointsource in an observation time of 1 year [92]. The dashed region corresponds to the 1 σ scatter between different realizations of Milky Way-sized halos. This band includes the variation in spatial distributions of IMBHs from one halo to the next as well as the variation in the individual properties of each IMBH in each realization. \nAlthough one would na¨ıvely expect that the fluxes scale with σv/m 2 χ , we note that the DM profile itself depends on m χ and σv , more precisely on the ratio σv/m χ [see Eq. (10)]. The maximum density is higher for the pessimistic case, and this partially compensates for the decrease in flux due to the prefactor σv/m 2 χ . It is easy to see this in the case γ = 1 from eq. (15) as, by virtue of Eq. 10, r cut ∝ ( σv/m ) -3 / 7 , and the final luminosity of the objects is thus proportional to ∼ ( σv ) 2 / 7 m -9 / 7 χ . \nThe number of detectable sources is very high, even in the pessimistic case, and either strong constraints on a combination of the astrophysics and particle physics of this scenario, or an actual detection, should be possible within the first year of operation of GLAST, which is expected to be launched in 2007. Depending on the specific scenario, EGRET may have observed some of these IMBH mini-spikes, which would still account only for a small fraction of the unidentified sources. \nWe show in Fig. 5 the integrated luminosity function of IMBHs in scenario A, for the same particle physics models shown in fig. 5. The lines and error bars all have the same meaning as those in Figure 4 for scenario B. In this case of scenario A, mini-spikes are weaker, but the number of black holes is larger by roughly an order of magnitude, so that GLAST may still detect between a few tens and several hundred sources, whereas EGRET may have seen only a few or none. \nIn figures 4 and 5 we have assumed that the main annihilation channel is b ¯ b . Although we have seen in the previous section that, depending on the nature of the \nFIG. 4: IMBHs integrated luminosity function, i.e. number of black holes producing a gamma-ray flux larger than a given flux, as a function of the flux, for our scenario B (i.e. for IMBHs with mass ∼ 10 5 M /circledot ). The upper (lower) line corresponds to m χ = 100 GeV, σv = 3 × 10 -26 cm 3 s -1 ( m χ = 1 TeV, σv = 10 -29 cm 3 s -1 ). For each curve we also show the 1σ scatter among different realizations of Milky Way-sized host DM halos. The figure can be interpreted as the number of IMBHs that can be detected from experiments with point source sensitivity Φ (above 1 GeV), as a function of Φ. We show for comparison the 5 σ point source sensitivity above 1 GeV of EGRET and GLAST (1 year). \n<!-- image --> \nDM particle, other channels may dominate and lead to different annihilation spectra. We see from these figures that the expected uncertainty, O (1), would have a small influence on the number of objects that GLAST should be able to detect, certainly smaller than the uncertainties associated with m χ and σv and typically smaller than, or comparable to, the 1σ scatter between different Milky Way halo realizations.', 'VI. DISCUSSION AND CONCLUSIONS': "We have studied the detectability of gamma-rays from DM annihilations in mini-spikes around IMBHs. The prospects of detection are summarized in figures 4 and 5, where we show the number of IMBHs that can be detected from experiments with point source sensitivity Φ (above 1 GeV), as a function of Φ. We found that the prospects of detection with GLAST are so promising that a large number of sources may be detected within its first year of operation. With respect to the case of a spike at the Galactic center, searching for annihilation radiation from mini-spikes has the obvious disadvantage that IMBHs are smaller than the SMBH at the Galactic \nFIG. 5: IMBHs integrated luminosity function in scenario A (i.e. for IMBHs with mass ∼ 10 2 M /circledot ). The upper (lower) line corresponds to m χ = 100 GeV, σv = 3 × 10 -26 cm 3 s -1 ( m χ = 1 TeV, σv = 10 -29 cm 3 s -1 ). For each curve we also show the 1σ scatter among realizations of Milky Way-sized halos. For the sake of comparison, we also show the point source sensitivity above 1 GeV for EGRET and GLAST. \n<!-- image --> \ncenter, and the mini-spikes grow from less dense initial profiles. However, there are also several advantages. \nFirst of all, it is likely that the vast majority of minispikes around unmerged IMBHs in the outer Galactic halo are not affected by the dynamical processes that tend to destroy the central spike, or to decrease significantly the DM density near the black hole. For instance, they lack stellar cusps and are rarely affected by tidal interactions. Furthermore, the prospects of detectability appear very promising and certainly less problematic than, say, annihilations from the Galactic center. In fact, the gamma-ray background is strongest in the direction of the Galactic center and substantially reduced when observations are performed off the disk and in particular at high Galactic latitudes. There is even reason to believe that the number density of IMBH may be enhanced at high Galactic latitudes [93]. \nMoreover, there are several known gamma-ray sources in the direction of the Galactic center, and the observation of a unique source, even coincident with the Galactic center would not necessarily imply a DM annihilation origin. On the contrary, the detection of tens or more gamma-ray sources with identical spectra, in particular identical cut-offs at the DM particle mass, and not associated with the Galactic disk or other luminous companions of the Milky Way, would provide a smoking-gun signature of DM annihilations. \nA natural place to search for IMBHs may be the known \ndwarf satellite galaxies about the Milky Way; however, the physics that govern the formation of these objects is still a topic of much debate and uncertainty, so such a search may be subject to the same drawbacks as searches for radiation near the center of the Galaxy. However, as we have already stressed, these IMBH formation scenarios have as a virtue that they predict that there may be hundreds of detectable objects within the Galactic halo, most of which would not be associated with the known population of dwarf satellite galaxies. An IMBH population similar to the one in the Milky Way halo should be present in Andromeda, given its similarity to the Galaxy in terms of size and mass. Moreover, the distance to the Andromeda IMBHs would be between ≈ 400 -1000 kpc, amounting to a factor of only a few more in distance than to the black holes of the outer MW halo. Hence the detection of such IMBHs in Andromeda may be possible in optimistic scenarios and may serve to demonstrate the ubiquity of such phenomena. \nAs a further implication, we found that the annihilation luminosity from any Milky Way-like halo may be dominated by annihilation around IMBHs in optimistic scenarios. As an example, we showed that in scenario B, the total luminosity of an individual mini-spike, in terms of annihilation radiation, may be comparable to the luminosity of the entire host halo. Therefore, such optimistic scenarios provide a significant 'boost factor' for the gamma-ray background due to DM annihilations in halos at all redshifts (e.g., Ref. [94], see also Ref. [95] for a comparison between the prospects of indirect detection from the Galactic center and the gamma-ray background), as well as an enhancement of anti-matter fluxes. A detailed study of these alternative indirect searches requires a full analysis of the redshift evolution of IMBHs and spikes in halos of all masses in the former case, and of the propagation of anti-particles in the Galaxy in the latter, and it is thus beyond the scope of this paper. \nInterestingly, the prospects of indirect detection in this scenario do not depend strongly on the particle physics parameters. In fact, while e.g. in the case of annihilations from the Galactic center the annihilation flux is proportional to σv/m 2 χ , the flux from mini-spikes is limited by the plateau in the number density due to DM annihilation itself For mini-spikes growing from γ = 1 profiles, we have shown that the annihilation flux is instead proportional to ( σv ) 2 / 7 m -9 / 7 χ . \nFinally, we stress that the detection of these sources would shed new light on the origin of IMBHs and SMBHs. Mini-spikes appear to be ideal targets for large-field-ofview experiments such as GLAST. Another promising experiment with a large field-of-view could be the Alpha Magnetic Spectrometer (AMS-02) instrument, if preliminary estimates of its sensitivity to gamma-rays are confirmed [96]. 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2012PThPh.128..153Y
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Bosenova Collapse of Axion Cloud around a Rotating Black Hole
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2012-01-01
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Motivated by possible existence of stringy axions with ultralight mass, we study the behavior of an axion field around a rapidly rotating black hole (BH) obeying the sine-Gordon equation by numerical simulations. Due to superradiant instability, the axion field extracts the rotational energy of the BH and the nonlinear self-interaction becomes important as the field grows larger. We present clear numerical evidences that the nonlinear effect leads to a collapse of the axion cloud and a subsequent explosive phenomena, which is analogous to the ``bosenova'' observed in experiments of Bose-Einstein condensate. The criterion for the onset of the bosenova collapse is given. We also discuss the reason why the bosenova happens by constructing an effective theory of a wavepacket model under the nonrelativistic approximation.
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https://arxiv.org/pdf/1203.5070.pdf
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{'Bosenova collapse of axion cloud around a rotating black hole': "Hirotaka Yoshino 1 and Hideo Kodama 1 , 2 \n1 Theory Center, Institute of Particles and Nuclear Studies, KEK, Tsukuba, Ibaraki, 305-0801, Japan \n2 Department of Particle and Nuclear Physics, Graduate University for Advanced Studies, Tsukuba 305-0801, Japan \nMotivated by possible existence of stringy axions with ultralight mass, we study the behavior of an axion field around a rapidly rotating black hole (BH) obeying the sine-Gordon equation by numerical simulations. Due to superradiant instability, the axion field extracts the rotational energy of the BH and the nonlinear self-interaction becomes important as the field grows larger. We present clear numerical evidences that the nonlinear effect leads to a collapse of the axion cloud and a subsequent explosive phenomena, which is analogous to the 'bosenova' observed in experiments of Bose-Einstein condensate. The criterion for the onset of the bosenova collapse is given. We also discuss the reason why the bosenova happens by constructing an effective theory of a wavepacket model under the nonrelativistic approximation.", '§ 1. Introduction': "Recently, it was pointed out that string theory may be probed through cosmology or astrophysics by observing phenomena caused by 'axions' 1), 2) (see Ref. 3) for a review). The axion usually refers to the QCD axion that was introduced to solve the strong CP problem by Peccei and Quinn, 4), 5) and the QCD axion is expected as one of the candidates of the dark matter. In addition to the QCD axion, in the context of string theory, the stringy axions or axionlike particles have been proposed and discussed. 1), 2) In string theory, many moduli arise when extra dimensions are compactified, and some of them are expected to behave as axionlike scalar fields with ultralight mass. The typical expected number of the axionlike particles is 10-100, and it leads to a generic landscape of stringy axions, the so-called ' axiverse '. \nThe ultralight axions cause the possibly observable phenomena in cosmology or astrophysics. Suppose that the decay constant f a of the axion is order of the GUT scale, f a ≈ 10 16 GeV. In the cosmological context, axions with mass from 10 -33 eV to 4 × 10 -28 eV affect the polarization of cosmic microwave background if the ChernSimons interaction is present, and those with mass from 4 × 10 -28 eV to 3 × 10 -18 eV may affect the matter power spectrum. On the other hand, in the astrophysical context, axions are expected to cause interesting phenomena around astrophysical black holes (BHs) if they have mass between 2 × 10 -20 eV and 3 × 10 -10 eV. We focus on axions around an astrophysical BH in this paper. \nSuppose an axion field exists around a rotating BH. Although some of the field would be absorbed by the BH, it is expected that the axion field forms a quasibound state which may be called the 'axion cloud'. Furthermore, the axion cloud is expected to grow by the superradiant instability. The superradiant instability is caused by the fact that the Killing vector field ( ∂ t ) a of the Kerr spacetime becomes \ntypeset using PTP T E X.cls 〈 Ver.0.9 〉 \nspacelike in the ergoregion, and therefore, the energy of the field can be negative. A mode with negative energy around the horizon is called a superradiant mode. If a quasibound state whose mode function satisfies the superradiant condition is occupied by the axion field, negative energy falls into the BH and the energy outside the horizon (and therefore, the amplitude of the field) increases in time. \nThe growth rate of the quasibound state of the axion cloud by the superradiant instability is characterized by the imaginary part γ of the angular frequency ω , where ω = ω 0 + iγ . The value γ/µ depends on the ratio of one half of the gravitational radius to the Compton wavelength of axion α g := ( GM/c 2 ) / ( /planckover2pi1 /µc ), or, α g = Mµ in the Planck units c = G = /planckover2pi1 = 1 (hereafter, the Planck units are used unless otherwise specified). The superradiant instability is effective for α g ∼ 1, and its typical time scale is τ ∼ 10 7 M for α g ∼ 1. For a solar-mass BH M = M /circledot (resp. a supermassive BH M = 10 9 M /circledot ), the superradiant instability effectively occurs if an axionlike field with mass µ ∼ 10 -10 eV (resp. µ ∼ 10 -19 eV) exists, and in that case, the typical time scale for the instability is 50 s (resp. 1600 year). Therefore, the time scale for instability is much shorter than the age of the universe and the superradiant instability should become really relevant to astrophysical phenomena. \nThe expected phenomena caused by the superradiant instability are discussed and summarized in Refs. 1), 2) ∗ ) . As the instability proceeds, the axion cloud extracts rotational energy from a BH and gradually becomes heavy (i.e., the number of axions increases). In Ref. 2), the gravitational wave emission was discussed from the viewpoint of the quantum theory. Since the structure of the axion cloud is analogous to the electron cloud of a hydrogen atom, the graviton emission by the level transition of axions can be discussed in analogy with the photon emission by electron's level transition. Another source of graviton emission is the pair annihilation of two axions. On the other hand, the nonlinear self-interaction of axions is also expected to cause important phenomena. In the case of the QCD axions, due to nonperturbative effects associated with instantons, the potential U ( Φ ) becomes periodic as typically described by the trigonometric function U ( Φ ) = f 2 a µ 2 [1 -cos( Φ/f a )]. The similar form of the potential can be expected for string axions because their masses are generated also by the instanton effects. Therefore, although the Klein-Gordon equation (i.e., U ( Φ ) = (1 / 2) µ 2 Φ 2 ) gives a good approximation for small Φ/f a , as the field grows large, the nonlinear effects become important. \nOne of the nonlinear effects is the mode mixing, which is expected to change the field configuration and affect the growth rate of the superradiant instability. Another interesting possibility is the 'bosenova'. The bosenova was observed in the experiments of the Bose-Einstein condensates (BEC) of Rb85. 7), 8) The interaction between atoms can be controlled in this system, and the interaction was switched from repulsive one to interactive one in that experiment. As a result, the BEC collapsed, but after that, a burst of atoms was observed. This phenomenon was studied also theoretically 9), 10), 11) and it was clarified that the implosion is caused by the nonlinear attractive interaction and the burst is induced mainly by atomic loss through three-body recombinations. Since the atomic loss weakens the attractive \ninteraction, the atoms begin to explode due to zero-point kinetic pressure. \nIn the case of the BH-axion system, we have to take account of the following two possibilities. The first possibility is that an explosive phenomena that is analogous to the bosenova happens as a result of the nonlinear effect. The second possibility is that the nonlinear effect saturates the growth by superradiant instability, as found in various instabilities of nonlinear systems, leading the system to a quasistationary state without explosive phenomena. We have to clarify which is the case, and if phenomena like the bosenova also occurs in the BH-axion system, the details and the observational consequence have to be studied. In order to clarify the strongly nonlinear phenomena, fully nonlinear simulations have to be performed, and this is the purpose of this paper. \nWe develop a three-dimensional (3D) code to simulate an axion field with a nonlinear potential in a Kerr spacetime. Here, the axion field is treated as a test field, and the background geometry is fixed to be the Kerr spacetime. In most cases, this approximation holds well. The setup of the problem is explained in more detail in Sec. 4.1. In short, our simulations indicate no evidence for saturation, and the bosenova is likely to happen in the final stage of the superradiant instability. \nThis paper is organized as follows. In the next section, we review the existing studies on the behavior of a massive scalar field and its superradiant instability focusing attention to the aspects closely related to our study. Section 3 explains the technical part, i.e., the formulation, our code, and code tests. In Sec. 4, we present the numerical results of our simulations. After presenting the results of typical two simulations, we discuss whether the bosenova actually happen by performing supplementary simulations. In Sec. 5, we discuss the reason why the bosenova happens in the BH-axion system by constructing an effective theory of an axion cloud model in the nonrelativistic approximation. Section 6 is devoted to summary and discussion. After summarizing our results, the similarity and difference between the bosenova phenomena in the BEC system and in the BH-axion system is discussed. We also roughly estimate whether gravitational radiation emitted in the bosenova can be detected by planned gravitational wave detectors. In Appendix A, the behaviour of the axion field generated by the nonlinear effect is studied using the Green's function approach, taking attention to the consistency with the results of our simulations.", '§ 2. Superradiant instability': 'This section is devoted to the review on massive scalar fields in a Kerr spacetime.', '2.1. Axion field in a Kerr spacetime': "The action for the axion field Φ in a spacetime of a metric g ab is \nS = ∫ d 4 x √ -g [ -1 2 g ab ∇ a Φ ∇ b Φ -U ( Φ ) ] , (2 . 1) \nwhere U ( Φ ) is the potential, i.e., U ( Φ ) = (1 / 2) µ 2 Φ 2 for the Klein-Gordon field and U ( Φ ) = f 2 a µ 2 [1 -cos( Φ/f a )] for the axion field with nonlinear self-interaction (i.e., the sine-Gordon field). Here, f a is the decay constant whose value depends on the \nmodel. For convenience, we normalize the amplitude of Φ with f a as \nϕ := Φ/f a . (2 . 2) \nThen, the field equation is \nwith ˆ U ( ϕ ) = U ( Φ ) /f 2 a . Here, ˆ U ' = µ 2 ϕ for the Klein-Gordon field and ˆ U ' = µ 2 sin ϕ for the axion field. Therefore, if the value of | ϕ | is sufficiently small, the axion field can be well approximated by the Klein-Gordon field. However, the nonlinear effect becomes important as ϕ | comes close to unity. \n| \n/square ϕ -ˆ U ' ( ϕ ) = 0 . (2 . 3) \n| The metric of the Kerr spacetime in the Boyer-Lindquist coordinates is given by \nds 2 = -( ∆ -a 2 sin 2 θ Σ ) dt 2 -2 a sin 2 θ ( r 2 + a 2 -∆ ) Σ dtdφ + [ ( r 2 + a 2 ) 2 -∆a 2 sin 2 θ Σ ] sin 2 θdφ 2 + Σ ∆ dr 2 + Σdθ 2 , (2 . 4) \nwith \nΣ = r 2 + a 2 cos 2 θ, ∆ = r 2 + a 2 -2 Mr. (2 . 5) \nHere, M is the Arnowitt-Deser-Misner (ADM) mass, and a is the ADM angular momentum per unit mass, a = J/M . In order to specify the rotation, the nondimensional parameter a/M is often used. The solutions of ∆ = 0 give the locations of the inner and outer horizons, r ± = M ± √ M 2 -a 2 , and the event horizon is located at r = r + . In the Kerr geometry, the equation for the axion field is \n-Fϕ ,tt -2 a ( r 2 + a 2 -∆ ) ϕ ,tφ + ∆ -a 2 sin 2 θ sin 2 θ ϕ ,φφ + ∆ ( ϕ ,θθ +cot θϕ ,θ ) +2 r∆ϕ ,r ∗ +( r 2 + a 2 ) 2 ϕ ,r ∗ r ∗ -Σ∆ ˆ U ' ( ϕ ) = 0 , (2 . 6) \nwhere \nHere, we introduced the tortoise coordinate by dr ∗ = [( r 2 + a 2 ) /∆ ] dr , or equivalently, \nF := ( r 2 + a 2 ) 2 -∆a 2 sin 2 θ. (2 . 7) \nr ∗ = r + 2 M r + -r -( r + log | r -r + | -r -log | r -r -| ) . (2 . 8) \nIn the tortoise coordinate r ∗ , the horizon is located at r ∗ = . \nThe scalar field in a Kerr spacetime has conserved quantities. If a Killing vector ξ a is present in a spacetime, we can define the conserved current P a = -T ab ξ b that satisfies ∇ a P a = 0. Here, T ab is the energy-momentum tensor (in the unit f a = 1) \n-∞ \nT ab = ∇ a ϕ ∇ b ϕ -1 2 g ab ( ∇ c ϕ ∇ c ϕ +2 ˆ U ( ϕ ) ) . (2 . 9) \nUsing this current P a , the conserved quantity C ( t ) ≡ C (0) can be introduced as \nC ( t ) := ¯ C ( t ) + ∆C ( t ) , (2 . 10) \nwith the quantity ¯ C ( t ) in the region r (in) ∗ ≤ r ∗ ≤ r (out) ∗ , \n¯ C ( t ) = ∫ Σ t P a n a dΣ, (2 . 11) \nand the integrated flux toward the horizon at r ∗ = r (in) ∗ , \n∆C ( t ) = ∫ r ∗ = r (in) ∗ P a s a dσ. (2 . 12) \nHere, we have assumed the absence of outgoing flux at the outer boundary r ∗ = r (out) ∗ . The integration of the first term ¯ C ( t ) is performed on t = const . slice Σ t in the range r (in) ∗ ≤ r ∗ ≤ r (out) ∗ with the past-directed timelike unit normal n a and the volume element dΣ , and it represents the conserved quantity contained in the region r (in) ∗ ≤ r ∗ ≤ r (out) ∗ . The second term ∆C ( t ) is the integrated flux, where the integration is performed on the hypersurface r ∗ = r (in) ∗ from time zero to t with the spacelike unit normal s a directing toward the horizon and the surface element dσ . The value of ∆C ( t ) indicates the total quantity that has fallen into the BH from time zero to t . Since the Kerr spacetime possesses two Killing vectors, ξ a = ( ∂ t ) a and ( ∂ φ ) a , there exist two conserved quantities: the energy E and the angular momentum J .", '2.2. Superradiant instability': "Here, we briefly review the superradiant instability of a massive Klein-Gordon field around a Kerr BH. If the nonlinear terms are absent, the separation of variables is available as follows. Setting ϕ = 2Re[ e -iωt R ( r ) S /lscriptm ( θ ) e imφ ], the equations for S /lscriptm ( θ ) and R ( r ) become \n1 sin θ d dθ ( sin θ dS /lscriptm dθ ) + [ -k 2 a 2 cos 2 θ -m 2 sin 2 θ + E /lscriptm ] S /lscriptm = 0 , (2 . 13) \nwhere \nand \nHere, S /lscriptm ( θ ) e imφ is the spheroidal harmonics, which coincides with the spherical harmonics in the case k = 0. The angular quantum numbers, /lscript and m , are integers /lscript = 0 , 1 , 2 , ... and -/lscript ≤ m ≤ /lscript . The eigenvalue E /lscriptm is E /lscriptm = /lscript ( /lscript +1) in the case of k = 0, while in the case k /negationslash = 0, it has to be evaluated numerically by the methods of Refs. 12),13) or by the approximate formulas. 14),15),16) \nd dr ( ∆ dR dr ) + [ K 2 ∆ -λ /lscriptm -µ 2 r 2 ] R = 0 , (2 . 14) \nK = ( r 2 + a 2 ) ω -am, (2 . 15) \nk 2 = µ 2 -ω 2 , (2 . 16) \nλ /lscriptm = E /lscriptm + a 2 ω 2 -2 amω. (2 . 17) \nFrom the equation (2 . 14) for the radial function R ( r ), the behavior of R ( r ) at r ∗ /M /greatermuch 1 is described as R ∼ r -1 exp( ± kr ). If Re[ ω ] < µ , the field is bounded \nby gravitational interaction and does not escape to infinity. On the other hand, the behavior of R ( r ) in the neighborhood of the horizon r ∗ /M /lessmuch -1 is R ∼ e ± i ˜ ωr ∗ , where the plus and minus signs correspond to the outgoing and ingoing modes, respectively. Here, ˜ ω is defined as ˜ ω = ω -mΩ H with the angular velocity of the horizon Ω H = a/ (2 Mr + ). \nHere, let us focus attention to the energy E of the Klein-Gordon field introduced in Sec. 2.1. Evaluating the energy density with respect to the tortoise coordinate r ∗ for the t = const . surface, we have d ¯ E/dr ∗ /similarequal 2 ω ˜ ω ( r 2 + + a 2 ) in the neighborhood of the horizon. Here, we have used the ingoing solution R ∼ e -i ˜ ωr ∗ for r ∗ /M /lessmuch -1. On the other hand, the energy flux F E := d ( ∆E ) /dt toward the horizon can be evaluated as F E /similarequal 2 ω ˜ ω ( r 2 + + a 2 ). Therefore, if waves satisfy the superradiant condition 0 < ω < mΩ H (i.e., ˜ ω < 0), the negative energy distributes in the neighborhood of the horizon and it falls into the BH 'at the speed of light' in the coordinates ( t, r ∗ ). \nThe negative energy of waves satisfying the superradiant condition leads to an interesting phenomena. Suppose waves satisfying the superradiant condition are incident to a rotating BH. A fraction of waves falls into the BH, and the rest is reflected back to infinity by the centrifugal potential barrier of the BH. Since the negative energy falls into the BH, the reflected waves have greater energy than the initial ingoing waves because of the energy conservation. In other words, reflected waves get amplified. This is called superradiance. The superradiance was proposed and analyzed for the massless Klein-Gordon field first by Zel'dovich. 17),18) \nUsing superradiance, Press and Teukolsky 19) proposed a mechanism to cause an instability of fields around a rotating BH, which is called the BH bomb. In this mechanism, a mirror is put around a BH. Waves satisfying the superradiant condition are reflected back and forth between the BH horizon and the mirror, and thus, continue to get amplified. As a result, the amplitude of waves exponentially grows in time. The mirror in the BH-bomb model seems to be artificial. However, it was pointed out by Damour et al. 20) that if the field has non-vanishing mass, the reflected waves can fall back to the BH because of the gravitational force on the rest mass. In other words, if the field is in a quasibound state, Re[ ω ] < µ , the superradiant instability occurs without putting a mirror. The instability of a massive Klein-Gordon field around a Kerr BH was analytically studied by Detweiler 21) and Zouros and Eardley. 22) \nDetweiler 21) analyzed the situation α g := Mµ /similarequal Mω /lessmuch 1. In this setup, the solution of the radial function R ( r ) can be obtained by the matching method. After the solutions for the distant region and the near-horizon region are obtained separately, they are matched to each other in an overlapping region. After the matching, the solution for a distant region is same as the wavefunction of the eigenstate of a hydrogen atom in quantum mechanics, since the equation for the scalar field is same as the Schrodinger equation for a hydrogen atom with the potential e 2 /r being replaced by α g /r . The result of the growth rate for the ( /lscript, m ) = (1 , 1) mode is γM = (1 / 24) α 9 g ( a/M ). \nZouros and Eardley 22) assumed α g /greatermuch 1 and analyzed with the WKB approximation. Introducing a function u = √ r 2 + a 2 R , the radial mode equation is rewritten \nFig. 1. The potential V ( ω, r ∗ ) (in the unit M = 1) in Eq. (2 . 18) for a quasibound state of the Klein-Gordon field for situation a/M = 0 . 99 and α g := Mµ = 0 . 4 (solid line). The horizontal dotted line indicates the value of ω 2 . Here, the imaginary part is ignored. There are four domains I, II, III, and IV, depending on the relation between V and ω 2 , and the quasibound state is formed in region III. Due to the tunneling effect, the waves gradually fall into the region I. Because the energy of waves takes a negative value in region I under the superradiant condition, the field in region III is amplified. \n<!-- image --> \nas the Schrodinger-type equation: \nd 2 u dr 2 ∗ + [ ω 2 -V ( ω, r ∗ ) ] u = 0 . (2 . 18) \nThe potential for a/M = 0 . 99 and α g = 0 . 4 is shown in Fig. 1. The potential V asymptotes to µ 2 from below for r ∗ → ∞ , and this potential rise plays the role of the mirror. The quasibound state is formed in the region III, and because of the tunneling effect, the mode function gradually escape into the region I as ingoing waves. If these ingoing waves toward the horizon satisfies the superradiant condition, the energy of the quasibound state increases and the wavefunction get amplified in the region III. Their result shows that the growth rate Mγ exponentially decreases as α g is increased. \nIn the region where the largest growth rate of instability is expected, α g ∼ 1, numerical calculations are required. These studies were done in Refs. 23), 24), 25), 26). The most detailed results have been reported by Dolan 26) by applying Leaver's continued fraction method 12) to this problem. The continued fraction method was originally developed to calculate the value of quasinormal frequencies numerically, and it enables us to obtain highly accurate values of ω for the quasibound state as well. The result is shown in Figs. 6 and 7 of Ref. 26). The largest growth rate is realized for ( /lscript, m ) = (1 , 1), a/M /similarequal 1, and α g /similarequal 0 . 4, and its value is γ/µ /similarequal 3 × 10 -7 . In Ref. 3), we also developed a code to calculate ω of the quasibound state and reproduced Dolan's result. As an example, the configuration of the field ϕ of the ( /lscript, m ) = (1 , 1) mode in the equatorial plane and in the ( ρ, z )-plane (where ρ := r sin θ and z := r cos θ ) are shown as contour plots in the left and right panels of Fig. 2, \nFig. 2. A snapshot for the contours of the Klein-Gordon field ϕ of the ( /lscript, m ) = (1 , 1) mode of the quasibound state in the case of a/M = 0 . 99 and α g := Mµ = 0 . 4 in the equatorial plane θ = π/ 2 (left panel) and in the ( ρ, z )-plane (right panel). Here, ρ := r sin θ and z := r cos θ , and the ( ρ, z )-plane is drawn for the azimuthal angle φ = π/ 5 and (6 / 5) π so that the plane crosses the peak of the field. \n<!-- image --> \nrespectively, for a/M = 0 . 99 and α g = 0 . 4.", '§ 3. Numerical method and code': 'In this section, we explain the technical part of our study. The formulation for solving the axion field around a Kerr BH is explained in Sec. 3.1, and the numerical techniques and code tests are summarized in Sec. 3.2.', '3.1. Numerical method': 'The most important point in simulations of fields in the Kerr background spacetime is to realize the sufficient stability. We found that if a simulation is performed in the Boyer-Lindquist coordinates with central difference method, a numerical instability immediately develops to crash the simulation. This is because the lines with constant spatial coordinates are spacelike in the ergoregion in the Boyer-Lindquist coordinates, and therefore, the frame is propagating superluminally. Although this problem may be avoided by adopting the upwind difference method, we have chosen another method with which greater stability is expected. This method is explained in Secs. 3.1.1 and 3.1.2. We also explain the boundary condition and how to regularize the equation at the poles in Secs. 3.1.3 and 3.1.4, respectively.', '3.1.1. ZAMO coordinates': "In our method, we realize the numerical stability by adopting the coordinates associated with the zero-angular-momentum observers (ZAMOs). The ZAMOs are observers such that they stay at fixed r and θ , but move in the φ direction so that their angular momenta are kept to be zero. Their four-velocity is given by u a = ∇ a t/ √ -∇ b t ∇ b t , which is timelike everywhere, and they rotate with the angular \nvelocity \nΩ ( r, θ ) = 2 Mar F , (3 . 1) \nwhere F is defined in Eq. (2 . 7). Using this angular velocity, we introduce the new coordinates ( ˜ t, ˜ φ, ˜ r, ˜ θ ) as \n˜ t = t, ˜ φ = φ -Ω ( r, θ ) t, ˜ r = r, ˜ θ = θ. (3 . 2) \nThe basis vector of the new time coordinate ˜ t is parallel to u a , and therefore, it is timelike everywhere. We call these coordinates the ZAMO coordinates. The equation for the axion field reads \n-Fϕ , ˜ t ˜ t + [ Σ 2 ∆ F sin 2 ˜ θ + ˜ t 2 ∆ ( ∆Ω 2 , ˜ r + Ω 2 , ˜ θ ) ] ϕ , ˜ φ ˜ φ +(˜ r 2 + a 2 ) 2 ϕ , ˜ r ∗ ˜ r ∗ +2˜ r∆ϕ , ˜ r ∗ + ∆ ( ϕ , ˜ θ ˜ θ +cot ˜ θϕ , ˜ θ ) -2 ˜ t∆ [ (˜ r 2 + a 2 ) Ω , ˜ r ϕ , ˜ r ∗ ˜ φ + Ω , ˜ θ ϕ , ˜ θ ˜ φ ] -˜ t∆ [ ( ∆Ω , ˜ r ) , ˜ r +( Ω , ˜ θ ˜ θ +cot ˜ θΩ , ˜ θ ) ] ϕ , ˜ φ -Σ∆ ˆ U ' ( ϕ ) = 0 , (3 . 3) \nin the ZAMO coordinates.", '3.1.2. Pullback of coordinates': "Because the angular velocity Ω of a ZAMO becomes larger as it is closer to the horizon, the ZAMO coordinates become distorted in time evolution. This is the shortcoming of the ZAMO coordinates because if the coordinates are distorted, the numerical error grows large, and also, the physical interpretation of the numerical results becomes difficult. We solve this problem by 'pulling back' the ZAMO coordinates. Namely, when the ZAMO coordinates become distorted to some extent, we introduce new ZAMO coordinates which are not distorted at that time (i.e., the new ZAMO coordinates agree instantaneously with the Boyer-Lindquist coordinates), and continue time evolution with the new coordinates. Iterating these processes, longterm evolution becomes feasible. Specifically, for nT P ≤ t ≤ ( n + 1) T P , we adopt the n -th ZAMO coordinates ( ˜ t ( n ) , ˜ φ ( n ) , ˜ r ( n ) , ˜ θ ( n ) ) by \n˜ t ( n ) = t, ˜ φ ( n ) = φ -Ω ( r, θ )( t -nT P ) , ˜ r ( n ) = r, ˜ θ ( n ) = θ. (3 . 4) \nThe numerical data of Φ and ∂Φ/∂ ˜ t in the new coordinates are generated by interpolation. In our numerical calculations, we adopt T P = M/ 4. If we list up the data of Φ at time t = nT P with n = 0 , 1 , 2 , ... , they can be regarded as the data in the Boyer-Lindquist coordinates.", '3.1.3. Boundary conditions': 'Since a simulation has to be performed in a finite coordinate region, the coordinate range of r ∗ is taken as r (in) ∗ ≤ r ∗ ≤ r (out) ∗ . Here, we discuss how to impose the inner and outer boundary conditions at r ∗ = r (in) ∗ and r (out) ∗ , respectively. \nFor a sufficiently small r (in) ∗ , we have ∆ /similarequal 0 at r = r (in) ∗ . Then, the equation for ϕ in the ZAMO coordinates, Eq. (3 . 3), becomes \n-ϕ , ˜ t ˜ t + ϕ ˜ r ∗ ˜ r ∗ /similarequal 0 . (3 . 5) \nTherefore, the in- and out- going modes are clearly separated, and we can impose the purely ingoing boundary condition in the standard manner. Typically, we adopt r (in) ∗ /M = 200. \nAt r = r (out) ∗ , we adopt the fixed boundary condition, ϕ = 0. When the axion field is in a bound state, this boundary condition gives a good approximation. If outgoing waves are generated, the outer boundary becomes reflective, which is quite artificial. In such a case, we avoid the problem by adopting sufficiently large r (out) ∗ . Typically, the outer boundary is located between r (out) ∗ /M = 200 and 1000 depending on the situation. \n-', '3.1.4. Regularization at poles': "Since the two poles ˜ θ = 0 and π are coordinate singularities, regularization of the equation is required at the poles. For this purpose, we introduce new coordinates ( x, y ) by \nx = ˜ θ cos ˜ φ, y = ˜ θ sin ˜ φ, (3 . 6) \nin the neighborhood of each pole. Rewriting Eq. (3 . 3) with these coordinates and taking the limit ˜ θ → 0 or π , we obtain \n-Fϕ , ˜ t ˜ t +(˜ r 2 + a 2 ) 2 ϕ , ˜ r ∗ ˜ r ∗ + ∆ [ ϕ ,xx + ϕ ,yy +2˜ r∆ϕ , ˜ r ∗ -Σ ˆ U ' ( ϕ ) ] = 0 . (3 . 7) \nHere, ϕ ,xx can be evaluated by the data at the grids on ˜ φ = 0 and π , and ϕ ,yy by the data at the grids on ˜ φ = π/ 2 and (3 / 2) π . Therefore, the data of grids at the poles can be evolved toward the next time step with this equation.", '3.2. Code and code checks': "Our code is a three-dimensional (3D) code of the ZAMO coordinates (˜ r ∗ , ˜ θ, ˜ φ ). The sixth-order finite differencing method is used in spatial directions, and time evolution is proceeded with the fourth-order Runge-Kutta method. Typically, we used the grid size ∆r ∗ /M = 0 . 5 and ∆θ = ∆φ = π/ 30. When the sphericalpolar coordinates are used, the Courant condition for the time step becomes severe and it has to be chosen so that ∆t /lessorsimilar min[( F/Σ∆ 1 / 2 ) θ =0 ∆θ∆φ ] from Eq. (3 . 3). Here we adopt the value of the time step as ∆t = (3 / 2 π ) ∆θ∆r ∗ in order to realize the sufficient stability of our simulations. In doing the 'pullback' of the ZAMO coordinates addressed in Sec. 3.1.2, the interpolation of data is necessary, and we applied the seventh-order Lagrange interpolation. \nIn order to validate the code, we have to perform test simulations. The code checks have been done in the three following manners, as explained one by one below.", '3.2.1. Comparison with semianalytic solution': 'The first check is to simulate time evolution of the quasibound state of the linear Klein-Gordon equation and compare the numerical data with the semianalytic solution. Here, we choose the BH with the rotation parameter a/M = 0 . 99 and the KleinGordon field of mass µ = 0 . 4 /M . The semianalytic solution ϕ = e -iωt R ( r ) S ( θ ) e imφ can be obtained by using the approximate formula for S /lscriptm ( θ ) 15) and numerically calculating ω and R ( r ) using the continued fraction method. 12),26) The time evolution \nFig. 3. The relation between the grid size ∆r ∗ (with unit M = 1) and the numerical error evaluated at t = 12 . 5 M . The error decreases as ∆r ∗ is increased, and the slope of the curve is /similarequal 5. This reflects our combined fourth- and sixth-order scheme. \n<!-- image --> \nwas performed up to t = 100 M , and the numerical data were confirmed to agree well with the semianalytic solution. \nThe imaginary part γ of frequency ω = ω 0 + iγ of the semianalytic solution gives the correct growth rate of the superradiant instability. In the present setup, it is calculated as γ/µ /similarequal 3 . 311 × 10 -7 by the continued fraction method. In order to check to what extent the superradiant instability is correctly realized in our numerical simulation, we calculated the energy ¯ E ( t ) in the region r (in) ∗ ≤ r ∗ ≤ r (out) ∗ [see Eq. (2 . 11) in Sec. 2.1 for the definition of ¯ E ( t )], and evaluated γ = ( d ¯ E/dt ) / 2 ¯ E /similarequal [ ¯ E ( t f ) -¯ E (0)] / [2 t f ¯ E (0)] with t f = 100 M . The numerical result performed in the grid sizes mentioned above is γ/µ /similarequal 3 . 255 × 10 -7 : The deviation from the value of the semianalytic solution is about 1.7%. Therefore, our code has the ability to describe the energy extraction by the superradiant instability fairly accurately.', '3.2.2. Convergence with respect to grid size': 'One of the standard tests of numerical simulations is to check whether the numerical solution converges as the grid size is made smaller. For this purpose, we adopt the numerical solution of ∆r ∗ /M = 1 / 6 as the reference solution, and evaluated the deviation of the numerical solutions with several grid sizes. Here, we adopted ϕ (0) = exp[( r ∗ / 30) 2 ] sin θ cos φ and ˙ ϕ (0) = 0 as the initial condition and evolved the data until t/M = 12 . 5 for the parameters α g := Mµ = 0 . 4 and a/M = 0 . 99. Figure 3 shows the relation between log 10 ∆r ∗ and log 10 (error). Since we use the sixth- and fourth-order schemes in the space and time directions, the curve is expected to have slope between four and six. Actually, the slope is ∼ 5 in this figure. This result reflects the adopted scheme, and supports the validity of our code.', '3.2.3. Conserved quantities': "As discussed in Sec. 2.2, we have the two conserved quantities, the energy E and the angular momentum J . In actual simulations, these quantities slightly change in \nFig. 4. The values of total energy and angular momentum normalized by the initial values, E ( t ) /E (0) and J ( t ) /J (0), as functions of time (the solid line and the dashed line, respectively). Deviation from unity indicates the amount of numerical error. The error is less than 0 . 04% at t/M = 1000. \n<!-- image --> \ntime because of numerical error. Therefore, the deviations of the values E ( t ) /E (0) and J ( t ) /J (0) from unity give indicators for the accumulated numerical errors. \nFigure 4 shows the values of E ( t ) /E (0) and J ( t ) /J (0) as functions of time t/M . Here, we show the results for the simulation of axion mass α g = 0 . 4 around a BH with a/M = 0 . 99 for initial amplitude ϕ peak (0) = 0 . 7 [i.e., simulation (B) of Sec. 4.2.2]. The deviation from unity is negligible for t/M /lessorsimilar 500. For t /greaterorsimilar 500, the deviations linearly increase. This is because the 'bosenova' happens and some part of the axion field distributes at a distant place. As a result, small error in the field value results in large errors of E ( t ) and J ( t ) because large volume element is multiplied there. Nevertheless the deviations from unity are less than 0 . 04% at t/M = 1000 for both E ( t ) /E (0) and J ( t ) /J (0). \nAs found above, we checked the validity of our code in three ways, and therefore, we can trust the results of our longterm simulations.", '§ 4. Numerical results': 'Now we present the numerical results. In Sec. 4.1, we describe the setup of the system and the initial conditions. In Sec. 4.2, we show the results of typical two simulations [referred as simulations (A) and (B)], for which the effect of nonlinearlity is weak and strong. respectively. This helps us to understand how nonlinearlity works in this system. Then, in Sec. 4.3, we discuss what actually happens in the final stage of the superradiant instability, taking special attention to whether the bosenova happens or not.', '4.1. Setup': "In order to study the nonlinear self-interaction of an axion field, we numerically solve the sine-Gordon equation /square ϕ -µ 2 sin ϕ = 0 in a Kerr spacetime. For simplicity, we consider an axion cloud with mass α g := Mµ = 0 . 4 around a Kerr BH with the rotational parameter a/M = 0 . 99. As the initial condition, we adopt the quasibound state solution to the linear Klein-Gordon field corresponding to the \nTable I. Performed two simulations, (A) and (B), in Sec. 4.2. 'KG bound state' means that the initial condition is adopted as the quasibound state of Klein-Gordon field of ( /lscript, m ) = (1 , 1) mode, and ϕ peak (0) indicates the initial amplitude. \n| Simulations | Initial condition | E/ [( f a /M p ) 2 M ] | nonlinearlity |\n|---------------|-----------------------------------------|--------------------------|-----------------|\n| (A) | KG bound state, ϕ (A) peak (0) = 0 . 60 | 1430 | weak |\n| (B) | KG bound state, ϕ (B) peak (0) = 0 . 70 | 1862 | strong | \n( /lscript, m ) = (1 , 1) mode. Namely, the configuration shown in Fig. 2 are used (but changing the amplitude of the oscillation). If nonlinear terms are absent (i.e., in the case of the Klein-Gordon field), the frequency is ω = ω 0 + iγ , where the real part is ω 0 M /similarequal 0 . 39 and the imaginary part (i.e., the growth rate by the superradiant instability) is γM /similarequal 1 . 32 × 10 -7 , which is the approximately largest possible growth rate. These are natural setups, because we consider the situation where the axion fields have grown due to the superradiant instability and, at least for small | ϕ | , such situations should be well approximated by the quasibound state of the Klein-Gordon field. As typical examples, we present the results of two simulations with different initial amplitude [simulations (A) and (B) in Secs. 4.2.1 and 4.2.2, respectively, see Table I]. In Sec. 4.3, we discuss what actually happens as a result of superradiant instability by performing supplementary simulations starting with initial conditions that is expected to be more natural compared to those of the simulations (A) and (B).", '4.2. Typical two simulations': "Now, we present the results of the simulations (A) and (B). \n4.2.1. \nSimulation (A): A weakly nonlinear case \nIn the simulation (A), we choose the initial amplitude to be ϕ peak (0) = 0 . 6. The effect of the nonlinearlity can be evaluated by ∆ NL := ( ϕ -sin ϕ ) /ϕ /similarequal ϕ 2 / 6, and for this setup, ∆ NL = 0 . 06 at the peak. Therefore, the nonlinear effects are weakly important for this situation. \nThe upper panel of Fig. 5 shows the value of the field at the peak, ϕ peak := sup[ ϕ ], and the lower panel shows the position r (peak) ∗ of the peak with respect to the tortoise coordinate r ∗ as functions of t/M . The value of ϕ peak oscillates with the period of about 700 M . The position of the peak also moves back and forth, and ϕ peak increases when r (peak) ∗ decreases, i.e., when the peak location approaches the horizon. Therefore, the change in the amplitude is mainly caused not by the superradiant instability but by the change of the peak position. Namely, when the peak position approaches the horizon due to nonlinear interaction, the field gets compacted in a small region around the BH, and therefore, the field is amplified because of the energy conservation. \nFigure 6 shows the energy flux F E := d ( ∆E ) /dt and the angular momentum flux F J := d ( ∆J ) /dt toward the horizon evaluated at r ∗ = -100 M , where ∆E and ∆J are integrated fluxes defined in Eq. (2 . 12). Initially, both F E and F J are negative, reflecting the fact that we have chosen the ( /lscript, m ) = (1 , 1) mode of the \nFig. 5. The peak value ϕ peak of the field ϕ (upper panel) and its location r (peak) ∗ with respect to the tortoise coordinate (lower panel) as functions of time observed in simulation (A) [i.e., ϕ peak (0) = 0 . 6]. The peak location moves back and forth periodically. When the peak location becomes close to the horizon, the value of ϕ peak becomes larger. \n<!-- image --> \nFig. 6. Fluxes F E and F J of energy and angular momentum, respectively, toward the horizon observed in simulation (A) [i.e., ϕ peak (0) = 0 . 6]. F E and F J are negative except for very short periods. Therefore, the energy and angular momentum are extracted from the BH. The nonlinear effect makes their values larger. \n<!-- image --> \nsuperradiant bound state as the initial condition. The nonlinear effect appears at t/M /greaterorsimilar 100. In that period, both F E and F J oscillate rapidly, and their mean values are negative. The absolute values of F E and F J become larger around t = 500 M . The primary nonlinear effects are the following two. The first effect is that it enhances the amplitude of the waves of the ( /lscript, m ) = (1 , 1) mode that fall into the BH scarcely changing the real part of the frequency ω 0 . The second effect is that it generates waves of the ( /lscript, m ) = (1 , -1) mode with frequency ω NL which is approximately same as that of waves of the ( /lscript, m ) = (1 , 1) mode, ω NL ≈ ω 0 . Although waves of the ( /lscript, m ) = (1 , -1) mode generate the positive energy flux to the horizon, it is very small in this case. Therefore, the first effect is much stronger than the second effect, and the rates of the extraction of energy and angular momentum are \n<!-- image --> \nFig. 7. The energy density dE/dr ∗ (left) and the angular momentum density dJ/dr ∗ (right) with respect to the tortoise coordinate r ∗ at time t/M = 0 and 1000 for simulation (A) [i.e., ϕ peak (0) = 0 . 6]. \n<!-- image --> \nenhanced. By the interference of the two modes, the small oscillation of F E and F J appear with frequency ≈ 2 ω 0 . The generation of waves of the ( /lscript, m ) = (1 , -1) mode becomes more significant in the strongly nonlinear case [the simulation (B)] as explained later. Although the generation of the ( /lscript, m ) = (1 , -1) mode may seem strange because naively we expect the nonlinearlity to produce modes in proportion to e ± in ( mφ -ωt ) with integer n from Re[ e i ( mφ -ωt ) ], the Green's function analysis in Appendix A supports it (see also Sec. 4.2.2). \nThe left and right panels of Fig. 7 show the energy density dE/dr ∗ and the angular momentum density dJ/dr ∗ with respect to the tortoise coordinate r ∗ , respectively, at t/M = 0 and 1000. There are two peaks for the curve of t/M = 1000 in each panel, the first peak near the horizon and the second peak at r ∗ /M /similarequal 140 (as can be seen in the inset). The locations of the first peaks of energy and angular momentum densities are shifted to small r ∗ values compared to t = 0. This means that most of the energy gets squeezed into a small region close to the horizon because of the nonlinear attractive self-interaction. Another effect of the nonlinearlity is that it transports a small fraction of energy and angular momentum to a region far from the BH, making the small second peak as seen in the inset of each panel.", '4.2.2. Simulation (B): A strongly nonlinear case': "In the simulation (B), we choose the initial amplitude to be ϕ peak (0) = 0 . 7. The parameter ∆ NL /similarequal ϕ 2 / 6 for the effect of the nonlinearlity is ∆ NL /similarequal 0 . 082 at the peak for this setup, and therefore, the nonlinear effect is larger compared to the simulations (A). The nonlinearlity in this situation is strong enough for causing the bosenova collapse. \nFirst, we would like to present some snapshots. Figure 8 shows the density plots of the axion field ϕ in the equatorial ( r cos φ, r sin φ )-plane ( θ = π/ 2) at t/M = 0, 150, 300, and 450. The initial condition t = 0 is the bound state of the KleinGordon field. In the time evolution, the axion cloud rotates counterclockwise and gradually becomes closer to the BH ( t = 150). At t = 300, the axion cloud is highly concentrated in a small region around the BH, and this is when the bosenova begins to happen. During the bosenova, the shape of the axion cloud becomes distorted \nFig. 8. Snapshots of density plot of the axion field ϕ in the equatorial ( r cos φ, r sin φ )-plane ( θ = π/ 2) at t/M = 0, 150, 300, and 450. The axion cloud is rotating counterclockwise. \n<!-- image --> \nand part of the cloud is scattered to the distant region and into the BH ( t = 450). The upper panel of Fig. 9 shows the value of the field at the peak ϕ peak = sup[ ϕ ], and the lower panel shows the position of the peak with respect to the tortoise coordinate r ∗ as functions of t/M . In contrast to the case (A), the value of ϕ peak increases only once around t = 300 M , and after that it fluctuates with short periods. The position of the peak r (peak) ∗ also approaches the horizon only once, and after that, it fluctuates around r ∗ = 10 M which is still fairly close to the horizon. \nIn order to understand the properties of the bosenova collapse, let us look at the field configuration focusing attention to the near horizon region. The left panel of Fig. 10 shows the density plot of the axion field ϕ at t = 500 M in the ( r ∗ /M,φ ) plane. The main part of the axion cloud is moving in the + φ direction. During the bosenova, ingoing waves that are different from those of the bound state of the ( /lscript, m ) = (1 , 1) mode are continuously generated from the cloud, as can be clearly seen in this figure. The cloud remains around r ∗ /similarequal 10 M , while the waves fall into the BH. The generated waves are of the ( /lscript, m ) = (1 , -1) mode in spite of the fact that the initial condition has just the ( /lscript, m ) = (1 , 1) mode. The generated waves have wavelength λ NL /similarequal 8 M in the tortoise coordinate r ∗ . This indicates ˜ ω NL /similarequal 0 . 785 M where ˜ ω := ω -mΩ H (see review in Sec. 2.2), and because the waves are in the m = -1 mode, their angular frequency is ω NL /similarequal 0 . 35 /M . This angular frequency is approximately same \nFig. 9. Same as Fig. 5 but for simulation (B) [i.e., ϕ peak (0) = 0 . 7]. The peak location r (peak) ∗ becomes fairly close to the horizon around t /similarequal 350 M , where ϕ peak reaches approximately four. This is when the bosenova begins to happen, and the behavior after that time is very different from (A): r (peak) ∗ continues small oscillation around r ∗ = 10 M with a short period, and correspondingly, ϕ peak fluctuates around 1.5. \n<!-- image --> \nFig. 10. Left panel: A snapshot of the field in the equatorial plane θ = π/ 2 at t = 500 M . Here, the magnitude of the field ϕ is shown by density plot in the plane ( r ∗ /M,φ ). Right panel: Snapshots of the field ϕ as a function of r ∗ /M at φ = 0 in the equatorial plane θ = π/ 2 for t/M = 0, 350, and 700. \n<!-- image --> \nas that of the bound state of the Klein-Gordon field, ω 0 /similarequal 0 . 39 /M . Since those waves violate the superradiant condition ω < mΩ H because m is negative, it carries the positive energy toward the horizon. \nIt is obvious that the generated ingoing waves originate from the nonlinear effect. However, at first glance, the generation of waves of the ( /lscript, m ) = (1 , -1) mode seems strange, because the nonlinear term ∼ ϕ 3 0 ∼ e 3 i ( φ -ω 0 t ) is unlikely to generate the observed waves whose behavior is ∼ e i ( φ + ω NL t ) . In Appendix A, we study the generation of waves by the nonlinear effect using the Green's function approach, and find that waves of the ( /lscript, m ) = (1 , -1) mode actually can be generated. Therefore, \nFig. 11. Same as Fig. 6 but for simulation (B) [i.e., ϕ peak (0) = 0 . 7]. After the bosenova happens at t /similarequal 350 M , the energy flux F E to the horizon is always positive while the angular momentum flux F J is negative. Therefore, the energy extraction stops while the angular momentum extraction continues. \n<!-- image --> \nthe waves of the ( /lscript, m ) = (1 , -1) mode found in our simulation are not numerical artifact. In short, due to the nonlinear effect, several modes of the bound states (discussed in Sec. 2.2) of frequency ω ≈ ± ω 0 are excited, and these modes include the modes with negative frequency. Since the ( /lscript, m ) = (1 , 1) mode of negative frequency is equivalent to the ( /lscript, m ) = (1 , -1) mode of positive frequency, waves of the ( /lscript, m ) = (1 , 1) mode can be observed. \nThe right panel of Fig. 10 shows the snapshots of the field on the φ = 0 line on the equatorial plane θ = π/ 2, at time t/M = 0, 350, and 700. At t/M = 350, the peak of the field becomes very high, and around this time, the bosenova begins to happen. At t/M = 700, the nonlinear generation of waves of ( /lscript, m ) = (1 , -1) mode continues, and the ingoing waves can be seen for r ∗ /lessorsimilar 0. \n- \nFigure 11 shows the energy flux F E and the angular momentum flux F J toward the horizon evaluated at r ∗ = -100 M . Although both F E and F J are negative initially, after the bosenova happens, the value of F E becomes positive at least up to t = 1000 M . Here, the dominant contribution to the flux comes from the waves of the ( /lscript, m ) = (1 , -1) mode generated by the nonlinear effect. Because those waves obviously violate the superradiant condition ω < mΩ H , the energy flux toward the horizon becomes positive. As a result, the extraction of energy is prevented by the bosenova in this simulation, and about 5.3% of energy falls into the BH by t/M = 1500. On the other hand, the value of F J continues to be negative, and the waves continue to extract the angular momentum from the BH. This is because the ingoing waves are in the m = -1 mode, and hence, carry negative angular momentum if energy is positive. The small fluctuations of F E and F J come from the interference of the ( /lscript, m ) = (1 , ± 1) modes, and the typical angular frequency of this oscillation is ≈ 2 ω 0 , similarly to the simulation (A). \n≈ The left and right panels of Fig. 12 show the energy density dE/dr ∗ and the angular momentum density dJ/dr ∗ with respect to the tortoise coordinate r ∗ , respectively, at t/M = 0, 750, and 1500. At late time, most of the energy is contained in the domain 0 /lessorsimilar r ∗ /M /lessorsimilar 30, and the energy and angular momentum densities \n<!-- image --> \nFig. 12. The energy density dE/dr ∗ (left) and the angular momentum density dJ/dr ∗ (right) with respect to the tortoise coordinate r ∗ at time t/M = 0, 750, and 1500 for simulation (B) [i.e., ϕ peak (0) = 0 . 7]. \n<!-- image --> \nhave similar shapes for t/M = 750 and 1500. The difference between t/M = 750 and 1500 can be seen in the domains r ∗ /M /lessorsimilar 0 and r ∗ /M /greaterorsimilar 100. The behavior of each of dE/dr ∗ and dJ/dr ∗ in the domain r ∗ /M /lessorsimilar 0 can be seen in the left inset of each panel. At t = 750 M , the energy density is positive and the angular momentum density is negative. This is consistent with the fact that the energy and angular momentum fluxes to the horizon are positive and negative, respectively, as seen in Fig. 11. At t = 1500 M , both two densities fluctuate around zero, and the mean values of energy and angular momentum densities are still positive and negative, respectively. This is because the nonlinear resonance becomes weak at this time, and the system settles down to a quasistationary state again. The behavior of each of dE/dr ∗ and dJ/dr ∗ at the distant place is shown in the right inset of each panel. At t = 750 M , some fraction of energy and angular momentum are distributed at a far region. Around r ∗ = 400 M , a small bump can be seen. This bump moves outward approximately at the speed of light. Therefore, a kind of 'explosion' happens in the bosenova. However, this explosion is very small because this bump has only ≈ 0 . 2% of the total energy. At t = 1500 M , more amount of energy and angular momentum can be seen at the distant place. Therefore, following the small explosion, the field energy gradually spreads out to the distant region. At t/M = 1500, about 16 . 6% of the total energy distributes in the region r ∗ /M ≥ 60. Except for the small bump moving at the speed of light, all field in the distant region seems to be gravitationally bounded. The simulation was performed up to t/M = 2000, and it was found that after t/M = 1500, some part of the energy at the distant place begins to fall back. \nIt is interesting to study how much energy of the axion field is converted from the ( /lscript, m ) = (1 , 1) mode to other modes. Unfortunately, numerical decomposition of the field ϕ into the modes is rather tedious because it requires Fourier transform from the time domain to the frequency domain, and also, in the case of the Kerr spacetime, the separation constant and the spheroidal harmonics depend on the frequency ω . Instead, we give rough estimate on how much energy is converted to the ( /lscript, m ) = (3 , ± 3) mode in an approximate way. In this approximation, we use the spherical harmonics instead of the spheroidal harmonics, and decompose the field into \nFig. 13. The estimated amount of energy E 33 of ( /lscript, m ) = (3 , ± 3) mode generated by the mode mixing in the bosenova as a function of time t/M . Here E 33 is normalized by the total energy E and shown in the unit of %. About 10% energy is converted into the ( /lscript, m ) = (3 , ± 3) mode. \n<!-- image --> \nthe form ϕ = ∑ 0 ≤ m ≤ /lscript ϕ /lscriptm , where ϕ /lscriptm := f /lscriptm ( t, r ) Y /lscriptm ( θ, φ ) + f /lscript -m ( t, r ) Y /lscript -m ( θ, φ ). The energy E /lscriptm of each ( /lscript, ± m ) mode is estimated by substituting ϕ /lscriptm into the formula for the energy. In this manner, we have studied the mode 0 ≤ l ≤ 4, and found that the modes with ( /lscript, m ) = (1 , ± 1) , (3 , ± 1), and (3 , ± 3) are nonzero and the other modes are approximately zero. \nFigure 13 shows the value of energy of the ( /lscript, m ) = (3 , ± 3) mode normalized by the total energy, E 33 /E , as a function of time. After the bosenova happens, the ( /lscript, m ) = (3 , ± 3) mode starts to grow and has about 10% of the total energy at t/M = 1000. On the other hand, before the bosenova, the mode mixing is fairly weak, and almost all fields are in the ( /lscript, m ) = (1 , 1) mode. We evaluated the value of E 33 /E also for the simulation (A) [i.e., ϕ peak (0) = 0 . 6], and found that the ( /lscript, m ) = (3 , ± 3) mode has at most 0 . 014% of the total energy. This result shows that the bosenova converts relatively large amount of the axion field from the ( /lscript, m ) = (1 , 1) mode to other modes. \nTo summarize, if the initial amplitude is sufficiently large, the bosenova happens for the axion cloud of quasibound state of the ( /lscript, m ) = (1 , 1) mode. The bosenova in this system is characterized by the following features. First, a small amount ( ≈ 0 . 2%) of energy comes out from the axion cloud approximately at the speed of light. After that, about 15% of energy gradually spreads out to the distant region, although it seems to be gravitationally bounded. Therefore, the bosenova of the BH-axion system is somewhat similar to the bosenova of BEC in experiments (In Sec. 6, we give a more detailed comparison). Second, about 5% of energy falls into the BH after the bosenova. This energy is carried by waves of the ( /lscript, m ) = (1 , -1) mode generated by the nonlinear effect. Therefore, in the BH-axion system, the 'explosion' happens both to distant place and to the horizon. Finally, once the bosenova happens, the mode mixing effectively occur. In addition to the generation \nFig. 14. Schematic picture of time evolution of the field amplitude and two possibilities of the final state of the superradiant instability. \n<!-- image --> \nof the ( /lscript, m ) = (1 , -1) mode, the /lscript = 3 mode was observed to get ≈ 10% of the total energy after the bosenova.", '4.3. Does the bosenova really happen?': 'In the two simulations performed above, we found the following: In the simulation (A), when the initial peak value is small, the nonlinear effect causes periodic changes in the peak location and the peak value, and enhances the energy and angular momentum extractions; In the simulation (B), when the initial peak value is large, the nonlinear effect causes an explosion, the bosenova. During the bosenova, waves of the ( /lscript, m ) = (1 , -1) mode violating the superradiant condition are generated, and thus, the energy flux toward the horizon becomes positive terminating the superradiant instability. \nIn this subsection, we discuss whether the bosenova happens as the result of the superradiant instability. In a realistic system, as the rotational energy of the BH is extracted, the field gradually gets amplified. In this sense, the simulation (B) may be artificial because we gave a quasibound state of large amplitude by hand and its initial condition may not be realized as the result of the superradiant instability. In particular, we have to take care of the possibility that the bosenova does not happen, as there are a lot of examples of nonlinear systems in which the nonlinear effects saturate the instabilities leading the systems to quasistable states. Figure 14 is a schematic picture depicting the two possibilities of time evolution: the bosenova and the saturation. \nIn order to discuss which is the case, we perform supplementary simulations as follows. In these simulations, we prepare the initial condition by applying the scale transformation to the result of the simulation (A) at t = 1000 M as ϕ (0) = Cϕ ( A ) (1000 M ) and ˙ ϕ (0) = C ˙ ϕ ( A ) (1000 M ). This is because in the presence of the nonlinear term, the energy gets confined in a smaller region near the BH compared to the case of the quasibound state of the massive Klein-Gordon field as found in Fig. 7, and the initial condition prepared by this procedure is expected to be more realistic and to approximate the nearly final state of the superradiant instability.', 'H. Yoshino and H. Kodama': "Fig. 15. The relation between time t and the amount of energy ∆E that has fallen into the BH by the time t . Here, the cases of C = 1 . 05, 1 . 08, and 1 . 09 are shown. \n<!-- image --> \nThe cases of C = 1 . 05, 1 . 08, and 1 . 09 were simulated. Here, the initial state for a larger C is expected to approximate the later state of Fig. 14. \nFigure 15 shows the relation between time t and the amount energy ∆E that has fallen by the time t . In other words, ∆E is the integrated energy flux toward the horizon from time zero to t : ∆E := ∫ t 0 F E dt . The gradient of each curve shows the flux F E toward the horizon: If it is negative (resp. positive), the negative (resp. positive) energy is falling into the BH. The curve of the case C = 1 . 05 is always negative, and at this stage, the energy continues to be extracted. In this case, the averaged rate of the superradiant instability is γ NL M ≈ 5 . 85 × 10 -7 , which is larger than that of the case of the linear Klein-Gordon field, γM = 1 . 30 × 10 -7 . This confirms that the nonlinear effect enhances the rate of superradiant instability before the bosenova. The bosenova does not happen at least by time t/M = 10000. \nNext, let us look at the case C = 1 . 09. In this case, the value of F E is negative until t/M /similarequal 4000. Here, the rate of energy extraction is γ NL M /similarequal 7 . 45 × 10 -7 , which is further larger than that of the case C = 1 . 05. However, around t/M /similarequal 4000, the value of F E becomes positive, and for t/M /greaterorsimilar 4500, the value of F E becomes positive and fairly large: the bosenova happens around this time. This case represents the example such that the bosenova happens after a certain period of energy extraction. Therefore, it is natural to consider that this simulation approximates what actually happens at the final stage of the superradiant instability. The burst of positive energy toward the horizon continues until t/M /similarequal 5300, and after that small explosions happen intermittently. \nIn the case of C = 1 . 08, the energy extraction continues from t/M = 0 to 5000. Around t/M = 5000 and 7500, small amounts of positive energy fall into the BH, and then, around t/M = 8000, a large positive ingoing energy flux is generated. This is the bosenova in this case. The bosenova in the case C = 1 . 08 happens later than the case C = 1 . 09, mainly because the initial amount of energy of the former is smaller than that of the latter, and therefore, a longer period of energy extraction is \nrequired. However, it should be noted that the bosenova of these two cases happen at different values of energy: The amount of energy when the positive flux is first generated is E/ [( f a /M p ) 2 M ] /similarequal 1633 and 1607 for the cases C = 1 . 09 and 1 . 08, respectively. Although the criterion for the occurrence of the bosenova is mainly determined by the energy amount, it would depend also on the detailed structure of the axion cloud. \nThe natural picture of the final stage of the superradiant instability is as follows. Before the bosenova, similarly to the case of C = 1 . 05, the energy continues to be extracted and the rate of the energy extraction is gradually enhanced. Then, at a certain critical point, where the energy of the axion cloud is E/ [( f a /M p ) 2 M ] ≈ 1620, a large amount of positive energy suddenly falls into the BH, and here, the bosenova happens like the simulations of C = 1 . 08 and 1 . 09. The answer to the question 'Does the bosenova really happen?' is ' Yes ,' because from our simulations, it is natural to consider that the bosenova actually happens as a result of the superradiant instability. In particular, we have obtained no evidence for the possibility that the nonlinear effect saturates the growth by the superradiant instability and leads the system to a quasistable state. \nIf we assume the decay constant f a to be the GUT scale, ≈ 10 16 GeV, the bosenova collapse happens when the energy of the axion cloud grows to be E /similarequal 1 . 6 × 10 -3 M , i.e., when the axion cloud gets energy of ≈ 0 . 16% of the BH mass. Therefore, if the value of f a is the GUT scale or smaller, the back reaction to the background spacetime, such as the change in the parameter in M and a of the BH or the distortion of the background geometry by the axion cloud, is negligible. On the other hand, if we assume f a ≈ 10 17 GeV, the bosenova collapse happens when E /similarequal 0 . 16 M , i.e., when the axion cloud gets energy of ≈ 16% of the BH mass. In such a situation, back reaction to the background geometry is significant, and the bosenova phenomena has to be studied by the method of numerical relativity. \nThe time evolution long after the bosenova is also an interesting subject, although this is beyond the scope of this paper. Looking at Fig. 15, in the case of C = 1 . 09, the small amount of positive energy intermittently falls into the BH. One possibility is that this phenomena continues and the axion cloud continue to extract and lose small amounts of energy. Another possibility is that the axion cloud loses almost all energy, and the superradiant instability happens from the beginning, leading to the bosenova collapse that has approximately same scale as the previous one. In order to clarify which is the case, a very-long-term simulation or construction of a good approximate model is necessary. \nWhen we consider a very-long-term evolution of the BH-axion system, taking account of changes in mass M and angular momentum J of the BH is also important. In our simulations, the energy is extracted from the BH in superradiant instability and falls back to the BH in the bosenova collapse. On the other hand, the angular momentum is extracted both in the superradiant instability and in the bosenova collapse. Therefore, the spin parameter a/M would gradually decrease in a verylong-term evolution. As a result, superradiant instability of the ( /lscript, m ) = (1 , 1) mode will stop when a/M is decreased to a certain value that is determined by the mass µ of the axion (see Fig. 7 of Ref. 26)). After that, superradiant instability of the \n( /lscript, m ) = (2 , 2) mode will become a primary factor in determining the evolution of the axion cloud.", '§ 5. Effective theory of axion cloud model': 'In this section, we discuss the reason why the bosenova happens in the BHaxion system by introducing an effective theory for this system. In this discussion, we model the axion cloud using a time-dependent Gaussian wavefunction. Also, we assume the non-relativistic approximation in which the gravity is treated by the Newtonian potential. The distribution is specified by the following three parameters: δ r (the width of the wavepacket along the r direction), δ ν (the width along the θ direction), and r p (the position of the peak with respect to the r coordinate). Under these approximations, we derive the effective action for the three parameters and find various properties of the bosenova which are consistent with the simulations.', '5.1. Effective action': 'The action for the axion field Φ is given by Eq. (2 . 1). In terms of the normalized field ϕ = Φ/f a , the action is rewritten as \nˆ S = ∫ d 4 x √ -g [ -1 2 ( ∇ ϕ ) 2 -µ 2 ( ϕ 2 2 + ˆ U NL ( ϕ ) )] , (5 . 1) \nwhere the nonlinear potential ˆ U NL is defined by \nWe introduce ψ as \nˆ U NL ( ϕ ) = 1 -ϕ 2 2 -cos ϕ = -∞ ∑ n =2 ( -1) n (2 n )! x 2 n . (5 . 2) \nϕ = 1 √ 2 µ e -iµt ψ + e iµt ψ ∗ ) . (5 . 3) \n( \n) Here, ψ is a slowly varying function under the non-relativistic approximation. Substituting this formula into Eq. (5 . 1), we have \nˆ S NR = ∫ d 4 x [ i 2 ( ψ ∗ ˙ ψ -ψ ˙ ψ ∗ ) -1 2 µ ∂ i ψ∂ i ψ ∗ + α g r ψ ∗ ψ -µ 2 ˜ U NL ( | ψ | 2 /µ ) ] , (5 . 4) \n˜ U NL ( x ) = -∞ ∑ n =2 ( -1 / 2) n ( n !) 2 x n . (5 . 5) \nwhere α g := Mµ and \nHere, the Newtonian approximation is adopted for gravity. \nNow, we assume the form of ψ as \nψ = A ( t, r, ν ) e iS ( t,r,ν )+ mφ , (5 . 6) \nwhere ν := cos θ and we set m = 1. The functions A ( t, r, ν ) and S ( t, r, ν ) are chosen to be the following form: \nA ( t, r, ν ) ≈ A 0 exp [ -( r -r p ) 2 4 δ r r 2 p -( ν -ν p ) 2 4 δ ν ] , (5 . 7) \nwhere \nT = N 2 µ [ p 2 +8 pPr p δ r 1 + δ r +4 P 2 r 2 p δ r 1 + 3 δ r 1 + δ r +4 π 2 ν δ ν r 2 p (1 + δ r ) ] (5 . 12) \nV Nµα 2 g = 1 2( α g µr p ) 2 (1 + δ r ) ( 1 + δ ν + 1 4 δ r + 1 4 δ ν ) -1 ( α g µr p )(1 + δ r ) -α -2 g ∞ ∑ n =2 ( -1 / 2) n ( n !) 2 n [ N ∗ √ δ r δ ν ( α g µr p ) 3 (1 + δ r ) ] n -1 , (5 . 13) \nwith N ∗ = ( α 3 g µ 2 / 4 π 2 ) N . Here, we have kept terms up to the first-order in δ ν because the wavepacket form, Eq (5 . 7), is not a very good approximation in the ν direction and taking account of higher-order terms in δ ν does not have an important meaning. The term depending on ˙ S 0 can be omitted since it just gives the conservation of N . The variables p , P , and π ν can be related to the conjugate momenta of the variables δ r , δ ν , and r p , and therefore, we can express the Lagrangian only in terms of δ r , δ ν , and r p , and their time derivatives. To summarize, the equivalent Lagrangian is \nL = T -V, (5 . 14) \nT = 1 2 A ˙ δ 2 r + B ˙ δ r ˙ r p + 1 2 C ˙ r 2 p + 1 2 D ˙ δ 2 ν , (5 . 15) \nA = 1 4 Nµr 2 p 1 + 45 δ r +198 δ 2 r +126 δ 3 r +45 δ 4 r +9 δ 5 r (1 + δ r ) 3 δ r (1 + 3 δ 2 r ) , (5 . 16a) \nB = 1 2 Nµr p -7 -30 δ r +54 δ 2 r +30 δ 3 r +9 δ 4 r (1 + δ r ) 2 (1 + 3 δ 2 r ) , (5 . 16b) \nwhere \nS ( t, r, ν ) ≈ S 0 ( t ) + p ( t )( r -r p ) + P ( t )( r -r p ) 2 + π ν ( t )( ν -ν p ) 2 + · · · . (5 . 8) \nδ r ( t ) is the width of the wavepacket along the r direction, δ ν ( t ) is the width along the ν direction (i.e., θ direction), and r p ( t ) is the position of the peak with respect to r coordinate. Since the center of the wavepacket always exists on the equatorial plane, the peak position with respect to ν coordinate is always zero, ν p ≡ 0. We define N as \n√ \nN = ∫ d 3 xA 2 ≈ 4 π 2 A 2 0 δ r δ ν r 3 p (1 + δ r ) . (5 . 9) \nHere, we ignored the inner cutoff of the integration range of r . In a similar manner, we perform the integration of the action (with respect to spatial coordinates) and derive the Lagrangian density as \nL = -˙ S 0 N + p ˙ r p N +(˙ p -2 P ˙ r p )2 r p δ r 1 + δ r N -˙ Pr 2 p δ r 1 + 3 δ r 1 + δ r N -˙ π ν δ ν N -H (5 . 10) \nwith \nH = T + V, (5 . 11) \n<!-- image --> \nFig. 16. Left panel: The behavior of the potential V as a function of α g µr p for N ∗ = 0 . 02,...,0 . 08 with 0 . 01 intervals in the case α g = 0 . 1. There is one stable point for N ∗ = 0 . 02, and as N ∗ is increased, another stable point appears for N ∗ ≈ 0 . 03. The outer stable point disappears for N ∗ ≈ 0 . 07 and the system moves to the inner stable point (i.e., the bosenova). For N ∗ = 0 . 08, there is only one stable point at α g µr p ≈ 0 . 025. Right panel: The position of the equilibrium point as a function of N ∗ in the case α g = 0 . 1. For 0 . 0304 /lessorsimilar N ∗ /lessorsimilar 0 . 071, there are two stable points and one unstable point, while only one stable point exists for N ∗ /lessorsimilar 0 . 0304 and /greaterorsimilar 0 . 071. \n<!-- image --> \nC = Nµ 1 + 6 δ r -26 δ 2 r +18 δ 3 r +9 δ 4 r (1 + δ r )(1 + 3 δ 2 r ) , (5 . 16c) \nD = 1 4 Nµr 2 p (1 + δ r ) δ ν . (5 . 16d) \nThe potential V is given in Eq. (5 . 13).', '5.2. Equilibrium point of the potential': 'Let us study the properties of the potential V . This potential is a function defined in the three-dimensional phase space ( δ r , δ ν , α g µr p ). For each N ∗ , the equilibrium point is determined by V ,δ ν = V ,δ r = V ,r p = 0. From this equilibrium condition, the following two simple relations are obtained: \nδ r = -1 + 4 δ 2 ν + √ 1 -8 δ ν +8 δ 2 ν +64 δ 3 ν +16 δ 4 ν 2( -2 + 4 δ ν +16 δ 2 ν ) , (5 . 17) \nα g µr p = 4 δ ν -1 2 δ ν + 1 4 δ r +1 . (5 . 18) \nIf we impose these two conditions (5 . 17) and (5 . 18), we have a line in the threedimensional phase space ( δ r , δ ν , α g µr p ), and along this line, the potential V can be regarded as the function of α g µr p . Let us study the behavior of this function V ( α g µr p ) varying the value of N ∗ . We also plot the value of α g µr p at the equilibrium point as a function of N ∗ . In the following, the two cases α g = 0 . 1 and 0 . 4 are shown. \n<!-- image --> \nFig. 17. The same as Fig. 16 but for α g = 0 . 4. The cases N ∗ = 1 . 0,...,1 . 5 are shown with 0 . 1 intervals for left panel. In this case, only one stable point exists for all N ∗ . Around N ∗ = 1 . 2, the position of the peak α g µr p rapidly changes to a smaller value. \n<!-- image -->', '5.2.1. The case α g = 0 . 1': 'Let us look at the case α g = 0 . 1. The left panel of Fig. 16 shows the relation between α g µr p and V along the line introduced above. The cases of N ∗ = 0 . 02,...,0 . 08 are depicted with 0 . 01 intervals. The right panel of Fig. 16 shows the relation between N ∗ and α g µr p at the equilibrium point. For N ∗ < 0 . 0304, there is only one stable point. But as the N ∗ is increased, the situation changes: At N ∗ /similarequal 0 . 0304, another stable point appears inside of the original stable point, and the two stable points exist for 0 . 0304 /lessorsimilar N ∗ /lessorsimilar 0 . 071. At N ∗ /similarequal 0 . 071, the outer stable point disappear, and only one stable point exists for N ∗ /greaterorsimilar 0 . 071. \nThe interpretation of this result is as follows. Because of the superradiant instability, the value of N ∗ is gradually increased and the shape of the potential gradually changes. The position of the wavepacket is around α g µr p ≈ 2 . 7 for small N ∗ , and it becomes smaller as N ∗ is increased. When N ∗ reaches ≈ 0 . 03, a new stable point appears inside of the original stable point. The system remains at the original outer stable point for a while, i.e., until N ∗ reaches ≈ 0 . 07. When N ∗ exceeds ≈ 0 . 07, the outer stable point disappears, and therefore, the system jumps from the original outer stable point to the inner stable point. Accordingly, the value of α g µr p jumps from ≈ 1 . 5 to ≈ 0 . 25. So, the phase transition occurs when the amplitude reaches some critical point, and this is interpreted as the bosenova collapse. This is consistent with our simulation results: In simulation (A), the peak position continues to oscillate in relatively distant region, while in simulation (B), the peak position remains in the neighborhood of the horizon after the bosenova.', '5.2.2. The case α g = 0 . 4': "We turn our attention to the case α g = 0 . 4. The left panel of Fig. 17 shows the value of V as a function of α g µr p along the line in the phase space ( δ r , δ ν , α g µr p ) introduced in the beginning of Sec. 5.2. The cases of N ∗ = 1 . 0,...,1 . 5 are depicted with 0 . 1 intervals. The right panel of Fig. 17 shows the relation between N ∗ and \nα g µr p at the equilibrium point. In this case, there is only one stable point for all values of N ∗ . Around N ∗ ≈ 1 . 2, the value α g µr p of the equilibrium point rapidly decreases as N ∗ is increased, and hence, the equilibrium point becomes located closer to the horizon. \nOur interpretation is as follows. In this axion cloud model, the phase transition does not occur for α g = 0 . 4: The situation with two stable equilibrium points occurs only for α g /lessorsimilar 0 . 3. But in our simulations for α g = 0 . 4, the bosenova suddenly happens. Therefore, the phase transition seen in the α g = 0 . 1 case gives a more correct picture. This discrepancy seems to arise because the axion cloud model discussed here is a model of rough approximation. Except for this point, however, the axion cloud model reproduces various characteristic features of the phenomena in the simulations. For example, Fig. 17 shows that as the superradiant instability progresses, the value of α g µr p of the peak position becomes smaller, and around the 'critical' value N ∗ ≈ 1 . 2, the position rapidly becomes very close to the horizon. These are quite consistent with the results of our numerical simulations.", '5.3. Small oscillations around the equilibrium point': 'It is also interesting to study small oscillations around the equilibrium point because it allows us to estimate the typical dynamical time scales of the BH-axion system. Here, we introduce the phase space parameter q i defined by \nq i = ( δ r , δ ν , α g µr p ) . (5 . 19) \nwith i = 1 , 2 , and 3. The equilibrium position is denoted by q (0) i and the deviation ∆q i from the equilibrium point is introduced by q i = q (0) i + ∆q i . Using the standard method in the classical mechanics, we can rewrite the Lagrangian in terms of ∆q i collecting only the second-order terms, and derive the equation of the harmonic oscillators, \n∆ q i = -Ω ij ∆q j . (5 . 20) \nThe solution of this equation can be written as a linear superposition of three normal modes, and their squared frequencies ω 2 EG are given by the eigenvalues of the matrix Ω ij . For each value of ω 2 EG , there exists an eigenvector describing the direction of oscillation of that normal mode in the phase space. \nIn the following, we discuss the cases N ∗ = 1 . 1 and 1 . 3 for α g = 0 . 4. Because the bosenova happens around N ∗ ≈ 1 . 2, the cases N ∗ = 1 . 1 and N ∗ = 1 . 3 are expected to approximate the state before and after the bosenova, respectively. \n5.3.1. The case α g = 0 . 4 and N ∗ = 1 . 1 \nIn this case, the eigenvalues of the matrix Ω ij are given by \n( ω EG µα 2 g ) 2 = 1 . 141 , 0 . 249 , 0 . 0166 , (5 . 21) \nwith the corresponding eigenvectors \n∆q i = 0 . 110 -0 . 027 0 . 994 , 0 . 075 0 . 724 0 . 686 , -0 . 378 -0 . 005 0 . 925 . (5 . 22) \nLet us focus attention to the third mode with ( ω EG /µα 2 g ) 2 /similarequal 0 . 0166. This mode represents the oscillation of the axion cloud in the direction of r and δ r , which scarcely changes the shape in the ν direction. The period of the oscillation of this mode is \n∆t ≈ 761 M. (5 . 23) \nThis is consistent with the longterm oscillation found in the simulation (A) in Sec. 4.2.1. The origin of the long period of the oscillation is that the effective potential V ( α g µr p ) becomes approximately flat and therefore the second-order derivative of this function becomes very small just before the bosenova happens. \n5.3.2. The case α g = 0 . 4 and N ∗ = 1 . 3 \nIn this case, the eigenvalues of the matrix Ω ij are \n( ω EG µα 2 g ) 2 = 14 . 06 , 5 . 59 , 0 . 175 , (5 . 24) \nwith the eigenvectors \n∆q = 0 . 218 -0 . 030 0 . 975 , 0 . 070 0 . 927 0 . 367 , -0 . 640 -0 . 085 0 . 763 . (5 . 25) \nEach eigenvalue in the case N ∗ = 1 . 1 is larger than the corresponding eigenvalue in the case N ∗ = 1 . 3. The period of oscillation of the first mode is \n∆t ≈ 26 M. (5 . 26) \nAlthough this period is longer than the period ∆t ≈ 10 M observed in simulation (B) in Sec. 4.2.2, this model explains that the typical dynamical time scale after the bosenova becomes shorter compared to that before the bosenova. The discrepancy would be because the axion cloud model discussed here is a rough approximate model; If this model is improved so that the two stable equilibrium positions appear also for α g = 0 . 4, it would give shorter periods of oscillations.', '§ 6. Summary and discussion': 'In this paper, we have studied the sine-Gordon field in a Kerr spacetime motivated by landscape of axionlike particles/fields, i.e. the axiverse. In order to calculate the evolution of a scalar field in a Kerr spacetime, we developed a 3D code that has the ability to describe the growth rate by superradiant instability of the linear Klein-Gordon field with /lessorsimilar 2% error (Sec. 3.2.1). Using this code, we have performed simulations for the scalar field mass α g = Mµ = 0 . 4 and the BH rotation parameter a/M = 0 . 99 (Sec. 4.2), where the initial condition is taken to be the quasistationary bound state of the ( /lscript, m ) = (1 , 1) mode of the Klein-Gordon equation. When the initial peak value is small (Sec. 4.2.1), the nonlinear effect causes periodic changes in the peak location and the peak value, and the nonlinear effect enhances the energy and angular momentum extractions. On the other hand, if the initial peak value is \nrelatively large (Sec. 4.2.2), the nonlinear effect causes a collapse of the axion cloud and a subsequent explosion, i.e. the bosenova collapse. During the bosenova collapse, a kind of resonance occurs to generate waves of the ( /lscript, m ) = (1 , -1) mode falling into the BH. Since these waves violate the superradiant condition, the energy flux toward the horizon becomes positive. Therefore, the energy extraction is terminated by the bosenova. \nIn Sec. 4.3, we have discussed whether the bosenova happens or not as a result of the superradiant instability taking account of the two possibilities, the bosenova collapse and the saturation of the superradiant instability. We performed additional simulations from improved initial conditions that are expected to be more realistic. In these simulations (the cases C = 1 . 08 and 1 . 09), the energy extraction continued for a long period of time and suddenly the bosenova collapse happened. This result supports the possibility that the bosenova collapse actually occurs as a result of the superradiant instability when the axion cloud gets energy of E ≈ 1600( f a /M p ) 2 M (for the present setup α g = 0 . 4 and a/M = 0 . 99). If the decay constant is order of the GUT scale, f a ≈ 10 16 GeV, the energy at the bosenova collapse is 0 . 16% of the BH mass. \nIn Sec. 5, we have discussed why the bosenova happens by constructing an effective theory in the nonrelativistic approximation. The axion cloud is assumed to be a time-dependent Gaussian wavepacket, and the effective theory is described by dynamics of three variables that specify its shape. The dynamics is determined by a potential V whose behavior depends on the amplitude of the wavepacket. When the value of α g is small (e.g., α g = 0 . 1 in Sec. 5.2.1), there are two (outer and inner) stable minima in the potential V for a small amplitude, and the outer minimum disappear when amplitude grows to some value. Therefore, the bosenova collapse is explained by the phase transition from the outer stable point to the inner stable point. The effective theory also indicates that for large values of α g , such phase transition does not occur and the bosenova is unlikely to happen. The dynamical time scales observed in our simulations before and after the bosenova also can be successfully explained by studying small oscillations around the local minimum. Therefore, the axion cloud model describes the BH-axion system fairly accurately. This result indicates that the critical amplitude for the onset of the bosenova collapse is primarily determined by the self-interaction of axions rather than the nonlinear gravity of the BH, although the nonlinear BH gravity is important in making the field amplitude larger by extraction of the rotational energy of the BH. \nHere, we compare the bosenova phenomena in the system of BEC atoms and in the BH-axion system. The action of the BH-axion system in nonrelativistic approximation, Eq. (5 . 4), has the same form as the action of the BEC atoms, e.g. Eq. (3) of Ref. 10), except that the parabolic potential is replaced by the Newton potential and the higher-order terms are included in the nonlinear potential. For this reason, the two phenomena are naturally expected to have similarity to each other. In fact, both of the growth of the amplitude in Fig. 9 in our simulation and the implosion of BEC atoms (e.g., Fig. 3 of Ref. 10) or Fig. 1 of Ref. 11)) are caused by nonlinear attractive interaction. But there exists qualitative difference between the two systems: The big implosions continue to happen in the BEC system, while \nthe big implosion happens only once in the BH-axion system. Also, the growth of the peak height in our system is not as sharp as the BEC system. The reason is that we are dealing with the ( /lscript, m ) = (1 , 1) mode while the typical BEC system is in the ( /lscript, m ) = (0 , 0) mode. Since the BEC system does not have the angular momentum, the BEC atoms concentrate to the center and the peak height can become large almost unboundedly, and this process can continue intermittently. On the other hand, since the axion cloud is rotating around the BH in our system, the centrifugal force prevents the axion cloud from collapsing to the center. For this reason, the growth of the peak height is limited. This point is also understood by looking at the effective potential. In Ref. 10), an effective theory for the BEC atoms was discussed using a time-dependent Gaussian wave function in a similar manner to Sec. 5. The effective potential of the BEC system behaves as f (0) = -∞ , and this enables the BEC atoms to concentrate at the center. On the other hand, the effective potential of the BH-axion system behaves as V (0) = ∞ , and this makes the inner stable point. Hence, the high concentration of axion cloud to the center is prohibited. (Compare Fig. 1 of Ref. 10) and Fig. 16 of this paper.) \nThe authors of Ref. 9), 10), 11) discussed the fact that the bursts after the implosions in the BEC system are caused by the two-body dipolar and the three-body recombination of the BEC atoms. The loss of atoms causes the decrease in the attractive interaction, and the burst is generated by the zero-point kinetic pressure. In numerical simulations, the loss of atoms are handled by introducing the phenomenological terms ( -i /planckover2pi1 / 2)( K 2 | ψ | 2 + K 3 | ψ | 4 ) ψ to the nonlinear Schrodinger equation (e.g., Eq. (1) of Ref. 11)). Although no such phenomenological terms are introduced in solving the axion field in our paper, part of the axion cloud was observed to spread out to the distant region. The reason would be that in the BH-axion system, infalling waves of the m = -1 mode are generated by excitation of the bound states, causing the loss of energy (and therefore, the attractive interaction) of the axion cloud. Therefore, in our system, fall of a fraction of the axion cloud to the BH would play the same role as the three-body recombination of atoms in the BEC system. \nFinally, we briefly discuss whether the bosenova can be observed by gravitational wave detectors. Studying gravitational waves emitted from an axion cloud is rather difficult since it requires quantum mechanical description, and the quadrupole formula cannot be directly applied (although the authors of Ref. 2) discussed the fact that the quadrupole formula corresponds to level transition of axion particles). The level transition from the ( /lscript, m ) = (1 , 1) mode is prohibited by the selection rule, and the two axion annihilation to a graviton is estimated to be rather small. 2) Therefore, the gravitational wave emission from the axion cloud scarcely affects the occurrence of the bosenova. On the other hand, it would be possible to discuss gravitational waves emitted in the bosenova collapse by the quadrupole formula (at least in the sense of order estimate), since the typical time scale of the bosenova is rather long, ∆t 500 M , and the nonrelativistic approximation can be applied. \nThe quadrupole moment Q ij of the axion cloud is estimated to be Q ij ∼ r 2 p E , where r p ∼ 10 M is the position of the peak with respect to the r coordinate. In the bosenova collapse, a burst of positive energy flux toward the horizon is generated, and about 5% of the total energy falls into the BH in the typical time ∆t ∼ 500 M . \n∼ \nIt causes decrease in Q ij through loss of the energy. Here, we ignored the shedding of the axion cloud to a far region because this process is slower than the burst to the horizon. Let us consider the situation where the decay constant f a is the GUT scale, and hence, the bosenova collapse occurs at the energy E ≈ 1 . 6 × 10 -3 M . Assuming the time dependence of energy as E = E 0 + ( ∆E/ 2)[cos( πt/∆t ) -1] (0 ≤ t ≤ ∆t , where the bosenova happens at t = 0 and ends at t = ∆t ) with ∆E ≈ 0 . 05 E , we have \nFrom this, the energy loss rate is estimated to be dE/dt ∼ ( ... Q ij ) 2 ∼ 10 -18 , and the total radiated energy is E rad ∼ 10 -15 M ∼ 10 -12 E . Therefore, the energy converted to gravitational waves is expected to be very small in the bosenova. In a similar manner, we can estimate the amplitude of gravitational waves emitted in this process as \n... Q ij ∼ r 2 p ... E ∼ r 2 p ∆E ( π ∆t ) 3 ∼ 10 -9 . (6 . 1) \nh ∼ Q ij r obs ∼ 10 -7 M r obs , (6 . 2) \nwhere r obs denotes the distance from the BH to an observer. \nFor the supermassive BH at the center of our galaxy, Sagittarius A ∗ , the mass and the distance from the Sun have been estimated to be M ≈ 4 . 5 × 10 6 M /circledot 27),28) and r obs ≈ 8 kpc. 28),29) For these values, we have M/r obs ∼ 10 -11 , and therefore, h ∼ 10 -18 . The frequency of gravitational waves emitted in this process is ∼ 1 /∆t ∼ 10 -4 Hz. The strain amplitude h rss := [ ∫ | h | 2 dt ] 1 / 2 of the gravitational wave burst in this process is h rss ∼ 10 -16 (Hz) -1 / 2 , and for the frequency 10 -4 Hz, this value is (by order one) above the threshold of the sensitivity of the future-planned spacebased gravitational wave detector, the LISA. 30) For the BH candidate Cygnus X1, the mass and the distance have been estimated to be M ≈ 8 . 7 ± 0 . 8 M /circledot 31) and r obs ≈ 1 . 86 +0 . 12 -0 . 11 kpc. 32) For these values, we have M/r obs ∼ 10 -16 , and therefore, h ∼ 10 -23 . The frequency of gravitational waves emitted in this process is ∼ 100 Hz. The strain amplitude is h rss ∼ 10 -24 (Hz) -1 / 2 , and for the frequency 100 Hz, this value is below the threshold of the sensitivity of the planned ground-based gravitational wave detectors, the Advanced LIGO, the Advanced Virgo, and the LCGT. 30) \nNote that the estimate here has been done for the parameters α g = Mµ = 0 . 4, a/M = 0 . 99, and f a = 10 16 GeV. For other parameters, e.g. if the value of f a is smaller, the detection of gravitational waves from the bosenova collapse is more difficult. However, if the value of f a is around the GUT scale or somewhat larger, we have the possibility of detection of gravitational wave bursts from the bosenova. Although we have discussed only bursts here, it is also interesting to study gravitational waves that originate from the oscillation of the axion cloud during the bosenova whose period is ∼ 10 M . Since this oscillation continues at least for ∼ 1000 M , the detection may be more plausible. Also, it is interesting to take account of the possibility of existence of unknown BHs in the neighborhood of the Sun, because all existing candidates for stellar mass BHs are X-ray binaries while many isolated BHs that cannot be seen by electromagnetic waves are expected to exist. \nFig. 18. The domain of integration shown in the Penrose diagram of a Kerr spacetime. \n<!-- image --> \nThe detailed studies on the gravitational wave emission in the BH-axion system during the bosenova are necessary in order to improve the above rough estimate and to obtain indication of the existence of axionlike particles or constraints on them. Another interesting observational possibility is that if the BH is immersed in the magnetic field and the axion field Φ has coupling to the electromagnetic field through the Chern-Simons interaction L aγγ = g aγγ Φ E · B , the axion cloud may radiate electromagnetic waves. Studying the feature of the electromagnetic radiation in this process and exploring the possibility of observing this phenomena are also interesting issues to be investigated.', 'Acknowledgements': 'H.Y. thanks Hajime Sotani, Hisa-aki Shinkai and Kunihito Ioka for helpful comments. This work was supported by the Grant-in-Aid for Scientific Research (A) (22244030).', 'Appendix A': "Green's function analysis \nIn this appendix, we study which modes are generated by the nonlinear effect when the amplitude of ϕ is relatively small using a perturbative approach with the Green's function method. In particular, we pay attention to whether waves of ( /lscript, m ) = (1 , -1) mode found in our simulations can be generated. We decompose the field as \nϕ ( x ) = ϕ 0 ( x ) + ∆ϕ, (A . 1) \nwhere ϕ 0 is the bound state of the ( /lscript, m ) = (1 , 1) mode of the Klein-Gordon equation: \nϕ 0 = 2Re [ e ( γ -iω 0 ) t P ( r ) S 1 1 (cos θ ) e iφ ] , (A . 2) \nand ∆ϕ denotes the deviation generated by the nonlinear effect. In the following, we consider the situation where ϕ 0 is relatively small, and take account of the order up to O ( ϕ 3 0 ) and ignore terms of O ( ϕ 4 0 ). Then, the sine-Gordon equation is approximated as \n( /square -µ 2 ) ∆ϕ = J, (A . 3) \nwith \nJ := -µ 2 6 ϕ 3 0 (A . 4) \nEquation (A . 3) is a linear equation with a source term J . In order to solve this type of equation, the Green's function method is useful. The Green's function is defined by \n( /square ' -µ 2 ) G ( x, x ' ) = δ 4 ( x, x ' ) , (A . 5) \nwhere x and x ' denote some points in the spacetime, and hereafter, prime ( ' ) indicates the coordinates x ' . Assuming the initial condition at t = 0 to be ∆ϕ = ∂ t ( ∆ϕ ) = 0, the solution of ∆ϕ is written in terms of the Green's function as \n∆ϕ = ∫ D ' d 4 x ' √ -g ( x ' ) G ( x, x ' ) J ( x ' ) . (A . 6) \nHere, for x = ( t, r ∗ , θ, φ ) and x ' = ( t ' , r ' ∗ , θ ' , φ ' ), the domain D ' is taken as the triangular region u ' ≤ u , v ' ≤ v , and t ' ≥ 0, where u and v are null coordinates, u = t + r ∗ and v = t r ∗ (see Fig. 18). \nThe Green's function can be constructed in terms of the eigenfunctions of the operator /square -µ 2 . The specific form is \n- \nG ( x, x ' ) = 1 (2 π ) 2 ∑ /lscript,m ∫ ∞ -∞ dωG ω /lscriptm ( r, r ' ) e -iω ( t -t ' )+ im ( φ -φ ' ) S m /lscript (cos θ ) ¯ S m /lscript (cos θ ' ) , \n(A . 7) \nwhere bar denotes the complex conjugate and \nG ω /lscriptm ( r, r ' ) = 1 W /lscriptmω [ θ ( r -r ' ) R + /lscriptmω ( r ) R -/lscriptmω ( r ' ) + θ ( r ' -r ) R -/lscriptmω ( r ) R + /lscriptmω ( r ' ) ] . (A . 8) \nHere, R + and R -are radial functions satisfying the boundary conditions \nR + /lscriptmω /similarequal { e ikr /r, r →∞ ; A [+] ω /lscriptm e i ˜ ωr ∗ + B [+] ω /lscriptm e -i ˜ ωr ∗ , r /similarequal r + , (A . 9) \nR -/lscriptmω /similarequal { A [ -] ω /lscriptm e -ikr /r + B [ -] ω /lscriptm e ikr /r, r →∞ ; e -i ˜ ωr ∗ , r /similarequal r + , (A . 10) \nwhere k = √ ω 2 -µ 2 and we impose Im[ k ] ≥ 0. The function R + is the solution satisfying the outgoing/decaying boundary condition at infinity, and the function R -is the solution satisfying the ingoing boundary condition at the horizon. Choosing R + and R -in this way, the solution ∆ϕ satisfies both of the ingoing boundary \ncondition at the horizon and the outgoing/decaying condition at infinity. W /lscriptmω is the Wronskian determined by \nW /lscriptmω ( R -, R + ) := ∆ -R + ∂ r R -+ R -∂ r R + ) . (A . 11) \n( \n) The value of W /lscriptmω is constant for arbitrary r , and it is calculated as \nW /lscriptmω ( R -, R + ) = 2 i ˜ ω ( r 2 + + a 2 ) A [+] ω /lscriptm = 2 ikA [ -] ω /lscriptm . (A . 12) \nSince we do not know the analytic expressions of R -and R + in the whole region r + ≤ r < ∞ , we cannot derive the exact solution. However, we can extract various properties of the solution of ∆ϕ . \nSubstituting the Green's function (A . 7) into Eq. (A . 6), we have the following formula: \n∆ϕ = ∑ /lscript,m ∫ ∞ -∞ dωe -iωt + imφ S m /lscript (cos θ ) [ R + /lscriptmω ( r ) X + /lscriptmω ( t, r ) + R -/lscriptmω ( r ) X -/lscriptmω ( t, r ) ] , (A . 13) \nwhere X + /lscriptmω and X -/lscriptmω are defined by \nX + /lscriptmω ( t, r ) = 1 W /lscriptmω ∫ r ∗ -v dr ' ∗ ∆ ' r ' 2 + a 2 ∫ v + r ' ∗ 0 dt ' e iωt ' R -/lscriptmω ( r ' ) J /lscriptmω ( t ' , r ' ) , (A . 14) \nX -/lscriptmω ( t, r ) = 1 W /lscriptmω ∫ u r ∗ dr ' ∗ ∆ ' r ' 2 + a 2 ∫ u -r ' ∗ 0 dt ' e iωt ' R + /lscriptmω ( r ' ) J /lscriptmω ( t ' , r ' ) , (A . 15) \nwith \nJ /lscriptmω = -1 (2 π ) 2 ∫ 2 π 0 dφ ∫ 1 -1 dν ( r 2 + a 2 ν 2 ) e -imφ S m /lscript ( ν ) U ' NL ( ϕ 0 ) , (A . 16) \nwhere ν := cos θ . Here, we introduce further approximation: we replace the spheroidal harmonics S m /lscript e imφ by the spherical harmonics P m /lscript e imφ . Although this approximation holds only when | a 2 k 2 | is small, this formula is assumed for all values of ω . In this approximation, J /lscriptmω becomes independent of ω , and therefore, we simply denote it by J /lscriptm . Then, the integration can be performed as \nJ /lscriptm = K m /lscript ( r ) e (3 γ -imω 0 t ) P ( r ) 3+ m 2 ¯ P ( r ) 3 -m 2 . (A . 17) \nwhere \nK ± 3 3 = µ 2 12 √ 210 π (9 r 2 + a 2 ) , (A . 18a) \nK ± 1 3 = µ 2 20 √ 14 π (3 r 2 -a 2 ) , (A . 18b) \nK ± 1 1 = 3 µ 2 140 π (7 r 2 + a 2 ) , (A . 18c) \nand K m /lscript = 0 for the other values of /lscript and m . \nNow, let us consider the point ( t, r ) in the neighborhood of the horizon, i.e., r /similarequal r + . For this point, it can be directly checked that X + /lscriptmω ( t, r ) /similarequal 0, and therefore, ∆ϕ satisfies the ingoing boundary condition at the horizon. On the other hand, X -/lscriptmω ( t, r ) becomes nonzero and can be written \nX -/lscriptmω ( t, r ) = 1 W /lscriptmω ∫ u r ∗ dr ' ∗ ∆ ' r ' 2 + a 2 e [3 γ + i ( ω -mω 0 )]( u -r ' ∗ ) -1 3 γ + i ( ω -mω 0 ) × R + /lscriptmω ( r ' ) K m /lscript ( r ' ) P ( r ' ) 3+ m 2 ¯ P ( r ' ) 3 -m 2 . (A . 19) \nSubstituting this formula into Eq. (A . 13), we get \n∆ϕ = ∑ /lscript,m e imφ S m /lscript (cos θ ) e imΩ H r ∗ 2 i ( r 2 + + a 2 ) × { e (3 γ -imω 0 ) u D /lscriptm ( u, r ∗ ) -∫ ∞ -∞ dω e -iωu ˜ ωA [+] ω /lscriptm [3 γ + i ( ω -mω 0 )] E ( ω ) /lscriptm ( u, r ∗ ) } , (A . 20) \nwith \nD /lscriptm ( u, r ∗ ) = ∫ ∞ -∞ dω 1 ˜ ωA [+] ω /lscriptm [3 γ + i ( ω -mω 0 )] × ∫ u r ∗ dr ' ∗ ∆ ' r ' 2 + a 2 e -[3 γ + i ( ω -mω 0 )] r ' ∗ R + /lscriptmω ( r ' ) K m /lscript ( r ' ) P ( r ' ) 3+ m 2 ¯ P ( r ' ) 3 -m 2 (A . 21) \nand \nE ( ω ) /lscriptm ( u, r ∗ ) = ∫ u r ∗ dr ' ∗ ∆ ' r ' 2 + a 2 R + /lscriptmω ( r ' ) K m /lscript ( r ' ) P ( r ' ) 3+ m 2 ¯ P ( r ' ) 3 -m 2 . (A . 22) \nHere, D /lscriptm ( u, r ∗ ) and E /lscriptm ( u, r ∗ ) are slowly varying functions with respect to u and r ∗ for r ∗ -M and u M . \n/lessmuch \n/greatermuch \n-Let us consider the first term of Eq. (A . 20). The dependence of the first term on t and φ is ∼ e 3 γt + im ( φ -ω 0 t ) , and this is a mode propagating to + φ direction. The integration of the second term is performed using the techniques of the complex analysis. Defining the contour C in a complex ω plane that goes along the real line from -∞ to ∞ and then clockwise along a semicircle centered at zero in the lower-half plane, the integral is rewritten as the sum of contribution from the poles and the integration along the branch cut. Since the branch cut integral typically gives a subdominant contribution, we focus attention to the poles. The singular points of the integrand appear at ω = mΩ H , ± µ , mω 0 +3 γ , and ω ( /lscriptmn ) BS satisfying A [+] ω ( /lscriptmn ) BS /lscriptm = 0 ( n = 1 , 2 , 3 , ... ). Among them, ω = mΩ H and ± µ become endpoints of the branch cut, and therefore, they are not poles. By applying the residue theorem to the poles ω = mω 0 + 3 iγ and ω ( /lscriptmn ) BS , the terms proportional to e 3 γt + im ( φ -ω 0 t ) and e i ( mφ -ω ( /lscriptmn ) BS t ) appear, respectively. Among these two, e 3 γt + im ( φ -ω 0 t ) represents waves propagating in the + φ direction. \nIn order to understand the behavior of e i ( mφ -ω ( /lscriptmn ) BS t ) , we have to evaluate ω ( /lscriptmn ) BS . ( /lscriptmn ) \nFrom Eq. (A . 9), the condition A [+] ω BS /lscriptm = 0 gives the mode satisfying the ingoing boundary condition at the horizon and the decaying/outgoing boundary condition at infinity simultaneously. This is the bound state discussed in Sec. 2. The typical mode of the bound state satisfies ω ( /lscriptmn )2 BS /similarequal µ 2 /similarequal ω 2 0 , since the gravitational binding energy is not so large. Then, typical value of ω BS is estimated as \nω ( /lscriptmn ) BS /similarequal ± ω 0 . (A . 23) \nRemember that the poles for the bound states typically appear in both right and left half complex planes. If we adopt m = 1, the behavior of e i ( mφ -ω ( /lscriptmn ) BS t ) becomes e i ( φ ∓ ω 0 t ) , and the waves of negative frequency ω ( /lscriptmn ) BS /similarequal -ω 0 propagate to the -φ direction (i.e., they are waves of the m = -1 mode).", 'References': "- 1) A. Arvanitaki, S. Dimopoulos, S. Dubovsky, N. Kaloper and J. March-Russell, Phys. Rev. D 81 (2010), 123530, arXiv:0905.4720.\n- 2) A. Arvanitaki and S. Dubovsky, Phys. Rev. D 83 (2011), 044026, arXiv:1004.3558.\n- 3) H. Kodama and H. Yoshino, arXiv:1108.1365.\n- 4) R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38 (1977), 1440.\n- 5) R. D. Peccei and H. R. Quinn, Phys. Rev. D 16 (1977), 1791.\n- 6) G. Mocanu and D. Grumiller, arXiv:1203.4681.\n- 7) S. L. Cornish, N. R. Claussen, J. L. Roberts, E. A. Cornell and C. E. 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Teor. fiz. 14 (1971), 270, JETP Lett. 14 (1971), 180.\n- 18) Y. B. Zel'dovich, Zh. Eksp. Teor. fiz. 62 (1972), 2076, Sov. Phys. JETP 35 (1972), 1085.\n- 19) W. H. Press and S. A. Teukolsky, Nature 238 (1972), 211.\n- 20) T. Damour, N. Deruelle and R. Ruffini, Lett. Nuovo Cim. 15 (1976), 257.\n- 21) S. L. Detweiler, Phys. Rev. D 22 (1980), 2323.\n- 22) T. J. M. Zouros and D. M. Eardley, Ann. of Phys. 118 (1979), 139.\n- 23) H. Furuhashi and Y. Nambu, Prog. Theor. Phys. 112 (2004), 983, gr-qc/0402037.\n- 24) M. J. Strafuss and G. Khanna, Phys. Rev. D 71 (2005), 024034, gr-qc/0412023.\n- 25) V. Cardoso and S. Yoshida, J. High Energy Phys. 0507 (2005), 009, hep-th/0502206.\n- 26) S. R. Dolan, Phys. Rev. D 76 (2007), 084001, arXiv:0705.2880.\n- 27) S. Gillessen, F. Eisenhauer, S. Trippe, T. Alexander, R. Genzel, F. Martins and T. Ott, Astrophys. J. 692 (2009), 1075, arXiv:0810.4674.\n- 28) A. M. Ghez et al. , Astrophys. J. 689 (2008), 1044, arXiv:0808.2870.\n- 29) F. Eisenhauer et al. , Astrophys. J. 597 (2003), L121, astro-ph/0306220.\n- 30) M. Pitkin, S. Reid, S. Rowan and J. Hough, Living Rev. Relativity 14 (2011), 5, arXiv:1102.3355.\n- 31) N. Shaposhnikov and L. Titarchuk, Astrophys. J. 663 (2007), 445, astro-ph/0703441. \n- 32) M. J. Reid, J. E. McClintock, R. Narayan, L. Gou, R. A. Remillard and J. A. Orosz, Astrophys. J. 742 (2011), 83, arXiv:1106.3688."}
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2017MNRAS.471.4256V
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Forming short-period Wolf-Rayet X-ray binaries and double black holes through stable mass transfer
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2017-01-01
| 36 | 0.51 | 161 |
['stars black holes', 'stars wolf rayet', 'astronomy x rays', '-', '-']
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[]
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We show that black hole high-mass X-ray binaries (HMXBs) with O- or B-type donor stars and relatively short orbital periods, of order one week to several months may survive spiral-in, to then form Wolf-Rayet (WR) X-ray binaries with orbital periods of order a day to a few days; while in systems where the compact star is a neutron star, HMXBs with these orbital periods never survive spiral-in. We therefore predict that WR X-ray binaries can only harbour black holes. The reason why black hole HMXBs with these orbital periods may survive spiral-in is: the combination of a radiative envelope of the donor star and a high mass of the compact star. In this case, when the donor begins to overflow its Roche lobe, the systems are able to spiral in slowly with stable Roche lobe overflow, as is shown by the system SS433. In this case, the transferred mass is ejected from the vicinity of the compact star (so-called isotropic re-emission mass-loss mode, or SS433-like mass-loss), leading to gradual spiral-in. If the mass ratio of donor and black hole is ≳3.5, these systems will go into common-envelope evolution and are less likely to survive. If they survive, they produce WR X-ray binaries with orbital periods of a few hours to one day. Several of the well-known WR+O binaries in our Galaxy and the Magellanic Clouds, with orbital periods in the range between a week and several months, are expected to evolve into close WR-black hole binaries, which may later produce close double black holes. The galactic formation rate of double black holes resulting from such systems is still uncertain, as it depends on several poorly known factors in this evolutionary picture. It might possibly be as high as ∼10<SUP>-5</SUP> yr<SUP>-1</SUP>.
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https://arxiv.org/pdf/1701.02355.pdf
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{'Forming short-period Wolf-Rayet X-ray binaries and double black holes through stable mass transfer': 'E.P.J. van den Heuvel 1 , 3 and S.F. Portegies Zwart 2 and S.E. de Mink 1 , 3 \n1 Astronomical Institute Anton Pannekoek, University of Amsterdam, P.O. Box 94249, 1090 GE, Amsterdam, The Netherlands 2 Leiden Observatory, Leiden University, PO Box 9513, 2300 RA, Leiden, The Netherlands 3 Kavli Institute for Theoretical Physics, UCSB Kohn Hall, Santa Barbara, CA 93106-4030, USA', 'ABSTRACT': "We show that black-hole High-Mass X-ray Binaries (HMXBs) with O- or B-type donor stars and relatively short orbital periods, of order one week to several months may survive spiral in, to then form Wolf-Rayet (WR) X-ray binaries with orbital periods of order a day to a few days; while in systems where the compact star is a neutron star, HMXBs with these orbital periods never survive spiral-in. We therefore predict that WR X-ray binaries can only harbor black holes. The reason why blackhole HMXBs with these orbital periods may survive spiral in is: the combination of a radiative envelope of the donor star, and a high mass of the compact star. In this case, when the donor begins to overflow its Roche lobe, the systems are able to spiral in slowly with stable Roche-lobe overflow, as is shown by the system SS433. In this case the transferred mass is ejected from the vicinity of the compact star (so-called 'isotropic re-emission' mass loss mode, or 'SS433-like mass loss'), leading to gradual spiral-in. If the mass ratio of donor and black hole is > ∼ 3 . 5, these systems will go into CE evolution and are less likely to survive. If they survive, they produce WR X-ray binaries with orbital periods of a few hours to one day. \nSeveral of the well-known WR+O binaries in our Galaxy and the Magellanic Clouds, with orbital periods in the range between a week and several months, are expected to evolve into close WR-Black-Hole binaries,which may later produce close double black holes. The galactic formation rate of double black holes resulting from such systems is still uncertain, as it depends on several poorly known factors in this evolutionary picture. It might possibly be as high as ∼ 10 -5 per year. \nKey words: stars: Wolf-Rayet stars, X-ray binaries, black holes, black-hole binaries", '1 INTRODUCTION': "Wolf-Rayet (WR) stars are hot and luminous evolved stars characterized by spectra with strong emission lines of He, C and/or N and O, produced by a dense high-velocity stellar wind. Their wind mass-loss rates are typically of order 10 -5 M glyph[circledot] yr -1 . Except for the most luminous WN stars, WR stars do not contain hydrogen; they are helium stars, as was first pointed out by Paczy'nski (1967, for recent reviews of WR star properties see Crowther 2007; Sander et al. 2012; McClelland & Eldridge 2016). WR X-ray binaries are composed of a helium star and a compact star, which can be a neutron star or black hole. Their existence was predicted by Tutukov & Yungelson (1973) and van den Heuvel & De Loore (1973), as the outcome of the later evolution of HighMass X-ray Binaries (HMXBs), when the evolved massive O- or B-type donor stars in these systems started to overflow their Roche lobes. (The classical definition of a HMXB \nis: an X-ray binary in which the mass-donor star is an O- or early B-type star, while in a WR X-ray binary the donor is a massive helium star; we follow here this nomenclature). Van den Heuvel and DeLoore pointed out that the outcome of this later evolution of a HMXB is expected to be a very close binary, consisting of a helium star (the helium core of the original HMXB donor star) and a compact star. They suggested that the peculiar 4.8 hour orbital period X-ray binary Cygnus X-3 is such a WR X-ray binary. This was confirmed 19 years later (van Kerkwijk et al. 1992): the companion of Cyg X-3 is a WR star of type WN5. When the 1973 prediction of the existence of WR X-ray binaries was made, it was thought that all HMXBs would produce such systems. Since the WR phase lasts some 400 000 years (Rosslowe & Crowther 2015), which is longer than the duration of the HMXB-phase (e.g.van den Heuvel 1994), one would expect that WR X-ray binaries would be more abundant in the Galaxy than the over 200 known HMXBs. However, apart \nfrom Cyg X-3, no other WR X-ray system has been found in the Galaxy, and the problem of the 'missing WR X-ray binaries' (Vanbeveren, van Rensbergen, & De Loore 1982; Lommen et al. 2005) has been with us for over 40 years. In recent years six more WR X-ray binaries have been discovered in other galaxies (Esposito et al. 2015); with the exception of one, all of these have short orbital periods, between 0.2 and 1.5 days; one system has a period of about 8 days. Two possible factors that may lead to the reduction of the predicted numbers of WR X-ray binaries in the Galaxy are: \n- (1) when the existence of WR X-ray binaries was predicted it was thought that all helium stars with masses > ∼ 3 -4 M glyph[circledot] would be observable as WR stars. However, more recent estimates of masses of WR stars, either in binary systems or from their absolute luminosities have shown that, in order to show the typical WR spectral characteristics produced by a dense high-velocity stellar wind outflow, the helium stars must have a mass > ∼ 10( ± 2) M glyph[circledot] (Crowther 2007; Sander et al. 2012). This implies that the hydrogenrich progenitor of the WR star most probably must have had a mass > ∼ 30( ± 5) M glyph[circledot] , and in any case > ∼ 20 M glyph[circledot] . \nOr: \n(2) Only very few HMXBs survive the spiral-in during the Roche-lobe overflow phase that follows the HMXB phase (we use here the expression 'spiral-in' for a drastic decrease of the orbital period by any kind of mechanism, not only by Common-Envelope (CE) evolution). \nHere we will argue that the latter factor is the key reason for the scarcity of WR X-ray binaries, and that HMXBs in which the compact star is a neutron star hardly ever survive spiral-in. (Only very wide neutron-star systems with donor masses below about 20 M glyph[circledot] may survive, see section 3). \nWe show that, on the other hand, if the compact star is a black hole, with a mass > ∼ 5 -10 M glyph[circledot] , the systems may survive spiral-in, and become close binaries consisting of a WR star and a black hole. For this reason, one expects in general that in WR X-ray binaries the compact star is a black hole. \nOur ideas about which systems may survive spiral-in and produce WR X-ray binaries were triggered by the realization that the peculiar X-ray binary SS433 has avoided going into Common-Envelope (CE) evolution, and that the donor star in this system is transferring mass to the compact star by stable Roche-lobe overflow (King & Begelman 1999; King et al. 2000). By analysing the properties of this system we realized that it is the high mass of the compact star in this system (4 . 3( ± 0 . 8) M glyph[circledot] ) in combination with the relatively low mass of its donor star (12 . 3( ± 3 . 3) M glyph[circledot] ) (Hillwig & Gies 2008), which allowed it to avoid Common-Envelope (CE) evolution and enables it to gently spiral-in without ever coalescing with its donor. As we consider SS433 to be a 'keystone' for understanding the formation of the WR X-ray binaries, we give in section 2 a brief overview of its properties and evolutionary state. The avoidance of its going into CE-evolution is - as we will argue - a consequence of the donor star having a radiative envelope (King et al. 2000), in combination with the donor star and accretor having a mass ratio less than 3.5. In section 3 we then examine for which donor masses, mass ratios and orbital periods HMXBs will, \nwhen they start Roche-lobe overflow, avoid going into CEevolution and may survive as WR X-ray binaries with short orbital periods. We also examine under which conditions they may still survive after having gone into CE-evolution. In this section some examples are given of how a number of well-known observed WR+O binaries with relatively short orbital periods are expected to evolve in the future, and are expected to produce WR X-ray Binaries and, as a final evolutionary state, close double black holes. In section 4 we attempt to estimate the birthrate of WR X-ray binaries in the Galaxy on the basis of our model, and find it to be still higher than observed and discuss possible ways to relieve this discrepancy. In section 5 we discuss the results and estimate the possible birthrate of double black holes based on our model.", '2.1 The evolutionary state and future evolution of SS433': 'The SS433 system consists of a Roche-lobe filling A47I supergiant donor star with an estimated mass of 12 . 3( ± 3 . 3) M glyph[circledot] and a luminosity of about 3800 L glyph[circledot] , plus a compact star with a mass of 4 . 3( ± 0 . 8) M glyph[circledot] , in a 13.1-day period binary (Hillwig & Gies 2008). The compact star is surrounded by an extended and luminous accretion disk, about an order of magnitude brighter than its A-supergiant companion. This disk ejects the famous precessing relativistic jets with a velocity of 0.265c, in which neutral hydrogen is ejected at a rate of some 10 -6 M glyph[circledot] yr -1 , while in a strong disk wind with a velocity of about 1500 km s -1 of order some 10 -4 M glyph[circledot] yr -1 is ejected, as is seen in the form of the stationary H α line and broad absorption lines (Fabrika 2004). The total mass loss from the disk is basically all the matter that the A-supergiant donor is transferring to the compact object by Roche-lobe overflow on its thermal timescale of ∼ 10 5 years (see also Begelman et al. 2006). \nThe observed radiative accretion luminosity of the compact star with its disk does not exceed the Eddington luminosity L Edd glyph[similarequal] 6 · 10 38 erg s -1 of the compact star (which corresponds to a real accretion rate onto the compact star of order only a few times 10 -8 M glyph[circledot] yr -1 ), although when seen along the jets the UV luminosity might be as large as perhaps 10 40 erg s -1 (Fabrika 2004), which would correspond to an accretion rate of order 10 -7 M glyph[circledot] yr -1 . This mass loss has been going on for thousands of years, as can be seen from the large radio nebula W50 that surrounds the system and has been produced by the precessing jets and the strong disk wind. Even though this mass transfer has been going on for thousands of years, the system has not entered in a Common-Envelope (CE) state. \nThe reason why SS433 has not gone into a CE phase is, as argued by King & Begelman (1999) and King et al. (2000), the fact that the A-supergiant star has a radiative envelope. If one takes away mass from a star with radiative envelope, this envelope responds by shrinking on a dynamical timescale, followed by a re-expansion on the thermal timescale of the envelope. As a result, this star can keep its \nradius close to that of its Roche lobe and will transfer matter to its companion on the thermal timescale of its envelope, without going into a CE phase. There is, however, an extra condition for keeping the Roche-lobe overflow stable, which was not mentioned in the above references, namely: the thermal timescale mass-transfer from a radiative donor envelope may itself become unstable, if the mass ratio of donor and companion star is larger than a value in the range 3 to 4 (e.g. Tout et al. 1997; Tauris et al. 2000; Hurley et al. 2000). \nFor the sake of argument we will assume here this limiting mass ratio to be 3.5. For mass ratios larger than about 3.5 the shrinking of the system due to the mass transfer goes so fast that the shrinking of the donor star cannot keep in pace with it, and the system will enter a CE phase. SS433 has indeed a mass ratio below 3.5 and therefore avoided going into a CE phase. Apart from systems with a donor with a radiative envelope and mass ratio larger than 3.5, also for systems in which the donor has a convective envelope, the formation of a Common Envelope is unavoidable. The reaction of a convective envelope to mass loss is expansion on a dynamical (pulsational) timescale, and thus the envelope becomes violently unstable, which leads to runaway mass transfer and the formation of a Common Envelope.', "2.2 Why systems like SS433 are so rare: the fate of HMXBs with a 'standard' neutron star companion": 'One may wonder what is so special about SS433 and why we do not see more SS433-like systems. We propose that the answer is: the very unusual combination for a HMXB of a rather low donor mass (presently ∼ 12 . 3 M glyph[circledot] and initially ∼ 14 to ∼ 15 M glyph[circledot] ) plus a quite massive compact star ( ∼ 4 . 3 M glyph[circledot] ). An 14 to 15 M glyph[circledot] initial donor mass and an orbital period of ∼ 13 . 1 days are typical for a Be/X-ray binary, the most common type of HMXB, containing a Bemission (Be) line star. There are some 200 Be/X-ray binaries known in our Galaxy and the Magellanic Clouds. Raguzova & Popov (2005) list 160 in our Galaxy and the two Magellanic Clouds and Reig (2011) lists 141 in our Galaxy plus the SMC alone. Since these papers appeared, Swift, INTEGRAL and other satellites have discovered several tens more, bringing the total presently known number to about 200. In all but one of the known Be/X-ray binaries, the compact stars are neutron stars which have a typical mass of about 1.4 M glyph[circledot] . Only one Be/X-ray binary is known to harbor probably a black hole companion, with a mass of 3.6 to 6.9 M glyph[circledot] (Casares et al. 2014). If the companion of a 14 M glyph[circledot] Be star is a 1.4 M glyph[circledot] neutron star, the mass ratio is 10, and the formation of a Common Envelope is unavoidable, and the two stars will merge (unless the orbital period is longer than 1 to2 years, which is the case for only a small fraction of the Be/X-ray binaries, see section 3). \nOnly if the compact star has a mass ∼ 4 M glyph[circledot] or larger, and the donor has a radiative envelope, the system will spiral in slowly and survive the SS433-like mass-transfer process. Therefore, out of the ∼ 200 Be/X-ray binaries, only the one system with an (alleged) black hole companion will in the future evolve like SS433 and survive, all the others will have merged after transferring only a small amount of mass (or, the small fraction of systems with orbital periods longer than about 1 to 2 years, will have produced a very close Helium \nstar plus neutron star system, after a short-lasting CommonEnvelope phase). So, the birthrate of SS433-like systems is at most about 0.5 per cent of the birthrate of Be/X-ray binaries. (For a model for the formation and future evolution of the possible Be/black-hole binary, see Grudzinska et al. 2015). \nThe simple reason why SS433 can stably survive this type of spiral-in process for ∼ 10 4 to perhaps ∼ 10 5 years is because of its unique combination - for a Be/X-ray binary progenitor - of a quite massive compact star and a relatively moderate-mass donor star. \nThe ∼ 4 . 3 M glyph[circledot] compact object in SS433 must be a low-mass black hole, because causality allows neutron stars to have masses not larger than about 3 M glyph[circledot] (Nauenberg & Chapline 1973; Kalogera & Baym 1996).', '3.1 Evolution of the orbit during SS433-like mass transfer': "As shown by King et al. (2000) and Begelman et al. (2006), in the case of Roche-lobe overflow from donor stars with a radiative envelope, a further condition for avoiding the formation of a Common Envelope is that the spherization radius R sp of the accreting compact object remains smaller than its Roche lobe, where R sp is given by (Shakura & Sunyaev 1973): \nR sp = -27 4 ˙ M donor ˙ M Edd R s , (1) \nwhere R s is the Schwarzschild radius of the compact object. In the case of SS433, -˙ M donor / ˙ M Edd is of order 10 4 and R s glyph[similarequal] 9 km, so R sp ∼ 6 · 10 5 km glyph[similarequal] 0 . 9 R glyph[circledot] , which is deep inside the Roche lobe of the compact star, and a CE will be avoided. In all HMXBs with orbital periods upward on one day the same will hold. So in all cases of HMXB systems with a donor with a radiative envelope and mass ratio less than about 3.5, one expects the system to go into normal Rochelobe overflow evolution similar to that of SS433. The 'SS433mode' of mass transfer is what we have in the past called 'isotropic re-emission'(e.g. Bhattacharya & van den Heuvel 1991; Soberman et al. 1997; van den Heuvel 1994; Tauris & van den Heuvel 2006; Massevitch & Yungelson 1975). \nWith the SS433-mode of mass transfer, followed by mass loss from the disk, which has the specific orbital angular momentum of the compact object, it is simple to calculate how the orbit of the system will change. In case that a fraction of the transferred matter is ejected from the compact star and its disk with the specific orbital angular momentum of this star, and a fraction (1 -β ) is accreted by this star, the orbital angular momentum loss leads to a change of the orbital radius a given by (e.g. see Tauris 1996; Soberman et al. 1997; Tauris & van den Heuvel 2006): \na/a o = q 0 +1 q +1 ( q 0 q ) 2 [ (1 -β ) q 0 +1 (1 -β ) q +1 ] -3 -2 / (1 -β ) , (2) \nwhere q is the mass ratio of donor and compact star, and \nsubscript zero indicates the initial situation at the onset of Roche-lobe overflow. \nFor the case in which in which β = 1, as is in fact the case in SS433, as the accreted amount is 10 -4 to 10 -3 times the transferred amount, this equation in the limit of β approaching unity simplifies to: \na/a 0 = ( q 0 +1 q +1 )( q 0 q ) 2 e -2( q 0 -q ) (3) \nUsing Keplers third law the corresponding equation for the change of the orbital period is: \nP/P 0 = ( q 0 +1 q +1 ) 2 ( q 0 q ) 3 e -3( q 0 -q ) . (4) \nIn the case of SS433, assuming the initial mass of the A-supergiant donor to have been ∼ 14 to 15 M glyph[circledot] , the mass of its helium core is about 3.5 M glyph[circledot] . This means that at the end of the Roche-lobe overflow phase q = 0 . 81, while at present q 0 = 2 . 86. Inserting these values into equation (4), one finds that at the end of the Roche-lobe overflow the orbital period of the system will be P glyph[similarequal] 5 . 60 days. So, SS433 will with these assumed component masses finish as a detached binary consisting of a 3.5 M glyph[circledot] helium star and a 4.3 M glyph[circledot] compact star. The entire process will take place on the thermal timescale of the envelope of the 10 M glyph[circledot] A-supergiant which is between ∼ 10 4 and ∼ 10 5 years. \nThe helium star in the resulting system may during helium shell burning go through a second mass-transfer phase and finally explode as a supernova, likely leaving a neutron star. If the system remains bound in response to the natal kick of the neutron star, a close eccentric binary will result, consisting of the present ∼ 4 . 3 M glyph[circledot] compact star plus a neutron star.", '3.2 Upper limiting orbital period for having a radiative envelope': "In order to determine the limiting orbital period for having a radiative envelope, we notice that for an effective temperature T eff > 8100 K (spectral type earlier than ∼ A7), stars have a deep radiative envelope (e.g. see Clayton 1968). If the luminosity of a donor star of a given mass M is known, its radius R can be found from the relation L = 4 πσR 2 T 4 eff , where σ is the Stefan-Boltzmann constant. If the mass M c of the compact companion of the star is known, and we set the stellar radius R equal to the radius R L1 of the Roche lobe of the donor, then the equation for the Roche lobe given by Eggleton (1983): \nR L1 = 0 . 49 a 0 . 6 + q -2 / 3 ln (1 + q 1 / 3 ) , (5) \nallows one to calculate the orbital radius a of the binary in which the donor fills its Roche lobe, as the mass ratio q = M donor /M c is known. Using Kepler's third law, one also calculate the corresponding orbital period P . \nTo calculate the maximum orbital periods up to which donor stars still have a radiative envelope, we used the luminosities of post-main-sequence evolutionary tracks, for solar metallicity, of rotating stars with masses up to 50 M glyph[circledot] given by Ekstrom et al. (2012), and the limiting effective temperature of T eff ∼ 8100 K. Post-main-sequence stars originating from stars more massive than about 50 M glyph[circledot] have very \nstrong stellar wind mass loss, or become Luminous Blue Variables, stars that experience strong eruptive mass loss episodes. Because of their strong mass loss they lose most of their hydrogen-rich envelope and always stay at effective temperatures above 8100K. Therefore these post-main sequence stars are expected to have radiative envelopes, so for them we used as maximum radius just their maximum post-main-sequence stellar radius. We made these calculations for compact companions with masses of 1.5 M glyph[circledot] , 5 M glyph[circledot] , 10 M glyph[circledot] and 15 M glyph[circledot] . Figures 1 and 2 give these upper limiting orbital periods as a function of donor mass for initial donors with masses between 9 M glyph[circledot] and 85 M glyph[circledot] . (In the mass range between 40 M glyph[circledot] and 60 M glyph[circledot] there are no tracks by Ekstrom et al. (2012) available. It is known from evolutionary calculations with similar assumed wind mass loss rates that for masses above 50 M glyph[circledot] the stars at the end of hydrogen burning have lost most of their H-rich envelopes and their radii drop rapidly as a function of mass. We have, for the sake of argument, assumed that in the mass range between 40 and 50 M glyph[circledot] the orbital periods at T eff = 8100K are constant and after that they linearly decrease towards the orbital period of the 60 M glyph[circledot] star). \nThe figures show that for donor masses up to about 50 M glyph[circledot] these upper limiting orbital periods range from about 50 to 400 days, and beyond 50 M glyph[circledot] they go down rapidly. Below these limiting curves, the donor stars in HMXBs will transfer mass to their compact companions according to the SS433-type of mass transfer, and the systems will not go into Common Envelope (CE) evolution, provided the mass ratio of donor and compact star is less than about 3.5. The regions where this SS433-like evolution will occur are indicated in the figure 2 by the blue-colored parts of the diagrams.(Notice that for calculating the curves of the limiting orbital periods, as well as the limiting donor masses for mass ratio 3.5, we used the real post-main-sequence masses of the stars, which are considerably reduced with respect to the intitial masses, due to stallar wind mass loss on the main sequence). To the right of these blue regions and below the radiative boundary periods, systems will go into Common Envelope evolution with a donor with radiative envelope. Above the radiative boundary periods they will go into Common Envelope evolution with a convective envelope (or if the period is too large they will not experience mass transfer at all). \nThe next question is: which systems with donor stars in this radiative-envelope regime, will survive as binaries after the onset of mass transfer by Roche-lobe overflow? It turns out that none of the systems with a 1.5 M glyph[circledot] compact star (neutron star) will, as was already mentioned above. Because of their mass ratios of far above the mass ratio upper limit of 3.5, they go into CE evolution and for initial donor masses below 40 M glyph[circledot] they all merge. This can be seen from the lower-limit curve for survival of CE evolution (the red curve) in figure 1. (The way this curve was calculated is explained in the next section). Although for donor masses larger than 40 M glyph[circledot] according to the figure systems might survive CE-evolution (grey hashed region), such massive donors with NS companions may not be very likely, for stellar evolution reasons. In figure 1 we have indicated also the ranges of orbital periods and masses of the bulk of the known supergiant HMXBs with neutron star companions, and of the B-emission X-ray binaries with neutron star companions. One observes that the vast majority of the \nFigure 1. Upper limiting orbital periods for having a donor star with a radiative envelope (blue line, circles, as explained in section 3.2), together with the formal lower limiting orbital periods for surviving CE evolution (red line, squares, as explained in section 3.3). This diagram shows the case of a 1.5 M glyph[circledot] neutron star companion (see figure 2 for more massive companions). In all cases considered here the mass ratio between donor and accretor is so large ( q > q limit = 3 . 5) that Roche-lobe overflow is expected to be unstable and lead to CE evolution, irrespective of whether the donor has a radiative envelope. Only systems with periods larger than the limiting period for CE-survival (above the red curve) are expected to survive spiral in and avoid coalescence. A small region of the parameter space allows for radiative donors that may survive CE inspiral (gray hashed region), but this is limited to donors with masses above 40 M glyph[circledot] . The typical orbital periods and donor masses of the observed supergiant HMXBs and the BeHMXBs are indicated. Over 95% of the NS-supergiant HMXBs do not survive SS433-like spiral in, and only the Be-HMXBs with very long orbital periods survive CE evolution and can be progenitors of binary pulsars. \n<!-- image --> \nknown supergiant HMXBs with neutron stars does not survive spiral-in. \nIt can be seen from this figure that among the known types of HMXBs only the B-emission X-ray binaries with a neutron star companion and orbital periods ranging from larger than 220 d with a 9 M glyph[circledot] donor to larger than 370 d with a 20 M glyph[circledot] donor (the red region of the diagram), will survive spiral in and can later form double neutron stars, after explosion of the helium star. This is a well-known result (e.g. see Taam 1996; Portegies Zwart & Spreeuw 1996; Dewi et al. 2005). \nWhile none of the systems in the radiative-donor regime with a 1.5 M glyph[circledot] neutron star survives the mass transfer, the systems with a 5 M glyph[circledot] , 10 M glyph[circledot] and 15 M glyph[circledot] compact companion star (black hole), in the blue-colored parts of figure 2, do survive SS433-like mass transfer and can produce heliumstar binaries with relatively short orbital periods: WR/Xray binaries. \nIt should be kept in mind that for the case with a 5 M glyph[circledot] , 10 M glyph[circledot] and 15 M glyph[circledot] compact star, the donors which we consider, overflow their Roche lobes after leaving the main sequence, which means that they evolve according to case B of close binary evolution (Kippenhahn & Weigert 1990). The lower limit for the orbital period for this case is about one week. \nFigure 2. Blue shaded region indicates systems that are expected to undergo stable mass transfer from a radiative donor and survive a 'SS433-like spiral in'. Top, middle and bottom panel show the range for a black-hole companion with a mass M c = 5, 10 and 15 M glyph[circledot] , respectively. (Compare with figure 1, where we showed the case of a M c = 1 . 5 M glyph[circledot] neutron star companion). The region is bounded by the condition that the mass ratio between the donor and the compact object is not too extreme, i.e. smaller than q limit = 3 . 5 (vertical dashed gray line). Systems in the part to the right of the blue region will go into CE evolution. The upper limiting period for mass transfer from a donor with a radiative envelope (blue line with circles, see section 3.2) and the the lowerlimiting orbital-period curves for survival of CE evolution as WR X-ray binaries are shown for two values of the CE parameters (light and dark red lines, see section 3.3). \n<!-- image --> \nAn example of a system that will survive SS433-like evolution is Cygnus X-1, which has 5.6 day orbital period, a 14 . 8 ± 1 . 0 M glyph[circledot] black hole and a 19 . 2 ± 1 . 9 M glyph[circledot] donor, which according to its luminosity must have started out as a 30 M glyph[circledot] star (Orosz et al. 2011). After SS433-like mass transfer the donor in Cygnus X-1 will leave a helium star of about 10 M glyph[circledot] , which might either leave a neutron star or a low-mass black hole, with an orbital period of about 9 days, depending on the direction and magnitude of the birth kick of the compact object. So Cyg X-1 will not terminate as a close system.", '3.3 Lower limiting orbital periods for surviving CE evolution': "Systems with orbital periods above the upper-limit line for radiative envelope in figures 1 and 2 will have convective envelopes and will, when their donor stars overflow their Roche lobes, go into CE evolution and spiral in, as described e.g. by Tutukov & Yungelson (1993), Portegies Zwart & Verbunt (1996), Lipunov et al. (1997) and later papers and recently by Belczynski et al. (2016) and Eldridge & Stanway (2016). \nThe same holds for systems with donors in the radiative region and mass ratios larger than about 3.5. We use here the formalism for the orbital change in the case of CE evolution as given by Webbink (1984) and de Kool (1990), which yields a ratio of the final and initial orbital radii a f and a i , respectively, given by: \na f /a i = M core M c /M donor M c +2 M env / ( α CE λr L ) (6) \nwhere M core is the mass of the helium core of the donor, M env ≡ M donor -M core is the mass of the hydrogen-rich envelope of the donor at the moment when Roche-lobe overflow begins, α CE is the so-called 'efficiency factor' of CE evolution which indicates the efficiency with which the release of orbital gravitational binding energy that occurs during spiral in of the compact star towards the core of the donor is converted into kinetic energy required to eject the common envelope, λ is a parameter that depends on the stellar mass density distribution and r L is the ratio of the Rochelobe radius R L and the orbital radius a i at the onset of CE evolution. \nThe value of r L is typically of order 0.5. There are many factors that influence the precise value of the product α CE λ , such as energy sources like recombination energy and accretion energy release (e.g. see the discussion in Tauris & van den Heuvel 2006; Portegies Zwart 2013, and in Ivanova et al. 2013; Kruckow et al. 2016). We assume here two values of the common envelope efficiency: α CE = 0 . 5 and α CE = 0 . 9 (e.g. Taam 1996), r L = 0 . 5, and - while we know this an oversimplification - for our calculations we assume λ = 0 . 5. \nTo calculate the outcome of CE evolution for stars in the mass range 10 to 85 M glyph[circledot] we used the rotating evolutionary models for solar metallicity of Ekstrom et al. (2012). These models were calculated with stellar wind mass loss, such that at the end of core-hydrogen burning, when the stars leave the main sequence, their masses are lower than their initial values, and the mass of the helium core is known. Using for M donor then these reduced post-main-sequence masses, and the corresponding M core values, one can calculate, for given values of M c and of the combination α CE λr L , what the values of the ratio a f /a i will be. \nGiven the mass M core of the helium core one knows the radius of this helium star (for these we used the interpolation formula from Onno Pols given in Tauris & van den Heuvel 2006). Together with the mass M c of the compact star this radius gives the minimum final orbital radius a f for systems that survive CE-evolution, as the helium star is not allowed to be larger than its Roche-lobe radius in the final system. If a f would be smaller than this minimum value, the system does not survive and merges. \nUsing the above-given values of α CE λ and r L , and the values of a f /a i calculated according to the above given recipe for each initial donor mass and M c , one can then calculate the lower-limits to the initial orbital radii a i required for the systems to survive CE evolution. These a i values give one also the minimum initial orbital periods for surviving CE evolution, as a function of initial donor mass and M c . \nIn figure 1 the lower limiting period curve for survival of CE evolution for α CE = 0 . 5 is indicated for systems with a neutron star companion with M c = 1 . 5 M glyph[circledot] . For the case of M c = 5M glyph[circledot] , 10 M glyph[circledot] and 15 M glyph[circledot] , the calculated lower limiting period curves for CE evolution with the above-given values of α CE λ = 0 . 25 and 0.45 are given. (The latter value was derived from orbits of post-CE binaries by Portegies Zwart & Verbunt 1996). Systems above these curves in the parts of the panels in figure 2, to the right of the blue-colored regions are expected to survive CE-evolution as WR/X-ray binaries with short orbital periods.", '3.4 Examples of the future evolution of some well-known WR+O spectroscopic binaries: formation of close WR X-ray binaries and of double black holes': 'Table 1 shows as an example how we expect seven wellknown observed massive WR+O spectroscopic binaries with well-determined masses and orbital periods to evolve in the future. The masses and orbital periods of these systems were taken from the catalogue of van der Hucht (2001), to which we refer for the original references for these systems. We have calculated the future evolution of these systems semi-empirically, as follows. We adopted the Conti-scenario (Conti 1976) for the evolution of WR stars. According to this model, WR stars begin as WN stars and then evolve with strong wind mass loss into WC/WO stars, that finally undergo core collapse (Crowther 2007). (For an alternative point of view, see Sander et al. (2012); McClelland & Eldridge (2016)). \nTo calculate the evolution of the observed WR+O binaries in Table 1, we made the following assumptions: \n- · On the basis of evolution calculations, such as by the Geneva group, we assumed, adopting the Conti scenario, the WR stars to spend 70% of their helium-burning lifetime as a WN star and 30% as a WC star.\n- · We assumed an observed WN star to be half-way its WNlifetime, that is: at 35% of its helium star lifetime. Similarly, we assumed observed WC stars to be at 85% of their helium star lifetime, so still to have 15% of this lifetime to go.\n- · We assumed their wind mass loss rates over their entire lifetime to correspond to the ones stars of their presently observed WR type. For these rates we used wind mass loss \nrates 1.4 times lower than assumed by Schaller et al. (1992), for the following reasons: Schaller et al. found for their adopted wind mass loss rates for solar metallicity that even stars up to 120 M glyph[circledot] finished with a mass of only 8 M glyph[circledot] . It is evident that this cannot be correct, since the Population I X-ray binary Cygnus X-1 is a 14.8 M glyph[circledot] black hole(Orosz et al. (2011),Ziolkowski (2012)). Assuming some 90 per cent of the final WC star (basically a CO core) to become the black hole ((e.g. Heger 2012)), the final mass of the WR progenitor of Cyg X-1 must have been 16.4 M glyph[circledot] . The formation of a black hole with the mass of Cyg X-1 is possible only if the real wind mass loss rates at solar metallicity are about 1.4 times lower than the ones used by Schaller et al. In that case the progenitor of Cyg X-1 was a star of about 80 M glyph[circledot] with solar metallicity. \n- · Also for the O-stars we used wind loss rates of 1.4 times lower than the ones used by Schaller et al. (1992).\n- · As total lifetime of the WR stars (massive helium stars) we used 400 000 yrs (Rosslowe & Crowther 2015). \nThe results of these calculations are listed in Table 1. Columns 2, 3 and 4 list the observed parameters of the systems. Column 5 lists the masses of the components after the WR star has terminated its evolution and has become a black hole, and subsequently the companion has evolved for 3 million years to leave the main sequence and start Rochelobe overflow. We calculated the masses in column 5 by taking into account the wind mass-loss of both stars, and by assuming that at the end of the life of the Wolf-Rayet star as a WC-type star, 90% of the mass of the WC star disappeared into the black hole (cf. Heger 2012; Sukhbold et al. 2016), meaning that just only the gravitational binding energy of the helium star is lost. As there is no mass lost, we have assumed that the black holes did not receive a natal kick. (We realize that our implicit assumption in these calculations that stars with initial masses larger than about 25 M glyph[circledot] always leave black holes, may not be fully realistic, since Sukhbold et al. (2016) (and also Ertl et al. and Muller in 2015, in papers referred to in this paper) find from their calculations that, quite arbitrarily, sometimes even very massive stars may still leave a neutron star as a remnant. Since the precise reasons why this happens is not fully understood, we have chosen not to take this into account here). \nA possible indication that WR stars tend to leave black holes is that, so far, no WR stars have been detected as progenitors of Type Ib,c supernovae (Smartt 2015). \nWe calculated how the orbits changed due to the wind mass loss and the core-collapse gravitational-mass loss, and assumed that after this the orbits tidally synchronized again. We then assumed the O-type companions of the BHs still to live another 3 million years as core-hydrogen burning stars, losing mass by stellar wind in this period. For the top six systems in the table, we subsequently calculated the spiral in of the resulting HMXB consisting of the O-star and the BH, using the SS433-type of spiral in. This resulted into the WR+BH systems with masses and orbital periods listed in columns (6) and (7), respectively. For the 7th system, the mass ratio of O-star and black hole is larger than 3.5 and we assumed this system to pass through CE-evolution. \nOne notices that several of the top-six WR X-ray binaries have orbital periods of order 1 to 2 days, and that their compact stars are black holes. After taking account of the \nwind mass loss of the WR stars in the WR X-ray systems during their further evolution and, again assuming that 90 per cent of the mass of the final WC star disappears into the black hole, one obtains masses and orbital periods of the double black hole binaries resulting from these systems listed in columns 8 and 9, respectively. (We assumed that the wind mass loss did not increase the orbital period of the WR X-ray binary, as the gravitational torque excerted by the black hole on the outflowing thick wind will cause loss of orbital angular momentum, which is expected to compensate the increase of orbital period due to the mass loss; if the latter torques would not occur, the final orbital periods will be about 1.8 times larger). \nOne observes from this table that some well-known WR+O spectroscopic binaries with orbital periods of order one week can produce close WR+BH X-ray binaries, and subsequently produce close double black hole binaries. \nSince the first three systems in Table 1 have a mass ratio of donor and black hole before spiral-in mass transfer very close to the 3.5 limit, it is not completely sure whether the mass transfer from the radiative envelope will not become unstable. To mark this uncertainty, we have a put a colon after the post-spiral-in orbital periods in Table 1. For the 4th to 6th systems in Table 1 the mass ratio before spiralin is below 3.5 and the transfer is expected to certainly be stable. The 4th and 5th systems, however, do not leave close WR X-ray binaries, and also not close double black holes. They leave WR X-ray binaries resembling the one observed system with an orbital period of about 8 days, mentioned in the Introduction. (In the VIIth catalogue of WR stars (van der Hucht 2001) there are 3 more systems (WR30, 113 and 141) that will evolve similarly and leave wide WR X-ray binaries and wide double BHs). \nIn the case of the first three systems in Table 1: if their mass transfer becomes unstable, a Common Envelope will form and the outcome of the evolution must be calculated using equation 6. It turns out that with the above assumed value of α CE λ = 0 . 25 -0 . 45, none of these three systems survives the CE-evolution, as in each case the Roche-lobe radius of the post-CE helium star is smaller than the radius of this star. \nOne sees that the orbital periods of WR XRBs produced by stable Roche lobe overflow are considerably larger than the one produced by CE evolution in the system of WR11.', '4.1 Predicted galactic numbers of WR X-ray binaries, compared to observations: still too few observed systems': "Rosslowe & Crowther (2015) estimate the number of WR stars in the Galaxy to be 1200( ± 200). WR X-ray binaries have in our model come from WR plus O-type binaries similar to the systems in Table 1. In these systems the WR star and the O-star have comparable luminosities, meaning that they are double-lined spectroscopic binaries (abbreviated as SB2), which started out as O-type binaries with components with roughly similar masses. van der Hucht (2001) found 13 of the 61 WR stars within 3 kpc from the sun to be SB2s, so \nTable 1. Anticipated future evolution of seven well-observed WR+O binaries. The orbital periods, spectral types and masses of the components were taken from the catalogue by van der Hucht (2001). The WR stars and O-stars in these systems are sufficiently massive to leave black hole remnants. The ways how the masses of these black holes (bold-print numbers) were calculated are explained in the text. It is assumed for the top six systems that the resulting O+BH systems will spiral-in following the SS433-recipe. In the three first systems, the mass ratio of O-donor and BH is very close to 3.5, such that it is not fully certain that the mass transfer will be stable (see text). To indicate this uncertainty, a colon was placed after the short orbital periods of the resulting WR-XRBs. Assuming the SS433-type mass transfer indeed to be stable, the masses of the double black holes, with their orbital periods, indicated in the last columns, will result. The very last column lists the GW-merger times of these systems (no times are given if the merger time is longer than the age of the Universe). In case the SS433-like mass transfer would not be stable, the WR/X-ray systems indicated with colons will go into CE evolution; in that case none of these systems will survive. The bottom line gives the anticipated evolution WR 11(Gamma-2 Velorum), which will in the second phase of mass transfer go into Common Envelope evolution and produce a very short-period WR/X binary and double black hole \n| | | Observed | Observed | HMXB at RLOF | WR X-ray binary | WR X-ray binary | Double black hole | Double black hole | Double black hole |\n|-----------|---------------|------------|-------------------------------|-------------------------------|-------------------------------|-------------------|-------------------------------|---------------------|---------------------|\n| Name | Spectrum | P [day] | Masses [ M glyph[circledot] ] | Masses [ M glyph[circledot] ] | Masses [ M glyph[circledot] ] | P [day] | Masses [ M glyph[circledot] ] | P [day] | t ( merge ) [Gyr] |\n| WR 127 | WN3 + O9.5V | 9.555 | 17 + 36 | 9.6 + 33 | 9.6 + 13.6 | 1.54: | 9.6 + 7.0 | 1.71 | 6.91 |\n| WR 21 | WN5 + O4-6 | 8.255 | 19 + 37 | 10.2 + 34 | 10.2 + 14.1 | 1.64: | 10.2 + 7.9 | 1.77 | 6.50 |\n| WR 62a | WN5 + O5.5 | 9.145 | 22 + 40.5 | 10.8 + 37 | 10.8 + 16.1 | 1.45: | 10.8 + 8.8 | 1.61 | 4.40 |\n| WR 42 | WC7 + O7V | 7.886 | 14 + 23 | 10.4 + 22 | 10.4 + 8.0 | 13.75 | 10.4 + 4.5 | 14.71 | - |\n| WR 47 | WN6 + O5V | 6.2393 | 51 + 60 | 18.1 + 46 | 18.1 + 25.8 | 7.96 | 18.1 + 10.4 | 8.59 | - |\n| WR 79 | WC7 + O5-8 | 8.89 | 11 + 29 | 9.0 + 27.4 | 9.0 + 10.1 | 2.44 | 9.0 + 5.4 | 2.64 | - |\n| WR 11(CE) | WC8 + O7.5III | 78.53 | 9.0 + 30 | 7.8 + 28.5 | 7.8 + 10.5 | 0.90 | 7.8 + 5.6 | 0.98 | 2.24 | \nabout 20 per cent, which would imply 240 SB2 WR-systems in the Galaxy. We particularly selected the 7 systems in Table 1, plus the 3 other systems mentioned in the foregoing section, for their masses and orbital periods such that they could potentially produce WR X-ray binaries. They therefore are not a representative sample of the SB2 WR+O binaries in van der Hucht's VIIth catalogue of WR stars, in which there are in total 31 SB2 systems. Assuming these ten systems to be representative for one third of all WR SB2 systems in the Galaxy, then 80 such systems may be expected to evolve similarly to the ten systems mentioned in the last section, and produce similar WR/X-ray binaries. As in WR+O binaries and in WR X-ray binaries the WR stars are expected to live equally long, one would then expect, in a steady state of star formation, that there are also 80 WR X-ray binaries in the Galaxy. However, we know only one such system in the Galaxy: Cygnus X-3, which has an orbital period of 0.2 days. This system, at some 8-10 kpc distance, is in absolute terms one of the brightest X-ray sources in the Galaxy; although only our side of the Galaxy has been well-surveyed for X-ray sources, it is unlikely that there are more than 3 similar systems in the entire Galaxy. The large absolute X-ray luminosity of Cyg X-3 may be due to its short orbital period, which implies that the compact star is moving in the low-velocity part of its stellar wind. As the accretion rate scales with the minus 4th power of the wind velocity times the minus 2nd power of the orbital radius, in systems with orbital periods of one day or more, like in Table 1, the compact stars are in the high-velocity part of the wind, and are at some 3 times larger orbital radius, such that their X-ray luminosities will be at least some three orders of magnitude lower than that of Cyg X-3, i.e. below ∼ 10 35 ergs/s. Since, however, we know none of such systems among the well-surveyed part of our Galaxy, up to some 6 kpc distance, which is one-fifth of the area of the galactic disk, the number in the entire galaxy is unlikely to be larger \nthan about 5. So, the total number of observable WR X-ray binaries in the Galaxy is unlikely to be larger than 8, which is at most only 10 per cent of the above expected number. In the following section we discuss possible ways out of this conundrum.", '4.2 Conceivable explanations for the large difference between expected and observed numbers of galactic black-hole WR X-ray binaries': "To calculate the parameters of the WR X-ray binaries in Table 1 a number of assumptions about the evolution of the WR+O binary predecessors were made, as described in section 3.4. We now critically examine the effects of changing a number of the underlying assumptions. \n- · (i) The assumption that for q < 3 . 5 the systems evolve with stable Roche-lobe overflow. If the real q-limit would be lower, e.g. q < 3, the six uppermost O + BH systems in Table 1 will all go into CE-evolution and will not survive. Also the three systems WR30, 113 and 141 mentioned in section 3.4 will not survive. Only the system WR 11 with its large orbital period will, going through CE-evolution, survive. This could reduce the expected number of WR Xray binaries by a factor of 10. It would then imply that there are some 8 WR X systems in the Galaxy with orbital periods like that of the descendent of WR 11: about one day.\n- · (ii) The assumption that the BH formed from a WC star has 90 per cent of the final mass of this star. Nelemans et al. (1999) argue that only 0.65 of the final mass of a helium star ends up in the BH. We repeated the calculations of the evolution of the systems in Table 1 with this assumption. The first three systems in Table 1, as well as WR 79 will now go into CE evolution and coalesce, while WR 42 and 47 go through 'isotropic re-emission' and terminate with \nperiods of about 4 days; WR 30, 113 and 141 evolve in a similar way and produce systems with still longer periods. The only system that in this case still survives as a shortperiod WR X-ray binary is WR11, which terminates with an orbital period of 2.13 hours if α CE = 0 . 50 and 4.98 hours if α CE = 0 . 90. \n- · (iii) Our assumption that the core collapses of the massive stars always leave black holes. As mentioned in section 3.4, the results of the core-collapse calculations of massive stars by Sukhbold et al. (2016) show that, while sometimes even stars in the mass range 15 to 20 M glyph[circledot] may leave a black hole, also in very massive stars the core collapse may in a number of cases still lead to the formation of a neutron star. If this is indeed the case, and would, for example, occur in half of the cases of the core collapses involved in the systems in Table 1, it would reduce the resulting number of WR X-ray binaries produced by these systems by a factor 2: if half of the systems produced a neutron star in the first core collapse in the system, these systems would merge on Roche-lobe overflow. And the produced number of double black holes would be reduced by a factor 4, as also in the second supernova half of the collapses would produce a neutron star.", '5 DISCUSSION - POSSIBLE GALACTIC FORMATION RATE OF DOUBLE BLACK HOLES ORIGINATING FROM WR X-RAY BINARIES': 'From the discussion in the last section, it appears that lowering the fraction of the final WR-star mass that goes to form the black hole does not solve the problem of the lack of Galactic WR X-ray binaries. On the other hand, we found that a more promising way to explain the low observed incidence of WR X-ray binaries is: a lower q-limit than q = 3 . 5 for stable mass transfer by Roche-lobe overflow. The latter limiting value was derived for binary systems with components less massive than 12 solar masses. It is therefore crucial to make more detailed binary evolution calculations for more massive black-hole HMXBs, in order to more precisely estimate this q-limit for massive systems (see also Pavlovskii et al. (2017)). \nAlso, it is important to calculate this limit for different values of the metallicity, as this type of evolution is expected to have been important also in the early universe. Further, we saw that if a sizeable fraction of the core-collapses of massive stars would still leave a neutron star instead of a black hole, this could further reduce the produced number of WR X-ray binaries by a factor two. This would then result in only 4 such system sin the Galaxy at any time. \nFinally, assuming that the solution of the low incidence of WR X-ray binaries in the galaxy is indeed a lower q-limit than 3.5, and combining this with half of the massive star core collapses producing neutron stars, one can make an estimate of the galactic formation and merger rate of close double black holes resulting from WR X-ray binaries. As mentioned above, the produced double black hole systems then resemble the systems resulting from WR11, which have orbital periods of around one day. Such systems will merge on a timescale of order ∼ 10 9 yr by the loss of gravitational radiation. As we expect half of the estimated 4 close WR X- \ny binaries to form a close double black hole in 400 000 yr (the WR lifetime), the galactic merger rate of close double black holes produced by this process is glyph[similarequal] 0 . 5 · 10 -5 per yr. The rate based on the system of Cygnus X-3 alone - assuming its WR star to be massive enough to leave a black hole- is of similar order, as its WR lifetime is again of order 400 000 years (see also Esposito et al. (2015) and references therein). Although the uncertainties on these estimated rates are expected to be at least an order of magnitude (each way), they nevertheless are interesting. The rates are an order of magnitude lower than the estimated galactic double-neutron-star merger rate (Kalogera et al. 2004).', 'ACKNOWLEDGMENTS': 'We thank Thomas Tauris for very useful comments. This work was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915. The CHARM is project P7/08 in the phase VII IA. It further was supported by the Netherlands Research Council NWO (grants #643.200.503, #639.073.803 and #614.061.608) by the Netherlands Research School for Astronomy (NOVA). 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2017MNRAS.467..524B
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Stellar-mass black holes in young massive and open stellar clusters and their role in gravitational-wave generation
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2017-01-01
| 33 | 0.49 | 161 |
['clusters open', 'clusters globular', 'stars kinematics and dynamics', 'stars black holes', 'methods numerical', 'gravitational waves', '-', '-']
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[]
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Stellar-remnant black holes (BH) in dense stellar clusters have always drawn attention due to their potential in a number of phenomena, especially the dynamical formation of binary black holes (BBH), which potentially coalesce via gravitational-wave radiation. This study presents a preliminary set of evolutionary models of compact stellar clusters with initial masses ranging over 1.0 × 10<SUP>4</SUP>-5.0 × 10<SUP>4</SUP> M<SUB>⊙</SUB>, and half-mass radius of 2 or 1 pc, which is typical for young massive and starburst clusters. They have metallicities between 0.05 Z<SUB>⊙</SUB> and Z<SUB>⊙</SUB>. Including contemporary schemes for stellar wind and remnant formation, such model clusters are evolved, for the first time, using the state-of-the-art direct N-body evolution program nbody7, until their dissolution or at least for 10 Gyr. That way, a self-regulatory behaviour in the effects of dynamical interactions among the BHs is demonstrated. In contrast to earlier studies, the BBH coalescences obtained in these models show a prominence in triple-mediated coalescences while being bound to the clusters, compared to those occurring among the BBHs that are dynamically ejected from the clusters. A broader mass spectrum of the BHs and lower escape velocities of the clusters explored here might cause this difference, which is yet to be fully understood. Among the BBH coalescences obtained here, there are ones that resemble the detected GW151226, LVT151012 and GW150914 events and also ones that are even more massive. A preliminary estimate suggests few 10-100 s of BBH coalescences per year, originating due to dynamics in stellar clusters that can be detected by the Laser Interferometer Gravitational-Wave Observatory (LIGO) at its design sensitivity.
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https://arxiv.org/pdf/1611.09357.pdf
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{'No Header': 'MNRAS 000 , 1-17 (2016)', 'Sambaran Banerjee 1 ; 2 ?': '- 1 Argelander-Institut fur Astronomie (AIfA), Auf dem Hugel 71, D-53121, Bonn, Germany\n- 2 Helmholtz-Instituts fur Strahlen- und Kernphysik (HISKP), Nussallee 14-16, D-53115 Bonn, Germany \nOctober 27, 2021', 'ABSTRACT': 'Stellar-remnant black holes (BH) in dense stellar clusters have always drawn attention due to their potential in a number of phenomena, especially the dynamical formation of binary black holes (BBH), which potentially coalesce via gravitational-wave (GW) radiation. This study presents a preliminary set of evolutionary models of compact stellar clusters with initial masses ranging over 1 : 0 GLYPH<2> 10 4 M GLYPH<12> GLYPH<0> 5 : 0 GLYPH<2> 10 4 M GLYPH<12> , and halfmass radius of 2 or 1 pc, that is typical for young massive and starburst clusters. They have metallicities between 0 : 05 Z GLYPH<12> GLYPH<0> Z GLYPH<12> . Including contemporary schemes for stellar wind and remnant formation, such model clusters are evolved, for the first time, using the state-of-the-art direct N-body evolution program NBODY7 , until their dissolution or at least for 10 Gyr. That way, a self-regulatory behaviour in the effects of dynamical interactions among the BHs is demonstrated. In contrast to earlier studies, the BBH coalescences obtained in these models show a prominence in triple-mediated coalescences while being bound to the clusters, compared to those occurring among the BBHs that are dynamically ejected from the clusters. A broader mass spectrum of the BHs and lower escape velocities of the clusters explored here might cause this difference, which is yet to be fully understood. Among the BBH coalescences obtained here, there are ones that resemble the detected GW151226, LVT151012, and GW150914 events and also ones which are even more massive. A preliminary estimate suggests few 10s-100s of BBH coalescences per year, originating due to dynamics in stellar clusters, that can be detected by the LIGO at its design sensitivity. \nKey words: open clusters and associations: general - globular clusters: general stars: kinematics and dynamics - stars: black holes - methods: numerical - gravitational waves', '1 INTRODUCTION': "The study of dynamical interactions of black holes (hereafter BH) in dense stellar systems is now nearly 30 years old. A key point of interest in this topic has always been the possibility of the generation of gravitational waves (hereafter GW) from dynamically-formed binary black holes (hereafter BBH). The interest in the topic has naturally got rejuvenated right after the first-time detection of GW from two BBH inspiral events by the Advanced LIGO detector, namely, the GW150914 (Abbott et al. 2016a,c), the GW151226 (Abbott et al. 2016b) and the marginal detection event LVT151012 (Abbott et al. 2016a). Generally speaking, for BBHs composed of stellar-remnant BHs such as the above detected ones, which are typically of GLYPH<24> 10 M GLYPH<12> GLYPH<0> GLYPH<24> 100 M GLYPH<12> , the frequency of the emitted GW dur- \ning their inspiral phase falls within the LIGO's detection band (Abbott et al. 2016a), placing such BBHs among the most promising sources for the LIGO. The bottom-line scenario of formation of dynamical (stellar-mass) BBHs in star clusters is straightforward: if a certain number of BHs receive sufficiently low natal kicks, during their formation via core-collapse (supernovae) of massive stars, that they remain bound to the gravitational potential of the cluster, they would segregate to the innermost regions of the cluster. Depending on the number of BHs retained in the cluster and their masses (which is GLYPH<24> 10 to GLYPH<24> 100 times the average stellar mass depending on their progenitor stars' wind; see below), the system of bound BHs might undergo a runaway mass segregation (mass-stratification or Spitzer instability; Spitzer 1987) to form a central and highly dense subsystem of BHs, where they would continuously interact. Otherwise, the dynamical friction of the dense stellar background would as well tend to keep the BHs centrally concentrated (as these \nBHs are much more massive than the normal stars, they would segregate towards the cluster's center simply due to dynamical friction, rather than being driven by two-body relaxation). \nSuch a dense BH-core serves as a constant resource for dynamically forming BBH, mainly via the three-body mechanism (Spitzer 1987; Heggie & Hut 2003). The subsequent frequent and super-elastic (Spitzer 1987) dynamical encounters of a BBH with other single BHs and BBHs serve as a recipe for (a) injecting kinetic energy (K.E.) into the BH sub-cluster and as well into the whole star cluster causing the latter to expand (Mackey et al. 2007, 2008), (b) ejecting single and binary BHs from the cluster depleting the BH population and (c) forming triple-BH systems within the BH-core. The ejected BBHs are typically dynamically tightened (hardened; Heggie 1975) and also eccentric, an adequate combination of which (Peters 1964) would lead to the inspiral via GW radiation and the coalescence of a BBH within the Hubble time. The triple-BHs that are bound to the clusters, on the other hand, would typically undergo Kozai-Lidov oscillations (Kozai 1962) leading to large eccentricity boost and hence GW inspiral and coalescence of the inner binary, provided this happens before the triple gets perturbed by an intruder. In other words, although a star cluster continues to eject single and binary BHs and form BH-triples until its BH reservoir is (nearly) depleted, the occurrence of a dynamical BBH inspiral is a probabilistic phenomenon. \nBeginning from 1990s, aspects of the above mechanism is studied at various levels of detail. Following preliminary but pioneering studies such as Kulkarni et al. (1993); Sigurdsson & Hernquist (1993); Portegies Zwart & McMillan (2000), more recent direct N-body ( e.g. , Banerjee et al. 2010; Aarseth 2012; Sippel & Hurley 2013; Wang et al. 2016; Hurley et al. 2016) and Monte-Carlo ( e.g. , Downing et al. 2010, 2011; Rodriguez et al. 2015; Morscher et al. 2015; Rodriguez et al. 2016a,b; Chatterjee et al. 2016a,b; Askar et al. 2016) calculations of model stellar clusters study the dynamicallydriven depletion of BHs, the resulting feedback onto the cluster and the BBH inspirals self consistently and in much more detail. Adopting somewhat simpler but realistic conditions, detailed semi-analytic studies of these aspects have also been performed recently (Breen & Heggie 2013a,b; Arca-Sedda 2016). Also, such a semi-analytic modelling, in the context of GLYPH<24> 10 7 M GLYPH<12> nuclear stellar clusters and with focus on stellarmass BBH inspirals produced by them, has been recently performed by Antonini & Rasio (2016). By nature, MonteCarlo calculations are restricted to massive clusters, initially GLYPH<24> 10 5 M GLYPH<12> GLYPH<0> 10 6 M GLYPH<12> , that are more representatives (or progenitors) of classical globular clusters (hereafter GC), although young clusters of up to GLYPH<24> 10 7 M GLYPH<12> are observed in nearby starburst galaxies ( e.g. , in the Antennae Larsen 2009; Johnson et al. 2015) and the impact on BBH inspiral rate due to age-spread among massive clusters has been explored very recently (Chatterjee et al. 2016b). The overall conclusion of such studies is that the dynamical BBH inspiral rate is GLYPH<25> 5 GLYPH<0> 10 yr GLYPH<0> 1 Gpc GLYPH<0> 3 , which would contribute to the detection of several 10s of BBH inspriral per year with the Advanced LIGO, given its proposed full sensitivity (Banerjee et al. 2010; Rodriguez et al. 2016a). These studies also infer that a GW150914-like BBH coalescence is intrinsically rare to \nbe produced from a star cluster compared to the other two events (Rodriguez et al. 2016b; Chatterjee et al. 2016b). \nAnother important corollary is that the BH population in massive clusters, although decays monotonically with time due to dynamical interactions, it hardly gets completely depleted even in a Hubble time, so that a substantial population of BHs would retain even in old globular clusters. In a nutshell, this is due to the energy generation via dynamical encounters in the BH-core that results in significant expansion of both the parent cluster and the BH-core itself, suppressing the BH-BH interaction rate. This 'self-regulation' causes the BH population to decline but exponentially; see below and also Morscher et al. (2015). This is consistent with the recent identification of stellar-mass BH candidates in the Galactic globular clusters M22 and 47 Tuc (Strader et al. 2012; Miller-Jones et al. 2015). \nThe present work focuses on the other end of the problem, namely, the role of intermediate-mass and open stellar clusters in generating BBH inspirals. In this paper, by intermediate-mass clusters, we will imply those within the mass range 10 4 M GLYPH<12> GLYPH<0> 10 5 M GLYPH<12> , the young ( . 100 Myr) versions of which are popularly called 'young massive clusters' (YMCs) and 'starburst clusters' (when . 4 Myr old; such young clusters with > 10 5 M GLYPH<12> are often denoted as 'super star clusters'). Clusters of < 10 4 M GLYPH<12> will be called open clusters; as such, there is no strict boundaries defined in the literature between these different 'types' of clusters. A reliable and self-consistent evolutionary modelling of clusters of such masses with realistic ingredients is possible only by direct N-body integration. Monte-Carlo calculations already become unsatisfactory for the corresponding typical particle numbers ( N . 1 : 7 GLYPH<2> 10 5 ), due to larger statistical fluctuations and various timescales becoming closer to the cluster's overall two-body relaxation time. \nAfter a recess of about a decade since a few notable initial studies on this topic (Kulkarni et al. 1993; Sigurdsson & Hernquist 1993; Portegies Zwart & McMillan 2000), Banerjee et al. (2010) have, for the first time, investigated the dynamical behaviour of a population of GLYPH<25> 10 M GLYPH<12> BHs in intermediate-mass compact (initial half-mass radius 1-2 pc) stellar clusters, where the dynamics of the BHs have been treated self-consistently using direct N-body calculations. As such, the above work is the first of its kind where the dynamics of the BHs in stellar clusters and the consequent impact on the cluster and the production of BBH coalescences have been studied explicitly. However, soon after, both observations of BH candidates in sub-solar metallicity regions in the Local Group and theoretical studies of the mass distribution of stellar remnants based on revised wind prescriptions for high-mass stars have suggested that stellar-remnant BHs can, in fact, be much more massive than the contemporarily accepted GLYPH<25> 10 M GLYPH<12> BHs, especially at low abundances (Belczynski et al. 2010). Of course, the existence of such stellar BHs is now confirmed after the GW150914 event which involves GLYPH<25> 30 M GLYPH<12> BHs (also by the LVT151012 event). Given the current knowledge on stellar winds in different evolutionary stages of massive stars as a function of metallicity, determining the remnant BH mass, it is likely that such GLYPH<25> 30 M GLYPH<12> BHs are formed in sub-solar metallicity regions (Belczynski et al. 2010; Spera et al. 2015). \nGiven that nearly all of the N-body studies of long-term BH dynamics in stellar clusters so far assumes GLYPH<25> 10 M GLYPH<12> BHs \nthat is representative of solar-like metallicity (Banerjee et al. 2010; Aarseth 2012; Sippel & Hurley 2013; Wang et al. 2016), it is now undoubtedly worthwhile to revisit the problem with revised BH masses. Especially, at lower metallicities, not only the stellar-remnant BHs would be substantially more massive, but also they would have a wider mass spectrum. This, in turn, would influence the nature of the dynamical interactions in a star cluster's 'BH-engine' and hence the latter's impact on the cluster and on the BBH coalescences from it, as we will see in the following sections. Since direct N-body integration (without force softening) tracks all sorts of dynamical encounters, and particularly the close ones, with full consistency and without any assumptions, it is the ideal approach for this study. In the present work, it is for the first time that model parsec-scale intermediate-mass and open star clusters of varying metallicities are evolved using direct N-body integration from their zero-age until the Hubble time (at least for 10 Gyr). The NBODY7 code, with modified stellar mass loss and remnant formation prescriptions adopting those of Belczynski et al. (2010) are used for this purpose; that way the BH-dynamics is studied as consistently and realistically as possible now. Given the long computing times, the present set of models cover the relevant parameter space only preliminarily and thus provide limited statistics. However, they would still comprise the state-of-the-art set of N-body calculations already suggesting intriguing and new conclusions, as we shall see in the following sections. Recently, lower-mass clusters of initially GLYPH<24> 10 3 M GLYPH<12> are evolved, using the direct N-body method containing similar model ingredients as in here (Sec. 2.2), for shorter evolutionary times of GLYPH<24> 100 Myr, to study the dynamical interactions involving BHs over young ages and low metallicities (Mapelli et al. 2013; Ziosi et al. 2014). It is also worth recalling the 'Dragon Simulations' (Wang et al. 2016) in this context, where relatively extended, GLYPH<25> 3 GLYPH<0> 8 pc-sized, much more massive clusters of N GLYPH<25> 10 6 stars (they can also be taken as representatives of galactic nuclear clusters) are evolved for nearly a Hubble time and the properties of the BH population are studied, through direct N-body calculations using the NBODY6++ program (Wang et al. 2015). \nThis paper is organized as follows: in Sec. 2, the model calculations are described (in Sec. 2.3) following a brief introduction to the NBODY7 code (Sec. 2.1) and its modifications for the present purpose (Sec. 2.2). In Sec. 3, the results of these calculations are discussed, with focus on the general dynamical behaviour of BHs and its impact on the parent cluster (Sec. 3.1), and as well on the dynamical production of BBHs and their coalescences via GW radiation (Sec. 3.2). These coalescences are compared with those detected by the LIGO (Sec. 3.2.1) and a preliminary estimate of the LIGO detection rate of dynamically-generated BBH mergers is suggested (Sec. 3.2.2). The inferences are summarized in Sec. 4.", '2.1 The NBODY7 N-body evolution program': "The NBODY7 code is an immediate descendant of the widelyused NBODY6 direct N-body evolution code (Aarseth 2003; Nitadori & Aarseth 2012). NBODY7 utilizes the Algorithmic Regularization Chain (ARC) of Mikkola & Tanikawa (1999) \n<!-- image --> \nFigure 1. Top: Remnant mass as a function of zero-age main sequence (ZAMS) mass for the Belczynski et al. (2008, 2010) stellar wind and remnant formation schemes adopted in this study, which is obtained from the variant of BSE that is integrated with NBODY7 (thin lines). They agree reasonably well with those from B10 (thick lines) for the respective metallicities ( Z = Z GLYPH<12> ; Z GLYPH<12> = 4 ; Z GLYPH<12> = 100 ; see B10), which have been obtained using their StarTrack program. Bottom: Mass distributions of BHs that remain bound to the Mcl (0) = 5 GLYPH<2> 10 4 M GLYPH<12> computed model clusters (Table 1), right after their formation, for Z = Z GLYPH<12> ; Z GLYPH<12> = 4 ; Z GLYPH<12> = 20 . \n<!-- image --> \ninstead of the the classic Chain Regularization in NBODY6 (Mikkola & Aarseth 1993; Aarseth 2003). This enables a more thorough and reliable treatment of multiple systems that continue to form dynamically in any dense environment and influence the dynamics, especially of those involving one or more massive objects like BHs. NBODY7 otherwise follows similar numerical strategies like NBODY6 , namely, a fourth-order Hermite integrator is used to accurately advance the trajectories of each star subjected to the resultant force from rest of the bodies. To ease the consequent / N 3 dependence in computing time, a neighbour-based scheme is utilized for computing the force contributions (Nitadori & Aarseth 2012) at the shortest time intervals (the 'irregular' force/steps). At longer time intervals (the 'regular' force/steps), all members in the system are included for the force evaluation. The inexpensive (but numerous) irregular forces are computed using parallel processing in regular single-node workstation CPUs, while the much more expensive regular force calculations are done on CUDA 1 -enabled \nFigure 2. Examples of computed evolution, for models with Mcl (0) GLYPH<25> 3 GLYPH<2> 10 4 M GLYPH<12> and rh (0) GLYPH<25> 2 pc having metallicities Z = 0 : 05 Z GLYPH<12> (left column) and Z = Z GLYPH<12> (right column); see Table 1. Top panels: time evolutions of the 1%, 2%, 5%, 10%, 20%, 30%, 40%, 50%, 62.5%, 75% and 90% Lagrange radii ( Rf denotes the ( f GLYPH<2> 100) % radius), Middle panels: evolutions of the numbers of BHs and NSs bound to the cluster, Bottom panels: evolutions of the total bound cluster mass (blue line) and the total masses of the BHs and NSs bound the cluster (black and green lines respectively). \n<!-- image --> \nhigh-performance GPUs 2 . The diverging gravitational forces during close passages and in binaries are dealt with twobody or KS regularization (Aarseth 2003) and higher-order multiples are treated with the ARC. \nLike its predecessors, NBODY7 utilizes the semi-analytic stellar evolution code BSE (Hurley et al. 2000, 2002) 3 to \nevolve each star and form their remnants. The stellar parameters or each star (including any tidal effect if it is a member of a binary or a multiple) are updated simultaneously with its trajectory integration. That way the effects \nevolving single stars is called SSE , which are available as separate standalone packages. The same wind mass loss and remnant formation recipes (see Sec. 2.2) can be applied to both. In the N-body code, the evolutionary subroutines of SSE and BSE are integrated into the various NBODY7 subroutines and we will simply denote the stellar-evolutionary part of NBODY7 as BSE . \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 3. The time evolution of the half-mass radius (50% Lagrange radius R 0 : 5 ), rh (top row), and the number of BHs bound to the cluster, N BH ; bound (bottom row), as a function of metallicity, Z, for computed model clusters with Mcl (0) GLYPH<25> 3 GLYPH<2> 10 4 M GLYPH<12> (left column) and Mcl (0) GLYPH<25> 5 GLYPH<2> 10 4 M GLYPH<12> (right column), each having rh (0) GLYPH<25> 2 pc (see Table 1). The 5% Lagrange radius, R 0 : 05 , is also plotted for the Mcl (0) GLYPH<25> 3 GLYPH<2> 10 4 M GLYPH<12> , Z = Z GLYPH<12> model (top left panel) to indicate that it has just arrived at its (second) core-collapsed phase. \n<!-- image --> \nof stellar evolutionary mass loss, via winds and supernovae, are naturally incorporated in a calculation. \nAn important aspect of NBODY7 is its general relativistic (GR) treatment, when an NS or/and a BH is a member of binary or a multiplet. The relativistic treatment, following the Post-Newtonian (PN) approach, is included in the ARC procedure (Mikkola & Merritt 2008). In principle, PN-1.5 (GR periastron precession), PN-2.5 (orbital shrinking due to GW radiation) and PN-3.5 (spin-orbit coupling) order terms are included in the ARC; see Aarseth (2012) for the key elements of the implementation in NBODY7 and Brem et al. (2013) for an alternative approach (in NBODY6++ ). This allows for onthe-fly GR orbital modifications and coalescences of relativistic subsystems (typically a binary or a triple containing one or more BH/NS) that are bound to the system. The latest implementation generally shows reasonable energy check (typical relative energy change GLYPH<24> 10 GLYPH<0> 6 ) even during extreme relativistic events such as a BBH coalescence within a triple. In the present computations (see Sec. 2.3), however, the PN terms up to order 2.5 are applied, as activating the spin terms would make these computations, which typically contain one or more relativistic subsystems nearly all of the time, much more slower 4 . The spin terms would have modi- \nfied the times of the BBH coalescences occurring within the cluster (see Secs. 1 & 3.2) to some extent, however, this is not critical in the present context due to the statistical nature of the dynamically-induced BBH coalescences. \nIn reality, a BBH would typically receive a substantial GW merger kick during its inspiral phase ( GLYPH<24> 100 GLYPH<0> 1000 km s GLYPH<0> 1 ; Campanelli et al. 2007; Hughes 2009), due to the presence of the BHs' spins. This would cause the newlyformed merged BH to escape from the cluster almost inevitably, and it would hardly have a chance to participate in dynamical encounters further. This situation is mimicked by applying a velocity kick onto the merged BH, immediately after a coalescence happens within the cluster (Sec. 3.2). To avoid large energy error, the applied kick is kept only marginally above the escape speed; GLYPH<25> 5 times the central RMS speed. This is still enough the eject the merged BH out of the cluster, in GLYPH<24> 10 dynamical times; in reality a BBH coalescence product would typically escape at much higher speed 5 . \ntic subsystems (a wide but eccentric BBH or a hierarchical BHtriple). This procedure is still under development and shows unstable behaviour when a large number of BHs are present (Sverre Aarseth, private communication), as in the present models, which is why the perturbed-PN has also not been applied here. \n5 In test calculations, it is found that even if the merged product", '6 S. Banerjee': 'Figure 4. The evolution of rh (top row) and N BH ; bound (bottom row), as a function of Mcl (0) , for computed models with Z = 0 : 05 Z GLYPH<12> (left column) and Z = Z GLYPH<12> (right column), each having rh (0) GLYPH<25> 2 pc (see Table 1). \n<!-- image --> \n<!-- image --> \nFigure 5. Left: The time evolution of the fraction of BHs bound to the cluster as a function of Z , for the computed models with Mcl (0) GLYPH<25> 5 GLYPH<2> 10 4 M GLYPH<12> , rh (0) GLYPH<25> 2 pc. Right: The time evolution of the fraction of BHs bound to the cluster as a function of Mcl (0) , for computed models with Z = 0 : 05 Z GLYPH<12> and rh (0) GLYPH<25> 2 pc (see Table 1). \n<!-- image -->', '2.2 The new wind prescription: remnant masses and natal kicks': "The masses of the BHs, and how many of them receive low natal kicks so that they can retain in the parent cluster at \nis retained, it does not necessarily take part in further relativistic coalescences. \ntheir birth, determines the effectiveness of the BH-engine (Sec. 1). For a given isolated progenitor star, the remnant BHmass is determined by the entire history of the wind mass loss until the pre-core-collapse stage and also on the material 'fallback' onto the remnant during the supernova. The BH's natal kick will be diminished by the amount of fallback; in particular there will be no natal kick if the fallback is 100%, i.e. , the entire pre-core-collapse star implodes into a BH (a 'failed supernova'; there might still be a small kick due to \nTable 1. Summary of the model stellar clusters in this work, whose evolutions are computed using NBODY7 . The columns from left to right respectively denote: (a) initial mass, Mcl (0) , of the model cluster, (b) initial half-mass radius, rh (0) , (c) metallicity, Z , (d) number of (triple-mediated) binary black hole (BBH) coalescences, N mrg ; in , that occurred within the clusters, (e) number of BBH coalescences (in BBHs ejected from the clusters), N mrg ; out , that occurred outside the clusters within the Hubble time. For the BBHs that have undergone coalescence, the masses of the corresponding binary members are indicated in parentheses in the columns (d) and (e). \n| Mcl (0)/ M GLYPH<12> | rh (0)/pc | Z = Z GLYPH<12> | N mrg ; in | N mrg ; out |\n|---------------------------------------|-------------|-------------------|----------------------------------------------------------------------------------------------------------------------------------|-------------------------------------------------------------------------------------------|\n| 5 : 0 GLYPH<2> 10 4 | 2.0 | 0.05 | 1 ( 24 : 3 M GLYPH<12> + 17 : 7 M GLYPH<12> ) | 1 ( 26 : 0 M GLYPH<12> + 42 : 8 M GLYPH<12> ) |\n| 5 : 0 GLYPH<2> 10 4 | 2.0 | 0.25 | 1 ( 34 : 5 M GLYPH<12> + 22 : 7 M GLYPH<12> ) | 0 |\n| 5 : 0 GLYPH<2> 10 4 | 2.0 | 1.00 | 3 ( 9 : 0 M GLYPH<12> + 7 : 5 M GLYPH<12> ) ( 10 : 6 M GLYPH<12> + 9 : 4 M GLYPH<12> ) ( 9 : 1 M GLYPH<12> + 9 : 0 M GLYPH<12> ) | 0 |\n| 3 : 0 GLYPH<2> 10 4 | 2.0 | 0.05 | 1 ( 38 : 1 M GLYPH<12> + 25 : 9 M GLYPH<12> ) | 2 ( 25 : 7 M GLYPH<12> + 13 : 8 M GLYPH<12> ) ( 23 : 6 M GLYPH<12> + 22 : 3 M GLYPH<12> ) |\n| 3 : 0 GLYPH<2> 10 4 | 2.0 | 0.25 | 0 | 2 ( 35 : 2 M GLYPH<12> + 20 : 3 M GLYPH<12> ) ( 15 : 7 M GLYPH<12> + 12 : 2 M GLYPH<12> ) |\n| 3 : 0 GLYPH<2> 10 4 | 2.0 | 1.00 | 1 ( 10 : 6 M GLYPH<12> + 9 : 0 M GLYPH<12> ) | 0 |\n| 1 : 5 GLYPH<2> 10 4 | 2.0 | 0.05 | 1 ( 49 : 4 M GLYPH<12> + 30 : 9 M GLYPH<12> ) | 0 |\n| 1 : 5 GLYPH<2> 10 4 | 1.0 | 0.25 | 0 | 0 |\n| 1 : 0 GLYPH<2> 10 4 | 2.0 | 0.05 | 0 | 0 |\n| 1 : 0 GLYPH<2> 10 1 : 0 GLYPH<2> 10 4 | 1.0 1.0 | 0.05 0.25 | 1 ( 43 : 6 M GLYPH<12> + 34 : 5 M GLYPH<12> ) 0 | 0 0 |\n| 0 : 7 GLYPH<2> 10 4 | 1.0 | 0.05 | 0 | 0 | \nthe escape of neutrinos). If the progenitor is initially in a close binary, its mass loss (or gain) and hence the BH mass can additionally be influenced by any mass transfer or by tidal heating effect from its companion. Unfortunately, to date, massive-stellar winds, the mechanisms of core-collapse supernovae and material fallback are still poorly understood or constrained. \nIn this work, we will adopt the semi-analytic remnant formation and wind prescriptions of Belczynski et al. (2008, 2010). These rather widely used prescriptions are based on empirically-determined wind mass loss formulae of Vink et al. (2001) for O/B-stars, metallicity-dependent winds (including suppression due to clumping) of Vink & de Koter (2005) for Wolf-Rayet (naked helium) stars and metallicityindependent (suppressed) wind for Luminous Blue Variable stars; see the formulae (6)-(9) and their explanations in Belczynski et al. (2010) (hereafter B10). After the wind mass loss until the formation of the pre-supernova core, the remnant (NS or BH) mass is determined based on the CO and FeNi core mass and the amount of fallback onto it as in Belczynski et al. (2008); see their Eqns. (1) & (2). \nSuch partly empirical and partly physically-motivated wind and remnant formation model combinations constitute perhaps the only known way to plausibly obtain GLYPH<25> 30 M GLYPH<12> BHs as observed in the GW150914 event, at GLYPH<25> Z GLYPH<12> = 4 metallicity that is plausible for the Local Universe (redshift z < 0 : 2 ) where this event must have occurred (Abbott et al. 2016a,c). Overall, the B10 winds are much weaker at any metallicity than the standard Hurley et al. (2000) (hereafter H2K) winds that is adopted by default in the BSE (see Sec. 2.1) routine, the latter yielding up to . 25 M GLYPH<12> BHs at metallicity as low as Z GLYPH<12> = 100 , that is unlikely to occur in the local uni- \nverse. That way the B10 wind prescription more plausibly yields the mass range of BHs inferred from the three LIGO events, than the H2K prescription (for the same Belczynski et al. 2008 remnant-formation prescription), which is why the former is preferred in this study. \nThe birth kicks of the remnants are assigned based on their type, mass and the amount of material fallback, as derived from the above prescriptions. The NSs produced from core-collapse supernovae are given large kicks of GLYPH<25> 265 km s GLYPH<0> 1 , as inferred from observations of radio pulsars in the Galactic field (Hansen & Phinney 1997; Hobbs et al. 2005). The BHs formed without or partial fallback are then assigned diminished kicks based on their final masses, that are scaled from the above NS kick assuming linear momentum conservation. For pre-supernova CO core mass of M CO = 7 : 6 M GLYPH<12> , the fallback is taken to become complete based on the studies by Fryer (1999); Fryer & Kalogera (2001) 6 ; the entire (pre-supernova) star is assumed to collapse directly into a BH for M CO GLYPH<21> 7 : 6 M GLYPH<12> when zero natal kick is assigned. NSs are also allowed to form via Electron Capture Supernovae (ECS; Podsiadlowski et al. 2004) as in Belczynski et al. (2008), which are of GLYPH<25> 1 : 26 M GLYPH<12> and are as well assigned zero natal kick. For the model clusters considered in this study (with masses M cl (0) GLYPH<20> 5 GLYPH<2> 10 4 M GLYPH<12> ; see Table 1), only the direct collapse BHs and ECS NSs will retain in the cluster at their birth. All core-collapse NSs and almost all other BHs will escape right after their formation, due to their high natal kicks.", '8 S. Banerjee': "The above 'new wind' and kick recipes have already been implemented in the StarTrack (Belczynski et al. 2002, 2008) semi-analytic stellar-evolutionary code and are now implemented in the BSE , that is integrated with the NBODY7 (in the current version of NBODY7 , different elements of the above wind prescriptions are available as options alternative to the default H2K recipe). As pointed out above, there are uncertainties in nearly all physical factors determining the remnant masses and their natal kicks. Therefore the presently adopted schemes can at most be taken as a physically plausible implementation of the present empirical and theoretical knowledge of stellar winds and remnant formation, which is also widely utilized ( e.g. , in Morscher et al. 2015; Chatterjee et al. 2016a). Note that a similar wind and remnant formation schemes are adopted by Spera et al. (2015) but in a different stellar evolution and population synthesis code, namely, the SEVN ; in this study, however, we will limit ourselves to the BSE as it is adapted to the NBODY7 . \nFig. 1 (top panel) shows the remnant mass as a function of zero-age main sequence (ZAMS) mass at different metallicities, Z , as obtained by the NBODY7 -adapted BSE for the presently assumed stellar wind and remnant formation recipes (thin lines). They agree reasonably well with those of B10 for the same values of Z (thick lines), which have been obtained using StarTrack . Fig. 1 (bottom panel) shows the mass distributions of the BHs that remain bound to the M cl (0) = 5 GLYPH<2> 10 4 M GLYPH<12> model clusters computed here (see Table 1) after their formation, i.e. , receive zero or low natal kicks (these mass distributions are obtained shortly after the last BH formed at GLYPH<25> 10 Myr, when most of the bound BHs are still segregating towrads the cluster's center but the unbound ones have already escaped through the tidal radius, and when the dynamical ejections of the BHs are yet to start; see below). As expected from the ZAMS mass-BH mass relations, lower Z would result in a wider BH mass spectrum. In all cases, only the BHs from the first mass bin are depleted due to natal kicks, the rest receive zero kicks; hence, the lower the Z is, the higher is the initial BH retention, in the present scheme.", '2.3 Model computations': "Table 1 lists the model computations for this study, which are done using NBODY7 . All the computed models are initially Plummer clusters with masses M cl (0) GLYPH<20> 5 GLYPH<2> 10 4 M GLYPH<12> and half-mass radius r h (0) = 2 pc; r h (0) = 1 pc are also used for a few lower mass models. Such masses and sizes are typical for Galactic and Local Group YMCs (Banerjee & Kroupa 2016; Portegies Zwart et al. 2010; Ryon et al. 2015); considering the masses of GCs, they represent intermediatemass and open stellar clusters (see Sec. 1). From the point of view of the study of BH dynamics and dynamical BBH coalescence, such mass range is relatively unexplored (see Sec. 1 and references therein), which is alone a good reason to study them here. However, reaching higher initial masses would still have been interesting (see Sec. 3.2), and not doing so here is mainly due to the computational costs involved (most models are initiated with r h (0) = 2 pc instead of 1 pc for the same reason). The initial mass function (IMF) of all the models are taken to be the canonical one, satisfying the observed maximum stellar mass-cluster mass relation (Wei- \ndner & Kroupa 2004; Kroupa et al. 2013). The metallicities of the models are varied between 0 : 05 Z GLYPH<12> GLYPH<20> Z GLYPH<20> Z GLYPH<12> ; GLYPH<25> 0 : 05 Z GLYPH<12> being the lowest metallicity observed in the Local Universe (S'anchez Almeida et al. 2015; Rubio et al. 2015). \nAll models are evolved for 10.0-13.7 Gyr or until they dissolve completely, beginning from their zero age. A solar neighbourhood-like, static external tidal field is applied for each model. Note that this external field is just a representative in this study, whose job is simply to remove any gravitationally unbound object from the cluster's membership, and its use is of course a simplification. In reality, a cluster would have gone through large changes in its environment over such long evolutionary times and hence in the tidal field it is subjected to (see, e.g. , Renaud et al. 2015); even our immediate cosmic neighborhood offers widely different environments to newly born clusters, e.g. , compare the external fields on to the YMCs of the Milky Way and of the Magellanic Clouds. However, the growth of the host galaxy is neighter fatal to the cluster nor alters the cluster's evolution drastically, as long as it is adiabatic (Renaud & Gieles 2015). Over most of their evolutionary time, the computed models underfill their tidal radii. \nAnother simplification is the assumption of a monolithic, gas-free structure of the model clusters right from their zero age, i.e. , neglecting their assembly phase and the effect of gas expulsion (Longmore et al. 2014; Banerjee & Kroupa 2015a). Depending on the duration of the assembly and gas dispersal phase, which, in turn, would govern the mass segregation of the massive stars and/or the stellar remnants, the dynamics of the BHs may or may not be affected significantly by the assembly process. The current assumption of a monolithic structure throughout is justified if the clusters undergo 'prompt assembly' (Banerjee & Kroupa 2015b). \nFinally, all the initial models contain only single stars. This is also a simplification done for the sake of computational ease. An appropriate initial binary fraction for O/Bstars would be 50%-70%, to be consistent with what is observed in starburst clusters (Sana & Evans 2011; Sana et al. 2013), which would be tedious to compute over such long evolutionary times. Recent Monte-Carlo calculations (using 5-10% O/B-stellar binary fraction; Chatterjee et al. 2016a) infer that the properties of dynamically-produced BBHs are nearly independent of the primordial binary content. This is because nearly all the BH progenitor binaries widen significantly due to stellar wind and supernova mass loss, to either get disrupted directly or get ionized by dynamical interactions with the surrounding stars, so that the majority of the BHs become single anyway. Hence, the lack of primordial binaries in our computed models would not pose a serious limitation as long as the dynamics of the BHs are concerned.", '3.1 The impact of stellar-mass black holes on star cluster evolution': "As outlined in Sec. 1, the primary impact of stellar-mass BH retention in a cluster just after their formation is to inject energy into the dense stellar environment due to the dynamical encounters in the BH-core (and also into the BH-core \nFigure 6. The time evolution of the half-mass radii of the whole cluster (blue line), the bound BHs (black line) and NSs (green line) for the computed model with Mcl (0) GLYPH<25> 5 GLYPH<2> 10 4 M GLYPH<12> , rh (0) GLYPH<25> 2 pc, Z = 0 : 05 Z GLYPH<12> . \n<!-- image --> \nFig. 3 compares the time evolution of the half-mass ra- \n<!-- image --> \nFigure 7. The same as in Fig. 6 but with Z = Z GLYPH<12> , for Mcl (0) GLYPH<25> 5 GLYPH<2> 10 4 M GLYPH<12> (left) and Mcl (0) GLYPH<25> 3 GLYPH<2> 10 4 M GLYPH<12> (right). rh (0) GLYPH<25> 2 pc for each. \n<!-- image --> \nitself). Generally, for a given mass and compactness of the parent cluster, the energy injection will be more efficient and consequently the expansion of the cluster will be larger with increasing number of post-birth retained BHs and as well with increasing mean BH mass. Both of these factors would generally boost the K.E. generated in dynamical interactions in the BH-core and the energy deposition onto the stellar environment via dynamical friction (see Sec. 1; Mackey et al. 2007, 2008). The strong and frequent dynamical encounters in the dense BH sub-cluster continue to eject BHs, mainly as single BHs and BBHs. \nFig. 2 demonstrates the evolution of M cl (0) GLYPH<25> 3 GLYPH<2> 10 4 M GLYPH<12> , r h (0) GLYPH<25> 2 pc models with Z = 0 : 05 Z GLYPH<12> and Z GLYPH<12> . As expected, the lowZ cluster expands by much larger extent than its solarZ counterpart, as seen in their Lagrange-radii plots (Fig. 2, top row), due to the larger retention of the BHs at birth (both in number and total mass; Fig. 2, middle and bottom rows) in the former case. The larger rate of expansion and the \ncorrespondingly larger rate of loss of stars across the tidal radius causes the lowZ cluster to dissolve in GLYPH<25> 11 Gyr, while the solarZ model would survive until the Hubble time. The lowZ model also continues to retain BHs in larger number and total mass as their dynamical ejections continue (Fig. 2, middle and bottom rows); see below for more on this point. The larger BH content, in turn, continues to expand the Z = 0 : 05 Z GLYPH<12> cluster until its point of dissolution, while the Z = Z GLYPH<12> cluster begins to re-collapse after GLYPH<25> 4 Gyr as its BHengine becomes weak enough due to only a few GLYPH<25> 10 M GLYPH<12> BHs remaining after this time; in fact the latter cluster undergoes (second) core collapse at GLYPH<25> 11 Gyr ( c.f. Fig. 2, top right panel). On the other hand, the much lower-mass ECS Ns (of GLYPH<25> 1 : 3 M GLYPH<12> each; see Sec. 2.2), that retain at birth and are yet to mass-segregate (see below), suffer a larger loss rate in the lowZ case (Fig. 2, middle and bottom rows) due to the correspondingly faster star removal. \nFigure 8. The distributions of orbital period (top), eccentricity (center) and mass ratio (bottom) of the dynamically-ejected binary black holes (BBHs) from all the computed models in Table 1. \n<!-- image --> \ndius (50% Lagrange radius), r h , and the number, N BH ; bound , of the BHs bound ( i.e. , those remaining within the cluster's tidal radius; except for a few which are on their way of escaping from the cluster, they are also gravitationally bound to the system) to clusters of fixed M cl (0) and r h (0) , but of varying Z . As can be expected based on the previous example, with decreasing Z , the by-birth retained N BH ; bound and mean BH mass increase, causing the cluster to expand at a higher rate (Fig. 3, top row). Also, with decreasing Z , an overall larger N BH ; bound is maintained; in fact the N BH ; bound ( t ) s for Z = 0 : 05 Z GLYPH<12> and Z = 0 : 25 Z GLYPH<12> closely follow each other for most of the time while that for Z = Z GLYPH<12> falls below significantly (Fig. 3, bottom row). This hints that at least for Z . 0 : 25 Z GLYPH<12> , \nthe number of BHs retaining over time within a cluster of a given initial mass and size is nearly independent of Z . However, for any given observing epoch t > 0 , a lowerZ (tidally under-full) cluster would generally be more expanded. \nIt would be also useful to compare r h ( t ) and N BH ; bound ( t ) , keeping Z and r h (0) fixed but varying M cl (0) , as done in Fig. 4. Except for M cl (0) = 5 GLYPH<2> 10 4 M GLYPH<12> and Z = 0 : 05 Z GLYPH<12> , which model continues to retain most BHs, the initial expansion of all clusters stall at some point when their BH-engines become weak enough due to the loss of BHs, and they begin and continue to contract until their dissolution. Because of generally smaller number (number and mean mass) of the BH content at all times, the collapse begins earlier with decreasing M cl (0) (increasing Z ), for fixed Z ( M cl (0) ) and r h (0) (also seen in Fig. 3). Note that the cluster dissolution times obtained here are exact only for the static and simplified external field assumed here (Sec. 2.3), and these times would be different under more realistic conditions. \nArelevant question here is that how the efficiency of the dynamical BH ejection depends on M cl (0) and Z (for fixed r h (0) )? This can be best described by plotting the time evolution of the fraction of the retaining BHs, as shown in Fig. 5. It can be seen that for a given M cl (0) and r h (0) , clusters with Z . 0 : 25 Z GLYPH<12> possess nearly the same efficiency of BH ejection at all evolutionary times whereas that with Z = Z GLYPH<12> is always more efficient in ejecting BHs (Fig. 5, left panel) 7 . This is counter-intuitive at the first glance, as lower Z clusters form more massive BHs and initially retain more of them in number, so that the BH-engine should become more efficient with decreasing Z . This is indeed the case: as demonstrated above, although a lowerZ cluster loses less mass over its young age from stellar winds and supernovae, it ultimately expands more and dissolves faster due to the work of its BHs. However, the larger extent of expansion and star loss causes the normal stellar density to drop faster, reducing the efficiency of dynamical friction throughout the cluster, and as well diluting the gradient of the central potential well offered by them. This causes expansion and dilution of the BH-core itself, suppressing the frequency and K.E. extraction in dynamical encounters within the BH-core, and hence its efficiency. In other words, the BH-dynamical heating of a cluster is self-regulatory, suppressing the relative BH ejection rate for more numerous and/or more massive BH retention at birth. This is as well manifested when Z and r h (0) are kept fixed and M cl (0) is varied (Fig. 5, right panel). Note that this self-regulatory behaviour is essentially a manifestation of the H'enon (1975) principle, according to which the central (dynamical) energy generation of a (post-corecollapse) cluster is controlled by the energy demands of the bulk of the cluster. This principle is as well applicable to the \n7 In these curves, the decline of N BH ; bound = N BH ; bound ; max after t & 100 Myr is due to the ejections of BHs via dynamical encounters. However, the drop at t GLYPH<24> 10 Myr (distinctly visible due to the use of logarithmic t -axis) is due to the formation of BHs with relatively low natal kicks but which are still unbound w.r.t. the cluster and are removed shortly when they cross the tidal radius (see above). These are among the least massive BHs that receive the scaled natal kicks (Sec. 2.2) and which are more numerous for Z = Z GLYPH<12> where the BHs have narrower mass range (Fig. 1). These BHs also cause the excursions of the BH half-mass radius evolution at t GLYPH<24> 10 Myr in Fig. 7 (see below). \nFigure 9. Filled squares: the mass ratios of the ejected BBHs against their respective cluster-evolutionary time of ejection, t ej , from the model clusters. Open circles: those ejected BBHs with GW coalescence time, GLYPH<28> mrg GLYPH<20> 13 : 7 Gyr (Hubble time) at their t ej s. Open triangles: the actual times of GW coalescence, t mrg GLYPH<17> t ej + GLYPH<28> mrg , of the above BBHs. Filled triangles: (triple-mediated) BBH coalescences within the clusters at their corresponding coalescence times, t mrg . All the symbols are colour-coded according to the BBH's total mass, M tot (vertical colour bar). Results from all the computed models in Table 1 are compiled here. \n<!-- image --> \nenergy generation due to dynamical encounters within the BH-core inside a stellar cluster, as recently done in the semianalytic study by Breen & Heggie (2013a). Nevertheless, the initial retention of more numerous and/or more massive BH sub-population, ultimately injects more energy (for a given compactness) into the parent cluster (and onto itself), increasingly prolongigng the BH retention , as seen from the above examples. For M cl (0) GLYPH<25> 3 GLYPH<2> 10 4 M GLYPH<12> and 5 GLYPH<2> 10 4 M GLYPH<12> , the Z = Z GLYPH<12> models are nearly deprived of their BHs by 10 Gyr whereas the lowerZ models continue to hold a significant number of BHs until the Hubble time or until the cluster's dissolution; c.f. Fig. 3. This result, therefore, suggests the presence of a significant population of BHs in present-day GCs as often envisaged in the literature (e.g., in Strader et al. 2012; Taylor et al. 2015; Bovill et al. 2016; Sollima et al. 2016; Peuten et al. 2016), which are typically of subsolar metallicity, irrespective of the strength of the external field under which they orbit . \nIn the present context, it would be useful to also consider the ECS NSs retaining in the clusters, since in GLYPH<24> Gyr old stellar systems they would be the second most massive objects. Fig. 6 compares the time evolution of the half-mass radius, r h ( t ) , of the overall cluster with that of the halfmass radius, r h ; BH ( t ) , of the BH subpopulation and of the half-mass radius, r h ; NS ( t ) , of the NS subpopulation, for the M cl (0) GLYPH<25> 5 GLYPH<2> 10 4 M GLYPH<12> , r h (0) GLYPH<25> 2 pc, Z = 0 : 05 Z GLYPH<12> model. After the initial central segregation of the BHs in GLYPH<25> 100 Myr, the BHcore could maintain a fluctuating but overall constant size for a few 100 Myr and then it expands with the cluster (see discussion above) until the Hubble time, r h ; BH ( t ) being always a factor of few smaller than r h ( t ) . This implies a continued centrally-concentrated (or BH-core) state of the BH sub-population within the cluster, despite significant dilu- \ntion of both with time. On the other hand, r h ; NS ( t ) > r h ( t ) always, especially for t & 50 Myr, which implies lack of mass segregation of NSs throughout the evolution. The initial large overshoot of r h ; NS ( t ) is due to the formation of NSs via core-collapse supernovae that gives them large kicks (see Sec. 2.2) and they are more likely found much outside the cluster but within the tidal radius ( GLYPH<24> 100 pc at that time), while escaping the system. The low-kick ECS NSs are born during t GLYPH<25> 56 : 1 GLYPH<0> 65 : 3 Myr ( t GLYPH<25> 40 : 2 GLYPH<0> 46 : 0 Myr) for Z = 0 : 05 Z GLYPH<12> ( Z = Z GLYPH<12> ) whose progenitors, of GLYPH<25> 6 : 8 GLYPH<0> 6 : 3 M GLYPH<12> ( GLYPH<25> 8 : 2 GLYPH<0> 7 : 7 M GLYPH<12> ) ZAMS, could not yet fully segregate and, in fact, the large mass loss until the remnant formation (the ECS NSs are only of GLYPH<25> 1 : 3 M GLYPH<12> ) is likely to reverse any partial segregation achieved by their progenitors. The central K.E. injection, due to the initial rapid segregation of the retaining BHs and the continued energetic dynamical encounters among them thereafter (see above), quenches the 'natural' two-body relaxation driven mass-segregation within the cluster, including that of the NSs. In the example of Fig. 6, the mass segregation is frozen until the Hubble time. For models of higher Z , where the BH-engine is weaker and the BHs are depleted relatively fast (see above), the mass segregation can revive; this is demonstrated in the Z = Z GLYPH<12> examples in Fig. 7 where the r h ; NS ( t ) falls below r h ( t ) after a few Gyr of evolution. Such stalling of mass segregation, due to the work of BHs, has also been demonstrated in the recent studies by Alessandrini et al. (2016); Peuten et al. (2016).", '3.2 Dynamically-formed binary black holes: gravitational-wave coalescence events': "Given the current interest in BBH coalescence events following their detection by the LIGO, it is undoubtedly worth- \nM cl (0) ≈ 3 × 10 4 ; r h (0) ≈ 2 pc; Z = 0 . 001 \n<!-- image --> \n4 \n40 60 80 100 N bound M cl (0) ≈ 5 × 10 M /circledot ; r h (0) ≈ 2 pc; Z = 0 . 02 Figure 10. Top: the M tot s of the ejected BBHs against their corresponding t ej s for the computed model with Mcl (0) GLYPH<25> 3 GLYPH<2> 10 4 M GLYPH<12> , rh (0) GLYPH<25> 2 pc, Z = 0 : 05 Z GLYPH<12> . The ejected BBHs with GLYPH<28> mrg GLYPH<20> 13 : 7 Gyr are indicated by the blue arrows and the BBH coalescence within the cluster is indicated by an orange arrow; c.f. Table. 1. The BBH coalescences, that resemble the LIGO-detected ones (see Sec. 1) in terms of their component masses, are indicated. Bottom: same as the top panel but for the model with Mcl (0) GLYPH<25> 5 GLYPH<2> 10 4 M GLYPH<12> , rh (0) GLYPH<25> 2 pc, Z = Z GLYPH<12> . All the BBH coalescences occur within the cluster, in this particular model; c.f. Table. 1. \n120 \n20 \ncoalescence inside cluster 0 0 while to investigate such events in the present model computations. This is of enhanced interest in this work, since YMCs and open-type clusters are dealt with here. If a power-law shape of the new-born clusters' mass function can be assumed for throughout the Universe, as observed in our nearby spiral and starburst galaxies (typically of index GLYPH<11> GLYPH<25> GLYPH<0> 2 , Gieles et al. 2006a,b; Larsen 2009), such clusters will be the most abundant ones, among those that survive for at least a few Gyr. Therefore, although expected to be lower than GCs per cluster, their overall contribution to the observable BBH inspiral rate in the Universe could be at least comparable to that from classical GCs, thereby potentially adding significantly to the BBH detection rate from the dynamical channel. This question remains rather unexplored until now. \nAll the models computed here continue to eject BBHs, which can quench only when 1 or 2 BHs remain bound (in the M cl (0) GLYPH<25> 3 GLYPH<2> 10 4 M GLYPH<12> , Z = Z GLYPH<12> model, even the final BH is ejected dynamically when the central density is increased, as the cluster approached core collapse, although such complete depletion of BHs is generally unlikely; see Fig. 2). \nHowever, the total number of ejected BBHs per cluster is much smaller than that is typical for more massive MonteCarlo based models (see Sec. 1 and references therein), as expected. Fig. 8 gives the distributions of orbital period, P , eccentricity, e , and mass ratio, q , for the escaped BBHs from all computed models combined. The majority of these BBHs have P GLYPH<24> 10 4 GLYPH<0> 10 5 days 8 . As characteristic of dynamically ejected binaries (via close encounters), the ejected BBHs are generally of high eccentricity, with the e -distribution peaked beyond e > 0 : 8 . The mass ratios of the ejected BBHs are typically of q > 0 : 5 , with the q -distribution function increasing towards q = 1 . This feature of the q -distribution is a signature of the dynamical formation of the BBHs within the cluster before getting ejected, in which process the pairing of BHs of comparable masses is energetically favourable. \nBH NS Except for the least massive ones, all cluster models computed here have produced BBHs that coalesce within a Hubble time (beginning from the clusters' zero age) due to GW emission, either still being bound to the cluster (which would typically occur due to Kozai mechanism in BH-triples; see Sec. 1) or being among the escaped BBHs. The secondlast column in Table. 1 shows the number, N mrg ; in , of BBH coalescences that occurred within each of the model clusters and also the corresponding component BH masses. The final column in Table. 1 gives, for each model, the number of ejected BBHs, N mrg ; out , and their component masses, that have GW merger time, GLYPH<28> mrg < 13 : 7 Gyr (Hubble time), at the time of ejection. Note that the bound-to-the-cluster coalescences happen in the computations on the fly (see Sec. 2.1), whereas, for the BBHs ejected from the clusters, the corresponding GLYPH<28> mrg s are estimated using the standard orbitaveraged GW-shrinkage formula by Peters (1964). \n2000 4000 6000 8000 10000 12000 t (Myr) An interesting fact, that immediately becomes apparent from Table 1, is that, in general, N mrg ; in > N mrg ; out , for models with M cl (0) > 10 4 M GLYPH<12> . In other words, YMCs and their derivative open clusters (as they evolve), are inherently more efficient in producing in-cluster, triple-mediated BBH coalescences than through ejecting BBHs . This is in contrast to what Monte-Carlo calculations of much more massive systems but with similar model ingredients find (see, e.g. , Morscher et al. 2015; Rodriguez et al. 2016a; Chatterjee et al. 2016a). This difference could be due to artefacts in the Monte-Carlo treatment itself, especially how multiple systems are treated there. On the other hand, this could as well be characteristic of the lower-mass systems that is dealt with here; because of lower density of stars and BHs in the present models, the dynamically-formed triples can last unperturbed for longer time, giving higher chance to their inner binaries to merge via Kozai oscillations. This becomes further apparent from the fact that for the models with M cl (0) GLYPH<20> 1 : 5 GLYPH<2> 10 4 M GLYPH<12> , BBH coalescences occur only within the clusters ( c.f. Table. 1). \n8 The majority of the ejected BBHs are from the Mcl (0) GLYPH<25> 3 GLYPH<2> 10 4 M GLYPH<12> and 5 GLYPH<2> 10 4 M GLYPH<12> clusters and hence the P -distribution and, particularly, its peak are more of the characteristics of the ejected BBHs from these clusters. Lower-mass clusters would generally eject wider BBHs and vice versa. The low number of ejected BBHs per cluster here makes the comparison among the BBH distributions corresponding to different Mcl (0) s and Z s less meaningful, which is otherwise advisable. \nFigure 11. Top panels: the M tot s of the in-cluster (triple-mediated; left) and the ejected (right) BBH coalescences against their corresponding t mrg s, for all the computed models in Table. 1. The colour-coding is according to the parent model cluster's Z (vertical colour bar). Bottom panels: the above BBHs plotted with their mass ratios in the Y-axis. In all the panels, the ranges in the Y-axis corresponding to the detected BBH coalescence events are indicated by the horizontal lines. \n<!-- image --> \nInterestingly, such prominence of in-cluster BBH coalescences, as seen here, also contrasts the results obtained from earlier direct N-body calculations of models of similar mass and size containing GLYPH<25> 10 M GLYPH<12> BHs, where the escaped BBH coalescences typically dominated over the in-cluster ones; see, e.g. , Banerjee et al. (2010). This is likely to be the result of a much broader BH mass distribution in the present models. The most massive couple of BHs would favourably become binary pair within the dense BH-core which would tend to prevent less massive BHs to pair, and would eject them preferably as singles via super-elastic scattering. This would suppress the number of ejected BBHs for a given (initial) mass and size of the cluster, and hence the coalescences among them. At the same time, the'bully'BH-binary, which would typically be the most massive object in the system, would be harder to eject dynamically, giving it enhanced opportunity to coalesce (or induce coalescence), if at all, within the cluster. In contrast, in a BH-core comprising of equal-mass or nearly equal-mass BHs, as in several previous studies, the dominance of a single BH pair would no more be energetically favourable, which would reverse the situation. To understand the role of (triple-induced) BBH coalescences within the cluster, it is necessary to do N-body calculations as in here in larger numbers and with even higher M cl (0) , which is planned for the near future (see also Kimpson et al. 2016; Haster et al. 2016 in this context). \nThe filled squares in Fig. 9 indicate the mass ratios ( q s) of the ejected BBHs from all the models computed here against their respective ejection times ( t ej s), which symbols are colour-coded according to their total masses, M tot . The q s corresponding to the in-cluster and after-ejection BBH coalescences are highlighted (by filled and empty triangles respectively) at the times, t mrg , of their occurrences (therefore, for the ejected BBH coalescences, t mrg GLYPH<17> t ej + GLYPH<28> mrg ). As already indicated by Fig. 8 (bottom panel), most ejected BBHs and all BBH coalescences have q > 0 : 5 . In-cluster coalescences can happen from as early as t mrg GLYPH<24> 100 Myr until GLYPH<24> 10 Gyr; more massive of those ( 60 M GLYPH<12> . M tot . 80 M GLYPH<12> ) typically happening within t . 1 Gyr. On the other hand, in the present sample, all coalescences among the ejected BBHs happen after t & 1 Gyr (including their t ej s). Note that the latter conclusion can be an artefact of the low number of (only 4) ejected coalescences within the Hubble time (see Fig. 9); single BHs and BBHs begin to get ejected as soon as the central BH sub-cluster becomes concentrated enough that three-body binaries start forming in them (see Sec. 1; this is also when the contraction of the BH sub-population stalls, see, e.g. , Fig. 6), from t GLYPH<24> 100 Myr.", '14 S. Banerjee': "3.2.1 Comparison with the detected binary black hole coalescences \nIt would undoubtedly be useful to compare the LIGOdetected events with the BBH mergers from the present set of computations. Fig. 10 plots the M tot s of the ejected BHHs vs. their t ej s for the M cl (0) GLYPH<25> 3 GLYPH<2> 10 4 M GLYPH<12> , r h (0) GLYPH<25> 2 pc, Z = 0 : 05 Z GLYPH<12> model (top panel) and the M cl (0) GLYPH<25> 5 GLYPH<2> 10 4 M GLYPH<12> , r h (0) GLYPH<25> 2 pc, Z = Z GLYPH<12> model (bottom panel). The ejected BBHs with GLYPH<28> mrg < 13 : 7 Gyr are indicated (blue arrows) and as well any coalescences within the clusters (orange arrows). Remarkably, two mergers in the M cl (0) GLYPH<25> 3 GLYPH<2> 10 4 M GLYPH<12> cluster resemble the events GW150914 and LVT151012, both in terms of their M tot s and also in individual component masses (Sec. 1 and references therein). \nThe M cl (0) GLYPH<25> 5 GLYPH<2> 10 4 M GLYPH<12> run is also remarkable in the sense that it has produced 3 BBH coalescences, all within the cluster and between 5 . t . 8 Gyr, when nearly 3/4th of its initially-retained BHs have already escaped (see Fig. 3, lower-right panel). All the mergers are of GW151226-type, in terms of their M tot s. This particular calculation is unique among the set in this study, where the lower mass GLYPH<25> 10 M GLYPH<12> BHs (since Z = Z GLYPH<12> ) allowed the cluster, and hence the BH sub-cluster (see Sec. 3.1), to remain sufficiently concentrated throughout its evolution. In particular, the cluster and its BH-core begin to collapse and boost their concentrations after t GLYPH<25> 3 Gyr (see Fig. 7, left panel), with a sufficient number of BHs ( GLYPH<25> 20 ) still retaining to continue forming BH-triples which are relatively uninterrupted. The BHs, in this case, are similarly massive which favour dynamical BBH formation (see above) and hence the formation of BH-triples within the core. This implies that intermediate-aged massive open clusters of solar-like metallicity serve as highly potential sites for dynamically generating GW151226-like BBH coalescences . Similar calculations over a wider parameter range is necessary to reassure this intriguing inference 9 . \nFig. 11 shows the M tot s (top row) and q s (bottom row) of the in-cluster (left column) and escaped (right column) BBH coalescences, at their respective t mrg s, from the present set of computations, which are compared with the limits of the detected events. All the symbols here are colour-coded according to the parent cluster's Z . As already seen above, the in-situ mergers occur from age as young as t GLYPH<25> 100 Myr up to at least 10 Gyr. Typically, lower Z clusters yield more massive coalescences, which occur at earlier t mrg (the negative trend in Fig. 11, top-left panel). This overall trend is due to the fact that more massive BHs segregate and interact dynamically earlier (see also Chatterjee et al. 2016b in this context). The q s of all in-situ BBH coalescences are > 0 : 6 (Fig. 11, bottom-left panel), since pairings within the cluster would preferably happen among BHs whose masses are close to each other (see above). The lower-mass BHs, which are dynamically processed later in time, are likely to have partners that are closer in mass (due to the IMF slope and the BH mass-ZAMS mass relation; see Sec. 2.2), giving rise to the positive trend of the in-situ mergers' q s with t mrg (Fig. 11, \nbottom-left panel). A similar trend follows for the q s of the escaped BBH coalescences; however, all the escaped coalescences happen after GLYPH<25> 1 Gyr (Fig. 11, bottom-right panel). Also, the escaped coalescences generally have lower q s than their in-cluster counterparts. There is no particular trend seen in the M tot s of the escaped BBH coalescences (Fig. 11, top-right panel) which is likely to be an artefact of the low number of them, in the present sample. \nAccording to Fig. 11, all of the clusters computed here have the potential to give rise to BBH coalescence events resembling the detected ones, by dynamical means. All of the detected events could have occurred either in-situ or after being ejected from their parent clusters. Finally, at the time of the coalescence, their parent cluster could either be a GLYPH<24> 100 Myr YMC (likely, if the event is in-situ) or be a few - 10 Gyr old intermediate-mass cluster/open cluster (likely for both in-situ and ejected events). Interestingly, BBH coalescences, with M tot exceeding the upper limit of GW150914 by GLYPH<25> 10 M GLYPH<12> , are also produced in the present computations, as seen in Fig. 11 (top-left panel). In fact these two in-cluster mergers take place the earliest ( t mrg GLYPH<24> 100 Myr) and in the least massive clusters ( M cl (0) GLYPH<25> 1 : 0 GLYPH<2> 10 4 M GLYPH<12> and 1 : 5 GLYPH<2> 10 4 M GLYPH<12> ; see Table 1). The lower velocity dispersion in the BH-core of such lower-mass systems would typically produce wider BBHs via three-body encounters, causing an overall weaker energy extraction in the BH-core and, thereby, making it harder for the most massive BHs to get ejected (although they begin to participate in the dynamical interactions the earliest); such clusters are otherwise most efficient in ejecting BHs, in the sense of Fig. 5 (right panel). This, combined with such clusters' shorter BH-segregation timescale and shorter twobody relaxation time of the BH-core, have resulted in such early and massive BBH coalescences. If BBH coalescences, substantially more massive than GW150914 (by GLYPH<24> 10 M GLYPH<12> ), are detected by the Advanced LIGO in the future, GLYPH<24> 10 4 M GLYPH<12> , low-metallicity, compact YMCs would be potential sites for them. \nIn making such comparisons, one should bear in mind the cosmological aspects of it. To detect an event today, one should take into account the time, t mrg , for the BBH coalescence to occur, counting from the zero age of the parent cluster, plus time light travel time, tL , for the merger GW signal to reach us from the location of the event. At its design sensitivity, the Advanced LIGO would detect the inspiral signal from a pair of 10 M GLYPH<12> BHs from maximum D 10 GLYPH<25> 1500 Mpc (comoving radial) distance (see, e.g. , Banerjee et al. 2010), which corresponds to a redshift of z GLYPH<25> 0 : 37 and a light travel time of tL GLYPH<25> 4 : 2 Gyr 10 (Wright 2006). For a pair of 40 M GLYPH<12> BBH coalescence (the most massive ones here; c.f. Table 1), the limiting distance is D 40 GLYPH<25> 4800 Mpc (Eqns. 6 & 7 of Banerjee et al. 2010), corresponding to z GLYPH<25> 1 : 68 and tL GLYPH<25> 9 : 9 Gyr. If the trends seen in Fig. 11 can be bought as a rough representative, then clusters of all ages can, in principle, be parents of the detected BBH merger events, depending on the epochs of their formation . However, the details of the cosmology must be taken into account to properly estimate the detection rate of BBH coalescences (see, e.g. , Belczynski et al. 2016; Chatterjee et al. 2016b), which is beyond the scope of this article. \n3.2.2 LIGO detection rate of binary black hole mergers: a preliminary estimate \nIt would still be useful to make a preliminary estimate of the GW-inspiral detection rate from dynamically-generated BBH coalescence events, by the LIGO at its design sensitivity, based on the qualitative aspects learnt from the present computations. We will assume that all the stellar clusters, contributing to the BBH mergers intercepted at the current epoch, are formed GLYPH<25> 10 Gyr ago (which is not necessarily true), i.e. , they would currently be either lowZ GCs or lowZ open clusters, if they still survive. Unlike in Banerjee et al. (2010), where it is assumed that nearly all BHs will be depleted by GLYPH<25> 3 Gyr of cluster evolution and also that most BBH mergers happen over this time period (based on the results from those calculations), typically a significant number of BHs continue to retain until the Hubble time or until dissolution, for lowZ systems, as seen in the present calculations (Sec. 3.1). Also, except for the least massive systems considered here, all clusters tend to produce BBH mergers (either in-situ or ejected) over all evolutionary ages (Sec. 3.2). Hence, not only the intermediate-aged clusters, but also the classical GCs of the Universe would actively contribute to present-day BBH mergers. Therefore, here we can take, as the space density of clusters, GLYPH<26> cl , the sum of those of both GCs and 'young populous clusters' (Portegies Zwart & McMillan 2000), i.e. , \nGLYPH<26> cl = 11 : 9 h 3 Mpc GLYPH<0> 3 ; \nwhich would contribute to the present-day BBH mergers. \nTo estimate a bare minimum rate, we will consider the spherical volume of radius D 10 (see Sec. 3.2.1), from within which essentially any BBH coalescence is, in principle, detectable by the LIGO, at its design sensitivity. If we conservatively assume only one BBH merger event per cluster, over the corresponding tL GLYPH<25> 4 : 2 Gyr (see Sec. 3.2.1), then the corresponding LIGO detection rate would be R LIGO GLYPH<25> 13 yr GLYPH<0> 1 . Over D 40 , this rate would scale up to R LIGO GLYPH<25> 425 yr GLYPH<0> 1 . However, the latter rate is an overestimation since not all BBH inspiral signals can be detected by the LIGO, from distances beyond D 10 , even at its full sensitivity. Instead, if one modestly assumes only one BBH inspiral per cluster over 10 Gyr (which time is close to the tL from D 40 ), that can be detected by the LIGO from within the D 40 limiting distance, then R LIGO GLYPH<25> 170 yr GLYPH<0> 1 , at the design sensitivity. Of course, larger contribution per cluster and contributions from clusters that are formed at later epochs, would increase both of the limits of R LIGO . Based on the above preliminary estimates, it can generally be taken that R LIGO , due to dynamical BBH coalescences, would lie between few 10s to few 100s per year, at the LIGO's design sensitivity. A more elaborate evaluation of R LIGO is planned for the near future.", '4 SUMMARY AND OUTLOOK': "The model cluster computations presented in this work, although comprise a preliminary set but with realistic ingredients (Secs. 2.1 & 2.2), provide new and intriguing inferences. They span an initial mass range of 1 : 0 GLYPH<2> 10 4 M GLYPH<12> . M cl (0) . 5 : 0 GLYPH<2> 10 4 M GLYPH<12> , a metallicity range of 0 : 05 Z GLYPH<12> GLYPH<20> Z GLYPH<20> Z GLYPH<12> and are of half-mass radius r h (0) GLYPH<25> 2 pc; for models of M cl (0) . 1 : 5 GLYPH<2> 10 4 M GLYPH<12> , r h (0) GLYPH<25> 1 pc is also assumed. All models \nare evolved by direct N-body method until their dissolution or otherwise at least for 10 Gyr. This allows one to study and compare YMC-like systems of varying Z , which evolve into open cluster-like systems, especially w.r.t. the role of stellar-mass BHs and the dynamically-triggered BBH coalescences. This mass range has not yet been explored in this way, although more massive GC-like systems have been studied often (Sec. 1 and references therein). \nThe BHs that form by direct collapse receive no birth kick and remain bound to the parent cluster right after their formation (Sec. 2.2). The initially-retained BHs segregate to the cluster center in GLYPH<24> 100 Myr, where they form a dense BH sub-cluster. The frequent and energetic dynamical encounters in this BH-engine (or BH-core) continue to inject energy into the dense normal-stellar bulk of the cluster, until the majority of the BHs are ejected via the strong dynamical encounters. The energy injection causes the parent cluster to expand, until the BH-engine is sufficiently weakened due to the depletion of BHs, after which the cluster begins to re-collapse (Sec. 3.1; Figs. 3 & 4). The BH-core also acts to 'freeze' the otherwise natural two-body relaxation-driven mass segregation process ( e.g. , that of the ECS NSs), which can resume only after most of the BHs are dynamically depleted (Figs. 6 & 7). LowerZ clusters would retain more massive and larger number of BHs at birth (Sec. 2.2; Fig. 1), which would have a more profound effect on the cluster. However, the energy injection process by the BHs seems to be self-regulatory: the more powerful BH-engine for a lowerZ system also acts to moderate the dynamical BH ejection, causing a larger fraction of BHs to be retained with time (Sec. 3.1; Fig. 5). As a result, the sub-solarZ clusters computed here still retain GLYPH<24> 10 BHs until the Hubble time or until shortly before their dissolution, whereas their solarZ counterparts are nearly deprived of their BHs by similar evolutionary times (Fig. 3). \nAll of the computed models of masses M cl (0) GLYPH<25> 5 GLYPH<2> 10 4 M GLYPH<12> and GLYPH<25> 3 GLYPH<2> 10 4 M GLYPH<12> and the lowestZ ones of masses M cl (0) . 1 : 5 GLYPH<2> 10 4 M GLYPH<12> produce BBH coalescences due to inspiral via GW radiation (Sec. 3.2; Table 1). They occur either via Kozai mechanism in the dynamically-produced BH-triples that are bound to the cluster or among the dynamically-ejected eccentric BBHs. For the present set of models, the majority of the BBH mergers happen triple-mediated, within the parent cluster. This is in contrast with earlier N-body calculations that contained BHs of masses similar to each other and also with Monte-Carlo calculations of more massive clusters having similar model ingredients as the present ones (see Sec. 3.2 and references therein). Given that dynamicallyformed subsystems are treated naturally and accurately in direct N-body calculations (Sec. 2.1), this is unlikely to be an artefact of the numerical methods applied here. As explained in Sec. 3.2, a much broader mass spectrum of the BHs is likely to favour in-situ BBH coalescences, at least over the mass range of the clusters computed here. It is important to reach even higher M cl (0) s using the direct N-body method, to properly understand the role of BH-triples and also the differences with the Monte-Carlo models and with the previous N-body models. Because of dynamical pairing, most of the BBHs and their coalescences have mass ratio q > 0 : 5 (Sec. 3.2; Fig. 9). Among the coalescences obtained here, there are ones which resemble well with one or other of the events detected by the LIGO until now (Sec. 3.2.1; Figs. 10 & \n11). Interestingly, the lowest massive clusters computed here have produced the most massive and the earliest (in-situ) BBH coalescences, that well exceed the M tot of GW150914; this might be a result of weaker dynamical encounters in these systems combined with their shorter relaxation times (see Sec. 3.2.1). Again, a larger number of such lower-mass, lowZ model computations is necessary to ascertain this, and to understand it better. A back-of-the-envelope conservative estimation is that, under its proposed design sensitivity, the LIGO should detect GLYPH<24> 10 GLYPH<0> GLYPH<24> 100 BBH inspiral events per year, due to dynamical interactions among stellar-mass BHs in star clusters alone (Sec. 3.2.2). \nThe primary uncertainties in the present calculations stem from the recipes of stellar wind, remnant formation and their natal kicks, that are adopted here (Sec. 2.2). Although such recipes can, in a sense, be called the state-of-the-art because of their relatively popular applications in the literature, nearly all aspects of stellar remnant formation (especially of BH formation; see Sec. 2.2) remain poorly understood or constrained to date. These uncertainties translate into those in the BHs' mass spectrum and initial retention in stellar clusters. Another factor that might have affected the current results to some extent is the lack of primordial binaries, which is done to make these computations feasible. Although Monte-Carlo calculations of much more massive clusters indicate near indifference of the BH dynamics to the presence of primordial binaries (Sec. 2.3 and references therein), the situation can be different for lower mass systems, where the dynamical interactions are generally less energetic. Furthermore, mass transfer and/or tidal interactions among massive-stellar and close primordial binaries would influence the masses of the BHs (Thomas Tauris, Philipp Podsiadlowski, private communications; see also De Mink et al. 2009; Marchant et al. 2016 in this context). Moreover, dynamically induced (Banerjee et al. 2012) or mass-transfer driven (De Mink et al. 2014) coalescences of massive-stellar binaries would form more massive merger products, which would yield more massive BHs. \nThe immediate next step would be to obtain an even more exhaustive set of model calculations by reaching even higher M cl (0) s and obtaining more evolutionary models for the lowest cluster masses (and perhaps explore even lower masses). These will also help to understand better the role of BH subsystems and their impact on the parent cluster's dynamics and BBH coalescence events, as discussed above and in the previous sections. Inclusion of primordial binaries would be feasible at least for the least-massive systems, which would be intriguing to do. Such steps are planned by the author for the near future.", 'ACKNOWLEDGEMENTS': "The author is indebted to Sverre Aarseth of the Institute of Astronomy, Cambridge, for his continuous efforts in improving NBODY6/7 , without which this study wouldn't have been possible. SB is thankful to Jarrod Hurley of the Swinburne University of Technology for reviewing the changes made in the BSE 's routines for this work, testing them independently, and also for interesting discussions. SB is thankful to Chris Belczynski of the Astronomical Observatory, University of Warsaw, for supplying the data corresponding to the ZAMS \nmass-remnant mass relations, as obtained from the StarTrack program (Fig. 1), and also for useful discussions. SB is thankful to the anonymous referee for helpful comments that improved some of the descriptions of the paper. 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2018EPJC...78..960O
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Black holes by gravitational decoupling
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2018-01-01
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We investigate how a spherically symmetric fluid modifies the Schwarzschild vacuum solution when there is no exchange of energy-momentum between the fluid and the central source of the Schwarzschild metric. This system is described by means of the gravitational decoupling realised via the minimal geometric deformation approach, which allows us to prove that the fluid must be anisotropic. Several cases are then explicitly shown.
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https://arxiv.org/pdf/1804.03468.pdf
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{'Black holes by gravitational decoupling': "J. Ovalle ab ∗ , R. Casadio cd † , R. da Rocha e ‡ , A. Sotomayor f § , Z. Stuchlik a ¶ \na Institute of Physics and Research Centre of Theoretical Physics and Astrophysics, Faculty of Philosophy and Science, Silesian University in Opava CZ-746 01 Opava, Czech Republic b Departamento de F'ısica, Universidad Sim'on Bol'ıvar, AP 89000, Caracas 1080A, Venezuela c Dipartimento di Fisica e Astronomia, Alma Mater Universit'a di Bologna via Irnerio 46, 40126 Bologna, Italy d Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, I.S. FLAG viale Berti Pichat 6/2, 40127 Bologna, Italy e Centro de Matem'atica, Computa¸c˜ao e Cogni¸c˜ao, Universidade Federal do ABC (UFABC) 09210-580, Santo Andr'e, SP, Brazil. f Departamento de Matem'aticas, Universidad de Antofagasta Antofagasta, Chile \nJuly 20, 2018", 'Abstract': 'We investigate how a spherically symmetric fluid modifies the Schwarzschild vacuum solution when there is no exchange of energy-momentum between the fluid and the central source of the Schwarzschild metric. This system is described by means of the gravitational decoupling realised via the minimal geometric deformation approach, which allows us to prove that the fluid must be anisotropic. Several cases are then explicitly shown.', '1 Introduction': "The study of black holes represents one of the most active areas of gravitational physics, from both a purely theoretical and the observational point of view. The interest black holes generate is due not only to their exotic nature, but also because they constitute ideal laboratories to study gravity in the strong field regime, and test general relativity therein. However, confronting theoretical predictions with observations is an arduous and complicated task. A formidable step in this direction is the \nrecent direct observation of black holes through the detection of gravitational waves, which opens a new and promising era for gravitational physics [1,2]. \nIt is well known that general relativity predicts surprisingly simple solutions for black holes, characterised at most by three fundamental parameters, namely the mass M , angular momentum J and charge Q [3]. The original no-hair conjecture states that these solutions should not carry any other charges [4]. Therefore, as the observations of systems containing black holes improve, the degree of consistency of these observations with the predictions determined according to the general relativistic solutions (with parameters M , J and Q ) will result in a direct test of the validity of general relativity in the strong field regime. There could in fact exist other charges associated with inner gauge symmetries (and fields), and it is now known that black holes could have (soft) quantum hair [5]. The existence of new fundamental fields, which leave an imprint on the structure of the black hole, thus leading to hairy black hole solutions, is precisely the scenario under study in this paper. \nPossible conditions for circumventing the no-go theorem have been investigated for a long time in different scenarios (see Refs. [6-15] for some recent works and Refs. [16-21] for earlier works). In particular, a fundamental scalar field φ has been considered with great interest (see Ref. [22] and references therein). In this work, we will take a different and more general approach than most of the investigations carried out so far and, instead of considering specific fundamental fields to generate hair in black hole solutions, we shall just assume the presence of an additional completely generic source described by a conserved energy-momentum tensor θ µν . Of course, this θ µν could account for one or more fundamental fields, but the crucial property is that it gravitates but does not interact directly with the matter that sources the (hairless) black hole solutions we start from. This feature may seem fanciful, but can be fully justified, for instance, in the context of the dark matter. Achieving this level of generality in the classical scheme represented by general relativity is a non-trivial task, and the gravitational decoupling by Minimal Geometric Deformation (MGDdecoupling, henceforth) is precisely the method that was developed for this purpose in Ref. [23]. \nThe MGD approach was originally proposed [24,25] in the context of the brane-world [26, 27] and extended to investigate new black hole solutions in Refs. [28, 29] (for some earlier works on the MGD, see for instance Refs. [30-33], and Refs. [34-50] for some recent applications). The MGD-decoupling has a number of ingredients that make it particularly attractive in the search for new spherically symmetric solutions of Einstein's field equations. The two main feature of this approach are the following [23]: \n- · Extending simple solutions into more complex domains. Wecan start from a simple spherically symmetric gravitational source with energy-momentum tensor ˆ T µν and add to it more and more complex gravitational sources, as long as the spherical symmetry is preserved. The starting source ˆ T µν could be as simple as we wish, including the vacuum indeed, to which we can add a first new source, say \nˆ T µν ↦→ ˜ T (1) µν = ˆ T µν + α (1) T (1) µν , (1.1) \nwhere α (1) is a constant that traces the effects of the new source T (1) µν . We can then repeat the process with more sources, namely \n˜ T (1) µν ↦→ ˜ T (2) µν = ˜ T (1) µν + α (2) T (2) µν , (1.2) \nand so on. In this way, we can extend straightforward solutions of the Einstein equations associated with the simplest gravitational source ˆ T µν into the domain of more intricate forms \nof gravitational sources T µν = ˜ T ( n ) µν , step by step and systematically. We stress that this method works as long as the sources do not exchange energy-momentum among them, namely \n∇ µ ˆ T µν = ∇ µ T (1) µν = . . . = ∇ µ T ( n ) µν = 0 , (1.3) \nwhich further clarifies that the constituents can only couple via gravity. \n- · Deconstructing a complex gravitational source. The converse of the above also works. In order to find a solution to Einstein's equations with a complex spherically symmetric energymomentum tensor T µν , we can split it into simpler components, say ˆ T µν and T ( i ) µν , provided they all satisfy Eq. (1.3), and solve Einstein's equations for each one of these parts. Hence, we will have as many solutions as are the contributions T ( i ) µν in the original energy-momentum tensor. Finally, by a straightforward combination of all these solutions, we will obtain the solution to the Einstein equations associated with the original energy-momentum tensor T µν . \nSince Einstein's field equations are non-linear, the MGD-decoupling represents a breakthrough in the search and analysis of solutions, especially when we deal with situations beyond trivial cases, such as the interior of self-gravitating systems dominated by gravitational sources more realistic than the ideal perfect fluid [51, 52]. Of course, we remark that this decoupling occurs because of the spherical symmetry and time-independence of the systems under investigation. \nIn analogy with the well-known electro-vacuum and scalar-vacuum, in this paper we will consider a Schwarzschild black hole surrounded by a spherically symmetric 'tensor-vacuum', represented by the aforementioned θ µν . Following the MGD-decoupling, we will separate the Einstein field equations in i) Einstein's equations for the spherically symmetric vacuum and ii) a 'quasi-Einstein' system for the spherically symmetric 'tensor-vacuum'. The MGD procedure will then allow us to merge the Schwarzschild solution for i) with the solution for the 'quasi-Einstein' system ii) into the solution for the complete system 'Schwarzschild + tensor-vacuum'. Like the case of the electrovacuum and (in some cases) scalar-vacuum, new black hole solutions with additional parameters q i besides the mass M can be obtained, each one associated with a particular equation of state for the 'tensor-vacuum'. Demanding the geometry is free of singularities and other pathologies, implies regularity conditions which show that not all of these parameters q i can be independent. \nThe paper is organised as follows: in Section 2, we first review the fundamentals of the MGDdecoupling applied to a spherically symmetric system containing a perfect fluid and an additional source θ µν ; in Section 3, new hairy black holes solutions are found by assuming the perfect fluid has sufficiently small support so that only θ µν exists outside the horizon; finally, we summarise our conclusions in Section 4.", '2 MGD decoupling for a perfect fluid': "Let us start from the standard Einstein field equations \nR µν -1 2 Rg µν = -k 2 T (tot) µν , (2.1) \nand assume the total energy-momentum tensor contains two contributions, namely \nT (tot) µν = T (m) µν + αθ µν , (2.2) \nwhere \nT (m) µν = ( ρ + p ) u µ u ν -p g µν (2.3) \nis the 4-dimensional energy-momentum tensor of a perfect fluid with 4-velocity field u µ , density ρ and isotropic pressure p . The term θ µν in Eq. (2.2) describes an additional source whose coupling to gravity is proportional to the constant α [53]. This source may contain new fields, like scalar, vector and tensor fields, and will in general produce anisotropies in self-gravitating systems. In any case, since the Einstein tensor satisfies the Bianchi identity, the total source in Eq. (2.2) must satisfy the conservation equation \n∇ µ T (tot) µν = 0 . (2.4) \nWe next specialise to spherical symmetry and no time-dependence. In Schwarzschild-like coordinates, a static spherically symmetric metric g µν reads \nds 2 = e ν ( r ) dt 2 -e λ ( r ) dr 2 -r 2 dθ 2 +sin 2 θ dφ 2 ) , (2.5) \n( \n) where ν = ν ( r ) and λ = λ ( r ) are functions of the areal radius r only, ranging from r = 0 (the star center) to some r = R (the star surface), and the fluid 4-velocity is given by u µ = e -ν/ 2 δ µ 0 for 0 ≤ r ≤ R . The metric (2.5) must satisfy the Einstein equations (2.1), which explicitly read \nk 2 ρ + αθ 0 0 ) = 1 r 2 -e -λ ( 1 r 2 -λ ' r ) , (2.6) \n( ( \n) k 2 -p + αθ 1 1 ) = 1 r 2 -e -λ ( 1 r 2 + ν ' r ) , (2.7) \nwhere f ' ≡ ∂ r f and spherical symmetry implies that θ 3 3 = θ 2 2 . The conservation equation (2.4) is a linear combination of Eqs. (2.6)-(2.8), and yields \n) k 2 ( -p + αθ 2 2 ) = e -λ 4 ( -2 ν '' -ν ' 2 + λ ' ν ' -2 ν ' -λ ' r ) , (2.8) \np ' + ν ' 2 ( ρ + p ) -α ( θ 1 1 ) ' + ν ' 2 α ( θ 0 0 -θ 1 1 ) + 2 α r ( θ 2 2 -θ 1 1 ) = 0 , (2.9) \nWe then note the perfect fluid case is formally recovered for α → 0. \nThe Eqs. (2.6)-(2.8) contain seven unknown functions, namely: two physical variables, the density ρ ( r ) and pressure p ( r ); two geometric functions, the temporal metric function ν ( r ) and the radial metric function λ ( r ); and three independent components of θ µν . This system of equations is therefore indeterminate and we should emphasise that the space-time geometry does not allow one to resolve for the gravitational source { ρ, p, θ µν } uniquely. \nIn order to simplify the analysis, and by simple inspection, we can identify an effective density \n˜ ρ = ρ + αθ 0 0 , (2.10) \nan effective radial pressure \nand an effective tangential pressure \n˜ p r = p -αθ 1 1 , (2.11) \n˜ p t = p -αθ 2 2 . (2.12) \nThese definitions clearly illustrate that θ µν could in general induce an anisotropy, \nΠ ≡ ˜ p t -˜ p r = α θ 1 1 -θ 2 2 ) , (2.13) \n( \n) inside a stellar distribution sourced by T (m) µν alone. The system of Eqs. (2.6)-(2.8) may therefore be formally treated as an anisotropic fluid [54,55]. \nThe MGD-decoupling can now be applied to the case at hand by simply noting that the energymomentum tensor (2.2) is precisely of the form (1.1), with ˆ T µν = T (m) µν , α (1) = α and T (1) µν = θ µν . The components of the diagonal metric g µν that solve the complete Einstein equations (2.1) and satisfy the MGD read [23] \ng µν = ˆ g µν = g (1) µν (2.14) \nfor µ = ν /negationslash = 1, and \ng 11 = ˆ g 11 + αg (1)11 , (2.15) \nso that only the radial component is affected by the additional source θ µν . This metric g µν is found by first solving the Einstein equations for the perfect fluid source T (m) µν , \nˆ G µν = -k 2 T (m) µν , ∇ µ T (m) µν = 0 , (2.16) \nand then the remaining quasi-Einstein equations for the source θ µν , namely \n˜ G µν = -k 2 θ µν , ∇ µ θ µν = 0 , (2.17) \nwhere the divergence-free quasi-Einstein tensor \n˜ G ν µ = G ν µ +Γ ν µ , (2.18) \nwith Γ ν µ a tensor that depends exclusively on g µν to ensure the divergence-free condition. For the spherically symmetric metric (2.5), it reads \nΓ ν µ = 1 r 2 ( δ 0 µ δ ν 0 + δ 1 µ δ ν 1 ) . (2.19) \nWe can then proceed by considering a solution to the Eqs. (2.16) for a perfect fluid [that is Eqs. (2.6)-(2.9) with α = 0], which we can write as \nwhere \nds 2 = e ξ ( r ) dt 2 -dr 2 µ ( r ) -r 2 ( dθ 2 +sin 2 θ dφ 2 ) , (2.20) \nµ ( r ) ≡ 1 -k 2 r ∫ r 0 x 2 ρ ( x ) dx = 1 -2 m ( r ) r (2.21) \nis the standard General Relativity expression containing the Misner-Sharp mass function m ( r ). The effects of the source θ µν on the perfect fluid solution { ξ, µ ρ, p } can then be encoded in the MGD undergone solely by the radial component of the perfect fluid geometry in Eq. (2.20). Namely, the general solution is given by Eq. (2.5) with ν ( r ) = ξ ( r ) and \ne -λ ( r ) = µ ( r ) + αf ∗ ( r ) , (2.22) \nwhere f ∗ = f ∗ ( r ) is the MGD function to be determined from the quasi-Einstein Eqs. (2.17), which explicitly read \nk 2 θ 0 0 = -f ∗ r 2 -f ∗ ' r , (2.23) \nk 2 θ 1 1 = -f ∗ ( 1 r 2 + ξ ' r ) , (2.24) \nk 2 θ 2 2 = k 2 θ 3 3 = -f ∗ 4 ( 2 ξ '' + ξ ' 2 +2 ξ ' r ) -f ∗ ' 4 ( ξ ' + 2 r ) . (2.25) \nWealso notice that the conservation equations for the additional energy-momentum tensor, ∇ µ θ µν = 0, yield \n( \nθ 1 1 ) ' -ξ ' 2 θ 0 0 -θ 1 1 ) -2 r ( θ 2 2 -θ 1 1 ) = 0 , (2.26) \n) which does not depend on the MGD function f ∗ . \n( \nIn the next section, we shall solve the above equations starting from the simplest vacuum solution given by the outer Schwarzschild metric, \nds 2 = ( 1 -2 M r ) dt 2 -( 1 -2 M r ) -1 dr 2 -d Ω 2 , (2.27) \ntherefore in a region of space where the perfect fluid ρ and p vanish.", '3 Black holes': 'When new paradigms beyond Einstein gravity are studied, the important question arises whether or not new black hole solutions exist. In order to address this point in general, we start from the results of the previous section and determine the MGD function f ∗ for the vacuum Schwarzschild solution (2.27). Figure 1 schematically shows the kind of system we deal with. The MGD metric will therefore read \nds 2 = ( 1 -2 M r ) dt 2 -dr 2 1 -2 M r + αf ∗ ( r ) -r 2 d Ω 2 , (3.28) \nwhere the MGD function f ∗ can be determined by imposing restrictions on the energy-momentum θ µν to close the system of Eqs. (2.23)-(2.25). \nIn the following we shall explore specific equations of state for θ µν and impose basic constraints on the causal structure of the resulting space-time in order to have a well-defined horizon structure. In particular, we recall that for the Schwarzschild metric (2.27), the surface r H = 2 M is both a Killing horizon (determined by the condition e ν = 0) and an outer marginally trapped surface (the causal horizon, in brief, determined by the condition e -λ = 0). For the MGD Schwarzschild metric (3.28), the component g tt = e ν is always equal to the Schwarzschild form in Eq. (2.27) and can only vanish at r = r H . This means that r H = 2 M is still a Killing horizon (which can also become a real singularity). However, the causal horizon is found at r = r h such that g rr ( r h ) = e -λ = 0, or \nr h [1 + αf ∗ ( r h )] = 2 M . (3.29) \nFigure 1: Spherically symmetric source θ µν in the vacuum ρ = p = 0. \n<!-- image --> \nWe should therefore require that r h ≥ 2 M , so that the surface r = r H is hidden behind (or coincides with) the causal horizon. Moreover, if r h > r H , the signature of the metric becomes (+ + --) for r H < r < r h , which one might want to discard as well, since not only the expansion of outgoing geodesics vanishes for r → r + h , but also ingoing geodesics never cross r = r h : in this case the surface r = r h would act as a border of the outer space-time manifold. To summarise, the MGD metric (3.28) represents a proper black hole only if the causal horizon coincides with the Killing horizon, that is when r h = r H = 2 M , and this is therefore the condition we shall require in the following.', '3.1 Isotropic sector': "Let us start by considering the case of isotropic pressure, so that \nθ 1 1 = θ 2 2 = θ 3 3 . (3.30) \nEqs. (2.24) and (2.25) then yield a differential equation for the MGD function, namely \nf ∗ ' ( ξ ' + 2 r ) + f ∗ ( 2 ξ '' + ξ ' 2 -2 ξ ' r -4 r 2 ) = 0 , (3.31) \nwhose general solution is given by \nf ∗ ( r ) = ( 1 -2 M r )( r -M /lscript iso ) 2 , (3.32) \nwhere /lscript iso is a constant with dimensions of a length. Hence, the MGD radial component for an isotropic deformation of the Schwarzschild exterior becomes \ne -λ = e ξ + αf ∗ = ( 1 -2 M r ) [ 1 + α ( r -M /lscript iso ) 2 ] , (3.33) \nwhich is clearly not asymptotically flat for r /greatermuch M 1 . We therefore conclude that the additional source θ µν cannot contain an isotropic pressure if we wish to preserve asymptotic flatness.", '3.2 Conformal sector': "The energy-momentum tensor for a conformally symmetric source must be traceless. Since θ 2 2 = θ 3 3 , we therefore assume \n2 θ 2 2 = -θ 0 0 -θ 1 1 , (3.34) \nso that the system (2.23)-(2.25) becomes \n-k 2 θ 0 0 = f ∗ r 2 + f ∗ ' r (3.35) \n-k 2 θ 1 1 = f ∗ ( 1 r 2 + ξ ' r ) , (3.36) \nwhere f ∗ is again MGD function and ξ the unperturbed Schwarzschild function. From Eq. (3.34), we find the radial deformation must satisfy the differential equation \nf ∗ ' ( ξ ' 2 + 2 r ) + f ∗ ( ξ '' + ξ ' 2 2 +2 ξ ' r + 2 r 2 ) = 0 , (3.37) \nand it is important to highlight that the conservation equation (2.26) remains a linear combination of the system (3.35)-(3.34). The general solution for Eq. (3.37) is given by \nf ∗ ( r ) = 1 -2 M/r 2 r -3 M /lscript c , (3.38) \nwith /lscript c a constant with units of a length. Thus the conformally deformed Schwarzschild exterior becomes \nwhere /lscript = α/lscript c , and its behaviour for r /greatermuch M is given by \ne -λ = ( 1 -2 M r )( 1 + /lscript 2 r -3 M ) , (3.39) \ne -λ /similarequal 1 -4 M -/lscript 2 r . (3.40) \nThe causal structure for this geometry is now more involved. We still have the Killing horizon of the Schwarzschild metric at r H = 2 M , but e -λ diverges for \nr c = 3 M 2 , (3.41) \nand there is a second zero of e -λ at \nr 0 = 3 M -/lscript 2 = r c -/lscript 2 . (3.42) \nWe can thus rewrite the radial metric component as \ne -λ = ( 1 -r H r ) ( r -r 0 r -r c ) . (3.43) \nNote that r c < r H but, depending on the sign ands size of /lscript , the second zero could occur inside or outside the critical radius r c and the Killing horizon r H . \nIn order to clarify the nature of the above solution, we compute explicitly the effective density \n˜ ρ = αθ 0 0 = -/lscript M 4 k 2 ( r -r c ) 2 r 2 , (3.44) \nthe effective radial pressure \n˜ p r = -αθ 1 1 = /lscript 2 k 2 ( r -r c ) r 2 , (3.45) \nand the effective tangential pressure \n˜ p t = -αθ 2 2 = /lscript ( r -M ) 4 k 2 ( r -r c ) 2 r 2 . (3.46) \nThe anisotropy is thus given by \nΠ = /lscript (3 r -4 M ) k 2 (2 r -3 M ) 2 r 2 . (3.47) \nThe first thing we notice is that the density and pressures are regular on both r H and r 0 , but diverge at r = r c < r H , which is therefore a real singularity, albeit hidden inside the Killing horizon. \nWe can then assume the black hole space-time is represented by the range r > r c , for which we must require that the region r c < r < r H has the proper signature, as discussed previously. This means that we must have r 0 ≤ r c , or \n/lscript > 0 , (3.48) \nwith /lscript = 0 of course representing the pure Schwarzschild geometry. We conclude that the conformal geometry in (3.39) represents a black hole solution with outer horizon r H = 2 M , and primary hairs represented by the parameter /lscript , which is constrained by the regularity condition (3.48). A solution similar to that in (3.39) was found in the context of the extra-dimensional brane-world [57].", '3.3 Barotropic equation of state': "If the additional source is a polytropic fluid, it should satisfy the equation of state \n˜ p r = K ˜ ρ Γ , (3.49) \nwith Γ = 1 + 1 /n , where n is the polytropic index and K > 0 denotes a parameter which contains the temperature implicitly and is governed by the thermal characteristics of a given polytrope. For instance, the ultrarelativistic degenerate Fermi gas has polytropic index n = 3, while the non-relativistic degenerate Fermi gas is found for n = 3 / 2. (for more details, see for instance Refs. [58-62]). However, due to the unknown nature of the source θ µν , we will include the possibility that K < 0. From Eqs. (2.10) and (2.11) with ρ = p = 0, we then obtain \n-αθ 1 1 = K αθ 0 0 ) Γ . (3.50) \n( \n) By using Eqs. (2.23) and (2.24) in the expression (3.50) we obtain a first order non-linear differential equation for the MGD function, \nf ∗' + f ∗ r = -1 K 1 / Γ ( k 2 r α ) 1 -1 / Γ ( f ∗ r -2 M ) 1 / Γ . (3.51) \nWe immediately notice that the right hand side is well-defined for a generic Γ only provided f ∗ / ( r -2 M ) > 0. \nIn order to proceed, we thus consider the simplest case Γ = 1, so that Eq. (3.50) becomes a barotropic equation of state. This corresponds to an isothermal self-gravitating sphere of gas and is thus more appropriate for our purpose. It is worth mentioning that this self-gravitating sphere can also describe the collisionless system of stars in a globular cluster. The geometric deformation for Γ = 1 and r > 2 M simplifies to \nf ∗ ( r ) = ( 1 -2 M r ) -1 /K ( /lscript p r ) 1+1 /K , (3.52) \nwhere /lscript p > 0 is a length, and the MGD radial component of the metric reads \ne -λ = ( 1 -2 M r ) [ 1 + α ( /lscript p r -2 M ) 1+1 /K ] , (3.53) \nagain for r > 2 M . We also note that asymptotic flatness at r →∞ requires K ≤ -1, with K = -1 yielding the pure Schwarzschild metric. \nNext, we note that the effective density is given by \nk 2 ˜ ρ = α Kr 2 ( /lscript p r ) 1+1 /K ( 1 -2 M r ) -1 -1 /K , (3.54) \nand diverges at r = 2 M unless -1 < K < 0. Of course, the effective pressure ˜ p r = K ˜ ρ also diverges at r = 2 M unless -1 < K < 0. The effective tangential pressure is given by \nk 2 ˜ p t = -αθ 2 2 = -α ( K +1) 2 Kr 2 ( 1 -M r )( /lscript p r ) 1+1 /K ( 1 -2 M r ) -2 -1 /K , (3.55) \nwhich also diverges at at r = 2 M unless -1 / 2 < K < 0. To summarise, the surface r = r H is a real singularity unless -1 / 2 < K < 0. However, this is not compatible with the asymptotically flat conditions, which requires K ≤ -1. We therefore conclude that the Killing horizon at r = r H = 2 M has become a real singularity, which is not hidden inside a larger horizon.", '3.4 Linear equation of state': "Now let us consider a generic equation of state in the form \nθ 0 0 = aθ 1 1 + b θ 2 2 , (3.56) \nwith a and b constants. The conformal case of Section 3.2 is represented by the set a = -1 and b = -2, whereas the polytropic Γ = 1 (barotropic) case of Section 3.3 is given by a = -1 /K and b = 0. Eqs. (2.23)-(2.25) then yield the differential equation for the MGD function \nf ∗ ' [ 1 r -b 4 ( ξ ' + 2 r )] + f ∗ [ 1 r 2 -a ( 1 r 2 + ξ ' r ) -b 4 ( 2 ξ '' + ξ ' 2 +2 ξ ' r )] = 0 , (3.57) \nwhose general solution for r > r H = 2 M is given by \nf ∗ ( r ) = ( 1 -2 M r )( /lscript r -BM ) A , (3.58) \nwhere /lscript is a positive constant with dimensions of a length, and \nA = 2( a -1) b -2 > 0 (3.59) \nB = b -4 b -2 , (3.60) \nwith b /negationslash = 2 and the condition A > 0 required by asymptotic flatness. Therefore the solution reads \ne -λ = ( 1 -2 M r ) [ 1 + α ( /lscript r -BM ) A ] , (3.61) \nwhich again shows the horizon at r H = 2 M , beside a possible divergence at r = r c and a second zero at r = r 0 , like in the previous cases. \nThe physical content of the system is again clarified by the explicit computation of the effective density \n˜ ρ = αθ 0 0 = -α k 2 r 2 ( /lscript r -BM ) A [ 1 -A ( r -2 M r -BM )] , (3.62) \nthe effective radial pressure \n˜ p r = -αθ 1 1 = α k 2 r 2 ( /lscript r -BM ) A , (3.63) \nand the effective tangential pressure \n˜ p t = -αθ 2 2 = -αA 2 k 2 r 2 /lscript ( /lscript r -BM ) A +1 ( r -M ) . (3.64) \nAgain, we see that the effective density and effective pressures diverge at \nr c = BM , (3.65) \nwhich represents a true singularity at 0 < r c < r H for 0 < B < 2, that is for \nb < 0 or b > 4 . (3.66) \nFor B > 2 (that is, 0 < b < 2), this singularity occurs outside the Killing horizon, r c > r H , and this case cannot be considered any further. Secondly, the effective density and effective pressures satisfy \n˜ p t = -1 2 ( r -M r -2 M ) (˜ ρ + ˜ p r ) , (3.67) \nshowing that ˜ p t < 0 when both ˜ ρ and ˜ p r are positive. We thus conclude that at least one of the thermodynamic variables will always be negative as long as the equation of state is linear. Moreover, the effective radial and tangential pressure are related by \n˜ p t = -A 2 ( r -M r -BM ) ˜ p r . (3.68) \nSince A > 0, we conclude that the two pressures always have opposite signs and one of them will be negative. On the other hand, the effective density and effective radial pressure are related by \n˜ ρ = [ A ( r -2 M r -BM ) -1 ] ˜ p r , (3.69) \nhence \n˜ ρ ∼ { -˜ p r for r ∼ 2 M ( A -1) ˜ p r for r /greatermuch 2 M . (3.70) \nSince A > 0, the asymptotic behaviour in Eq. (3.70) demands 0 < A ≤ 1 in order to ensure that the density does not change its sign in the region 2 M < r < ∞ [the pressure (3.63) always has the same sign inside this region]. We conclude that the dominant energy condition ˜ ρ ≥| ˜ p r | cannot be satisfied with a linear equation of state of the form displayed in Eq. (3.56). Nonetheless, the effective density is positive everywhere if α < 0 2 . \nFor 2 < b < 4 one has r c < 0 and there is no extra singularity beside the usual Schwarzschild one at r = 0. In this case, we must demand that no second zero r 0 > 0 of e -λ exists, otherwise the space-time signature would become unacceptable inside a portion of r > 0. This condition is immediately satisfied if α > 0, for any A > 0, that is for a > 1. We next notice that there is a second zero of e -λ at \nr 0 = BM + /lscript ( -α ) 1 /A > r c , (3.71) \nwhen α < 0. To have a proper black hole solution, this second zero r 0 must lie inside the relevant singularity. If 2 < b < 4, the relevant singularity occurs at r = 0 an we must have \nr 0 ≤ 0 , (3.72) \nthat is, if /lscript and | α | are small enough to satisfy \n/lscript ( -α ) 1 /A ≤ -BM . (3.73) \nOtherwise, if b < 0 or b > 4, the relevant singularity occurs at 0 < r c < r H , but r 0 > r c makes this case unsuitable. \nThe final conclusion is thus that the linear equation of state (3.56) always produces black holes (with a Schwarzschild singularity at r = 0) if 2 < b < 4 and a > 1, provided α > 0 or α < 0 and Eq. (3.73) holds.", '3.5 A particular solution with no extra singularity': 'The reader can see that Eq. (3.56) leads to a system very rich in possibilities, whose generic solutions 3 are given in Eqs. (3.61)-(3.64), and whose general analysis is detailed throughout Eqs. (3.65)(3.73). The main feature of these solutions is that they do not satisfy the dominant energy condition. In this respect, let us recall that the energy conditions are a set of constraints which are usually imposed on the energy-momentum tensor in order to avoid exotic matter sources, hence they can be viewed as sensible guides to avoid unphysical situations. However, it is well-known that these \nFigure 2: Case b = 3. Metric components for α = -0 . 7 and a = 1 . 4 (black lines) compared to the Schwarzschild component g -1 rr (gray line). The mass M = 1. \n<!-- image --> \nenergy conditions might fail for particular classical systems which are still reasonable [63]. In our case we are dealing with a gravitational source θ µν whose main characteristic is that it only interacts gravitationally with the matter that, by itself, would source the (hairless) black hole solution (2.27). Hence, one should not exclude a priori that such matter is a kind of exotic source (as indeed the conjectured dark matter is expected to be). \nOf all the possible solutions, we shall here analyse the particular case b = 3 (with a > 1 for asymptotic flatness) as an example of space-time which does not contain any extra singularity beside the usual Schwarzschild one at r = 0. The radial metric component is obtained from Eq. (3.61) and reads \ne -λ = ( 1 -2 M r )[ 1 + α ( r + M ) 2 ( a -1) ] , (3.74) \nwhich makes it immediately clear that there is no second divergence. In fact, the effective density is given by 4 \n˜ ρ = αθ 0 0 = α k 2 r 2 [ 2 a ( r -2 M ) -3( r -M ) ( r + M ) 2 a -1 ] , (3.75) \nthe effective radial pressure by \n˜ p r = -αθ 1 1 = α k 2 r 2 ( r + M ) 2 ( a -1) , (3.76) \nand the effective tangential pressure by \n˜ p t = -αθ 2 2 = -α ( a -1) ( r -M ) k 2 r 2 ( r + M ) 2 ( a -1) . (3.77) \nThe reader can easily check that the deformed Schwarzschild metric (3.28) with g rr = e -λ in Eq. (3.74), along with the source terms in Eqs. (3.75)-(3.77), solve the complete Einstein equations (2.6)-(2.8) with ρ = p = 0. \nCombining the expressions (3.75) and (3.76), we get \n˜ ρ = [ 2 a ( r -2 M ) -3( r -M ) ( r + M ) ] ˜ p r , (3.78) \nFigure 3: Case b = 3. Effective source terms { ˜ ρ, ˜ p r , ˜ p t } × 10 3 for α = -0 . 7 and a = 1 . 4. The horizon r H = 2 M and the mass M = 1. \n<!-- image --> \nfrom which we obtain the asymptotic behaviours \n˜ ρ ∼ -˜ p r for r ∼ 2 M (2 a -3) ˜ p r for r /greatermuch 2 M . (3.79) \nTherefore, when α < 0, the pressure ˜ p r < 0 and the effective energy will always be positive for r > 2 M whenever a ≤ 3 / 2. Figs. 2 and 3 show the corresponding metric elements and density and pressures in (3.75)-(3.77) respectively for α = -0 . 7 and a = 1 . 4.', '4 Conclusions': "By making use of the MGD-decoupling approach, we have presented in detail how the Schwarzschild black hole is modified when the vacuum is filled by a generic spherically symmetric gravitational fluid, described by a 'tensor-vacuum' θ µν , which does not exchange energy-momentum with the central source. For this purpose, we have separated the Einstein field equations into i) the Einstein equations for the spherically symmetric vacuum ρ = p = 0 and ii) the 'quasi-Einstein' system in Eqs. (2.23)-(2.25) for the spherically symmetric 'tensor-vacuum' θ µν . Following the MGD procedure, the superposition of the Schwarzschild solution found in i) plus the solution for the 'quasi-Einstein' system in ii), has led to the solution for the complete system 'Schwarzschild + tensor-vacuum.' \nThe quasi-Einstein system (2.23)-(2.25) was solved by providing some physically motivated equations of state for the source θ µν . In this respect, four different scenarios were considered, namely, i) the isotropic θ 1 1 = θ 2 2 ; ii) the conformal θ µ µ = 0; iii) the polytropic αθ 1 1 = K ( αθ 0 0 ) Γ and iv) the generic linear equation of state in (3.56). In the isotropic case, we only found a metric which is not asymptotically flat for r → ∞ , which means that the tensor-vacuum for a black hole cannot be isotropic as long as its interaction with regular matter is purely gravitational. On the other hand, the conformal case leads to the hairy black hole solution in Eq. (3.39), whose primary hairs is represented by the length /lscript , which is constrained by the regularity condition (3.48). Among all polytropic equations of state, we have only considered the barotropic Γ = 1, which represents a tensor-vacuum made of an isothermal self-gravitating sphere of gas. This leads to \nthe exterior solution in Eq. (3.53) endowed with the parameters { M,α,/lscript p , K } . Since the Killing horizon r = r H = 2 M becomes a real singularity, this solution may represent the exterior of a self-gravitating system of mass M and radius R > r H but not a black hole solution. \nFinally, we have analysed the generic linear equation of state in Eq (3.56), which includes both the conformal and barotropic fluids as particular cases. This leads to the solution in Eq. (3.61), showing that even a simple linear equation of state may yield hairy black hole solutions with a rich geometry described by the parameters { M,α,/lscript, a, b } , where { α, /lscript, a, b } represents a potential set of charges generating primary hairs. In this context, a particular black hole solution with primary hairs { α, a } was found in Eq. (3.74), whose main characteristic is the absence of other singularities in the region 0 < r < ∞ . \nAll the black holes solutions mentioned above have the horizon at r H = 2 M and primary hairs represented by a number of free parameters. However, these parameters can be restricted by demanding i) the correct asymptotic behaviour and ii) regularity conditions for black hole solutions free of pathologies. In this respect, there are always a potential singularity r c and a possible second horizon r h in our solutions. In order to have a proper black hole, it is necessary that r c ≤ r H to avoid a naked singularity, and r h = r H to have a metric with a proper signature. We emphasize that r h > r H yields both g tt and g rr positive inside the region r H < r < r h . All these conditions yields restrictions on potential primary hairs. For instance, the linear equation of state (3.56) always produces black holes if 2 < b < 4 and a > 1, provided α > 0 or α < 0 and Eq. (3.73) holds. \nWe have shown that different characteristics of the gravitational source lead to different hairy black hole solutions. Therefore, the compatibility between some of these solutions and the observations could determine the main features of the tensor-vacuum, and eventually the fundamental field(s) that constitute it. Finally, we would like to emphasize that the non-existence of an isotropic tensor-vacuum that does not exchange energy-momentum with regular matter favours scenarios with Klein-Gordon type fields φ , which naturally induce anisotropy in the Einstein field equations. These scalar fields are found in a large number of alternative theories to general relativity.", '5 Acknowledgements': 'J.O. and S.Z. have been supported by the Albert Einstein Centre for Gravitation and Astrophysics financed by the Czech Science Agency Grant No.14-37086G. R.C. is partially supported by the INFN grant FLAG and his work has been carried out in the framework of GNFM and INdAM and the COST action Cantata . R.dR. is grateful to CNPq (Grant No. 303293/2015-2), and to FAPESP (Grant No. 2017/18897-8) for partial financial support. A.S. is partially supported by Project Fondecyt 1161192, Chile.', 'References': "- [1] B. P. Abbott et al. [LIGO Scientific and Virgo Collaborations], Phys. Rev. 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2018PASJ...70S..37G
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Galaxy interactions trigger rapid black hole growth: An unprecedented view from the Hyper Suprime-Cam survey
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2018-01-01
| 38 | 0.53 | 161 |
['galaxies active', 'galaxies evolution', 'galaxies interactions', '-', '-']
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[]
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Collisions and interactions between gas-rich galaxies are thought to be pivotal stages in their formation and evolution, causing the rapid production of new stars, and possibly serving as a mechanism for fueling supermassive black holes (BHs). Harnessing the exquisite spatial resolution (∼0{^''<SUB>.</SUB>}5) afforded by the first ∼170 deg<SUP>2</SUP> of the Hyper Suprime-Cam (HSC) survey, we present our new constraints on the importance of galaxy-galaxy major mergers (1 : 4) in growing BHs throughout the last ∼8 Gyr. Utilizing mid-infrared observations in the WISE all-sky survey, we robustly select active galactic nuclei (AGN) and mass-matched control galaxy samples, totaling ∼140000 spectroscopically confirmed systems at i < 22 mag. We identify galaxy interaction signatures using a novel machine-learning random forest decision tree technique allowing us to select statistically significant samples of major mergers, minor mergers / irregular systems, and non-interacting galaxies. We use these samples to show that galaxies undergoing mergers are a factor of ∼2-7 more likely to contain luminous obscured AGN than non-interacting galaxies, and this is independent of both stellar mass and redshift to z < 0.9. Furthermore, based on our comparison of AGN fractions in mass-matched samples, we determine that the most luminous AGN population (L<SUB>AGN</SUB> ≳ 10<SUP>45</SUP> erg s<SUP>-1</SUP>) systematically reside in merging systems over non-interacting galaxies. Our findings show that galaxy-galaxy interactions do, on average, trigger luminous AGN activity substantially more often than in secularly evolving non-interacting galaxies, and we further suggest that the BH growth rate may be closely tied to the dynamical time of the merger system.
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https://arxiv.org/pdf/1706.07436.pdf
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{'Galaxy Interactions Trigger Rapid Black Hole Growth: an unprecedented view from the Hyper Suprime-Cam Survey': 'Andy D. GOULDING 1 , Jenny E. GREENE 1 , Rachel BEZANSON 1 , Johnny GRECO 1 , Sean JOHNSON 1 , Alexie LEAUTHAUD 2 , Yoshiki MATSUOKA 3 , Elinor MEDEZINSKI 1 and Adrian M. PRICE-WHELAN 1 \n- 1 Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ 08544, USA.\n- 2 Department of Astronomy and Astrophysics, University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA 95064, USA\n- 3 Research Center for Space and Cosmic Evolution, Ehime University, Matsuyama, Ehime 790-8577, Japan\n- ∗ E-mail: [email protected] \nReceived ; Accepted', 'Abstract': 'Collisions and interactions between gas-rich galaxies are thought to be pivotal stages in their formation and evolution, causing the rapid production of new stars, and possibly serving as a mechanism for fueling supermassive black holes (BH). Harnessing the exquisite spatial resolution ( ∼ 0.5 arcsec) afforded by the first ∼ 170 deg 2 of the Hyper Suprime-Cam (HSC) Survey, we present our new constraints on the importance of galaxy-galaxy major mergers (1:4) in growing BHs throughout the last ∼ 8 Gyrs. Utilizing mid-infrared observations in the WISE All-Sky survey, we robustly select active galactic nuclei (AGN) and mass-matched control galaxy samples, totaling ∼ 140,000 spectroscopically confirmed systems at i < 22 mag. We identify galaxy interaction signatures using a novel machine-learning random forest decision tree technique allowing us to select statistically significant samples of major-mergers, minormergers/irregular-systems, and non-interacting galaxies. We use these samples to show that galaxies undergoing mergers are a factor ∼ 2 -7 more likely to contain luminous obscured AGN than non-interacting galaxies, and this is independent of both stellar mass and redshift to z < 0 . 9 . Furthermore, based on our comparison of AGN fractions in mass-matched samples, we determine that the most luminous AGN population ( L AGN /greaterorsimilar 10 45 erg s -1 ) systematically reside in merging systems over non-interacting galaxies. Our findings show that galaxy-galaxy interactions do, on average, trigger luminous AGN activity substantially more often than in secularly evolving non-interacting galaxies, and we further suggest that the BH growth rate may be closely tied to the dynamical time of the merger system. \nKey words: galaxies: active; galaxies: evolution; galaxies: interacting', '1 Introduction': "The connections between galaxy-galaxy interactions and the triggering and/or presence of accreting supermassive black \nholes (hereafter, active galactic nuclei; AGN) are a matter of significant on-going debate. In the broad scope of galaxy evolution, there are many compelling theoretical reasons to expect \na connection between the encounters of two (or more) gas-rich galaxies with similar (1: < 4 -5 ) stellar masses, and the accretion of material onto at least one of the BHs present in these systems (e.g., Volonteri et al. 2003; Hopkins et al. 2005; Di Matteo et al. 2005; Springel et al. 2005). Concurrent BH growth and the rapid production of new stars (e.g., Somerville et al. 2008; Angl'es-Alc'azar et al. 2013) can naturally give rise to known correlations between the BH mass and galaxy properties, such as the bulge mass and stellar velocity dispersion (e.g., Magorrian et al. 1998; Ferrarese & Merritt 2000; Tremaine et al. 2002; Gultekin et al. 2009; McConnell & Ma 2013). Furthermore, self-regulation of the AGN activity, due to socalled 'quasar mode' feedback processes, can serve as a violent mechanism capable of disrupting the on-going star-formation by depositing energy back into the galaxy merger, heating the gas and/or even expelling material out into the wider dark matter halo. Indeed, AGN feedback is a widely accepted solution for the formation of massive quiescent early-type galaxies, and the build-up of the red-sequence, in cosmological simulations. \nAparticularly persuasive argument for a connection between the most rapidly growing BHs and major-mergers is that galaxy interactions provide a simple solution to the 'angular momentum problem'. In principle, growing a BH requires only a source of cool gas to fuel the nucleus, supplies of which are typically plentiful in the host galaxy. However, continuously transporting significant quantities of this material from the gas reservoirs in the host, down to scales in which it can accrete onto the BH, while simultaneously dissipating the specific angular momentum of the gas, is a non trivial issue. Models of galaxy-galaxy mergers show that tidal forces between the galaxies can cause gas to be subject to substantial gravitational torques, resulting in the efficient loss of angular momentum, ultimately causing substantial gas flow towards the BH, and igniting a powerful AGN (e.g., Barnes & Hernquist 1991; Mihos &Hernquist 1996; Di Matteo et al. 2008; Angl'es-Alc'azar et al. 2017). Despite the theoretical successes of AGN-galaxy coevolution models, observational evidence for a connection between merging galaxies, galaxy instabilities, and the enhanced presence of AGN activity, is still inconclusive. \nRecent observations of luminous ( L AGN > 10 46 erg s -1 ) dust reddened z ∼ 1 -2 quasars have revealed these AGN to be overwhelmingly hosted by galaxy mergers (e.g., Urrutia et al. 2008; Glikman et al. 2012, 2015), possibly suggesting that the most luminous BH growth is increasingly likely to be triggered by galaxy interactions (e.g., Treister et al. 2012; Fan et al. 2016). However, others do not observe a rise in the incidence of mergers at the highest AGN luminosities (e.g., Schawinski et al. 2012; Villforth et al. 2014). Furthermore, at similar redshifts, AGN with more moderate luminosities ( L AGN ∼ 10 43 -10 44 erg s -1 ) also appear no more likely to show interaction signatures than non-AGN systems (e.g., Cisternas et al. 2011; \nSchawinski et al. 2011; Kocevski et al. 2012). Similarly, at z < 1 , large-scale galaxy interaction signatures such as mergers, and galaxy-scale bars and instabilities, do not appear to significantly boost the likelihood of hosting a lower luminosity AGN ( L AGN /lessorsimilar 10 43 erg s -1 ; Athanassoula 1992; Ho et al. 1997; Regan & Mulchaey 1999; Cisternas et al. 2015; Cheung et al. 2015; Goulding et al. 2017). By contrast, others find evidence supporting a correlation between merging and AGN in some more nearby galaxies (e.g., Koss et al. 2010; Ellison et al. 2011b; Silverman et al. 2011; Ellison et al. 2013; Satyapal et al. 2014a; Hong et al. 2015; Weston et al. 2017). \nTypically, studies have focused on selecting large samples of AGN, and then comparing the host galaxies of these AGN to non-AGN systems. However, AGN activity is a stochastic process that is believed to vary on timescales far shorter than changes related to galaxy-wide processes (e.g., morphology, star-formation). AGN variability may therefore cause dilution of would-be significant correlations between average BH accretion and on-going star-formation (e.g., Chen et al. 2013; Hickox et al. 2014) and/or stellar mass (Yang et al. 2017). Further complicating the observational view-point, the importance of galaxy interactions for triggering AGN may also be obscuration dependent, as well as merger-stage dependent (e.g., Kocevski et al. 2015; Koss et al. 2016; Weston et al. 2017; Ricci et al. 2017). More often than not, these previous investigations were hampered by the ability to sample significant numbers of AGN and mergers that cover large dynamic ranges in AGN luminosity, and for the more high-redshift studies, the ability to accurately identify AGN or robustly detect the presence of galaxy interaction signatures. Progress in the field can therefore be made by bridging the gap between the low and high-redshift studies, through the construction of large samples of merging and nonmerging galaxies with deep high-resolution imaging to z ∼ 1 , which simultaneously encompass statistically significant populations of moderate to extremely luminous AGN. \nUsing dedicated telescopes, wide-format surveys such as the Sloan Digital Sky Survey (SDSS) carried out comprehensive multi-band imaging surveys of significant fractions of the sky. These surveys have been incredibly successful in characterizing the properties of extremely large galaxy/AGN samples (e.g., Strateva et al. 2001; Vanden Berk et al. 2001; Strauss et al. 2002; Eisenstein et al. 2005; Ross et al. 2013). Owing to mirror size and total integration times these surveys were necessarily limited to the relatively nearby Universe ( z < 0 . 2 ), while still encompassing large survey volumes of V ∼ 0 . 2( h -1 Gpc) 3 . Following in the footsteps of SDSS, the next generation of wide-format imaging surveys, capable of providing SDSS-like volumes and imaging quality out to z = 1 are beginning to take shape. The on-going Hyper Suprime-Cam (HSC) survey (Aihara et al. 2017b) is now providing an unprecedented new view of the Universe. The combination of the wide field of view \nand large 8.2 meter mirror diameter provided by the Subaru Telescope gives the HSC survey exquisite sensitivity and resolving power. Upon completion, the Wide survey layer of HSC will image ∼ 1400 deg 2 in grizy to a depth of i ∼ 26 mag and with a typical i -band seeing of ∼ 0 . 5 '' , less than half that of the median seeing in SDSS ( ∼ 1 . 4 '' ). Given that the angular diameter increases by only a factor ∼ 2 . 5 from z =0 . 2 to z =1 , HSC is now allowing the exploration of galaxy morphologies with SDSS-like precision in SDSS-like survey volumes out to z ∼ 1 . \nHere we harness the unprecedented sensitivity of the first 170 deg 2 of the HSC survey combined with complementary all-sky data available from Wide-field Infrared Survey Explorer (WISE; Wright et al. 2010) to explore the incidence of midinfrared (mid-IR; λ ∼ 3 -100 µ m) identified AGN in merging galaxies out to z ∼ 0 . 9 as a function of the AGN host galaxy properties. In Section 2 we define our spectroscopic sample of massive galaxies that have been observed as part of the HSC survey. In Section 3, we outline the modeling of the spectral energy distributions for the sample to determine their restframe photometry and intrinsic properties, in order to match our galaxy sample in color and stellar mass, and we utilize the ALLWISE catalog to identify those galaxies containing luminous AGN. In Section 4, we describe our novel implementation of a machine learning algorithm to identify interacting and non-interacting galaxies by harnessing the HSC imaging. In Section 5 we present the incidence of AGN in interacting and non-interacting galaxies, finding that AGN are, on average, at least a factor /greaterorsimilar 3 more abundant in merging systems, and the most luminous AGN at fixed stellar mass are preferentially found in merging galaxies. In Section 6 we discuss the implication of our results, and outline a framework, linking the observed AGN fractions to the dynamical time of the merger system. Our concluding remarks are presented in Section 7. All magnitudes are in the AB system, unless otherwise stated. Throughout we assume a standard flat Λ CDM cosmology with H 0 =70 km s -1 Mpc -1 and Ω M =0 . 3 .", '2.1 The Hyper Suprime-Cam Survey': "The Hyper Suprime-Cam (HSC) Survey is an ambitious 300 night imaging survey undertaken as part of the Subaru Strategic Program (SSP; Aihara et al. 2017b). The HSC survey is designed to provide nested wide-field multi-band imaging over a total of ∼ 1400 deg 2 using the HSC instrument on the Subaru 8.2m telescope on Mauna Kea in Hawaii. HSC is constructed of 116 (104 science detectors) Hamamatsu Deep Depletion 2K × 4K CCDs, with a 1.77 deg 2 field-of-view (FOV) and has an instrumental Point-Spread Function (PSF) of D 80 < 0.2 '' (80% enclosed light fraction) over the entire FOV across all imaging filters. \nFig. 1. Top: Distribution of spectroscopically identified galaxies in our parent sample as a function of spectroscopic survey (SDSS-II Legacy, SDSS-III BOSS, PRIMUS, VIPERs, GAMA and Wiggle-Z). Middle: Fractional contribution of a given spectroscopic survey to our parent galaxy sample as a function of redshift for all galaxies within our parent sample, and for the starforming galaxy sample after applying the UVJ cut shown in Fig 3. Bottom: Spectroscopic sample as a function of i -band magnitude. \n<!-- image --> \nThe HSC survey consists of three survey layers: the Wide layer covers a solid angle of 1400 deg 2 in grizy filters to a depth of r ≈ 26 mags (5 σ , point source). The 27 deg 2 Deep layer reaches r ≈ 27 mags, with the addition of three narrowband filters at λ ∼ 3870 ,8160 and 9210 ˚ A, and the 3.5 deg 2 Ultradeep layer is a further ≈ 1 mag fainter than Deep , allowing detection of Ly α emitters to z ≈ 7 . In addition, the HSC Survey fields were carefully constructed to overlap with existing multi-wavelength survey fields, e.g., millimeter data from the Atacama Cosmology Telescope (ACT); X-ray data from multiple XMM-Newton and Chandra; near-/mid-infrared imaging surveys such as VIKING/VIDEO; UKIDSS; Spitzer and WISE ; and optical spectroscopic surveys such as SDSS Legacy/BOSS, PRIMUS (Coil et al. 2011), VIPERS (Guzzo et al. 2014), GAMA ((Liske et al. 2015), Wiggle-Z (Drinkwater et al. 2010), COSMOS (Lilly et al. 2009), and DEEP2 (Newman et al. 2013). For the specific region centers of the three layers that make up the HSC-SSP survey, we refer the reader to Aihara \net al. 2017b.", '2.3 Selecting bright ( i < 22 mag) galaxies in HSC': "The data used throughout this manuscript is based on an internal release of the Wide layer data, release S16A, and covers ∼ 170 deg 2 . Basic data processing, including bias and background subtraction, flat-fielding, astrometric calibration, individual exposure co-addition, and object detection was performed using hscPipe v4.0.1 , which is an HSC-specific derivative of the Large Synoptic Survey Telescope (LSST) processing pipeline. For further details regarding hscPipe and the HSC-SSP data releases, see Bosch et al. (2017) and Aihara et al. (2017a). For our analyses we use the source catalogs, images, and other relevant data derived from the co-added HSC images produced by hscPipe . The image co-adds are shifted to a common World Co-ordinate System (WCS), and have a pixel scale of 0 . 168 '' .", '2.2 Outline of this manuscript': "Our primary goal is to constrain the effects of gas-rich merging of galaxies on the growth of BHs out to z /lessorsimilar 1 . To achieve this we require: \n- 1. deep, high spatial resolution optical imaging (from HSC) of a large parent sample of galaxies spectroscopically confirmed to be in the redshift range 0 . 1 <z < 0 . 9 , from which we can robustly identify large sub samples of interacting and non-interacting galaxies. See Section 2.3.\n- 2. the intrinsic properties of the parent galaxy sample, such as rest-frame colors, stellar masses, star-formation rates, derived using SED fitting (see Section 3.1). These measurements allow us to perform property-matched tests between different interaction-state systems, such as stellar mass matching, and rest-frame color matching, using diagnostics such as the UVJ diagram (see Section 3.2).\n- 3. a homogeneous obscuration independent selection of AGN within our parent sample. We achieve this through the use of mid-infrared color diagnostics performed on photometry obtained from the WISE All-sky survey. See Section 3.3.\n- 4. an accurate automated method for classifying signatures of recent/on-going merger events. We achieve this through a novel implementation of a Random Forest machine learning algorithm, trained on a large sample of visually identified mergers and non-mergers within the HSC i -band imaging. See section 4.2. \nIn Section 5, we present the results of our investigation, and show conclusive statistical evidence that BHs hosted by merging galaxies are at least three times more likely to be rapidly growing at high-Eddington ratios than a mass-matched sample of non-interacting galaxies. This suggests not only that AGN are triggered by merging, but also the rapid growth of the BH(s) can be sustained during the merger event. \nIn this section we describe our selection techniques in order to construct samples of interacting and non-interacting galaxies with firm spectroscopic redshifts. In Section 5, we use these galaxy samples to assess the importance of galaxy interactions on the growth of BHs to z < 1 . Our parent galaxy sample contains all objects with i < 22 . 3 Kron magnitudes in the S16A data release of the HSC survey. We set the detect is tract inner , detect is patch inner , detect is primary and is extended data flags on the sample, as we require the most complete galaxy sample available within the HSC database 1 . Our choice of flux limit derives from our ability to (1) recover the source morphologies and robustly identify interacting galaxy features such as disturbances, irregular morphologies, tidal tails and bridges out to z < 0 . 9 (a detailed analysis of the HSC galaxy morphologies and comparison to Hubble Space Telescope data will be presented in a future publication; Goulding et al. in prep.), (2) the completeness of spectroscopic catalogs within the survey regions; and (3) the addition of a systematic uncertainty of ± 0 . 3 mags due to difficulty in measuring photometry of merging systems. \nEach source extracted from the HSC database is then crossmatched to within < 1 '' with the publicly available spectroscopic redshift (specz ) catalogs pertaining to the survey skyregions. The median separation between the HSC and the specz position is ∼ 0 . 13 '' . Specifically, we harness specz measurements from the SDSS Legacy Catalog (complete to r < 17 . 77 ), the SDSS-DR12 BOSS survey (color-selected galaxies, and approximately stellar mass limited; see Dawson et al. 2013; Maraston et al. 2013; Reid et al. 2016; Leauthaud et al. 2016), the GAMA-DR2 survey (complete to r < 19 . 0 ), the PRIMUS survey (complete to i < 22 . 5 ; Coil et al 2011), the WiggleZ Dark Energy survey ( 20 . 0 <r < 22 . 5 ; Drinkwater et al. 2009) and the first data release of the VIMOS Public Extragalactic Redshift Survey (VIPERs; i < 22 . 5 ; Garilli et al. 2014; Guzzo et al. 2014). \nOur requirement of a detected i < 22 . 3 mag source in HSCS16A WIDE, as well as a publicly available spectroscopic redshift (within at least one of the aforementioned surveys), and HSC imaging with a well characterized point spread function, results in a combined area of ∼ 170 deg 2 , and a galaxy catalog containing 140,158 unique galaxies at 0 . 1 < z < 0 . 9 . Similar to our brightness limit, the imposed redshift limits are based on the bright photometry limit for HSC (see Bosch et al. 2017), the targeted completeness limits for the spectroscopic redshift sur- \n- 1 We do not make use of the HSC Bright Star Masks. Due to the substantially poorer PSF in WISE, galaxies with HSC photometry that are contaminated by foreground stars are typically not identified in WISE, and hence our excluded during our HSC-WISE cross-match outlined in Section 3.3. Furthermore, the inclusion of the HSC Bright Star Mask would potentially mask bright/large galaxies, which would bias our sample against the most massive merging systems. \nveys, and our ability to accurately determine the morphological classifications, and identify low-surface brightness tidal-tails of the systems from the WIDE-depth (i < 25.9 mag) HSC imaging at higher redshifts. \nIn Figure 1 we present the breakdown of the spectroscopic redshift distributions for our parent sample as a function of the redshift survey from which the specz originated, and as function of the source brightness in the i -band. It is clear that at any given redshift our total sample is dominated by 2-3 of the redshift surveys. For example, at lower redshifts ( z < 0 . 3 ), our sample is mainly composed of objects selected from the SDSSLegacy and GAMA surveys, while at redshifts 0 . 4 <z< 0 . 6 , the sample is driven by SDSS-BOSS systems, with roughly equal sub-dominant contributions from VIPERs and PRIMUS galaxies. 2 While our sample is by selection heterogeneous, and contains a range of selection functions for the different surveys, we will demonstrate directly in Section 5.2 that our results are not sensitive to the details of the different samples.", '3.1 SED modeling using FAST': 'In this section, we use a suite of available photometry, in conjunction with the spectroscopic redshifts to derive the UV to IR spectral energy distributions (SEDs) for our galaxies in our specz sample, defined in Section 2.3. We use SED modeling to derive physical properties such as the stellar mass ( M ∗ ), starformation rate (SFR), and dust extinction ( A V ). We use these derived measurements in Section 5 to produce M ∗ -matched samples of interacting and non-interacting galaxies. \nIn order to accurately constrain the galaxy light blueward of the 4000 ˚ A break, for objects in our lowest redshift bin z < 0 . 3 , we require u -band photometric measurements. For this purpose we choose to harness the available Petrosian magnitude ugriz photometry from the 12th data release (DR12) of the SDSS survey, which is complete to r /lessorsimilar 22 . 4 mags. All of the galaxies in our main specz galaxy sample are detected in at least the g , r and i bands in SDSS-DR12. At faint magnitudes ( i SDSS , Petro > 20 . 5 ), the uncertainties in the SDSS photometry become large, due to the sensitivity limit of the SDSS observations. Given the depth of the HSC data, the inclusion of HSC photometry will serve to increase the precision of our SED modeling for faint systems. Hence, for sources with \ni SDSS , Petro > 20 . 5 magnitudes, we supplement the observedframe optical data with HSC grizy Kron-magnitude photometry. \nTypically, extended sources with i /greaterorsimilar 20 . 5 mags have HSC i -band photometry that is consistent ( ± 0 . 05 mags) with the photometry from SDSS. However, some sources with i SDSS , Petro > 20 . 5 mag still have significantly discrepant photometric measurements between SDSS and HSC ( | i HSC , kron -i SDSS , Petro | > 0 . 15 mags). Such a difference between the SDSS and HSC photometry is well beyond the typical statistical uncertainty quoted for the photometry in either survey ( σ SDSS ∼ 0 . 09 ; σ HSC ∼ 0 . 04 ), and points towards a photometric measurement issue for a given object in the HSC pipeline. 3 Hence, we choose not to include the additional HSC photometry for these objects. We note that the exclusion of the HSC photometry for some objects does not significantly affect our SED fitting procedure or its derived measurements, as we show in Figure 2. \nTo place more stringent constraints on the stellar mass of our galaxy sample, it is prudent to measure the stellar-light centered around the rest-frame near infrared, hence we include available Y JHK photometry from the Deep Extragalactic Survey (DES) and the Large Area Survey (LAS), which are part of the 10th data release of the UKIRT InfraRed Deep Sky Surveys (UKIDSS), which covers the specz surveys considered throughout this manuscript. Specifically, we use the Petrosian magnitudes available within UKIDSS-DES and LAS, which we correct for aperture biases between SDSS/HSC and UKIDSS. Based upon our comparison of the UKIDSS-DR10 catalog photometry with our own SDSS/HSC aperture-matched photometry, which we extracted directly from the UKIDSS imaging for a random subset of the sources in our specz catalog, we determined that a flux-dependent correction to the UKIDSS Petro photometry of +0 . 02 -0 . 05 magnitudes, produces adequately matched photometric measurements between the three catalogs. We further confirmed our aperture corrections by harnessing the aperture-matched catalog of GAMA/SDSS and UKIDSS sources (Hill et al. 2011), finding a similar systematic average offset between the UKIDSS-DR10 Petro measurements and GAMA/SDSS-UKIDSS Petro-mags of +0 . 03 mags. \nAt higher redshifts ( z > 0 . 65 ), the rest-frame near-IR moves into the mid-IR, hence, we also include the 4-band W 1 -4 mid-infrared photometry from the WISE All-Sky Survey where AGN emission may also be prominent. In Section 3.3 we use \n∼ \nFig. 2. Examples of FAST produced spectral energy distribution (SED) fits while harnessing available photometry SDSS ( ugriz ; filled circles), HSC ( grizy ; filled stars), UKIDSS LAS Y JHK (filled triangles) and WISE ( W 1 W 2 ; filled squares) and spectral redshifts. SED fits are shown with (solid line) and without (dashed line) the inclusion of the HSC grizy photometry. Top panel provides the normalized filter response used throughout our analysis. Lower panel provides the stacked rest-frame SEDs (solid lines) and the 1σ spread in the SED models (dotted-lines) for the obscured-AGN (red) and the star-forming galaxies (blue) in three bins of stellar mass: log ( M ∗ /M /circledot ) ∼ 10 ,10.5 and 11.0. \n<!-- image --> \nthe WISE mid-IR photometry to build our AGN sub-samples using WISE color-color diagnostics. \nWe use the publicly available IDL code, FAST (Kriek et al. 2011) to model the optical-IR SED of each object to derive physical properties, as well as the appropriate k-corrections required to produce rest-frame photometry for each galaxy. FAST searches over a grid of models and uses χ 2 -statistics to determine the best solution. Throughout the fitting we assume an exponentially declining star-formation history with SFR ∼ exp[ -t/τ ] and a characteristic time-scale of log ( τ ) = 7 . 0 -10 . 0 yr, a Chabrier initial mass function, assuming stellar ages in the range, log (age) = 8 -10 . 1 , and the high-resolution stellar population synthesis models of Bruzual & Charlot (2003). Furthermore, we use a Calzetti et al. (2000) dust reddening curve, and allow for extinction in the range A V =0 . 2 -4 . 0 and derive templates for metalicities of { 0 . 008 , 0 . 02(solar) , 0 . 05 } . To determine the uncertainties for the fitted parameters we perform 500 Monte Carlo realizations of this FAST setup, and quote the 67th percentile of the simulations. \nThe Bruzual & Charlot SPS templates do not include contributions from AGN. Hence, to further ensure that we do not overestimate the stellar mass of sources containing a mid-IR detected AGN, for known AGN (see Section 3.3) we fit the WISE photometry with a power-law, and following Azadi et al. (2017), we subtract this AGN continuum from the WISE photometry to estimate the galaxy-only continuum. In Figure 2 we show examples of the two best-fit SED templates to our suite of photometry assuming the inclusion (solid line) or exclusion (dotted line) of the HSC photometry. 4 Qualitatively, the best-fit templates appear extremely similar across a large wavelength range ( λ ∼ 3000 ˚ A-4 µm ). Indeed, we find that the difference between the derived M ∗ , SFR and A V measurements between the two photometry sets are all consistent at the 1 σ uncertainty level determined directly from our Monte Carlo realizations. In turn, this suggests that for the subset of significantly extended and/or well-resolved galaxies currently lacking reliable photometry in HSC, the exclusion of these photometric points does not affect our ability to measure the galaxy properties using FAST. Furthermore, we assessed systematic bias towards the M ∗ measurements between the (obscured) AGN and non-AGN galaxies. In the lower panel of Fig. 2 we provide the rest-frame stacked SEDtemplates for sources predicted to have M ∗ ∼ 10 10 , 10 10 . 5 , and 10 11 M /circledot for AGN and non-AGN. We show that in each instance that the stacked SED templates are similar for AGN and non-AGN systems in each individual mass bin. This suggests that scattered light from the obscured AGN is not present or not adversely affecting the stellar mass estimates in these systems. Hence, we conclude that there is no significant systematic \nFig. 3. Rest-frame U-V versus V-J diagrams for all spec-z galaxies in our parent sample. Rest-frame AB photometry is derived from best-fit SED templates produced from FAST. Column panels: galaxy sample split in three redshift bins, 0 . 1 < z < 0 . 3 , 0 . 3 < z < 0 . 6 and 0 . 6 < z < 0 . 9 . Rows: redshift bins separated by two stellar mass bins, log M ∗ /M /circledot < 10 . 5 and log M ∗ /M /circledot /greaterorsimilar 10 . 5 . \n<!-- image --> \nbias between the stellar mass estimates for AGN and non-AGN galaxies.', '3.2 Rest-frame photometry and UVJ selection': "As well as matching our interacting and non-interacting galaxy samples based on their stellar masses, it is also prudent to consider matching on galaxy color. In this section we use the derived rest-frame photometric information to separate our specz galaxy sample using a typical star-forming/quiescent classification diagram, which harnesses the apparent rest-frame color bi-modality between star-forming and non-star-forming galaxies. \nFollowing the procedure outlined in the previous section, we determine rest-frame photometry directly from the best-fit SPS template. We apply a simple Gaussian noise model to the bestfit template with the noise amplitude matched to the average 1σ uncertainty of the measured photometric data, and use the known specz to produce a rest-frame simulated SED. The addition of the Gaussian noise provides a more realistic estimate of the measurement of the photometry that would be found from real observations, and that may otherwise not be captured in the discretized SPS models. We convolve the simulated SED with rest-frame U , V and J filters, as well as determining appropriate K-corrections for all of the input photometry, to produce rest-frame measurements. \nIn Figure 3, we show the rest-frame U -V vs V -J colorcolor diagram (hereafter, UVJ diagram) for our specz galaxy sample. We separate quiescent and star-forming (SF) galaxies using the proposed boundaries computed by Williams et al. (2009). The implementation of this two color cut allows us to identify dust-reddened star-forming galaxies, i.e., those with U -V > 1 . 6 that may otherwise be tagged as quiescent using \ntypical rest-frame color-magnitude diagrams. We confirm that in three separate and distinct redshift bins, z ∼ 0 . 1 -0.3, 0.30.6 and 0.6-0.9, that the SF galaxies are typically less massive systems (with M ∗ < 3 × 10 10 M /circledot ) than their quiescent counterparts. At these lower masses, our sample is over-whelmingly dominated by SF systems by factors of ∼ 6-10 at z > 0 . 3 , while in the lowest redshift bin, the samples of quiescent and SF galaxies are more comparable, owing mainly to the spectroscopic coverage from GAMA-DR2. Of the more massive galaxies with M ∗ > 3 × 10 10 M /circledot , our spec-z sample is dominated by quiescent galaxies by factors of ∼ 2-5, driven mainly by the abundance of massive 'red' galaxies targeted for spectroscopic follow-up in SDSS-BOSS. The more massive population of SF galaxies, that reside outside of the quiescent boundary in the UVJ diagram, have systematically higher U -V and V -J colors, than the lower-mass star-forming galaxies. This suggests at least 1 magnitude of optical extinction towards the galaxy continuum in these objects. Our final specz sample contains 75,886 and 62,657 quiescent and SF galaxies, respectively.", '3.3 Mid-IR AGN Selection using the (ALLWISE) WISE All-Sky Survey': 'While most AGN are intrinsically luminous in any given wavelength band, the homogeneous selection of an unbiased population of AGN from survey data is not straightforward. Intervening gas and dust, as well as dilution of the AGN signatures by host galaxy light, are the most common sources of observation selection bias. Indeed, many studies have now shown that no one wave-band can identify all AGN (Alexander et al. 2008; Donley et al. 2008; Hickox et al. 2009; Juneau et al. 2011; Mendez et al. 2013; Goulding et al. 2014; Trump et al. 2015; Azadi et al. 2017). Moreover, in the previous section we determined that the most massive SF galaxies in our specz sample are likely obscured by A V > 1 mags, further hampering AGN detections. However, even in the presence of significant dust attenuation, relatively unbiased detections of luminous AGN may be made at mid-IR wavelengths. AGN emission produced directly from the optical/UV luminous accretion disk or from the X-ray emitting corona may be absorbed and reprocessed by dust which surrounds the central BH. This dust-rich torus isotropically re-emits at mid-IR wavelengths, which is relatively insensitive to further absorption at larger radial distances from the AGN. \nMid-IR AGN identifications can be made either through the detection of high-ionization emission lines in mid-IR spectroscopy (e.g., Diamond-Stanic et al. 2009; Goulding & Alexander 2009), or through the establishment of the presence of an AGN-produced powerlaw continuum using mid-IR photometry. More specifically, a wide-variety of AGN selection techniques have been proposed that harness color-color \ndiagrams developed using 2, 3 or 4-band mid-IR photometry taken using the IRAC instrument on the NASA Spitzer Space Telescope (e.g., Lacy et al. 2004; Stern et al. 2005; AlonsoHerrero et al. 2006; Donley et al. 2012, or more recently using WISE (e.g., Jarrett et al. 2011; Stern et al. 2012; Mateos et al. 2012), which we harness here. Mid-IR color-color selection is particularly effective at identifying high-luminosity AGN ( L AGN /greaterorsimilar 10 43 erg s -1 ), where the contrast between the AGN and the host-galaxy is high. However, IR color-color diagrams may fail to readily identify AGN accreting at low Eddington ratios (e.g., Donley et al. 2012; Mateos et al. 2012; Hainline et al. 2016).', '3.3.1 Identifying WISE counterparts to galaxies in HSC': "Due to the all-sky nature of the WISE survey, HSC is covered in its entirety by 4-band cryogenic mid-IR observations at 3.4, 4.6, 12 and 22 µm . We matched the positions of our HSC specz sample to the objects present in the the ALLWISE release using the same method outlined in D'Abrusco et al. (2013) and Goulding et al. (2014), which we briefly outline here. We used expanding radial apertures of ∆ r =0 . 1 '' to search for all WISE counterparts to HSC sources to a distance of r = 4 '' . By randomly shifting the centroid of the HSC source by ± 20 '' , we computed the likelihood of spurious counterparts as a function of radial distance from the true HSC source position. We determined that the optimal maximum matching radius for HSC and WISE is r ( <R ) ∼ 1 . 6 '' . At r > 1 . 6 '' the probability of including spurious counterparts into our sample exceeds that of a real HSC-WISE match. \nWe identified 103,406 HSC galaxies in our specz sample that have at least one counterpart in WISE. For the ∼ 0 . 6 % of HSC sources with multiple WISE counterparts within 1.6 '' , we chose the WISE source with the smallest separation to the HSC galaxy. Over 97% of the HSC-WISE matches are at separations of r < 1 '' , and the distribution of the matching radii are characterized by a log-normal, peaked at 0.18 '' , and with a FWHM of ∼ 0 . 35 dex. The peak at 0.18 '' is consistent with the astrometric precision of the WISE sources (Wright et al. 2010).", '3.3.2 AGN identification using mid-IR color selection': "In Figure 4 we use the [3.4]-[4.6], [4.6]-[12.0] mid-IR colorcolor diagram to identify galaxies with a significant contribution from a central AGN. Specifically, we include all galaxies that are detected in the WISE [3.4], [4.6] and [12.0] bands, with a signal-to-noise (S/N) /greaterorsimilar 4 for the [3.4] and [4.6] bands. The longer wavelength bands in WISE are significantly less sensitive (by at least a factor 2) than the [3.4] and [4.6] bands. Hence, we marginally relax our S/N threshold to S/N > 2 . 5 in order to consider a source detected in the [12.0] band. We note this cut is still more conservative than the S/N > 2 used to identify objects throughout the ALLWISE catalog. 41,990 galaxies in our \nFig. 4. WISE infrared [3.4]-[4.6] versus [4.6]-[12.0] Vega magnitude color-color diagram of HSC galaxies with WISE counterparts. All objects have detections in the [3.4] and [4.6] bands with S/N /greaterorsimilar 4 and in the [12.0] band with S/N /greaterorsimilar 2 . 5 . Green dotted box shows the 2-color AGN selection region of Mateos et al. (2012) and black dashed line the 1-color AGN selection cut of Stern et al. (2012). Panels columns are split by redshift, using the same cuts as in Fig. 3, rows indicate galaxies separated according to their position in the UVJ diagram. In the 0 . 1 < z < 0 . 3 panels, simulated color-color tracks derived from SED templates (Polletta et al. 2007) are shown. These tracks begin at z =0 . 1 (filled symbols) and finish at z =0 . 9 (open symbols). Individual templates are for a 13 Gyr Elliptical (star), a ULIRG (downward triangle), a starburst (diamond), and S0 (upward triangle), Sb (square) and Sd (circle) spiral galaxies. \n<!-- image --> \nHSC specz sample are detected in 3-bands using WISE with our S/N cuts, and an additional 44,389 galaxies are detected in only [3.4] and [4.6] WISE bands. \nWe use the two-color IR-AGN wedge defined by Mateos et al. (2012) to identify [3.4], [4.6] and [12.0] detected objects that have powerlaw-like continua, indicative of the presence of a radiatively-efficient AGN. For galaxies that are not detected in the 3 WISE bands considered in Fig. 4, we also use the single color cut of Stern et al. (2012) to identify additional AGN (dashed line in Fig 4). This has the advantage of allowing us to boost source statistics due to the relative insensitivity of the longer wavelength WISE bands, as well as including the abundance of heavily obscured AGN that reside in UltraLuminous IR galaxies with [4.6]-[12.0] > 3 . 5 that may otherwise be excluded by the 2-color wedge. Mid-IR AGN selections are suspected to be contaminated by low-metallicity strongly star-forming dwarf galaxies in the low-redshift universe (e.g., Hainline et al. 2016), and by hot strongly dust-obscured galaxies beyond z > 2 . In Fig. 4, we additionally show simulated color-color tracks that are derived from the SED templates of \nPolletta et al. (2007) in our considered redshift range. These tracks typically lie below or outside of the AGN selection regions used throughout, and hence, contamination to our AGN selection from non-AGN interlopers is likely to be minimal (see also Goulding et al. 2014). \nOf the ∼ 41,990 galaxies in our matched HSC-WISE 3-band sample, 3,125 are selected as AGN using the Mateos et al. (2012) WISE selection method. An additional 665 galaxies are selected as AGN using the Stern et al. single color cut i.e., a total of 3,790 WISE-selected AGN. We find that if we separate the galaxies based on their position in the UVJ diagram, the WISE-AGN are over-whelmingly hosted in star-forming galaxies. Indeed, only ∼ 4% of the WISE-AGN in our specz sample are hosted in quiescent galaxies. This clear separation of mid-IR AGN residing in star-forming galaxies over quiescent galaxies serves to highlight the previously observed connection between star-formation rate and BH accretion rate (e.g., Chen et al. 2013; Hickox et al. 2014). \nUnlike more traditional color-magnitude diagrams (Strateva et al. 2001; Baldry et al. 2004), the separation of star-forming \nand quiescent galaxies through UVJ diagnostics is relatively insensitive to dust extinction in the host. As a result, dusty starforming galaxies are still robustly identified using UVJ while they may otherwise be classified as quiescent or green-valley systems using color-magnitude diagrams. The observed separation of AGN in Fig. 4 using UVJ may explain why many AGN have previously been believed to be an interesting population of transitioning 'green-valley' galaxies that lie between the blue cloud and red sequence (Bell et al. 2004; Faber et al. 2007; Nandra et al. 2007; Hasinger 2008; Silverman et al. 2008; Mendez et al. 2013). In reality, it would appear that these luminous mid-IR AGN are merely hosted in dusty star-forming systems with reddened optical colors. \nThe lack of mid-IR AGN observed in quiescent galaxies does not suggest that there are no accreting BHs in these systems. The vast majority of radio-loud AGN are known to be hosted in massive quiescent galaxies (e.g., Best et al. 2005; Hickox et al. 2009; Goulding et al. 2014; Delvecchio et al. 2017), though the majority of these radio AGN lack the signatures of a radiatively efficient accretion disk, which would be observed in the mid-IR. Furthermore, in galaxy group or cluster environments, evidence of AGN feedback due to radio emission from the BH present in the brightest cluster galaxy (so called, maintenance mode feedback) has long been established (e.g., Best et al. 2005; Rafferty et al. 2006; McNamara & Nulsen 2007; Kauffmann et al. 2008; Fabian 2012). In these systems, powerful radio lobes inject mechanical energy back into the intracluster medium, which in turn prevents the efficient cooling gas, and are believed to be responsible for restricting the formation of new stars in quiescent galaxies. \nGiven the apparent paucity of mid-IR AGN in quiescent galaxies, the contribution of these systems to the rapid growth of BHs must be negligible in comparison to the AGN present in star-forming galaxies. Hence, for all further analyses presented here, we neglect the inclusion of quiescent galaxies in our specz sample, as identified using the UVJ diagnostic diagram, due to the systematic lack of mid-IR AGN in these systems. Furthermore, by removing relatively quiescent systems through our UVJ selection, our morphological analyses that are designed to identify merging features (described in Section 4) are not subject to degeneracies arising from the existence of extremely long-lived stellar shells that are readily identified in early-type systems located within dense environments, and are unrelated to gas-rich mergers.", '3.3.3 Separation of Obscured & Unobscured AGN': 'AGN identifications made at mid-IR wavelengths are relatively independent of obscuration. Following simple AGN unification, Type-1 AGN are those where the accretion disk can be viewed almost directly, with very little intervening gas or dust, while a Type-2 AGN is viewed edge-on, and therefore has the disk \nemission and broad-line region hidden from the line-of-sight by an optically thick torus surrounding the central BH. However, as this torus isotropically reradiates the AGN emission at IR wavelengths, a mid-IR AGN selection results in a mixture of both Type-1 and Type-2 AGN. While the emission from both these AGN populations dominate their SEDs at mid-IR, the characteristic tail of the AGN accretion disk, which is typically observed in the UV/optical, remains absent for only the Type2 AGN. Hence, studies have revealed that a simple observedframe optical-IR color cut reliably separates unobscured Type1 AGN from their obscured counterparts (see Hickox et al. 2007; Hickox et al. 2011; Chen et al. 2015). \nIn Fig. 5 we present the distributions of our mid-IR selected AGN sample in their observed-frame i SDSS -[4 . 6] WISE color. In a similar vein to Hickox et al. (2007), we find that these optical-IR colors can be characterized by two distinct Gaussian distributions, peaking at i SDSS -[4 . 6] WISE ∼ 0 . 5 and 1.9. Similar to Hickox et al. (2007), we cut our AGN sample into obscured and unobscured sub-samples using optical-IR color. We use a cut of i SDSS -[4 . 6] WISE =1 . 1 , which is based on the intersection of the Gaussian distributions. This serves to maximize the number of AGN with the correct Type-1/2 classification, while minimizing contaminants. We find 2,552 and 1,238 sources with optical-IR colors that are red-ward and blue-ward, respectively, of our i SDSS -[4 . 6] WISE = 1 . 1 cut. Inspection of the SDSS spectroscopy for a subset of the AGN with i SDSS -[4 . 6] WISE /lessorsimilar 1 . 1 confirms the presence of broad Hβ , Hγ emission lines and/or a strong blue disk continuum. \nWe further show in Fig. 5 that the obscured and unobscured AGN do not follow similar distributions when considered in optical-IR color versus absolute i -band magnitude space. There is an additional population of low redshift, lowluminosity ( M i > -20 . 5 mags), extremely blue objects with i SDSS -[4 . 6] WISE /lessorsimilar -0 . 5 that are not mirrored in the obscured AGN population. These may be a set of AGN hosted in very low-mass galaxies (e.g., Satyapal et al. 2014b; Secrest et al. 2015; Sartori et al. 2015) or a population of low-metalicity blue dwarf galaxies ( M ∗ /lessorsimilar 5 × 10 9 M /circledot ) with powerful young starburst regions. These starbursts produce red colors in WISE that are similar in practice to emission from AGN (e.g., Hainline et al. 2016). \nBy contrast, there appears to be a population of luminous obscured AGN at M i /lessorsimilar -23 . 8 mags that are not present in our Type-1 AGN sample. At these luminosities, Type-1 AGN will most likely saturate the HSC detector for relatively nearby systems and/or appear similar to bright point sources at higher redshifts. These systems are therefore preferentially removed from our sample during our initial catalog selection by setting the is extended flag. Such dominance of the AGN over the host galaxy would hinder and bias our determination of the host galaxy properties during the SED-fitting process (e.g., stellar \nFig. 5. Observed optical-IR color versus absolute magnitude diagram used for separating our mid-IR selected AGN into obscured and unobscured subsamples (gray-scale contours). In i -[4.6] color, the AGN sample shows a distinct bimodality, with unobscured Type-1 AGN exhibiting bluer colors of i -[4 . 6] < 1 . 1 . Overlaid are the Type-2 AGN shown with rainbow colors to represent the source spectroscopic redshift. Right panel provides the histogram of the optical-IR color (black solid line) that is well characterized by the summation of two Gaussians, a type-1 AGN population (blue dashed) and an obscured AGN population (red dashed). The dotted lines is a simple cut that separates the AGN populations with minimum contamination. \n<!-- image --> \nmasses are known to be over-estimated for Type-1 AGN) and during our morphological analysis presented in the next section. Hence, to ensure the most unbiased measurements of the AGN host galaxies, we select only the 2,552 obscured AGN with M ∗ > 5 × 10 9 M /circledot for all further analyses that compare the AGN/galaxy properties.', '4 Identifying interacting and merging galaxies within HSC images': 'In this section we harness the exquisite sensitivity and spatial resolution afforded to us by the HSC survey to provide a basic morphological classification for each galaxy in our specz sample. Using parametric and non-parametric metrics, combined with a novel implementation of a Random Forest Machine Learning algorithm, we separate our specz galaxy sample into subsamples of major-mergers, minor-mergers and irregulars, and non-interacting galaxies.', '4.1 Profile fitting with GALFIT': "Image analysis techniques have been developed to produce parametric measures that are capable of separating galaxies by their morphological type. Using a-priori knowledge of a galaxy's structural properties - early-type galaxies have \nsmooth, elliptical isophotes, while late-type galaxies tend to be more disk-dominated with flatter light-profiles - it has been shown that even simple one or two-dimensional decompositions of the light profiles are capable of separating galaxies by their Hubble-type (e.g., Kormendy et al. 2009; Simard et al. 2011). \nIn order to analyze the size, morphology and stellar-light distribution of the galaxies in our sample we begin by fitting a single 2-dimensional Sersic profile (S'ersic 1963) using GALFIT (Peng et al. 2002) to the HSC i -band images. We extracted 100 × 100 kpc postage stamps from the co-added data products produced by hscPipe , along with the associated variance image and data mask. Point spread function (PSF) images are extracted from the pipeline products on a source-by-source basis. Within hscPipe , the PSF images are computed using the PSFEx software (Bertin 2011) from 41 × 41 pixel images of nearby stars to determine the size and ellipticity of the PSF for each visit. These PSFs are then co-added to replicate the average PSF of the co-added image. The median PSF size for our sample is ∼ 0 . 6 '' . See Bosch et al. (2017) for further details on the computation of the PSF images. \nTo measure a background level for each image, we used the full HSC catalog, which is sensitive to sources with i ∼ 27 mags, to identify and mask all objects that lie within the postage stamp image based on their catalog shape measurements and their Kron radii. We additionally applied the byte mask to those pixels previously flagged by hscPipe as erroneous. We fit a simple 2-dimensional linear profile to the non-masked pixels to assess any overall background gradient within the image and determine a mean background level in each pixel. We fill all areas within the background image that were previously masked with Poisson noise determined by the mean background level predicted for the individual masked pixels. The measured background level was included as an input to GALFIT, and held fixed throughout the fitting procedure. \nTo create an input mask image for GALFIT, we masked all sources within the 100 × 100 kpc postage stamp that had integrated magnitudes at least 3 mag fainter than the target galaxy (i.e., a factor 1:15 fainter in flux). All areas identified in the hscPipe bad-pixel mask were also masked, and all bright point sources were masked with shapes based on the ellipticities and radii of the co-added PSF. For all remaining unmasked extended objects within the image, we included an additional Sersic profile into the GALFIT fit centered at the position of the additional galaxy. Hence, during the GALFIT fitting procedure, will simultaneously model all bright galaxies with the postage stamp image. Our choice to mask objects determined to be at least a factor ∼ 15 fainter than the target objects allows us to simultaneously model all components of possible major or minor mergers to at least mass ratios of 1 : 10 (i.e., allowing for variability in the mass-to-light ratio). \nWe next extracted a sub-image of 50 × 50 kpc centered \nFig. 6. Four examples of the imaging analysis described in Section 4 performed on our spec-z HSC galaxy sample. Large panels are K-corrected (pseudorestframe) 3-color images; smaller inset panels are the best-fit Sersic model calculated using GALFIT (upper) and the residual image ( i - model) with red color gradients for increasingly positive residuals and blue gradients for increasingly negative (lower). Labels provide the measures of asymmetry ( A img ), smoothness/clumpiness ( S img ), concentration index ( C img ) and Gini index ( G img ) calculated from the i -band image, as well the asymmetry ( A resid ), smoothness/clumpiness ( S resid ) and Residual Flux Fraction (RFF) calculated from the residual image. Interaction probability ( P merge ) determined from our implementation of a Random Forest Machine Learning algorithm in also given (see Section 4.2). \n<!-- image --> \naround the target galaxy along with the respective mask and variance images. This sub-imaging approach has the advantage of limiting the computation time with GALFIT, while also maintaining that any large (unrelated) sources, which may have significantly overlapping isophotes with the region immediately surrounding the target galaxy but may have centroids outside the sub-image, will still be appropriately masked or have a Sersic profile assigned during the fitting process. The source image, variance image and PSF model were all used as inputs for GALFIT. \nIn the upper-right sub-panels of Fig. 6 we provide examples of the best-fit 2-dimensional Sersic profile that were fit to the target galaxy. For each galaxy, we extract the best fit parameters for the Sersic profile, namely the Sersic index, n , and the characteristic effective radius, R e . R e is provided as a pixel length within GALFIT, which we convert to a physical scale in kiloparsecs for all further analyses. In the next section, we use the Sersic parameters and Sersic-profile subtracted images (residuals) to compute metrics in order to identify interacting and non-interacting galaxies.", '4.2.1 Parametric & non-parametric morphology indicators': "Many image analysis techniques have been developed to automatically separate merging systems from non-interacting and/or isolated galaxies, to varying degrees of success and accuracy. These methods often make use of parameterizing the structures present in the image of a given galaxy. In the previous section, we applied a 2-dimensional Sersic profile to HSC postage stamp images, which was a simple parametric approach for modeling the galaxy light distribution. \nA tangential approach is to use non-parametric indices, which have been developed to assess the distribution of light within an image in order to separate/quantify a galaxy's Hubble class and/or interaction stage (e.g., Patton et al. 2000, 2002; Lin et al. 2004; De Propris et al. 2007; Lin et al. 2008; Robaina et al. 2010; Bluck et al. 2012; Glikman et al. 2015 for a recent review see Conselice 2014). Typical non-parametric indices make use of the light concentration, asymmetry and smooth- \nness/clumpiness (hereafter, CAS measurements; see Bershady et al. 2000; Conselice 2003), as well as other measures involving the Gini index and the second-order moments of the light distributions (see Abraham et al. 2003; Lotz et al. 2004, 2008). \nIn the same spirit as these non-parametric indices, studies have now begun to develop new metrics that implicitly incorporate parametric measurements, resulting in hybrid parametric/non-parametric indices. For example, the residual flux fraction (RFF; Hoyos et al. 2011, 2012) measures the fluctuation of counts in residual images of galaxies once a simple best-fit Sersic profile has been subtracted. Residual images increase the contrast of concentrated structures, as well as enhance low-surface brightness features. Taken together, analysis of the residuals may better reveal interaction signatures between galaxies that may otherwise be missed in the original images. \nPrevious studies have determined that simple cuts on asymmetry and smoothness ( A> 0 . 35 and A > S ; Conselice 2003) or with the Gini and M 20 parameters ( G> -0 . 14 × M 20 +0 . 33 ; Lotz et al. 2004) can produce a reliable ( ∼ 50 %) separation of galaxies undergoing mergers in relatively nearby massive systems. With the advent of new generations of telescopes and deep surveys, like HSC, we are now able to resolve faint merger signatures in large galaxy samples that were previously too weak to identify. However, as sensitivity to low surface brightness material increases, it becomes necessary to fine-tune our selection algorithms to identify features of interest, particularly as long-lived tidal debris, low surface brightness galaxies, and the outer parts of spiral galaxies may all trigger the same indicators (e.g., Greco et al. 2017). \nProgress can be made by considering all of the information that can be extracted from a combination of each of these different parametric and non-parametric structure measures. Here we use a novel implementation of a Machine Learning technique to provide a statistical measure of the interaction state of a given system. \nAs morphology 'features' for our machine learning algorithm, we measure the CAS parameters for each galaxy in our HSC specz sample, as well as the Gini and RFF indices. For the precise formulation of these parameters we refer the reader to Section 2.3 of the review by Conselice (2014) and Hoyos et al. (2012). We measure each of these indices on the 50 × 50 kpc i -band postage stamp galaxy images. Following Hoyos et al. (2012), we also compute the asymmetry and smoothness/clumpiness parameters on the residual flux images (i.e., the i -band image after subtraction of the best-fit Sersic model for the galaxy determined following the method outlined in the previous section). These non-parametric indices are combined with the parametric measurements of the best-fit Sersic profiles to provide a suite of morphological parameters (hereafter, 'features') that we use to determine the interaction state of the galaxy through 'automated classification'.", '4.2.2 Training a Random Forest Classifier': "The goal of automated classification frameworks is to determine a model that describes some in-hand data for a set of objects whose 'science classification' is known a-priori. This model is then applied a new set of objects, whose classifications are unknown, and then used to predict a class or probability of a given classification for each new object. Several forms of data-driven automated classification schemes have been used to solve an abundance of astrophysical problems, such as Gaussian mixture models, Bayesian networks, neural networks, and support vector machines (e.g., Goldstein et al. 2015; Moolekamp & Mamajek 2015; Williams et al. 2016; Melchior & Goulding 2016; Avestruz et al. 2017). A conceptually simple, extremely efficient, and yet powerful classification method, which is becoming popular throughout astronomy, is that of decision-tree learning. \nDecision trees are supervised non-parametric classifiers that remain efficient even when attempting to capture complicated feature-based structures. They naturally handle multiple classification schemes, and are relatively robust to outliers. However, tree models tend to have high variance. Due to the hierarchical structure of the trees, even small changes in the top levels of a training tree, induced by random selection of the variables used to split nodes, can produce vastly different trees on subsequent nodes. Also, while large trees will, by design, always fit the training data very well, a specific large tree may not generalize well to test data. This process is akin to over-fitting in simple regression. \nNoise in the final classifications can be reduced by considering multiple decision trees for a given dataset, so-called 'Random Forests'. Random Forest classifiers fit multiple decision trees to bootstrap subsamples of a given training set. The final classification for an object is then the average of the classifications produced by the individual bootstrap decision trees, which naturally provides a (pseudo-)probability for the classification (driven by the input training data) while controlling for over-fitting of the data. \nTo build our training sample, we visually classified the 50 × 50 kpc k-corrected 3-band HSC images for a random sample of 5,900 galaxies in our specz sample that were deemed to be star-forming based on their position in the UVJ diagram. The specific visual classification scheme involved the identification of (1) irregular/disturbed/torqued morphologies, (2) double-nuclei/late-stage merger, (3) evidence for interaction with a distinct companion galaxy, (4) regular morphologies with no evidence for recent interaction, or (5) too-small to conclusively identify. To normalize the responses of the seven expert classifiers, we averaged the individual visual classifications for a test subsample of 600 galaxies, and then weighted the responses accordingly for the remaining visual classifications. For galaxies that were clearly undergoing or had recently \nFig. 7. Example of a decision tree within our implementation of a Random Forest machine learning algorithm. The Random Forest is constructed from our representative sample of 5,900 visually classified galaxies. Each decision tree is formed from a bootstrap resampling of a subsample of 4,500 visually classified galaxies and is trained to identify objects based on three morphological classifications (1: non-interacting [orange]; 2: major/late-stage merger [purple]; 3: minor-merger/irregular [green]). Nodes are gradient color-coded depending on the purity of the classification decision (light colors have low purity, dark colors have high purity). \n<!-- image --> \nFig. 8. Distributions of probabilistic merger-state classifications assigned by our implementation of Random Forest Machine Learning algorithm to 1,400 visually classified galaxies. The Random Forest was trained on an independent sample of 4,500 visually classified galaxies randomly selected from our main HSC spec-z sample. Left: probability of being an isolated galaxy ( P Isolated ); Center: probability of being an irregular galaxy or a minor-merger ( P irregular / minor -merger ); Right: probability of being a major merger ( P Major -Merger ). Distributions are split as a function of their visual classification (isolated; irregular/minor-merger; major merger), see Section 4.2 for further details. \n<!-- image --> \nTable 1. RandomForestClassifier Initiation Parameters \n| Parameter | Value |\n|-------------------|--------------------------------------------------------------|\n| input features | C img ; A img ; S img ; G img ; A res ; S res ; RFF; R e ; n |\n| n estimators | 1000 |\n| criterion | gini |\n| max features | √ 9 |\n| max depth | 15 |\n| min samples split | 12 |\n| bootstrap | True |\n| warm start | False |\n| class weight | balanced | \nundergone an interaction, our expert classifiers were in strong agreement that at least one of the interaction classifications was valid. However, we noted significant variance among the experts when attempting to separate these different signatures of galaxy-galaxy interactions. As such, we elected to consolidate our visual classifications for interacting systems, as we determined that this provided a cleaner separation between interacting and non-interacting galaxies. 5 \nExamples of four systems determined to be major-mergers from their visual classifications are shown in Fig. 6. Each of these systems are clearly at different stages of merging. In terms of the interaction classification outlined by Veilleux et al. (2002), these galaxies would be classified as IIIa: wide binary (right column), IIIb: close binary (top-left) and IV: Merger (bottom-left). 6 These four examples exhibit relatively wide ranges in parameters such as their smoothness/clumpiness ∼ 0 . 1 -0 . 8 (typical values in the range -0.5-1.5), but have narrow ranges in Gini ( ∼ 0 . 6 ) and RFF ( ∼ 0 . 3 -0 . 4 ). The role of our Random Forest implementation will be to search for correlations between the visual classifications and the specific values/ranges of these features. \nOur visually-classified training sample was split to provide an input of 4,500 galaxies used to construct the decision trees, and an independent subsample of 1,400 galaxies to test the output classifications of the Random-Forest classifier. We used the publicly available Python-based RandomForestClassifier code provided as part of the scikit-learn package (Pedregosa et al. 2011) to build the decision trees. The input features for the decision tree construction (see Table1) were the concentration \n( C img ), asymmetry ( A img ), smoothness/clumpiness ( S img ) and Gini ( G img ) indices measured from the HSC i -band images, the RFF, asymmetry ( A res ) and smoothness/clumpiness ( S res ) indices measured from the residual (galaxy-Sersic model) image, and the Sersic index and R e measured from the best-fit model. \nTo avoid importance bias of a particular input feature, we first normalize the distributions of each feature to have mean zero and unity variance before inputting to the Random Forest generator. The Random Forest is initiated with the parameters shown in Table 1, and then trained to identify galaxies based upon the three morphological classifications assigned during our visual classifications: 1: non-interacting (inclusive of Stage I pre-mergers); 2: major-merger (inclusive of Stage II-IV mergers); 3: minor-mergers (inclusive of Stage V irregulars). The final assigned classification is then the average of the 'votes' from each of the 1000 decision trees, i.e., the fraction of trees that assign a classification of 'isolated' is P isolated . \nAn example of one of the 1000 decision trees in the Random Forest is shown in Fig. 7. After experimentation, the branches are pruned to not allow depths beyond 15 nodes, though most branches terminate before this as we set a minimum threshold of > 12 sources for a new node to be created. In the example presented in Fig. 7 we find that in the initial node of the tree (leftmost box in the diagram), that a relatively neutral cut in RFF (0.0076 in normalized units) ultimately results in a strong overall distinction between interacting and non-interacting galaxies. All subsequent nodes leading upwards and away from the initial node (i.e., training objects with RFF ≤ 0 . 0076 ) are, in general, colored orange, denoting non-interacting galaxies. By contrast, subsequent nodes leading downwards and away from the initial node (i.e., training objects with RFF > 0 . 0076 ) are more likely to result in nodes containing interacting galaxies (colored either purple:major-merger or green:irregular/minor-merger). \nFurthermore, in Fig. 7 we show that minor-mergers are difficult to distinguish from major mergers and non-interacting galaxies. The minimum node value (i.e., the number of connecting nodes required to reach a node from the initial [leftmost] node) of an irregular/minor-merger classification is 4, with the majority of the irregular/minor-merger leaves not being identified until node > 7 . From a decision tree stand-point, minor-mergers then become a sub-category of the more dominant isolated and major-merger classifications, making their robust identification complex. \nUsing our training visual classification sample, we additionally calculated the importance of the input features that went into producing our Random Forest classifier. The importance can be thought of as the fraction of useful information that is used by the classifier during the construction of a decision tree, with the sum of importances, I , over all features equaling unity. The most important features, averaged over all trees, were S res , A img and RFF, each with I ∼ 0 . 17 -0 . 21 , while the C img was \nFig. 9. Fraction of sources in our test set of 1,400 visually classified galaxies as a function of the probability of a particular system being a merger. Merger probabilities are computed during the implementation of a pythonbased Random Forest Machine Learning algorithm trained on an independent set of 4,500 visually classified galaxies in our main HSC spec-z sample. Dashed, dotted and solid lines are those galaxies visually classified to be major mergers (flux-ratio > 1 : 4 ), minor mergers 1 : 4 -10 , and major or minor mergers, respectively. Red lines provide the fraction of objects with a given P merge that are determined to not be the given merger classification (i.e., for major mergers, contaminant populations are non-interacting galaxies and minor-mergers). \n<!-- image --> \nthe least useful with I ∼ 0 . 03 .", '4.2.3 Testing the Random Forest Classifier': "To assess the ability of our Random Forest for providing reliable probabilistic classifications to the remainder of our HSC spec-z galaxy sample, we applied the trained Random Forest to our 'test sample' of 1,400 visually classified galaxies that were not used during the training of the classifier. In Fig. 8 we provide the distributions of the classification probabilities for our test sample separated by their visual classifications. For each of classification probability, the distribution of the true visual classified objects peak at higher probability values. Indeed, it is clear from the P isolated histograms that we can cleanly recover a sample of isolated galaxies with a cut of P isolated > 0 . 7 , with little or no contamination from interacting galaxies. However, this of course does not recover the full population of isolated systems, as this population of objects begins to mix significantly with objects towards lower values of P isolated . This is also mirrored in the distributions of P minor -merger and P major -merger . \nIn Fig. 9 we further explore contamination to a major merger sample when applying a threshold in P merger . As expected, we find galaxies assigned to have high values of P merger by the Random Forest are increasingly more likely to actually be major mergers based on their visual classification. We find that while \na threshold of P merger > 0 . 33 would provide a sample that is ∼ 90% complete towards major mergers, ∼ 30 %of the sample would be contaminated by non-interacting galaxies and minormergers/irregulars. Based on our Random Forest and limited training sample, we cannot yield a truly pure sample of major mergers that is more than ∼ 39% complete. However, in the range 0 . 32 <P merger < 0 . 67 it is clear we suffer from only mild contamination ( ∼ 10% ), and only ∼ 1 / 3 of the sample contamination arises from isolated galaxies. Hence, a cut of P merger > 0 . 32 yields a relatively clean sample of interacting systems (i.e., minor + major merger), while a cut of P merger > 0 . 46 yields a sample of major-mergers that is over > 75% complete and suffers less than ∼ 10% contamination, the majority of which arises due to minor-mergers and irregulars, which themselves may have somewhat ambiguous visual classifications given the almost arbitrary demarcations that are made between the visual classes. Finally, we also tested for any effect to the classifications due to the presence of an unobscured AGN, which may have been incorrectly classified as a Type-2 AGN from our IRoptical color-cut. While we do include a PSF model during our GALFIT analysis, we found that the presence of a Type-1 AGN still marginally steepens the Sersic index, significantly increases the concentration index, and lowers the asymmetry value. These are each due to the galaxy light being partially contaminated by the AGN. Irrespective of whether the source was visually classified as an non-interacting or major-merger, we found this typically lowered the value of P merger , resulting in the source being more likely to be classified as a non-interacting galaxy. As such, we note here that the presence of Type-1 AGN in our sample may artificially increase the merger fraction in noninteracting galaxies, and hence these will dilute the signal from AGN being intrinsically preferentially hosted in major mergers in the next section. In the next section, we use these automatic classification probabilities to construct relatively robust samples of non-interacting isolated galaxies, major mergers, and a set of interacting (irregular + minor-merger + major-merger) galaxies, and investigate the incidence of AGN in these systems.", '5 Results': 'Despite the theoretical successes of BH-galaxy co-evolution models to explain observed present-day galaxy populations, observational evidence for the presence of an on-going merger and the concurrent rapid growth of BHs, which is now a required ingredient of galaxy formation simulations, remain elusive. In this section we use the morphological/interaction probabilities derived using our implementation of a Random Forest machine learning algorithm to assess the incidence of AGN in carefully constructed statistically-significant samples of major mergers, minor mergers and irregulars, and non-interacting galaxies.', '5.1 Incidence of AGN in interacting and non-interacting galaxies': 'AGN activity is a highly stochastic process, with changes in accretion rate that typically occurs on time-scales that are much shorter than longer lived galactic processes, such as changes in stellar mass, star-formation rate, or even merger-stage. Thus, it is more robust to probe the average AGN property (i.e., averaging over BH accretion variability) as a function of the longer timescale galactic process (see Hickox et al. 2014). Hence, we now investigate the average incidence of AGN based on host-galaxy interaction stage by harnessing three galaxy samples: 1) major mergers; 2) all interacting galaxies (including major merger, minor mergers, and irregular systems), and 3) non-interacting galaxies.', '5.1.1 Selecting galaxies & AGN in bins of interaction type': 'In Section 4.2 we determined that a threshold of P major -merger > 0 . 46 in our trained Random Forest classifier recovers ∼ 75% of the major-mergers present in our visually-classified test sample. A sample of major-mergers defined by such a cut suffers contamination at the ∼ 7 % level from minor-mergers and irregular galaxies, and < 3 % from non-interacting galaxies. Applying this threshold in P major -merger provides a clean sample of 4,449 gas-rich major mergers at 0 . 1 <z < 0 . 9 . In Fig. 10, we show a random set of examples of major mergers identified by our Random Forest Classifier, which reside in a subregion of the HSC survey. The diversity of the sample of major mergers in their interaction state, mass ratio, number of systems, and redshift is clearly apparent. This is mainly driven by our large training sample of 4,500 visually classified systems covering wide ranges in galaxy properties (including interaction state), as well as the sensitive HSC imaging that is capable of detecting the low surface brightness emission associated with merging, which may be otherwise missed in shallower wide field surveys. \nWe define two additional galaxy samples: (1) a set of galaxies at all stages of interaction, which include major mergers, minor mergers and irregular systems; and (2), a control set of non-interacting/isolated galaxies. Following our testing in the previous section, we invoke a threshold of P merger > 0 . 32 or P minor -merger > 0 . 40 to define the set of interacting galaxies, which yields an interacting sample of 5,594 systems. The noninteracting star-forming galaxies are defined by P isolated > 0 . 7 , which provides a sample of 12,513 galaxies, with /lessmuch 1 % contamination from interacting systems. \nIn Fig. 11 we show the distributions of the three samples in their WISE [3.4]-[4.6] color and stellar mass, separated by three redshift bins ( 0 . 1 <z < 0 . 3 , 0 . 3 <z < 0 . 6 , and 0 . 6 <z < 0 . 9 ). The [3.4]-[4.6] color is indicative of the dust temperature, and incident SF or AGN activity. Additionally, we highlight each galaxy that has a significant AGN contribution to the mid- \nFig. 10. Examples of K-corrected 3-color 50 × 50 kpc HSC images of the 4,449 gas-rich major-merger candidates at 0 . 1 < z < 0 . 9 selected to have P major -merger > 0 . 46 based on our Random Forest Classifier described in Section 4.2. These merger candidates cover a wide-range in interaction state, stellar mass ratio and redshift. \n<!-- image --> \nIR continua, and thus has a relatively high [3.4]-[4.6] color ( /greaterorsimilar 0 . 7 ). The AGN were previously identified based on their 2 or 3 band mid-IR colors, as measured from their WISE photometry (see Section 3.3); crucially, the AGN selection was performed independently of the morphology and interaction state of the galaxy. \nPrevious observations have suggested that the AGN fraction (above some particular threshold in AGN luminosity) rises steeply as a function of stellar mass (e.g., Xue et al. 2010; Aird et al. 2012; Bongiorno et al. 2012; Mullaney et al. 2012). Thus, it is important to consider stellar mass matched samples when assessing the relative incidence of AGN in our morphological/interaction-state samples. Here we provide the quoted mass-completeness limits towards star-forming galaxies of the spectroscopic surveys, from which our main specz sample derives. As shown previously in Fig. 1, in the lowest redshift bin, our sample is dominated by galaxies identified in the GAMA-DR2 survey, which is complete to M ∗ ∼ 5 × 10 9 M /circledot for star-forming galaxies with restframe g -i < 0 . 5 (Taylor et al. 2011) at the median redshift of the bin ( z ∼ 0 . 2 ). Furthermore, at z ∼ 0 . 5 , our sample is mainly comprised ( ∼ 75 %) of galaxies drawn the PRIMUS and VIPERS surveys, which for starforming galaxies at z ∼ 0 . 5 are complete to M ∗ ∼ 2 × 10 9 M /circledot \n(Moustakas et al. 2013) and M ∗ ∼ 3 × 10 9 M /circledot (Davidzon et al. 2013), respectively. We note that while these surveys claim completenesses toward star-forming galaxies of M ∗ ∼ 5 × 10 9 M /circledot , we ultimately do not require our sample to be complete in stellar mass, as the AGN fractions we present in the proceeding sections are compared in relatively between interacting and non-interacting galaxies.', '5.1.2 WISE Completeness Corrections': 'In each of our considered redshift bins, our star-forming parent sample is roughly mass complete at around M ∗ ∼ 5 × 10 9 M /circledot . However, as we advance in redshift, we systematically miss low-mass galaxies in our sample due to the flux limit of the WISE survey. This is observed in Fig. 11 as a deficit of sources in the 0 . 3 <z< 0 . 6 and 0 . 6 <z< 0 . 9 bins that have blue [3.4][4.6] colors and M ∗ /lessorsimilar 2 × 10 10 M /circledot . This subsection outlines our statistical corrections for this incompleteness brought about by the WISE-HSC cross-match. \nBy comparison of the fractions of WISE detected and nondetected galaxies at 0 . 3 <z < 0 . 6 , we find that for galaxies in our specz sample with M ∗ ∼ 5 × 10 9 M /circledot , ∼ 85 % are not detected by WISE. Conversely, at the same redshift, only ∼ 5% of the galaxies with M ∗ ∼ 10 11 M /circledot are not detected in WISE. \nFig. 11. Lower Panels: WISE [3.4]-[4.6] color (vega) as a function of stellar mass (in logarithmic units of M /circledot ) for three morphologically selected samples - non-interacting galaxies ( P isolated > 0 . 7 ; orange; bottom row), major mergers ( P major -merger > 0 . 46 ; purple; center row), and all interacting galaxies (major-merger + minor-merger + irregulars; P major -merger > 0 . 32 ∨ P minor -merger > 0 . 4 ; red; top row). For merging/interacting systems, reported stellar masses are those of the most massive galaxy in the pair. Panels columns are split by redshift, using the same cuts as in Fig. 3. Galaxies identified to host IR-luminous Type 2 AGN using the diagnostics presented in Figs. 4 and 5 are shown with star symbols and colors denoting morphological classification. Dotted lines illustrate the AGN selection boundaries for galaxies with M ∗ /greaterorsimilar 5 × 10 9 M /circledot and [3.4]-[4.6] > 0.8 (Stern et al. 2012) Top Panels: Histograms of AGN fraction as a function of stellar mass constructed in redshift bins matched to the lower panel. AGN fractions in a given redshift bin are separated by morphological classification - non-interacting galaxies (orange dots); major mergers (purple dashed); all interacting galaxies (red solid). The AGN fractions are also corrected for incompleteness of non-WISE detections in the two highest redshift bins. We find in each redshift bin that galaxies undergoing mergers are a factor ∼ 2 -3 more likely to contain AGN than non-interacting galaxies, and this is independent of stellar mass. \n<!-- image --> \nIf we compare these fractions of WISE non-detections to our lowest redshift bin ( 0 . 1 <z < 0 . 3 ), we find that only ∼ 3% and ∼ 2%, respectively, of the galaxies are not detected in WISE in the same mass bins. This demonstrates that we are essentially complete towards WISE detections at low redshift, and that the loss of low-mass blue WISE color sources as a function of redshift can be almost entirely attributed to the flux limit in WISE. \nFrom Fig. 11 we can deduce that the loss of low-mass blue WISE color sources is relatively independent of morphology classification. Two-sample K-S tests show that there is no significant evidence for a difference between M ∗ distributions of the non-interacting, major merger and all-merger samples within a given redshift bin. \nIR-luminous AGN are selected to be more luminous and redder in WISE than non-active galaxies. As such, incompleteness may not affect AGN in the same way as non-active galaxies. To test this, we can compare the raw AGN fraction (i.e., no corrections for underlying completeness) between two redshift and stellar mass bins. For galaxies with M ∗ ∼ 10 11 M /circledot (a stellar-mass bin which is unaffected by WISE-completeness issues) the AGN fractions at 0 . 1 < z < 0 . 3 and 0 . 3 < z < 0 . 6 are consistent with each other: ∼ 2 . 4 % and ∼ 2 . 3 %, respectively, and we therefore see no evolution in AGN fraction between the redshifts in this mass bin. However, for galaxies with M ∗ ∼ 5 × 10 9 M /circledot , the AGN fractions increase by almost an order of magnitude from ∼ 1 . 9 % at 0 . 1 < z < 0 . 3 to 17 . 2 % at 0 . 3 < z < 0 . 6 . Such a large jump in AGN fraction in a small redshift range is wholly unphysical, and strongly suggests that while we are not detecting non-active galaxies in WISE for galaxies with M ∗ ∼ 5 × 10 9 M /circledot , we do still identify those galaxies containing AGN. Thus, in order to calculate the AGN fractions presented in the following sections, we need only statistically account for the non-active WISE undetected objects in each M ∗ bin as a function of z . Indeed, we show in the next section that by making the assumption that we are HSC-WISE cross-match only misses non-active galaxies, and correcting for this incompleteness, the AGN fractions are found to be similar in each mass bin across the full redshift range considered. \nBased on our parent sample, in each redshift bin we compute the M ∗ distributions of the galaxies that are not detected in WISEduring our WISE-HSC cross-match. Under the informed assumption that none of these systems contain AGN, we use the ratio of the M ∗ -distributions between the WISE detected and non-detected to systems to normalize the AGN fractions presented in the next subsection. 7 As our lowest redshift bin is rel- \nely complete towards WISE detections, our computed corrections in this redshift bin are factors of ∼ 1 . 02 -1.03. However, these corrections become large, /greaterorsimilar 10 , in the lowest mass bins at higher redshift. Ultimately, these completeness corrections allow us to qualitatively compare AGN fractions across the redshift bins. But crucially, as the corrections are applied irrespective of the interaction classification, they do not effect our conclusions when comparing fractional differences between AGN fractions at fixed M ∗ and z .', '5.1.3 AGN fractions at fixed M ∗ and z': 'For a given redshift bin, we construct 3 equal stellar mass bins with width 0.7 dex (a factor ∼ 5 ) for each of our morphological samples. Accurate photometric measurements, and hence, stellar mass measurements, for merging galaxies are non-trivial for the most distant sources in our sample. As stated previously (see Section 2.3), based on our comparison between HSC and Hubble Space Telescope data in the COSMOS field, we determined that the photometric measurements for distant mergers to be accurate to ± 0 . 3 mags, resulting in a factor ∼ 1 . 3 -1 . 5 uncertainty in derived M ∗ (dependent on typical mass-to-light ratios). As such, we conservatively construct our coarse M ∗ bins in Fig. 11 to mitigate the effects of uncertainty in M ∗ . \nIn the upper-panel of Fig. 11, we show the fraction of objects in the mass-matched bins that are determined to be mid-IR AGN ( f AGN ). In the 0 . 3 < z < 0 . 6 and 0 . 6 < z < 0 . 9 bins, these AGN fractions are corrected using our derived M ∗ completeness functions. We find that the AGN fractions in our major merger and all-merger samples are a factor ∼ 2 -7 higher than those for non-interacting galaxies. This result appears to be consistent across the redshift range considered here, in that f AGN is systematically higher for interacting galaxies over noninteracting galaxies, and this in observed in each of the redshift bins. We find that this enhancement in f AGN is significant at the 3.5-5.4 σ level for the sources with M ∗ < 10 11 M /circledot , dropping to 1.7-2.2 σ in the high-mass bin for each individual redshift slice considered here. When combining the data across the redshift bins, the significance of this result increases to 3.3-8.0 σ across all stellar masses. \nTo further test the robustness of this result to the misidentification of AGN in our sample (i.e., normal star-forming galaxies with unusually red mid-IR colors that could mimick an AGNsignature), we used a stricter W1-W2 color cut of > 1 . 0 to classify a source as an AGN, and re-calculated the AGN fractions. While this cut greatly affected the number of AGN being identified, particularly in the lowest-redshift bin, we found that the increase in f AGN for interacting galaxies was still significant in six of the nine z -M ∗ bins. \nThe fact that we observe an increase in f AGN in each redshift bins suggests this result is independent of our M ∗ complete- \ngiven redshift bin. \nness corrections. Indeed, we find that if we do not implement a completeness correction, f AGN remains enhanced by a similar factor in merging/interacting systems. For interacting systems, we find a marginal increase in f AGN with M ∗ that is more pronounced with increasing redshift. However, we find that the fractional difference of f AGN for mergers and non-interacting galaxies does not appear to be conditional on M ∗ . \nConsistent with the merging/interacting galaxies, at z > 0 . 3 , there is a marginal enhancement in f AGN for the non-interacting systems with the largest M ∗ ( > 10 11 M /circledot ) over the lower mass non-interacting galaxies. However, we note that only 2 AGN are identified in the highest mass bin for the isolated systems at z < 0 . 3 ; these poor source statistics would prevent us from significantly identifying a similar rise at high masses, as observed in the higher redshift bins. Although we observe a rise in AGN fraction related to M ∗ for isolated systems, crucially, these measurements do not exceed the AGN fractions found for interacting galaxies at the same M ∗ .', '5.2 Testing for observational bias and heterogeneity in our specz sample': 'As discussed in Section 2.3, our parent galaxy sample is constructed from a heterogeneous set of spectroscopic redshift surveys. While the majority of the surveys targeted all galaxies to a given brightness threshold and within a particular region of the sky, spectroscopic redshifts may still not have been measured for some objects. This can be due to observing difficulties, signal to noise effects, lack of emission features etc., and hence are individually incomplete at some level to all galaxies within the sky region. Moreover, spectroscopic surveys such as SDSS-BOSS invoke optical color cuts to pre-select galaxies in a given redshift range, which results in complex selection/incompleteness effects. Thus, each spectroscopic redshift survey has its own unique set of selection biases, which become imprinted onto the main parent galaxy sample considered throughout our analyses. Here we test whether the spectroscopic surveys are biasing the AGN fractions measured in the previous section and presented in Figure 11. \nIn Figure 1 we showed that our parent sample is dominated by objects drawn from 2-4 different spectroscopic surveys for each of the three redshifts bins (i.e., 0 . 1 <z< 0 . 3 ; 0 . 3 <z< 0 . 6 ; 0 . 6 < z < 0 . 9 ) considered throughout. To test whether one of the spectroscopic surveys excessively contributes to the measured AGN fractions in any of the redshift bins presented in Figure 11, and hence may be causing a bias in the AGN fraction at those redshifts, we systematically removed all objects pertaining to one particular redshift survey and recomputed the AGN fractions for that redshift bin. For example, in the lowest redshift bin at 0 . 1 < z < 0 . 3 , it is clear that our parent sample is mainly drawn from objects presented in the SDSS- \nLegacy and GAMA surveys. Hence, we removed all galaxies (irrespective of morphology) drawn from the SDSS-Legacy survey and recomputed the AGN fractions at 0 . 1 < z < 0 . 3 for the three morphology/interaction categories for a single stellar mass bin. 8 Even after removing the SDSS-Legacy survey objects, at 0 . 1 <z < 0 . 3 , we found fully consistent AGN fractions with those presented in Fig. 11: f AGN , Major -Merger ∼ 0 . 031 , f AGN , All -Merger ∼ 0 . 029 and f AGN , Non -Interacting ∼ 0 . 0065 . We repeated this test at 0 . 1 < z < 0 . 3 by removing all objects drawn from the GAMA survey, and again found a factor ∼ 3 . 5 increase in the interacting AGN fractions over the noninteracting sources. We continued this test in each redshift bin for each redshift survey, and consistently found that AGN are more prevalent in interacting galaxies and major mergers than non-interacting systems. The only marginal bias we observed during this test was with the exclusion of SDSS-BOSS galaxies, where we found that the AGN fractions difference between non-interacting and interacting galaxies increased from a factor ∼ 3 to a factor ∼ 6 at 0 . 3 < z < 0 . 6 . This suggests a possible bias against the targeting of merging galaxies and/or AGN in the SDSS-BOSS survey, and hence, the f AGN presented in Fig. 11 at 0 . 3 < z < 0 . 6 may be marginally conservative, and the true difference between mergers and non-interacting galaxies is likely to be larger.', '5.3 The most luminous AGN preferentially reside in merging galaxies': 'Several recent studies have identified a possible correlation between AGN luminosity and galaxies undergoing mergers. These studies found high merger fractions ( ∼ 85% ) in luminous L AGN > 10 46 erg s -1 dust reddened quasars (e.g., Urrutia et al. 2008; Glikman et al. 2012), consequently leading to the intriguing suggestion that merger fraction is dependent on AGN bolometric luminosity (Treister et al. 2012). This has been substantiated by the identification of a strong positive trend of increasing merger fraction spanning over 3 decades in L AGN , which may persist beyond z ∼ 2 (e.g., Kocevski et al. 2015; Del Moro et al. 2016; Fan et al. 2016). However, others have found less convincing evidence for a connection between merger fraction and the most luminous AGN activity, particularly in Type-1 quasars (e.g., Villforth et al. 2017). Furthermore, at more moderate luminosities, L AGN < 10 44 erg s -1 , this connection appears to be weaker, with merger fractions remaining relatively constant (at ∼ 20% ) with decreasing L AGN . These seemingly contradictory results are consistent with a picture in which mergers drive the high-Eddington growth of the most massive BHs ( L AGN /greaterorsimilar 10 44 erg s -1 ), but with secular/internal processes becoming increasingly dominant at lower Eddington ratios and/or \nFig. 12. Lower Panels: AGN fraction as a function of stellar mass after removal of the upper quartile ( f AGN , -UQ ) of sources with the highest L AGN (solid histograms). Hashed regions provide the AGN fraction determined after randomly removing 25% ( f AGN , -Rand25 ) of the AGN in a given stellar mass bin, irrespective of L AGN . This fraction is recomputed 10,000 times using a jack-knife re-sampler, and the bounds provide the 90th percentile range of the samples. Color coding is the same as Fig 11. Upper Panels: Residual AGN fraction between the f AGN , -UQ and f AGN , -Rand25 (i.e., ∆ f AGN = f AGN , -Rand25 -f AGN , -UQ ) as a function of stellar mass. In general, we find that the most luminous AGN systematically reside in the interacting/merging galaxies, hence their positive residual signatures. Conversely, fewer luminous AGN reside in non-interacting galaxies, as we show that f AGN , -UQ is systematically larger than the expected null result value, f AGN , -Rand25 . \n<!-- image -->', 'lower BH masses.': 'We can test such a scenario using our interacting and noninteracting galaxy samples by assessing whether the most bolometrically luminous AGN are preferentially hosted in interacting galaxies over non-interacting galaxies. If the most luminous AGN are biased regarding the morphologies/interaction state of the host, then the AGN fractions for interacting/noninteracting galaxies, presented in Fig. 11, should have an additional dependency on L AGN at fixed stellar mass. However, and crucially, if indeed the most luminous AGN are systematically more likely to reside in merging galaxies, then f AGN , interacting and f AGN , non -interacting would not have the same dependency on L AGN , at fixed stellar mass. \nWe test this hypothesis by measuring the change in f AGN between those values presented in Fig. 11, and the f AGN calculated after removing the upper-quartile of the most luminous AGN present in a given M ∗ bin (hereafter, f AGN , -UQ ) 9 For consistency, we use the same bins of M ∗ and z as those \npresented in Fig. 11. Similar to our analysis in Section 5.1, by measuring changes in f AGN between interacting and noninteracting galaxies, we naturally control for BH accretion variability, which likely occurs on shorter timescales than merger events. \nAGN bolometric luminosities for our sample are computed by fitting powerlaw slopes to the mid-IR photometry of the AGN, and using the best-fit slope to predict the rest-frame 6 µm continuum luminosity, which is shown to be a robust indicator of L AGN (e.g., Lutz et al. 2004; Fiore et al. 2009; Chen et al. 2017). The AGN considered here cover a relatively wide range, with luminosities of L AGN ∼ 3 × 10 43 -2 × 10 46 erg s -1 . \nIn each M ∗ bin, we find that the AGN fraction decreases substantially for the major-merger and interacting galaxy samples by factors of ∼ 0 . 7 -2 after removing the upper-quartile of the most luminous AGN, i.e. ∆ f AGN = f AGN -f AGN , -UQ ∼ 0 . 01 -0 . 03 . While the decrease in f AGN for the non-interacting galaxies was in some cases consistent with ∼ 0 . This provides tentative evidence that the most luminous AGN do preferentially reside in merging galaxies. \nWe can give these results a stronger statistical footing by simulating the ∆ f AGN had we not preferentially removed the upper quartile of the most luminous AGN, but instead we had randomly removed 25% of the AGN from a given M ∗ bin (i.e., independent of L AGN ). This is achieved through 10,000 Jack-knife re-samplings of f AGN for each morphology/interaction sample after removing a random set of 25% of the AGN in each M ∗ bin at each redshift ( f AGN , -Rand25 ). In Fig. 12 we plot the residual between f AGN , -Rand25 and f AGN , -UQ . The hashed regions represent the 90th percentiles of the Jack-knife samples. Across all three redshift ranges, we show a general trend of positive ∆ f AGN for the interacting galaxies (i.e., f AGN , -Rand25 , mergers > f AGN , -UQ ), and negative residuals for the non-interacting galaxies (i.e., f AGN , -Rand25 , non -interacting < f AGN , -UQ ). Hence, we have good statistical evidence at the > 90 % level, that the most luminous AGN systematically reside in the interacting galaxies at fixed stellar mass. \nIn Fig. 12, we further show that at z > 0 . 3 the ∆ f AGN appears to diverge with increasing M ∗ , suggesting that a larger fraction of the most luminous AGN reside in interacting galaxies at higher M ∗ , with an additional marginal preference for the most luminous AGN residing in major-mergers. At z < 0 . 3 we find consistent (at the 90th percentile) residual f AGN values between the interacting and non-interacting galaxies in the highest M ∗ bin, suggesting no statistically significant preference for luminous AGN between the interaction classifications. We note that this result may also be driven by the small number of AGN in non-interacting galaxies at high M ∗ in our sample, preventing us from measuring a strong systematic difference. However, overall we show that fewer luminous AGN reside in \nnon-interacting galaxies, with a strong preference for the most rapidly growing BHs to be generally hosted in major mergers.', '6.1 Evidence for stochastic BH growth during major-merger events': "Based on our population analysis (Section 5.1), we can robustly conclude that, on average, those galaxies that are currently undergoing or have recently undergone some form of a merger/interaction are a factor ∼ 2 -7 more likely to be rapidly growing their central BHs than more isolated and noninteracting star-forming galaxies. Furthermore, whilst we have not attempted to address the question of whether all luminous AGN events must be triggered by mergers (such a study would require a thorough understanding of incompleteness effects), our results clearly indicate a systematic enhancement of f AGN in major mergers over non-interacting galaxies and/or even minor-mergers. This could suggest that significant BH growth phase(s) are linked specifically to a major merger scenario. \nPrevious studies that have investigated merger-AGN connections have produced mixed results (e.g. Gabor et al. 2009; Cisternas et al. 2011; Schawinski et al. 2011; Kocevski et al. 2012; Treister et al. 2012; Ellison et al. 2013; Villforth et al. 2014; Kocevski et al. 2015), often finding that the host galaxies of AGN are similar to those of non-AGN (i.e., no strong enhancement in merger fractions of AGN over non-AGN). These results are seemingly at odds with our population analysis that shows, on average, AGN are markedly more likely to occur during a major-merger than in an isolated galaxy. We suggest that these apparent differences can be reconciled by considering the relative time-scales of the luminous AGN activity, the dynamical time of a major-merger, and the time spent as an isolated galaxy. \nMotivated by galaxy merger simulations, here we outline a framework that closely ties BH accretion rate variability to the dynamical time of the galaxy merger. Given that AGN activity is known to vary on timescales much shorter than galaxy processes (e.g., Novak et al. 2011; Gabor & Bournaud 2013), and on average, BH accretion rate is linked to available gas supply (e.g., Hickox et al. 2014), we make the ansatz that the act of merging further enhances AGN variability as galaxy merging strongly affects the inflow of gas, which can serve to fuel the AGN. For example, on first pericentric passage, gravitational torques may be sufficient to induce a short period of rapid BH growth ( < 50 Myrs). After first passage, interaction signatures, such as tidal tails, may still be evident as the galaxies move to maximum separation (lasting ∼ 200 -400 Myrs). However, internally, the galaxies may partially relax, limiting fuel to the BH, and causing accretion to slow, and the wide-separation merger may no longer be observationally identified as an AGN. \nIndeed, the fraction of AGN is seen to fall dramatically at large projected separations (e.g., Ellison et al. 2013; Satyapal et al. 2014a; Ricci et al. 2017). However, new episodes of significant AGN activity may then be re-ignited on subsequent passages until coalescence of the galaxies. In such a scenario, the AGN activity would seemingly occur sporadically throughout the merger, but with the overall AGN light curve being strongly correlated with the merger's dynamical time. \nWithin our proposed framework, during the merger there may be multiple periods of observable AGN activity, and seemingly non-AGN activity. Including a close connection with the dynamical time forces the non-AGN phases to last substantially longer during first and second passage, but with the non-active phases become shorter-lived as the galaxies begin to coalesce. In accordance with the results of previous investigations, comparisons of the merger rates of AGN to merger rates in nonAGN should produce similar fractions, as the AGN is not always 'active' during the entire merger event. As the merger begins to reach coalescence, the probability to observe the AGN increase dramatically as the AGN episodes occur more frequently, and may result in the maximal growth phase of the BH, as predicted in merger simulations. Such a scenario would simultaneously explain the apparent increase in merger rate with AGN luminosity (e.g., Urrutia et al. 2008; Glikman et al. 2012; Treister et al. 2012), and our finding of the most luminous AGN preferentially residing within merging systems. \nFurthermore, in Figure 11, we show that /lessorsimilar 10 % of majormergers contain luminous (obscured mid-IR) AGN, suggesting that the total AGN duty-cycle over the course of the merger may only be ∼ 50 -100 Myrs. Moreover, in Figure 11, we show that luminous AGN activity still occurs during isolated, secular evolution phases (i.e., a major-merger may not be an absolute requirement for luminous AGN activity to occur). Taken together, these two results suggest that merger-AGN studies primarily selected on the basis of AGN activity would need to be large in number in order to observe the relatively small ∼ 10 %enhancement in mergers relative to non-mergers in an AGN-selected sample (e.g., Villforth et al. 2014). However, we show that when considering a time-averaged look at the AGN duty cycle, the probability of luminous AGN occurring in major-mergers is clearly enhanced over non-interacting galaxies by at least a factor ∼ 3 . \nOverall, our results suggest that, on average, mergers do trigger AGN significantly more often than in secularly evolving galaxies above a particular luminosity threshold ( L AGN /greaterorsimilar 10 44 erg s -1 ), but that the BH does not necessarily need to be growing at a significant rate throughout the entire merger phase. However, given that we have focused on the growth of obscured AGN only, we have implictly not considered a possible evolutionary scenario between Type 2 and Type 1, which may also be linked with merger stage. This is beyond the scope \nof this investigation, but may be possible with future wideformat high-resolution imaging capable of identifying Type-1 AGN, while also providing the ability to extract galaxy properties such as morphology. Finally, at lower BH masses and/or lower AGN luminosities, secular processes may be more important for driving BH growth, with major-mergers becoming subdominant. Indeed, an enhancement in AGN fraction of only a factor ∼ 0 . 5 -2 is seen in Seyfert-like luminosity z < 0 . 1 galaxy pairs in SDSS (e.g., Ellison et al. 2011a).", '7 Summary & Conclusions': 'In this paper, we have investigated the effect of the merging of gas-rich galaxies on the growth of BHs out to z /lessorsimilar 1 . We have used the exquisite imaging quality afforded to us by the HSC instrument on the Subaru Telescope to identify merging and noninteracting galaxies across the first 170 deg 2 of the HSC survey. We used publicly available archival data within the HSC survey regions to identify spectroscopically confirmed galaxies in the redshift range 0 . 1 < z < 0 . 9 (see Section 2.3), and performed SED fitting (see Section 3.1) to derive their internal properties. We used photometry from the all-sky WISE mid-IR survey to identify the galaxies in our sample containing luminous AGN (see Section 3.3), and used the sensitive and high spatial resolution HSC imaging to implement a Random Forest machine learning algorithm to robustly identify large samples of merging and non-interacting galaxies (see Section 4.2). We use our morphological classifications in conjunction with the mid-IR AGN identifications to place constraints on the average incidence of luminous AGN in merging versus non-interacting galaxies. Our conclusions are the following: \n- 1. Based on stellar mass matched samples of galaxies, BHs hosted in merging galaxies are a factor ∼ 2 -7 more likely to be rapidly growing than in non-interacting galaxies. This result is found to be consistent in three separate redshift bins ( 0 . 1 <z< 0 . 3 ; 0 . 3 <z< 0 . 6 ; 0 . 6 <z< 0 . 9 ), and is relatively independent of stellar mass.\n- 2. Our parent sample of galaxies is drawn heterogeneously based on spectroscopic redshift confirmations from a variety of dedicated surveys. We investigated the likelihood of our AGN fractions being driven by the source selection induced by any one of these spectroscopic surveys by systematically removing each individual redshift survey from our parent sample and recomputing the AGN fractions as a function of morphology. We determined that our results are not driven by spectroscopic selection, finding fully consistent AGN fractions throughout.\n- 3. Several previous studies have suggested a strong link between galaxy merging and the most luminous AGN. We tested this result by assessing whether the most bolometrically luminous AGN are systematically hosted in merging \ngalaxies over non-interacting systems. At any given stellar mass bin, we found that the upper-quartile of the most luminous AGN preferentially reside in merging galaxies over non-interacting galaxies. We use these results to suggest that a major merger between two galaxies is sufficient to induce a flow of cool gas towards the central BH in one or both galaxies, and this is systematically more likely to trigger a significant AGN event that in an isolated galaxy alone. \n- 4. To place our findings into the wider context of AGN-galaxy co-evolution, and reconcile our conclusions with seemingly contradictory results within the recent literature, we outline a coherent framework that closely ties the variable AGN light curve to the dynamical time of the merger event. Our proposed framework requires that AGN accretion undergoes several distinct peaks in luminosity over the lifetime of the merger, with BH fueling linked to the close passage and interaction of the merging galaxies. The substantial time spent at wide pair separations, when the BH is not growing at an appreciable rate, serves to explain previous findings that highlight similarities between the fractions of AGN and nonAGN in merger states. \nOverall, our morphological investigation of 0 . 1 <z< 0 . 9 galaxies identified in the first 170 deg 2 of the HSC survey, provides conclusive evidence that luminous AGN are systematically more likely (by at least a factor /greaterorsimilar 3 ) to occur in majormergers when compared to non-interacting galaxies. Our results suggest that, on average, mergers do trigger AGN significantly more often than in secularly evolving galaxies. However, the BH need not be growing at an appreciable rate throughout the entire merger phase.', 'Acknowledgments': 'We thank the anonymous referee for their considered report, which allowed us to clarify and improve several aspects of this manuscript. ADGand JEG gratefully acknowledge support from the National Science Foundation under Grant Number AST-1613744. YM was supported by JSPS KAKENHI Grant No. JP17H04830. The authors thank Lisa Kewley, Robert Lupton, Jim Bosch and Bob Armstrong for enlightening conversations. This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration. The Hyper SuprimeCam (HSC) collaboration includes the astronomical communities of Japan and Taiwan, and Princeton University. The HSC instrumentation and software were developed by the National Astronomical Observatory of Japan (NAOJ), the Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), the University of Tokyo, the High Energy Accelerator Research Organization (KEK), the Academia Sinica Institute for Astronomy and Astrophysics in Taiwan (ASIAA), and Princeton University. Funding was contributed by the FIRST program from Japanese Cabinet Office, the Ministry of Education, Culture, Sports, Science and Technology (MEXT), the Japan Society for the Promotion of \nScience (JSPS), Japan Science and Technology Agency (JST), the Toray Science Foundation, NAOJ, Kavli IPMU, KEK, ASIAA, and Princeton University.', 'References': "Abraham, R. G., van den Bergh, S., & Nair, P. 2003, ApJ, 588, 218 Aihara, H., Armstrong, R., Bickerton, S., et al. 2017a, ArXiv e-prints, arXiv:1702.08449 \nAihara, H., Arimoto, N., Armstrong, R., et al. 2017b, ArXiv e-prints, arXiv:1704.05858 Aird, J., Coil, A. 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2024NatAs...8..126B
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Evidence for heavy-seed origin of early supermassive black holes from a z ≈ 10 X-ray quasar
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2024-01-01
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Observations of quasars reveal that many supermassive black holes (BHs) were in place less than 700 Myr after the Big Bang. However, the origin of the first BHs remains a mystery. Seeds of the first BHs are postulated to be either light (that is, 10−100 M<SUB>⊙</SUB>), remnants of the first stars, or heavy (that is, 10−10<SUP>5</SUP> M<SUB>⊙</SUB>), originating from the direct collapse of gas clouds. Here, harnessing recent data from the Chandra X-ray Observatory, we report the detection of an X-ray-luminous massive BH in a gravitationally lensed galaxy identified by the James Webb Space Telescope at redshift z ≈ 10.3 behind the cluster lens Abell 2744. This heavily obscured quasar with a bolometric luminosity of ~5 × 10<SUP>45</SUP> erg s<SUP>−1</SUP> harbours an ~10<SUP>7</SUP>−10<SUP>8</SUP> M<SUB>⊙</SUB> BH assuming accretion at the Eddington limit. This mass is comparable to the inferred stellar mass of its host galaxy, in contrast to what is found in the local Universe wherein the BH mass is ~0.1% of the host galaxy's stellar mass. The combination of such a high BH mass and large BH-to-galaxy stellar mass ratio just ~500 Myr after the Big Bang was theoretically predicted and is consistent with a picture wherein BHs originated from heavy seeds.
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https://arxiv.org/pdf/2305.15458.pdf
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{'Evidence for heavy seed origin of early supermassive black holes from a z ∼ 10 X-ray quasar': "' Akos Bogd'an 1* , Andy D. Goulding 2 † , Priyamvada Natarajan 3,4,5 † , Orsolya E. Kov'acs 6 , Grant R. Tremblay 1 , Urmila Chadayammuri 1 , Marta Volonteri 7 , Ralph P. Kraft 1 , William R. Forman 1 , Christine Jones 1 , Eugene Churazov 8 and Irina Zhuravleva 9 \n1* Center for Astrophysics ❘ Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA. 2 Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA. 3 Department of Astronomy, Yale University, 52 Hillhouse Avenue, New Haven, CT 06511, USA. 4 Department of Physics, Yale University, P.O. Box 208121, New Haven, CT 06520, USA. 5 Black Hole Initiative, Harvard University, 20 Garden Street, Cambridge, MA 02138. 6 Department of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University, Brno, 611 37, Czech Republic. 7 Institut d'Astrophysique de Paris, Sorbonne Universit'e, CNRS, UMR 7095, 98 bis bd Arago, 75014 Paris, France. 8 Max Planck Institut fur Astrophysik, Karl-Schwarzschild-Str.1, 85741 Garching bei Munchen, Germany. 9 Department of Astronomy and Astrophysics, The University of Chicago, Chicago, IL 60637, USA . \n*Corresponding author(s). E-mail(s): [email protected]; † These authors contributed equally to this work.", 'Abstract': "Observations of quasars reveal that many supermassive black holes (BHs) were in place less than 700 million years after the Big Bang. However, the origin of the first BHs remains a mystery. Seeds of the first BHs are postulated to be either light (i.e., 10 -100 M ⊙ ), remnants of the first stars or heavy (i.e., 10 4 -10 5 M ⊙ ), originating from the direct collapse of gas clouds. Harnessing recent data from the Chandra X-ray Observatory , we report the detection of an X-ray-luminous massive BH in a gravitationally-lensed galaxy identified by JWST at z ≈ 10 . 3 behind the cluster lens Abell 2744. This heavily-obscured quasar with a bolometric luminosity of L bol ∼ 5 × 10 45 erg s -1 harbors a M BH ∼ 10 7 -10 8 M ⊙ BH assuming accretion at the Eddington limit. This mass is comparable to the inferred stellar mass of its host galaxy, in contrast to what is found in the local Universe wherein the BH mass is ∼ 0 . 1% of the host galaxy's stellar mass. The combination of such a high BH mass and large BH-to-galaxy stellar mass ratio just ∼ 500 Myrs after the Big Bang was theoretically predicted and is consistent with a picture wherein BHs originated from heavy seeds. \nData from JWST is rapidly transforming our understanding of the early universe by enabling the detection of large samples of faint, distant galaxies deep into the epoch of reionization. Early studies are hinting at a higherthan-expected abundance of galaxies in the early Universe[1-6]. Cluster gravitational lenses, nature's telescopes, further augment JWST 's sensitivity by bringing into view low-mass, faint galaxies at the highest redshifts. As part of an Early Release Science Program[7] and a Cycle 1 Treasury Program[8], JWST has peered through the Hubble Frontier Fields cluster lens Abell 2744 at z = 0 . 308, revealing a high galaxy density[1, 2] at z > 9. In this field, multiple independent methods have ascertained accurate photometric redshifts for several z ∼ (9 -15) galaxies, and their physical properties, such as their stellar mass and star-formation rate, have been derived from fitting their spectral energy distributions (SEDs)[1, 2]. \nTo address the fundamental open question about the origin of the first BHs, we deployed an innovative approach to probe galaxies and their central BHs at cosmic dawn, utilizing the achromatic, magnifying power of the extremely well-calibrated cluster lens Abell 2744 with deep Chandra X-ray observations. X-rays are a ubiquitous signature of BH accretion, and hard X-ray observations can detect accreting BHs even if they are surrounded by large columns of absorbing gas and dust. To this end, we search for X-ray emission associated with high redshift accreting BHs in JWST -detected lensed galaxies in the Abell 2744 field using 1 . 25 Ms deep Chandra imaging observations. \nFig. 1 A 4 . 2 σ Chandra X-ray detection of a source cospatial with UHZ1: Panel (a) 1.25 Msec Chandra X-ray image of 0 . 5 -7 keV emission associated with the galaxy cluster lens Abell 2744. The image has been smoothed with an adaptive Gaussian kernel, and a logarithmic stretch has been applied. The 15 '' × 15 '' white box is centered on the location of UHZ1 and shows the zoomed-in field of view of the adjacent panel. Panel (b) Chandra 2 -7 keV band X-ray image of the 15 '' × 15 '' region surrounding UHZ1. Black contours show the JWST /NIRCam F200W morphology of the UHZ1 galaxy candidate at z ≈ 10 . 32. The solid white circle has a 1 '' radius, corresponding to both the off-axis Chandra PSF ( ≈ 88% encircled counts fraction) at this location on the image, as well as the source spectral extraction region as described in the text. The dashed white circle has an inner and outer radius of 3 '' and 6 '' respectively, marking the background spectral extraction region. North is up and East is left. \n<!-- image --> \n<!-- image --> \nTotal 2-7 keV X-ray counts \nThe processing and analysis of the Chandra X-ray data in Abell 2744 were carried out with CIAO [9]. We reprocessed the observations using the chandra repro tool to apply the latest calibration data. The individual observations were merged and we generated images in the 0 . 5 -7 keV, 0 . 5 -2 keV, and 2 -7 keV bands with the merge obs task. For each energy range, maps of the point spread function were generated, weighted, and coadded. The absolute astrometry of each observation was corrected using a set of bright X-ray point sources and by employing the wcs match and wcs update tools before merging the X-ray images. Details on the data analysis are presented in the Methods Section. \nTo probe the presence of high-redshift X-ray sources magnified by Abell 2744, we investigated a sample of 11 JWST -detected, gravitationallylensed galaxies with high signal-to-noise and z > 9 photometric redshifts[1, 2]. Using the JWST positions of the z > 9 galaxies, we performed photometry on the merged Chandra X-ray images. To minimize the contribution of the intracluster medium (ICM) of Abell 2744, we extract source regions with a 1 '' radius centered on each galaxy. The background is measured using local regions around each source to account for the varying foreground ICM level of Abell 2744. We used circular annuli around each source with 3 '' -6 '' radii (Figure 1). \n<!-- image --> \n<!-- image --> \nJWST \nChandra overlays of UHZI \nFig. 2 JWST and Chandra images of UHZ1: Panel (a) The JWST NIRCam image of the surroundings of UHZ1, and a zoom-in NIRCam image of UHZ1 in Panels (b and c). Panel (d) JWST images of UHZ1 in seven filters. The galaxy is detected in all JWST bands except for F115W. The non-detection in the bluest F115W band clearly indicates the dropout nature of the galaxy and suggests that it is located at z ≈ 10. The source is extended, with a potentially disturbed morphology evocative of late-stage mergers at lower redshift. A bright nuclear region is apparent in the F150W and F200W bands, and the contrast of this nucleus against the galaxy outskirts decreases for the redder bands. (e) A JWST / Chandra overlay showing a 4 . 2 σ excess of X-ray counts cospatial with UHZ1. (f) The same Chandra 2 -7 keV Chandra image, this time with UHZ1 represented as black contours. The size of the X-ray source is consistent with a point source. The location, luminosity, and spectral characteristics of the source suggest that it is a heavily obscured quasar residing in the z = 10 . 3 galaxy, UHZ1. North is up and East is left. \n<!-- image --> \nOf this sample of 11 JWST galaxies, we detect a statistically significant Xray source associated with UHZ1 (RA=0:14:16.096, Dec=-30:22:40.285); this galaxy is magnified[2] by a factor of µ = 3 . 81 +0 . 41 -0 . 56 . No other galaxies are located in the vicinity of UHZ1 that could be associated with the X-ray source (Figure 2). We note that of the galaxy sample, UHZ1 has the highest lensing \nmagnification factor. Based on the deep JWST imaging data, three methods have provided the best fit photometric redshift of z ≈ 10 . 3 +0 . 6 -1 . 3 , owing primarily to its robust non-detection in the F115W band[2], as well as further non-detections in Hubble Space Telescope (HST) F606W and F814W bands. Moreover, the redshift probability distribution function of UHZ1 does not reveal a secondary peak, making UHZ1 a robust high-redshift galaxy candidate.[2]. The Chandra X-ray source associated with UHZ1 has 42 total and 20.6 net counts within the extraction region in the 2 -7 keV band (restframe energy range of 22 . 6 -79 . 1 keV). The probability of finding 42 counts with a background expectation of 21.4 counts is 2 . 73 × 10 -5 , which is equivalent to a 4 . 2 σ detection significance assuming a Poisson distribution (Figures 1 and 2). We emphasize that the particular choice of the background region does not influence the detection of the X-ray source. The source is undetected in the 0 . 5 -2 keV band (rest-frame energy range of 11 . 3 -22 . 6 keV) with the 1 σ upper limit of < 2 . 4 net counts. The non-detection can be explained by the heavily absorbed nature of the source (see below and the Methods Section). The Chandra data of other JWST -detected high-redshift galaxies will be discussed in a follow-up study. To further probe the detection significance of the X-ray source, we performed an additional experiment that probed the distribution of the counts in the vicinity of UHZ1. This test, which is described in the Methods Section, suggests that the X-ray source associated with UHZ1 is detected at the 4 . 4 σ level. \nWe establish the characteristics of the X-ray source by spectral fitting the Chandra X-ray data (Figure 3). We extract spectra from the source and background regions, where the latter describes the emission from the ICM that dominates the local background. We fit the background annulus with an optically-thin thermal plasma model ( apec in XSpec ) with the temperature and normalization being free parameters. We fixed the line-of-sight column density at the Galactic value[10] ( N H = 1 . 35 × 10 20 cm -2 ), the metallicity at 0 . 3 Z ⊙ , and the redshift of Abell 2744 at z = 0 . 308. We measure a best-fit temperature for the ICM of kT = 10 . 9 ± 1 . 9 keV. To fit the spectrum of the source region, we utilize a two-component model. The first component is the apec model established from the background annulus with all parameters fixed and normalization adjusted to the smaller source aperture. The second component describes the heavily absorbed X-ray source and implements a model with a zeroth-order absorbed continuum combined with a Compton-scattered continuum using the MyTorus library[11]. We fix the slope of the power law at Γ = 1 . 9, a disk inclination of 85 · , and a Compton-scattering fraction of 2%, values that are typical of rapidly accreting, heavily absorbed high-redshift AGN[12]. We find a best-fit cstat = 32 . 5 with 30 degrees of freedom, which implies an acceptable fit. Further details of the spectral fitting procedure are provided in the Methods Section. \nWe obtain a best-fit column density of N H ≈ 8 +inf -7 × 10 24 cm -2 and a corresponding intrinsic 2 -10 keV luminosity of L X , int ≈ 9 × 10 45 erg s -1 after correcting for the µ = 3 . 81 lensing magnification at the location of UHZ1, \nFig. 3 Chandra X-ray spectral energy distribution and model fits: Observed-frame X-ray photons extracted from the merged Chandra ACIS data in the source aperture containing UHZ1 and the foreground lensing cluster, Abell 2744 (gray crosses; binned to a minimum of 2 counts for plotting purposes). The dotted gray line provides the thermal plasma model (APEC with kT ∼ 10 . 9 keV) established from a surrounding background annulus and rescaled to match the source aperture area. The solid gray curve provides the best-fit total model of the combined cluster+AGN X-ray emission constructed utilizing the MyTorus library. The observed X-ray spectrum is inconsistent with a pure plasma model - solid blue line shows the best-fit pure AGN model after subtraction of the cluster emission; this model reproduces the background-subtracted broad-band X-ray photometry extracted using forced photometry at the position of UHZ1 (0 . 5 -1 . 5 keV photometric point is a 3 σ upper limit due to the non-detection of X-ray emission after background subtraction). For illustration purposes, we further provide the best-fit pure AGN components for a range of assumed column densities ( N H = { 0.1,0.3,1.0,3.0 }× 10 24 cm -2 ). \n<!-- image --> \nand accounting for the source extraction region including ≈ 85% of the source counts given the Chandra point spread function. These results imply that a heavily obscured, most likely Compton-thick, accreting BH is present in UHZ1. Due to the small number of photons, the spectral fitting is somewhat degenerate between L X and N H , with larger N H columns requiring significantly larger values of L X , and within 2 σ uncertainties, values as low as N H > 10 22 cm -2 \nare also allowable within the fit, which would imply a 2 σ lower limit for the 2 -10 keV luminosity of L X , int > (2 -4) × 10 43 erg s -1 . \nWe use our X-ray spectral measurements to derive the physical properties of the accreting BH in UHZ1. Our best-fit 2 -10 keV luminosity, combined with the appropriate bolometric correction factor[13] of L bol /L 2 -10keV ∼ 73, implies a BH mass of M BH ≈ 6 × 10 9 M ⊙ , assuming Eddington-limited accretion. However, this high-mass estimate is driven by (and as noted previously) an L X measurement that is strongly degenerate with N H , and is hence, likely unreasonable given the inferred stellar mass of the galaxy. Therefore, we conservatively adopt the lower 1 σ uncertainty measurement for the column density of N H = 2 × 10 24 cm -2 , which provides a sensible balance between the steep degenerate fits required for very high values of N H , while still providing a measurement of L X that is allowable across a wide range of potential N H . This yields an intrinsic 2 -10 keV luminosity of L X , int ≈ 1 . 9 × 10 44 erg s -1 . Utilizing L bol /L 2 -10keV = 21 at this L X [13], we predict a BH mass of M BH ≈ 4 × 10 7 M ⊙ . Allowing for the quoted factor ∼ 2 uncertainty in the L bol /L 2 -10keV relation[13], combined with potential sub-Eddington accretion, we conservatively adopt a mass estimate for the BH in UHZ1 at z = 10 . 3 in the range of 10 7 -8 M ⊙ . Taken together, these combined JWST and Chandra observations provide unambiguous evidence that UHZ1 harbors a rapidly-accreting and heavily-obscured supermassive BH already in place when the Universe was only 500 Myrs old. \nA range of seeding scenarios operating within the first few hundred million years of the Universe has been proposed to account for the rapid assembly of early BHs, such as the one detected in UHZ1. They can be grouped broadly into 'light seed' and 'heavy seed' models. The light seed scenario[18] involving the collapse of the first generation - Population III - stars predicts initial BH seeds with 10 -100 M ⊙ . The remnant's BH mass estimate suffers from considerable uncertainty stemming from our current poor understanding of the initial mass function (IMF) of Population III stars[19, 20]. Such light seeds are unlikely to grow into massive BHs given their sub-optimal spatial locations and small capture radius[21, 22]. Alternately, heavy seed models invoking the formation of massive (10 4 -10 5 M ⊙ ) BH seeds through the direct collapse of pristine, massive gas clouds or pre-galactic disks[18, 23-26]. While these heavy seeds from direct collapse require rare physical conditions, simulations suggest that they are available in the early Universe[27-29]. There have also been additional proposals that suggest formation scenarios for intermediate mass BHs post the formation of Population III stars in the early Universe that blurs this broad dichotomy of light and heavy seeds. After the formation of Population III stars (that will inevitably result in the formation of light seeds), the subsequent episode of star formation is expected to produce dense nuclear star clusters. These environments could, in turn, serve as further new incubators for the formation of intermediate-mass BHs facilitated by stellar and gas dynamical processes that occur in them; from a range of stellar dynamical interactions that can occur in these dense environments[30] as well as extremely rapid \nFig. 4 Sketch of the growth of BHs with different initial seed masses and accretion rates: BHs formed via the light seed scenario with 10 -100 M ⊙ mass can only reach 10 4 -10 5 M ⊙ by z = 10 . 3 if they accrete at their Eddington limit (blue shaded region), which falls short by 2 -4 orders of magnitude of the BH mass estimated for UHZ1. Implausibly high sustained accretion at a rate of at least twice the Eddington limit would be required for light seeds to reach the BH mass close to that of UHZ1 (blue-hatched region). However, for light seeds continuous accretion at the Eddington limit or above for several hundred of million years is highly unlikely as noted by [14]. Heavy seed models with 10 4 -10 5 M ⊙ initial BH masses can grow to the mass of the BH powering UHZ1 by z = 10 . 3 assuming accretion at the Eddington limit (tan shaded region). All over-plotted models assume a radiative efficiency of 10% and continuous accretion. We also show the location of the three previously known highest redshift quasars at z ∼ 7 . 5, which were identified in large-area optical surveys[1517]. The systematic uncertainty (not shown) on the BH mass of these quasars is ∼ 0 . 5 dex. The mass range shown for UHZ1 corresponds to the derived estimate as noted in the text. \n<!-- image --> \namplified growth of an embedded light seed from wind-fed accretion[31, 32]. Because information about the initial seeding of BHs is mostly erased during the complex growth and subsequent evolution of BHs over cosmic time[33, 34], the detection of the X-ray quasar in UHZ1 at z ≈ 10 . 3 opens up a new, exciting frontier. \nPrevious SED fitting of the JWST photometric data[2] suggests that the galaxy has a stellar mass of M ⋆ = 0 . 4 +1 . 9 -0 . 2 × 10 8 M ⊙ , which makes it comparable to the inferred BH mass. This implies a strikingly different BH-to-host galaxy stellar mass ratio than observed in the local Universe, where the mass of the central BH is roughly 0 . 1% of the stellar mass. Such a high BH-galaxy mass ratio is predicted by theoretical studies of high-redshift galaxies seeded with heavy initial BHs[35]. Heavy seeds and their host galaxies are expected to inevitably transition through such an Outsize Black Hole Galaxy (OBG) stage at early times, before feedback-regulated efficient stellar assembly takes over, \neventually leading to the flipping of the mass ratio at later cosmic times[27, 35]. We note that the current JWST SED fitting for UHZ1[2] assumes that the restframe UV/optical emission derives solely from the stellar component modeled with a Salpeter initial mass function (IMF) and that the accreting BH does not contribute appreciably to the SED in the JWST bands. By contrast, the restframe UV/optical SED for OBGs is predicted to be dominated by accreting BH, as these transient sources are postulated to harbor a growing heavy initial seed with the stellar light contribution from stars modeled with a Population III IMF[35]. \nAccording to current BH formation theories, seed BHs may form as early as 200 million years after the Big Bang, implying that the BH in UHZ1 had only ∼ 300 million years to grow from its initial seed mass. To interpret the observed properties of the BH powering the X-ray quasar in UHZ1, we trace the mass assembly history of initial BH seeds starting from light and heavy seeds from z = 25 to the final inferred BH mass for UHZ1 of ∼ 10 7 -8 M ⊙ . We note that in this work, the growth history of seeds is tracked from birth to the epoch at which UHZ1 is detected, namely z = 10 . 3, and no claims are currently made for the future growth of this source over cosmic time. As shown in Figure 4, an initially light seed with a mass of 10 -100 M ⊙ , needs to be consistently accreting at more than twice the Eddington rate throughout; while a heavy seed with an initial mass of 10 4 -5 M ⊙ reaches the final mass of the BH powering UHZ1 by accreting at just the Eddington rate. We note from Figure 4 that UHZ1, by virtue of its inferred mass and redshift, currently offers more discriminating power viz-a-viz initial seeding models and hence more compelling evidence for heavy initial BH seeds than the three currently known z ∼ 7 . 5 quasars. Even in the very small regions probed, JWST may now be finding evidence for substantial episodes of BH growth at z > 8[36, 37]. The detection of UHZ1, and potentially other sources, may likely represent only the tip of the iceberg in terms of uncovering the accreting BH population at these early cosmic times. However, as demonstrated in cosmological simulations[38, 39], the most massive BH detected at very early epochs does not necessarily go on to produce the highest mass BH at later times as the growth history depends strongly on the details of the environment of these sources. Therefore, these early sources like UHZ1 may not necessarily all be the progenitors for the observed optically-detected quasars harboring 10 9 -10 M ⊙ BHs at z ∼ 6 -7. With additional data that is forthcoming and dedicated follow-up studies, with a better census, abundance estimates will soon be possible. \nFor a light initial BH seed in UHZ1, regular accretion at more than twice the Eddington rate for a period of 300 million years requires gas to be continuously delivered to the nucleus after efficient removal of its angular momentum. A gas reservoir in excess of 10 7 -10 8 M ⊙ would be needed to continuously feed the BH at this rate for several hundred million years. Numerical studies[40, 41], including the state-of-the-art Renaissance simulation suite, demonstrate that this is likely infeasible given the spatial location of light seeds given their Population III origin. By tracing the growth history of ∼ 15 , 000 light seeds, \nacross three differing over-density zoom-in regions with parsec scale resolution, populated by mini-haloes ranging in mass from 10 6 -10 9 M ⊙ , no significant growth by accretion was found. Even the most active BHs grew in mass by at most 10% over the seed mass across several hundred million years.[21]. Though many mini-haloes contained dense, cool gas clumps to accrete from, light seeds rarely inhabited such regions. In fact, star formation was found to compete with BH accretion in terms of gas consumption; and the resultant feedback processes in turn effectively suppressed BH growth. Furthermore, as light seeds are predicted to be produced from supernovae explosions of Population III stars, they are typically born into evacuated, low gas density environments, bereft of gas, potentially leading to their stunted growth[21]. \nWhile super-Eddington accretion is ubiquitously realized in idealized General Relativistic Magneto Hydro-dynamics simulations, these simulations do not adopt cosmological initial conditions, since their domain of integration extends only to ∼ 1000 gravitational radii; furthermore, they do not track the evolution of BHs on timescales of several hundred million years[42]. Additionally, cosmological simulations indicate that the feedback resulting from super-Eddington accretion rates curtails BH growth rather than amplifies it, in a variety of astrophysical environments, including the birth site of heavy seeds[43-45]. Therefore, starting with an initially light seed to reach the inferred BH mass of UHZ1 by z ≈ 10 . 3 requires multiple finely tuned conditions that have been demonstrated to be implausible in current cosmological simulations. Additionally, physical conditions at z = 10 . 3 are markedly different, as the Universe has not yet reionized, in comparison to z ∼ 6 -7. Therefore, studies of the sizes of proximity zones of quasars studied at those later epochs [46] that have been used to make a case for episodic super-Eddington accretion for a small fraction of the population may not capture the conditions at z = 10 . 3. Besides, simulations tailored to study high-redshift quasar absorption spectra also find that proximity zones sizes are highly epoch-dependent and compactness could result from the presence of Damped Lyman-alpha or Lyman Limit systems along the line of sight in the IGM rather than telegraph information about accretion physics [47]. Meanwhile, as we demonstrate, an initially heavy seed growing even modestly at the Eddington rate or even slightly below can comfortably reach the inferred mass of UHZ1.", 'The Chandra data': "The high spatial resolution Chandra data of Abell 2744 is the backbone of the analysis presented in this work. The processing and analysis of the data were carried out with standard CIAO [9] tools. In this work, we used 60 individual Chandra observations that are listed in Table 1. The total exposure time of all observations is 1.25 Ms. The first step of the analysis was to reprocess the individual observations using the chandra repro tool to apply the latest calibration data. Observations with Chandra 's ACIS-I detector are not prone to background flares, which was verified for each Abell 2744 observation. Although Chandra 's absolute pointing accuracy is better than 0 . 4 '' , we implemented the wcs match and wcs update tools to further improve the accuracy of the astrometry. To this end, we created a frame transformation for each observation by utilizing a set of bright X-ray sources in the Abell 2744 field, thereby minimizing the aspect difference. The aspect-corrected observations were co-added with the merge obs task using ObsID 8477 as the reference coordinate system. With this tool, we generated merged images in the 0 . 5 -7 keV, 0 . 5 -2 keV, and 2 -7 keV energy ranges. Similarly, we created maps of the point spread function (PSF) for each observation and each energy range, which were then weighted, and co-added. The merged images were used to carry out the X-ray photometry. To probe the Chandra PSF at the location of the UHZ1 X-ray source, we utilized the MARX ray tracing simulation suite (https://space.mit.edu/cxc/marx/). We applied the simulate psf tool and simulated the PSF for each observation assuming the E = 2 . 5 keV peak energy of the photons. Based on the co-added simulated PSF maps, we found that an r = 1 '' circular region encircles ≈ 88% of the source counts at the location of UHZ1. To probe the distribution of the X-ray photons associated with UHZ1, we extracted the number of source and background counts as a function of radius from r = 0 . 5 '' to r = 1 . 5 '' and derived the distribution of net (i.e. background subtracted) counts. We found that the distribution of X-ray photons associated with UHZ1 is in excellent agreement with the PSF model, which confirms that the X-ray source in UHZ1 is a point source and that the r = 1 '' region includes ≈ 88% of the source counts. \nAs discussed in the main body of the paper, we carry out the X-ray photometry using a 1 '' circular region centered on UHZ1[2]. The applied background region is a circular annulus with 3 '' and 6 '' inner and outer radii, respectively. The X-ray photometry was performed using the dmextract tool. We detected 42 counts from the source region with a background expectation of 21.4 counts, implying the presence of 20.6 net counts. Given the underlying and spatiallyvariable emission from the intracluster medium (ICM) of Abell 2744, applying a local background region is most suitable. Because spatial variations in the level of the ICM emission around UHZ1 may influence the observed detection significance, we probed the stability of the ICM emission in the vicinity of the \ngalaxy. Therefore, we measured the 2 -7 keV band X-ray surface brightness level of the emission using six circular annuli with a width of 2 '' between 2 '' to 14 '' radii. We found that the X-ray surface brightness of the emission is fairly uniform and its level exhibits < 3% variations in these annuli, which is comparable with the statistical uncertainties. Given the low-level spatial variation, we conclude that our particular choice of the background region does not influence our ability to detect the X-ray source co-spatial with UHZ1. Additionally, we also investigated the statistical uncertainty that stems from the finite number of counts in the background. Given the number of background counts in the 3 '' -6 '' annulus, we derive that the uncertainty associated with Poisson statistics is ≈ 4 . 4%. Given the background expectation of 21.6 counts in the source region, this implies an uncertainty of 0 . 95 counts. \nBecause the aim of this work is to search for X-ray sources associated with JWST -detected galaxies, we probed the astrometric consistency between Chandra and JWST . We identified a sample of bright X-ray sources with JWST counterparts and derived the offset between the X-ray and near-infrared centroids. Based on the individual offsets, we computed the average shift and the dispersion in the Abell 2744 field. We found that the average offset is δ = 0 . 33 '' with a dispersion of rms = 0.14, less than the Chandra pixel size. The small offset highlights the consistency between the Chandra and JWST astrometry and demonstrates that inaccuracies in the absolute astrometry do not present a systematic uncertainty in the detection of the X-ray source at the position of UHZ1.", 'Spectral fitting of the X-ray data': "X-ray counts were extracted from the merged Level 2 events file using the CIAO package specextract along with associated response matrices. Two apertures were used for the photon extraction, 1) an r = 0 . 8 '' circular aperture at the position of UHZ1, and 2) an annulus with an inner radius of 3 '' and outer radius of 6 '' centered at UHZ1. The annulus region represents a 'clean' region containing only emission arising from the foreground cluster gas, while the smaller source region contains emission from both the foreground cluster and UHZ1. Using the grppha package, we grouped the extracted photons to contain at least one count per spectral bin for the use of Cash statistics during the fitting process[48]. \nWe proceed by first fitting the cluster-only X-ray emission using XSPEC version 12.13 assuming an APEC model attenuated by an additional photoelectric absorber arising from material in the Milky Way with N H = 1 . 35 × 10 20 cm -2 [10]. We froze the previously known redshift of the cluster to z = 0 . 308 and set an abundance of 0.3 Solar[49]. We find a best-fit temperature of kT = 10 . 9 ± 1 . 9 keV and normalization of (17 . 23 ± 0 . 57) × 10 -6 with C-stat = 277.7 (for 323 degrees of freedom). This fit is consistent with temperature measurements of the intracluster medium around UHZ1[50]. We use this best-fit cluster-gas model as a fixed and known background in the smaller circular source aperture after renormalizing to account for the differences in \nObs. Date 2022-11-26 2023-01-28 2022-05-04 2022-09-08 2022-09-27 2022-09-30 2022-11-18 2022-09-17 2022-04-24 2022-09-02 2022-09-08 2022-05-04 2022-11-26 2022-08-01 2022-07-12 2022-10-09 2022-06-12 2022-05-04 2022-05-18 2022-11-11 2022-01-18 2022-09-09 2022-09-24 2022-09-26 2022-11-15 2022-12-02 2023-01-27 2023-01-28 2023-01-28 2023-01-29 \nDetector \n(ks) \nt \nA CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I \nexp 18.66 14.32 15.18 21.62 17.02 27.87 28.71 24.75 13.40 13.90 21.80 12.32 37.59 33.64 27.46 27.72 24.75 12.61 31.75 18.15 11.74 21.95 9.78 9.78 25.15 19.65 12.42 11.93 13.21 9.78 \nObs. ID \nObs. Date \nDetector \n25938 25939 25942 25944 25945 25948 25951 25953 25954 25956 25957 25958 25963 25967 25968 25969 25970 25971 25972 25973 26280 27347 27449 27450 27556 27575 27678 27679 27680 27681 \n2001-09-03 2007-09-10 2006-11-08 2007-06-10 2007-06-14 2022-12-02 2022-09-06 2022-11-08 2022-09-23 2022-09-25 2022-04-19 2022-04-18 2022-09-03 2022-10-15 2022-09-03 2022-09-13 2022-06-13 2022-06-13 2022-06-14 2022-09-04 2022-09-07 2022-09-02 2022-05-03 2022-08-26 2022-11-15 2022-04-23 2022-05-05 2022-04-21 2023-01-26 2022-11-27 \nA CIS-S A CIS-I A CIS-I A CIS-I A CIS-I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I A CI S -I \nt exp (ks) \n24.82 8.07 18.62 45.91 27.81 9.78 24.46 36.80 22.61 19.31 16.85 15.37 19.64 28.20 21.08 20.63 25.29 30.51 31.37 10.94 21.79 23.60 15.87 27.72 19.92 14.58 14.08 19.30 12.92 30.78 \nObs. ID \n2212 7712 7915 8477 8557 25278 25279 25907 25908 25910 25911 25912 25913 25914 25915 25918 25919 25920 25922 25923 25924 25925 25928 25929 25930 25931 25932 25934 25936 25937 \nFig. 5 Best-fit statistical uncertainties and posterior draws of the demagnified intrinsic X-ray luminosity ( L X , int ) and gas column density ( N H ) of UHZ1. One, 2, and 3 σ (solid, dashed, and dotted lines, respectively) contours are constructed from the X-ray spectral fits to the Chandra data. Black lines show the statistical fits for a fixed AGN spectral slope of Γ = 1 . 9 and assuming fixed best-fit (rescaled) normalization and plasma temperature for the background APEC component. Light-gray color represents a fixed slope of Γ = 2 . 3. Blue lines are the posterior draws from a Bayesian fit employing broad informative Gaussian priors for Γ, n APEC and kT. The mode of the posterior is shown with a blue cross. For values of N H ≳ 2 × 10 24 (equivalent to the lower 1 σ contours), a strong steep degeneracy between L X , int and N H is clearly evident, we thus adopt this threshold to mitigate these degeneracies, resulting in an adopted L X ∼ 1 . 9 × 10 44 erg/s. \n<!-- image --> \naperture sizes. Employing this pre-determined model directly to the photons extracted from the source aperture we find a poor fit with cstat=42.2 and 32 degrees of freedom. We note that if we do not assume a known value, and instead fit for the gas temperature, the required temperature is pegged at the maximally allowed temperature ( kT ∼ 67 keV) for the APEC model, owing to the presence of additional high energy photons in the source aperture. \nWe next fit the source aperture using a combination of the pre-determined aperture-corrected cluster emission and a redshifted absorbed power-law component to represent a distant point source. The redshift of the model was set to z = 9 . 99 i.e., the photometric redshift solution determined of UHZ1[2] using the updated EaZY spectral template library specifically designed for use with JWST data[51]. However, we found that a simple redshifted photoelectric absorber model ( zphabs ) requires an extremely high column density in excess \nof N > 10 25 cm -2 , and thus we instead opt to use the MyTorus library[11] to model the absorption as these models are better suited to the complex emission arising in highN H environments. We assume a relatively minimal model with a zeroth-order absorbed continuum combined with a Comptonscattered continuum. Given the low-photon statistics, we set a power-law slope of Γ = 1 . 9, a disk inclination of 85 · , and a Compton-scattering fraction of 2%, all of which are typical of rapidly accreting heavily absorbed AGN[12]. We find a best-fit cstat=32.5 with 30 degrees of freedom, i.e., a ∆C-stat = 9.7 over the APEC -only model. For the two additional degrees of freedom required, the APEC+AGN model passes an F-test at the > 99% significance level over the simple APEC-only model and thus provides very strong evidence for the presence of an additional X-ray point source. \nThe best-fit column density is found to be N H ≈ 8 +inf -7 × 10 24 cm -2 , with an intrinsic 2 -10 keV luminosity of L X , int ≈ 9 × 10 45 erg s -1 (using the magnification factor of µ = 3 . 81 established for UHZ1[2]). However, both of these measurements are degenerate in the current data, particularly in the N H > 10 24 cm -2 regime (see Figure 5). An analysis of the C-stat space shows that within 2 σ , values as low as N H = 10 22 cm -2 are allowable, owing mainly to the diminished soft-energy response of Chandra's ACIS detector. Moreover, we determine that due to the high-energy restframe coverage, for column densities of N < 10 24 cm -2 , the allowable range of L X , int is very stable, ∼ (2 -4) × 10 43 erg s -1 , which we consider as the lower limit on the X-ray luminosity. \nWe determined that the precise parameterization of the presumed AGN component has a mild effect on our subsequent L X and N H measurements. For example, we tested the effect of our assumption of Γ = 1 . 9 on our derived values. From X-ray analysis of z ∼ 6 -7 quasars, a marginal steepening of the spectral slope to Γ ∼ 2 . 3 was found [17]. Assuming this steeper value, we find no significant change in our measured N H , but find a systematic shift to larger L X , int by a factor ∼ 1.4. Thus, within our uncertainties, we find no significant difference. Furthermore, significantly larger values for the scattering fraction are ruled out by the lack of detected soft photons. Similarly, lower inclination angles i < 75 degrees are also disfavored due to insufficient soft emission. Moreover, the precise values of disk inclinations in the range i = 75-90 degrees have a weak systematic effect on N H , but these variations are well within our uncertainties. \nWe further note that fixing the parameters of the APEC model, used to parameterize the foreground cluster emission, to the best-fit values found during the background analysis may also have an effect on the measurements of the AGN component. In an attempt to simultaneously fold in all of these uncertainties, we opted for an additional Bayesian treatment of the data. We applied physically-motivated informative priors on the AGN spectral slope parameterized as P (Γ) ∼ N (1 . 9; 0 . 15), the APEC normalization with P ( n APEC ) ∼ N (3 . 5 × 10 -7 ; 0 . 2 × 10 -7 ) and plasma temperature with P ( kT ) ∼ N (10 . 9; 1 . 9). These priors were designed to reflect the previous \nknowledge of AGN spectra as well as the highly constrained fit to the background spectra. By contrast, flat uninformative priors were applied to both L AGN and N H . We performed a Markov-Chain Monte-Carlo using a GoodmanWeare algorithm with 10 walkers, 10 6 draws, and a burn-in of 10 5 per chain. We find that the posteriors of the draws for the APEC component are dominated by the priors, as is expected. We note that using an uninformative set of priors on the APEC component tends to a luminosity that is a factor ∼ 3 greater than the background expectation and to an unfeasible temperature of kT > 60 keV. The joint posterior distribution for N H and L AGN are provided in Figure 5, and show remarkable consistency with our previous and more simple frequentist fit, albeit now with marginally larger 1σ uncertainties around the peak of the posterior driven by the non-fixed nature of Γ and the background APEC model. The lower-1 σ values for N H and L X are consistent between the Bayesian and frequentist fits, and show the same steep degeneracy between N H and L X towards higher values. Interestingly, we find that the allowable range of N H is more constrained than found previously, owing to the ability of the AGN spectral slope to becoming marginally flatter, as well as the allowance of a cooler plasma temperature, to balance the soft-Xrays. The Bayesian analysis places a 95% lower limit on the column density of N H ∼ 7 × 10 23 cm -2 , and L X ∼ 4 × 10 43 erg s -1 , which corresponds to a BH mass of M BH ∼ 6 × 10 6 M ⊙ assuming accretion at the Eddington limit. Because BHs formed via light seeds can only reach 10 4 -10 5 M ⊙ by z = 10 . 3 if they accrete at their Eddington limit, the overall conclusions presented in the main body of the paper remain the same even if we assume the 95% lower limit on the X-ray luminosity. Finally, we note that significantly more sensitive data taken with the next generation of X-ray telescopes will be required to provide a more in-depth analysis of the X-ray AGN properties of UHZ1.", 'Verifying the statistical significance of the detection': "Using the r = 1 '' source extraction region we found that the X-ray source associated with UHZ1 is detected at the 4 . 2 σ statistical confidence level. To further probe the detection significance, we performed an additional statistical test. From the 2 -7 keV band Chandra X-ray image, we cut out a 25 '' × 25 '' region. We construct a background map within this rectangular region, by making the ansatz that a point source with r = 1 '' radius exists at the location of each given pixel in the map. To avoid contamination by the bright source associated with UHZ1, we masked an r = 2 '' region around this Xray source. Given the size of the selected region, there are ≈ 2500 potential aperture positions. We measured the number of total counts associated with each region and present the distribution of the counts in Figure 6. Outside of the masked region containing UHZ1, none of the regions include 42 counts (i.e., the detected number of photons for the X-ray source in UHZ1). The bestfit Poisson mean of the count distribution is 19.9 counts, which is marginally lower than the Gaussian background expectation for UHZ1 ( ∼ 21 . 4). Based on the best-fit, we conclude that the probability of observing 42 or more counts \nFig. 6 Distribution of background counts extracted from individual r = 1 '' apertures in a 25 '' × 25 '' box centered on the position of UHZ1. The central r = 2 '' region is masked to exclude contaminating X-ray photons from UHZ1 itself. The mean Gaussian background ( ∼ 21 . 4 photons; blue dotted line) is found to be in good agreement with the best-fit Poisson mean ( ∼ 19 . 9). The best-fit Poisson distribution shows that the probability of detecting a point source with 42 counts (red dash line) is 1 . 05 × 10 -5 , and hence, we conclude that UHZ1 is robustly detected with a 4 . 2 -4 . 4 σ significance. \n<!-- image --> \nfrom a Poisson distribution with µ = 19 . 9 is P ( X ≥ 42) = 1 . 05 × 10 -5 , which is equivalent to a 4 . 4 σ detection. This result is in agreement with the value derived in the main body of the paper, where we obtained a 4 . 2 σ detection significance for UHZ1 using standard aperture photometry methods. Thus, this independent analysis quantitatively and statistically confirms the robust nature of the X-ray detection.", 'Star-formation and clumping cannot be the origin of X-ray emission associated with UHZ1': "Although the characteristics of the X-ray source associated with UHZ1 are consistent with those of a heavily obscured AGN, we explore if the source may have a different origin. We explore two possibilities: (1) star formation associated with UHZ1; (2) gas clumping in the intracluster medium of Abell 2744. \nIn actively star-forming galaxies the total X-ray luminosity originating from X-ray binaries is proportional to the star-formation rate of galaxies. This relationship is robust and has been calibrated for local[52] and for distant galaxies[53, 54]. The observed L X -SFR relation can be described with a linear scaling relation and it shows an increasing trend with higher redshift due to the lower metallicity of galaxies. Using the redshift-dependent L X -SFR scaling relation[54] and the star formation rate of UHZ1 (SFR = 4 . 4 M ⊙ yr -1 ), we predict a 2 -10 keV band X-ray luminosity of ∼ 4 × 10 41 erg s -1 . This value falls substantially short of the observed X-ray luminosity by about 2 -4 orders of magnitude from the measured X-ray luminosity of UHZ1. As a caveat, we note that the redshift dependence of the L X -SFR relation has only been tested to z ∼ 4, implying that a steeper redshift dependence is feasible for galaxies in the early universe. However, an L X -SFR relation that is two orders of magnitude higher than this relation would be inconsistent with stacking observations of the Chandra Deep Field South[55] and the RELICS clusters[56]. \nWe further investigated the possibility that the excess X-ray emission is associated with the intracluster medium of Abell 2744, and is potentially in projection with UHZ1. In this scenario, the emission does not originate from a distant point source, but from a small, dense clump of gas in the intracluster medium. However, dense gas clumps are predicted to be cooler than the surrounding intracluster medium[57]. This, however, is inconsistent with the observed hard energy spectrum extracted from the source aperture. For example, for a gas temperature of kT ∼ 5 keV (roughly half of that of the observed temperature of the intracluster medium), we would expect to detect a comparable number of counts in the 0 . 5 -2 keV and 2 -7 keV bands. This is inconsistent with the Chandra observations, as the source is robustly undetected below E < 2 keV. Indeed by contrast, and as discussed in the previous section, the observed X-ray spectrum requires a substantially hotter (kT ∼ 67 keV) temperature in order to fit with a simple thermal plasma, not cooler. Thus, we conclude that the excess X-ray emission cannot be associated with the intracluster medium of Abell 2744. \nData Availability: The JWST data of Abell 2744 is publicly available at MAST (http://archive.stsci.edu). The Chandra data is available upon request. \nAuthor contributions: ' A.B. is the Principal Investigator of the Chandra proposal, analyzed the Chandra observations, led the analysis and drafted the paper. A.D.G and P.N. contributed equally. A.D.G. carried out the spectral fitting of the X-ray source, played a major role in writing the manuscript, and provided figures. P.N. led the theoretical interpretation and played a major role in writing the manuscript. O.E.K. contributed to the analysis of the Chandra data and its interpretation and provided a figure. G.R.T. contributed figures and text to the manuscript. U.C. contributed to the sample selection and data analysis. M.V. contributed to the interpretation and text of the manuscript. R.P.K. contributed to the interpretation of the paper. W.R.F., C.J., E.C., and I.Z. reviewed the manuscript and contributed to the text. \nAcknowledgements: We thank the anonymous reviewers for their constructive reports that have allowed us to clarify and improve several aspects of the manuscript. This research has made use of data obtained from the Chandra Data Archive and software provided by the Chandra X-ray Center (CXC) in the application packages CIAO and Sherpa. The authors thank helpful discussions with Fabio Pacucci, Angelo Ricarte, and Peter Edmonds. ' A.B., G.R.T., R.P.K., C.J., and W.R.F. acknowledge support from the Smithsonian Institution and the Chandra X-ray Center through NASA contract NAS8-03060. A.D.G. acknowledges support from NSF/AAG grant No. 1007094. P.N. acknowledges support from the Black Hole Initiative at Harvard University, which is funded by grants from the John Templeton Foundation and the Gordon and Betty Moore Foundation. O.E.K. is supported by the GA ˇ CR EXPRO grant No. 21-13491X. \nCompeting Interests: The authors declare that they have no competing financial interests. \nCorrespondence: Correspondence and requests for materials should be addressed to ' AB (email: [email protected]).", 'References': "- [1] Castellano, M., Fontana, A., Treu, T., Santini, P., Merlin, E., Leethochawalit, N., Trenti, M., Vanzella, E., Mestric, U., Bonchi, A., Belfiori, D., Nonino, M., Paris, D., Polenta, G., Roberts-Borsani, G., Boyett, K., Bradaˇc, M., Calabr'o, A., Glazebrook, K., Grillo, C., Mascia, S., Mason, C., Mercurio, A., Morishita, T., Nanayakkara, T., Pentericci, L., Rosati, P., Vulcani, B., Wang, X., Yang, L.: Early Results from GLASS-JWST. III. Galaxy Candidates at z 9-15. Astrophys. J. 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2015PhRvL.115l1102B
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Fast and Accurate Prediction of Numerical Relativity Waveforms from Binary Black Hole Coalescences Using Surrogate Models
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2015-01-01
| 57 | 0.45 | 160 |
['-', '-', '-', '-', 'methods numerical', 'methods numerical', '-', 'waves', '-', '-', 'methods numerical', '-', '-', '-', '-']
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[]
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Simulating a binary black hole coalescence by solving Einstein's equations is computationally expensive, requiring days to months of supercomputing time. Using reduced order modeling techniques, we construct an accurate surrogate model, which is evaluated in a millisecond to a second, for numerical relativity (NR) waveforms from nonspinning binary black hole coalescences with mass ratios in [1, 10] and durations corresponding to about 15 orbits before merger. We assess the model's uncertainty and show that our modeling strategy predicts NR waveforms not used for the surrogate's training with errors nearly as small as the numerical error of the NR code. Our model includes all spherical-harmonic <SUP>-2</SUP>Y<SUB>ℓm</SUB> waveform modes resolved by the NR code up to ℓ=8 . We compare our surrogate model to effective one body waveforms from 50 M<SUB>⊙</SUB> to 300 M<SUB>⊙</SUB> for advanced LIGO detectors and find that the surrogate is always more faithful (by at least an order of magnitude in most cases).
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| 7 |
https://arxiv.org/pdf/1502.07758.pdf
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{'Fast and accurate prediction of numerical relativity waveforms from binary black hole coalescences using surrogate models': "Jonathan Blackman, 1 Scott E. Field, 2 Chad R. Galley, 1 B'ela Szil'agyi, 1 Mark A. Scheel, 1 Manuel Tiglio, 3 and Daniel A. Hemberger 1 \n1 TAPIR, Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA 2 Center for Radiophysics and Space Research, Cornell University, Ithaca, New York 14853, USA 3 Center for Astrophysics and Space Sciences, Center for Computational Mathematics, San Diego Supercomputer Center, University of California San Diego, 9500 Gilman Drive, La Jolla, California 92093-0424, USA (Dated: October 9, 2015) \nSimulating a binary black hole (BBH) coalescence by solving Einstein's equations is computationally expensive, requiring days to months of supercomputing time. Using reduced order modeling techniques, we construct an accurate surrogate model, which is evaluated in a millisecond to a second, for numerical relativity (NR) waveforms from non-spinning BBH coalescences with mass ratios in [1 , 10] and durations corresponding to about 15 orbits before merger. We assess the model's uncertainty and show that our modeling strategy predicts NR waveforms not used for the surrogate's training with errors nearly as small as the numerical error of the NR code. Our model includes all spherical-harmonic -2 Y glyph[lscript]m waveform modes resolved by the NR code up to glyph[lscript] = 8 . We compare our surrogate model to Effective One Body waveforms from 50-300 M glyph[circledot] for advanced LIGO detectors and find that the surrogate is always more faithful (by at least an order of magnitude in most cases). \nSince the breakthroughs of 2005 [1-3], tremendous progress in numerical relativity (NR) has led to hundreds of simulations of binary black hole (BBH) coalescences [4-10]. This progress has been driven partly by data analysis needs of advanced ground-based gravitational wave detectors like LIGO [11] and Virgo [12]. Recent upgrades to these detectors are expected to yield the first direct detections of gravitational waves (GWs) from compact binary coalescences [13]. \nDespite the remarkable progress of the NR community, a single high-quality simulation typically requires days to months of supercomputing time. This high computational cost makes it difficult to directly use NR waveforms for data analysis, except for injection studies [4, 9], since detecting GWs and inferring their source parameters may require thousands to millions of accurate gravitational waveforms. Nevertheless, a first template bank for nonspinning binaries in Advanced LIGO has been recently constructed from NR waveforms [14]. Furthermore, NR waveforms have been used successfully in calibrating inspiral-merger-ringdown (IMR) effectiveone-body (EOB) [15-21] and phenomenological [22-25] models. These models have free parameters that can be set by matching to NR waveforms and are suitable for certain GW data analysis studies [26]. However, these models can have systematic errors since they assume a priori physical waveform structure and are calibrated and tested against a small set of NR simulations. \nIn this Letter, we present an ab initio methodology based on surrogate [27, 28] and reduced order modeling techniques [29-33] that is capable of accurately predicting the gravitational waveform outputs from NR without any phenomenological assumptions or approximations to general relativity. From a small set of specially selected non-spinning BBH simulations performed with \nFIG. 1. Top : The + polarization (2 , 2) mode prediction for q = 2, the surrogate model's worst prediction over q from a 'leave-one-out' surrogate that was not trained with this waveform (see below). Our full surrogate, trained on the entire data set, is more accurate. Bottom : Phase δϕ 2 , 2 and waveform differences between the surrogate and highest resolution (Lev4) SpEC waveforms. Also shown is the SpEC numerical truncation error found by comparing the two highest resolution (Lev4 and Lev3) waveforms. \n<!-- image --> \nthe Spectral Einstein Code (SpEC) [34-36], we build a surrogate model that can be used in place of performing SpEC simulations. The techniques are general, however, and directly apply to other NR codes or even analytical waveform models. The surrogate model constructed here generates non-spinning BBH waveforms with mass ratios q ∈ [1 , 10], contains 25-31 gravitational wave cycles before peak amplitude, and includes many sphericalharmonic modes (see Table II and its caption). These choices are made based on available NR waveforms and are not limitations of the method. Our surrogate model has errors close to the estimated numerical error of the \ninput waveforms. An example comparing the surrogate output to an NR waveform can be seen in Fig. 1. This simulation took 9 . 3 days using 48 cores but only ∼ 0 . 01 sec for the surrogate evaluation of the (2,2) mode. \nPrevious work [27, 37] built surrogates for EOB waveforms; building and assessing surrogate models of NR waveforms have unique challenges associated with input waveforms that are expensive to compute. We summarize next the construction of our model, focusing on steps not addressed in [27] but are required for NR surrogates. \nParametric samplingTypically, a surrogate model is trained on a dense set of waveforms known as the training set . In the case of NR, we cannot afford to generate a large number of waveforms. Instead, we generate a dense set of non-spinning waveforms using an EOB model [18], as implemented in [38], which contains the ( glyph[lscript], m ) = { (2 , 2) , (2 , 1) , (3 , 3) , (4 , 4) , (5 , 5) } spin-weight -2 spherical-harmonic modes and captures robust features of NR waveforms. The EOB training set waveforms are computed for times in [ -2750 , 100] M ( M is the total mass), which is the interval over which we build our surrogates. \nNext, on this training set we apply a greedy algorithm to expose the most relevant mass ratio values [39, 40]. The algorithm proceeds from a linear basis constructed from i waveforms already chosen. The L 2 norms of the differences between the training set waveforms and their projection onto this basis are computed. The waveform with the largest such error is added to the basis as its i + 1 element. SpEC simulations of non-spinning BBH mergers are then performed for these mass ratios. The resulting NR waveforms are used to build our surrogates without any further input from the EOB model. \nWe seeded the greedy algorithm with 5 publicly available SpEC simulations of non-spinning BBH mergers [10, 19] (see Table I), and the next 17 (ordered) mass ratio values are the algorithm's output based on the EOB model. The final ∼ 10 mass ratios are included to improve the surrogate if necessary, since we can assess the surrogate model's accuracy only after it is built. Our method for building surrogates is hierarchical [27, 40]; additional NR waveforms can be included to improve the model's accuracy. \nGenerating the NR waveformsTable I summarizes the 22 SpEC simulations used in this paper. See, e.g., Ref. [35] for the numerical techniques used in SpEC. The numerical resolution is denoted by 'Lev i ', where i is an integer that controls the local truncation error in the metric and its derivatives allowed by adaptive mesh refinement (AMR) in SpEC; larger numbers correspond to smaller errors (the error threshold scales like e -i ) and more computationally-expensive simulations. The scaling of global quantities (e.g. waveform errors) with i is difficult to estimate a priori . Two to five levels of resolution are simulated for each mass ratio. To achieve quasi-circular orbits, initial data are subject to an itera- \nTABLE I. Properties of the highest resolution SpEC simulations used for building BBH waveform surrogates. The quantity e -5 is the orbital eccentricity divided by 10 5 [43]. The duration T/M and number of orbits (Orbs) are also given. The SpEC simulations are available in the public waveform catalog [10] under the name 'SXS:BBH: ID .' \n| # ID q | e - 5 T/M | Orbs # ID | q | e - 5 | T/M Orbs |\n|-----------------|---------------|---------------|-----------------|---------------|---------------|\n| 1 180 1 . 00 5 | . | 1 9867 28 . 2 | 12 191 2 . | 51 65 6645 22 | . 5 |\n| 2 181 6 . 00 5 | . | 8 7056 26 . 5 | 13 192 6 . 58 4 | . | 0 5149 21 . 1 |\n| 3 182 4 . | 00 12 3840 15 | . 6 | 14 193 3 . 50 3 | . | 0 5242 19 . 6 |\n| 4 183 3 . 00 4 | . | 8 4008 15 . 6 | 15 194 1 . | 52 74 5774 19 | . 6 |\n| 5 184 2 . | 00 15 4201 15 | . 6 | 16 195 7 . | 76 22 5226 21 | . 9 |\n| 6 185 9 . | 99 31 5817 24 | . 9 | 17 196 9 . | 66 23 5330 23 | . 1 |\n| 7 186 8 . | 27 16 5687 23 | . 7 | 18 197 5 . | 52 25 5061 20 | . 3 |\n| 8 187 5 . 04 3 | . | 0 4807 19 . 2 | 19 198 1 . | 20 17 6315 20 | . 7 |\n| 9 188 7 . | 19 15 5439 22 | . 3 | 20 199 8 . 73 8 | . | 5 5302 22 . 6 |\n| 10 189 9 . | 17 13 6019 25 | . 2 | 21 200 3 . | 27 36 5507 20 | . 2 |\n| 11 190 4 . 50 2 | . | 5 5199 20 . 1 | 22 201 2 . | 32 15 5719 20 | . 0 | \nFIG. 2. The relative error, | h 22 i -h 22 i +1 | / | h 22 i +1 | , of successive resolutions SpEC Lev i for the (2,2) mode of simulation 19 in Table I. Top : Waveform output as directly given by SpEC ('Unaligned'). Bottom : 'Aligned,' which involves a multimode peak alignment scheme described by Eq. (2) followed by a rotation of the binary around the z -axis to align the waveform phases at t i = -2750 M . Our surrogate is built from NR waveform data after alignment, and so this measurement of truncation error is the most relevant for surrogate model building. \n<!-- image --> \ntive eccentricity reduction procedure resulting in eccentricities glyph[lessorsimilar] 7 × 10 -4 [41-43]. \nSpEC numerically solves an initial boundary value problem defined on a finite computational domain. To obtain waveforms at future null infinity I + , we use the Cauchy characteristic extraction (CCE) method [4448]. Using the PittNull code [44-46], we compute the Newman-Penrose scalar Ψ 4 at I + and finally obtain the gravitational wave strain h through two temporal integrations. We minimize the low-frequency, noise-induced \n'drifts' [47] by using frequency cut-offs. 1 \nFigure 2 shows the convergence typically observed in our simulations when using AMR. Because AMR makes independent decisions for different Lev i , a particular subdomain may sometimes have the same number of grid points for two different values of Lev i at a given time, and the subdomain boundaries do not necessarily coincide for different Lev i . Thus, plots like Figure 2 sometimes show anomalously small differences between particular pairs of numerical resolutions (for instance Lev2 vs. Lev3 near t = -3500 M in the top panel of Figure 2). See Sec. IIIB of [35]. Nevertheless, the waveform differences generally decrease quickly with increasing resolution. Let \nδh glyph[lscript],m ( q ) ≡ ‖ h glyph[lscript],m 1 ( · ; q ) -h glyph[lscript],m 2 ( · ; q ) ‖ 2 ∑ glyph[lscript],m ‖ h glyph[lscript],m 2 ( · ; q ) ‖ 2 (1) \nbe the disagreement between two waveform modes h glyph[lscript],m 1 and h glyph[lscript],m 2 where ‖ h glyph[lscript],m ( · ; q ) ‖ 2 = ∫ dt | h glyph[lscript],m ( t ; q ) | 2 . We estimate the numerical truncation error of each mode when h 1 and h 2 are waveforms computed at the two highest resolutions. The full waveform 2 error for a given mass ratio is δh ( q ) = ∑ glyph[lscript],m δh glyph[lscript],m ( q ). We report numerical truncation errors after an overall simulation-dependent time shift and rotation (which we shall refer to as surrogate alignment , described in the next section), which are physically unimportant coordinate changes. The resulting estimated numerical truncation errors of the dominant (2 , 2) modes, using our surrogate alignment scheme, are shown in Fig. 3 (black circles). \nAdditional error sources are non-zero eccentricity in the (intended to be circular) NR initial data, and an imperfect procedure for integrating Ψ glyph[lscript],m 4 to obtain h glyph[lscript],m ≡ A glyph[lscript],m exp( -iϕ glyph[lscript],m ). These both cause small oscillations in the waveform amplitudes A glyph[lscript],m ( t ) and phases ϕ glyph[lscript],m ( t ) [47, 50] that we model following [50]. We also compute the error in the strain integration scheme by comparing Ψ glyph[lscript],m 4 to two time derivatives of h glyph[lscript],m , as well as estimates for numerical errors in the CCE method [48]. For the (2 , 2) mode, these additional errors are negligibly small compared to SpEC truncation errors (cf. Fig. 3). \nPreparing NR waveforms for surrogate modelingWe apply a simulation-dependent time shift and physical rotation about the z -axis so that all the modes' phases are aligned. This reveals the underlying parametric smoothness in q that will be useful for building a surrogate. Our \nFIG. 3. Numerical truncation errors (black) dominate all other sources of error for the (2,2) mode, except for simulation 1 ( q = 1), where the truncation errors are already very small. For some weaker modes, systematic amplitude oscillations primarily due to eccentricity may become more relevant. \n<!-- image --> \ntime shifts set each waveform's total amplitude \nA ( t ; q ) 2 ≡ ∫ S 2 d Ω | h ( t, θ, φ ; q ) | 2 = ∑ glyph[lscript],m | h glyph[lscript],m ( t ; q ) | 2 , (2) \nto be maximum at t = 0. After enforcing this alignment scheme we interpolate the waveform mode amplitudes and phases onto an array of uniformly spaced times in [ -2750 , 100] M , with ∆ t = 0 . 1 M . Finally, we align the initial gravitational wave mode phases by performing a simulation-dependent, constant (in time) physical rotation about the z -axis so that ϕ 2 , 2 ( t i ) = ϕ 2 , -2 ( t i ), which fixes a physical rotation up to multiples of π . We resolve the ambiguity by requiring ϕ 2 , 1 ( t i ) ∈ ( -π, 0]. Waveform truncation errors, after performing this surrogate alignment scheme, are shown in Fig. 2. In what follows, we call 'truncation error after surrogate alignment' simply 'truncation error.' \nBuilding the surrogateEach m > 0 mode, h glyph[lscript],m ( t ; q ), is modeled separately while (due to reflection symmetry about the orbital plane) m < 0 modes are evaluated using h glyph[lscript], -m ( t ; q ) = ( -1) glyph[lscript] h glyph[lscript],m ( t ; q ) ∗ . We model all m glyph[negationslash] = 0 modes but keep only those yielding smaller surrogate errors δh glyph[lscript],m compared to setting the mode to zero. Table II lists our modeled modes and their errors. \nOur complete surrogate waveform model is defined by h S ( t, θ, φ ; q ) = ∑ glyph[lscript],m h glyph[lscript],m S ( t ; q ) -2 Y glyph[lscript]m ( θ, φ ) where \nh glyph[lscript],m S ( t ; q ) = A glyph[lscript],m S ( t ; q ) e -iϕ glyph[lscript],m S ( t ; q ) , X glyph[lscript],m S ( t ; q ) = N X ∑ i =1 B glyph[lscript],m X,i ( t ) X glyph[lscript], m i ( q ) , X = { A,ϕ } . (3) \nUnlike Ref. [27], we construct a reduced basis representation for the waveform amplitudes and phases separately, instead of the waveforms themselves [37]. Here, the { B glyph[lscript]m X,i } N X i =1 are computed off-line from the SpEC waveforms [27]. At a set of N X specially selected times { T glyph[lscript]m X,i } N X i =1 , which are the empirical interpolant nodes [27, 51], the functions X glyph[lscript]m i ( q ) ≈ X glyph[lscript]m ( T glyph[lscript]m X,i ; q ) approximate the parametric variation of the amplitudes and \nphases (via fitting). A thorough discussion of surrogate model building steps is presented in [27]. When evaluating the surrogate at a particular mass ratio, the fits are evaluated first to determine the amplitudes and phases at their respective interpolating times { T glyph[lscript]m X,i } N X i =1 . The remaining operations yield the surrogate model prediction, h S ( t, θ, φ ; q ). \nTo find each X glyph[lscript]m i ( q ) we perform least-squares fits to the 22 data points, { X glyph[lscript]m ( T glyph[lscript]m X,i ; q j ) } 22 j =1 . All fits except odd m mode amplitudes use 5th degree polynomials in the symmetric mass ratio, ν = q/ (1 + q ) 2 . For odd m modes, the amplitude approaches 0 and its derivative with respect to ν diverges as q → 1 (or ν → 1 / 4). Consequently, we use A glyph[lscript]m i ( ν ) = ∑ 5 n =1 / 2 , 1 a glyph[lscript]m n (1 -4 ν ) n to account for this behavior. The waveform phases of odd m modes at q = 1, which are undefined, are excluded when fitting for each ϕ glyph[lscript]m i ( q ). \nAssessing surrogate errorsWe next assess the surrogate's predictive quality. To quantify the error in the surrogate model itself, as opposed to its usage in a data analysis study, we do not minimize the errors over relative time and phase shifts here. \nA first test is a consistency check to reproduce the 22 input SpEC waveforms used to build the surrogate. These errors are shown in Fig. 4 (red squares) and are comparable to or smaller than the largest SpEC truncation errors (black circles). \nA more stringent test is the leave-one-out crossvalidation (LOOCV) study [52]. For each simulated mass ratio q i , we build a temporary trial surrogate using the other 21 waveforms, evaluate the trial surrogate at q i , and compare the prediction with the SpEC waveform for q i . Hence, the trial surrogate's error at q i should serve as an upper bound for the full surrogate trained on all 22 waveforms. Repeating this process for all possible 20 LOOCV tests 3 results in Fig. 4 (blue triangles). Despite the i th trial surrogate having no information about the waveform at q i , the errors remain comparable to the largest SpEC truncation errors. The LOOCV errors are typically twice as large as the full surrogate ones confirming the former as bounds for the latter. Relative errors for selected modes are shown in Table II. While weaker modes have larger relative errors, their power contribution is small enough that the error in the full surrogate waveform, δh , is nearly identical to the SpEC resolution error. \nA third test is to compare the surrogate waveforms to those of a second surrogate, built from the second highest resolution SpEC waveforms. The resulting comparison is shown in Fig. 4 (cyan line). These errors are comparable to SpEC waveform truncation errors (black circles). \nTABLE II. Relative mode errors, reported as 10 3 × ‖ h glyph[lscript],m S ( q ) -h glyph[lscript],m ( q ) ‖ 2 / ‖ h glyph[lscript],m ( q ) ‖ 2 , from the leave-one-out surrogates. Only those modes which contribute greater than 0 . 02% to the full waveform's time-domain power are used in the computation of the max and mean, except for 'All' which is just δh . Our surrogate also includes the (3 , 1), (4 , [2 , 3]), (5 , [3 , 4 , 5]), (6 , [4 , 5 , 6]), (7 , [5 , 6 , 7]), and (8 , [7 , 8]) modes. Weaker modes typically have relative errors between 1% and 35%. \n| ( glyph[lscript], m ) | Surrogate Max Mean Max Mean | 0.07 | NR | ( | glyph[lscript], m ) | Surrogate Max Mean Max Mean | | NR | |\n|-------------------------|-------------------------------|--------|-------|------|-----------------------|-------------------------------|-------|------|------|\n| (2 , 2) | 0.36 | | 0.36 | 0.08 | (3 , 2) | 100 | 17 | 1.7 | 0.43 |\n| (2 , 1) | 29 | 3.4 | 4.1 | 0.54 | (4 , 4) | 7.4 | 2.2 | 20 | 2.1 |\n| (3 , 3) | 53 | 4.1 | 11 | 0.94 | All | 0.42 | 0.12 | 0.4 | 0.1 | \nFIG. 4. Waveform differences between the two highest SpEC resolutions (black circles), surrogates built from the two highest SpEC resolutions (cyan line), the full surrogate and SpEC (red squares), and leave-one-out trial surrogates and SpEC (blue triangles). The largest surrogate error is for q = 2, for which the (2 , 2) mode is shown in Fig. 1. \n<!-- image --> \nWe find that the surrogate building process is robust to resolution differences. Furthermore, the surrogate can be improved using NR waveforms of higher accuracy. \nWe perform a final test and construct surrogates using the first N selected mass ratios (from Table I) as input waveforms, leaving 22 -N mass ratios with which to test. We find the total surrogate error decreases exponentially with N and is comparable to the SpEC truncation error after using 15 waveforms. Some modes (e.g., (2 , 2)) are fully resolved after as few as 7 waveforms. \nComparison to EOBFor data analysis purposes, we compare our surrogate with EOBNRv2 [19] and SEOBNRv2 [21] models (generated from a current implementation 4 in LAL [38]). In Fig. 5, we show the unfaithfulness \n1 -max δϕ,δt Re ∫ ∞ 15Hz df ˆ ˜ h 1 ( f ; θ, ϕ ) ˆ ˜ h ∗ 2 ( f ; θ, ϕ + δϕ ) e 2 πifδt S n ( f ) (4) \nFIG. 5. Unfaithfulness, from Eq. (4), comparing SpEC with our surrogate, EOBNRv2, and SEOBNRv2 models using all available m glyph[negationslash] = 0 modes. Dashed lines show the unfaithfulness for (2 , 2) modes only. All waveforms are Planck-tapered [54] for t ∈ [ -2750 , -2500] M and t ∈ [50 , 90] M . For the full multi-modal waveforms, we maximize the unfaithfulness over θ and ϕ for the worst-case scenario. We use the '+' polarization, which is non-zero for all ( θ, ϕ ). Left : The shaded regions contain all 22 mass ratios, while the dashed lines maximize over mass ratio. The vertical grey line is the minimum total mass ( ≈ 115 M glyph[circledot] ) ensuring all (2 , 2) modes start with ≤ 15Hz at the end of the first tapering window. Right : Unfaithfulness for a 115 M glyph[circledot] binary. \n<!-- image --> \nof the surrogate and the two EOB models against the NR waveforms. Here, ˆ ˜ h is the normalized Fourier transform of h (such that a waveform's unfaithfulness with itself gives 0), and S n ( f ) the advanced LIGO zero-detuned high power sensitivity noise curve [53]. The surrogate is more faithful than both EOB models for all cases considered. Since SEOBNRv2 only provides (2 , ± 2) modes, it performs worst for large total masses where additional modes become important. All models predict the (2 , 2) mode with an unfaithfulness < 1% for q ∈ [1 , 10] at 115 M glyph[circledot] , however the EOB models are limited by the availability of subdominant modes. \nDiscussionWe have built a surrogate model for NR non-spinning BBH merger waveforms generated by SpEC. On a standard 2015 single core computer, all 77 modes with 2 ≤ glyph[lscript] ≤ 8 are evaluated in ≈ 0 . 5 sec ( ≈ 0 . 01 sec for a single mode) providing a factor of ∼ 10 6 -8 speedup compared to SpEC. Importantly, this is achieved with only a small loss in accuracy. Like other data-driven modeling strategies, our surrogate is valid only within the training intervals, namely, q ∈ [1 , 10] and t/M ∈ [ -2570 , 100]. Therefore, within the training intervals, our surrogate model generates BBH merger waveforms that are equivalent to SpEC outputs up to numerical error and a small modeling error. \nNR surrogates can be used for multiple-query applications in gravitational wave data analysis such as detectorspecific template-bank (re-)generation and parameter estimation. Our surrogate, and more generally the results of this paper, open up the exciting possibility of perform- \ning, for example, parameter estimation with multi-modal NR waveforms (with hybridization, if needed). Parameter estimation studies seeking to incorporate model error may benefit from the surrogate's relatively straightforward characterization and assessment of uncertainty from a combination of the surrogate's and SpEC's systematic and numerical errors. We anticipate NR surrogate modeling to complement traditional strategies [1524, 26] by providing unlimited high-fidelity approximations of NR waveforms with which to calibrate, refine and make comparisons. Building NR surrogates of precessing BBH merger waveforms, which may be modeled from the parameters specially selected in [55], offer a promising avenue for modeling the full 7 dimensional BBH parameter space. The surrogate model described in this paper is available for download at [56, 57]. \nWe thank Mike Boyle, Alessandra Buonanno, Collin Capano, Jan Hesthaven, Jason Kaye, Geoffrey Lovelace, Lee Lindblom, Tom Loredo, Christian Ott, Yi Pan, Harald Pfeiffer, Rory Smith, and Nicholas Taylor for many useful discussions throughout this project. This work was supported in part by NSF grants CAREER PHY0956189, PHY-1068881, PHY-1005655, PHY-1440083, PHY-1404569, and AST-1333520 to Caltech, NSF grants PHY-1306125 and AST-1333129 to Cornell University, NSF grant PHY-1500818 to the University of California at San Diego, NSF grants PHY-1208861 and PHY1316424 to the University of Maryland (UMD), NSERC of Canada, and the Sherman Fairchild Foundation. Computations were performed on the Zwicky cluster at Caltech, which is supported by the Sherman Fairchild Foundation and by NSF award PHY-0960291. Portions of this research were carried out at the Center for Scientific Computation and Mathematical Modeling cluster at UMD. \n- [1] F. Pretorius, Phys. Rev. Lett. 95 , 121101 (2005), arXiv:gr-qc/0507014 [gr-qc].\n- [2] M. Campanelli, C. Lousto, P. Marronetti, and Y. Zlochower, Phys. Rev. Lett. 96 , 111101 (2006), arXiv:grqc/0511048 [gr-qc].\n- [3] J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and J. van Meter, Phys. Rev. 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Tiglio, Phys.Rev.Lett. 113 , 021101 (2014), arXiv:1401.7038 [grqc].\n- [56] 'Simulating eXtreme Spacetimes,' (), http://www. black-holes.org/surrogates/ .\n- [57] 'Gwsurrogate,' https://pypi.python.org/pypi/ gwsurrogate/ ."}
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2010ApJ...719.1315K
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Constraints on Black Hole Growth, Quasar Lifetimes, and Eddington Ratio Distributions from the SDSS Broad-line Quasar Black Hole Mass Function
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2010-01-01
| 11 | 0.48 | 160 |
['black hole physics', 'galaxies active', 'galaxies luminosity function;mass function', 'stars luminosity function;mass function', 'galaxies nuclei', 'galaxies statistics', 'galaxies quasars', '-']
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[]
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We present an estimate of the black hole mass function of broad-line quasars (BLQSOs) that self-consistently corrects for incompleteness and the statistical uncertainty in the mass estimates, based on a sample of 9886 quasars at 1 < z < 4.5 drawn from the Sloan Digital Sky Survey (SDSS). We find evidence for "cosmic downsizing" of black holes in BLQSOs, where the peak in their number density shifts to higher redshift with increasing black hole mass. The cosmic mass density for black holes seen as BLQSOs peaks at z ~ 2. We estimate the completeness of the SDSS as a function of the black hole mass and Eddington ratio, and find that at z > 1 it is highly incomplete at M <SUB>BH</SUB> <~ 10<SUP>9</SUP> M <SUB>sun</SUB> and L/L <SUB>Edd</SUB> <~ 0.5. We estimate a lower limit on the lifetime of a single BLQSO phase to be t <SUB>BL</SUB> > 150 ± 15 Myr for black holes at z = 1 with a mass of M <SUB>BH</SUB> = 10<SUP>9</SUP> M <SUB>sun</SUB>, and we constrain the maximum mass of a black hole in a BLQSO to be ~3 × 10<SUP>10</SUP> M <SUB>sun</SUB>. Our estimated distribution of BLQSO Eddington ratios peaks at L/L <SUB>Edd</SUB> ~ 0.05 and has a dispersion of ~0.4 dex, implying that most BLQSOs are not radiating at or near the Eddington limit; however, the location of the peak is subject to considerable uncertainty. The steep increase in number density of BLQSOs toward lower Eddington ratios is expected if the BLQSO accretion rate monotonically decays with time. Furthermore, our estimated lifetime and Eddington ratio distributions imply that the majority of the most massive black holes spend a significant amount of time growing in an earlier obscured phase, a conclusion which is independent of the unknown obscured fraction. These results are consistent with models for self-regulated black hole growth, at least for massive systems at z > 1, where the BLQSO phase occurs at the end of a fueling event when black hole feedback unbinds the accreting gas, halting the accretion flow.
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https://arxiv.org/pdf/1006.3561.pdf
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{'CONSTRAINTS ON BLACK HOLE GROWTH, QUASAR LIFETIMES, AND EDDINGTON RATIO DISTRIBUTIONS FROM THE SDSS BROAD LINE QUASAR BLACK HOLE MASS FUNCTION': 'Brandon C. Kelly 1,2,3 , Marianne Vestergaard 4,5 , Xiaohui Fan 5,6 , Philip Hopkins 7 , Lars Hernquist 3 , Aneta Siemiginowska 3 \nDraft version September 24, 2018', 'ABSTRACT': "We present an estimate of the black hole mass function (BHMF) of broad line quasars (BLQSOs) that self-consistently corrects for incompleteness and the statistical uncertainty in the mass estimates, based on a sample of 9886 quasars at 1 < z < 4 . 5 drawn from the Sloan Digital Sky Survey. We find evidence for 'cosmic downsizing' of black holes in BLQSOs, where the peak in their number density shifts to higher redshift with increasing black hole mass. The cosmic mass density for black holes seen as BLQSOs peaks at z ∼ 2. We estimate the completeness of the SDSS as a function of black hole mass and Eddington ratio, and find that at z > 1 it is highly incomplete at M BH /lessorsimilar 10 9 M /circledot and L/L Edd /lessorsimilar 0 . 5. We estimate a lower limit on the lifetime of a single BLQSO phase to be t BL > 150 ± 15 Myr for black holes at z = 1 with a mass of M BH = 10 9 M /circledot , and we constrain the maximum mass of a black hole in a BLQSO to be ∼ 3 × 10 10 M /circledot . Our estimated distribution of BLQSO Eddington ratios peaks at L/L Edd ∼ 0 . 05 and has a dispersion of ∼ 0 . 4 dex, implying that most BLQSOs are not radiating at or near the Eddington limit; however the location of the peak is subject to considerable uncertainty. The steep increase in number density of BLQSOs toward lower Eddington ratios is expected if the BLQSO accretion rate monotonically decays with time. Furthermore, our estimated lifetime and Eddington ratio distributions imply that the majority of the most massive black holes spend a significant amount of time growing in an earlier obscured phase, a conclusion which is independent of the unknown obscured fraction. These results are consistent with models for self-regulated black hole growth, at least for massive systems at z > 1, where the BLQSO phase occurs at the end of a fueling event when black hole feedback unbinds the accreting gas, halting the accretion flow. \nSubject headings: galaxies: active - galaxies: mass function - galaxies: statistics \n- 3 Harvard-Smithsonian Center for Astrophysics, 60 Garden St, Cambridge, MA 02138\n- 4 Freja Fellow, Dark Cosmology Centre, The Niels Bohr Institute, University of Copenhagen 5\n- 5 Department of Astronomy, University of Arizona, Tucson, AZ 85721 \n- 7 Miller Fellow, Department of Astronomy, University of California, Berkeley, CA", '1. INTRODUCTION': "Understanding how and when supermassive black holes (SMBHs) grow is currently of central importance in extragalactic astronomy. Observations have established correlations between SMBH mass and host galaxy spheroidal properties, such as luminosity (e.g., Kormendy & Richstone 1995; McLure & Dunlop 2001, 2002), stellar velocity dispersion ( M BH -σ relationship, e.g., Gebhardt et al. 2000; Merritt & Ferrarese 2001; Tremaine et al. 2002), concentration or Sersic index (e.g., Graham et al. 2001; Graham & Driver 2007), bulge mass (e.g., Magorrian et al. 1998; Marconi & Hunt 2003; Haring & Rix 2004), and binding energy (e.g., Aller & Richstone 2007; Hopkins et al. 2007c). These correlations imply that the evolution of spheroidal galaxies and the growth of SMBHs are intricately tied together, where black holes grow by accreting gas, possibly fueled by a major merger of two gas-rich galaxies, until feedback energy from the SMBH expels gas and shuts off the accretion process (e.g., Silk & Rees 1998; Fabian 1999; Begelman & Nath 2005; Murray et al. 2005; Hopkins et al. 2009b). This 'self-regulated' growth of black holes has been successfully applied in smoothed particle hydrodynamics simulations (Di Matteo et al. 2005; Springel et al. 2005; Johansson et al. 2009), and has motivated numerous models linking the SMBH growth, the quasar phase, and galaxy evolution (e.g., Haehnelt et al. 1998; Kauffmann & Haehnelt 2000; Haehnelt & Kauffmann 2000; Wyithe & Loeb 2003; Volonteri et al. 2003; Cattaneo et al. 2005; Di Matteo et al. 2008; Somerville et al. 2008; Hopkins et al. 2008a; Croton 2009; Sijacki et al. 2009; Booth & Schaye 2009; Shen 2009, and references therein). Within this framework, the broad line quasar 8 phase persists after feedback energy from the black hole 'blows' the gas away (e.g., Hopkins et al. 2005a, 2006b). The broad line quasar phase is expected to persist until the accretion rate drops low enough to switch to a radiatively inefficient accretion flow (e.g., Churazov et al. 2005; Hopkins et al. 2009a). \nWhile major-mergers of gas-rich galaxies may fuel quasars at high redshift, and grow the most massive SMBHs, alternative fueling mechanisms are likely at lower redshift and fainter luminosities. Mergers alone do not appear to be sufficient to reproduce the number of X-ray faint AGN (e.g., Marulli et al. 2007), and accretion of ambient gas (e.g., Ciotti & Ostriker 2001; Hopkins & Hernquist 2006), may fuel these fainter, lower M BH AGN at lower z , resulting in an alternative growth mechanism for these SMBHs. Indeed, many AGN are observed to live in late-type galaxies out to z ≈ 1 (e.g., Guyon et al. 2006; Gabor et al. 2009), and the X-ray luminosity function of AGN hosted by late-type galaxies suggests that fueling by minor interactions or internal instabilities represents a non-negligible contribution to the accretion history of the Universe (Georgakakis et al. 2009). Furthermore, feedback from the SMBH may continue to affect its environment long after its growth through so-called 'radio mode' feedback (Croton et al. 2006; Bower et al. 2006; Sijacki et al. 2007). Observations qualitatitively support a model where the fueling mechanism for black hole growth depends on the mass of the host dark matter halo, but regardless of the fueling mechanism, black hole feedback and accretion follow a similar evolutionary path (e.g., Hickox et al. 2009; Constantin et al. 2009). \nObservationally, a significant amount of recent work has utilized the argument of Soltan (1982) to indirectly map the growth of all SMBHs (e.g., Salucci et al. 1999; Yu & Tremaine 2002; Shankar et al. 2004, 2009; Marconi et al. 2004; Yu & Lu 2004; Hopkins et al. 2007b; Merloni & Heinz 2008). Work along this line has used the correlations between M BH and host galaxy spheroidal properties to infer the local distribution of M BH for inactive black holes, which are assumed to be the relics of past AGN activity. The distribution of M BH as a function of redshift is then estimated by stepping backward from the local distribution of M BH , employing a continuity equation describing the 'flow' of black hole number density through bins in M BH (e.g., Small & Blandford 1992). The quasar luminosity function is used as a constraint on the rate of change in the SMBH mass density, because it traces the accretion of matter onto black holes, modulo the accretion efficiency and the bolometric correction. From this work, it has generally been inferred that black hole growth is dominated by periods of near Eddington accretion, with the most massive SMBHs growing first, and that many SMBHs have non-zero spin. \nAn alternative to the technique of Soltan (1982) for estimating the SMBH mass function has been used by Siemiginowska & Elvis (1997) and Hatziminaoglou et al. (2001). These authors used a model for thermal-viscous accretion disk instabilities (Siemiginowska et al. 1996) to calculate the expected distribution of luminosity at a given black hole mass. Based on this calculated distribution, they use the quasar luminosity function to constrain the quasar black hole mass function. Siemiginowska & Elvis (1997) found evidence for black hole 'downsizing', with the peak of the quasar mass function shifting toward lower masses at lower redshift. \nCorrelations between the SMBH mass, width of the broad emission lines, and luminosity of the quasar continuum have made it possible to estimate M BH for broad line quasars (BLQSOs) (e.g., Vestergaard & Peterson 2006; Kelly & Bechtold 2007), albeit with considerable statistical uncertainty of ∼ 0 . 4 dex and various systematic effects (e.g., Krolik 2001; Vestergaard & Peterson 2006; Greene & Ho 2006; Marconi et al. 2008; Denney et al. 2009). This offers an alternative to estimating SMBH mass functions based on the Soltan (1982) argument, because the mass function may be estimated directly at all redshifts, and because the distribution of quasar emission line widths provides an additional observational constraint on the mass function. Estimates of M BH obtained from the broad emission lines have been used to estimate the distribution of quasar black hole masses and Eddington ratios at a variety of redshifts (e.g., McLure & Dunlop 2004; Vestergaard 2004; Kollmeier et al. 2006; Netzer & Trakhtenbrot 2007; Shen et al. 2008; Fine et al. 2008; Kelly et al. 2008; Trump et al. 2009; Labita et al. 2009a). \nThe BLQSO black hole mass function (BHMF) maps the comoving number density and evolution of active supermassive black holes contained within broad line AGN, and is therefore a complete census of the population of these SMBHs over cosmic time. The BLQSO BHMF is important for a number of reasons, including the following: \n- · At high redshift SMBHs with masses M BH ∼ 10 9 are already in place by z ∼ 6 (e.g., Fan et al. 2001b; Jiang et al. 2007), and therefore the highz BLQSO BHMF places important constraints on the formation and growth of seed SMBHs (e.g., Volonteri et al. 2003, 2008; Lodato & Natarajan 2007).\n- · If, after a fueling event, the growth of the SMBH persists until it becomes massive enough such that feedback energy begins to unbind the gas, the active SMBH will be seen as a BLQSO shortly before entering quiescence, and its fractional mass growth will not be significant during this time period (Hopkins & Hernquist 2006). The BLQSO BHMF thus (1) is related to the distribution of spheroidal binding energies in the central regions of the galaxy (Hopkins et al. 2007a,c; Younger et al. 2008), and (2) gives the nearly instantaneous increase in the AGN relic SMBH mass density.\n- · Because mass is a fundamental physical quantity of SMBHs, we can use the BLQSO BHMF to estimate the duty cycle for broad line quasar activity as a function of mass by comparing the number density of all SMBHs with those seen as BLQSOs. The duty cycle can then be converted into an estimate of the lifetime of BLQSO activity, which is of significant importance for understanding the origin of BLQSO activity. This cannot be done using the quasar luminosity function.\n- · The distribution of luminosities at a given BLQSO SMBH mass depends on the quasar lightcurve (Yu & Lu 2004; Hopkins & Hernquist 2006; Yu & Lu 2008; Hopkins & Hernquist 2009), which is a function of both evolution in the rate at which fuel is supplied to the accretion disk, and the time-dependent behavior of the accretion disk (Siemiginowska & Elvis 1997; Hatziminaoglou et al. 2001). Thus, understanding the distribution of L at a given M BH , or alternatively the distribution of Eddington ratio, gives insight into the BLQSO fueling mechanism and accretion physics. \nThe BLQSO BHMF therefore provides an important observational constraint on models of SMBH growth, the onset and duration of quasar activity, quasar feedback, and galaxy evolution. \nThere have been several estimates of the BHMF of BLQSOs calculated directly from the mass estimates derived from the broad emission lines (Wang et al. 2006; Greene & Ho 2007; Vestergaard et al. 2008; Vestergaard & Osmer 2009; Kelly et al. 2009b, hereafter KVF09). In particular, Vestergaard & Osmer (2009) found evidence for cosmic 'downsizing' of black hole mass, in that the most massive SMBHs are more common at high redshift, consistent with previous work on mapping black hole growth (e.g., Marconi et al. 2004; Merloni 2004; Shankar et al. 2004; Merloni & Heinz 2008), conclusions based on the quasar luminosity function (e.g., Steffen et al. 2003; Ueda et al. 2003; Croom et al. 2004; La Franca et al. 2005; Hasinger et al. 2005; Hopkins et al. 2007b; Silverman et al. 2008) and the local active BHMF (Heckman et al. 2004). However, a major concern with previous estimates of the BLQSO BHMF is uncorrected incompleteness and the additional broadening caused by the statistical errors in mass estimates (Kelly & Bechtold 2007; Shen et al. 2008, KVF09). Because the massive end of the BLQSO BHMF falls off with increasing M BH , the intrinsic uncertainty scatters more sources into higher M BH bins than lower ones, potentially having a significant effect on the estimated number density of the most massive SMBHs. Furthermore, even if a sample is complete in luminosity, it is not necessarily complete in M BH , and the completeness in M BH depends on the unknown Eddington ratio distribution. Motivated by these issues, and the fact that the importance of the BLQSO BHMF demands that its determination be statistically rigorous, KVF09 developed a Bayesian statistical technique for estimating the BLQSO BHMF that self-consistently corrects for the incompleteness in M BH and the statistical uncertainties in the estimates of M BH . In this work we apply the technique of KVF09 to the SDSS BLQSO sample of Vestergaard et al. (2008) in order to estimate the black hole mass function of SMBHs that reside in BLQSOs, and discuss the implications for SMBH growth, quasar lifetimes, and Eddington ratio distributions.", '2. THE DATA': "Our sample is drawn from the Sloan Digital Sky Survey (SDSS) Data Release 3 (DR3) quasar sample as presented by Richards et al. (2006) and Vestergaard et al. (2008). Richards et al. (2006) used 15,180 quasars from the SDSS DR3 to determine the quasar optical luminosity function over 0 . 3 < z < 5. Vestergaard et al. (2008) obtained black hole estimates for 14,434 of the quasars presented in Richards et al. (2006) using scaling relationships between the width of the broad emission lines, continuum luminosity, and black hole mass. The details of the sample and fitting process are described in Richards et al. (2006) and Vestergaard et al. (2008). For completeness, we briefly review the spectral modeling used by Vestergaard et al. (2008) to extract the relevant quantities. \nVestergaard et al. (2008) modeled the observed quasar spectra using a power-law continuum, an optical-UV iron line spectrum based on I Zw I (Vestergaard & Wilkes 2001; V'eron-Cetty et al. 2004), a Balmer continuum, and host galaxy template (Bruzual & Charlot 2003) for objects at z < 0 . 5. The continuum luminosities used for the mass estimates are based on the power-law continuum fits. The emission lines used for the mass estimates in this work are Mg II and C IV, and were modeled using multiple Gaussian functions so to reproduce the line profile. Contributions from the narrow emission line region were subtracted from Mg II when appropriate, and line profiles with strong absorption were ignored. The end result of this analysis was a set of emission line FWHM and continuum luminosities, from which black hole mass estimates were calculated. \nWe have performed a few additional cuts to the sample presented by Vestergaard et al. (2008) before obtaining our final sample. First, we remove the sources located at z < 1. We do this because the quasar sample contains a significant number of extended sources below z < 1, and therefore their i -band magnitudes sometimes contain a \nsignificant contribution from the host galaxy (see the discussion in Richards et al. (2006)). This reduces the effective flux limit at z < 1, creating an artificial second peak in the redshift distribution. While we could attempt to empirically correct the selection function to account for this, we find it easier to simply limit our analysis to 1 < z < 4 . 5. Finally, in order to make our analysis more robust against uncertainty in the selection function, we omit any sources for which the value of the selection function of Richards et al. (2006) is less than 0 . 01, and force all values of the selection function to be zero that are < 0 . 01. After making these cuts, we were left with a sample of 9886 quasars.", '3. THE STATISTICAL MODEL AND DATA ANALYSIS': 'We use an expanded version of the statistical model outlined in KVF09 to estimate the BLQSO BHMF. For completeness, we describe the important aspects of the technique developed by KVF09, and the reader is referred to KVF09 for further details regarding the technique and its derivation. Qualitatively, the technique of KVF09 assumes parameteric forms for the BHMF and the distribution of luminosities at a given M BH . The average value of the broad line mass estimates at a given M BH is fixed to be consistent with the scaling relationships reported by Vestergaard & Peterson (2006) and Vestergaard & Osmer (2009); we assume that these scaling relationships produce unbiased estimates of M BH . The BHMF and distribution of L at a given M BH , in combination with a selection function, imply an observed distribution of z, L, and FWHM. The technique of KVF09 attempts to recover the BHMF and distribution of L at a given M BH by matching the observed distribution of z, L, and FWHM (or, equivalently, the mass estimates) that is implied by the model to the actual observed distribution. The posterior probability distribution is used to quantify how well the implied distributions match the observed distributions. As a result, the technique of KVF09 estimates the probability distribution of the BHMF, and of the distribution of L at a given M BH , given the observed data set.', '3.1. Model for the Joint Distribution of M BH , L, FWHM , and z': 'We model the BLQSO BHMF as a mixture of K log-normal distributions: \nφ (log M BH , log z ) = N ( dV dz ) -1 K ∑ k =1 π k 2 π | Σ k | 1 / 2 exp [ -1 2 ( y -µ k ) T Σ -1 k ( y -µ k ) ] , (1) \nwhere ∑ K k =1 π k = 1. In this work we choose K = 4, based on our previous experience in working with simulated data sets (KVF09), and because we did not notice a significant difference in the results obtained using larger values of K . Here, N is the total number of BLQSOs in the Universe that could be seen by an observer on Earth at the time of the survey, y = (log M BH , log z ), µ k is the 2-element mean vector for the k th Gaussian functions, Σ k is the 2 × 2 covariance matrix for the k th Gaussian function, and x T denotes the transpose of x . In addition, we denote π = ( π 1 , . . . , π K ) , µ = ( µ 1 , . . . , µ K ), and Σ = (Σ 1 , . . . , Σ K ). The variance in log M BH for Gaussian function k is σ 2 m,k = Σ 11 ,k , the variance in log z for Gaussian function k is σ 2 z,k = Σ 22 ,k , and the covariance between log M BH and log z for Gaussian function k is σ mz,k = Σ 12 ,k . The parameters for the mass function are N,π,µ, and Σ. \nThe distribution of luminosity density at a given black hole mass and wavelength λ is assumed to also follow a mixture of J log-normal distributions: \np (log L λ | M BH ) = J ∑ j =1 γ j 2 πσ 2 l,j exp [ -1 2 ( log λL λ -α 0 ,j -α m,j (log M BH -9) σ l,j ) 2 ] . (2) \nThis represents an extension of the model of KVF09, which only used J = 1 log-normal distribution. We made this extension to incorporate additional flexiblity in p ( L λ | M BH ), ensuring that the luminosity distribution, and therefore the black hole mass incompleteness correction, is robust to the particular parameteric form. For each log-normal distribution, the parameters for the distribution of L λ at a given M BH are γ j , α 0 ,j , α m,j , and σ l,j . We used J = 3 log-normal distributions, as we did not notice a signficant change in the estimated values of p ( L λ | M BH ) when using J ≥ 3. We further assess the robustness of our assumed form for p ( L λ | M BH ) in § 3.3, and show that our results should not be signficantly altered if the true form of p ( L λ | M BH ) is a power-law, as might be expected from some physical models for BLQSO lightcurves. \n√ \nIn our analysis we use the luminosity density at 1350 ˚ A in order to minimize bias at the highest redshifts introduced from extrapolating the power-law continuum beyond the spectral window. At z /greaterorsimilar 1 . 8 the rest frame λ = 1350 ˚ A falls within the observed spectral window for the SDSS sources used in this work, while a smaller redshift window is available when using the luminosity densities calculated at other wavelengths. Furthermore, the z ∼ 1 distributions of bolometric luminosity did not exhibit any significant difference when using the luminosity density at 1350 ˚ A, as compared to that calculated using the luminosity density at λ > 1350 ˚ A, suggesting that significant biases at lower z are not introduced by extrapolating the power-law continuum. \nWe can connect the parameters in Equation (2) to the distribution of Eddington ratio and bolometric correction. The monochromatic luminosity is related to the Eddington luminosity ratio Γ Edd 9 and bolometric correction C λ as \nλL λ = 1 . 3 × 10 38 Γ Edd C λ M BH M /circledot [erg s -1 ] . (3) \nTherefore, Equation (2) implies that for the j th log-normal distribution we are assuming that on average log(Γ Edd /C λ ) = α 0 ,j -47 . 11 + ( α m,j -1) log M BH , with a Gaussian scatter about this mean value of standard deviation σ l,j . We do not make any formal attempt to prohibit Equation (2) from allowing values of L/L Edd > 1, as this would require us to make an assumption about the bolometeric correction. However, as we will show in § 4.5, our estimate for p ( L λ | M BH ) implies only a negligible fraction of BLQSOs with L/L Edd > 1, assuming a constant bolometric correction of C 1350 = 4 . 3 (Vestergaard & Osmer 2009). \nThe distribution of emission line widths at a given luminosity density and black hole mass is modeled as a log-normal distribution: \np \n√ \n(log FWHM BL | L BL , M BH ) = 1 2 π ( σ 2 BL + σ 2 FWHM ) exp { -1 2 (log FWHM BL -β BL 0 +1 / 4 log L BL -1 / 2 log M BH ) 2 σ 2 BL + σ 2 FWHM } . (4) \nHere, FWHM BL is the line width for a particular broad emission line, L BL is the luminosity density used as an estimate for the broad line region size for a particular broad emission line, σ FWHM is the measured uncertainty in FWHM due to measurement error, and β BL 0 and σ BL are the parameters for Equation (4) for a particular broad emission line. We do not make any attempt to correct for radiation pressure on the broad emission line clouds (Marconi et al. 2008), as its importance and effects are currently poorly understood (e.g., for a discussion see Vestergaard & Osmer 2009). \nEquation (4) implies that on average FWHM ∝ M 1 / 2 BH /L 1 / 4 BL , or equivalently M BH ∝ L 1 / 2 BL FWHM 2 BL . Therefore, we can use the results obtained for the broad emission line scaling estimates of M BH to fix β 0 . We use the mass scaling relationship for C IV that is presented by Vestergaard & Peterson (2006), and the relationship for Mg II that is presented by Vestergaard & Osmer (2009). As noted in KVF09, these scaling relationships imply β MgII 0 = 10 . 61 and β CIV 0 = 11 . 33, and we fix β 0 to these values. \n0 In addition, the dispersion in FWHM BL at a given L BL and M BH can be related to the scatter in the mass estimates based on the scaling relationships. Vestergaard & Peterson (2006) find the statistical uncertainty in the broad line mass estimates to be 0.36 dex for C IV. Therefore, because FWHM ∝ M 1 / 2 BH , σ BL (CIV) ≈ 0 . 18 dex. Likewise, the intrinsic uncertainty in the broad line mass estimate for Mg II is found to be ∼ 0 . 4 dex (Vestergaard et al., in preparation), and therefore σ BL (MgII) ≈ 0 . 2 dex. However, there have been indications from previous analysis of flux limited surveys, which probe higher redshifts and a narrower range in luminosity than that exhibited by the objects with reverberation mapping data, that the intrinsic scatter in the mass estimates may be smaller than ≈ 0 . 4 dex (Kollmeier et al. 2006; Shen et al. 2008; Fine et al. 2008; Steinhardt & Elvis 2010a,b). This may be caused by correlated scatter in the mass estimates with luminosity (e.g., Shen & Kelly 2010) or redshift, or a dependence on σ BL with luminosity or redshift. Both of these possibilities would decrease the dispersion in the scatter in the mass estimates when only probing a smaller range in L , or higher redshifts. In addition, Marconi et al. (2008) find a smaller scatter in the masses estimated using H β when one corrects the reverberation mapped masses for radiation pressure. Because of the possibility for a smaller scatter in the mass estimates, we perform our analysis with both σ BL fixed to ≈ 0 . 2 dex, and with σ BL as a free parameter.', '3.2. The Posterior Distribution and Fitting Technique': "The technique for estimating the BHMF developed by KVF09 takes a Bayesian approach for performing statistical inference, meaning that it calculates the probability distribution of the mass function, given the observed data. All the information regarding the BHMF and the parameters for the statistical model is contained within the posterior probability distribution, which is defined as the probability distribution of the model, given the observed data. KVF09 derived the posterior distribution for the statistical model described in the previous section. Denoting the model parameters for the shape of the BHMF as θ = ( π, µ, Σ , γ, α 0 , α m , σ l ), the posterior distribution is \np ( θ | FWHM , L, z ) ∝ p ( θ ) [ p ( I = 1 | θ )] -n n ∏ i =1 p (FWHM i , L λ,i , z i | θ ) , (5) \nwhere the number of data points is n , p ( θ ) is the prior on θ , and p ( I = 1 | θ ) is the probability as a function of θ that an BLQSO makes it into the SDSS DR3 catalogue. We note that if the dispersion in the mass estimates, σ BL , is also treated as a free parameter, then θ also contains σ BL . The joint distribution of FWHM , L, and z , p (log FWHM i , log L λ,i , log z i | θ ), is given by Equations (30)-(40) in KVF09, modified to use the mixture form for p ( L λ | M BH ). We use the the Mg II line at 1 < z < 1 . 6, both the Mg II and the C IV line at 1 . 53 < z < 1 . 6, and the C IV line at z > 1 . 6. We do not use H β because we limit our analysis to z > 1. At z ∼ 0 . 8 H β is shifted into the water vapor bands, which tends to decrease the FWHM accuracy; Mg II is similarly affected at higher redshifts. In addition, we see systematic changes in the Mg II FWHM distribution above a redshift of 1.6, suggesting the presence of biases, which need further investigation (M. Vestergaard et al., in preparation). The posterior distribution for the BHMF normalization, given θ and n , is given by Equation (16) in KVF09. \nThe inclusion probability as a function of θ is calculated by averaging the SDSS DR3 selection function over the distribution of L λ and z (see Eq.(46)-(49) in KVF09). In order to simplify our analysis we ignore the lower limit of FWHM > 1000 km s -1 on the emission line width for the SDSS DR3 sample. The distribution of FWHM for the \nSDSS DR3 falls off before reaching FWHM = 1000 km s -1 , and it does not appear that imposing the lower limit results in a non-negligible fraction of the BLQSO population being missed. Therefore, any correction for the lower limit in FWHM is negligible, and we ignore it. In this case, the inclusion probability is \np ( I = 1 | θ ) = Ω 4 π ∫ ∞ L λ =0 ∫ 4 . 5 z =1 s ( L λ , z ) L λ z ln 10 × K ∑ k =1 π k J ∑ j =1 γ j N (log L λ | ¯ l kj ( z ) , V l,kj ) N (log z | µ z,k , σ 2 z,k ) dz dL λ (6) \n ¯ l kj ( z ) = α o,j + α m,j µ m,k + α m,j σ mz,k σ 2 z,k (log z -µ z,k ) (7) \nV l,kj = α 2 m,j σ 2 m,k (1 -ρ 2 mz,k ) + σ 2 l,j . (8) \nHere, Ω = 1622 deg 2 is the effective sky area of the SDSS DR3 sample (Richards et al. 2006), s ( L λ , z ) is the SDSS selection function, N ( x | µ, σ 2 ) denotes a Normal distribution with mean µ and variance σ 2 , as a function of x , µ m,k and µ z,k are the mean values of log M BH and log z for the k th Gaussian function, respectively, and ρ mz,k is the correlation between log M BH and log z for the k th Gaussian function. Note that ¯ l k ( z ) and V l,k define the mean and variance in log L λ at a given redshift for the k th Gaussian function. The integral in Equation (6) is over 1 < z < 4 . 5 because we have removed the sources at z < 1, and there are no useable broad emission lines at z /greaterorsimilar 4 . 5. \nEquation (6) is in terms of the selection function with respect to the luminosity density. As mentioned above, we use the luminosity density at 1350 ˚ A in this work. However, Richards et al. (2006) report their selection function in terms of the i -band magnitude. We can convert the selection function of Richards et al. (2006) to be in terms of the BLQSO power-law continuum L 1350 through the equation \ns ( L 1350 , z ) = ∫ ∞ -∞ s ( i, z ) p ( i | L 1350 , z ) di, (9) \nwhere p ( i | L 1350 , z ) is the distribution of i -band magnitude at a given L 1350 and z , and s ( i, z ) is the SDSS DR3 selection function in terms of i and z . We model the distribution of i magnitudes at a given L 1350 in different redshift bins as a student's t distribution with mean that depends linearly on log L 1350 : \np ( i | L 1350 , z ) = Γ[( ν ( z ) + 1) / 2] Γ( ν ( z ) / 2) σ i ( z ) ν ( z ) π ( 1 + 1 ν ( z ) ( i -A i ( z ) -B i ( z ) log L 1350 σ i ( z ) ) 2 ) -( ν ( z )+1) / 2 (10) \nThe student's t distribution converges to the normal distribution for ν → ∞ ; for finite ν the t-distribution is more heavy tailed than the normal distribution, and we have found it to better describe the distribution of i -magnitudes at a given L 1350 and z . The parameters A i ( z ) , B i ( z ) , σ i ( z ) , and ν ( z ) are fit by maximizing their posterior probability distribution, given the observed set of i magnitudes and L 1350 . The posterior distribution is given by inserting the assumed distributions into Equation (40) of Kelly (2007), and for simplicity we assume a simple flux limit of i = 19 . 1 for z < 2 . 7 and i = 20 . 2 for z > 2 . 7 (e.g., Richards et al. 2006). At z < 2 . 7 the redshift bins used in Equation (10) have width ∆ z = 0 . 1, while at z > 2 . 7 the redshift bins have width ∆ z = 0 . 3. \n√ \nIn this work we constrain the parameters of our statistical model to be within certain limits, but in general assume a uniform prior on θ . Our prior is uniform on the parameters within the limits 44 . 5 < α 0 < 46 . 5 , -1 < α m < 3 , 0 . 1 < σ l < 2 , 7 < µ m,k < 10 , log 1 < µ z,k < log 4 . 5 , 0 . 1 < σ m,k , σ z,k < 2 , and -0 . 98 < ρ mz,k < 0 . 98. We also assume a uniform prior on π and γ , subject to the elements of π and γ summing to unity. The limits on α m were chosen because we did not consider it realistic that L 1350 would depend on M BH outside of the range of dependencies spanning L 1350 ∝ 1 /M BH to L 1350 ∝ M 3 BH , and the limits on α 0 were chosen to restrict the average value of L/L Edd to be within 0 . 01 < L/L Edd < 1 for BLQSOs with M BH = 10 9 M /circledot , assuming a bolometeric correction of C 1350 = 4 . 3. The limits on σ l were chosen because ∆ log L ≈ 0 . 1 is comparable to the grid spacing on which the selection function was computed by Richards et al. (2006), and we did not consider it likely that the dispersion in L 1350 at a given M BH would be greater than 2 dex. The limits on the BHMF parameters were chosen so as not to extrapolate the BHMF very far beyond the detectable range of M BH . As such, in this work we limit our analysis of the BHMF to M BH /greaterorsimilar 10 7 M /circledot . \nAs mentioned before, there exists the possibility that, in the range of L and z we probe, the uncertainty in the mass estimates may be smaller than the commonly quoted ∼ 0 . 4 dex. If the error in the mass estimates is correlated with luminosity, then the variance in the mass estimates at a given luminosity and mass is reduced to \nV ar (log ˆ M BL | L ) = V ar (log ˆ M BL )(1 -ρ 2 BL ) . (11) \nHere, ˆ M BL denotes the broad line mass estimate, V ar (log ˆ M BL ) is the variance in the mass estimates about the true mass, typically thought to be ∼ 0 . 4 2 dex 2 , and ρ BL is the correlation with luminosity in the scatter in the mass estimates about the true mass. Because we probe a somewhat narrow range in L λ , when we estimate the parameter \nσ BL in Equation (4), we are really estimating 2 σ BL ≈ √ V ar (log ˆ M BL | L ), and we therefore need to also construct a prior for σ BL . We do this by first noting that Vestergaard & Peterson (2006) estimated the dispersion in the scatter in the broad line mass estimates about the reverberation mapping estimates using 27 AGN. Therefore, the appropriate prior distribution for V ar (log ˆ M BL ) is the posterior probability distribution of the variance in the mass estimates, given the reverberation mapping sample. Since we assume that the scatter in the mass estimates about the true mass is log-normal, the probability distribution of their variance follows using a standard result from Bayesian statistics, and is a scaled inverse χ 2 distribution with 26 degrees of freedom (e.g., Gelman et al. 2004). However, there are currently no constraints on the value of ρ BL , and we use a uniform prior on its value. Our prior on σ BL is then calculated by combining these two priors according to Equation (11). This results in a broad prior which peaks at V ar (log ˆ M BL | L ) ≈ 0 . 4 2 dex 2 , and falls off slowly to zero as V ar (log ˆ M BL | L ) → 0 and V ar (log ˆ M BL | L ) → 0 . 6 2 . \nIn addition to the above constraints, we also impose the prior constraint that the number density of BLQSO SMBHs must never exceed the local number density of all SMBHs. In principle, this constraint could be violated if a large number of 'wandering' black holes are present. These wandering black holes would be ejected from galactic nuclei in the late stages of a merger due to asymmetric emission of graviational radiation (e.g., Volonteri 2007). However, this constraint would only be violated if the binary black hole system is ejected after or during the broad line quasar phase. While this would certainly be a very intriguing result, we do not test it here; indeed, our results are unaffected by this prior constraint as the estimated BHMF is always below the local value for all random draws from the posterior probability distribution. \nAs described in KVF09, we do not work with Equation (5) directly, but instead use a Markov Chain Monte Carlo (MCMC) sampler algorithm to obtain a set of random draws of θ , distributed according to Equation (5). Because the posterior distribution described by Equation (5) is multimodal due to ambiguities in labeling the Gaussian functions, we label the BHMF Gaussian functions in order of increasing implied mean flux, and the p ( L | M BH ) Gaussian functions in order of increasing mean luminosity at M BH = 10 9 M /circledot . In addition, in order to make our MCMC sampling algorithm robust against additional possible multimodality, we include parallel tempering (e.g., Liu 2004) in our Markov Chain Monte Carlo (MCMC) algorithm to facilitate sampling from the different modes. For each value of θ we obtained from our MCMC random number generator, we also obtain a value of the BHMF normalization, N , by drawing from a negative binomial distribution with parameters n and p ( I = 1 | θ ) (KVF09). The random realizations of θ and N then define a sample of random realizations of the BHMF obtained from the posterior probability distribution of the BHMF, given the observed set of mass estimates, luminosity densities, and redshifts. We ran our MCMC sampler for 2 × 10 5 iterations each, keeping every 20 th iteration.", '3.3. Robustness to Incorrect Assumptions Regarding the BLQSO BHMF and Eddington Ratio Distribution': "Although we have assumed a flexible parameteric form for the BLQSO BHMF and p ( L | M BH ), this form is inconsistent with more physically-motivated BLQSO p ( L | M BH ), such as might be expected from a power-law decay in the BLQSO accretion rate (e.g., Yu et al. 2005; Hopkins & Hernquist 2006, also see discussion in § 5.1). Instead, our assumed log-normal mixture form for p ( L | M BH ) was motivated by mathematical convenience, as some of the integrals necessary for evaluation of the posterior distribution can be done analytically (KVF09). If we were to use a form which required numerical integration, we would have to perform over ten billion numerical integrals in our MCMC sampler, which is computationally prohibitive. Moreover, we also used the mixture form to allow flexibility in the estimated p ( L | M BH ), so that our assumed form should be able to approximate many different forms for p ( L | M BH ). \nIn order to assess the impact of our chosen parameteric form on the inferred BHMF, we simulated a data set where the distribution of Eddington ratios was assumed to have a power-law form p (Γ Edd | M BH ) ∝ Γ -(1+1 /β ) Edd with β = 2 (see discussion in § 5.1). The distribution of bolometric corrections was log-normal with geometric mean C 1350 = 5 and dispersion of 0.2 dex, and the BHMF normalization was N = 2 × 10 6 ; all other aspects of the simulation were done in the same manner as described in § 6.1 of KVF09. \nThe results are illustrated in Figure 1, where we compare the true and estimated BHMF for the simulated sample at z = 2, and the true Eddington ratio distribution with that inferred assuming the statistical model described in § 3.1. Here, and elsewhere in this paper, in addition to a single 'best-fit' mass function, we also plot 100 random draws of the mass function from its probability distribution, generated by our MCMC random number generator. The spread and density of the random draws of the BHMF, and any quantities derived from it, give a visual representation of the uncertainty in these quantities, with the spread on the random draws constraining the BHMF. Because we use 100 random draws, the probability of a quantity falling within a certain area on a plot can be estimated by counting the number of random draws that intersect that area. For this example, the BLQSO BHMF inferred assuming a mixture of J = 3 log-normal distributions for p ( L λ | M BH ) is able to recover the true BHMF and Eddington ratio distributions, at least when the true distribution of L/L Edd is p (Γ Edd ) ∝ Γ -1 . 5 Edd . Although this test is far from exhaustive, we consider it reasonable to conclude that our flexible form for the BHMF and p ( L λ | M BH ) is able to accurately approximate the true forms, and therefore our results are robust against errors resulting from using an incorrect parameteric form for these distributions. \n<!-- image --> \nFig. 1.(a) True BHMF for the simulated sample described in § 3.3, having a power-law distribution of Eddington ratios over 0 . 01 < L/L Edd < 1 (thick red line), compared with the BHMF estimated assuming a mixture of J = 3 log-normal distributions of Eddington ratios (thin black lines); each of the thin solid lines denotes a random draw from the probability distribution of the BHMF, given the observed data and assumptions outlined in § 3.1. Here, and in all figures in this work, we plot 100 random draws of the function of interest, so the probability that the BHMF, say, has a certain value in a given range can be estimated by counting the number of random draws of the BHMF that fall within that range. For this example, the BHMF estimated assuming a mixture of log-normal distributions for L/L Edd is able to recover the true BHMF, implying that our mixture form is robust against mispecification of the parameteric model. (b) True Eddington ratio distribution for the same simulated sample (thick red line), compared with the estimated distribution assuming a mixture of J = 3 log-normal distributions (thin black lines). In this plot we have convolved the power-law form of the distribution of L/L Edd with the scatter in the bolometeric correction used in this simulation, to incorporate the error in the estimated Eddington ration distribution introduced from assuming a constant bolometeric correction. The mixture of log-normals form is able to adequately approximate the true distribution of L/L Edd . \n<!-- image --> \nFig. 2.Distribution of z, L 1350 , and the mass estimates for our sample (histograms), compared with the distributions implied by our best-fit BHMF and distribution of L 1350 at a given M BH (red squares). In the top row of plots we fix the statistical error in the mass estimates to be 0 . 4 dex, while in the bottom row we allow it to be a free parameter. The error bars denote the 1 σ uncertainties on the implied distributions, although often they are smaller than the red squares. The statistical model described in § 3.1 is able to reproduce the observed distributions only if we allow the standard deviation in the statistical error in the mass estimates to be a free parameter, implying that the standard value of ∼ 0 . 4 dex is too large for this luminosity and redshift range. \n<!-- image --> \nWe used our Bayesian method to derive the BLQSO BHMF from the SDSS DR3 quasar sample, both holding the magnitude of the scatter in mass estimates fixed to 0 . 4 dex, and treating the amplitude of the scatter as a free parameter. However, before discussing the results, we first evaluate how well the statistical model described in § 3.1 fits our sample. In order to do this, we compare the distributions of redshift, luminosity, and broad line mass estimates of our sample to the distributions implied by our model, as described in KVF09. Figure 2 compares the observed distributions of z , λL λ (1350 ˚ A), and ˆ M BL to those implied by the statistical model described in § 3.1. The implied distributions were calculated by first simulating a sample of M BH and z from a BHMF randomly output from the MCMC sampler. Then, for each value of M BH , we simulated a value of λL λ (1350 ˚ A) and mass estimate ˆ M BL . Finally, we applied the selection function given by Equation (9) to the simulated data set. This was repeated for each realization of the BHMF obtained from the MCMC output, in order to account for the uncertainty in our estimated model parameters. \nWe are not able to fit the mass estimate distributions if we keep the statistical scatter in the mass estimates fixed to 0.4 dex. In particular, this predicts a distribution of mass estimates that is too broad compared to the actual \ndistribution. In fact, the observed dispersion in the mass estimates for our SDSS sample is ≈ 0 . 35 dex, smaller than that expected even if all objects had the same mass. However, if we allow the dispersion in the mass estimate error to be a free parameter, we are able to obtain a good match to the data. Our best-fit value for the scatter in the mass estimates at a given mass is 0 . 18 ± 0 . 01 dex for Mg II and 0 . 13 ± 0 . 01 for C IV. Our result that the scatter in the mass estimates for high L and z must be smaller than ∼ 0 . 4 dex is consistent with what has been found in previous work (Kollmeier et al. 2006; Shen et al. 2008; Fine et al. 2008), and our best-fit values of the standard deviations in the mass estimate errors are consistent with the upper limits recently calculated by Steinhardt & Elvis (2010b). \nThe origin of this smaller scatter is unclear, and there may be several different possibilities. One possibility, as mentioned earlier and by Shen et al. (2008) and Shen & Kelly (2010), is that the error in the mass estimates may be correlated with luminosity. If this is true, then an error of ∼ 0 . 4 dex represents the error in the mass estimates when averaging over a broad range in L , as was done for the reverberation mapping sample, while a smaller scatter of ∼ 0 . 15 dex represents the error in the mass estimates when one is limited to a more narrow range in L , as the SDSS quasar sample is. It is unclear why the error in the mass estimates would be correlated with luminosity, but one possible source unaccounted for is radiation pressure. Marconi et al. (2008) argue that virial mass estimates should be corrected for radiation pressure. They find a correction that implies a steeper dependence on luminosity than ˆ M BL ∝ L 0 . 5 , especially for sources with high L/L Edd . Under their model, if one does not correct for radiation pressure then one will tend to underestimate the mass with increasing L , producing a correlation between the error in the mass estimates with luminosity, and possibly producing a smaller scatter in the mass estimate errors over a narrow range in luminosity. It is interesting to note that Marconi et al. (2008) find a smaller scatter in the mass estimates of ∼ 0 . 2 dex when correcting for radiation pressure for the reverberation mapped sample, which covers a larger range in luminosity. However, more work is need to understand the importance of radiation pressure, and if it can produce the observed smaller scatter in the mass estimates over the range in luminosity we probe. \nAnother possibility is that the error in the mass estimates may not be correlated with L , but the dispersion in the errors may decrease with increasing L or z . Unfortunately, the number of AGN with M BH estimated from reverberation mapping is too small to test this, and dominated by sources at lower L and z . The third possibility is that the virial mass estimates are biased at high L and z , at least for Mg II and C IV, and are only marginally related to the actual masses. The R -L relationship is well established for H β , and does not require a large extrapolation to the luminosities probed in our sample, and thus we do not expect H β -based mass estimates to be significantly biased in this range (Vestergaard 2009). The situation is less clear for Mg II and C IV. There is only one reliable time lag for the Mg II emission line (Metzroth et al. 2006). An R -L relationship has been estimated for the C IV line over a broad range in luminosity and redshift (Kaspi et al. 2007), including those probed in our study, and the R -L relationship is similar for both H β and C IV. Unfortunately, the C IV R -L relationship is estimated from only eight data points, and further work is needed in order to understand the Mg II- and C IV-based mass estimates and their errors. \nThroughout the rest of this work will we focus on the results obtained from allowing the dispersion in the mass estimate error to be a free parameter, a value of ∼ 0 . 4 dex is clearly ruled out. However, we note that we have performed the same analysis for both cases, and while the quantitative details change, the scientific conclusions are unaffected by treating the amplitude of the scatter as a free parameter. The only exception is that we infer a much more narrow distribution of Eddington ratio if we fix the amplitude of the scatter in the mass estimates to be ∼ 0 . 4 dex. In addition, the results reported in this section highlight the need for more reverberation mapping studies in order to better understand the nature of the errors in the mass estimates.", '4.2. The Black Hole Mass Function for Broad Line AGN': "Figure 3 shows the BLQSO BHMF at several redshifts. Our estimated BHMF is compared with an estimate of the local mass function of all SMBHs, and the BHMF reported by Vestergaard et al. (2008), obtained from binning up the broad line mass estimates. Following Merloni & Heinz (2008), the local BHMF was computed to be near the middle of the uncertainty range reported by Shankar et al. (2009) by convolving a Schechter function with a log-normal distribution with standard deviation 0.3 dex, chosen to be consistent with the scatter about the M BH -σ relationship; the parameters for the Schechter function are those reported by Merloni & Heinz (2008). Also, we note that early type galaxies dominate the local BHMF at M BH /greaterorsimilar 4 × 10 7 M /circledot (Yu & Lu 2008). \nIn order to focus on the region of the BHMF that is robust against uncertainties in the selection function, as well as against uncertainty on the Eddington ratio distribution, we estimate the black hole mass completeness for our sample as a function of z . Our best estimate of the SDSS completeness as a function of M BH and z is shown in Figure 4, and the 10% completeness limit is marked by a vertical line in Figure 3. The black hole mass completeness depends on both the completeness in luminosity, and the assumed distribution of L at a given M BH . As can be seen from Figure 4, at z /greaterorsimilar 2 the SDSS quasar sample is only ≈ 10% complete at M BH ∼ 10 9 M /circledot , becoming more incomplete at lower masses. At masses much lower than M BH ∼ 10 9 M /circledot , the estimated mass function almost completely depends on extrapolation from the set of BHMFs and Eddington ratio distributions that fit the observed data well, constrained by our assumed parameteric forms. Therefore, we stress that below M BH ∼ 10 9 M /circledot the BLQSO BHMF must be interpreted with caution, and in this work we will try to focus on what we can infer from the high mass end of the mass function. \nWe also estimate the completeness of our sample as a function of L/L Edd , assuming a constant bolometric correction of C 1350 = 4 . 3. The estimated completeness depends on the selection function and the estimated BHMF, as the selection function depends on luminosity, which is defined by the BHMF at a given L/L Edd . The Eddington ratio completeness \nFig. 3.BLQSO BHMF (thin solid lines) obtained using our Bayesian approach, compared with the local BHMF fo all SMBHs (dashed line), and the BHMF from Vestergaard et al. (2008, solid red line with points); as in Figure 1, each thin solid line denotes a random draw of the BHMF from its probability distribution. The thick green line is the median of the BHMF random draws, and may be considered our 'best-fit' estimate. The vertical line marks the mass at which the SDSS DR3 sample becomes 10% complete. \n<!-- image --> \n<!-- image --> \nFig. 4.Estimated completeness in black hole mass (left) and Eddington ratio (right) for the SDSS DR3 quasar sample of Richards et al. (2006), calculated using our assumed distribution of luminosity at a given black hole mass (Eq.[2]). The red, green, and blue lines denote the 10% , 50% , and 90% completeness levels, respectively. The SDSS sample is highly incomplete at M BH /lessorsimilar 10 9 M /circledot and L/L Edd /lessorsimilar 0 . 5. Note that the incompleteness at the highest masses is due to the upper flux limit of the SDSS. \n<!-- image --> \nis also shown in Figure 4. The SDSS is only ∼ 50% complete for sources radiating at the Eddington limit, and /lessorsimilar 10% complete for sources radiating at /lessorsimilar 10% of Eddington. This heavy incompleteness is due to the fact that, for our estimated BLQSO BHMF, half of BLQSOs have black holes that are not massive enough to make the SDSS flux limit even if they radiate at Eddington. We further discuss the Eddington ratio distribution in Section § 4.5. \nThe difference between the binned estimate of the BLQSO BHMF, calculated from the estimate of Vestergaard et al. (2008), and our estimate, is the result of the difference in statistical methodology employed by Vestergaard et al. (2008) and our work. First, the statistical error in the broad line mass estimates results in a broader inferred BHMF when one simply bins up these estimates (e.g., Kelly & Bechtold 2007, KVF09). Second, the 1 /V a technique corrects \nFig. 5.Evolution in the number density of BLQSO SMBHs for four different values of M BH . A larger fraction of the most massive SMBHs are seen as BLQSOs at higher redshift, an effect commonly referred to as black hole downsizing. Downsizing of BLQSO SMBHs has also been seen in the LBQS (Vestergaard & Osmer 2009). Symbols are as in Figure 3. The wiggles seen in the lowest mass bin are an artifact of fitting a Gaussian mixture model, unlikely to be real, and probably due to small unaccounted for errors in the selection function, which can become magnified due to the the fact that we are incomplete in this mass bin. \n<!-- image --> \nfor incompleteness in flux, but not black hole mass. As a result, the 1 /V a corrections only partially correct for incompleteness in M BH , and the estimated binned BHMF will still suffer from incompleteness, especially at the low mass end. The two effects combined result in a systematic shift in the estimated BHMF toward higher M BH (Shen et al. 2008, KVF09). However, the Bayesian approach outlined in KVF09 is able to self-consistently correct for these two effects in a statistically rigorous manner, conditional on the survey selection function and assumptions implicit within the statistical model. In spite of these differences in methodology, our estimated BHMF agrees fairly well with that of Vestergaard et al. (2008) at the high mass end, considering the differences in the methodology, and the two do not strongly diverge until values of M BH where the SDSS is highly incomplete. The poorer agreement between the two estimates at z < 2 is due to the fact that the BHMF is estimated using the Mg II emission line in this redshift range, which we find to have a higher statistical error than that of C IV. As a result, the our correction to the BHMF due to the error in the mass estimates is greater at these redshifts. \nIn Figure 5 we show the evolution in the comoving number density of SMBHs in BLQSO at four different masses. As is evident, the number density of higher mass BLQSO SMBHs peaks at higher redshift, where the number density of BLQSO SMBHs of M BH ∼ 5 × 10 8 M /circledot peaks at z ∼ 1 . 5, and the number density for M BH ∼ 5 × 10 9 M /circledot peaks at z ∼ 2 . 5. This result is consistent with what has commonly been referred to in the literature as black hole 'downsizing', where the most massive SMBHs are active, and grow, at earlier epochs than lower mass black holes, an effect also seen by Vestergaard & Osmer (2009) and Steinhardt & Elvis (2010a).", '4.3. The Broad Line Quasar Black Hole Mass Density, and Constraints on their Duty Cycle and Average Lifetime': "In Figure 6 we show the comoving mass density of SMBHs that reside in BLQSOs, as a function of redshift, ρ QSO ( z ). The peak in the cosmic mass density of BLQSO SMBHs occurs at z ∼ 2. Our constraint on the location of the peak in ρ QSO ( z ) is consistent with the peak in the mass density derived from the Large Bright Quasar Survey and highz sample of Fan et al. (2001a), as calculated by Vestergaard & Osmer (2009). Previous work has not found any evidence for evolution in the Eddington ratio distribution at z > 1 for the most massive BLQSOs (e.g., McLure & Dunlop 2004; Vestergaard 2004; Kollmeier et al. 2006; Vestergaard & Osmer 2009). If there is no significant evolution in the Eddington ratio distribution at z > 1 for these systems, then we would expect the peak in their luminosity density to coincide with the peak in their black hole mass density. The luminosity density of quasars peaks at z ∼ 2 (e.g., Wolf et al. 2003; Hopkins et al. 2007b), matching the peak in black hole mass density observed in our work. \nWe can use the quasar BHMF to place constraints on the fraction of black holes that are seen in the BLQSO phase \nFig. 6.Evolution in the mass density of SMBHs seen as BLQSOs, symbols as in Figure 3. For comparison, recent estimates of the local mass density of all SMBHs suggest ρ BH ( z = 0) ∼ 4 × 10 5 M /circledot Mpc -3 (e.g. Yu & Lu 2008). The mass density of SMBHs in BLQSOs peaks at z ∼ 2. \n<!-- image --> \nas a function of M BH ; this quantity is commonly referred to as the BLQSO 'duty cycle'. Denote the duty cycle as δ ( M BH , z ). Then \nδ ( M BH , z ) ≡ φ QSO ( M BH , z ) φ BH ( M BH , z ) . (12) \nHere, φ BH ( M BH , z ) is the BHMF of all SMBHs at a given redshift. Ignoring mergers of SMBHs, we can compute a lower-limit to the duty cycle by comparing the BHMF at a certain redshift with the local SMBH number density, since φ BH ( M BH , z ) ≤ φ BH ( M BH , 0). In Figure 7 we compute the lower limit of the BLQSO duty cycle at z = 1 for SMBHs with M BH > 5 × 10 8 M /circledot . The duty cycle at z = 1 is constrained to be δ /greaterorsimilar 0 . 01 at M BH ∼ 10 9 M /circledot , falling steeply to δ /greaterorsimilar 10 -5 at M BH ∼ 10 10 M /circledot . The decrease in the duty cycle with increasing black hole mass may be another reflection of SMBH downsizing, with the most massive BLQSO SMBHs being active at earlier cosmic epochs. Alternatively, it may be due to the fact that the most massive SMBHs spend a shorter amount of time in the broad line phase, as expected from some simulations of black hole feedback (e.g., Hopkins et al. 2006a). However, we note that the local number density of the most massive SMBHs is poorly constrained and subject to considerable systematic uncertainty (Lauer et al. 2007), and therefor duty cycle of BLQSOs with M BH ∼ 10 10 M /circledot may be subject to considerable systematic error. Indeed, the observed fall-off in the lower limit on the duty cycle for the most massive systems may simply be due to the fact that we are overestimating the number density of local SMBHs. \nThe quasar lifetime is not well constrained empirically (e.g., Martini 2004), but we can use our estimated BLQSO BHMF to estimate the BLQSO lifetime. For simplicity, we assume that all BLQSOs of a given mass have a single lifetime, t BL . Furthermore, if a SMBH undergoes multiple episodes of BLQSO activity, then we assume that t BL is the same for each episode. These assumptions are unlikely to be true, but for the purposes of our work we may still think of t BL as a 'typical' BLQSO lifetime. Because the duty cycle is the probability of observing a SMBH as a BLQSO at a given M BH and redshift we can relate t BL to δ ( M BH , z ). The probability of observing a SMBH as a BLQSO at redshift z is the product of the probability that a SMBH becomes a BLQSO along our line of sight at some point in its evolution with the probability that that SMBH is a BLQSO between cosmic ages t ( z ) -t BL and t ( z ), normalized by the probability that that SMBH is a BLQSO at t ≤ t ( z ). The normalization results from the additional assumption that if a SMBH of mass M BH exists at t ( z ), and if it goes through at least one BLQSO episode as some point in its growth, then it had to have gone through at least one BLQSO episode at t < t ( z ) in order to grow to M BH by t ( z ). This is a reasonable assumption, at least for the most massive system, since all SMBHs at t ( z ) had to grow from much lower mass seeds, and if BLQSO activity occurs for a SMBH at some point in its life, BLQSO activity would \nFig. 7.Lower bound on the duty cycle for BLQSO activity at z = 1 as a function of M BH , symbols as in Figure 3. The duty cycle for M BH ∼ 10 9 M /circledot BLQSO SMBHs at z ∼ 1 is δ /greaterorsimilar 0 . 01, falling to δ /greaterorsimilar 10 -5 for M BH ∼ 10 10 M /circledot . The decrease in the duty cycle with increasing black hole mass is likely due to either cosmic downsizing or a shorter BLQSO phase for the most massive SMBHs. \n<!-- image --> \nhave occurred during this growth period. In other words, the assumption implicit in the normalization correction is that SMBH seeds of mass, say, M BH ∼ 10 9 M /circledot , do not simply emerge and then undergo BLQSO activity at a cosmic epoch later than t ( z ). The practical result of this assumption is that the BLQSO duty cycle of, say, SMBHs with M BH ∼ 10 9 M /circledot , can decrease from δ ∼ 1 at z ∼ 7 to δ ∼ 10 -3 at z ∼ 1. \nDenote the time that a BLQSO phase is initiated in a SMBH as τ . Then, the probability that a SMBH BLQSO is seen between t ( z ) -t BL and t ( z ) is the same as the probability that the time that a BLQSO phase was initiated for that source occured at t ( z ) -t BL < τ < t ( z ). Note that this allows for multiple BLQSO episodes, as multiple phases of BLQSO episodes simply alter the probability of observing a SMBH as a BLQSO at a certain redshift. The duty cycle is thus related to t BL as \nδ ( M BH , z ) = Pr ( BLQSO | M BH ) ∫ t ( z ) t ( z ) -t BL p ( τ | M BH , BLQSO ) dτ t ( z ) 0 p ( τ | M BH , BLQSO ) dτ , (13) \n∫ \nwhere, Pr ( BLQSO | M BH ) is the probability that a SMBH with a mass of M BH is a BLQSO at some point in its evolution, and p ( τ | M BH , BLQSO ) is the probability distribution of BLQSO episode initiation time for a given black hole mass. Note that Pr ( BLQSO | M BH ) is not the probability that an object is currently seen as a BLQSO, but is the probability that it appears as a BLQSO to an observer on Earth at some point in its life. As mentioned above, p ( τ | M BH , BLQSO ) is sufficiently general to include both multiple and single episodes of BLQSO activity, although the actual form of p ( τ | M BH , BLQSO ) will depend on the distribution of the number of BLQSO episodes a SMBH goes through. All values of M BH in Equation (13) refer to the mass of the SMBH at redshift z . \nIf the distribution of τ does not evolve significantly over a time t BL , then the integral in the numerator of Equation (13) can be approximated as p ( τ | M BH , BLQSO ) t BL . Likewise, if p ( τ | M BH , BLQSO ) is approximately constant over t BL , then the distribution of BLQSO initiation times will be similar to the distribution of BLQSOs at the time we observed them. This is a reasonable assumption so long as t BL is short compared to the timescale for a significant change in the process that initiates BLQSO activity (e.g., mergers and other fueling events). Making this assumption, and rearranging Equation (13), it follows that \nt BL ≈ δ ( M BH , z ) ∫ t ( z ) 0 p ( t ' | M BH , BLQSO ) dt ' p ( t ( z ) | M BH , BLQSO ) Pr ( BLQSO | M BH ) (14) \nFig. 8.Probability distribution, given the observed data and assumptions of our analysis, of the lower bound on the BLQSO lifetime (Eq.[14]) calculated at z = 1 for M BH = 10 9 M /circledot . The upper axis gives the lifetime in terms of the Salpeter timescale, which is t Salp = 4 . 3 × 10 7 yrs for a radiative efficiency of /epsilon1 r = 0 . 1. The lower-bound on the BLQSO lifetime is equal to the BLQSO lifetime if (1) all SMBHs of M BH = 10 9 M /circledot go through a BLQSO phase and (2) if the number density of these SMBHs at z = 1 is not significantly less than the local number density. There is evidence that the latter condition is true (e.g., Merloni & Heinz 2008), while the former condition is expected if the SMBH's growth is self-regulated (Hopkins & Hernquist 2006). \n<!-- image --> \nwhere p ( t ( z ) | M BH , BLQSO ) is the probability distribution of BLQSOs as a function of cosmic age, t ( z ). The term p ( t | M BH , BLQSO ) is related to the BHMF for BLQSOs according to \n∣ \np ( t ( z ) | M BH , BLQSO ) = ∣ ∣ ∣ dt dz ∣ ∣ ∣ -1 [ φ ( M BH , z ) ∞ 0 φ ( M BH , z ) dM BH ] . (15) \n∣ ∣ From Equation (14) it follows that for the special case where all SMBHs experience a BLQSO phase, and where BLQSOs initiation times are uniformly distributed over the age of the Universe, then t BL = δ ( M BH , z ) t ( z ). This is the definition of a quasar 'lifetime' commonly found in the literature; however, as the assumption of a uniform distribution of BLQSO initiation times is inconsistent with the BLQSO BHMF or luminosity function, t BL will in general not equal δ ( M BH , z ) t ( z ) (see also Hopkins & Hernquist 2009). \n∣ \n∫ \nBecause we have assumed that t BL is the same for all BLQSOs of a given mass, and therefore must be the same at all redshifts, Equation (14) may be calculated at any redshift. However, in reality we can only estimate a lower limit to t BL . This is because we do not know the fraction of SMBHs that will experience a BLQSO phase, although if the SMBHs growth is self-regulated we might expect Pr ( BLQSO | M BH ) ≈ 1. However, if some SMBHs of mass M BH can never be seen as BLQSOs, possibly due to orientation-dependent obscuration, then Pr ( BLQSO | M BH ) < 1. In addition, as discussed above, we can only calculate a lower limit to δ ( M BH , z ) by comparing the BLQSO BHMF at redshift z with the local BHMF of all SMBHs. Moreover, implicit in these calculations is the assumption that none of these quantities depend strongly enough on mass to vary significantly during the growth that occurs in the BLQSO phase. In Figure 8 we show the probability distribution of the BLQSO age of SMBHs with M BH = 10 9 M /circledot , calculated from Equation (14) at z = 1. We calculate t BL at M BH = 10 9 M /circledot because our sample is reasonably complete at this mass, especially at this redshift. In addition, we chose z = 1 because the local BHMF better approximates the z = 1 BHMF than the BHMF at z > 1, therefore giving the tightest lower bound on the duty cycle at z = 1. We estimate t BL = 150 ± 15 Myr as a lower bound on the age of the BLQSO phase for SMBHs of M BH = 10 9 M /circledot . This estimate is roughly consistent with other estimates of quasar lifetimes (e.g., Yu & Tremaine 2002; Martini 2004; Gon¸calves et al. 2008; Hopkins & Hernquist 2009). \n<!-- image --> \nFig. 9.Probability distribution of M BH for the most massive SMBH that could be observed as a BLQSO at 1 < z < 4 . 5 (left), and the probability distribution for the redshift that this SMBH would be seen as a BLQSO (right). Here, we do not interpret M Max BH to represent a hard physical upper limit to M BH , but rather it is the result of a finite number of black holes drawn from a mass function. The constraints on M Max BH that we have obtained are consistent with previous observational work, but this is the first time that rigorous constraints on M Max BH and its redshift have been obtained. The most likely value for M Max BH is M BH ≈ 2 . 5 × 10 10 M /circledot , and this SMBH would likely be seen as a BLQSO at z /greaterorsimilar 2. \n<!-- image --> \nThe probability distribution for the maximum mass of a SMBH in a BLQSO may place important constraints on models of SMBH growth, and may be calculated directly from the BHMF. In the context of our work, we do not consider the maximum mass of a SMBH to be caused by a hard upper limit, above which it is impossible to make a more massive black hole, but rather the result of a finite number of black holes drawn from a mass function. The probability that the maximum mass of a SMBH in a sample of N BLQSOs is less than M is simply given by the probability that all N SMBHs have M BH < M : \nPr ( M Max BH < M| N ) = [ Pr ( M BH < M )] N (16) \nNote that N is the total number of BLQSOs that could be observed in an all-sky survey with no flux limit; i.e., N is the normalization of the BLQSO BHMF. The term Pr ( M BH < M ) is calculated from the BHMF as \nPr ( M BH < M ) = 1 N ∫ M 0 ∫ ∞ 0 φ ( M BH , z ) ( dV dz ) dz dM BH . (17) \nThe probability distribution of M Max BH is then found by differentiating Equation (16) with respect to M and evaluating the result at M = M Max BH : \np ( M Max BH | N ) = [ Pr ( M BH < M Max BH ) ] N -1 ∫ ∞ 0 φ ( M BH , z ) ( dV dz ) dz. (18) \nThe posterior probability distribution for M Max BH , given the observed data, can be calculated by averaging Equation (18) over the MCMC output. \nIn Figure 9 we show the posterior probability distribution of the maximum SMBH in a BLQSO at 1 < z < 4 . 5. Here, M Max BH should be interpreted as the maximum mass of a SMBH in a BLQSO that would be observed in an all-sky survey without a flux limit over the redshift interval 1 < z < 4 . 5. Thus, M Max BH is a lower bound on the mass of the most massive SMBH in the Universe. In addition, in Figure 9 we also show the probability distribution of the redshift at which the quasar with M Max BH would be found. As can be seen from these figures, the maximum mass of a SMBH in a BLQSO at 1 < z < 4 . 5 is M Max BH ∼ 3 × 10 10 M /circledot . The probability distribution for the redshift of M Max BH is rather broad, but we constrain the redshift for the BLQSO hosting this SMBH to be z /greaterorsimilar 2. \nOur results are in agreement with what others have found using samples of broad line mass estimates (e.g., Vestergaard 2004; Vestergaard et al. 2008; Netzer et al. 2007), although Labita et al. (2009a,b) find a smaller value of M Max BH ∼ 5 × 10 9 M /circledot . However, in the model of Labita et al. (2009a) M Max BH is a parameter determining the shape of the mass function, and is not the actual maximum black hole mass of a sample of objects drawn from the distribution of black hole mass; i.e., there is nothing in the statistical model of Labita et al. (2009a) that prevents objects with M BH > M Max BH . As such, the actual realized maximum mass in a large sample of objects will be greater than the value of M Max BH estimated using the model of Labita et al. (2009a), as we have found. Considering this, our results are consistent with those of Labita et al. (2009a,b). \nThese highly massive black holes represent the extremes of black hole growth, and thus are important for constraining models of black hole growth. Furthermore, these massive black holes offer the best chance of probing the faint end of the BLQSO Eddington ratio distribution. Therefore, it is of use to investigate how large and deep a survey must be \nFig. 10.Expected number counts of BLQSOs at 1 < z < 4 . 5 with black hole masses 5 × 10 8 < M BH /M /circledot < 5 × 10 9 (solid line) and 5 × 10 9 < M BH /M /circledot < 5 × 10 10 (dashed line) in a 1 deg 2 survey as a function of limiting magnitude. One would need to search an angular area Ω > 10 deg 2 in order to expect to find at least one BLQSO with M BH ∼ 10 10 M /circledot . In addition, one does not become complete at M BH ∼ 5 × 10 8 M /circledot until i ∼ 23. \n<!-- image --> \nin order to find an useful number of these objects. In Figure 10 we show the expected number of high mass SMBHs in BLQSOs at 1 < z < 4 . 5 in a 1 deg 2 survey as a function of limiting i -magnitude, assuming our best-fit statistical model. We stress that these are the expected number counts for the true black hole masses, and not the mass estimates. Because the errors in the mass estimates will scatter more sources into higher mass bins than into lower mass bins, the number of sources with estimated mass in a mass bin will be larger than the actual number of sources. \nAny survey with a flux limit of i < 19 should be able to detect BLQSOs with M BH /greaterorsimilar 5 × 10 9 M /circledot ; however, the survey must have an area of Ω /greaterorsimilar 10 deg 2 to expect to detect at least one of these objects. Similarly, in order to get a large number of BLQSOs with M BH /greaterorsimilar 5 × 10 8 M /circledot , one needs a deep, large area spectroscopic survey. For example, the BLQSO spectroscopic samples from the COSMOS survey (Scoville et al. 2007) cover ≈ 2 deg 2 at i < 24 (Trump et al. 2007) and i < 22 . 5 for zCOSMOS (Merloni et al. 2010), and, ignoring redshift incompleteness, are therefore expected to contain ∼ 60 BLQSOs at 1 < z < 4 . 5 with masses 5 × 10 8 < M BH /M /circledot < 5 × 10 9 , but none with masses M BH > 5 × 10 9 M /circledot .", '4.5. Implied Eddington Ratio Distributions': "As discussed in § 3.1, we also estimate the distribution of the ratio of the Eddington ratio to the bolometric correction at 1350 ˚ A, Γ Edd /C 1350 . Under the assumption that Γ Edd /C 1350 follows a mixture of log-normal distributions, with mean values that depends linearly on log M BH , and that C 1350 = 4 . 3 for all sources (Vestergaard & Osmer 2009), our model implies that the geometric mean Eddington ratio increases with M BH according to: \n〈 L L Edd 〉 geo = 0 . 12 ± 0 . 01 ( M BH M /circledot ) 0 . 48 ± 0 . 04 . (19) \nThe dispersion in L/L Edd decreases with increasing M BH , having a value of ∼ 0 . 4 dex at M BH ∼ 10 8 M /circledot , and decreasing to ∼ 0 . 3 dex at M BH ∼ 10 9 M /circledot . Both the dispersion and slope are smaller than what was found by KVF09, who analyzed a sample of BLQSOs from the Bright Quasar Survey at z < 0 . 5, suggesting possible evolution in the Eddington ratio distribution. However, the statistical significance of a difference in the slopes is marginal at best, as the differences are only significant at ∼ 2 σ . \nIn Figure 11 we show the implied distribution of BLQSO Eddington ratios assuming a constant bolometric correction \nFig. 11.Inferred Eddington ratio distribution, assuming the mixture of log-normals form of Equation (2) and a bolometric correction of C 1350 = 4 . 3. The vertical line marks the 10% completeness limit for the SDSS DR3 sample at z = 1; Figure 4 shows that the 10% completeness limits are similar for z ∼ 1 , 3 , and 4, but shallower for z ∼ 2. Our inferred Eddington ratio distribution peaks near L/L Edd ∼ 0 . 05 and has dispersion of ∼ 0 . 4 dex, although the peak fall below the 10% completeness limit and should be interpreted with caution. Our inferred Eddington ratio distribution is shifted toward lower values of L/L Edd and has a higher dispersion than previous work, which did not correct for incompleteness. Our estimated Eddington ratio distribution suggests that most SMBHs in BLQSOs are not radiating near their Eddington limit. \n<!-- image --> \nof C 1350 = 4 . 3 (Vestergaard & Osmer 2009). As can be seen, the estimated distribution of quasar Eddington ratios peaks at L/L Edd ∼ 0 . 05. However we note that the location of the Eddington ratio peak falls below the 10% completeness limit of the SDSS, and thus the exact location of the peak is highly uncertain. Our inferred Eddington ratio distribution is broader and shifted towards smaller values of L/L Edd than what has been found in previous work, which were not able to fully correct for incompleteness in black hole mass and Eddington ratio (e.g., McLure & Dunlop 2004; Vestergaard 2004; Kollmeier et al. 2006). However, it is consistent with what Shen et al. (2008) found using a similar approach to ours. The major differences between our work and that of Shen et al. (2008) is that they assume a power-law distribution of M BH , and they constrain the BHMF and Eddington ratio distribution by visually matching the model distributions to the observed distributions. Our results, as well as the results of Shen et al. (2008), show that the narrow distribution in quasar Eddington ratios seen in previous work, and peaking at L/L Edd ∼ 0 . 25, is due to uncorrected incompleteness. Indeed, deeper samples of BLQSOs have observed Eddington ratio distributions that are broader and peak at lower values (Gavignaud et al. 2008; Trump et al. 2009). Therefore, we conclude that most BLQSOs are not radiating at or near the Eddington limit, and that there is a large dispersion in Eddington ratio for BLQSOs. \nRecently, Steinhardt & Elvis (2010a) have analyzed the SDSS DR5 sample of mass estimates calculated by Shen et al. (2008), and concluded that there is a dirth of objects at high Eddington ratio, and that there is a redshift-dependent systematic decrease in Eddington ratio with increasing black hole mass for BLQSOs in the high L/L Edd tail of the distribution. They have called this effect a sub-Eddington boundary. While we also find evidence that BLQSOs radiating near the Eddington limit are rare, as did KVF09, the dependence on black hole mass is seemingly in contrast to what we find here, in that we find that the average Eddington ratio increases with M BH , assuming a constant bolometeric correction. The most important difference between their work and ours is the fact that Steinhardt & Elvis (2010a) analyze seperate redshift bins, while we do not model a redshift dependence in the Eddington ratio distribution. Steinhardt & Elvis (2010a) did not perform any calculations for the whole sample, averaged over all redshifts, making a more direct comparision difficult, and thus it is unclear how discrepant our conclusions are. \nAnother potential contributor to this discrepancy is the handling of the statistical error in the mass estimates. Statistical errors (e.g., measurement error) have the effect of flattening slopes and correlations when one analyzes the quantity contaminated by the error (e.g., Akritas & Bershady 1996; Kelly 2007; Kelly & Bechtold 2007), due to the \nfact that the distribution of the estimated quantity is a biased estimate of the distribution of the quantity of interest. Because the mass estimates are contaminated by statistical error, the dependence of luminosity on the mass estimates will be flatter than the intrinsic dependence of luminosity on M BH . In particular, the statistical error will cause some masses to be underestimated, and some to be overestimated. The objects which appear to have the highest masses at a given luminosity will have the highest overestimates, and thus will have their Eddington ratios underestimated. As a result, there will be an apparent dirth of high Eddington ratio objects at the highest masses. Similarly, there will be an apparent excess of high Eddington ratio objects in the low mass bins. Our method estimates the intrinsic dependence of luminosity on M BH by, in a sense, 'deconvolving' the distribution of the mass estimates and luminosity with the distribution of the error in the mass estimates; this is also why we find a somewhat steeper decline in the mass function at the highest masses, as compared to the estimate reported by Vestergaard et al. (2008). \nAlthough the statistical errors in the mass estimates can have the affect described above, possibly contributing to the different conclusions, it is unlikely that the statistical errors alone are sufficient to account for the discrepancy, and other possibilities include differences in the methods and scaling relationship employed for estimating the masses, and other differences in the methods of data analysis employed. However, further investigation is beyond the scope of our work. In addition, we note that our samples only overlap at 1 < z < 2 10 , and we cannot compare with the low redshift results of Steinhardt & Elvis (2010a), where they find the greatest evidence for these effects. \nAlthough we have used a flexible form for p ( L | M BH ), our estimated distribution for L/L Edd may be affected by significant systematics, as it relies on the assumption of a constant bolometric correction. There is evidence that the bolometric correction depends on both Eddington ratio (Vasudevan & Fabian 2007; Young et al. 2010) and black hole mass (Kelly et al. 2008), and the increase of L/L Edd with M BH we find may instead be a reflection of a decrease in the bolometeric correction with M BH . Indeed, Kelly et al. (2008) find that the ratio of optical/UV flux to X-ray flux increases with M BH , implying that the bolometeric correction to the UV flux decreases with M BH , and therefore we would infer an increase in L/L Edd with M BH assuming a constant bolometeric correction, as we do. In addition to these issues, we also note that the SDSS is at least partially incomplete at all Eddington ratios at z > 1 11 , as shown in Figure 4, and further work is needed using deeper surveys to confirm these results.", '5.1. Connection with Quasar Lightcurves': "A significant amount of recent work has suggested that quasar feedback regulates the growth of SMBHs and affects the large-scale evolution of its host galaxy. Understanding the quasar lightcurve is thus of fundamental importance for understanding black hole growth and feedback, as well as placing constraints on the physics of quasar accretion flows. The distribution of luminosity at a given black hole mass can be related to the quasar lightcurve, and therefore one can use the estimated distribution of luminosity at a given black hole mass to place some empirical constraints on models for quasar lightcurves. We address this in the following section, and argue that our estimated Eddington ratio distribution is consistent with models where the BLQSO phase represents the final stages of a SMBH's growth, in which the QSO lightcurve is expected to decay until the object no longer appears as a BLQSO. \nFor a given fueling episode, denote the quasar lightcurve as L ( t ), and the mass fueling rate to the SMBH as a function of time as ˙ M ( t ). The fueling rate, ˙ M ( t ), gives the rate at which matter is externally supplied to the BLQSO accretion disk. The probability distribution of quasar luminosity at a given M BH , ˙ M, and t is p ( L | M BH , ˙ M,t ), and the probability distribution of the fueling rate at a given M BH and t is p ( ˙ M | M BH , t ). The distribution p ( L | M BH , ˙ M,t ) is determined by the physics of the accretion disk, since the quasar luminosity is produced by viscous stresses in the accretion disk. For example, Siemiginowska & Elvis (1997) calculate p ( L | M BH , ˙ M,t ) implied by the model of Siemiginowska et al. (1996) for a thermal-viscous accretion disk stability. The distribution p ( ˙ M | M BH , t ), on the other hand, depends on the physics and stochastic nature of the fueling mechanism and quasar feedback. \nBecause the quasar luminosity is determined by the physics of the accretion disk, it is reasonable to assume that given M BH and ˙ M , L is independent of time. This does not imply that the quasar lightcurve does not vary with time, as it certainly does, but rather that knowing the value of t does not give us any additional information on the value of L when we already know M BH and ˙ M at a given t . We therefore drop the explicit dependence of p ( L | M BH , ˙ M,t ) on time. \nThe distribution of luminosity at a given M BH is calculated as \np ( L | M BH ) = ∫ ∞ 0 p ( L | ˙ M,M BH ) [∫ t o + t BL t 0 p ( ˙ M | t, M BH ) p ( t | M BH ) dt ] d ˙ M, (20) \nwhere p ( t | M BH ) is the probability of seeing a BLQSO at time t , given M BH , and t BL is the length of time that a quasar would be classified as a BLQSO. Here we have defined the broad line phase of quasar activity to start at t = t 0 . The term p ( t | M BH ) is related to the model for black hole growth, since p ( t | M BH ) ∝ p ( M BH | t ) p ( t ). Because we observe quasars randomly during their BLQSO phase, p ( t ) is uniform and p ( t | M BH ) ∝ p ( M BH | t ). In general, BLQSOs with larger values of M BH are more likely to be seen later in their lifetime, i.e., at larger values of t . \nAs an alternative to Equation (20), p ( L | M BH ) may be directly calculated from the amount of time that a BLQSO spends above a given luminosity, as a function of M BH . In this case, the term p ( L | M BH ) is directly related to the 'luminosity-dependent' lifetime interpretation advocated by Hopkins et al. (2005b,c), where the quasar 'lifetime' is the amount of time that a quasar spends above a given luminosity. Assume that p ( t | M BH ) ∝ 1 for BLQSOs, and denote T ( L | M BH ) to be the amount of time a BLQSO spends above a luminosity L , as a function of M BH . Then \np ( L | M BH ) ∝ ∣ ∣ dT ( L | M BH ) dL ∣ ∣ ∣ . (21) \n∣ \n∣ ∣ ∣ For example, Hopkins & Hernquist (2009) suggest that p ( L | M BH ) can be well approximated as a Schechter function, which generalizes the distributions implied by several common models of quasar lightcurves often found in the literature.", '5.1.1. Effects of Evolution in the Quasar Fueling Rate': "Recent work on quasar fueling and feedback suggests that the BLQSO phase occurs at the end of the fueling event, during which feedback energy from the AGN is able to unbind the surrounding gas, removing the obscuring material and halting the black hole's growth. Within the context of this model, the SMBH does not experience a large fractional increase in mass beyond the value of M BH that is observed during its time as a BLQSO. Therefore, we approximate p ( t | M BH ) as being uniform and independent of M BH , i.e., p ( t | M BH ) = 1 /t BL . In addition, this model predicts that during this so-called 'blow-out' phase, the fuel supply is set by the evolution of the blastwave caused by the injection of energy from the SMBH, and is expected to be of the form ˙ M ( t ) ∝ t -β (Hopkins & Hernquist 2006; Hopkins et al. 2006a). Alternatively, if the fuel supply is suddenly removed, then the accretion rate during the broad line phase is due to evolution of the viscous accretion disk. The fueling rate also has a power-law form under viscous evolution of the disk, but with a different value for β (Cannizzo et al. 1990; Pringle 1991). Under these models, it is reasonable to approximate the fuel supply as a deterministic process, ˙ M ( t ) ∝ t -β . If the evolution of the fuel supply is a deterministic and one-to-one process, and assuming that p ( t | M BH ) ∝ 1 /t BL , then p ( ˙ M | M BH ) is \np ( ˙ M | M BH ) = 1 t BL dt ( ˙ M,M BH ) d ˙ M , (22) \nwhere t ( ˙ M,M BH ) is the time implied by ˙ M ( t, M BH ), i.e., t ( ˙ M,M BH ) is the inverse function of ˙ M ( t, M BH ). If the quasar lightcurve is a many-to-one function, then the right side of Equation (22) is replaced by a sum of derivatives, as in Equation (15) of Yu & Lu (2004). Inserting Equation (22) into Equation (20) and dropping the time integral gives 12 \np ( L | M BH ) = 1 t BL ∫ ∞ 0 p ( L | ˙ M,M BH ) dt ( ˙ M,M BH ) d ˙ M d ˙ M. (23) \nFor models where ˙ M ( t ) ∝ t -β , a power-law distribution of fueling rates follows from Equation (22), p ( ˙ M | M BH ) ∝ ˙ M -(1+1 /β ) (see also Eq. 43 in Yu & Lu 2008). Therefore, assuming ˙ M ( t ) ∝ t -β gives \np ( L | M BH ) ∝ ∫ ∞ 0 p ( L | ˙ M,M BH ) ˙ M -(1+1 /β ) d ˙ M. (24) \nThe feedback-driven model of Hopkins & Hernquist (2006) predicts a value of β ∼ 2, while the disk evolution model predicts β ∼ 1 . 2 (Cannizzo et al. 1990; Yu et al. 2005; King & Pringle 2007), implying that p ( ˙ M | M BH ) ∝ ˙ M -1 . 5 or p ( ˙ M | M BH ) ∝ ˙ M -1 . 67 , respectively.", '5.1.2. Effects of Time-Dependent Accretion Disks': 'It is apparent from Equation (24) that the effect of time-dependent accretion disk behavior is to create a distribution of luminosities at a given M BH that is flatter than that expected from a simple power-law decay in the fueling rate. While the BLQSO fueling rate may be approximated as a deterministic process, quasar lightcurves are observed to exhibit aperiodic and stochastic variations across all wavelengths (for a review see Ulrich et al. 1997). Therefore, the quasar lightcurve will be stochastic, i.e., the value of the luminosity is not completely determined by M BH , and ˙ M . From Equation (24) it can be observed that if we assume a deterministic relationship between L and ˙ M , i.e., a time-steady accretion disk, as much previous work on SMBH growth and feedback has assumed, then L ∝ ˙ M and the quasar lightcurve simply traces the fueling rate evolution. In this case, p ( L | M BH , ˙ M ) is a delta function and p ( L | M BH ) ∝ L -(1+1 /β ) . However, the time-dependent and stochastic nature of the accretion disk emission broadens the observed luminosity function beyond that implied solely from the BLQSO fueling evolution. \nAlthough quasars are observed to vary at all wavelengths, we focus our remaining discussion on optical variability, since we are interested in the distribution of optical luminosities in this work. Kelly et al. (2009a) found that quasar optical lightcurves on timescales /lessorsimilar 7 yrs in the rest frame of the quasar are well-described by a Gaussian process \non the logarithmic scale (see also Koz/suppresslowski et al. 2010), with characteristic timescale consistent with accretion disk thermal time scales. They suggested that quasar variability on these time scales is due to a turbulent magnetic field in the accretion disk, which drives changes in the radiation energy density of the disk, as also seen in 3-dimensional magneto-hydrodynamic simulations of radiation-pressure dominated accretion disks (Hirose et al. 2008). Extrapolation of their best-fit lightcurves imply that flux variations for this particular process have a standard deviation ∼ 0 . 1 dex on timescales long compared to the disk thermal time scale, independent of M BH . If this is the only source of quasar optical variability, then p ( L | M BH , ˙ M ) is a gaussian distribution centered at L opt = C opt /epsilon1 r ˙ Mc 2 with standard deviation ∼ 0 . 1 dex. Here C opt is the bolometric correction to the optical luminosity (which likely depends on M BH and ˙ M ), /epsilon1 r is the radiative efficiency, and c is the speed of light. This process only produces a small amount of broadening in the luminosity distribution, relative to that implied solely from the BLQSO fueling evolution. Therefore, p ( L | M BH ) ∝ L -(1+1 /β ) is a reasonable approximation when the fuel rate declines as a power-law, so long as there are no other variability components in the accretion disk beyond that observed by Kelly et al. (2009a). However, this does not avoid the issue of a non-constant bolometric correction. \nVariability on longer time scales may be driven by accretion disk instabilities. Accretion disks based on the standard α -prescription for viscosity (Shakura & Sunyaev 1973) are subject to a number of accretion disk instabilities that can potentially have a significant effect on the quasar lightcurve. At high accretion rates ( ˙ m /greaterorsimilar 0 . 025) a radiationpressure instability may operate on time scales /greaterorsimilar 10 4 yrs (Czerny et al. 2009), while at lower accretion rates a thermal-viscous ionization instability (e.g., Lin & Shields 1986; Siemiginowska et al. 1996; Menou & Quataert 2001; Janiuk et al. 2004) may operate on time scales /greaterorsimilar 10 6 yrs. The ionization instability has been invoked to explain the outbursts seen in dwarf novae and soft X-ray transients (for a review see Lasota 2001), and the radiation pressure instability has been invoked to explain the outbursts seen in the microquasar GRS 1915+105 (e.g., Nayakshin et al. 2000; Janiuk et al. 2000). Moreover, Goodman & Tan (2004) suggest that stars may also form in AGN accretion disks due to fragmentation in the outer edges of the disk. While it is apparent that accretion disks around galactic sources exhibit instabilities, it is currently unclear if and how these instabilities operate in AGN, as the time scales involved are too long to observe transitions between quiescence and outburst. Furthermore, theoretical predictions of lightcurves based on these instabilities vary, and the existence and importance of disk stabilities can depend on how the viscosity is parameterized (e.g., Stella & Rosner 1984; Szuszkiewicz 1990; Merloni 2003), how much of the accretion energy is dissipated in a hot corona (e.g., Svensson & Zdziarski 1994), or if there is an outflow or an advection dominated accretion flow (ADAF) (e.g., Hameury et al. 2009). \nThe location in the disk where the instability occurs is important, as p ( L | ˙ M,M BH ) is with respect to the optical flux in our model. Various instabilities are expected to operate in different regions of the disc, sometimes in a stochastic manner, and thus may or may not significantly affect the optical flux. The term p ( L | ˙ M,M BH ) is also conditional on the source being seen as a BLQSO, and if the disk instabilities cause the ionizing continuum to disappear during quiescence, thus causing the broad emission lines to disappear as well, then p ( L | ˙ M,M BH ) is only with regard to the distribution of optical luminosities during a disk outburst. Considering these issues, and the current uncertainty regarding the importance of disk instabilities, we consider it beyond scope of this paper to fully investigate the effects of time-dependent behavior in the accretion disk. However, in light of this discussion, disk instabilities can potentially have a significant effect on the distribution of luminosity at a given M BH (Siemiginowska & Elvis 1997). Thus, the distribution of luminosity at a given M BH is likely altered beyond a simple power-law expected if the optical flux is trivially related to the external fueling rate. \nIn spite of these issues related to the detailed physics of the accretion disk, our estimated p ( L | M BH ) is qualitatively consistent with models which predict the BLQSO phase to occur while the quasar accretion rate is decaying, as we find that high Eddington ratio objects are rare, and that the number density of BLQSOs increases steeply toward lower values of L/L Edd . In addition, although our estimated Eddington ratio distribution continues to rise toward lower L/L Edd , our sample is too incomplete to rigorously probe the low Eddington ratio region of the distribution, and our estimated L/L Edd distribution in this region is heavily dependent on our parameteric form used to extrapolate beyond L/L Edd /lessorsimilar 0 . 1. Thus, we cannot test if the Eddington ratio distribution continues to rise until L/L Edd ∼ 10 -2 , after which a steep decay in p ( L/L Edd ) might occur due to the disappearance of the broad emission lines (e.g. Churazov et al. 2005; Trump et al. 2009).', '5.2. Implications for the Growth of Supermassive Black Holes': "Our results in this work are broadly in agreement with recent observational and theoretical work suggesting that SMBH growth and spheroid formation have a common origin. In particular, models where mergers dominate black hole growth predict that major mergers of gas-rich galaxies initiate a burst of star formation (Mihos & Hernquist 1994a, 1996), during which the black hole undergoes Eddington-limited obscured growth. Eventually the black hole becomes massive enough for radiation-driven feedback to unbind the surrounding gas, halting the accretion flow and revealing the object as a BLQSO (e.g., Hopkins et al. 2006b). As the activity further declines, the remnant will redden and become quiescent, satisfying the black hole-host galaxy correlation and leaving a dense stellar remnant from the starburst (Mihos & Hernquist 1994b). This dense stellar remnant is identifiable as a second component in the light profiles of elliptical galaxies (Hopkins et al. 2008b, 2009c,d). Techniques based on the argument of Soltan (1982) have concluded that the SMBH accretion rate density of the Universe peaks at z ∼ 2 (e.g., Marconi et al. 2004; Merloni & Heinz 2008; Shankar et al. 2009), in agreement with the observed peak in the SMBH luminosity density of \nthe Universe at z ∼ 2 (e.g., Wolf et al. 2003; Hopkins et al. 2007b). We have found that the peak in the BLQSO cosmic mass density also occurs at z ∼ 2, in broad agreement with models where the SMBH undergoes Eddington-limited growth up to the BLQSO phase, at least on a cosmological level. \nIn addition, our results suggest that a lower bound on the typical length of time for which a massive ( M BH ∼ 10 9 M /circledot ) SMBH could be seen as a BLQSO during a single BLQSO episode is t BL /greaterorsimilar 120 Myr, i.e., a few Salpeter times. If all M BH ∼ 10 9 M /circledot SMBHs go through a BLQSO phase, and if the number density of SMBHs with M BH ∼ 10 9 M /circledot does not significantly increase from z = 1 to z = 0, then Equation (14) is no longer a lower bound and t BL ∼ 150 Myr represents an estimate of the lifetime of a single BLQSO phase. Indeed, if the SMBH's growth is self-regulated then there is good reason to assume that most SMBHs go through a BLQSO phase at the end of their growth (Hopkins & Hernquist 2006), as the obscured/unobscured dichotomy represents an evolutionary sequence, at least at z > 1. Furthermore, recent work employing the argument of Soltan (1982) suggests that the number density of M BH ∼ 10 9 M /circledot SMBHs does not significantly increase from z = 1 to z = 0 (Merloni & Heinz 2008). These two considerations imply that t BL ∼ 150 Myr should represent a reasonable estimate of the length of the BLQSO phase. \nOur estimated BLQSO lifetime of t BL ∼ 150 Myr is longer than the value of t BL ∼ 10-20 Myr predicted by Hopkins et al. (2006b). Part of this discrepancy may be due to the fact that they defined a BLQSO as having a B-band luminosity brighter than some fraction of the host galaxy's, while we define a BLQSO as simply having broad emission lines along our line of sight. Therefore, if a BLQSO has a decaying lightcurve, as assumed in the model of Hopkins et al. (2006b), then by our definition a SMBH may still be considered a BLQSO even after the B-band luminosity has fallen below its host's. This would imply that the predicted BLQSO lifetime under the definition of Hopkins et al. (2006b) would be shorter than our estimated value, as is the case. However, it is unclear if the lifetime of the BLQSO should be a factor of ∼ 10 shorter under the definition of Hopkins et al. (2006b). Unfortunately, we cannot invert our estimated Eddington ratio distribution to a quasar lightcurve to test this, as the inversion is not unique, and we would have to assume a distribution of host galaxy B-band luminosity during the BLQSO phase at z ∼ 1, which is poorly constrained; as Hopkins et al. (2006b) point out, uncertainty in the ratio of host galaxy to quasar B-band luminosity also introduces considerable systematic uncertainty in their predicted BLQSO lifetime. Moreover, our estimated lifetime is subject to the assumptions outlined in § 4.3, which may introduce additional systematic uncertainties of a factor of a few. Considering this, a more quantitative comparison between our estimate and the prediction from Hopkins et al. (2006b) is difficult. \nOur estimated BLQSO BHMF suggests a typical value of M BH ∼ 10 8 M /circledot for BLQSOs. Theoretical estimates of the mass distribution of SMBH seeds suggets that typical values for the seed mass should be M BH ∼ 10 5 -10 6 M /circledot at z ∼ 10 (Lodato & Natarajan 2007; Pelupessy et al. 2007; Volonteri et al. 2008), and grow to M BH ∼ 10 8 M /circledot by z ∼ 3 (Volonteri & Natarajan 2009), consistent with our results. If the black hole's growth is Eddington-limited, then at least several Salpeter times are required to grow a seed black hole to M BH ∼ 10 9 M /circledot from a M BH ∼ 10 6 M /circledot seed, which is inconsistent with our estimated BLQSO lifetime. Furthermore, we find that most SMBHs in BLQSOs are not radiating near the Eddington limit, with a typical value being L/L Edd ∼ 0 . 1 at M BH ∼ 10 9 M /circledot , as we might expect if BLQSOs are seen during a phase with a decaying fueling rate. Because most BLQSOs are radiating at considerably less than the Eddington limit, this therefore suggests that a factor of ∼ 10 longer is needed to grow these SMBHs from seeds of M BH ∼ 10 6 M /circledot , i.e., a growth time scale of ∼ 70 Salpeter times. This corresponds to the age of the universe at z ∼ 2, implying that sources observed at z > 2 had to have been accreting at higher rates earlier in their growth. \nAn inferred growth time scale of ∼ 70 Salpeter times is significantly longer than our estimated BLQSO lifetime at z ∼ 1, implying one of a few explanations. First, it implies that if we assume that BLQSO SMBHs spend all of their growth as a BLQSO, then our assumption that all SMBHs go through a BLQSO phase along our line of sight is incorrect. In other words, this implies that some of the SMBH population that is growing at any redshift could never be observed by us to be a BLQSO at any time. Because the lifetime is related to what we can calculate from the BHMF by Equation (14), and if the SMBH spends all of its growth as a BLQSO, then an inferred SMBH growth time scale of t BL ∼ 70 t salp implies that only ∼ 4% of SMBHs ever go through a phase where we could observe them as BLQSOs at any time. This is an unrealistically low number, and thus we conclude that t BL is shorter than the growth time. Furthermore, it also illustrates that even if we can never observe a large fraction of active SMBHs, say ∼ 50 -75%, possibly due to orientation-dependent obscuration, then our estimated t BL is only underestimated by a factor of ∼ 2-4. \nAnother possibility is that SMBHs in BLQSOs at z ∼ 1 built up their mass via numerous fueling events of length t BL /greaterorsimilar 150 Myr. In this case one could grow a SMBH from a seed mass of M BH ∼ 10 6 M /circledot at z ∼ 10, say, to a mass of M BH ∼ 10 9 M /circledot by z ∼ 1, without the need for an earlier phase of obscured and accelerated growth. Similarly, we may have incorrectly assumed that the lifetime of the BLQSO phase is short compared to the time scale needed for any significant change in the BLQSO triggering time distribution, as this assumption was needed in order to use the observed distribution of BLQSOs as an estimate of their triggering distribution. This assumption may be incorrect if BLQSOs undergo a single long growth phase from z ∼ 10 to z ∼ 1. However, this possibility only exists for z /lessorsimilar 1 . 7, since at higher redshifts the growth time for objects radiating at L/L Edd ∼ 0 . 1 is longer than the lookback time to z ∼ 10, and thus BLQSOs that are observed at z /greaterorsimilar 1 . 7 would have had to experience an earlier phase of obscured growth at an enhanced accretion rate. Moreover, if SMBHs that are seen as BLQSOs at z ∼ 1 with M BH ∼ 10 9 M /circledot are grown in a single long BLQSO episode, or numerous repeated ones of t BL ∼ 150 Myr, this would also imply that BLQSOs that are seen at z /lessorsimilar 1 . 7 would have had a different fueling mechanism than BLQSOs that are seen at z /greaterorsimilar 1 . 7, and it is unclear why this should be true. Therefore, we conclude that while part of the mass of SMBHs in \nBLQSOs may have been accumulated via multiple BLQSO episodes, it is unlikely that most of these SMBHs did not also experience a phase of obscured growth at an enhanced accretion rate. \nIf most SMBHs go through a BLQSO phase along our line of sight at some point in their growth, the fact that our estimated BLQSO lifetime is short compared to the SMBH growth time implies that SMBHs spend a significant amount of their time growing in a non-BLQSO phase, such as an obscured phase. Note that this argument is unaffected by the fact that at any z we miss a considerable fraction of the SMBH population that is growing (e.g., due to obscuration), so long as these missed SMBHs undergo a BLQSO episode at some point in their life. The conclusion that a significant amount of growth occurs in an earlier obscured phase was also reached by Treister et al. (2010), and is expected from self-regulation models models where mergers or other triggering mechanisms fuel and initiate Eddington-limited accretion and obscured SMBH growth, until the SMBH becomes massive enough to unbind the ambient gas, revealing it as a BLQSO for t BL ∼ 150 Myr. Within this interpretation, the BLQSO black hole mass density shown in Figure 6 is proportional to the rate at which the relic SMBH mass increases as a function of redshift. Furthermore, considering that the growth time scale for a SMBH radiating at L/L Edd ∼ 0 . 1 to grow from 10 6 M /circledot to 10 9 M /circledot corresponds to the age of the universe at z ∼ 2, we also conclude that SMBHs accrete at a significantly higher rate during the earlier obscured phase, as compared to the BLQSO phase. \nIf the SMBHs growth is self-regulated, then the final mass of the SMBH after a fueling event is set by the binding energy of the bulge regardless of the fueling mechanism (Younger et al. 2008). In order to buildup a mass as large as M Max BH , multiple fueling episodes would likely be necessary, as is seen in cosmological simulations (e.g., Li et al. 2007; Di Matteo et al. 2008; Sijacki et al. 2009). However, it is unlikely that M Max BH could be increased significantly beyond the observed value as additional fueling mechanisms would likely result in only a small increase in the binding energy of the bulge. Therefore, additional fueling episodes would immediately result in the SMBH unbinding the accreting gas, preventing significant additional growth. Indeed, if our estimated BLQSO lifetime of t BL ∼ 150 Myr is correct for SMBHs with M BH ∼ 10 9 M /circledot , and if the SMBH is radiating at a typical Eddington ratio of L/L Edd ∼ 0 . 1 during the BLQSO phase, then it would only accrete an additional ∼ 10 8 M /circledot , assuming a radiative efficiency of /epsilon1 r = 0 . 1. If the distribution of Eddington ratios is a power-law with p (Γ Edd ) ∝ Γ -1 . 5 Edd , as suggested by the discussion in § 5.1.1, then the typical Eddington ratio would be L/L Edd /lessorsimilar 0 . 1, suggesting even less growth during the BLQSO phase. Moreover, additional growth via black hole mergers, as might be expected from 'dry' mergers of galaxies, is also unlikely to lead to significant additional growth, as the mass of the additional SMBH is likely to be negligible compared to M Max BH . And finally, additional growth through radiatively inefficient low-accretion rate modes does not contribute significantly to the black hole's final mass (Hopkins et al. 2006c; Cao 2007). Therefore, our estimated value of M Max BH ∼ 3 × 10 10 M /circledot should be representative of the most massive SMBH for both active and inactive SMBHs. \nIn this work we have found that the most massive SMBH that could be seen as a BLQSO is M Max BH ∼ 3 × 10 10 M /circledot and would most likely be observed at z /greaterorsimilar 2. These constraints on M Max BH are consistent with recent cosmological simulations of SMBH growth, as well as expectations from black hole feedback models. Cosmological simulations that follow the growth of SMBHs in bright z ∼ 6 quasars have been able to grow SMBHs to M BH ∼ 10 9 M /circledot by z ∼ 4 (Li et al. 2007; Di Matteo et al. 2008) and M BH ∼ 2 × 10 10 M /circledot by z ∼ 2 (Sijacki et al. 2009). Similarly, considerations based on quasar feedback and self-regulated SMBH growth, combined with the local distribution of bulge velocity dispersion, also suggest a value of M Max BH ∼ 10 10 M /circledot (Natarajan & Treister 2009). \nOur results are consistent with previous data-based work that has attempted to map the distribution and growth of SMBHs. Most previous observational work has mapped SMBH growth by using the M BH -σ relationship to infer the local BHMF, and then stepped backward in time using the quasar luminosity function to infer the contribution to the BHMF from accretion (e.g., Soltan 1982; Yu & Tremaine 2002; Shankar et al. 2004; Marconi et al. 2004; Merloni & Heinz 2008). In addition, there have been attempts to predict the BHMF and its evolution for all SMBHs, or all active SMBHs (e.g., Volonteri et al. 2003; Sijacki et al. 2007; Hopkins et al. 2008a; Di Matteo et al. 2008; Sijacki et al. 2009; Shen 2009). While not directly comparable to these studies, as we focus on broad line quasars, our results and conclusions are qualitatively consistent with previous observational and theoretical work in that we find evidence for self-regulated SMBH growth, black hole downsizing, and BLQSO lifetimes of t BL ∼ 150 Myr.", '6. SUMMARY': "Our main results are: \n- · We have, for the first time, obtained an estimate of the black hole mass function for broad-line quasars that self-consistently corrects for incompleteness and the statistical uncertainty in the mass estimates derived from the broad emission lines in a statistically rigorous manner. Our estimated BHMF was obtained using data from the SDSS DR3 quasar sample.\n- · The standard deviation in the statistical error of the broad line mass estimates is less than the commonly used ∼ 0 . 4 dex within the range of luminosity and redshift probed in our analysis. This may be due to correlation between the error in the mass estimates and luminosity and/or redshift, or a dependence of the standard deviation of the error on luminosity and/or redshift. When we treat the standard deviation in the statistical error as a free parameter, we estimate that the the Mg II-based mass estimates scatter about the reverberation mapping estimates with an amplitude of ≈ 0 . 18 dex, while the C IV-based estimate scatter with an amplitude of ≈ 0 . 13 dex. \n- · We find evidence for cosmic downsizing among BLQSOs, where the number density of BLQSOs peaks at higher redshift with increasing black hole mass.\n- · We find that the comoving mass density of SMBHs in BLQSOs peaks at z ∼ 2. We use our estimate for the BHMF to place constraints on the duty cycle, δ ( M BH , z ), and lifetime, t BL , for BLQSOs. The duty cycle at z = 1 is constrained to be δ /greaterorsimilar 0 . 01 at M BH ∼ 10 9 M /circledot , falling to δ /greaterorsimilar 10 -5 at M BH ∼ 10 10 M /circledot . We estimate the lifetime of the BLQSO phase for SMBHs of M BH = 10 9 M /circledot at z = 1 to be t BL = 150 ± 15 Myr. However, we will have underestimated the BLQSO lifetime if there is a population of M BH = 10 9 M /circledot SMBHs that never experience a BLQSO phase along our line of sight, or if the local number density of these black holes is significantly larger than the z = 1 number density of these black holes. We argue that our estimated BLQSO lifetime, in combination with the estimated Eddington ratio distribution, suggests that most of a SMBH's growth occurs when it is not seen as a BLQSO and accreting at a higher rate, and that BLQSO activity represents a short phase that most SMBHs go through, consistent with self-regulated growth models.\n- · We estimate that the most massive SMBH that could be seen as a BLQSO is M BH ≈ 3 × 10 10 M /circledot . This SMBH would most likely be seen as a BLQSO at z > 2. While largely in agreement with previous work, we have for the first time obtained statistically rigorous constraints on the value of M Max BH and its redshift.\n- · Assuming a constant bolometric correction of C 1350 = 4 . 3 (Vestergaard & Osmer 2009), our inferred distribution of Eddington ratios peaks at L/L Edd ∼ 0 . 05 and has a dispersion of ∼ 0 . 4 dex. Compared to previous work, our inferred Eddington ratio distribution is broader and shifted toward lower values of L/L Edd , showing that previous estimated distributions of L/L Edd were significantly affected by incompleteness. We therefore provide evidence that most BLQSOs are not radiating at or near the Eddington limit, and that there is a large dispersion in Eddington ratio for BLQSOs. In addition, we also find that the number density of BLQSOs increases steeply toward lower values of L/L Edd , consistent with models where the BLQSO phase occurs when the fuel supply is dwindling or halted.\n- · The evolution of the cosmological SMBH mass density for BLQSOs tracks the evolution in the cosmological accretion rate density of SMBHs estimated from variations of the Soltan (1982) argument (Marconi et al. 2004; Merloni & Heinz 2008; Shankar et al. 2009). This result, in combination with our estimated BLQSO lifetime and Eddington ratio distribution, are qualitatively consistent with models of self-regulated SMBH growth, with the BLQSO phase occuring at the end of the SMBHs obscured Eddington-limited growth (e.g., Hopkins & Hernquist 2006; Hopkins et al. 2006b). \nWe thank Yue Shen, Priyamvada Natarajan, Martin Elvis, and Charles Steinhardt for helpful discussions and comments on this paper, Mark Ammons and Aleks Diamond-Stanic for helpful discussions, and the anonymous referee for a careful reading and comments that lead to improvement of this work. BK acknowledges support by NASA through Hubble Fellowship grants #HF-01220.01 and #HF-51243.01 awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555. MV acknowledges financial support through grants HST-AR-10691, HST-GO-10417, and HST-GO-10833 from NASA through the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS5-26555. XF acknowledges support by NSF grant AST-0806861, a Packard Fellowship for Science and Engineering, a John Simon Guggenheim Memorial Fellowship and the Max Planck Society. The Dark Cosmology Centre is funded by the Danish National Research Foundation.", 'REFERENCES': "Akritas, M. G., & Bershady, M. A. 1996, ApJ, 470, 706 Aller, M. C., & Richstone, D. O. 2007, ApJ, 665, 120 Begelman, M. C., & Nath, B. B. 2005, MNRAS, 361, 1387 Booth, C. M., & Schaye, J. 2009, in press at MNRAS(arXiv:0904.2572) Bower, R. G., Benson, A. J., Malbon, R., Helly, J. C., Frenk, C. S., Baugh, C. M., Cole, S., & Lacey, C. G. 2006, MNRAS, 370, 645 Bruzual, G., & Charlot, S. 2003, MNRAS, 344, 1000 Cannizzo, J. K., Lee, H. 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2010CQGra..27a5010N
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Noncommutative geometry-inspired dirty black holes
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2010-01-01
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We provide a new exact solution of the Einstein equations which generalize the noncommutative geometry-inspired Schwarzschild metric, we previously obtained. We consider here a more general relation between the energy density and the radial pressure and find new geometries describing a regular 'dirty black hole'. We discuss strong and weak energy condition violation and various aspects of the regular dirty black hole thermodynamics.
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https://arxiv.org/pdf/0902.4654.pdf
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{'P Nicolini 1 ‡ and E Spallucci 2': "1 Physics Department, CSU Fresno, Fresno, CA 93740-8031, USA and Dipartimento di Matematica e Informatica, Consorzio di Magnetofluidodinamica, Universit'a degli Studi di Trieste and I.N.F.N., Via Valerio 12, 34127 Trieste, Italy 2 Dipartimento di Fisica, Universit'a degli Studi di Trieste and I.N.F.N., Strada Costiera 11, 34014 Trieste, Italy \nE-mail: [email protected], [email protected] \nAbstract. We provide a new exact solution of the Einstein equations which generalizes the noncommutative geometry inspired Schwarzschild metric, we previously obtained. We consider here more general relations between the energy density and the radial pressure and find a new geometry describing a regular 'dirty black hole'. We discuss strong and weak energy condition violations and various aspects of the regular dirty black hole thermodynamics. \nPACS numbers: 04.20.Jb, 04.70.Bw, 04.70.Dy", '1. Introduction': "In a recent series of papers we obtained exact solutions of the Einstein equations describing both neutral and charged black holes free of curvature singularities in the origin [1, 2, 3, 4, 5, 6, 7, 8]. We reached these conclusions after a long path starting from an original approach to noncommutative geometry which is based on coordinate coherent states [9, 10, 11, 12, 13, 14, 15, 16, 17, 18], in alternative to the mathematically correct, but physically hard to implement, 'star-product' formulation. \nHowever, our results are 'model-independent' in the sense that our approach improve the short-distance behaviour of the Einstein equations by taking into account the presence of a quantum gravity induced minimal length, whatever it is. \nString Theory, Noncommutative Geometry, Generalized Uncertainty Principle, etc., all point out the existence of a lower bound to distance measurements. Thus, the very concept of 'point-like' particle becomes physically meaningless and must be replaced with its best approximation consistent with the tenets of quantum mechanics, i.e. a minimal width Gaussian distribution of mass/energy. Solving the Einstein equations for a static, minimal width, mass-energy distribution centered around the origin, we found black hole type solution smoothly interpolating between deSitter spacetime at short distance and Schwarzschild geometry at large distance. The characteristic length scale of this system is given by the matter distribution width √ θ . \nThe geometric and thermodynamics features of the solution can be summarized as follows: \n- i) there is no curvature singularity in r = 0; the center of the black hole is a regular 'ball' of deSitter vacuum accounting for the short-distance quantum fluctuations of the spacetime manifold, whatever is their physical origin;\n- ii) even in the neutral case, there exist an outer and an inner horizon. At the end of the Hawking evaporation the two horizons coalesce into a single, degenerate horizon, corresponding to an extremal black hole;\n- iii) the Hawking temperature reaches a finite maximum value and the drops down to zero in the extremal configuration. \nTo achieve this regular behaviour the choice of the matter equation of state is instrumental. Stability of the solution is guaranteed by choosing \nρ ( r ) = -p r ( r ) (1) \nwhere, ρ ( r ), p r ( r ) are the Gaussian matter density and radial pressure, respectively. The tangential pressure p ⊥ ( r ) was determined by ρ ( r ) and p r ( r ) through hydrodynamic equilibrium equation. The condition (1) has the same form of the vacuum equation of state. Thus, one can expect that near the origin the metric will be deSitter with an effective cosmological constant Λ ∝ G N ρ (0). On the other hand, if the width of the matter distribution is √ θ , at large distance one sees a small sphere of matter with radius about √ θ . Thus, Birkhoff theorem assures the metric to be Schwarzschild. In the intermediate region the metric is neither deSitter, nor Schwarzschild, and can be analytically written in terms of lower incomplete gamma function. \nIn this paper we are going to relax equation (1) and look for a new solution describing a dirty black hole. The starting point is the minimal width, Gaussian, mass/energy distribution \nρ ( r ) = M ( 4 πθ ) 3 / 2 exp ( -r 2 4 θ ) (2) \nwhich we obtained from our coordinate coherent states approach to noncommutative geometry. From this perspective the θ parameter is a length squared quantity defining the scale where spacetime coordinates become non-commuting (quantum) objects. However, in a more general framework the distribution (2) represents the most localized energy density which is compatible with the existence of a minimal length, whatever it is. Thus, the width of the bell-shaped function (2) can be consistently related to the string length √ α ' , the TeV Quantum Gravity scale, etc. In a recent paper [19] a gaussian source has been used to model phantom energy supported wormholes. \nThe constant M is the total mass energy given by \nM ≡ 4 π ∫ ∞ 0 drr 2 ρ ( r ) . (3) \nThe mass distribution (2) is the component T 0 0 of the energy momentum tensor. Before proceeding we have to define the remaining components. We model our source through a fluid-type T µ ν of the following form: \nT µ ν = Diag ( -ρ ( r ) , p r ( r ) , p ⊥ ( r ) , p ⊥ ( r ) ) (4) \nwhere, p r is the radial pressure and p ⊥ the tangential pressure. In order to be a viable source for the Einstein equations, the condition T µ ν ; µ = 0 must be satisfied by ρ , p r , p . \nWe are going to solve Einstein equations assuming the metric to be spherically symmetric, static, asymptotically flat. Thus, we can write the general form of the line element in terms of two independent functions Φ ( r ) and m ( r ) as \n⊥ \nds 2 = -e 2Φ( r ) dt 2 + dr 2 1 -2 m ( r ) /r + r 2 ( dψ 2 +sin 2 ψdφ 2 ) (5) \nwhere m ( r ) is the shape function and Φ( r ) is the red-shift function [20, 21]. Both unknown functions must be determined by the Einstein equations. To this purpose, we now briefly recall a recent solution one obtains by assuming in place of (1) the condition \np r = -1 4 πr 3 m ( r ) . (6) \nwhere \nm ( r ) ≡ 4 π ∫ r 0 duu 2 ρ ( u ) . (7) \nThus, the resulting line element reads \nds 2 = -dt 2 + dr 2 1 -4 Mγ ( 3 / 2 ; r 2 / 4 θ ) / √ πr + r 2 ( dϑ 2 +sin 2 ϑdφ 2 ) (8) \nwhich describes a wormhole geometry. Its properties have been recently investigated in [22] and we are not going to repeat them here. We have recalled this wormhole solution since it turns out to be useful to understand the procedure we shall follow in the next section. Indeed to derive the dirty black hole solution, we are going to relax the condition (6) in favor of a more general one motivated by physical requirements.", '2. Dirty Black Hole': "The term 'dirty black holes' refers to black hole solutions of the Einstein equations in interaction with various kind of matter fields § , some remarkable examples are: \n- · gravity + electromagnetism + dilaton [23, 24, 25, 26];\n- · gravity + electromagnetism + Abelian Higgs field [30];\n- · gravity + electromagnetism + axion [27, 28, 29];\n- · gravity + electromagnetism + dilaton+ axion [31];\n- · gravity +axion + non-Abelian gauge fields [36];\n- · gravity + non-Abelian gauge fields [32, 33, 34, 35]; \nIn our model we do not select any specific kind of field theory, but we simply introduce a smeared energy/pressure distribution, which is a classical parametrization for the energy/pressure 'stored' in some kind of field. However, the condition which provides the relation between energy and radial pressure is crucial to determine the new solution. Among the huge variety of possible choices of this condition, we require that the pressures p r and p ⊥ satisfy the following physical conditions: \n- (i) p r and p ⊥ must be asymptotically vanishing;\n- (ii) p r and p ⊥ must be finite at the horizon(s);\n- (iii) p r and p ⊥ must be finite at the origin. \nThe condition i. assures that the solution matches the Minkowski space at infinity while the conditions ii. and iii. warrant the regularity at the horizon(s) and at the origin, a fact that is in the spirit of all the previous solutions generated by a smeared source term. \nThe line element we are looking for is of the form: \nds 2 = -e 2Φ( r ) ( 1 -2 m ( r ) /r ) dt 2 + dr 2 1 -2 m ( r ) /r + r 2 ( dψ 2 +sin 2 ψdφ 2 ) (9) \nwhich can be obtained form (5) by the following local rescaling of the red-shift function Φ( r ) -→ Φ( r ) + 1 2 ln ( 1 -2 m/r ). Therefore the Einstein equation read \n- § This kind of solutions are alternatively referred to as ' hairy ' black holes. \ndm dr = 4 π r 2 ρ , (10) \n1 2 g 00 dg 00 dr = m ( r ) + 4 π r 3 p r r ( r 2 m ( r )) , (11) \ndp r dr = -1 2 g 00 dg 00 dr ( ρ + p r ) + 2 r ( p ⊥ -p r ) (12) \n- \nwhere, g 00 ≡ -e 2Φ( r ) ( 1 -2 m ( r ) /r ). \nAccording to the recipes i., ii and iii, pressures regularity requirements are satisfied by the following condition replacing the (6) \nρ ( r ) + p r ( r ) ≡ -√ θ ( 1 -2 m/r ) dρ dr = 1 2 √ θ r ρ ( 1 -2 m/r ) (13) \nthat can be rewritten as \np r ( r ) = -ρ ( r ) [ 1 -r 2 √ θ ( 1 -2 m/r ) ] . (14) \nThis is the simplest choice, satisfying the physical conditions above. In principle, one can still consider further conditions, involving additional terms ∼ ( r/ √ θ ) m ( 1 -2 m/r ) n in the square brackets on the r.h.s.. In such a case, one ends up with a more complicated solution, endowed only with subleading contributions to the solution we are going to derive, in the three physically meaningful regions (infinity, horizon(s) and origin). \nFrom (12) one finds that the angular pressure p ⊥ is \np ⊥ ( r ) = p r ( r ) + r 2 dp r dr + 1 4 √ θ ρ ( r ) ( m ( r ) + 4 π r 3 p r ) . (15) \nWe notice that both p r and p ⊥ enjoy the above physical conditions at the origin, horizon(s) and asymptotically. From Eqs. (11) and (14) one has \nd Φ dr = 4 πr ( ρ + p r ) 1 -2 m/r = 2 πr 2 √ θ ρ ( r ) (16) \nwith the boundary condition Φ ( ∞ ) = 0 in order to reproduce Minkowski geometry at infinity. The solution reads \nΦ( r ) = MG √ πθ γ ( 3 / 2 , r 2 / 4 θ ) -MG 2 √ θ (17) \nwhere, for the sake of clarity we momentarily re-introduced the Newton constant G . \nAs a result the noncommutative Schwarzschild 'dirty' black hole is described by the following line element \nds 2 = -exp [ -MG √ θ ( 1 -2 √ π γ ( 3 / 2 , r 2 / 4 θ ) )] (1 -2 Gm ( r ) /r ) dt 2 + dr 2 1 -2 Gm ( r ) /r + r 2 d Ω 2 (18) \nBefore discussing thermodynamical properties, let us check the regularity of the geometry at the origin. From the Einstein equations one finds \nR (0) = -8 πGT µ µ (0) = 8 πG [ ρ (0) -p r (0) -2 p ⊥ (0) ] . (19) \nFrom (13), (14), (15) we see that all the pressures approach -ρ (0) near the origin. Thus, we find \nR (0) = 32 πGρ (0) = 4 GM √ πθ 3 / 2 ≡ 4Λ eff (20) \nwhich is the Ricci scalar for a deSitter metric characterized by a positive effective cosmological constant Λ eff . The same conclusion can be obtained by expanding the line element near r = 0: \nds 2 = -e -M/ √ /θ ( 1 -Λ eff 3 r 2 ) dt 2 + ( 1 -Λ eff 3 r 2 ) -1 dr 2 + r 2 ( dψ 2 +sin 2 ψ dφ 2 ) (21) \nwhere, the constant e -M/ √ θ can be reabsorbed into a rescaled time coordinate in order to have the ordinary deSitter line element. Also in the case of the dirty black hole there is no curvature singularity and the central black hole geometry results to be a deSitter spacetime. \nRegarding the horizon(s), we have to study the equation M = U ( r H ) where the 'effective potential' U ( r H ) is defined to be \nU ( r H ) = 4 √ π r H γ ( 3 / 2 , r 2 H / 4 θ ) . (22) \nFor a positive given value of M , the (22) can admit two, one, or no solution. We studied in detail the solutions of this equation in [1] and we are not going to repeat all the discussion here. It may be useful to summarize the main conclusion, i.e. the existence of a lower bound for a black hole mass given by M 0 ≈ 1 . 9 √ θ/G . Thus, we find again three possible cases: \n- · M > M 0 non-extremal dirty black hole with two horizons r + > r -; the outer horizon r + is said to be of canonical type , while the r -is a Cauchy horizon.\n- · M = M 0 extremal dirty black hole with one degenerate horizon r + = r -≡ r 0 ≈ 3 √ θ ; the function g rr does not change sign at the horizon, which is said to be of non-canonical type .\n- · 0 < M < M 0 dirty minigravastar, no horizons. \nAn interesting feature of the above line element regards the gravitational redshift z ≡ ∆ λ/λ , where λ is is the wavelength of the electromagnetic radiation at the source and ∆ λ is the difference between the observed and emitted wavelengths. This quantity now depends both on the redshift function Φ( r ) and the shape function m ( r ). The redshift measured by an asymptotic observer turns out to be \nz = e MG 2 √ θ ' 1 -2 √ π γ ( 3 / 2 ,r 2 / 4 θ ) ' (1 -2 Gm ( r ) /r ) -1 / 2 -1 (23) \nFigure 1. The asymptotic redshift z = ∆ λ/λ versus the radius r/ √ θ . On the right part of the figure, the curves are for the mass M = 2 . 5 √ θ/G , while on the left are for M = M 0 . The dashed curves corresponds to Schwarzschild case, while the solid ones are for the noncommutative Schwarzschild (thin) and the noncommutative dirty case (thick). We can observe that, for M = 2 . 5 √ θ/G all the curves coincide and we can conclude that noncommutative effects are not yet important, while for smaller masses in the vicinity of the extremal configuration the curves become distinct. We can conclude that the shape function m ( r ) lowers the redshif, as in the case of a smaller gravitational field, while the redshif function Φ( r ) slightly increases the values of z . \n<!-- image --> \nwhich is a clearly divergent quantity approaching the horizon, while asymptotically vanishes. On the other hand, in the absence of horizons i.e. the minigravastar case, the redshift z is finite even at the origin. \nIn the framework of 'wormhole engineering' exotic matter is advocated in order to violate energy conditions and avoid the tunnel to collapse. Thus, weak (WEC) and/or strong energy (SEC) conditions violation are an important issue to discuss in the black hole case, as well. Indeed, an eventual violation would mark a substantial departure from the behaviour of any kind of classical matter. This is what we expect as the Gaussian distribution, which is the corner stone of our approach and was originally derived in [12] in a quantum framework. The figure (3) shows the new features of the dirty black hole. \nLet us start from the dashed curve representing the plot of the function ρ + p r +2 p ⊥ . Where it is negative, the SEC is violated. This occurs well for r < 2 . 8 √ θ , which is well inside the event horizon located at r + ≈ 5 √ θ , as in the case of the 'clean' noncommutative geometry inspired Schwarzschild black hole. Both in the case of dirty and clean solutions, the SEC is a short-distance effect caused by the underlying non-commutative geometry. Spacetime fluctuations provide an effective gravitational \nFigure 2. The asymptotic redshift z = ∆ λ/λ versus the radius r/ √ θ for the minigravastar case, i.e. M < M 0 . The dashed curve correspond to the Schwarzschild line element and exhibit a divergent behavior due to the presence of horizons. The thin solid curve corresponds to the noncommutative Schwarzschild case, while the thick solid curve is for the noncommutative dirty case. As a general prescription, the shape function m ( r ) provides a regular peak at r = r 0 , while the redshif function Φ( r ) slightly increases the values of z with respect the noncommutative Schwarzshild case for all r . \n<!-- image --> \nrepulsion smearing out the classical curvature singularity. On the other hand, outside the black hole matter behaves in a ' classical ' manner. However, the memory of the short-distance effects is still present in the black hole temperature as it will be shown below. The new distinctive feature of the dirty black hole is that WEC are violated as well, while in the clean case they are preserved. Violation occurs, once again, inside the horizon and more specifically where the two solid curves in figure (3) lower below the r-axis. The presence of some 'dirt' on the event horizon causes a suppression of the Hawking temperature with respect to the corresponding clean case, in agreement with the general result shown in [21]. \nThe Hawking temperature, mentioned above, is given by ( G = 1 , κ B = 1): \nT H = 1 4 πr + e Φ( r + ) [ 1 -r 3 + 4 θ 3 / 2 e -r 2 + / 4 θ γ (3 / 2; r 2 + / 4 θ ) ] , (24) \nT H is defined for any r + ≥ r 0 . The profile of T H ( r + ) resembles that found in [15] and implies a 'SCRAM phase' at the terminal phase of the evaporation, namely a cooling \nΦ( r + ) ≡ √ πr + 8 √ θγ ( 3 / 2 , r 2 + / 4 θ ) ( 2 √ π γ ( 3 / 2 , r 2 + / 4 θ ) -1 ) (25) \nFigure 3. The energy conditions for M = 3 . 0 √ θ/G versus r/ √ θ . The thick solid curve is for the functions ρ + p ⊥ , while the thin one corresponds to the function ρ + p r , which vanishes at the horizons. The violation of the weak energy condition occurs when either one of the solid curves have negative values. The dashed curve is the function ρ + p r +2 p ⊥ , whose negative values take place where the strong energy condition is violated. \n<!-- image --> \ndown to an asymptotic absolute zero configuration. The correction due to the prefactor e Φ plays a non-trivial role only near the peak of the temperature by lowering of the maximum value, as it can be seen in figure (4). This is in agreement with the choice of the equation (13) and the consequent violations of WEC in a region inside the event horizon r + [21]. \nOnce the Hawking temperature is known let us look at the area/entropy law. From the first law of black hole thermodynamics, i.e. dM = T H dS , we can write the infinitesimal entropy variation dS in terms of the effective potential U ( r + ): \ndS = 1 T H ∂U ∂r + dr + (26) \nIn order to integrate (26) in the correct way, we must take into account that the extremal, zero temperature, black hole configuration has zero thermodynamical entropy. Thus, the integration range starts from the radius r 0 of the degenerate horizon, and runs up to a generic radius r + > r 0 . With this choice of the integration range we find \nwhere, \nS = π 3 / 2 2 [ r 2 + γ ( 3 / 2 , r 2 + / 4 θ ) e -Φ( r + ) -r 2 0 γ ( 3 / 2 , r 2 0 / 4 θ ) e -Φ( r 0 ) ] -∆ S (27) \n∆ S ≡ π 3 / 2 2 ∫ r + r 0 drr 2 d dr [ e -Φ( r ) γ ( 3 / 2 , r 2 / 4 θ ) ] (28) \nFigure 4. The Hawking temperature T H versus the horizon radius r + . The dashed curve corresponds to Schwarzschild temperature, while from top to bottom the solid curves are for the noncommutative Schwarzschild (thin) and the noncommutative dirty case (thick). We can observe that, while the temperature decreases for the dirty case, the remnant radius is unchanged. \n<!-- image --> \nThe first term in (27) differs from the corresponding quantity in the clean case only for the presence of the exponential factors. On the other hand, the correction ∆ S is characteristic of the dirty black hole, and vanishes in the large distance limit. Indeed, for r + , r 0 >> √ θ we recover the classical Area/Entropy law \nS → 1 4 ( A + -A 0 ) (29) \nAt first glance, the reader could question why the celebrated Bekenstein-Hawking (29) results to be valid only asymptotically, and not for any value of r + . The key observation is that nowadays everybody is taking for granted that the classical entropy of any black hole 'is' one forth of the area of event horizon, measured in Planck units, up to eventual logarithmic corrections from unspecified quantum gravity effects. Thus, nobody cares anymore to 'recover' the area law from basic principles. Rather, the main interest is to match the classical result with the statistical interpretation of entropy in terms of black hole quantum micro-states. We would like to notice that the consistency between the geometric and thermodynamical definition of the black hole entropy requires that in any case it must be possible to derive the Area/Entropy law from the first law (26), as we did above, and to assume a priori it is valid. It is a common feature of regular black holes to display a more complex relation between horizon area and entropy, rather than a simple proportionality law. This effect can be traced back, once again, to the presence of a minimal length or, equivalently, to the granular nature of the 'quantum' \nspacetime. This effect, together with the existence of a finite leftover by the Hawking process, hints to a possible resolution of the information paradox worth of a more indepth investigation.", '3. Conclusions': "In this paper we extended our previous investigation of black hole solutions of Einstein equations with a Gaussian source. Gaussian distributions are widely used in physics and it is surprising that nobody considered before the gravitational effects of this kind of sources. From our vantage point, we used this distribution to model the physical effects of short-distance fluctuations of noncommutative coordinates. Generally covariant divergence free condition on the energy-momentum tensor in the Einstein equations allows various kind of physically acceptable conditions between the energy density and radial pressure. In our case the component T 00 is assigned and the other ones must be determined in a consistent way. After recalling the wormhole spacetime geometry of the type conjectured in [22], we investigated a new dirty black hole type solution. The novelty of this solution with respect to the corresponding 'clean' case is displayed in figure (3). While in the original noncommutative black hole [1] only the strong energy condition is violated, in the dirty case weak energy conditions are broken as well. The former violation removes the curvature singularity in the origin, the latter decreases the Hawking temperature in agreement with the general result by Visser [21]. \nIf LHC will turn to be an effective black hole factory, all the regular objects we have investigated so far are expected to contribute to the cross section production.", 'Acknowledgments': 'P.N. was partially supported by a CSU Fresno International Activities Grant. P.N. is supported by the Helmholtz International Center for FAIR within the framework of the LOEWE program (Landesoffensive zur Entwicklung Wissenschaftlich-Okonomischer Exzellenz) launched by the State of Hesse. P.N. would like to thank both the Max Plank Institute for Gravitational Physics (Albert Einstein Institute), Golm, Potsdam, Germany and the CERN Theory Division for the kind hospitality during the final period of work on this project.', 'References': '- [1] Nicolini P, Smailagic A and Spallucci E 2006 Phys. Lett. B 632 547\n- [2] Rizzo T G 2006 J. High Energy Phys. JHEP09(2006)021\n- [3] Ansoldi S, Nicolini P, Smailagic A and Spallucci E 2007 Phys. Lett. B 645 261\n- [4] Spallucci E, Smailagic A and Nicolini P 2009 Phys. Lett. B 670 449\n- [5] Casadio R and Nicolini P 2008 J. High Energy Phys. JHEP11(2008)072\n- [6] Ansoldi S Spherical black holes with regular center: a review of existing models including a recent realization with Gaussian sources Preprint arXiv:0802.0330 [gr-qc]\n- [7] Nicolini P 2009 Int. J. Mod. Phys. A 24 1229 \n- [8] Arraut I, Batic D and Nowakowski M 2009 A non commutative model for a mini black hole Preprint arXiv:0902.3481 [gr-qc]\n- [9] Smailagic A and Spallucci E 2002 Phys. Rev. D 65 107701\n- [10] Smailagic A and Spallucci E 2002 J. Phys. A: Math. Gen. 35 L363\n- [11] Smailagic A and Spallucci E 2003 J. Phys. A: Math. Gen. 36 L467\n- [12] Smailagic A and Spallucci E 2003 J. Phys. A: Math. Gen. 36 L517\n- [13] Smailagic A and Spallucci A 2004 J. Phys. A: Math. Gen. 37 1 [Erratum-ibid. A 37 7169]\n- [14] Nicolini P, Smailagic A and Spallucci E 2005 The fate of radiating black holes in noncommutative geometry Preprint arXiv:hep-th/0507226\n- [15] Nicolini P 2005 J. Phys. A: Math. Gen. 38 L631\n- [16] Spallucci E, Smailagic A and Nicolini P 2006 Phys. Rev. D 73 084004\n- [17] Banerjee R, Chakraborty B, Ghosh S, Mukherjee P and Samanta S 2009 Found. Phys. 39 , 1297\n- [18] Nicolini P and Rinaldi M 2009 A minimal length versus the Unruh effect Preprint arXiv:0910.2860 [hep-th]\n- [19] Sushkov S V 2005 Phys. Rev. D 71 043520\n- [20] Morris M S and Thorne K S 1988 Am. J. Phys. 56 395\n- [21] Visser M 1992 Phys. Rev. D 46 2445\n- [22] Garattini R and Lobo F S N 2009 Phys. Lett. B 671 146\n- [23] Gibbons G W and Maeda K I 1988 Nucl. Phys. B 298 741\n- [24] Ichinose I and Yamazaki H 1989 Mod. Phys. Lett. A 4 1509\n- [25] Yamazaki H and Ichinose I 1992 Class. Quantum Grav. 9 257\n- [26] Garfinkle D, Horowitz G T and Strominger A 1992 Phys. Rev. D 43 3140 [Erratum-ibid. D 45 3888]\n- [27] Allen T J, Bowick M J and Lahiri A 1990 Phys. Lett. B 237 47\n- [28] Campbell B A, Kaloper N and Olive K A 1991 Phys. Lett. B 263 364\n- [29] Lee K M and Weinberg E J 1991 Phys. Rev. D 44 3159\n- [30] Dowker F, Gregory R and Traschen J H 1992 Phys. Rev. D 45 2762\n- [31] A. D. ShapereA D, Trivedi S and Wilczek F 1991 Mod. Phys. Lett. A 6 2677\n- [32] Galtsov D V and Ershov A A 1989 Phys. Lett. A 138 160\n- [33] Straumann N and Zhou Z H 1990 Phys. Lett. B 243 33\n- [34] Bizon P 1990 Phys. Rev. Lett. 64 2844\n- [35] Bizon P and Wald R M 1991 Phys. Lett. B 267 173\n- [36] Lahiri A 1992 Phys. Lett. B 297 248'}
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2009ApJ...698..766M
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Accretion onto "Seed" Black Holes in the First Galaxies
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2009-01-01
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['black hole physics', 'cosmology theory', 'galaxies active', 'hydrodynamics', 'galaxies quasars', 'radiation', 'astrophysics']
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The validity of the hypothesis that the massive black holes in high redshift quasars grew from stellar-sized "seeds" is contingent on a seed's ability to double its mass every few 10 million years. This requires that the seed accrete at approximately the Eddington-limited rate. In the specific case of radiatively efficient quasi-radial accretion in a metal-poor protogalactic medium, for which the Bondi accretion rate is often prescribed in cosmological simulations of massive black hole formation, we examine the effects of the radiation emitted near the black hole's event horizon on the structure of the surrounding gas flow. We find that photoheating and radiation pressure from photoionization significantly reduce the steady-state accretion rate and potentially render the quasi-radial accretion flow unsteady and inefficient. The time-averaged accretion rate is always a small fraction of the "Bondi" accretion rate calculated ignoring radiative feedback. The pressure of Lyα photons trapped near the H II region surrounding the black hole may further attenuate the inflow. These results suggest that an alternative to quasi-radial, radiatively efficient Bondi-like accretion should be sought to explain the rapid growth of quasar-progenitor seed black holes.
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https://arxiv.org/pdf/0809.2404.pdf
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{"ACCRETION ONTO 'SEED' BLACK HOLES IN THE FIRST GALAXIES": "MILOŠ MILOSAVLJEVI ' C 1 , VOLKER BROMM 1 , SEAN M. COUCH 1 , AND S. PENG OH 2 Submitted to the Astrophysical Journal", 'ABSTRACT': "The validity of the hypothesis that the massive black holes in high redshift quasars grew from stellar-sized 'seeds' is contingent on a seed's ability to double its mass every few ten million years. This requires that the seed accrete at approximately the Eddington-limited rate. In the specific case of radiatively efficient quasiradial accretion in a metal-poor protogalactic medium, for which the Bondi accretion rate is often prescribed in cosmological simulations of massive black hole formation, we examine the effects of the radiation emitted near the black hole's event horizon on the structure of the surrounding gas flow. We find that photoheating and radiation pressure from photoionization significantly reduce the steady-state accretion rate and potentially render the quasiradial accretion flow unsteady and inefficient. The time-averaged accretion rate is always a small fraction of the 'Bondi' accretion rate calculated ignoring radiative feedback. The pressure of Ly α photons trapped near the H II region surrounding the black hole may further attenuate the inflow. These results suggest that an alternative to quasiradial, radiatively efficient Bondi-like accretion should be sought to explain the rapid growth of quasar-progenitor seed black holes. \nSubject headings: black hole physics - cosmology: theory - galaxies: active - galaxies: formation hydrodynamics - quasars: general - radiation mechanisms", '1. INTRODUCTION': "The origin and the early growth of massive black holes remains poorly understood. The massive black holes in quasars ( M BH ∼ 10 8 -10 10 M glyph[circledot] ), active galactic nuclei (AGN; M BH > 10 5 M glyph[circledot] ), and the quiescent nuclei of nearby galaxies may have started out as stellar-mass ( M BH < 100 M glyph[circledot] ) 'seed' black holes (e.g., Madau & Rees 2001; Menou et al. 2001; Islam et al. 2003). Is this plausible, that is, could the seed black holes have grown rapidly enough in the cosmic time available to them (e.g., Haiman & Loeb 2001; Volonteri et al. 2003; Tanaka & Haiman 2008, see also Haiman & Quataert 2004, Djorgovski et al. 2008, and references therein)? The rate at which a seed massive black hole can accrete is limited by the local density and the thermal structure of the protogalactic medium and by the effects of the radiation emitted near the event horizon on the accretion flow. Cosmological hydrodynamic simulations suggest that gravitational collapse produces dense central gas concentrations in protogalaxies (e.g., Springel et al. 2005; Li et al. 2007; Pelupessy et al. 2007; Wise & Abel 2007a,b; Wise et al. 2008; Di Matteo et al. 2008; Greif et al. 2008). Atomic densities have been found to reach n ∼ 10 4 cm -3 (e.g., Greif et al. 2008), and as much as n ∼ 10 6 cm -3 averaged over the central parsec around the potential minimum (Bromm & Loeb 2003; Wise & Abel 2007b; Wise et al. 2008). On spatial scales that are resolved in the simulations, gas is sufficiently concentrated to enable rapid accretion onto a seed black hole. An exception are the first hundred million years after the seed was formed, during which the surrounding gas density is lowered by the radiative feedback from the black hole's progenitor star (see, e.g., Johnson & Bromm 2007; Alvarez et al. 2006, 2008). \nGiven an ample gas supply, will rapid accretion be inhibited by radiative effects? A reassessment of an accreting black hole's ability to control its own gas supply is needed to im- \nprove the realism of the treatment of black hole accretion in cosmological simulations. Existing cosmological simulations modeling the growth of seed black holes do not resolve the spatial scales on which some of the radiative processes may alter the accretion. The simulations also do not resolve the fine structure of the dense, turbulent, and possibly multiphase protogalactic medium in which the black holes are embedded. Semianalytic prescriptions are normally adopted for the accretion rate, but these prescriptions normally do not take into account the radiative feedback; it is normally assumed that a black hole, residing in a pressure-supported primordial gas cloud, can accrete steadily at the Bondi rate subject to the Eddington limit (e.g., Volonteri & Rees 2005; Alvarez et al. 2006; Johnson & Bromm 2007; Pelupessy et al. 2007; Di Matteo et al. 2008; Greif et al. 2008). Here, we will evaluate the applicability of this assumption in view of the local radiative feedback that is present if the black hole accretes in a radiatively efficient fashion. We restrict our analysis to the early growth of quasar-progenitor 'seed' black holes, which are occasionally referred to as 'miniquasars,' where a stellar mass or an intermediate-mass black hole (10 2 M glyph[circledot] glyph[lessorsimilar] M BH glyph[lessorsimilar] 10 5 M glyph[circledot] ) accretes from a metal and dust poor environment. \nThe black hole's growth rate is particularly sensitive to the detailed thermal state of the irradiated accretion flow. This can be seen by noticing that the accretion rate, for quasiradial accretion, is influenced by the conditions at the sonic radius \nr s ∼ 3 × 10 14 M 2 T s , 5 cm , (1) \nwhere T s = 10 5 T s , 5 K is the temperature of the photoionized and photoheated flow at the sonic radius and M BH = 100 M 2 M glyph[circledot] is the black hole mass. The sonic radius is normally unresolved in cosmological simulations of accretion onto black holes in the intermediate range of masses. 1 It was \n1 The recent, highly-resolved simulation of primordial protostar formation by Yoshida et al. (2008), which has the requisite spatial resolution to resolve a sonic radius if it exists, only proceeds to the point where the initial hydrostatic core is formed, and does not treat the subsequent accretion flow onto the growing core. \nrecognized early that photoheating and photoionization pressure may prohibit steady radiatively efficient accretion (e.g., Shvartsman 1971; Buff & McCray 1974; Hatchett et al. 1976; Ostriker et al. 1976). It was suggested that accretion can still proceed at rates disallowed by the steady state solutions if cycles of rapid gas inflow, during pauses in accretion near the event horizon, alternate with photoheating or photoionization pressure-driven outflows (Buff & McCray 1974; Ostriker et al. 1976; Cowie et al. 1978; Stellingwerf & Buff 1982; Begelman 1985). Such quasiperiodic cycling is seen in onedimensional simulations of Compton-heated accretion onto M BH glyph[greaterorsimilar] 10 8 M glyph[circledot] black holes in galaxy clusters, where the black hole accretes from a hot, ionized, and pressure supported atmosphere (Ciotti & Ostriker 1997, 2001, 2007; Sazonov et al. 2005). Recently, Ricotti et al. (2008) revisited the problem of irradiated quasiradial accretion in the context of primordial black hole growth following the cosmic recombination (see, e.g., Ricotti 2007, and references therein), and suggested that the accretion duty cycle is determined by the periodic formation of an H II region surrounding the black hole. \nIt was further recognized that the formal existence of steady-state, spherically symmetric accretion solutions is sensitive to the treatment of boundary conditions far from the sonic radius (e.g., Bisnovatyi-Kogan & Blinnikov 1980), and that these accretion flows can be locally thermally unstable (e.g., Stellingwerf 1982; Krolik & London 1983) and should break down into time-dependent two-phase structure, containing a warm ionized phase and a hot, coronal phase (e.g., Krolik et al. 1981). Wang et al. (2006) claimed that Compton heating in the vicinity of a seed massive black hole reduces the radial accretion rate to a small fraction of the Eddingtonlimited rate; we here suggest, however, that thermal runaway may engender the Compton-heated coronal phase only in metal-rich flows, where photoionization heating of the incompletely stripped oxygen drives gas heating beyond ∼ 10 5 K (see, e.g., Kallman & McCray 1982, and § 2.1 below). \nThe impact of the radiation field produced near the event horizon on the accretion flow and on the state of the interstellar medium of the protogalaxy and that of the intergalactic medium (e.g., Dijkstra et al. 2004; Kuhlen & Madau 2005; Zaroubi et al. 2007; Thomas & Zaroubi 2008; Ripamonti et al. 2008; Spaans & Meijerink 2008) is sensitive to the shape of the spectral energy distribution (SED) of the central source. The SEDs of rapidly accreting low-mass massive black holes ( M BH ∼ 10 5 -10 6 M glyph[circledot] ) exhibit significantly larger X-ray (2 keV) to optical spectral ratios than the AGN containing more massive rapidly accreting black holes (Greene & Ho 2007), as is expected if a fraction of the radiation is produced in a geometrically thin disk. On the low mass end, if the microquasar SEDs (e.g., Remillard & McClintock 2006, and references therein) are an adequate prototype, the seed black hole SEDs may contain energetically significant components extending into the hard X-rays. The luminosity-weighted average spectrum of AGN containing intermediate mass black holes could differ substantially (see, e.g., Venkatesan et al. 2001; Madau et al. 2004) from the scaled average quasar spectrum of Sazonov et al. (2004). The ability of X-rays to escape the protogalaxy affects their contribution to the soft X-ray background (e.g., Venkatesan et al. 2001; Dijkstra et al. 2004; Salvaterra et al. 2005) and the infrared background (Cooray & Yoshida 2004). Another difference between the first AGN and the starburst or post-starburst AGN is related to the differences in metallicity of the accretion flows. The thermal phase structure of the interstellar medium exposed to \nUV and X-ray radiation is sensitive to metal abundances, especially for T < 10 4 K (e.g., Donahue & Shull 1991) and for T glyph[greaterorsimilar] few 10 4 K(e.g., Kallman & McCray 1982, and § 2.1). \nThe role of radiative feedback in the formation of the first massive black holes resembles the radiative regulation in the formation of the first massive protostars (Omukai & Palla 2001, 2003; Omukai & Inutsuka 2002; Omukai & Yoshii 2003; Tan & McKee 2004; McKee & Tan 2008), though, of course, the central sources have very different spectra. In protostars, an H II region forms around the protostar and the protostellar disk that feeds its growth; persistent accretion onto the protostar may be quenched by radiation pressure. The protostellar accretion is characterized by a lower radiative efficiency and a higher accretion rate than the black hole accretion for the same accretor mass. The growing protostar is embedded in a supersonically collapsing and very dense envelope from inception, whereas here we assume that the seed black hole is born in the collapse of a massive star without such an envelope. The presence of an infalling envelope implies that the H II region is initially trapped near the protostar, inside the radius where the envelope infall velocity turns from subsonic to supersonic. For accretion from a diffuse medium onto a seed black hole, the H II region is inevitably extended and the flow crossing the ionization front is highly subsonic; this marks a crucial difference with the protostellar accretion scenarios. \n× \nIgnoring radiative effects, accretion onto the black hole is quasiradial on certain length scales (e.g., r s glyph[lessorsimilar] r glyph[lessorsimilar] 10 pc) if turbulence in the gas is weak (Krumholz et al. 2006) and if the gas is not rotationally supported on these scales. If the accretion flow is shock-free (an unlikely condition) and possesses small net rotation, the buildup of vorticity near r ∼ r s may reduce the accretion rate by ∼ 60% (Krumholz et al. 2005). Quasiradial accretion may be expected even when the baryons in a protogalaxy initially form a rotating disk, because selfgravity in the gas on scales of the protogalactic disk destabilizes rotational equilibria to convert disk-like configurations into quasiradial, stratified, pressure supported, and possibly turbulent configurations. The gas distribution could still be rotationally supported on larger scales, where, e.g., dark matter dominates gravity, and on much smaller scales, where the black hole dominates gravity. This description may apply to high-redshift protogalaxies (see, e.g., Oh & Haiman 2002; Volonteri & Rees 2005; Wang et al. 2006), and so here, we focus on angular momentum-free accretion and defer examining the role of angular momentum to a subsequent paper. \nThe quasiradial gas flow can either be steady and radial, or unsteady and characterized by alternating inflow, outflow, and nonradial motions. In view of these possibilities, this work is organized as follows. In § 2 we attempt, and fail, to construct a steady, radial solution for accretion at high accretion rates and high radiative efficiencies. In § 3 we provide a qualitative analysis of time-dependent, episodic accretion, and attempt to estimate the average accretion rate. In § 4 we discuss the consequences of the presence of cold, inhomogeneous, and turbulent gas in the vicinity of the black hole. In § 5 summarize our main conclusions. In Table 1 we present an overview of our notation.", '2. THE PROSPECT OF TIME-INDEPENDENT ACCRETION': 'In this section we set out to test the model, ubiquitous in semi-analytic and semi-numerical studies of massive black hole evolution, in which the accretion onto a black hole from the protogalactic medium is steady and quasiradial. In § 2.1', "MILOSAVLJEVI ' C, ET AL.": "TABLE 1. INDEX OF NOTATION \n| Quantity | Symbol | Note |\n|----------------------------------------------------------------------------------------------------------------------------------------------------------|---------------------------------|-------------------------------------------|\n| Acceleration due to gravity | a grav | GM BH / r 2 |\n| Acceleration due to radiation pressure | a rad | - see text (§ 2.5) |\n| | | - α |\n| Spectral index of the SED | α | F ν , f ν ∝ ν |\n| Case B recombination rate for hydrogen | α B | ≈ 2 . 6 × 10 - 13 T - 1 4 cm 3 s |\n| Collisional ionization rate for hydrogen Abundance of species i relative to hydrogen | α ion | · · · |\n| glyph[epsilon1] | χ i | ≡ ni / n H |\n| Logarithmic slope of T s as a function of | δ | T s ∝ glyph[epsilon1] δ = L / ˙ Mc 2 |\n| Radiative efficiency | glyph[epsilon1] glyph[epsilon1] | φ r ion / r s = 1 |\n| Critical efficiency for radiation pressure suppression at low efficiencies Ionization potential of species i | crit Ei | · · · |\n| | f duty | 2 |\n| | f epi | ≡〈 L 2 〉 / 〈 L 〉 |\n| Duty cycle for episodic accretion | | ≥ 1 |\n| Density enhancement in the ionized gas in episodic accretion Fraction of photon energy going to photoionizations in the ionized gas | f ion | ∼ 1 / 3 |\n| Fraction of photon energy reprocessed to Ly α | f Ly | ∼ 2 / |\n| Fraction of L that reaches the edge of the H II region | α f res | 3 |\n| | f | see text (§ 2.5) ∼ 1 + M 2 |\n| Pressure enhancement due to turbulence in the neutral gas Adiabatic index | | = 5 / 3 |\n| | turb γ | |\n| Luminosity in units of the Eddington luminosity | glyph[lscript] | L / L |\n| Luminosity of isothermal accretion ignoring radiative heating | glyph[lscript] | Edd see text (§ 2.3) |\n| Ly α -pressure-limited accretion rate | Bondi | see text (§ 2.7) |\n| | glyph[lscript] crit , Ly α | |\n| Peak luminosity in episodic accretion | glyph[lscript] max | · · · |\n| Luminosity of steady-state photoheated accretion | glyph[lscript] s . s . | see text (§ 2.3) |\n| Luminosity emitted at r glyph[lessorsimilar] r disk | L | · · · = 4 π GM BH mpc / |\n| Eddington luminosity for Thomson scattering Mean molecular mass of the ionized gas | L Edd | σ T ∼ 0 . 6 |\n| Turbulent Mach number of the neutral gas | µ | · · · |\n| Mass of the black hole | M | · · · |\n| | M BH | |\n| Central accretion rate | ˙ M | · · · |\n| | ˙ M | see text (§ 2.3) |\n| Central accretion rate ignoring radiation | Bondi n cusp | see text (§ 2.4) |\n| Density at the H II region's edge in a protogalactic density cusp Density of the neutral gas and ambient density Particle density within the H II region | n HI , n n | · · · f n T / T |\n| Maximum density for high ionization at the sonic radius - | HII n max , ion | turb HI HI HII see text (§ 2.3) |\n| H -dissociating photon production rate | ˙ N γ , H - | see text (§ 4.2) |\n| Lyman-Werner photon production rate | ˙ N LW | see text (§ 4.2) |\n| Total ionization rate in the H II region | ˙ N ion | see text (§ 2.2) |\n| Total recombination rate in the H II region | ˙ N rec | |\n| Number of Ly α reflections on H II region's walls to escape | N φ | see text (§ 2.2) |\n| | reflect ψ | see text (§ 2.7) see text (§ 2.5) |\n| Dimensionless photoionization pressure acceleration in the H II region Dimensionless radiation pressure acceleration at the H II region's edge | | see text (§ 2.5) = n HI mpkT HI |\n| Pressure of the neutral gas | P gas | |\n| Pressure of the Ly α radiation Radius of the H II region in a protogalactic density cusp | | |\n| | P Ly α r ion , cusp | see text (§ 2.7) see text (§ 2.4) |\n| Bondi radius ignoring radiative heating and acceleration Sonic radius | r B | see text (§ 3.2) v ( r c r |\n| | | s) = s( s) |\n| Disk radius | r s | glyph[lessmuch] r |\n| Radius where thermal time equals inflow time | r disk | s max { t heat , t cool } = r / v |\n| Radius of the H II region Width of the H2 photodissociation shell | r equi r ion | see text (§ 2.2) see text (§ 4.2) |\n| Width of the neutral shell surrounding the H II region | ∆ r diss ∆ r shell | see text (§ 2.7) 14 - |\n| Line-center Ly α scattering cross section | σ 0 | ≈ 5 . 9 × 10 - T 1 / 2 4 see text (§ 2.5) |\n| Flux-averaged photoionization cross section of species Photon number-averaged photoionization cross section | ¯ σ i ˜ σ i | see text (§ 2.5) |\n| i H2 formation suppression factor due to H - photodissociation | S | see text (§ 4.2) see text (§ 2.7) |\n| Ly α line-center optical depth of the neutral shell Bremsstrahlung cooling time | τ 0 t | see text (§ 2.3) |\n| Compton heating time | Brems t C t | see text (§ 2.3) · · · |\n| Cooling time Heating time | cool t heat | · · · |\n| Photoionization heating time Inflow time at the sonic radius | t photo t s t Salp | see text (§ 3.1) ∼ r s / c s( r s) 2 |\n| | T | = glyph[epsilon1] M BH c / L |\n| Salpeter mass-exponentiation time scale Temperature of the ambient neutral gas | HI | Edd · · · |\n| Temperature within the H II region Temperature at the sonic radius | T HII T s | · · · see text (§ 1) |\n| Density enhancement over isothermal accretion with T HII = T s Radial inflow velocity Dimensionless inflow velocity | Υ v w | glyph[greaterorsimilar] 1 · · · ≡ v / c |\n| Ionization parameter | | s( r s) 2 |\n| | ξ Ξ | ≡ L / r n ≡ ξ / 4 π kT |\n| | | HII |\n| | | c ∼ r / r |\n| Dimensionless radius Metallicity | | s |\n| | y | |\n| Dimensionless ionization parameter | | |\n| | Z | · · · | \nwe review the standard theory of photoionized radial accretion, and pay particular attention to the dependence of the structure of the flow on the metallicity of the accreting gas. We find that at low metallicities, the flow evades thermal runaway and heating to the Compton temperature at least until it passes the sonic radius. In § 2.2 we estimate the size of the H II region surrounding the black hole and derive conditions under which the self-gravity within the ionized sphere can be ignored. In § 2.3, we take a closer look at the state of the gas as it passes the sonic radius and check whether it is in local thermal and statistical equilibrium. In § 2.4 we justify our reference choice for the ambient density in the neutral medium surrounding the H II region. This justification is necessary because the central gravitational collapse in a protogalaxy will yield a wide range of densities, yet most of our estimates depend on a specific choice of density. In § 2.5 we study the effects of the photoionization radiation pressure within the H II region on the structure of the steady-state accretion flow. We find that, depending on the parameters, photoionization radiation pressure in the outer parts of the H II region may prevent accretion at near the Eddington-limited rate. In § 2.6, we briefly address the case of radiatively-inefficient accretion. In § 2.7 we estimate the pressure of Ly α line radiation produced in and near the H II region and confined by resonance line scattering in its vicinity. We find that Ly α radiation pressure can exceed the thermal gas pressure in the H II region, and this presents an additional challenge to strictly stationary quasiradial solutions.", '2.1. Photoionized Quasiradial Accretion': "We ignore the angular momentum of the gas, which is assumed to be free of metals and dust, on radial length scales glyph[greaterorsimilar] r s and assume that the accretion flow becomes rotationally supported and collapses into a hypothetical geometrically thin disk only at radii glyph[lessorsimilar] r disk glyph[lessmuch] r s. Furthermore, we assume that the disk accretes onto the black hole with a high radiative efficiency glyph[epsilon1] , as is expected for thin-disk accretion, such that the bolometric outward radiation flux passing through radius r s is F ( r s) = glyph[epsilon1] ˙ M ( r s) c 2 / 4 π r 2 s . We assume that, absent feedback effects, the density scale-height is much larger than any other length scale under consideration, so that the ambient medium has effectively constant density. We restrict our attention to the accretion flow at radii r > r s, where we assume that the flow is quasiradial and exposed to the radiation emitted by the disk at r ∼ 0. Without angular momentum, the radial support against the black hole's gravity must arise from gas pressure gradients and the radiation pressure force. Conservation laws yield the relation (see, e.g., Ostriker et al. 1976; Lamers & Cassinelli 1999) \ndv dr ( v 2 -γ k T µ mp ) = v ( 2 γ k T r µ mp + a tot ) + ( γ -1)( H -C ) , (2) \nwhere v is the radial inflow velocity, which is positive when the flow is directed inward, γ the adiabatic index that defines the relation of internal energy density to pressure, k the Boltzmann constant, µ the mean molecular mass, mp the proton mass, T the gas temperature, a tot = a rad + a grav the sum of the accelerations due to gravity and radiation pressure, H the photoheating rate, and C the cooling rate (both per unit mass). \nAs we show in § 2.2, there exists a radius r equi outside of which the heating time t heat ∼ kT / ( γ -1) µ mpH and cooling time t cool ∼ kT / ( γ -1) µ mpC are much shorter than the inflow time ∼ r / v . Since photochemical time scales in dense, ionized \ngas are generally short, outside this radius the gas reaches an approximate local thermal and statistical equilibrium, and so the last term in equation (2) that is proportional to H -C can be dropped. An estimate that we provide at the end of § 2.2 below suggests that for metal-poor accretion, the radius r equi is usually at most slightly larger than the radius r s at which the flow becomes supersonic. We ignore this complication and assume r equi glyph[lessorsimilar] r s. \nThe flow is not adiabatic, but we let c s ≡ ( γ kT /µ mp ) 1 / 2 denote the usual adiabatic sound speed. Continuity at the sonic radius r s where v = c s requires that 2[ c s( r s)] 2 r -1 s + a grav( r s) + a rad( r s) = 0, and because the self gravity of the gas is negligible, a grav( r s) = -GM BH / r 2 s . Since the gas is almost fully ionized at r ∼ r s, the radiation pressure at the sonic radius is mainly due to electron scattering. Then a rad( r s) = -glyph[lscript] ( r s) a grav( r s), where glyph[lscript] ( r ) ≡ L ( r ) / L Edd is the ratio of the total luminosity to the Eddington luminosity for all opacities. Since under a wide range of conditions (see, e.g., Blondin 1986), the flow at the sonic radius is optically thin to electron scattering, provided that the small accretion disk inside the sonic radius does not shadow and reprocess to low frequencies a substantial fraction of the central luminosity, we can assume that L ( r s) = L (0). Then, in a steady state, the luminosity can be related to the total mass flux into the central source, L (0) = glyph[epsilon1] ˙ Mc 2 = 4 πglyph[epsilon1] r 2 s c s( r s) µ n ( r s) mpc 2 , where n ( r ) is the gas number density. \nFor a fixed SED, T eq( Ξ ) is sensitive to the metallicity for 10 5 K glyph[lessorsimilar] T glyph[lessorsimilar] 10 6 K because helium line cooling and Bremsstrahlung cooling dominate the cooling rate at any metallicity (with a comparable contribution from iron at high temperatures in the metal-enriched case), while oxygen and iron photoionization heating are by far the most important heating processes in the metal rich case (e.g., Kallman & McCray 1982). Figure 1 shows that for the f ν ∝ ν -1 . 5 spectrum for 0 . 1 Ryd < h ν < 10 3 Ryd, at near-solar metallicities, the Compton-heated hot coronal phase appears at Ξ ∼ 10 (as is well known, see, e.g., Krolik et al. 1981), whereas in a gas with subsolar metallicity Z glyph[lessorsimilar] 0 . 1 Z glyph[circledot] , the coronal phase does not appear until the ionization parameter reaches Ξ ∼ 10 3 . The optically thin T eq( Ξ ) becomes independent of metallicity at Z 0 . 01 Z glyph[circledot] . \nLet Ξ ≡ L / 4 π r 2 nkTc denote the dimensionless ionization parameter introduced in Krolik et al. (1981), which is the ratio of the radiative momentum flux to the gas pressure. Then at the sonic radius Ξ s = glyph[epsilon1]γ c / c s( r s) (see straight lines in Fig. 1). In a chemical equilibrium determined purely by two-body collisional processes and photoionization, the equilibrium abundances χ i ≡ ni / n of all species are functions of the temperature and F ν / n only, where F ν is the radiation flux at frequency ν such that ∫ F ν d ν = F = L / 4 π r 2 . On the other hand, the temperature T eq arising from the equilibrium of photoheating and two-body collisional cooling is a function of χ i and F ν / n only. Therefore, for a particular SED f ν = F ν / F , the equilibrium temperature is determined only by L / 4 π r 2 n , i.e., T eq lies in one-to-one relation with Ξ and we can write T eq = T eq( Ξ ; f ν ). For particularly hard spectra, instead of being one-to-one, the function T eq( Ξ ; f ν ) can be multivalued in a certain range of Ξ . Then, the temperature of a fluid element depends on its thermal history. Since thermodynamic perturbations at constant Ξ are isobaric, the possibly multivalued function T eq( Ξ ) at fixed f ν determines the thermal phase structure. \n∼ \nIn Figure 1, we also show the relations Ξ = glyph[epsilon1] c ( γµ mp / kT ) 1 / 2 that must hold at the sonic radius for three values of the radia- \nFIG. 1.- Metallicity dependence of the local thermal and statistical equilibrium temperature of a photoionized gas under optically thin conditions as a function of the ionization parameter Ξ (see text). The functions T eq( Ξ ) for four gas metallicities, expressed in units of the solar metallicity, were calculated with the photoionization code XSTAR (Kallman & Bautista 2001) for an f ν ∝ ν -1 . 5 spectrum between 0 . 1 Ryd and 1 , 000 Ryd. The curves T eq( Ξ ) are single valued and thus all equilibria are stable for the particular choice of spectrum, but they need not all be stable for harder spectra. The plot shows that T eq( Ξ ) is nearly independent of metallicity for Z glyph[lessorsimilar] 0 . 01 Z glyph[circledot] , and that a hot coronal phase appears at Ξ ∼ 10 3 in a gas with Z glyph[lessorsimilar] 0 . 1 Z glyph[circledot] and at Ξ ∼ 10 in a more metal rich gas. The straight lines are the relations Ξ = glyph[epsilon1] c ( γµ mp / kT ) 1 / 2 that must hold at the sonic radius for values of the radiative efficiency, from left to right, of glyph[epsilon1] = 0 . 025, glyph[epsilon1] = 0 . 1, and glyph[epsilon1] = 0 . 4. The temperature at the sonic radius is found at the intersection with the T eq( Ξ ) curve. \n<!-- image --> \ne efficiency: glyph[epsilon1] = 0 . 025, glyph[epsilon1] = 0 . 1, and glyph[epsilon1] = 0 . 4, which, speculatively, might be expected for a rapidly rotating black hole with a geometrically thin retrograde disk, a nonrotating black hole with a thin disk, and a rapidly rotating black hole with a geometrically thin prograde disk, respectively (see, e.g., Novikov & Thorne 1973; Zhang et al. 1997; Beckwith et al. 2006, 2008; Noble et al. 2008, and references therein). The temperature at the sonic radius is found at the intersection with the T eq( Ξ ) curve, that is, we have the implicit relation T s = T eq[ Ξ s; f ν ( r s)] = T eq[ glyph[epsilon1] c ( γµ m p / kT s) 1 / 2 ; f ν ], which can be solved for the temperature at the sonic radius T s as a function of glyph[epsilon1] and f ν (e.g., Ostriker et al. 1976). Evidently, in metal poor gas, the equilibrium temperature at the sonic radius is a strong function of the radiative efficiency; highly radiatively efficient accretion is susceptible to thermal runaway, where the ionization state converges to full ionization as the Compton heating overtakes thermal evolution and the gas becomes fully ionized (see, e.g., Krolik 1999, and references therein). Note, however, that the gas may not attain the Compton temperature if the inflow time becomes shorter than the heating time for combined Compton and photoionization heating (see § 2.2). Given T s and the sonic radius determined from the relation involving c s( r s) and the forces acting on the gas, we have that r s = (1 -glyph[lscript] ) µ mpGM BH / 2 γ kT s. For example, for power law SEDs F ν ∝ ν -1 . 5 and radiative efficiencies glyph[epsilon1] ∼ 0 . 1, typical temperatures at the sonic radius are T s ∼ 10 5 K, and so the sonic radii are r s ∼ 3 10 14 M 2 cm. \n∼ We can write the usual ionization parameter ξ ≡ L / r 2 n = 4 π kTc Ξ of Tarter et al. (1969) 2 as ξ = 4 π v µ mpL / ˙ M , which in the optically thin limit becomes ξ = 4 π v µ mp glyph[epsilon1] c 2 . In this limit, \n× \nthe temperature T eq, which is in one-to-one relation with ξ , is also in one-to-one relation with the velocity, T eq( v ; f ν , glyph[epsilon1] ), and so one can write equation (2) as a differential equation with a single unknown function v ( r ). When the flow is isothermal, we have the well-known asymptotic solution far from the black hole \nv ∼ e 3 / 2 ( r r s ) -2 c s( r s) ( r glyph[greatermuch] r s , isothermal) . (3) \nIf the temperature of the photoionized flow can decrease with radius and thus the infalling gas can acquire momentum before it heats to ∼ T s, the asymptotic velocity v ( r ) far from the black hole can exceed the isothermal value given in equation (3) by a factor of several. We thus write \nv ∼ 4 . 5 Υ ( r r s ) -2 c s( r s) ( r glyph[greatermuch] r s) , (4) \nwhere Υ glyph[greaterorsimilar] 1. \nIdeally, we would like to match this solution to the conditions far from the black hole, where the density and the total pressure are n ∞ and P ∞ . One could attempt to set the boundary conditions n ( r 1) = n ∞ and n ( r 1) kT eq( r 1) = P ∞ ≡ n ∞ kT ∞ at some radius r 1, where the last relation defines T ∞ . This may indeed be possible at very low densities or very low radiative efficiencies. At high densities and efficiencies, however, the photoionization equilibrium at some well defined radius r ion glyph[greatermuch] r s abruptly transitions into a neutral state, i.e., on its way toward the black hole the gas passes a quasistationary ionization front. Gas density, velocity, temperature, and pressure may be discontinuous at the ionization front; furthermore, the warm or cold neutral gas in the immediate vicinity of the ionized region may be supersonically turbulent, in which case the gas density is inhomogeneous and the ram pressure of turbulent flows cannot be neglected. We proceed to an attempt to determine under which conditions is a steady, strictly time-independent accretion across the stationary ionization front possible.", '2.2. Size of the H II Region around the Black Hole': "We will assume that the ionized region is surrounded by warm, partially-ionized gas, though in reality, the accreting black hole and its ionization sphere may be embedded in a cold, molecular, and supersonically turbulent medium. However, since the UV radiation from the black hole and the He II recombination radiation from the photoionization annulus (see § 4.2 below) will dissociate molecules in the vicinity of the H II region, a layer of warm, atomic gas should surround the ionization sphere even if the molecular phase exists at somewhat larger optical depths. We further assume that far from the sonic radius, where the black hole's gravity can be ignored, the ionized gas is in gas pressure equilibrium with the surroundings, i.e., \nT HII n HII = f turb T HI n HI ( r glyph[greatermuch] r s) , (5) \nwhere n HII is the total density of ions and electrons in the H II region, n HI is the atomic density in the neutral gas, T HII and T HI are the respective temperatures, and f turb ≥ 1 is a factor quantifying the degree of pressure enhancement due to turbulence in the neutral gas. Pressure equilibrium will be violated in non-steady-state, episodic accretion (§ 3). \nDepending on the SED of the central source, the photoionization rate will be dominated by primary photoionizations or by secondary photoionizations carried out by photoelectrons \n(e.g., Shull & van Steenberg 1985; Xu & McCray 1991; Dalgarno et al. 1999). Taking into account only the ionization of hydrogen from the ground state, the total rate of photoionization in the annulus can be written ˙ N ion = f ion L / E H, where E H = 13 . 6 eV, L = 10 40 L 40 erg s -1 is the luminosity shortward of E H, and f ion ∼〈 E H / h ν 〉 is the average fraction of the energy of an absorbed photon that goes into photoionization. For an almost fully ionized gas and a power law spectrum f ν ∝ ν -α , this fraction equals f ion = ( α -1) /α for α> 1; in what follows, it should be borne in mind that f ion depends on the shape of the SED. \nIn a photoionization equilibrium ˙ N ion equals the total hydrogen recombination rate in the ionized gas ˙ N rec ∼ 4 3 π r 3 ion α B ( T HII) n H + ne , where α B ( T ) ∼ 2 . 6 × 10 -13 T -1 4 cm 3 s -1 is the approximate hydrogen recombination coefficient in the on-the-spot approximation at temperatures 10 4 K glyph[lessorsimilar] T < 10 5 K(Ferland et al. 1992, quoted in Jappsen et al. 2007), and n H + ∼ ne ∼ 1 2 n HII are the ion and electron densities, respectively. Though not entirely justified in Strömgren spheres, the on-the-spot approximation is reasonable unless the density profile within the ionized region is sharply peaked toward the center (Ritzerveld 2005), as is assumed not to be the case here. \nEquating the ionization rate to the recombination rate we obtain \nr ion ∼ 4 . 4 × 10 18 f 1 / 3 ion L 1 / 3 40 T HII , 4 . 7 f 2 / 3 turb n 2 / 3 5 T 2 / 3 HI , 3 . 7 cm , (6) \nwhere T HII = 5 × 10 4 T HII , 4 . 7 K, T HI = 5 × 10 3 T HI , 3 . 7 K, and n HI = 10 5 n 5 cm -3 (we justify our choice of the reference density in § 2.4 below). If we express the luminosity in terms of the dimensionless ratio glyph[lscript] ≡ L / (4 π GM BH mpc /σ T) of the luminosity to the Eddington luminosity (here and henceforth, for Thomson scattering), the radius of the H II region becomes \nr ion ∼ 4 . 7 × 10 18 glyph[lscript] 1 / 3 f 1 / 3 ion M 1 / 3 2 T HII , 4 . 7 f 2 / 3 turb n 2 / 3 5 T 2 / 3 HI , 3 . 7 cm . (7) \nThe ionization radius is normally much larger than the sonic radius, \nr ion r s ∼ 2 × 10 4 glyph[lscript] 1 / 3 f 1 / 3 ion T HII , 4 . 7 T s , 5 f 2 / 3 turb n 2 / 3 5 T 2 / 3 HI , 3 . 7 M 2 / 3 2 . (8) \nIn Figure 2 we provide a schematic illustration of the structure of the H II region surrounding an accreting seed black hole. \nSelf-gravity of the gas inside the H II region is negligible compared to that of the black hole, 4 3 π r 3 n H + mp glyph[lessmuch] M BH, when glyph[lscript] glyph[lessmuch] 0 . 054 f -1 ion f turb n 5 T HI , 3 . 7 T -2 HII , 4 . 7 ( r / r ion) -3 which is satisfied for r glyph[lessmuch] r ion, but need not always be true at the very edge of the H II region. There, however, the combined gravity of the black hole and the gas contained within r ion are negligible compared to gas pressure gradients; the ratio of the Jeans length inside the H II region to its radius is normally much larger than unity, λ J / r ion ∼ 28 f -1 / 3 ion f 1 / 6 turb glyph[lscript] -1 / 3 n 1 / 6 5 T 1 / 6 HI , 3 . 7 M -1 / 3 2 , and thus self-gravity of the ionized gas can be ignored.", '2.3. Conditions at the Sonic Radius': 'We will assume throughout that the gas is almost fully ionized at the sonic radius, i.e., that r s glyph[lessmuch] r ion. This condition places an upper limit on the density of the neutral gas just \nFIG. 2.- Schematic representation (not to scale) of the structure of the H II region surrounding an accreting seed black hole. For normal protogalactic densities, the radius of the H II region r ion (eq. [7]) is a few orders of magnitude larger than the sonic radius r s (eq. [1]). Gas is in local thermal and statistical equilibrium outside the radius r equi (§ 2.3), which may be slightly larger or smaller than r s. The ionized gas contains a small, nonzero neutral fraction (eq. [20]), which gives rise to photoionization radiation pressure. The ionized region is surrounded by a shell of atomic gas of thickness ∆ r diss where molecule photodissociation is efficient (§ 4.2). The outer, molecular shell may be supersonically turbulent. \n<!-- image --> \noutside the H II region, n glyph[lessmuch] n max , ion, where \nn max , ion = 2 . 7 × 10 11 f 1 / 2 ion glyph[lscript] 1 / 2 T 3 / 2 HII , 4 . 7 T 3 / 2 s , 5 f turb T HI , 3 . 7 M 2 cm -3 , (9) \nand we have assumed that T HII , 4 . 7 also represents the temperature of the ionized gas at the sonic radius. This shows that for an isotropic central radiation source, the gas at the sonic radius is guaranteed to be ionized unless the accreting gas has densities far in excess of those expected for the diffuse protogalactic medium and instead characteristic of self-gravitating, star-forming cores. \nThe central luminosity can be related to the mass accretion rate by substituting equation (4) into glyph[lscript] ≡ glyph[epsilon1] r 2 v µ nc σ T / GM BH to obtain the dimensionless luminosity for steady state accretion through the H II region \nglyph[lscript] s . s . ∼ 0 . 001 glyph[epsilon1] -1 f turb Υ M 2 n 5 T HI , 3 . 7 T HII , 4 . 7 T 3 / 2 s , 5 , (10) \nwhere glyph[epsilon1] = 0 . 1 glyph[epsilon1] -1 is the radiative efficiency. Evidently, the mass accretion can be strongly suppressed by the heating of the ionized gas near the sonic radius. The accretion rate is much smaller than the Bondi accretion rate ˙ M Bondi = e 3 / 2 π G 2 M 2 BH mpn HI / c 3 s , HI that would be calculated ignoring photoionization and photoheating altogether, i.e., for radiatively-inefficient accretion \nglyph[lscript] Bondi ∼ 5 . 2 glyph[epsilon1] -1 M 2 n 5 f 3 / 2 turb T 3 / 2 HI , 3 . 7 (no photoheating/ionization) , (11) \nwhere have assumed an isothermal equation of state. \nTo justify our assumption in § 2.1 that the photoionized gas is in local thermal and statistical equilibrium, we calculate \nthe ratio of the Bremsstrahlung cooling time t Brems ∼ 2 . 5 × 10 11 T 1 / 2 n -1 H s at the sonic radius to the inflow time at the sonic radius \nto obtain \nt s ∼ r s c s( r s) ∼ 1 . 45 M 2 T 3 / 2 s , 5 yr , (12) \nt Brems t s ∼ 77 T 2 s , 5 T HII , 4 . 7 f turb Υ T HI , 3 . 7 n 5 M 2 . (13) \nCooling due to the recombination of H II and He III is comparable to and only slightly stronger than that due to Bremsstrahlung at T s glyph[lessorsimilar] 2 10 5 K. \nWe also estimate the ratio of the Compton heating time at the sonic radius t C ∼ 0 . 0675 GM µ 2 mempc /glyph[lscript] k 2 T s T C, where T C ∼ 10 7 T C , 7 K is the Compton temperature, to the inflow time (see also Fig. 2 in Sazonov et al. 2005), \n× \nt C t s ∼ 0 . 01 T 1 / 2 s , 5 glyph[lscript] T C , 7 . (14) \nAt the relatively high gas densities considered here, which are required for a rapid growth of seed massive black holes in protogalaxies, the Compton cooling of the photoionized gas by the cosmic microwave background photons (see, e.g., Ricotti et al. 2008) can be ignored. \nSubstituting equation (10) in equation (14) we obtain for ratio of the Compton heating time to the infall time \nt C t s ∼ 11 T 2 s , 5 T HII , 4 . 7 glyph[epsilon1] -1 f turb Υ n 5 M 2 T C , 7 T HII , 3 . 7 . (15) \nEquations (13) and (15) suggest that heating and cooling times at the sonic radius can be longer than the inflow time and that the ionized gas may not be in local thermal and statistical equilibrium at all radii r glyph[greatermuch] r s, especially if the accretion occurs below the Eddington-limited rate. However, the ionized gas should be in equilibrium at only a somewhat larger radius because the infall time increases rapidly with radius, r / v ∝ r 3 . \n∝ So far we have ignored the dependence of the temperature at the sonic radius on the radiative efficiency, which as Figure 1 shows, can be strong. The precise form of the function T s( glyph[epsilon1] ), defined by the intersection of the T eq( Ξ ) curve and the T s( Ξ ) line, depends sensitively on the metallicity of the gas and on the SED of the central source and we do not attempt to model it in general. If the dependence can be approximated with a power low in a range of efficiencies, T s ∝ glyph[epsilon1] δ , from equations (13), (10), and (15) we obtain \nt Brems t s ∝ glyph[epsilon1] 2 δ , glyph[lscript] ∝ glyph[epsilon1] 1 -3 δ / 2 , t C t s ∝ glyph[epsilon1] 2 δ -1 . (16) \nIf, e.g., δ ∼ 1 for glyph[epsilon1] glyph[greaterorsimilar] 0 . 1, we find that with an increasing efficiency it becomes more difficult for the gas to achieve local thermal equilibrium at the sonic radius, but if the equilibrium is achieved, the central luminosity, and especially the accretion rate that is proportional to ˙ M ∝ glyph[lscript]/glyph[epsilon1] ∝ glyph[epsilon1] -3 δ / 2 , decrease with increasing efficiency. \nCiotti & Ostriker (1997, 2001, 2007) and Sazonov et al. (2005) have studied spherically-symmetric accretion of hot interstellar medium onto an X-ray quasar in an elliptical galaxy and have identified a limit cycle driven by Compton heating. The quasar heats the interstellar medium to temperatures exceeding the virial temperature, which leads to an outflow and quenching of central accretion. This model differs \nfrom ours in that a metal-enriched environment is assumed, so that photoionization equilibrium temperature reaches the Compton temperature already at Ξ ∼ 50, whereas in our metal-poor model the transition to the Compton temperature occurs at higher values of the photoionization parameter, Ξ ∼ 10 3 . Therefore, in the model of Sazonov et al. (2005), the ionized gas can reach the Compton temperature well outside the sonic radius. Another crucial difference is the assumed ionized gas density profile far from the sonic radius: the hot gas surrounding a quasar was assumed to be hydrostatically confined by the gravity of the host galactic stellar spheroid such that its density declines steeply with radius, n HII ∝ r -2 , which implies that Ξ is roughly independent of radius and so the Compton-heated equilibrium can exist at arbitrarily large radii. In our model, since the H II region surrounding a seed black hole is confined by external pressure at radii r glyph[greatermuch] r s and is roughly isothermal (consistent with the very weak dependence of T eq on the dimensionless ionization parameter for 10 glyph[lessorsimilar] Ξ glyph[lessorsimilar] 10 3 ; see Fig. 1), the density inside it is approximately independent of radius. Then Ξ ∝ r -2 , and the gas is progressively farther from being able to heat the Compton temperature at radii much larger than the sonic radius.', '2.4. Protogalactic Density at the Edge of the H II Region': "We will now pause our investigation of radiative feedback effects to clarify our choice of the density of the ambient neutral gas surrounding the H II region. Rapidly growing protogalaxies contain gas with a wide range of densities, and thus care must be taken to appropriately specify the gas density on scales relevant for regulation of accretion onto a seed black hole by radiative feedback effects. In the simplest picture, central gravitational collapse of the gas at the center of a protogalaxy produces a distribution in which gas density is spherically symmetric and a function of radius only, n ( r ). If the gas is approximately isothermal and quasi-hydrostatic in the presence of turbulence, n ( r ) ∝ r -2 . If we ignore the gravity of any seed black hole and let f turb denote a radius-independent turbulent pressure enhancement, the density scales as \nn ( r ) ∼ f turb c 2 s 2 π G mp r 2 ∼ 6 × 10 4 f turb T HI , 3 . 7 ( r 1 pc ) -2 cm -3 . (17) \nEquation (17) is a good approximation to the density 'cusp' profile resulting from central gravitational collapse in a M halo ∼ 10 8 M glyph[circledot] cosmological halo in the simulations of Bromm & Loeb (2003) and in the simulations of Wise et al. (2008), who found turbulent Mach numbers M∼ 3 in the cusp, which would imply f turb 10. \nIf a seed black hole is located near the center of the density cusp, radiative effects may prevent the gravitational collapse from proceeding to the arbitrarily large densities. We will here assume that radiative effects prevent collapse at radii smaller than the radius of the H II region, i.e., that the density profile of the neutral gas has a 'core' on scales ∼ r ion such that density within the H II is roughly uniform far from the sonic radius. For self-consistency, we substitute n = n ( r ion) from equation (17) in equation (7) and solve for r ion, and in turn for \n∼ \nn ( r ion), to obtain \nr ion , cusp ∼ 3 . 8 × 10 17 f 4 turb T 4 HI , 3 . 7 f ion glyph[lscript] T 3 HII , 4 . 7 M 2 cm , n cusp ∼ 4 . 5 × 10 6 f 2 ion glyph[lscript] 2 T 6 HII , 4 . 7 M 2 2 f 7 turb T 7 HI , 3 . 7 cm -3 . (18) \nThese estimates are but crude self-consistency conditions and suffer from a strong sensitivity to the turbulent Mach number and other parameters. The estimate of gas density at the edge of the H II region given in equation (18) can only be used as a rough guide for the range of densities that should be addressed in the ensuing analysis. Our choice of the reference density, 10 5 cm -3 , is compatible with the self-consistency conditions in a protogalaxy in a cosmological halo of M halo ∼ 10 8 M glyph[circledot] for f ion ∼ 1 3 , ignoring any recent supernova activity in the center of the halo, which can drastically reduce the central density (see, e.g., Wada & Venkatesan 2003; Kitayama & Yoshida 2005; Wise & Abel 2008). \nEquations (18) also suggest that 'minihalos' with masses M halo ∼ 10 6 M glyph[circledot] and Mach numbers M glyph[greaterorsimilar] 1 will be fully ionized out to r glyph[greaterorsimilar] 100 pc if, somehow, glyph[lscript] max ∼ 1 is realized at an early instant prior to the expansion of the H II region, and the interior of the H II region has not had chance to heat beyond 10 4 K. This is seen in the simulations of radiative feedback during accretion onto seed black holes in cosmological minihalos of Alvarez et al. (2008).", '2.5. Continuum Radiation Pressure': "Here we estimate the effects of the continuum radiation pressure in the interior of the H II region. The continuum radiation pressure acceleration is a rad = ¯ σ H L χ HI / 4 π r 2 mpc , where χ HI is the abundance of neutral hydrogen in the ionized gas, and ¯ σ H = F -1 ∫ σ H , ν F ν d ν is the frequency-averaged mean absorption cross section. In a highly ionized gas, most of the photoelectron energy goes into heating, so in ionization balance \nχ HI = α B ( T HII) L ˜ σ H / (4 π r 2 ne E H) + α ion( T HII) , (19) \nwhere ˜ σ H = F -1 E H ∫ ( h ν ) -1 σ H , ν F ν d ν and α ion( T ) is the collisional ionization rate. At temperatures glyph[lessmuch] 10 5 K, photoionization typically dominates collisional ionization. The neutral hydrogen abundance then becomes \nχ HI ∼ 7 . 6 × 10 -4 f 2 / 3 ion glyph[lscript] 1 / 3 f 1 / 3 turb M 1 / 3 2 n 1 / 3 5 T 1 / 3 HI , 3 . 7 ( r r ion ) 2 , (20) \nwhich implies that at radii not much smaller than r ion, the neutral abundance is glyph[greaterorsimilar] 10 -6 , and in this regime, since ¯ σ H ∼ 10 6 σ T, radiation pressure due to photoionization exceeds that due to Thomson scattering. The acceleration due to the former is then given by \na rad = ¯ σ H ˜ σ H α B ( T HII) ne E H mp c . (21) \nNote that, unlike in the case of Thomson scattering, this is independent of distance from the black hole because here the abundance of absorbers increases with the square of the radius, χ HI ∝ r 2 . The cross section ratio is ¯ σ H / ˜ σ H ∼ (3 + α ) / (2 + α ) for power-law spectra F ν ∝ ν -α . We compare a rad to the black hole's gravity a grav = -GM BH / r 2 to find, \na rad | a grav | ∼ 980 f 2 / 3 ion glyph[lscript] 2 / 3 f 1 / 3 turb T 1 / 3 HI , 3 . 7 n 1 / 3 5 M 1 / 3 2 ( r r ion ) 2 (22) \nwhere we have assumed a fiducial α = 1 . 5 spectrum, implying f ion ∼ 1 3 and ¯ σ H / ˜ σ H ∼ 9 7 . The relative importance of photoionization radiation pressure depends on distance, and dominates at larger radii. This shows that for a wide range of parameters, for near-Eddington accretion, the radiation pressure in the H II region at radii r glyph[lessorsimilar] r ion greatly exceeds the black hole's gravity, which implies that steady state accretion may occur at a reduced rate. Inward accretion is still possible if a positive pressure gradient is set up to counteract the radiation pressure force. \nTo estimate the reduction of the steady-state accretion rate as a result of the photoionization radiation pressure in the H II region, we simplify the mathematical problem by ignoring any heating of the gas near the sonic radius and setting T s = T HII = const; this simplification, as we shall see, will turn out to be unreasonably optimistic. Then, after a change of variables, \nw ≡ v c s , HII , y ≡ r ( GM BH 2 c 2 s , HII ) -1 ≈ r r s , (23) \nequation (2) describing momentum conservation in the accretion flow becomes \ndw dy ( w 2 -1) = w ( 2 y -2 y 2 + φ ) , (24) \nwhere φ is a dimensionless parameter proportional to the photoionization radiation pressure acceleration, \nφ ≡ GM BH 2 c 4 s , HII a rad ∼ 2 × 10 -5 f turb n 5 M 2 T HII , 3 . 7 T 4 HII , 4 . 7 . (25) \nEquation (24), subject to the regularity condition imposed at the sonic radius, w ( y s) = 1, where y s = φ -1 [(1 + 2 φ ) 1 / 2 -1] ≈ 1, can be solved in closed form in terms of the transcendental Lambert W ( x ) function, defined implicitly via x = We W , to obtain \nw ( y ) = √ √ √ √ -W { -( y y s ) -4 exp [ -2( φ y 2 + 2) y + 2( φ y 2 s + 2) y s -1 ] } ≈ y -2 exp ( -φ y 2 -3 y / 2 + 2 y ) , (26) \nwhere in the second line we applied the expansion W ( x ) = x + ( x 2 ). \nFor φ glyph[lessmuch] 1, the ratio of density n ∝ ( r 2 v ) -1 ∝ ( y 2 w ) -1 at radius r glyph[greatermuch] r s to that at the sonic radius has asymptotic form \nn ( r ) n ( r s) ∼ e φ r / r s [ n ( r ) n ( r s) ] φ =0 . (27) \nThis result is exact for isothermal accretion, but here, we adopt it as an optimistic estimate of the density reduction due to photoionization radiation pressure in the case of realistic, nonisothermal accretion. \nWe expect significant reduction in the accretion rate when n HII( r s) glyph[lessmuch] n HII( r ion) or, according to equation (27), when the quantity \nφ r ion r s ∼ 0 . 4 f 1 / 3 ion f 1 / 3 turb glyph[lscript] 1 / 3 T 1 / 3 HI , 3 . 7 T s , 5 n 1 / 3 5 M 1 / 3 2 T 3 HII , 4 . 7 (28) \nO \nis comparable to or greater than unity. The dimensionless accretion rate can be eliminated by substituting equation (10) that relates the accretion rate to the conditions at the sonic radius, to obtain \nφ r ion r s ∼ 0 . 04 f 1 / 3 ion f 1 / 3 turb Υ 1 / 3 glyph[epsilon1] 1 / 3 -1 T 2 / 3 HI , 3 . 7 T 1 / 2 s , 5 n 2 / 3 5 M 2 / 3 2 T 10 / 3 HII , 4 . 7 . (29) \nThus we find that accretion will proceed in one of the two distinct regimes. When \nφ r ion r s glyph[lessmuch] 1 , (30) \nradiation pressure does not affect the accretion rate even if the combined gravitational and radiative acceleration is outward in parts of the H II region. However, when \nφ r ion r s glyph[greaterorsimilar] 1 , (31) \nthe ionized gas assumes a density gradient at radii r glyph[greatermuch] r s such that the density increases outward, toward the edge of the H II region. In this case, central accretion is suppressed at least by the factor \nglyph[lscript] ∝ exp ( -φ r ion r s ) , (32) \nand probably more, because the central drop in density will lead to a photoionization equilibrium at a higher temperature, thereby increasing gas temperature near the sonic radius. Therefore, for φ r ion / r s glyph[greaterorsimilar] 1, radiation pressure acting on the ionized gas imposes a more restrictive limit on the accretion rate than the mere photoheating near the sonic radius. The two limits are complementary: while the photoheating limit is more restrictive for small black hole masses, ambient densities, and turbulent Mach numbers, the radiation pressure limit is more restrictive in the very opposite regime. A combination of these two limits suggests that a steady state solution for near-Eddington accretion does not seem to exist. 3 \nA fraction f res < 1 of the luminosity emitted by the central source passes unabsorbed through the H II region and reaches its edge, where most of it is absorbed in the partially ionized gas. Momentum deposition by the residual luminosity leads to a density drop by the factor of exp( -ψ ) where \nψ ≡ f res L 4 π r ion γ n HII k T HII c ∼ ( 0 . 033 f res T HII , 4 . 7 f ion T s , 5 ) φ r ion r s . (33) \nSince T s ≥ T HII, radiation pressure in the partially-ionized edge of the H II region is always less important than in its interior.", '2.6. Accretion at Low Radiative Efficiencies': "In this work we focus entirely on the regime in which central accretion is radiatively efficient, e.g., glyph[epsilon1] glyph[greatermuch] 0 . 01, as it should be the case when the radiation is emitted by a geometrically thin disk. In this regime, heating of the gas at the sonic radius to glyph[greaterorsimilar] 10 5 K, and perhaps even to the Compton temperature at high metallicities, seems inevitable. We hope to generalize our analysis to radiatively inefficient accretion, \nwhich should be accompanied by outflows and shadowing, in a subsequent investigation. Here, we will make only a brief mention of this alternate regime. \nIf we ignore the mechanical and radiative effects of the outflows and the shadowing, which is an implausible and inconsistent assumption, a change in the thermodynamic behavior occurs at glyph[epsilon1] glyph[lessorsimilar] 10 -3 , namely, at such low efficiencies the gas at the sonic radius and throughout the H II region can remain at temperatures well below 10 5 K (see, e.g., Buff & McCray 1974; Hatchett et al. 1976; Krolik et al. 1981; Kallman & McCray 1982; Krolik & Kallman 1984; Lepp et al. 1985; Donahue & Shull 1991). If, e.g., T s ∼ T HII ∼ 10 4 K and Υ ∼ 1, we find that, still, under a wide range of conditions characteristic of protogalactic clouds, an extended ( r ion glyph[greatermuch] r s) H II region exists around the seed black hole. In this case, photoionization radiation pressure suppresses central accretion when φ r ion / r s glyph[greaterorsimilar] 1, which can be expressed as a condition on the radiative efficiency; the suppression takes place when glyph[epsilon1] glyph[greaterorsimilar] glyph[epsilon1] crit where \nglyph[epsilon1] crit ∼ 0 . 005 f ion f 2 turb T 2 HI , 3 . 7 n 2 5 M 2 2 . (34) \nFor glyph[epsilon1] glyph[greaterorsimilar] glyph[epsilon1] crit, the ionized gas may not be able to heat above 10 4 K, but the photoionization radiation pressure suppresses accretion well below the rate glyph[lscript] s . s . derived in equation (10). For glyph[epsilon1] glyph[lessmuch] glyph[epsilon1] crit, accretion is limited by conditions at the sonic radius (§ 2.3). The accretion rate may become close to the 'Bondi' rate calculated ignoring radiative effects altogether (eq. [11]) if the neutral gas surrounding the H II region is weakly supersonically turbulent, but the ionized gas inside it is not. \nWe would like to reiterate that the analysis presented in this subsection is somewhat unrealistic, because radiativelyinefficient accretion flows are probably accompanied by outflows that can radically alter the hydrodynamic, thermodynamic, and chemical structure of the region surrounding a seed black hole. Also, the radiation field produced by a radiatively-inefficient accretion flow is potentially highly anisotropic.", '2.7. Ly α Radiation Pressure': "Another source of pressure is from the Ly α photons that are produced throughout the H II region and can become trapped within the ionized region and the surrounding neutral shell (see, e.g., McKee & Tan 2008, and references therein). Let f Ly α ∼ 2 3 (e.g., Osterbrock 1989) denote the fraction of the total power emitted by the central source above E H that is reprocessed into Ly α photons. Consider a shell of neutral gas at radii r ion < r < r ion + ∆ r shell, where ∆ r shell denotes the thickness of the shell. For shells with ∆ r shell glyph[lessorsimilar] r ion, the linecenter optical depth of the shell τ 0 = σ 0( T ) n H ∆ r shell, where σ 0( T ) = 5 . 9 × 10 -14 T -1 / 2 4 cm 2 , is large \nτ 0 ∼ 4 × 10 10 f 1 / 3 ion glyph[lscript] 1 / 3 M 1 / 3 2 n 1 / 3 5 T HII , 4 . 7 f 2 / 3 turb T 7 / 6 HI , 3 . 7 ( ∆ r shell r ion ) . (35) \nSince the shell is very optically thick, a photon injected at the edge ( r = r ion) near line center will scatter across the ionized region many times before escaping the shell. The ionized gas within the H II region has some residual optical depth due to the presence of a small neutral fraction. Under the conditions considered here, the ionized gas is optically thick to the photons in the core of the Ly α line, and is marginally optically thin to the photons in the wings. We ignore this complication and treat the ionized gas as optically thin. \nTo estimate the number of times a photon injected at the edge of the neutral gas crosses the H II region, we assume that ∆ r shell glyph[lessorsimilar] r ion, and carry out a simple Monte Carlo calculation of a photon's frequency diffusion before it escapes the shell. For this, we employ the accurate expressions for the transmission coefficient and the reflection frequency redistribution function at the shell edge that Hansen & Oh (2006) obtained from Monte-Carlo resonant line scattering calculations. In the dust-free limit, the number of times a photon injected near the line core reflects against the walls of the ionized region is accurately approximated with \nN reflect ≈ 0 . 609 ( a τ 0) 0 . 659 e -0 . 00607 [ln(2120 . 0 a τ 0)] 2 , (36) \nwhere a ( T ) = ν L / 2 ν Dop = 4 . 72 × 10 -4 T -1 / 2 4 is the ratio of the natural to the Doppler line width. This is close to the asymptotic formula due to Adams (1972, 1975), N reflect ∼ 15 ( τ 0 / 10 5 . 5 ) 1 / 3 , in the optically thick limit (quoted from Dijkstra & Loeb 2008). At T =10 4 K, for τ 0 = (10 9 , 10 10 , 10 11 ), the numbers of reflections take values N reflect ∼ (250 , 610 , 1400). Rayleigh scattering contributes to the opacity negligibly for τ 0 glyph[lessorsimilar] 10 10 . Destruction of Ly α photons by stimulated two photon emission is negligible in the range of optical depths that we consider. Destruction by H -ions in the partially ionized shell surrounding the H II region is potentially important (see § 4.2), but because of significant uncertainties, we ignore it and adopt Adams' formula in what follows. \nIf the radiation pressure drives a coherent expansion (outflow) in the neutral shell, this may promote photon escape and reduce P Ly α / P gas. If the outflow contains a velocity gradient of magnitude ∆ v , Bonilha et al. (1979, see also Bithell 1990) estimate that N scatter is reduced by a factor ∼ (1 + 0 . 04 | ∆ v / v Dop | 3 / 2 ) -1 , where v Dop ∼ c s ∼ 10 km s -1 is the thermal or turbulent Doppler velocity. The same reduction factor should apply to N reflect. Thus, even for a highly supersonic outflow ∆ v / v Dop ∼ 8, the number of reflections to escape and the Ly α radiation pressure to which it is proportional will drop by only a half. \nThe energy density in Ly α photons can be approximated with (see, e.g., Bithell 1990; Haehnelt 1995) \nU Ly α ∼ 3 f Ly α L N reflect 4 π ( r + ∆ r shell) 2 c . (37) \nFrom this, the radiation pressure P Ly α = 1 3 U Ly α scale height is ∼ 1 2 r ion. At depth equal to one scale height we have, with the fiducial choice f Ly α = 2 3 , \nP Ly α P gas ∼ 3 . 8 glyph[lscript] 4 / 9 f 10 / 9 turb M 4 / 9 2 n 4 / 9 5 f 5 / 9 ion T 1 / 18 HI , 3 . 7 T 5 / 3 HII , 4 . 7 . (38) \nFor near-Eddington accretion the Ly α pressure can significantly exceed the gas pressure and impart an outward impulse to the neutral gas surrounding the H II region. This result appears to imply that even at relatively small glyph[lscript] , the pressure of the trapped resonance line radiation remains in excess of the gas pressure. \nIf we speculatively require P Ly α < P gas for strictly stationary accretion, we invert equation (38) to derive an upper limit on the accretion rate, glyph[lscript] <glyph[lscript] crit , Ly α , where \nglyph[lscript] crit , Ly α = 0 . 05 f 5 / 4 ion T 1 / 8 HI , 3 . 7 T 15 / 4 HII , 4 . 7 f 5 / 2 turb M 2 n 5 . (39) \nBecause of the strong dependence on T HII and our not having sought, within the confines of the present work, a selfconsistent dynamical solution for gas flow in the presence of resonance line scattering radiation, we are not able determine whether the continuum photoionization pressure or the resonance line scattering is more constraining to the maximum accretion rate for strictly stationary quasiradial accretion that can be achieved. Additional sources of uncertainty affecting any attempt to estimate the impact of resonance line radiation pressure are the topology of the density field beyond the spherically symmetric approximation, as the radiation may 'leak out' even through a relatively small hole in the neutral shell (see, e.g., Hansen & Oh 2006; McKee & Tan 2008) and the dynamical stability of shells accelerated by resonance line radiation (Mathews 1992).", '3.1. Mechanics of Inflow and Outflow': "Here we revoke the assumption of strictly stationary flow and attempt to describe a sequence of periodically recurring stages in the seed black hole's accretion cycle. Let glyph[lscript] max < 1 denote the peak luminosity in units of the Eddington luminosity. The peak luminosity is reached due a sudden infall of material; the thermodynamic state of the infalling gas is assumed to be set by a central luminosity well below the peak luminosity. The infalling gas reacts to the sudden rise in central luminosity on a finite time scale. Facing a sudden rise in the central luminosity, the accreting gas heats by photoionization and gets accelerated outward by the radiation pressure. The photoionization heating time t photo = 4 π r 2 ( γ -1) -1 kT HII / 1 2 χ H ¯ σ H L , where ¯ σ H is the average photoionization cross section and χ H is the neutral hydrogen abundance (see § 2.5), is given by \nt photo ∼ 2 . 9 T 3 HII , 4 . 7 f turb f epi , 1 n 5 T HI , 3 . 7 yr , (40) \nwhere we have included a potentially large density enhancement f epi = 10 f epi , 1 ≥ 1, due to, e.g., infall or a lack of pressure equilibrium, over the steady-state density of the H II region. The photoionization heating time scale is thus rather short and comparable to the inflow time at r r s. \nFollowing the sudden increase in central luminosity, material in the outer part of the H II region, r ∼ r ion, is accelerated outward by the radiation pressure if a rad glyph[greaterorsimilar] P gas , HII / r µ mpn HII, which implies the condition (cf. eq. [28]) \n∼ \n3 . 3 f 1 / 3 ion f 1 / 3 turb f epi , 1 glyph[lscript] 1 / 3 max T 1 / 3 HI , 3 . 7 n 1 / 3 5 M 1 / 3 2 T 2 HII , 4 . 7 glyph[greaterorsimilar] 1 . (41) \nPhotoionization heating and radiation pressure will act to erase the infall-induced central density enhancement and drive the central density and the accretion rate down, toward the limits imposed by thermodynamics (§ 2.3) and radiation pressure (§ 2.5). The accretion flow remains directed inward close to the sonic radius, but far from the sonic radius, the inflow gives way to an outflow. \nIf the outflowing material reaches radii ∼ r ion, it encounters a shell of denser hot gas that has been photoevaporated from edge of the H II region. The hot shell is overpressured by a factor ∼ T HII / f turb T HI and is expanding in both radial directions. As the central density and luminosity drop, radiation pressure is not able anymore to accelerate the ionized gas against the positive pressure gradient in the exterior of the H II region. Then the pressure gradient resulting from the rapid \ndecline of radiation pressure and from the photoevaporative heating accelerates the gas near the edge of the H II region inward, in an implosion back toward the black hole. The gravitational acceleration is subdominant compared to the pressure gradient acceleration, | a pres | glyph[greaterorsimilar] a rad , max glyph[greatermuch] | a grav | (see eq. [22]) until the infalling gas approaches to within a few sonic radii from the black hole, where, by definition, gravity becomes competitive with pressure. The returning gas acquires a velocity similar to the local sound speed of the photoevaporated gas, which is perhaps somewhat larger than c s , HII( r ion), and returns to the vicinity of the black hole on a time scale \nt return ∼ r ion c s , HII \n∼ 4 × 10 4 glyph[lscript] 1 / 3 max f 1 / 3 ion M 1 / 3 2 T 1 / 2 HII , 4 . 7 f 2 / 3 turb n 2 / 3 5 T 2 / 3 HI , 3 . 7 yr . (42) \nThe return completes the accretion cycle. The return time may be shorter by the factor of a few than this estimate if the overpressuring during to photoevaporation, which we ignored in equation (42), is taken into account. \nThe pressure of the Ly α scattering radiation trapped within, and in the neighborhood, of the H II region may further reduce the average accretion rate by imparting an outward impulse to partially ionized gas and thus possibly significantly extending the return time t return. In view of the lingering concerns regarding the transfer of Ly α radiation that we have raised in § 2.7, we ignore it in the following estimate of the duty cycle.", '3.2. Estimates of the Duty Cycle': 'If, ultimately, after the central density and luminosity have decreased substantially, thermodynamics imposes the most stringent limit on the accretion rate, we might expect that the luminosity drop toward the steady-state rate ∼ glyph[lscript] s . s . derived in equation (10). Because of the sensitivity to the various parameters of our model, we are not able to generally determine whether the luminosity reaches glyph[lscript] s . s . in time t return separating consecutive infall episodes. For the simplicity of the remaining analysis of episodic accretion, we assume that the minimum luminosity is indeed ∼ glyph[lscript] s . s . and that the decline is exponential, \nglyph[lscript] ( t ) = glyph[lscript] max ( glyph[lscript] s . s . glyph[lscript] max ) t / t return . (43) \nThen, the time-average luminosity is given by \n〈 glyph[lscript] 〉 = glyph[lscript] max ln( glyph[lscript] max /glyph[lscript] s . s . ) , (44) \nand the duty cycle, which we define as f duty ≡〈 glyph[lscript] 2 〉 / 〈 glyph[lscript] 〉 2 (see, e.g., Ciotti & Ostriker 2001), is in the limit glyph[lscript] s . s . glyph[lessmuch] glyph[lscript] max given by \nf duty ∼ 2 ln( glyph[lscript] max /glyph[lscript] s . s . ) ∼ 2 〈 glyph[lscript] 〉 glyph[lscript] max . (45) \nThe weak logarithmic dependence on the ratio of the maximum to the minimum luminosity is deceptive; while we generally expect a duty cycle in the range f duty ∼ 0 . 2 -1, the average accretion rate depends on the peak rate. We can attempt to generalize equation (10) to model the peak rate by replacing T s with T HII (to mimic an initial absence of heating at the sonic radius) and by including the density enhancement factor f epi to obtain \nglyph[lscript] max ∼ 0 . 03 glyph[epsilon1] -1 f turb f epi , 1 Υ M 2 n 5 T HI , 3 . 7 T 5 / 2 HII , 4 . 7 , (46) \nwhich would imply a duty cycle of f duty ∼ 0 . 6 (for f epi ∼ 10) and an average accretion rate that is a factor of ∼ 9 times higher than the steady state rate. \nThe duty cycle estimated here differs from the one derived by Ricotti et al. (2008), f duty , ROM = ( r B / r ion) 1 / 3 , where for a stationary black hole r B ≡ GM BH / c 2 s , HI ∼ 2 . 4 × 10 16 M 2 T -1 HI , 3 . 7 cm is the Bondi radius ignoring radiative feedback (Bondi & Hoyle 1944). This estimate assumes that the accretion is efficient only when r ion ≤ r B. In our picture, r B does not have a unique physical meaning because the accretion flow structure is strongly modified by photoionization, radiative heating, and the radiation pressure; episodic accretion through the H II region can proceed even when r ion r B. \nThe peak accretion rate glyph[lscript] max and the average rate 〈 glyph[lscript] 〉 during episodic accretion can exceed the limits imposed by photoheating and radiation pressure because of the presence of the finite inertia of dense infalling shells and the lack of pressure equilibrium in the gas photoionized from the inner edge of the H II region during luminosity maxima. We are not able to determine with certainty whether the accretion from a weakly turbulent quasiuniform density medium will be episodic, in which case it will proceed at the rate given by equations (44) and (46), or whether it will be steady at the rate given in equation (10) subject to radiation pressure-suppression derived in § 2.5. We proceed to discuss the implications of inhomogeneity and turbulence in the environment of a seed black hole. \nglyph[greatermuch]', '4.1. Accretion of Self-Shielding Clumps': 'As a protogalaxy grows, the baryonic inflow velocity into its center increases. Cosmological hydrodynamic simulations show that baryons accreting along the filaments of the cosmic web can remain cold until they reach the central region of the protogalaxy. Supersonic baryonic inflow, in protogalaxies with virial velocities glyph[greaterorsimilar] 10 km s -1 , drives turbulence in the galaxy. The resulting central turbulent Mach numbers measured in the simulations of ∼ 10 8 M glyph[circledot] cosmological halos are ∼ 3 (Wise et al. 2008; Greif et al. 2008). Supersonic turbulence is expected in view of the short cooling time t shock , cool ∼ 10 n -1 5 yr of the gas heated at the termination shocks of the cold inflows (for somewhat higher velocity, 50 km s -1 shocks, e.g., Shapiro & Kang 1987; Kang & Shapiro 1992, see also Gnat & Sternberg 2008 for dependence on metallicity, which is weak for Z glyph[lessorsimilar] 0 . 1 Z glyph[circledot] ). If the shocked gas cools on a dynamical time to temperatures ∼ 10 4 K, molecules form and the gas quickly cools further. Therefore, a fraction of the dense gas mass that is collecting at the center of the protogalaxy may reside at temperatures substantially below the T HI ∼ 5 , 000 K that we have somewhat arbitrarily taken as the temperature floor thus far. \nNear an accreting seed black hole, molecule formation could be enhanced even in the absence of dust. Hard Xrays emitted by the black hole could maintain a high electron fraction in the neutral gas surrounding the H II region, which would catalyze molecule formation. Molecules can also form if traces of metals and dust are present. The cooling to temperatures characteristic of the molecular phase leads to an increase of turbulent Mach numbers. In supersonic turbulence, the local density exhibits intermittent strong fluctuations around the mean. Simulations of isothermal turbulence have shown that the density probability density function is a normal distribution in ln ρ with mean 〈 ln ρ 〉 = -1 2 σ 2 and disper- \n2 = ln(1 + b 2 M 2 ), where M is the turbulent Mach number and b ≈ 0 . 26 for unmagnetized turbulence (e.g., Kritsuk et al. 2007, and references therein). For example, for M = 10, ∼ 1% of the volume contains densities in excess of the average density by a factor of 10 or greater. Dense clumps can enter the ionized sphere and remain self-shielded from photoionization and photodissociation. Since from equation (6) the critical radius for photoionization is proportional to n -2 / 3 , a clump with an overdensity of 10 can withstand photoionization to radii 0 . 2 r ion. \nThe presence of neutral clumps in the ionized sphere may fundamentally alter the structure of the accretion flow if they are dense enough to survive photoionization and photoevaporation, which requires densities violating the condition in equation (9). Such dense clumps may form as a product of turbulent fragmentation and may be self-gravitating and on their way to turn into star-forming cores, provided that they are not tidally disrupted by and accreted onto the black hole (e.g., Bonnell & Rice 2008). During accretion minima when glyph[lscript] glyph[lessmuch] glyph[lscript] max, dense clumps may withstand photoionization and photoevaporation even at r glyph[lessorsimilar] r s. In this case, they could accrete directly into the small optically thick accretion disk around the black hole. If the clumps occupy a substantial solid angle as seen from the black hole, the gas in their shadows recombines, cools, and accelerates toward the black hole. \n∼ \nWe thus speculate that with the increase of turbulent Mach numbers and the progression of ever higher degree of clumping in the gas, the severity of the radiative feedback discussed in § 2 and § 3 decreases. The feedback-limited accretion may proceed at only a small fraction of the rate corresponding to the Eddington limit until turbulent inhomogeneities reach the level at which the densest clumps are self-shielded at ∼ r s even at glyph[lscript] max ∼ 1. Then, unless the gas supply is depleted by star formation and supernovae, the seed black hole may be able to grow efficiently and double its mass on the Salpeter time scale t Salp ∼ 5 × 10 7 glyph[epsilon1] -1 yr. Central stellar velocity dispersions found in massive black hole-hosting stellar systems in the local universe are > 30 km s -1 (e.g., Barth et al. 2005, and references therein); if massive black holes are not found in systems with smaller dispersions, this may be interpreted as a hint that efficient accretion commences only when the baryonic gravitational potential well depth around the black hole exceeds a critical minimum value.', '4.2. Photodissociation Sphere around the Black Hole': 'The existence of dense clumps in the neighborhood of the H II region may be contingent on the presence of molecules. The radiation from the black hole likely contains a photodissociating component below the Lyman edge. However, for a central source with a hard spectrum, two photon emission from the 2 1 S → 1 1 S transition in the recombination of He II is a guaranteed source of H2-dissociating photons. Photons in the Lyman-Werner (LW) bands of hydrogen are produced at the rate ˙ N LW ∼ 1 3 f ion , He L / E He (Johnson & Bromm 2007), where f ion , He is the fraction of photon energy that goes into helium ionization in mostly neutral gas, and E He = 24 . 6 eV. Some LW radiation may also be produced by the central source itself (Kuhlen & Madau 2005). If the gas surrounding the black hole is dust-free and molecule synthesis is catalyzed primarily by H -, one can estimate the dissociation depth ∆ r diss in the neutral gas via \n4 π 3 ( r 3 diss -r 3 ion ) k H2 n H -n H = ˙ N LW , (47) \nwhere r diss ≡ r ion + ∆ r diss, k H2 is the rate for the reaction H -+ H → H2 + e -, and n H -is the equilibrium abundance of the H -ions. Adopting the temperature T = 1 , 000 K we have k H2 ≈ 1 . 2 × 10 -9 cm 3 s -1 and n H -∼ 7 × 10 -7 ne (see, e.g., Oh & Haiman 2002, and references therein). In writing equation (47), we have ignored the potential enhancement of H2 formation rate in a supersonically turbulent gas, where molecule formation is particularly efficient in intermittent overdensities (see, e.g., Pavlovski et al. 2002; Glover & Mac Low 2007). \nThe maximum ionization fraction χ e ≡ ne / n H allowed if H2 is to be dissociated in a sphere of radius 2 r ion is obtained by substituting ∆ r diss = r ion in equation (47) and solving for χ e , to obtain the condition \nχ e < 2 . 4 × 10 -4 f 2 turb T 2 HI , 3 . 7 f ion T 3 HII , 4 . 7 , (48) \nwhere we have set f ion , He = 0 . 06. If the SED of the black hole extends into the hard X-rays, an electron fraction violating the condition in equation (48) may be maintained in the neutral shell surrounding the H II region. This suggests that, even in an environment with primordial composition, the H II region associated with a seed black hole may be surrounded by only a thin ( ∆ r diss glyph[lessmuch] r ion) photodissociation region where gas cooling and clumping is reduced (see also Johnson et al. 2007). Photodissociation may be even less important if the protogalaxy has been enriched with trace quantities of metals and dust (e.g., Omukai et al. 2008), as gas cooling can proceed even in the absence of molecular hydrogen in that case. Also, if the accretion is episodic (§ 3), molecular gas can form during the quiescent periods in which central accretion is temporarily diminished or suspended by the radiative feedback. \nWe have ignored the photodissociation of H -, which reduces the H2 formation rate by a factor S -1 , where S = 1 + ˙ N γ , H -σ H -/ 4 π r 2 k H2 n H, ˙ N γ , H -is the total dissociating photon number luminosity, and σ H -is the average photodissociation cross section (see, e.g., Glover et al. 2006; Glover 2007; Yoshida et al. 2007; Chuzhoy et al. 2007, and references therein). In our estimate of ˙ N γ , H -, we will ignore the multiplying effect of Ly α trapping (§ 2.7) on the H --photodissociating photon number density; a more accurate approach, which would be too involved to include here, would be to solve for Ly α resonance line and X-ray continuum transfer in the H -photodissociation region self-consistently. Chuzhoy et al. (2007) estimate that for reprocessed ionizing radiation, the average cross section per recombination photon times the average number of photons per recombination varies in the range 〈 σ H -〉 ∼ (1 . 6 -3 . 4) × 10 -17 cm 2 . Setting ˙ N γ , H -∼ ˙ N rec, where ˙ N rec ∼ (4 π/ 3) α B ( T HII) n H + ne is the hydrogen recombination rate inside the H II region, we find \nS ∼ 1 + 0 . 43 glyph[lscript] 1 / 3 f 1 / 3 ion f 4 / 3 turb M 1 / 3 2 n 1 / 3 5 T 4 / 3 HI , 3 . 7 T 2 HII , 4 . 7 ( r r ion ) -2 , (49) \nwhich shows that H2 formation suppression is marginally important in the case of an accretion from a high density cloud. The suppression widens the photodissociation region surrounding the H II region. \nHaving raised the possibility of a multiplying effect of Ly α trapping on the photodissociation of H -, we remark that the destruction of Ly α photons by H -, which is a processes that have ignored in § 2.7, may have to be considered in a selfconsistent study of Ly α transfer in a gas with primordial composition. Because of significant uncertainties in the ionization \nfraction and the H -abundance in the neutral shell, which both depend on the SED are not calculated accurately in our simplified model, we attempt only a crude estimate. According to equation (7) of Hansen & Oh (2006), Ly α photons scattering through a shell of column depth N are destroyed before escaping if \nN > 6 . 7 × 10 19 T 1 / 4 HI , 3 . 7 χ 3 / 4 H -( σ Ly α , H -10 -21 cm 2 ) -3 / 4 cm -2 , (50) \nwhere σ Ly α , H -∼ 4 . 4 × 10 -18 cm -2 is the H -photodissociation cross section by Ly α , and χ H -is the H -abundance. Setting N ∼ n H ∆ r we find that Ly α photons are destroyed before escaping the shell of width ∆ r when the ionization fraction in the shell is \nχ e > 0 . 0024 S f 8 / 9 turb T 11 / 9 HI , 3 . 7 glyph[lscript] 4 / 9 f 4 / 9 ion n 4 / 9 5 M 4 / 9 2 T 4 / 3 HII , 4 . 7 ( ∆ r r ion ) -4 / 3 . (51) \nThis crude estimate suggests that for accretion in a dense, dust-free medium with a central X-ray source that maintains a high ionization fraction in the neutral gas, Ly α photons may be destroyed by H -absorption before penetrating the shell. The issue warrants a self-consistent treatment of resonance line scattering and continuum radiation transfer coupled to the chemistry of the H II region and the surrounding neutral shell.', '5. CONCLUSIONS': "We model quasiradial accretion onto seed massive black holes in metal and dust poor protogalaxies with the goal of evaluating the common assumption that the M BH glyph[greaterorsimilar] 100 M glyph[circledot] seed black holes accrete at the Bondi accretion rate moderated by the Eddington limit. After considering radiative feedback effects in the neighborhood of an accreting black hole, we are able to derive the following conclusions: \nThe local thermal and statistical equilibrium temperature of a photoionized gas is a strong function of the metallicity of the gas at radii from the black hole where gas becomes captured by the black hole. The photoionized gas outside the sonic radius is normally in equilibrium, but, particularly for metallicities Z glyph[lessorsimilar] 0 . 1 Z glyph[circledot] , the gas passing through the sonic radius may not be in equilibrium. For radiative efficiencies glyph[epsilon1] glyph[lessorsimilar] 0 . 1, the gas will not complete thermal runaway that would heat it to the Compton temperature prior to crossing the sonic radius \nDue to the radiative heating of the gas, the sonic radius is much smaller than the Bondi radius evaluated in terms of the temperature of the protogalactic medium far from the black hole; the corresponding gasdynamical accretion rate is thus reduced to a small fraction of the Bondi value. The gas temperature at the sonic radius is also a strong function of the radiative efficiency of the accretion near the event horizon of the black hole. \nThe black hole is surrounded by an H II region, which is further surrounded by a photodissociation region if the ambient protogalactic medium contains molecules. The radius of the H II region in dense ( n glyph[greaterorsimilar] 10 5 cm -3 ) clouds, that must be present to support near-Eddington accretion, is small ( r ion ∼ 1 pc) compared to the size of the protogalaxy. The sonic radius is smaller by orders of magnitude than the radius \nof the H II region. This vast separation of scales renders selfconsistent simulations of accretion onto seed black holes from a protogalactic environment particularly challenging, because to appropriately capture the radiative effects, the simulations must at least resolve the sonic radius. \nPhotoionization radiation pressure from the continuum radiation produced near the event horizon may further diminish the accretion rate. Compared to the gravitational acceleration, the acceleration due to radiation pressure is the strongest in the bulk of the H II region, i.e., not too close to the black hole where the equilibrium neutral fraction is negligible. We have derived a simple criterion for accretion rate suppression by the photoionization radiation pressure. \nA significant fraction of the luminosity of the black hole is converted into Ly α resonance line radiation; resonance line scattering keeps this radiation trapped in the neighborhood of the H II region. Radiation pressure from the trapped radiation can exceed the external, confining gas pressure. We do not self-consistently solve for a stationary accretion flow in the presence of resonance line radiation pressure, but our estimates do suggests that the resonance line pressure could be detrimental to stationary accretion at high accretion rates. \nAs an alternative to strictly stationary accretion, we outline a model for episodic accretion in which short episodes of sudden, rapid accretion alternate with longer periods of accretion reduced by photoheating and radiation pressure. In this model, the outflow and central rarefaction induced by photoheating and radiation pressure acceleration in the bulk of the HII region drives down the central accretion rate rapidly. The time-average accretion rate is then set by the peak accretion rate, divided by the logarithm of the number of e -foldings in the outflow-induced accretion rate decay. Since the peak accretion rate need not be limited by the thermodynamic and kinematic constraints that apply to steady-state solutions, the episodic accretion rate may exceed the steady-state rate by a large factor. \nOur idealized treatment here already indicates how extremely complex the self-consistent accretion problem is. To more fully explore the time-dependent accretion flow onto a seed black hole, numerical simulations are clearly needed. Of particular importance is to study the effect of angular momentum, the impact of turbulence, and the possible emergence of a multi-phase medium in the infalling gas. We will report on the results from such simulations and on a comparison with the analysis presented in this work elsewhere (Milosavljevi'c et al. 2008, Couch et al., in preparation). If our result that the initial accretion onto stellar seed black holes is greatly reduced in the presence of radiative feedback holds up even in fully three dimensional radiation-hydrodynamical simulations, the need for massive black hole formation by radiatively-inefficient accretion or by direct collapse of massive primordial gas clouds might be significantly increased (e.g., Bromm & Loeb 2003; Begelman et al. 2006, 2008; Djorgovski et al. 2008). \nWe thank an anonymous referee for detailed and very helpful comments and suggestions. V. B. and M. M. acknowledge support from NSF grant AST-0708795. S. P. O. acknowledges support from NASA grant NNG06GH95G.", 'ACCRETION ONTO SEED BLACK HOLES': "Beckwith, K., Hawley, J. F., & Krolik, J. H. 2006, preprint (astroph/0605295) \nBeckwith, K., Hawley, J. 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2004MNRAS.353.1035M
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The anti-hierarchical growth of supermassive black holes
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2004-01-01
| 7 | 0.47 | 160 |
['accretion', 'accretion disks', 'black hole physics', 'galaxies active', 'galaxies evolution', 'galaxies quasars', 'astrophysics']
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[]
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I present a new method to unveil the history of cosmic accretion and the build-up of supermassive black holes (SMBHs) in the nuclei of galaxies, based on observations of the evolving radio and (hard) X-ray luminosity functions of active galactic nuclei (AGN). The fundamental plane of black hole activity discovered by Merloni, Heinz & Di Matteo, which defines a universal correlation among black hole mass (M), 2-10keV X-ray luminosity and 5-GHz radio luminosity is used as a mass and accretion rate estimator, provided a specific functional form for the dependency of the X-ray luminosity on the dimensionless accretion rate mdot is assumed. I adopt the local black hole mass function (BHMF) as derived from the velocity dispersion (σ) distributions of nearby galaxies coupled with the M-σ relation as a boundary condition to integrate backwards in time the continuity equation for the evolution of SMBH, neglecting the role of mergers in shaping up the BHMF. Under the most general assumption that, independently on M, black hole accretion proceeds in a radiatively efficient way above a certain rate, and in a radiatively inefficient way below it, the redshift evolution of the BHMF and the black hole accretion rate (BHAR) function (i.e. the distribution of the Eddington scaled accretion rates for objects of any given mass) are calculated self-consistently. The only tunable parameters are the overall efficiency of extracting gravitational energy from the accreting gas, ɛ, and the critical ratio of the X-ray to Eddington luminosity, L<SUB>2-10keV,cr</SUB>/L<SUB>Edd</SUB>≡x<SUB>cr</SUB>, at which the transition between accretion modes takes place. For fiducial values of these parameters (ɛ= 0.1 and x<SUB>cr</SUB>= 10<SUP>-3</SUP>), I found that half (~85 per cent) of the local black hole mass density was accumulated at redshift z < 1 (z < 3), mostly in radiatively efficient episodes of accretion. The evolution of the BHMF between z= 0 and z~ 3 shows clear signs of an anti-hierarchical behaviour: while the majority of the most massive objects (M>~ 10<SUP>9</SUP>) were already in place at z~ 3, lower mass ones mainly grew at progressively lower redshift, so that the average black hole mass increases with increasing redshift. In addition, the average accretion rate decreases towards lower redshift. Consequently, sources in the radiatively inefficient regime of accretion only begin to dominate the comoving accretion energy density in the Universe at z < 1 (with the exact value of z depending on x<SUB>cr</SUB>), while at the peak of the BHAR history, radiatively efficient accretion dominates by almost an order of magnitude. I will discuss the implications of these results for the efficiency of accretion on to SMBH, the lifetimes of quasars and duty cycles, the history of AGN feedback in the form of mechanical energy output and, more generally, for the cosmological models of structure formation in the Universe.
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https://arxiv.org/pdf/astro-ph/0402495.pdf
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{'Andrea Merloni': 'Max-Planck-Institut fur Astrophysik, Karl-Schwarzschild-Strasse 1, D-85741, Garching, Germany', 'ABSTRACT': 'I present a new method to unveil the history of cosmic accretion and the build-up of supermassive black holes in the nuclei of galaxies, based on observations of the evolving radio and (hard) X-ray luminosity functions of active galactic nuclei. The fundamental plane of black hole activity discovered by Merloni, Heinz & Di Matteo (2003), which defines a universal correlation among black hole mass ( M ), 2-10 keV X-ray luminosity and 5 GHz radio luminosity is used as a mass and accretion rate estimator, provided a specific functional form for the dependency of the X-ray luminosity on the dimensionless accretion rate ˙ m is assumed. I adopt the local black hole mass function as derived from the velocity dispersion ( σ ) distributions of nearby galaxies coupled with the M -σ relation as a boundary condition to integrate backwards in time the continuity equation for the supermassive black holes evolution, neglecting the role of mergers in shaping up the black hole mass function. Under the most general assumption that, independently on M , black hole accretion proceeds in a radiatively efficient way above a certain rate, and in a radiatively inefficient way below, the redshift evolution of the black hole mass function and the black hole accretion rate function (i.e. the distribution of the Eddington scaled accretion rates for objects of any given mass) are calculated selfconsistently. The only tunable parameters are the overall efficiency of extracting gravitational energy from the accreting gas, /epsilon1 , and the critical ratio of the X-ray to Eddington luminosity, L 2 -10keV , cr /L Edd ≡ x cr , at which the transition between accretion modes takes place. For fiducial values of these parameters ( /epsilon1 = 0 . 1 and x cr = 10 -3 ), I found that half ( ∼ 85 %) of the local black hole mass density was accumulated at redshift z < 1 ( z < 3 ), mostly in radiatively efficient episodes of accretion. The evolution of the black hole mass function between z = 0 and z ∼ 3 shows clear signs of an anti-hierarchical behaviour: while the majority of the most massive objects ( M > ∼ 10 9 ) were already in place at z ∼ 3 , lower mass ones mainly grew at progressively lower redshift, so that the average black hole mass increases with increasing redshift. Also, the average accretion rate decreases towards lower redshift. Consequently, sources in the radiatively inefficient regime of accretion only begin to dominate the comoving accretion energy density in the universe at z < 1 (with the exact value of z depending on x cr ), while at the peak of the black hole accretion rate history, radiatively efficient accretion dominates by almost an order of magnitude. I will discuss the implications of these results for the efficiency of accretion onto SMBH, the quasars lifetimes and duty cycles, the history of AGN feedback in the form of mechanical energy output and, more generally, for the cosmological models of structure formation in the universe. \nKey words: accretion, accretion disks - black hole physics - galaxies: active - galaxies: evolution - quasars: general', '1 INTRODUCTION': "In the last decade, supermassive black holes (hereafter SMBH) in the nuclei of galaxies have been discovered at ever increasing pace, and are now believed to reside in most of (perhaps all) the bulges of present-day galaxies (Kormendy & Richstone 1995; Richstone et al. 1998). Moreover, there is evidence of a correlation between the mass of the central black holes and either the mass and luminosity (Kormendy & Richstone 1995; Magorrian et al. 1998; McLure & Dunlop 2002; Marconi & Hunt 2003) or the velocity dispersion (Ferrarese & Merritt 2000; Gebhardt et al. 2000; Merritt & \nFerrarese 2001; Tremaine et al. 2002) of its host bulges. This has led to the recognition that the formation and growth of SMBH and of their host galaxies are related processes, and that understanding their evolutionary history can provide fundamental insight into the theories of structure formation in the universe and into the physical nature of AGN feedback (Silk & Rees 1998; Fabian 1999; Kauffmann & Haehnelt 2000; Umemura 2001; Granato et al. 2001; Cavaliere & Vittorini 2002; Wyithe & Loeb 2003; Granato et al. 2004). \nWithin this framework, the most important question that needs to be answered is when and how the mass currently locked up in \nSMBHwas assembled. In practice, this corresponds to asking what is, at any given redshift z , the number of black holes per unit comoving volume per unit (base 10) logarithm of mass, i.e. the black hole mass function φ M ( M,z ) (hereafter BHMF). In the simplest case of purely accretion driven evolution (i.e. assuming mergers do not play an important role in shaping the BHMF), the simultaneous knowledge of a black hole accretion rate distribution function φ ˙ m ( M,z ) , would completely determine the evolutionary solution, as the two distributions must be coupled via a continuity equation (Small & Blandford 1992; Marconi et al. 2004, hereafter M04), to be solved given the appropriate boundary conditions. Moreover, if black holes mainly grew by accretion, a direct link should exist between the quasar (QSO) and active galactic nuclei (AGN) phenomena, signatures of actively accreting phases, and nearby SMBH tracing directly the entire accumulated mass. In fact, previous comparisons between direct estimates of the local black hole mass density, ρ BH , 0 ≡ ρ BH ( z = 0) ≡ ∫ ∞ 0 Mφ M ( z = 0) dM , and the total energy density liberated by powerful quasars over the cosmic history, as originally proposed by Soltan (1982), seem to consistently suggest that SMBH mainly grew while they were active (Salucci et al. 1999; Fabian & Iwasawa 1999; Yu & Tremaine 2002, hereafter YT02; Elvis, Risaliti & Zamorani 2003; Cowie et al. 2003; M04). \nFrom the theoretical point of view, several attempts have been made to link the evolution of the SMBH population to analytic and semi-analytic models of structure formation (Efstathiou & Rees 1988; Haenelt & Rees 1993; Haenelt, Natarayan & Rees 1998; Haiman & Loeb 1998; Cattaneo, Haenelt & Rees 1999; Kauffmann & Haenelt 2000; Monaco, Salucci & Danese 2000; Cavaliere & Vittorini 2000; Volonteri, Haardt & Madau 2003; Wyithe & Loeb 2003; Hatziminaoglou et al. 2003; Bromley, Somerville & Fabian 2004; Mahamood, Devriendt & Silk 2004). A somewhat different approach has been followed recently by Di Matteo et al. (2003), who made use of large cosmological hydrodynamical simulations to predict the evolutionary history of supermassive black holes growth and activity. \nThe path from cosmological CDM structure formation models (and/or simulations) to the predicted AGN evolving luminosity functions, however, is dotted with uncertainties regarding a number of important physical processes. In general, semi-analytic modelers need to make specific assumptions (and introduce parameters) to describe gas cooling in dark matter halos, the fueling of the central nuclear black holes, the physics of the merger events and the nature of the various feedback mechanisms (from star formation and/or from AGN). In order to constrain some of these parameters, all the above models, then, need to be tested against observationally determined signatures of SMBH growth history. The usual test-benches are: the evolution of the AGN luminosity functions (in any specific band), the local black holes mass function, as derived from the M -σ relation, the slope and intercept of the M -σ relation itself and the spectrum and intensity of the X-ray background light, known to be produced by the sum of individual AGN. \nHere I would like to propose an alternative approach which is capable to provide a self-consistent evolutionary picture for supermassive black holes. Such an approach is largely based on observed data and only on a minimal number of theoretical assumptions (and parameters). These assumptions are only needed to describe the physics of the innermost accretion process, through which black holes shine, and not the more complex interplay between growing SMBHand their galactic environment. This is an advantage for two reasons: first of all, the physics of black hole accretion is reasonably well understood, both theoretically and phenomenologically; second, the physical properties of the innermost part of an accre- \ntion flow, where the dynamics is almost completely dominated by the strong gravitational field of the central black holes, should not depend on cosmology and redshift. \nThe approach followed here is similar in spirit to those of Small & Blandford (1992); Salucci et al. (1999); YT02; M04. More specifically, I will make use of the local black hole mass function and of the evolution of AGN luminosity functions to constrain the evolutionary history of SMBH. The main novelty of the present work is the realization that simultaneous radio and (hard) X-ray observations of accreting black holes can provide tight constraints on both the mass and the accretion rate of an active black hole, through the so-called 'fundamental plane' relationship for active black holes (Merloni, Heinz & Di Matteo 2003, hereafter MHD03). From this, the history of SMBH growth can be followed in detail up to the redshift at which reliable X-ray and radio luminosity functions can be obtained. \nThe structure of the paper is the following: in section 2 I will describe how the knowledge of the X-ray and radio luminosity functions of local AGN, and of the local SMBH mass function can be used to obtain a complete census of the local SMBH population and activity distribution in the form of an accretion rate distribution function. A necessary ingredient to perform this calculation is the functional form that relates the observed X-ray luminosity of an accreting black hole to its mass and accretion rate. In section 3 I will describe how it is possible to use our theoretical and phenomenological knowledge of the different modes of accretion onto a black hole to obtain such a relation. In section 4, the method to calculate the redshift evolution of both black hole mass and accretion rate functions is described, and its basic assumptions clearly spelled out. Section 5 is then devoted to the analysis of the most important results of the calculations and of the main properties of the evolving SMBH population between z = 0 and z ∼ 3 . A more general discussion of the implication of these results is presented in section 6. Finally, I draw my conclusions in section 7. \nThroughout this paper, we adopt a background cosmological model in accordance with the Wilkinson Microwave Anisotropy Probe ( WMAP ) experiment. The model has zero spatial curvature, a cosmological constant, Ω Λ = 0 . 71 a cosmological constant H 0 = 70 km s -1 , dominated by cold dark matter with Ω m = 0 . 29 and Ω b = 0 . 047 (Spergel et al. 2003).", '2 THE IMPORTANCE OF ESTIMATING THE CONDITIONAL RADIO/X-RAY AGN LUMINOSITY FUNCTION': "Large optical surveys carried out in recent years (Boyle et al. 2000; Fan et al. 2001; Wolf et al. 2003) probe the evolution of the QSO luminosity function up to high redshift, and all agree in establishing a strong rise in their activity from the local universe up to redshift z ∼ 2 and a decline above z ∼ 3 . However, inferring the properties of the entire class of accreting supermassive black holes from samples selected in a single waveband can be misleading, in particular in the case of the optical band, where obscuration effects can be significant. Indeed, Barger et al. (2003) have clearly shown that optically selected broad-line AGN and QSO form only about one third of the X-ray background: hard X-ray selected samples, therefore, provide a more direct probe of SMBH activity (see also Cattaneo & Bernardi 2003). Recent works by Hosokawa (2004) and M04 have also demonstrated that the redshift evolution of the hard X-ray luminosity function better describes the growth history of accreting supermassive black holes. Discrepancies with the results obtained \nfrom optically selected QSO luminosity functions for the average accretion efficiency and the local black hole mass density can be understood by taking into account the luminosity dependence of both obscuration and bolometric corrections in the different bands (see e.g. Ueda et al. 2003; Cattaneo & Bernardi 2003; Hosokawa 2004; M04). \nSimilarly, hard X-ray emission reveals fundamental properties of an accreting black hole: In a recent paper (MHD03) it has been shown that if we define the instantaneous state of activity of a black hole of mass M (in units of solar masses), by the radio (at 5 GHz, L R ) and hard (in the 2-10 keV band, L X ) X-ray luminosity of its compact core, and represent such an object as a point in the threedimensional space ( log L R , log L X , log M ), all black holes (either of stellar mass or supermassive) will lie preferentially on a plane (the 'fundamental plane' of black hole activity), described by the following equation: \nlog L R = (0 . 60 +0 . 11 -0 . 11 ) log L X +(0 . 78 +0 . 11 -0 . 09 ) log M +7 . 33 +4 . 05 -4 . 07 . (1) \nEquation (1) can be inverted to relate BH masses to observed nuclear radio and X-ray luminosities: \nlog M /similarequal 1 . 28(log L R -0 . 60 log L X ) -9 . 34 ± 1 . 06 (2) ≡ g (log L R , log L X ) . \nThis is an entirely empirical relation, and as such is independent on any accretion (or jet) physical model. It shows, however, that disc and jet emission from active black holes of any mass, from galactic X-ray binary sources to the most powerful quasars, are physically and observationally correlated phenomena. Moreover, as the fundamental plane relationship is obeyed by the intrinsic hard Xray luminosities, it is basically unaffected by absorption, and therefore largely independent on the validity of any specific unification scheme for AGN. \nOne of the consequences of this relationship is that, in an ideal case, the conditional radio/X-ray luminosity function of active black holes, i.e. the number of sources per unit co-moving volume per unit logarithm of radio and X-ray luminosity, Ψ C ( L R , L X ) , could be used to reconstruct the mass function of the underlying black hole population. The importance of such a possibility should not be underestimated. For example, it could provide an alternative way to study the demography of the SMBH population (at any redshift) to be compared with what obtained from the M -σ relation (which the Sloan Digital Sky Survey will largely contribute to, see e.g. McLure & Dunlop 2004), or with any other analysis based on different mass estimators (see e.g. Vestergaard 2004). In alternative, such comparisons can be used to test the redshift evolution of the fundamental plane relationship itself. \nAlthough the future goal for solving the problem at hand should therefore be identified with the study of large multiwavelength (X-ray and radio in particular) SMBH samples, nevertheless I will argue here that it is still possible to make some progress with the currently available pieces of information. In fact, the lack of the exact knowledge of the conditional radio/X-ray AGN luminosity function can be (at least partially) superseded, given the two separate radio, φ R ( L R , z ) , and X-ray, φ X ( L X , z ) , luminosity functions at redshift z , and an independent estimate of the black hole mass function, φ M ( M,z ) at the same redshift. By taking into account the fundamental plane relationship, we have that the conditional luminosity function Ψ C has to satisfy the following integral constraints: \nφ X ( L X ) d log L X = ∫ ∞ L R , min Ψ C ( L X , L R ) d log L R (3) \nφ R ( L R ) d log L R = ∫ ∞ L X , min Ψ C ( L X , L R ) d log L X (4) \nφ M ( M ) d log M = ∫∫ log M<g< log M + d log M Ψ C ( L X , L R ) d log L R d log L X , (5) \nwhere g ( L R , L X ) is defined in equation (2). In the above formulae, the lower end of the X-ray and radio luminosity functions, L X , min and L R , min , should be chosen in such a way as to give the same total number of objects with mass larger than a certain minimal value M min , as computed by integrating the BHMF. \nIn the following, I will make the assumption that any function Ψ C ( L X , L R ) that satisfies equations (3), (4) and (5) can be regarded as the true conditional radio/X-ray luminosity function. To begin with, I will show in the next section how to use the above formalism to derive informations about the state of activity of the local SMBH population, in the specific form of its accretion rate distribution function.", '2.1 The census of the local population': 'Using the integral constraints provided by eqs. (3-5), it is possible to deduce a conditional radio/X-ray AGN luminosity function at redshift zero. In order to do that, a specific choice of the observed luminosity (and mass) functions needs to be done. I will adopt the following: \n- · For the hard X-ray luminosity function (HXLF) I use the recently estimated one of Ueda et al. (2003). This is arguably the most complete luminosity function in the 2-10 keV spectral band, spanning the luminosity range of 10 41 . 5 -10 46 . 5 erg s -1 , corrected for absorption.\n- · The 5GHz radio luminosity function (RLF) of AGN is here derived from the lower frequency one of Willott et al. (2001), obtained from the 3CRR, 6CE and 7CRS complete samples, assuming for the radio spectral index a constant value α R = 0 . 7 to rescale the luminosities. Although other determinations of the local AGN RLFat higher frequencies are available (see e.g. Sadler et al. 2002), the Willott et al. one is probably the most accurate to date up to relatively high redshifts, and this will be instrumental in studying the growth history of the SMBH population (see section 4).\n- · The local black hole mass function is instead estimated following Aller and Richstone (2002). They derived it from the local luminosity functions of galaxies of different morphologies given in Marzke et al. (1994), together with the empirical relationships among total luminosity, bulge luminosity and velocity dispersion. However, differently from Aller & Richstone (2002), we do take into account the effects of the scatter in the M -σ relation 1 , in the same way as described in Yu & Lu (2004). Although the overall effect of considering such a scatter affects the value of the total black hole mass density only marginally (a dispersion of 0.27 dex results in an increase of the local black hole mass density of a factor of 1.2, see YT02), nevertheless, the shape of the local black hole mass function is strongly affected, as demonstrated by M04. \nStarting with an initial guess for Ψ C , we proceed via successive iterations, minimizing the differences between the projections \n1 I will assume throughout the paper that the black hole mass-velocity dispersion relation is given by log M BH = 8 . 18 +4 . 02 log( σ/ 200km s -1 ) , as discussed in Tremaine et al. (2002). \nof the conditional luminosity function onto the X-ray and radio luminosity axes and the observed luminosity functions, until we obtain a conditional LF that simultaneously satisfies the integral constraints given by eq. (3), (4) and (5). \nOnce such an estimate of the conditional luminosity function is found, it is possible to derive the local distribution of the second fundamental physical parameter that characterizes any active black hole: its accretion rate in units of Eddington luminosity, \n˙ m ≡ /epsilon1 ˙ Mc 2 /L Edd (6) \n( /epsilon1 is the accretion efficiency, see section 3.1). Such an inversion, however, is model dependent, as it depends on the choice of which spectral energy distribution should correspond to any specific couple of fundamental parameters M and ˙ m . In practice, it corresponds to the choice of the accretion mode of a SMBH of given mass and 2-10 keV luminosity. This can be done by choosing a specific functional form L X = L X ( M, ˙ m ) , as I will describe in more detail in the next section. \nFor the moment, suffice it to say that the main results concerning the local SMBH population are summarized by the black solid lines in Figures 4 and 6, showing the z = 0 . 1 black hole mass and accretion rate functions. The population of local black holes is dominated by sources shining, in the X-ray band, below 10 -3 of the Eddington luminosity (assuming a 10% efficiency, see below). This is in agreement with the average value found using the X-ray luminosity function of Seyfert 1 galaxies by Page (2001). A more detailed view of the local accreting SMBH population can be obtained by studying the mean dimensionless accretion rate as a function of black hole mass, defined as: \n〈 ˙ m ( M ) 〉 = ∫ ∞ M ˙ m ( M ) φ M ( M ) dM ∞ M φ M ( M ) dM . (7) \nIn figure 1 such a quantity is plotted versus the SMBH mass at z = 0 . 1 . It is clear that the mean accretion rate is a strong function of black hole mass, with small black holes accreting at a higher rate. Similarly, if we define the instantaneous growth rate of a SMBH of mass M as M/ 〈 ˙ M ( M ) 〉 , we see in figure 2 that the growth time is very large (about one order of magnitude larger than the Hubble time) for the more massive holes, a result confirmed by the SDSS optical study of 23,000 local AGN (Heckman et al. 2004). This implies that the very high mass SMBH must have formed at significantly higher redshift, as we will see in detail in section 5, where the redshift evolution of the SMBH population will be calculated. \n∫ \nBefore entering into a detailed description of the results on the redshift evolution of the SMBH population obtained using the local distributions as a boundary condition, I will discuss in the next section the main theoretical assumptions underlying such a calculation.', '3.1 Accretion efficiency': 'The accretion efficiency /epsilon1 , appearing in equation (6) represents the efficiency with which gravitational energy of the matter infalling onto the black hole can be extracted, regardless of it being transformed into radiation or not. It is therefore an upper limit to the radiative efficiency, /epsilon1 rad , and a function of the inner boundary condition of the accretion flow only. The closest the innermost stable circular orbit (ISCO) of the accreting gas is to the event horizon, the higher the accretion efficiency. In the classical general rel- \nFigure 1. The mean accretion rate (in units of Eddington) 〈 ˙ m ( M ) 〉 as a function of black hole mass for local ( z = 0 . 1 ) accreting black holes. \n<!-- image --> \nFigure 2. The instantaneous growth rate of local supermassive black holes, defined as M/ 〈 ˙ M ( M ) 〉 is plotted as a function of black hole mass M . \n<!-- image --> \nativistic case of test particles, the position of the ISCO depends on the dimensionless angular momentum of the black holes, and the corresponding efficiency varies between /epsilon1 /similarequal 0 . 06 for nonrotating holes and /epsilon1 /similarequal 0 . 42 for maximally rotating Kerr black holes (Novikov & Thorne 1973). Recently, both YT02, comparing the local black hole mass function with the black hole mass density accreted during luminous QSO phases, and Elvis, Risaliti & Zamorani (2002), comparing the X-ray background intensity with the local SMBH mass density, came to the conclusion that most supermassive black holes must rapidly spinning, i.e. they must have accreted at an average radiative efficiency higher than the canonical value of 10%, so that /epsilon1 > ∼ /epsilon1 rad > ∼ 0 . 1 . Such a conclusion, however, have been questioned by M04 on the basis of a revised estimate of the local SMBH mass function and of the hard X-ray luminosity function of AGN. \nIn what follows, I will assume that the accretion efficiency is a constant, regardless of the nature of the specific accretion mode of each SMBH. The radiative efficiency, instead, will be a function of the accretion rate, expressed though the dependence of the observ- \nable 2-10 keV luminosity on ˙ m and of the bolometric correction, as described below.', '3.2 Accretion mode transition and bolometric corrections': "The relevance of the different (theoretical) accretion modes for the various AGN populations is still a matter of open debate. Here I will follow the approach of MHD03 and try to maximize the amount of information on the issue by comparing active black holes of different masses. \nGalactic (stellar mass) black holes in X-ray binaries, either of transient or persistent nature, are commonly observed undergoing so-called transitions, i.e. dramatic changes in their spectral and variability properties (I will adopt here the terminology of McClintock & Remillard 2004, which the reader is referred to for a recent review). There are at least three well defined spectral states. In the low/hard state the emission is dominated by a hard X-ray power-law with an exponential cutoff at about few 100 keV. The spectrum of the thermal dominant (or high/soft) state, instead, is dominated by a thermal component likely originated in a standard Shakura & Sunyaev (1973) accretion disc, while in the steep powerlaw (or very high) state, usually associated with a source's highest flux level, both a thermal and a steep power-law component substantially contribute to the spectrum. Maccarone (2003) has shown that the hard-to-soft transition in these systems generally occurs at X-ray luminosities (in the 2-10 keV band) of about x cr /similarequal 5 × 10 -3 . This transition is also accompanied by a 'quenching' of the steady radio emission observed in the low/hard state. \nGiven the many similarities between the high-energy spectra of galactic black holes and AGN, a similar phenomenology has long been searched for in active supermassive black holes. The reader is referred to MHD03 for a thorough discussion on the scale-invariant properties of black holes coupled accretion/jet system, and on how to constrain theoretical accretion models for the different classes of objects on the basis of the observed fundamental plane correlation coefficients. There we showed that, for black holes of any mass characterized by x ≡ L X /L Edd < ∼ 0 . 01 , the fundamental plane relation is consistent with the most general theoretical relation between radio emission, mass and accretion rate expected from synchrotron emitting jets (regardless of their detailed geometrical and kinematical properties), provided that the X-ray emitting flow is radiatively inefficient (RIAF; for a recent review about RIAF, see Narayan 2002 or Quataert 2003). In this case, L X ∝ ˙ m 2 . 3 , and the radio luminosity satisfies: L R ∝ ˙ m 1 . 38 M 1 . 38 = ˙ M 1 . 38 , i.e., L R scales with the physical accretion rate only. \nOn the other hand, the most luminous sources, as those falling into the standard definition of Quasars and broad lined AGN, must be accreting at an higher rate, close to the Eddington one. Also their spectral energy distribution, usually dominated by the so-called Big Blue Bump (BBB: quasi thermal UV emission most likely from an optically thick standard accretion disc, see e.g. Malkan 1983; Laor 1990), indicates that above a certain critical X-ray to Eddington ratio x cr , an accretion mode transition should take place to what is usually described as a standard, geometrically thin and optically thick accretion disc (Shakura & Sunyaev 1973). More recently, Maccarone, Gallo & Fender (2003), analysing the same AGN sample of MHD03, have found evidence of a connection between the radio-quiet AGN and X-ray binaries in the thermal dominant state. Also the transition luminosity, x cr , has been found to be consistent with the hard-to-soft transition of galactic black holes. \nThe scaling of the the hard X-ray luminosity with the accretion \nrate in this high ˙ m regime, L X ∝ ˙ m q , is not straightforwardly predicted by the standard accretion disc theory, as the origin of the hard X-ray emission itself is not self-consistently predicted by the theory (but see, for example, Merloni 2003). On the other hand, observational studies of the spectral energy distributions of a large number of QSO and AGN and comparisons of X-ray and optical luminosity functions of AGN can be used to put constraints on the value of q (Ueda et al. 2003). \nIn light of these facts, I will adopt here the simplest possible functional form for the L X /L Edd vs. ˙ m function of a broken power-law, bridging the low accretion rate (radiatively inefficient) regime and the high accretion rate one. For radiatively efficient sources, we should assume, by definition, that the bolometric luminosity is simply proportional to the accretion rate, L bol ∝ ˙ m . Then, using the fitting formula for the X-ray to bolometric luminosity ratio of M04, I obtain x ∝ ˙ m 0 . 76 . Summarizing, the universal accretion mode function adopted here has the following scalings: \nx ∝ { ˙ m 2 . 3 x < ∼ x cr ˙ m 0 . 76 x > x cr (8) \nThe overall normalization is found by imposing continuity at x cr and that in the radiatively efficient regime log x/ (0 . 76 log ˙ m ) = -1 . 5 , such that AGN at the peak of the mass distribution have Xray to optical ratios consistent with observations (see e.g. Vignali, Brandt & Schneider 2003).", '4 THE REDSHIFT EVOLUTION OF THE BLACK HOLE MASS AND ACCRETION RATE FUNCTIONS': 'In section 2.1 I have briefly described the general method by which to derive the local supermassive black holes mass and accretion rate distribution functions given the luminosity functions of AGN in both radio and X-ray bands (down to a sufficiently low luminosity in each band in order to match the integrated number densities). From these, the redshift evolution of the SMBH population can be computed integrating backwards the continuity equation that describes SMBH evolution driven by accretion only (Small & Blandford 1992; Steed & Weinberg 2004; Hosokawa 2004): \n∂φ M ( M,t ) ∂t + ∂ [ φ M ( M,t ) · 〈 ˙ M ( M,t ) 〉 ] ∂M = 0 , (9) \nwhere the mean accretion rate as a function of black hole mass and time, 〈 ˙ M 〉 can be calculated directly from the accretion rate distribution function at time t . By setting the right hand side of equation (9) to zero, I have implicitly assumed that mergers are unimportant for the black holes growth history. A thorough examination of the possible consequences for SMBH growth of the inclusion of additional merger or direct formation terms in equation (9) can be found in YT02, Hosokawa (2004) or Menou & Haiman (2004). \nIn practice, starting from the BHMF and the accretion rate function at a given redshift z , it is possible to derive the new black hole mass function at redshift z + dz , φ M ( M,z + dz ) , by just subtracting the mass accreted in the time interval dt = dz ( dt/dz ) calculated according to the accretion rate function of redshift z . This new BHMF can then be used together with the radio and Xray luminosity functions at the same redshift, φ R ( L R , z + dz ) and φ X ( L X , z + dz ) , to obtain the new conditional luminosity function Ψ C ( L R , L X , z + dz ) , and therefore the new accretion rate function, and so on. Thus, the local BHMF, as determined independently from the galaxy velocity dispersion distribution and the M -σ relation, and the local accretion rate distribution function, as derived \nfrom the conditional radio/X-ray LF and a specific accretion modes scenario, i.e. the function x ( ˙ m ) , can be used together as a boundary condition to integrate eq. (9) up to the redshift where the HXLF and the RLF of AGN can be reliably estimated. At each redshift, I discard all SMBH whose mass has decreased below M min = 10 6 M /circledot , so that nothing can be said, within such a scheme, about the formation of the SMBH seeds. Furthermore, at every redshift, the values of L X , min and L R , min (see equations 3, 4) are increased to take into account the loss of SMBH below our threshold. In this way, at every redshift the number of objects above a certain minimal mass is always equal to the number of radio and X-ray sources above the corresponding limiting luminosities. \nThe final result will depend on the following assumptions: \n- · The X-ray and the Radio luminosity functions of AGN, if extrapolated down to low enough luminosity, do indeed describe the same class of objects (SMBH), and this class of objects is assumed to be described by the mass function φ M . Apart from physical arguments invoking the same inner engine for radio and X-ray selected AGN, there are indeed strong similarities between the cosmic evolution of radio sources and of X-ray and optically selected AGN (Dunlop 1998; Willott et al. 2001), a further hint of the correctness of this assumption.\n- · The fundamental plane of black hole activity is the same at all redshifts, and the correlation coefficients are the same as those observed at redshift zero. As remarked in the introduction, this is a natural consequence of the fact that the fundamental plane is a relationship among physical quantities in the innermost region of the coupled accretion flow-jet system. There, the strong black hole gravitational field dominates, and cosmological evolution is negligible. However, it should be noted here that the radio emission from steep spectrum AGN, whose RLF we adopt here, may be sensitive to properties of the interstellar and intergalactic medium which should indeed depend on redshift. Ideally, the redshift evolution of the AGN radio cores luminosity function should be used, as the fundamental plane (1) is a relationship between core luminosities only.\n- · The evolution with redshift of both φ R and φ X is known. This is certainly more accurate for the hard X-ray luminosity function, and in the following we adopt the luminosity dependent density evolution (LDDE) model of the 2-10 keV luminosity function derived by Ueda et al. (2003). The analytic approximation to the HXLF is described in the Appendix. \nOnthe other hand, the high redshift evolution of the RLF is much less certain. The best estimate to date is probably that by Willott et al. (2001). This is based on three redshift surveys of flux-limited samples of steep spectrum sources selected at low frequencies. By selecting only steep spectrum sources the authors made sure that the effect of strongly beamed sources (which have typically flat spectrum) were minimized, especially for the highest redshift sources. However, the fundamental plane relationship was determined by including both flat and steep spectrum sources (but excluding beamed objects, see MHD3), and the selection was made at higher frequencies. Despite this potential source of uncertainty, I will here adopt the Willott et al. (2001) parameterization, assuming a uniform radio spectral index α R = 0 . 7 to rescale the fluxes. Previous studies of redshift evolution of the RLF of flat and steep spectrum sources (Dunlop & Peacock 1990) have shown that although steep spectrum sources dominate number counts, the two populations display similar redshift evolution. The analytical approximation to the RLF of Willott et al. (2001) used here can be found in the Appendix. \nFigure 3. Redshift evolution of the comoving black hole mass density ρ BH (top panel), of the average X-ray to Eddington ratio, 〈 L X /L Edd 〉 (middle panel), and of the average SMBH mass, in units of solar masses (bottom panel). The dashed horizontal line on the middle panel marks the adopted value of the critical accretion rate, x cr , separating radiatively inefficient accretion from radiatively efficient one: most of the SMBH growth, therefore, took place during episodes of radiatively efficient accretion (see also Fig. 5). \n<!-- image --> \nFigure 4. Redshift evolution of the SMBH mass function (BHMF), from redshift 3.5 till redshift 0.1. Different colors and symbols correspond to different redshift bins. \n<!-- image --> \n- · The function L X /L Edd = x ( ˙ m ) does not depend on the black hole mass and is expressible as broken power-law: at low accretion rates black holes accrete in a radiatively inefficient way, and x ∝ ˙ m 2 . 3 , while at high accretion rates BH are radiatively efficient, with the bulk of the emission being radiated in the optical/UV bands, so that x ∝ ˙ m 0 . 76 (see section 3). \nThe whole history of supermassive black hole growth can then be reconstructed from the evolution of X-ray and radio AGN luminosity functions with just three free parameters: the accretion efficiency /epsilon1 , the value of the critical ratio x cr that separates the radiatively inefficient and efficient regimes, and the corresponding critical accretion rate or, equivalently, an X-ray bolometric correction.', '5 RESULTS': 'The family of all possible outcomes of the calculation outlined above is obtained by varying parameters in the relatively narrow range of physically and phenomenologically realistic values: 0 . 06 < /epsilon1 < 0 . 42 and 10 -4 < ∼ x cr < ∼ 10 -2 . However, the uncertainties on the observed luminosity functions, the high-redshift radio one in particular, do not allow to put tight constraints on these parameters. More interesting, instead, is the overall trend emerging from this calculation for the cosmic evolution of the SMBH population and of its activity level. For this reason, here I discuss in detail the results of a calculation performed assuming /epsilon1 = 0 . 1 and x cr = 10 -3 , leaving a discussion of the consequences of a different choice of parameters to section 5.1. \nFigure 3 shows, in the upper panel, the redshift evolution of the comoving black hole mass density from z = 3 . 5 to z = 0 . The local black hole mass density is ρ BH ( z = 0) = 2 . 6 × 10 5 M /circledot Mpc -3 , a value consistent with that obtained by YT02, as already discussed by Aller and Richstone, (2002), while the SMBH mass density at redshift 3 is about six times lower: ρ BH ( z = 3) = 4 . 5 × 10 4 M /circledot Mpc -3 . Half of the total black hole mass density was accumulated at redshift z < 1 . This is consistent with the most recent results from X-ray Background (XRB) studies. After the deep Chandra and XMM surveys have revealed the redshift distribution of the obscured sources that most contribute to the XRB (Alexander et al. 2001; Barger et al. 2002; Mainieri et al. 2002; Rosati et al. 2002; Hasinger 2003; Fiore et al. 2003), the newest synthesis models seem to suggest that indeed a substantial fraction of the locally measured mass density of SMBH (maybe up to 50%) was accumulated in (obscured) 2 low-mass AGN (with M < ∼ 10 8 M /circledot ) at z < 1 (see e.g. Gandhi & Fabian 2003, or Fabian 2003 and references therein). This is also consistent with the general picture for the BHMF evolution discussed below. \nThe middle panel of Figure 3 shows the evolution of the average X-ray to Eddington rate, defined as: \n〈 L x L Edd ( z ) 〉 ≡ 〈 x ( z ) 〉 ≡ ∫ ∞ x min xφ x ( z ) dx ∫ ∞ x min φ x ( z ) dx . (10) \nHere φ x ( z ) is the X-ray to Eddington ratio function, representing the number of sources per unit co-moving volume per unit logarithm of the X-ray to Eddington ratio. It is simply related to the accretion rate function thorough the monotonic function of equation (8): φ x ( z ) = φ ˙ m ( z ) d ˙ m/dx . In the same panel, the dashed horizontal line marks the value of x cr . The average accretion rate increases by almost an order of magnitude from redshift zero until z ∼ 2 , where the luminosity density of the hard X-ray selected sources peaks, and then levels off. Supermassive black holes were more active in the past, in the sense that their average dimensionless accretion rate was higher at higher redshift. A similar conclusion was drawn by Small & Blandford (1992), who adopted a phenomenological approach close to the one followed here, and by Haiman & Menou (2002) and Menci et al. (2003) in the framework of semi-analytic models for structure formation in cold dark matter universes (see discussion below, § 6). \n2 It is worth emphasizing that both the fundamental plane relationship and the HXLF used here rely on absorption corrected 2-10 keV luminosities, and are therefore unaffected by Compton thin absorption. Therefore throughout the paper no distinction is made (and is possible) between obscured and unobscured sources. \nFigure 5. Comparison of the evolution of comoving black hole accretion rate densities computed according to different prescriptions. The thick solid line shows the results from this work, calculated for /epsilon1 = 0 . 1 and x cr = 10 -3 . The thin triple dotted-dashed line, instead, shows the results derived from the hard X-ray luminosity function alone, assuming radiative efficiency for all sources and a fixed bolometric correction (see e.g. M04). Thin dashed line shows the evolution of the comoving accretion rate density calculated from the evolution of the QSO luminosity function of Boyle et al. (2000) with the bolometric correction of Elvis et al. (1994). Thick dashed and dot-dashed lines in the same plot represent the accretion rate density for sources above and below the critical rate, respectively. \n<!-- image --> \nThe bottom panel of Fig. 3, shows the average black hole mass, \n〈 M ( z ) 〉 ≡ ∫ ∞ M min Mφ M ( z ) dM ∫ ∞ M min φ M ( z ) dM , (11) \nwhere at every redshift we have taken M min = 10 6 M /circledot . The antihierarchical (Granato et al. 2001, 2004; M04) nature of supermassive black holes growth is summarized by this plot, showing the increase of the average black hole mass with increasing redshift. \nMore specifically, the evolution of the shape of the black hole mass function, which in turn determines the evolution of 〈 M 〉 , is shown in Figure 4. As opposed to the standard picture of hierarchical mass build up of dark matter halos in CDM cosmologies, supermassive black holes growing by accretion between z ∼ 3 and now have a mass function which is more and more dominated by largest mass objects at higher and higher redshift, at least up to the limit where we can trust the evolution of our parametrized Xray and radio luminosity functions. As it is indeed emerging from the study of the QSO population of the SDSS (Vestergaard 2004; McLure & Dunlop 2004), most of the more massive black holes ( M > 10 9 ) were already in place at z ∼ 3 . \nThe history of accretion activity is instead summarized in figure 5, where I plot the comoving black hole accretion rate (BHAR) density, (thick solid line). The thick dashed and dot-dashed lines in the same plot represent the accretion rate density for sources above and below the critical rate, respectively. For comparison, also plotted are the evolution of the comoving accretion rate density calculated by M04 from the hard X-ray luminosity function alone, assuming high radiative efficiency for all sources (thin triple dotted-dashed line); and the corresponding quantity, calculated instead from the evolution of the QSO luminosity function of Boyle et al. (2000) with the bolometric correction of Elvis et al. (1994) (thin \n∼ \nFigure 6. Redshift evolution of the SMBH accretion rate function, from redshift 3.5 till redshift 0.1. Different colors and symbols correspond to different redshift bins. \n<!-- image --> \ndashed line). The accretion rate history is therefore dominated by sources with high radiative efficiency, which explains why many authors were indeed able to explain most of the local SMBH population as remnants of bright AGN phases (see e.g. YT02). However, due to the progressive decrease of the average accretion rate with time, as shown in Fig. 3, the population of accreting supermassive black holes at z < ∼ 0 . 5 is dominated by radiatively inefficient sources. For this reason, the evolution of the BHAR density from z = 0 to z = 1 is less rapid than what would be inferred assuming that all growing SMBH are high ˙ m objects. The consequences of this fact for the history of AGN feedback are discussed in section 6.2. \nFinally, in Figure 6, the evolution of the black hole accretion rate (expressed here as X-ray to Eddington ratio, L X /L Edd ) function is plotted for different redshift intervals, from z = 0 . 1 to z = 3 . 5 . While the number of sources accreting at low rates increases monotonically with decreasing redshift, the situation is different for rapidly accreting objects. High accretion rate sources (which should correspond to QSO and bright AGNs) rapidly increase in number with increasing redshift. The cut-off redshift, above which the number of sources declines again, is a function of the typical X-ray to Eddington ratio, being lower for lower accretion rate sources. \nThe combined evolution of mass and accretion rate functions derived here is the cause of the strong trend observed in deep Xray selected samples (Cowie et al. 2003; Hasinger 2003; Fiore et al. 2003), where progressively lower luminosity AGN reach their maximal space density at progressively lower redshifts.', '5.1 Exploring the parameter space: constraints on accretion efficiency': 'As discussed already, recent works by YT02, Elvis et al. (2002), M04, have shown that a direct comparison between local SMBH mass density and the AGN/QSO energy density integrated over cosmic time can be used to constrain accretion physics and, in particular the mean radiative efficiency of all accreting SMBH throughout their history. In doing so, the obvious inequality should be satisfied, that the local ρ BH , 0 be always larger than (or at most equal to) the total mass density accreted onto active black holes. If this is not \nFigure 7. Shaded area is the region of the parameter space ( /epsilon1 , x cr ) which must be exclude, as the total black hole mass density at z = 0 is smaller than the total calculated mass density accreted over cosmic time. \n<!-- image --> \nthe case, the simplest solution to the problem is to assume a higher accretion (and radiative) efficiency, as indeed argued by Elvis et al. (2002) and YT02. \nAsimilar line of reasoning can of course be applied to our calculations. Following the evolution of the BHMF backwards in time, we can search for the values of the free parameters /epsilon1 and x cr for which the SMBH mass density becomes negative in a finite time. For the specific x ( ˙ m ) and bolometric correction adopted (see § 3), I found the acceptable region is bounded below by the following empirical relation: /epsilon1 /similarequal 0 . 2 log( x cr ) + 0 . 7 , which is shown in Fig. 7, with the shaded area representing the excluded region of the parameter space. The mean radiative efficiency of SMBH that these calculations yield is itself a function of /epsilon1 and x cr , as it is shown in figure 8. Obviously, the higher the critical X-ray to Eddington rate where the transition occurs, the lower the average radiative efficiency is for any given value of /epsilon1 , as a larger number of objects of any given L X will be in the radiatively inefficient regime of accretion at any time. \nFigures 7 and 8 seem to suggest that a relatively high average accretion efficiency is to be preferred, implying a non zero average spin for the SMBH population, unless x cr is lower than 10 -3 . It should be stressed, however, that such a constraint also depends crucially on the value of the local black hole mass density 3 , and on the adopted bolometric correction, and that the uncertainties in the HXLF and, to a larger extent, in the RLF adopted here should affect the exact determination of the accepted region of the parameter space. \nIn any case, it is worth stressing that the qualitative behaviour of the BHMF evolution, being driven essentially by the evolution of the shape of the two luminosity functions, is not modified by changes of /epsilon1 and/or x cr . In particular, the anti-hierarchical character of the solution found is a robust result of the approach presented here. \nFigure 8. The average radiative efficiency, /epsilon1 rad for the evolving SMBH population between redshift 3.5 and 0, as a function of the accretion efficiency /epsilon1 . Shown are the calculations done for three values of the critical rate: log( x cr ) = -3 . 5 (dashed line); -3 (fiducial case, solid line) and -2 . 5 (dot-dashed line). Only values of /epsilon1 that are above the line /epsilon1 /similarequal 0 . 2log( x cr ) + 0 . 7 (see figure 7) are shown. \n<!-- image -->', '6 DISCUSSION': "In the previous section I have shown and discussed what is possible to deduce about the evolution of the supermassive black hole mass function from the coupled evolution of HXLF and RLF. The limits of this approach are a consequence of the extreme difficulty of obtaining high redshift luminosity functions in the radio and X-ray band. Little can be said then, about the sources evolution at higher redshifts, where there are hints of a decline of AGN activity with redshift. \nTheoretically, the high z evolution should be mainly driven by the gravitational growth of structures, with the AGN activity triggered by mergers, according to the standard hierarchical picture (Efstathiou & Rees 1988; Cavaliere & Vittorini 2000; Wyithe & Loeb 2003; Di Matteo et al. 2003; Menci et al. 2004). In those phases of early structure formation and evolution, there should be plenty of gas available for accretion due to the frequent galaxy merging events which can effectively destabilize it and make it reach the central SMBH sphere of influence. Black hole growth is essentially limited by the Eddington limit (the feast , according to Small & Blandford 1992, or the self-limited regime , Cavaliere & Vittorini 2000). The shape of the black hole mass function inferred at z ∼ 3 (see Fig. 4), suggesting that higher mass black holes were already in place at that time, requires a very rapid growth and very high accretion rates in the high density peaks of the cosmic density distribution at early times, also consistent with the trend of ˙ m shown in Figure 3. \nOn the other hand, the rapid, anti-hierarchical, evolution of the SMBH population between z = 0 and z ∼ 2 . 5 has always been difficult to incorporate into standard CDM models for structure formation. It is interesting, in this context, to compare the main qualitative, anti-hierarchical, picture emerging for growing black holes with the evolution of the galaxy population. As for SMBH, the most striking and robust evolutionary trend observed for the galaxy population is the rapid increase of the global star formation rate (SFR) between z = 0 and z ∼ 1 -2 (Lilly et al. 1996; Madau et al. 1996). Indeed, Di Matteo et al. (2003) and M04 have already shown that BHAR and SFR histories have broadly similar shapes, \nwith an approximately constant ratio of about few times 10 -3 , and this has also been recently measured directly by the SDSS (Heckman et al. 2004). Cowie at al. (1996) have shown that such an evolution is caused by the smooth decline with redshift (from z ∼ 1 to the present) of the rest-frame K band luminosity (an indicator of the total stellar mass) of galaxies that undergo rapid star formation. This phenomenon, called 'down-sizing' is the exact analogous of the anti-hierarchical growth of supermassive black holes described here, and has been recently confirmed by Kauffmann et al. (2004), by studying the environmental dependence of the relation between star formation and stellar mass in a large number of SDSS galaxies. \nWhat is the physical origin of such a common behavior? The combined effects of the decrease in the galaxy interaction rate in the era of groups and clusters formation (see e.g. Cavaliere & Vittorini 2000), the expansion of the universe affecting the gas cooling efficiency (Hernquist & Springel 2003; Di Matteo et al. 2003), the progressive depletion of the cold gas reservoirs within galaxies needed to power accretion and strong feedback from both stars and AGN (see e.g. Wyithe & Loeb 2003; Granato et al. 2004), should all contribute to the observed evolution at redshift below 3, which can be defined as that of supply-limited accretion (Cavaliere & Vittorini 2000), or the famine after the feast (Small & Blandford 1992). Clearly, the aim of this paper is not to explore all the above theoretical issues and predictions, rather to provide a detailed picture to compare those predictions with. Therefore, the rest of the discussion will be devoted to two specific results on the lifetimes and duty cycles of active black holes and on the possible implication of the derived SMBH growth history for the AGN feedback evolution.", '6.1 Lifetimes and duty cycles of active black holes': "In all models that try to derive the properties of the SMBH population from the observed QSO evolution, a key element is represented by the typical quasar lifetime or by the almost equivalent activity duty cycle (see Martini 2003 and references therein). However, the significance of these parameters is limited to the standard case in which, on the basis of an observed luminosity function in a specific waveband, one tries to derive the distribution of either BH masses or accretion rates. Usually, a constant Eddington ratio is assumed in this case, which implies that QSO are considered as on-off switches. Then, the duty cycle is simply the fraction of black holes active at any time, and the lifetime is the integral of the duty cycle over the age of the universe. \nThe picture discussed here is different, in that a broad distribution of Eddington rates is not only allowed, but actually calculated for the SMBH population at every redshift. When this is the case, a more meaningful definition of activity lifetime is needed. I follow Steed & Weinberg (2004) formulation, by first defining the mean Eddington rate for object of mass M 0 at redshift z = 0 〈 ˙ m ( M 0 , z ) 〉 and then introducing the mean accretion weighted lifetime of a SMBH with a given mass today : \nτ ( M 0 , z ) = ∫ z ∞ 〈 ˙ m ( M 0 , z ' ) 〉 dt dz ' dz ' , (12) \nThe ratio of τ ( M 0 , z ) to the Salpeter time, t S = /epsilon1M c 2 /L Edd = ( /epsilon1/ 0 . 1)4 . 5 × 10 7 yrs, gives the mean number of e -folds of mass growth for objects with mass M 0 up to redshift z . The ratio of τ ( M 0 , z ) to the Hubble time t Hubble ( z ) = H ( z ) -1 , instead, is a measure of the activity duty cycle of SMBH. \nAs I follow here a phenomenological approach based on observed luminosity functions, and no information is therefore available on the formation and early growth of the first black holes, it \nFigure 9. Partial mean accretion weighted lifetimes of SMBH with mass today M 0 , calculated for three different redshift intervals: 0 < z < 1 (solid line), 1 < z < 2 (dashed line) and 2 < z < 3 (dot-dashed line). The horizontal dotted line is the Salpeter time for accretion efficiency of 10%. The accretion weighted lifetime for BH of any given mass between 0 < z < 3 is the sum of the three. \n<!-- image --> \nis interesting here to calculate 'partial' lifetimes in a given redshift interval ∆ z = ( z i , z f ) : \n∆ τ ( M 0 , ∆ z ) = ∫ z f z i 〈 ˙ m ( M 0 , z ' ) 〉 dt dz ' dz ' ; (13) \nIn Figure 9, I show ∆ τ ( M 0 , ∆ z ) for three redshift intervals: 0 < z ≤ 1 ; 1 < z ≤ 2 and 2 < z ≤ 3 . The accretion weighted lifetime for BH of any given mass between 0 < z < 3 is of course just the sum of the three. The anti-hierarchical nature of mass buildup in actively accreting AGN and QSOs is again clearly illustrated by this plot. In fact, the major growth episode of a SMBH must coincide with the period when ∆ τ > t S . This happens at z < 1 for M 0 < ∼ 10 7 . 6 , between redshift 1 and 2 for 10 7 . 6 < ∼ M 0 < ∼ 10 8 . 2 , and at 2 < z < 3 for 10 8 . 2 < ∼ M 0 < ∼ 10 8 . 4 . Supermassive black holes with masses larger than M 0 ∼ 10 8 . 5 today, must have experienced their major episodes of growth at redshift higher than 3. Black hole of lower mass today, which are also accreting at the higher rates in the local universe (see section 2.1) drop below an hypothetic seed mass (here fixed at 10 4 M /circledot , but these results do not depend strongly on this value) and effectively 'disappear' at higher redshift. This reflects the obvious impossibility of working out the initial condition of black hole growth from the local population evolved backwards once an object has exponentiated its mass just a few times. \nIt also interesting to note that the objects that dominate the SMBHmassfunction today, i.e. those in the range of masses around 10 7 . 5 M /circledot , where M 0 φ M ( M 0 , z = 0) peaks, mainly grew around z ∼ 1 , which is when most of the X-ray background light we see today was emitted (Hasinger 2003). \nThe ratio ∆ τ ( M 0 , ∆ z ) / ∆ t (∆ z ) , where ∆ t (∆ z ) is the time elapsed in the redshift interval ∆ z , is an indication of the 'partial' duty cycle of a black hole of mass M 0 today in that particular epoch. This is shown in figure 10 for the same redshift intervals of fig. 9. \nFigure 10. Partial mean accretion weighted duty-cycles of SMBH with mass today M 0 , calculated for three different redshift intervals: 0 < z < 1 (solid line), 1 < z < 2 (dashed line) and 2 < z < 3 (dot-dashed line). \n<!-- image -->", '6.2 On the kinetic energy output and the history of AGN feedback': 'The main novelty of the constraints on black hole growth history presented here is the opportunity to determine both mass and accretion rate for each and every source, thanks to the fundamental plane relationship. As I discussed in section 3, different modes of accretion are expected at different ˙ m . In fact, not just the radiative output of an accreting black hole scales differently with ˙ m for different accretion modes, but also the total kinetic energy carried by the jets/outflows that are responsible for radio emission. It is possible, therefore, to derive a parallel history of the (mostly unseen) mechanical power output from supermassive black holes growth. \nTo derive a scaling of the jet kinetic power, W jet , with mass and accretion rate for sources above and below x cr I will proceed in the following way. I will fist assume that the jet kinetic power at injection is carried by internal energy, and I assume equipartition between the jet magnetic field and the total pressure in the disc (see Heinz & Sunyaev 2003). Then W jet ∝ P rel R 2 S ∝ B 2 M 2 , where P rel represents the pressure in relativistic particles at the base of the jet and R S = 2 GM/c 2 is the Schwarzschild radius. The scaling of the magnetic field, instead, can be directly inferred from the inclination of the fundamental plane. Equation (11) of MHD03 relates the observed correlation coefficients of the fundamental plane relation to the slope of the electron distribution in the jet, p , the observable radio spectral index α R , the logarithmic derivatives of the magnetic field intensity with respect to mass and accretion rate and the index q of the L X -˙ m relation (see section 3). From that equation we have, assuming p = 2 and α R = 0 : \n∂ ln B ∂ ˙ m = 0 . 35 q ξ RX /similarequal 0 . 21 q ∂ ln B ∂M = 0 . 35( ξ RX + ξ RM ) -1 /similarequal -0 . 51 , (14) \nwhere we have used the results of MHD03 ξ RX = 0 . 6 and ξ RM = 0 . 78 . \nThus, we obtain the expected result that, for radiatively inefficient flows ( q /similarequal 2 . 3 ) the total jet kinetic power is proportional to the physical accretion rate only W jet ∝ ˙ mM ∝ ˙ M (see Falcke & Biermann 1996; Heinz & Sunyaev 2003; Fender, Gallo & Jonker 2003). Whether such an output dominates the energy budget (ac- \nFigure 11. Redshift evolution of the total integrated mechanical power W jet /c 2 from accreting black holes per unit co-moving Mpc (triple dotteddashed line). For comparison, the evolution of the total black hole accretion rate density (solid line) is shown together with the two separate contributions from the sources accreting above(dashed line) and below (dot-dashed line) the critical rate, all taken from Fig. 5. \n<!-- image --> \ng to the so-called ADIOS picture, Blandford & Begelman 1999, and as proposed also by Fender, Gallo & Jonker 2003) or not depends however on the dynamics of the innermost disc-jet coupling and, from the observationally point of view, on the radiative efficiency of the jets. On the other hand, SMBH accreting above the critical rate x cr , for which we have assumed q = 0 . 76 , should obey the scaling W jet ∝ M ˙ m 0 . 33 . \nGiven the above scaling relations for W jet , it is possible to calculate the mechanical power output from each accreting black hole and the total integrated one, as a function of redshift. For the sake of simplicity I assume that indeed the total kinetic power of the jet/outflow from radiatively inefficient black holes dominates as a sink of energy. Therefore, the calculated W jet should be considered an absolute upper limit to the jet/outflow kinetic power. The results are shown in Figure 11. \nBecause the relative contribution of the mechanical energy output is much larger in low accretion rate sources, the history of W jet does not follow that of the most luminous sources, but instead exhibit a much weaker evolution, both at low and high redshifts. Also, because the number of low accretion rate sources increase with time (due to the overall decrease of the average accretion rate, see previous section), the peak of the mechanical power output lies at a much lower redshift than the peak of the radiative energy output history of AGN, which should have interesting implications for the formation and dynamical evolution of clusters of galaxies.', '7 CONCLUSIONS': 'I have presented a new method to study the growth of accreting supermassive black holes, based on the simultaneous evolution of the AGN radio and hard (2-10 keV) X-ray luminosity functions. The method is based on the locally observed trivariate correlation between black hole mass, X-ray and radio luminosity (the so-called fundamental plane of black hole activity, MHD03). Thanks to this correlation, it is possible for the first time to break the degeneracy between luminosity, mass and accretion rate that affected all previ- \nous attempts to study the evolution of the supermassive black holes population by looking at the evolution of a single AGN luminosity function. \nHere, the redshift evolution of the SMBH mass function between z = 0 and z ∼ 3 is evaluated integrating backwards in time a continuity equation for the black hole population in which the role of mergers is neglected. The local black hole mass function, estimated by applying the correlation between black hole mass and velocity dispersion to the velocity distribution function of local galaxies, is used as a boundary condition for the continuity equation. The solution to this equation is uniquely determined once the accretion efficiency is specified, together with the function L X /L Edd ( ˙ m ) , that links the observed Eddington scaled hard X-ray luminosity of an accreting black hole to its accretion rate. For the latter, I have assumed a very general form of a double power-law, corresponding to a radiatively inefficient regime at low accretion rates, and a radiatively efficient one at high ˙ m . \nBy comparing the local observed BHMF with the total mass accreted onto SMBH over their history, it is possible to put simultaneous constraints on the accretion efficiency and on the critical value of the accretion rate, x cr , at which the transition takes place between the two accretion modes. \nThe main results of this work are the following. For fiducial values of the parameters ( /epsilon1 = 0 . 1 and x cr = 10 -3 ), half ( ∼ 85 %) of the local black hole mass density was accumulated at redshift z < 1 ( z < 3 ), mostly in radiatively efficient episodes of accretion. Qualitatively (i.e. independently on the values of these two parameters), the evolution of the black hole mass function between z = 0 and z ∼ 3 shows clear signs of an anti-hierarchical behaviour. This is a purely phenomenological assessment, and reflects the fact that, while the majority of the most massive objects ( M > ∼ 10 9 ) were already in place at z ∼ 3 , lower mass ones mainly grew at progressively lower redshift, so that the average black hole mass increases with increasing redshift. On the other hand, the average accretion rate decreases towards lower redshift. Therefore, sources in the RIAF regime of accretion only begin to dominate the comoving accretion energy density in the universe at z < 1 (with the exact value of z depending on x cr ), while at the peak of the black hole accretion rate history, radiatively efficient accretion dominates by almost an order of magnitude. By carefully evaluating the contributions to the total black hole accretion energy density from the different modes of accretion as a function of redshift, I show how to derive a physically motivated AGN feedback history. This, as well as the anti-hierarchical behaviour of SMBH growth described above, may be of some importance for cosmological models of structure formation in the universe.', 'ACKNOWLEDGMENTS': 'I thank Xuelei Chen, Tiziana Di Matteo, Sebastian Heinz and Susumu Inoue for helpful discussions, and the anonymous referee for his/her useful comments.', 'REFERENCES': "Alexander D. M., Brandt W. N., Hornschemeier A. E., Garmire G. P., Schneider D. P., Bauer F. 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A., 2001, MNRAS, 322, 536 \nWolf C., Wisotzki L., Borch A., Dye S., Kleinheinrich M., Meisenheimer K., 2003, A&A, 408, 499 \nYu Q. & Lu Y., 2004, ApJ, in press. astro-ph/0311404 Yu Q. & Tremaine S., 2002, MNRAS, 335, 965 (YT03)", 'APPENDIX A: ANALYTICAL APPROXIMATION OF THE X-RAY LUMINOSITY FUNCTION': 'I adopt here the functional form for the HXLF described in Ueda et al. (2003), which has the following analytic approximation: \nφ X ( L X , z ) = φ X ( L x , 0) η ( z, L X ) , (A1) \nwhere the local X-ray luminosity function is expressed as smoothly connected double power-law: \nφ X ( L X , z = 0) = A [( L X /L ∗ ) γ 1 +( L X /L ∗ ) γ 2 ] -1 , (A2) \nwhile the evolutionary part is expressed as \nη ( z, L x ) = { (1 + z ) p 1 z < z c ( L X ) η ( z c )[(1 + z ) / (1 + z c ( L X ))] p 2 z ≥ z c ( L X ) (A3) \nand \nz c ( L x ) = { z ∗ c L X ≥ L a z ∗ c ( L X /L a ) α L X < L a (A4) \nFor a Λ CDM universe, the best fit parameters are: A = 5 . 04 ± 0 . 33 × 10 -6 h 3 70 Mpc -3 ; L ∗ = 10 ( 43 . 94 +0 . 21 -0 . 26 ) h -2 70 erg s -1 ; γ 1 = 0 . 86 ± 0 . 15 ; γ 2 = 2 . 23 ± 0 . 13 ; p 1 = 4 . 23 ± 0 . 39 ; p 2 = -1 . 5 (fixed); z ∗ c = 1 . 9 (fixed); L a = 10 44 . 6 h -2 70 erg s -1 (fixed); α = 0 . 335 ± 0 . 070 .', 'APPENDIX B: ANALYTICAL APPROXIMATION OF THE RADIO LUMINOSITY FUNCTION': 'The 5GHz RLF is derived from the low-frequency (151 MHz) one of Willott et al. (2001), assuming a fixed radio spectral index of α R = 0 . 7 to rescale the luminosities. \nThe analytical approximation is that of model C of Willott et al.(2001), and is given by a sum of two differently evolving populations (a low and a high luminosity one): \nφ R , 151 ( L 151 , z ) = φ R , l + φ R , h . (B1) \nThese two populations evolve differently with redshift, according to the following expressions: \nφ R , l = φ 0 R , l ( L 151 L l , ∗ ) -α l exp ( -L 151 L l , ∗ ) (1 + z ) k l ; z < z l0 φ 0 R , l ( L 151 L l , ∗ ) -α l exp ( -L 151 L l , ∗ ) (1 + z l0 ) k l ; z ≥ z l0 (B2) \nφ R , h = φ 0 R , h ( L 151 L h , ∗ ) -α h exp ( -L h , ∗ L 151 ) f h ( z ) . (B3) \nThe high luminosity redshift evolution f h ( z ) is given by: \nf h ( z ) = exp [ -1 2 ( z -z h0 z h1 ) 2 ] z < z h0 exp [ -1 2 ( z -z h0 z h2 ) 2 ] z ≥ z h0 (B4) \n The best-fitting parameters to the observed luminosity function are: log φ 0 R , l = -7 . 12 +0 . 10 -0 . 11 , α l = 0 . 539 ± 0 . 02 , log L l , ∗ = 26 . 10 +0 . 08 -0 . 09 , z l0 = 0 . 71 ± 0 . 10 , k l = 4 . 30 +0 . 57 -0 . 55 , log φ 0 R , h = -6 . 20 +0 . 09 -0 . 11 , α h = 2 . 27 +0 . 12 -0 . 11 , log L h , ∗ = 26 . 95 +0 . 11 -0 . 10 , z h0 = 1 . 91 ± 0 . 16 , z h1 = 0 . 56 ± 0 . 05 , z h2 = 1 . 38 +0 . 52 0 . 28 . This paper \nhas been typeset from a T E X/ L A T E X file prepared by the author.'}
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2011ApJ...735..110B
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Evidence for Low Black Hole Spin and Physically Motivated Accretion Models from Millimeter-VLBI Observations of Sagittarius A*
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2011-01-01
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['accretion', 'accretion disks', 'black hole physics', 'galaxy center', 'astronomy submillimeter', 'techniques interferometric', '-']
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Millimeter very long baseline interferometry (mm-VLBI) provides the novel capacity to probe the emission region of a handful of supermassive black holes on sub-horizon scales. For Sagittarius A* (Sgr A*), the supermassive black hole at the center of the Milky Way, this provides access to the region in the immediate vicinity of the horizon. Broderick et al. have already shown that by leveraging spectral and polarization information as well as accretion theory, it is possible to extract accretion-model parameters (including black hole spin) from mm-VLBI experiments containing only a handful of telescopes. Here we repeat this analysis with the most recent mm-VLBI data, considering a class of aligned, radiatively inefficient accretion flow (RIAF) models. We find that the combined data set rules out symmetric models for Sgr A*'s flux distribution at the 3.9σ level, strongly favoring length-to-width ratios of roughly 2.4:1. More importantly, we find that physically motivated accretion flow models provide a significantly better fit to the mm-VLBI observations than phenomenological models, at the 2.9σ level. This implies that not only is mm-VLBI presently capable of distinguishing between potential physical models for Sgr A*'s emission, but further that it is sensitive to the strong gravitational lensing associated with the propagation of photons near the black hole. Based upon this analysis we find that the most probable magnitude, viewing angle, and position angle for the black hole spin are a = 0.0<SUP>+0.64 + 0.86</SUP>, \theta ={68^\circ }^{+5^\circ +9^\circ }_{-20^\circ -28^\circ }, and \xi ={-52^\circ }^{+17^\circ +33^\circ }_{-15^\circ -24^\circ } east of north, where the errors quoted are the 1σ and 2σ uncertainties.
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https://arxiv.org/pdf/1011.2770.pdf
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{'EVIDENCE FOR LOW BLACK HOLE SPIN AND PHYSICALLY MOTIVATED ACCRETION MODELS FROM MILLIMETER VLBI OBSERVATIONS OF SAGITTARIUS A*': 'AVERY E. BRODERICK 1 VINCENT L. FISH 2 SHEPERD S. DOELEMAN 2 ABRAHAM LOEB 3 1 Canadian Institute for Theoretical Astrophysics, 60 St. George St., Toronto, ON M5S 3H8, Canada; [email protected] 2 Massachusetts Institute of Technology, Haystack Observatory, Route 40, Westford, MA 01886. 3 Institute for Theory and Computation, Harvard University, Center for Astrophysics, 60 Garden St., Cambridge, MA 02138. Draft version May 1, 2018', 'ABSTRACT': "Millimeter very-long baseline interferometry (mm-VLBI) provides the novel capacity to probe the emission region of a handful of supermassive black holes on sub-horizon scales. For Sagittarius A* (Sgr A*), the supermassive black hole at the center of the Milky Way, this provides access to the region in the immediate vicinity of the horizon. Broderick et al. (2009) have already shown that by leveraging spectral and polarization information as well as accretion theory, it is possible to extract accretion-model parameters (including black hole spin) from mm-VLBI experiments containing only a handful of telescopes. Here we repeat this analysis with the most recent mm-VLBI data, considering a class of aligned, radiatively inefficient accretion flow (RIAF) models. We find that the combined data set rules out symmetric models for Sgr A*'s flux distribution at the 3 . 9 σ level, strongly favoring length-to-width ratios of roughly 2.4:1. More importantly, we find that physically motivated accretion flow models provide a significantly better fit to the mm-VLBI observations than phenomenological models, at the 2 . 9 σ level. This implies that not only is mm-VLBI presently capable of distinguishing between potential physical models for Sgr A*'s emission, but further that it is sensitive to the strong gravitational lensing associated with the propagation of photons near the black hole. Based upon this analysis we find that the most probable magnitude, viewing angle, and position angle for the black hole spin are a = 0 . 0 + 0 . 64 + 0 . 86 , θ = 68 · + 5 · + 9 · -20 · -28 · , and ξ = -52 · + 17 · + 33 · -15 · -24 · east of north, where the errors quoted are the 1 σ and 2 σ uncertainties. \nSubject headings: black hole physics - Galaxy: center - techniques: interferometric - submillimeter - accretion, accretion disks", '1. INTRODUCTION': "Despite being invoked to power a variety of energetic astrophysical phenomena, the detailed structure and dynamics of black hole accretion flows remain a central problem in astrophysics. Moreover, using electromagnetic observations to probe the structure and dynamics of the black hole spacetimes requires a substantial understanding of the physical processes that determine the fate of the accreting matter. Only recently has it become possible to probe this physics via large-scale computational simulations. Nevertheless, ab initio calculations are beyond our present capability, requiring numerous simplifying, and in some cases unphysical, assumptions. This is evidenced by the number of models proffered to explain the various properties of accreting black hole candidates. In turn, this ambiguity complicates efforts to use electromagnetic observations to probe the structure and dynamics of the spacetime surrounding the black hole. \nBy virtue of its proximity, the supermassive black hole at the center of the Milky Way, associated with the bright radio point source Sagittarius A* (Sgr A*), provides an unparalleled opportunity to study black hole accretion in detail. For this reason, Sgr A* may serve as an exemplar of the larger class of supermassive black holes specifically, and of black holes in general. Presently, the best estimates of the mass and distance of Sgr A* come from the observations of orbiting stars. These have yielded M = 4 . 3 ± 0 . 5 × 10 6 M /circledot and D = 8 . 3 ± 0 . 4kpc, respectively, where both include the systematic uncertainties (Ghez et al. 2008; Gillessen et al. 2009b,a). The mass is necessarily confined to within the periapse of nearby stars, giving a maximum radius of roughly 10 2 AU /similarequal 3 × 10 3 GM / c 2 , ruling out many extended objects. These represent the best mass \nmeasurement for any known black hole to date. \nIn addition to the dynamical observations, a wealth of spectral and polarization data exists for Sgr A*. From these it is apparent that Sgr A* is unlike many active galactic nuclei, being vastly underluminous, emitting a bolometric luminosity of roughly 10 36 erg, approximately 10 -9 of Eddington. This is especially small in light of the considerable amount of gas within the black hole's sphere of influence, presumably available for accretion (Loeb & Waxman 2007; Cuadra et al. 2008). As a result it is widely accepted that Sgr A*'s accretion flow is qualitatively different from those in its active analogs, though perhaps indicative of the roughly 90% of black holes that are presently not in an active phase. \nNevertheless, the existing spectral and polarization data has produced a canonical set of components all models for Sgr A* include: populations of thermal and nonthermal electrons, nearly equipartition magnetic fields. Less certain is the structure of the emission region. This is evidenced by the variety of models that have been proposed (e.g., Narayan et al. 1998; Blandford & Begelman 1999; Falcke & Markoff 2000; Yuan et al. 2002, 2003; Loeb & Waxman 2007). Despite being able to reproduce the observed features of Sgr A*, these differ dramatically in the morphology of the emitting region. As a consequence, many of the theoretical ambiguities can be immediately addressed by direct probes of the spatial distribution of the emitting plasma surrounding the central supermassive black hole. \nThe spectrum of Sgr A* peaks near millimeter wavelengths, implying a transition from optically thick to optically thin emission. The location of this emission is currently debated, however the presence of short-timescale variability at millime- \nter, near-infrared and X-ray wavelengths implies that optically thin emission is dominated by contributions arising in the immediate vicinity of the black hole. Furthermore, at millimeter wavelengths the blurring due to interstellar electron scattering is subdominant. Thus, at wavelengths of 1 . 3mm and below it is possible to image the emitting region surrounding Sgr A*. \nEven with the strong gravitational lensing in the vicinity of the horizon, imaging the immediate vicinity of the black hole requires extraordinary resolutions. The silhouette cast by the horizon on the surrounding emission is roughly 53 ± 2 µ as 1 . At the present time, this resolution is accessible only via millimeter-wavelength very-long baseline interferometry (mmVLBI). VLBI observations of Sgr A* at 1 . 4mm using the Institut de Radioastronomie Millimétrique (IRAM) 30m telescope at Pico Veleta and one of the 15m dishes at Plateau de Bure, produced the size estimate of 110 ± 60 µ as, with the large uncertainties due to limited calibration accuracy (Krichbaum et al. 1998). \nThe first successful mm-VLBI observation of Sgr A* with Earth-scale baselines was performed in April, 2007, during which visibilities were measured on the 4 . 6 × 10 3 km baseline between Mauna Kea, Hawaii to Mount Graham, Arizona (Doeleman et al. 2008). By fitting these with a gaussian model, Doeleman et al. (2008) found a typical intrinsic source size of 37 + 5 -3 µ as 2 (after correcting for the sub-dominant broadening due to interstellar electron scattering), smaller than the black hole silhouette. \nSince that time a number of groups have analyzed the 2007 mm-VLBI data using various physically motivated accretion models for the emission region (Broderick et al. 2009; Huang et al. 2009; Mo'scibrodzka et al. 2009; Dexter et al. 2010), inferring from these efforts the black hole spin vector. Despite finding generally similar results, these have been limited by the lack of multiple long baseline observations and the limited north-south coverage obtained. Recently, a second, and considerably larger set of mm-VLBI observations have been reported (Fish et al. 2010), providing the opportunity to revisit, and substantially improve, constraints upon the black hole spin and accretion physics. \nHere we report upon the first effort to do this using a physically motivated accretion model, similar to that described in Broderick et al. (2009), that fits the known spectral and polarization properties of Sgr A*. In addition to improving the resulting parameter estimation, it is now possible to identify statistical signatures of both the asymmetry of the image and the importance of the underlying physics that governs the image morphology. Section 2 summarizes the full set of mm-VLBI observations we consider. Section 3 describes the models we consider and how the resulting visibility data is produced. How models are compared and the parameter estimates are produced is discussed in Section 4. The fitting process and results are presented in Section 5, and our best estimates for the black hole spin vector can be found in Section 6. Section 7 describes the implications for different potential future observations. Finally concluding remarks are collected in Section 8.", '2. SUMMARY OF MILLIMETER-VLBI OBSERVATIONS': 'In the analysis presented here we make full use of the recent observations described in Fish et al. (2010) and Doeleman et al. (2008). In both cases, observations targeting Sgr A* were made at 1 . 3mm using the Submillimeter Telescope (SMT) on Mt. Graham in Arizona, 10m dishes in the Combined Array for Research in Millimeter-wave Astronomy (CARMA) at Cedar Flat, California, and the James Clerk Maxwell Telescope (JCMT) located on Mauna Kea, Hawaii.', '2.1. April 2007': 'Doeleman et al. (2008) report upon measurements obtained on the nights of the April, 11 & 12, 2007, using the JCMT, SMT and a single CARMA dish. 19 visibility amplitudes were obtained on the CARMA-SMT and JCMT-SMT baselines, with an upper limit on April 11th, 2007 along the JCMTCARMA baseline. The locations of these observations on the u -v plane are indicated in the lower-left panel of Figure 1, labeled 2007. Signal-to-noise ratios typical of the short and long baselines are 8 and 4, respectively. \nDuring this time, observations the single-dish flux was estimated via the full CARMA array, operating as a stand-alone instrument, to be 2 . 4 ± 0 . 25Jy. This is similar to the visibility amplitudes obtained on the CARMA-SMT baselines and consistent with a single, compact gaussian component (Doeleman et al. 2008). This flux is anomalously low in comparison to the typical 1 . 3mm flux of ∼ 3Jy, and was taken as evidence for Sgr A* appearing in a quiescent state. This interpretation is supported by the lack of a significant difference between analyses of each day separately (Broderick et al. 2009). \nFull details of the observations, calibration and data processing can be found in Doeleman et al. (2008).', '2.2. April 2009': "Fish et al. (2010) report upon more recent observations performed on the nights of April, 5-7, 2009, corresponding to the 95, 96, and 97 days of 2009. These made use of the JCMT, SMT, and two CARMA dishes, operated as independent VLBI stations. 54 visibility amplitudes were obtained on JCMTSMT and CARMA-SMT baselines on all days, and to both of the JCMT-CARMA baselines on days 96 and 97. Positions of the observations on each day are indicated in the upper panels of Figure 1, labeled 2009.95, 2009.96, and 2009.97. Signal-tonoise ratios typical of the short and long baselines are 17 and 5, respectively. Thus, this second data set represents a significant improvement in both the number and precision of the data obtained. \nIn addition to the VLBI baselines, the presence of two independent CARMA dishes in the array allowed the measurement of very-short baseline visibilities, probing angular scales ∼ 10 '' . These found substantially more correlated flux density than the CARMA-SMT baselines did, inconsistent with a single compact gaussian component. The interpretation of the difference in correlated flux density between the inter-CARMA baselines and the CARMA-SMT baselines is presently unclear, and it may be possible for multiple geometric models (e.g., annular rings, extended double source) to fit the data. Within the context of our analysis, we will assume that this difference is due to a separate large-scale component not present during the 2007 observations. This is indirectly supported by the fact that the source sizes inferred from the mid and long baseline data are unchanged despite the variations in the visibility magnitudes (Fish et al. 2010). Therefore, we \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 1. Locations in the u -v plane of the visibilities observed during the 2007, 2009.95, 2009.96, 2009.97 epochs. Also shown are the combined set of observations. Finally, for references the combined set is compared to the potential baselines from existing and upcoming sub-mm telescopes. Each baseline is color-coded according to the associated two sites. In all plots, detections are denoted by green circles and upper-limits by red triangles. \n<!-- image --> \ndo not consider the inter-CARMA data further here, restricting ourselves to modeling the compact component observed with the longer baselines. \nOn days 95 and 96 the short-baseline flux densities are consistent with each other, with inferred single dish fluxes of 2 . 15 ± 0 . 06Jy, which while somewhat lower than those obtained in 2007, justify treating these as a similar quiescent period. This is not the case for day 97, which exhibited a 30%-40% increase in the luminosity of the compact component. Note that during the 3hr observing periods on days 96 and 97 there is no evidence for rapid changes in the CARMASMT visibility amplitudes, implying that during each Sgr A* was stable; i.e., the process responsible for the brightening occurred between observing periods and is stable on timescales of hours. As a consequence, we will treat the visibilities obtained on each day as due to a stationary source, though with properties that vary from day to day. \nFull details of the observations, calibration and data processing can be found in Fish et al. (2010).", '2.3. Combined Data Set': 'Combined, the 2007 and 2009 mm-VLBI measurements may be separated into 4 observational epochs: that containing the entire 2007 observations (2007), those on day 95 of 2009 (2009.95), those on day 96 of 2009 (2009.96), and those on day 97 of 2009 (2009.97). The combined coverage in the u -v plane is shown in the lower-middle panel of Figure 1. The long base- \nlines (JCMT-CARMA and JCMT-SMT) are oriented primarily in the east-west direction, extended roughly 3 . 6G λ . Nevertheless, the combined data set also extends roughly 2G λ in the north-south direction, providing substantial angular coverage in the u -v plane for the first time. \nIn Section 7 we will discuss the implications out analysis has for future observations. However, we note here that the baselines considered in the 2007 and 2009 mm-VLBI experiments are a small subset of the baselines that are possible with existing mm and sub-mm telescopes. Figure 1 shows the combined visibility data set in comparison to baselines associated with other potential mm-VLBI stations. These include stations in Chile (e.g., the Atacama Pathfinder EXperiment, Atacama Submillimeter Telescope Experiment, and Atacama Large Millimeter Array; APEX, ASTE, and ALMA, respectively), Mexico (Large Millimeter Telescope; LMT), Spain (Pico Veleta; PV), France (Plateau de Bure; PdB), and at the South Pole (South Pole Telescope; SPT). These both, extend the region covered in the u -v plane, and provide additional complementary short and intermediate baselines, primarily along the north-south directions. To date, visibilities on only a handful of potential baselines have been measured.', '3. VISIBILITY MODELING': "Our primary goal is to use physically motivated models of Sgr A*'s accretion flow to infer the properties of the central \nsupermassive black hole and its surrounding matter. To do this we compare both physical and phenomenological models of Sgr A* to the mm-VLBI visibilities. This requires the computation of model visibilities. Given a trial image intensity distribution, I ( α,β ), where α and β are angular coordinates, we may compute the visibilities in the standard fashion: \nV ( u , v ) = ∫ ∫ d α d β e -2 π i ( α u + β v ) / λ I ( α,β ) . (1) \nHere we describe three classes of model images: those associated with radiatively inefficient accretion flows (RIAFs) of the form discussed in Broderick & Loeb (2006a), symmetric and asymmetric gaussians. We also summarize the effects of interstellar electron scattering.", '3.1. Radiatively Inefficient Accretion Flows': "We employ a suite of radiatively inefficient accretion flow (RIAF) models, first described in Broderick & Loeb (2006a), and based upon those of Yuan et al. (2003). Here these models, which henceforth we refer to as BL06, are summarized. \nSgr A* transitions from an inverted, presumably optically thick spectrum to an optically thin spectrum near millimeter wavelengths. This implies that near 1.3mm Sgr A* is only becoming optically thin, and thus absorption in the surrounding medium is likely to be important. This transition does not occur isotropically, happening at longer wavelengths for gas that is receding and at shorter wavelengths for gas that is approaching. Therefore, properly modeling the structure and relativistic radiative transfer is crucial to producing high fidelity images. \nAlthough Sgr A* is vastly sub-Eddington, its bolometric luminosity, roughly 10 36 ergs -1 , is still large in absolute terms, Like many AGN, in the radio Sgr A* exhibits the nearly-flat, power-law spectrum associated with non-thermal synchrotron sources, with the power emitted ( ν L ν ) peaking at millimeter wavelengths. As a consequence, it has been widely accepted that Sgr A* is accretion powered, implying a minimum accretion rate of 10 -10 M /circledot yr -1 . It is presently unclear how this emission is produced, evidenced by the variety of models that have been proposed (e.g., Narayan et al. 1998; Blandford & Begelman 1999; Falcke & Markoff 2000; Yuan et al. 2002, 2003; Loeb & Waxman 2007). Models in which the emission arises directly from the accreting gas have been subsumed into the general class of RIAFs, defined by the generally weak coupling between the electrons, which radiate rapidly, and the ions, which efficiently convert gravitational potential energy into heat (Narayan et al. 1998). This coupling may be sufficiently weak to allow accretion rates substantially in excess of that required to explain the observed luminosity with a canonical AGN radiative efficiency of 10%. However, the detection of linear polarization in Sgr A* above 100GHz (Aitken et al. 2000; Bower et al. 2001, 2003; Marrone et al. 2006) and subsequent measurements of the Faraday rotation measure (Macquart et al. 2006; Marrone et al. 2007), have implied that the accretion rate near the black hole is much less than the Bondi rate, requiring the existence of large-scale outflows (Agol 2000; Quataert & Gruzinov 2000). \nRelating the outflow to the properties of the accretion flow requires an ab initio calculation that is presently not possible. Nevertheless, a number of authors have studied this relationship in the context of a variety of simplifying assumptions, with large-scale general-relativistic magnetohydrodynamic and radiative-hydrodynamic simulations playing a central role (De Villiers et al. 2005; McKinney 2006; Hawley & \nFigure 2. Comparison of the spectrum of the most probable accretion model and the observed SED of Sgr A*. Orange circles are from Yuan et al. (2004), and references therein, for which the errorbars are indicative of the variability. Yellow squares are coincident flux measurements from Marrone (2006), for which the errorbars are indicative of the intrinsic measurement error. The green bar shows the flux range of the compact component inferred from fitting accretion models to the mm-VLBI data (Section 5. In addition to the most probable model, spectra are shown for emission from a rapidly rotating black hole ( a = 0 . 998) as seen nearly face on ( θ = 1 · , green dotted) and edge on ( θ = 90 · , purple dotted), indicating the range of variation within the image library. Finally, for reference the contribution from the thermal (red dash) and nonthermal (blue long-dash) are shown. Note in particular that at 1.3mm the nonthermal contribution is not negligible. \n<!-- image --> \nKrolik 2006; Beckwith et al. 2008; McKinney & Blandford 2009; Tchekhovskoy et al. 2010; Dexter et al. 2010; Penna et al. 2010; Kurosawa & Proga 2009). In these it has been found that the structure and dynamics of the outflow critically depends upon the initial conditions. The applicability of the MHD prescription to Sgr A* is still unclear, where the accretion rate is sufficiently low that non-MHD effects may become important (Sharma et al. 2006, 2007). More importantly, most of these approaches do not model the electron heating (beyond ad hoc prescriptions) and none model the production of nonthermal electrons (see, e.g., Mo'scibrodzka et al. 2009; Dexter et al. 2010; Shcherbakov et al. 2010). Furthermore, simulations are computationally expensive to produce. For these reasons we adopt a simple, self-similar model for the accretion flow which includes substantial mass loss. \nFor concreteness, as in Broderick & Loeb (2006a), we follow Yuan et al. (2003) and employ a model in which the accretion flow has a Keplerian velocity distribution, a population of thermal electrons with density and temperature \nne , th = n 0 e , th ( r r S ) -1 . 1 e -z 2 / 2 ρ 2 (2) \nand \nTe = n 0 e ( r r S ) -0 . 84 , (3) \nrespectively, and a toroidal magnetic field in approximate ( β = \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 3. Best fit images and visibilities for the three models considered: symmetric gaussian (top), asymmetric gaussian (middle) and BL06 accretion flow (bottom). For each model we show the intrinsic flux distribution (left), flux distribution after interstellar electron scattering (center) and the visibility amplitudes (right). For reference the locations of the observed visibilities (over all epochs) are shown by the white points. In all plots the intensity scales linearly with I and V . \n<!-- image --> \n10) equipartition with the ions (which are responsible for the majority of the pressure), i.e., \nB 2 8 π = β -1 ne , th mpc 2 r S 12 r . (4) \nIn all of these, r S = 2 GM / c 2 is the Schwarzschild radius, ρ is the cylindrical radius and z is the vertical coordinate. Inside of the innermost-stable circular orbit (ISCO) we assume the gas is plunging upon ballistic trajectories. In principle the plunging gas can still radiate, though in practice it contributes little to the overall emission due to the large radial velocities it develops. In the case of the thermal quantities the radial structure was taken from Yuan et al. (2003), and the vertical structure was \ndetermined by assuming that the disk height is comparable to ρ . Note that all of the models we employ necessarily have the spin aligned with the orbital angular momentum of the accretion flow. For the regions that dominate the mm emission, this assumption is well justified due to disk precession and viscous torques, though it may be violated at large distances. \nThermal electrons alone are incapable of reproducing the nearly-flat spectrum of Sgr A* below 43GHz. Thus it is necessary to also include a nonthermal component. As with the thermal components, we adopt a self-similar model for a population of nonthermal electrons, \nne , nth = n 0 e , nth ( r r S ) -2 . 02 e -z 2 / 2 ρ 2 , (5) \nwith a power-law distribution corresponding to a spectral index of 1 . 25 and cut off below Lorentz factors of 10 2 (consistent with Yuan et al. 2003). The radial power-law index was chosen to reproduce the low frequency spectrum of Sgr A*, and is insensitive to the black hole properties due to the distant location of the long-wavelength emission. \nThe primary emission mechanism at the wavelengths of interest is synchrotron, arising from both the thermal and nonthermal electrons. We model the emission from the thermal electrons using the emissivity described in Yuan et al. (2003), appropriately altered to account for relativistic effects Broderick & Blandford (see, e.g., 2004). Since we perform polarized radiative transfer via the entire complement of Stokes parameters, we employ the polarization fraction for thermal synchrotron as derived in Petrosian & McTiernan (1983). In doing so, we have implicitly assumed that the emission due to thermal electrons is isotropic, which while generally not the case is unlikely to change our results significantly. For the nonthermal electrons, we follow Jones & O'Dell (1977) for a power-law electron distribution, with an additional spectral break associated with the minimum electron Lorentz factor. For both emission components the absorption coefficients are determined directly via Kirchhoff's law. Images are then produced using the fully relativistic ray-tracing and radiative transfer schemes described in Broderick & Loeb (2006a,b) and Broderick (2006). An example image and associated visibilities are shown in the bottom line of Figure 3. \nBecause Yuan et al. (2003) neglected relativistic effects and assumed spherical symmetry, it is not directly applicable here. For these reasons, as in Broderick & Loeb (2006a), the coefficients ( n 0 e , th , T 0 e , n 0 e , nth ) were adjusted to fit the radio spectral energy distribution (SED) of Sgr A*, shown in Figure 2. The 1 . 3mm and 0 . 43mm points (orange squares) were measured coincidentally, and the errors represent the intrinsic measurement error (Marrone 2006). All other points (red circles) are taken from Yuan et al. (2004) (and references therein) and were obtained by averaging over multiple epochs. As a result the errorbars represent the range of variability, and are correspondingly larger. \nAs in Broderick et al. (2009) we systematically fit Sgr A*'s SED at a large number of positions in the spin-inclination ( a -θ , where θ is the viewing angle relative to the disk axis) parameter space, specifically at values of a ∈ { 0 , 0 . 1 , 0 . 2 , ..., 0 . 9 , 0 . 99 , 0 . 998 } for all θ ∈ { 1 · , 10 · , 20 · , ..., 80 · , 90 · } , producing a tabulated set of the coefficients ( n 0 e , th , T 0 e , n 0 e , nth ) at 120 points in the a -θ parameter space. In all cases it was possible to fit the SED with extraordinary precision, with reduced χ 2 < 0 . 3 in all cases and typically reduced χ 2 /similarequal 0 . 17. This is likely a consequence of employing the variability-determined errorbars on the non-coincident flux measurements. Over the a -θ plane the χ 2 was remarkably uniform, implying that on the basis of the spectra alone it is difficult to constrain the parameters of our simple model. Nevertheless, the dynamical properties of the disk manifest themselves in the breadth of the sub-millimeter bump. Models with low spin and/or low θ have little Doppler shifting, and correspondingly narrow bumps. In contrast, models with large spins ( a > 0 . 9) viewed edge-on ( θ > 80 · ) have broad bumps, and are responsible for the relatively larger, though still small, χ 2 . From the tabulated values, the coefficients are then obtained at arbitrary a and θ using high-order polynomial interpolation. \nAn example spectrum, resulting from the above procedure, is shown in Figure 2, corresponding to a = 0 . 01 and θ = 66 · . \nIn addition, the individual contributions from the thermal and nonthermal components are also shown (though the crossabsorption is neglected). The necessity of the nonthermal electrons is clearly illustrated at long wavelengths, where the thermal contribution is negligible. The transition from nonthermally dominated to thermally dominated occurs near 2mm, though the precise location depends upon a and θ . However, note that at no point can the nonthermal component be neglected. Specifically, at 1 . 3mm the nonthermal component is still responsible for roughly 30% of the emission. As a result, efforts to model the millimeter image of Sgr A* without accounting for the nonthermal component can produce order unity systematic errors in parameter estimation. \nDuring the mm-VLBI observations Sgr A*'s flux varied by roughly 30%. We model this as a variable accretion rate, moving the electron density normalization up and down. In practice, we reduced the electron density normalization by an amount sufficient to produce a total flux of 2 . 5Jy, and then multiplied the resulting images by a correction faction during the mm-VLBI data analysis. Because the source is not uniformly optically thin, this is not strictly correct, though this makes a small change to the images themselves. For the purpose of the mm-VLBI data analysis (described below) we produced 9090 images, with flux normalized as described above, at a ∈ { 0 , 0 . 01 , 0 . 02 , ..., 0 . 98 , 0 . 99 , 0 . 998 } for each θ ∈ { 1 · , 2 · , ..., 89 · , 90 · } . We then produce models with arbitrary position angles, ξ , by rotating the image on the sky. For this purpose we define ξ such that at 0 · the projected spin vector points north, and as ξ increases points progressively more eastward 3 . \nThe Faraday rotation measures observed in Sgr A* are produced at much larger radii than those of interest in direct imaging experiments. Nevertheless, it is worth noting that the models employed here are broadly consistent with the polarization observations, though breaks in the radial power-laws which define the properties of the thermal electron component may be required at large spins. \nFor the purposes of fitting the mm-VLBI visibilities, during each observation epoch this model has 4 parameters: spin ( a ), viewing angle ( θ ), position angle ( ξ ) and flux normalization ( V 00). When we analyze multiple epochs together the parameters defining the orientation of the system ( a , θ , ξ ) will be held fixed, while those corresponding to the time-variable accretion rate ( V 00) will be allowed to vary, though a full discussion will have to await Section 5.", '3.2. Gaussian Flux Distributions': 'For comparison we consider two gaussian flux distributions. These differ from the accretion flow model described above in that they are purely phenomenological, without any clear physical motivation and thus not constrained at all by the spectral and polarization properties of Sgr A*. As a result, we might expect these to be intrinsically less likely than physically motivated models that are already chosen to be consistent with these properties. Nevertheless, we will consider them on equal footing with the BL06 model discussed above. For reasons that will become clear, we consider both symmetric and asymmetric gaussian intrinsic flux distributions. \nWe may write the asymmetric gaussian flux distribution as \nI = V 00 exp ( -α 2 M 2 σ 2 M -α 2 m 2 σ 2 m ) , (6) \n3 Note that this is opposite the definition employed in Broderick et al. (2009). \nwhere α M , m and σ M , m are the angular coordinates and widths in the major/minor axis directions. This is fully defined once σ M , m and the position angle of the major axis is given. However, we choose to parametrize the asymmetric gaussian in terms of a single width, σ , an anisotropy parameter, A , and the position angle: \nI = V 00 exp ( -α 2 2 σ 2 -A α 2 2 σ 2 cos2 ϖ ) , (7) \nwhere α = √ α 2 M + α 2 m and ϖ is the angular coordinate measured from the position angle. The σ and A are related to the σ M , m by \n1 σ 2 = 1 2 σ 2 m + 1 2 σ 2 M and A σ 2 = 1 2 σ 2 m -1 2 σ 2 M . (8) \nExample gaussian images, with associated visibilities, are shown in the top two lines of Figure 3. \nClearly, for isotropic configurations (i.e., σ M = σ m ), A =0 and σ = σ M , m . More generally, σ m /σ M = √ (1 -A ) / (1 + A ). Thus, this parametrization has the virtue of separately emphasizing size (via σ ) and asymmetry (via A ) in the image. Note that this model has precisely the number of free parameters as the accretion flow model described above: those describing the image morphology, ( σ, A , ξ ), and the flux normalization, V 00, for each epoch. \nWhile the symmetric gaussian models are obviously a subset of the asymmetric gaussian models (corresponding to when A vanishes), we must be careful to distinguish the number of free parameters. In this case, A and ξ are superfluous, and for each epoch we have only 1 parameter.', '3.3. Interstellar Electron Scattering': 'The effect of interstellar electron scattering in the direction of Sgr A* have been carefully characterized empirically by a number of authors. This has been found to be consistent with convolving the source with an asymmetric gaussian, with major axis nearly aligned with east-west, and a λ 2 wavelength dependence. We employ the model from Bower et al. (2006), which has major axis oriented 78 · east of north, with associated full width at half-maximum for the major and minor axes given by \nFWHM ES M = 1 . 309 ( λ 1cm ) 2 mas , FWHM ES m = 0 . 64 ( λ 1cm ) 2 mas , (9) \nrespectively. In practice, the interstellar electron scattering convolution was effected in the u -v plane, where the convolution reduces to a multiplication.', '4. BAYESIAN DATA ANALYSIS': 'In fitting the observed visibilities we have two primary goals: choosing among various possible model flux distributions and estimating the parameters of these models. Both of these may be naturally accomplished within the context of Bayesian analysis. Here we briefly summarize how we do this. \nWe define the likelihood, p ( V | q ), for observing the visibilities V given the model parameters q , as described in Broderick et al. (2009). From this we obtain the log-likelihood, which we refer to as χ 2 : \nχ 2 ≡ -2log p ( V | q ) + C , (10) \nwhere the normalization constant depends only upon the particulars of the data and, since we will only be interested in comparing identical data sets, will henceforth be ignored. When only detections are considered this reduces trivially to the standard definition of χ 2 . Here it differs only due to the presence of an upper-limit upon the visibility along the CARMAJCMT baseline during the 2007 epoch.', '4.1. Model Comparison': "To compare the significance of different models we make use of the Bayesian Information Criterion (BIC) and the Akaike Information Criterion (AIC). Both of these are discussed in detail within the context of astrophysical observations by Liddle (2007) and Takeuchi (2000), and references therein. Thus, here we only define and summarize the properties of these statistics. \nIn terms of the smallest effective χ 2 for a given model, the BIC is defined by \nBIC ≡ χ 2 min + k ln N , (11) \nwhere k is the number of model parameters and N is the number of data points. Note that this is simply the χ 2 statistic penalized for models with large numbers of parameters. Assuming that the data points are independent (likely true) and identically distributed (nearly true), this is related to the posterior probabilities of two different models, M 1 and M 2, by \np ( M 1 | V ) p ( M 2 | V ) = p ( M 1) p ( M 2) e -(BIC1 -BIC2) / 2 , (12) \nwhere p ( M 1 , 2) is the prior on model M 1 , 2. Thus, up to the unknown priors, the BIC is a measure of the posterior probability for a given model. If we further assume that p ( M 1) = p ( M 2), the difference in BIC's gives the relative posterior probabilities directly. Therefore the model with the lowest BIC is preferred. Similarly, the AIC is defined by \nAIC ≡ χ 2 min + 2 k + 2 k ( k + 1) N -k -1 (13) \nwhere we have included a correction appropriate for when N is small (Burnham & Anderson 2002, 2004). This is very similar to the BIC: the χ 2 statistic penalized by a factor depending upon the number of model parameters, though with a somewhat different penalty. Unlike the BIC, the AIC is not directly related to the posterior probability of a given model. Rather it is an approximate measure of the difference between the true data distribution and the modeled data distribution. Nevertheless, it is possible to interpret the AIC in terms of a model likelihood in a way identical to the BIC. \nFor either criterion, lower values are preferred, with the relative significance given by \nwij = e -(IC i -IC j ) / 2 , (14) \nwhere IC may be replaced by BIC or AIC. The ∆ IC are conventionally judged on the Jeffrey's scale, which sets ∆ IC > 5 as 'strong' and ∆ IC > 10 as 'decisive' evidence against the model with the higher IC. However, here we describe these in terms of the typical σ as well, with model i being excluded at the n σ level if there exists a model j for which wij is less than the associated cumulative normal probability (e.g., 1 σ implies that wij < 0 . 32, 2 σ implies that wij < 0 . 05, 3 σ implies that wij < 0 . 003, etc.).", '4.2. Parameter Estimation': 'For the accretion flow model we have the additional problem of identifying the most likely model parameters. The procedure we use to estimate the posterior probabilities is identical to that described in Broderick et al. (2009). In particular, we assume flat priors on all of the visibility normalizations, V00 , the spin magnitude, a , and an isotropic prior on the spin direction ( θ,ξ ). Since we are primarily interested in the estimates for the black hole spin, we present the posterior probability of a marginalized over the V00 , p ( a ). We also construct marginalized posterior probability distributions of a , θ , and ξ in the normal way (for specific definitions of p ( a ), p ( θ ), and p ( ξ ), see Broderick et al. 2009).', '5. MODEL FITTING': 'A number of important implications follow from computations of the relevant χ 2 for the three image models described in Section 3. These include whether or not we are justified in comparing mm-VLBI observations obtained at different times, the symmetry of the image and the importance of physics for reproducing the measured visibilities.', '5.1. Consistency of the 2007 & 2009 Epochs': "The dynamical timescale of the Sgr A* is comparable to the orbital period at the ISCO, as measured at Earth, and ranges from roughly 4 min to 30 min, depending upon spin. As a result it is not at all clear that we may ignore the possibility of variability when attempting to model mm-VLBI observations spanning many nights, let alone years. Broderick et al. (2009) took special pains to ensure that the visibilities measured on the two consecutive days were consistent with a single, static underlying flux distribution. This was done by comparing fits to the individual days. Here we repeat this analysis for the 2009 observations as well. \nIn 2007, at 2 . 4Jy, the luminosity of Sgr A* was anomalously low and stable over the two observation days. This is not the case during the 2009 observation, during which Sgr A* exhibited a dramatic brightening on the third day. During the preceding two days the luminosity of Sgr A*'s compact component, corresponding to scales smaller than 10 2 r S, was significantly smaller than that associated with the 2007 observations. Thus it is clear from the outset that we are not justified in comparing a single, static model to the observations. Instead, we begin with the ansatz that the morphology of Sgr A*'s image is fixed, with the flux variability being driven by changes in the accretion rate on day-to-day timescales. While this period is considerably larger than the 30 min timescale over which the properties of the accretion flow may change, it is justified in part by the stability of Sgr A*'s luminosity on these scales as well as the intrinsically short duration ( ∼ 2hr) of the observations each night. Therefore, we separate the data into 4 epochs: 2007, 2009.95, 2009.96 and 2009.97, corresponding to the data obtained in 2007 and on days 95, 96 and 97 of 2009, respectively. For all epochs we keep the parameters that define the image morphology fixed, e.g., ( a , θ, ξ ), ( σ, A , ξ ), or σ , but allow the overall flux normalization to vary from epoch to epoch. Upon fitting each epoch separately, and all epochs together, we may ask if the resulting parameter likelihoods are consistent with each other, i.e., check if our ansatz is self consistent. We will remark upon this further in the sections describing the fits for the individual models; however here it is sufficient to note that in all cases we find that the epochs are indeed consistent with a single underlying image morphology. \nFigure 4. Symmetric gaussian χ 2 's as a function of σ , at the most likely V00 , for the 2007, 2009.95, 2009.96, and 2009.97 epochs, as well as when all epochs are combined. The minimum χ 2 , number of visibility observations ( N ) and fit parameters ( k ) are listed in each plot. In all cases the range shown corresponds to min( χ 2 ) -1 to min( χ 2 ) + 6. For references the 1 σ error estimate is shown by the dotted errorbars for each data set. \n<!-- image -->", '5.2. Symmetric Gaussian': 'We begin with the symmetric gaussian model, which is primarily sensitive to the characteristic size of the image. In this case for each epoch there are 2 parameters: σ and V 00. The minimum χ 2 is shown as a function of σ for each epoch in Figure 4. With the exception of epoch 2009.96, the reducedχ 2 is comparable to unity. Over all epochs the σ with the highest likelihood varies over 1 . 5 µ as, well within the single-epoch uncertainty (defined by the region in which χ 2 is within unity of the minimum value). In particular, there are no trends distinguishing either the much earlier 2007 epoch or the considerably brighter 2009.97 epoch. Thus we conclude that the characteristic size of Sgr A* did not vary substantially from one epoch to the next, despite considerably changes in its luminosity. \nThe best fit intrinsic source size is σ = 15 . 8 ± 0 . 2 µ as (FWHM=37 . 2 ± 0 . 5 µ as) with the flux normalization provided in Table 1, ranging from 2 . 07Jy to 2 . 81Jy over the various epochs. The associated reducedχ 2 is 1 . 16, with 66 degrees of freedom. Upon convolving with the interstellar electron scattering, which along the general direction of the baselines SMT-JCMT and CARMA-JCMT baselines has a FWHM of width of 22 µ as, we find the FWHM of the broadened image is 43 . 2 ± 0 . 6 µ as. This is in excellent agreement with the inferred size of 43 + 5 -3 found by Doeleman et al. (2008) on the basis of the 2007 epoch alone 4 . It is also in excellent agreement with the inferred sizes of 41 . 3 + 1 . 8 -1 . 4 , 44 . 4 + 1 -1 , and 42 . 6 + 1 -1 found by Fish et al. (2010) for epochs 2009.95, 2009.96, and 2009.97, respectively 4 . The associated intrinsic and scatter-broadened image, with the associated visibilities are shown in the left, center and \n4 Here we have quoted the 1 σ errors upon their result in order to provide a direct comparison. \nTable 1 Model Fitting Results Summary \n| Model | k | χ 2 | DoF | χ 2 / DoF | V 2007 00 | V 2009 . 95 00 | V 2009 . 96 00 | V 2009 . 97 00 | BIC | w BIC i , BL 06 | AIC | w AIC i , BL 06 |\n|---------------------|------------------|------------------|------------------|------------------|------------------|------------------|------------------|------------------|------------------|-------------------|------------------|-------------------|\n| Estimated Errors | Estimated Errors | Estimated Errors | Estimated Errors | Estimated Errors | Estimated Errors | Estimated Errors | Estimated Errors | Estimated Errors | Estimated Errors | Estimated Errors | Estimated Errors | Estimated Errors |\n| Symmetric Gaussian | 5 | 76.78 | 66 | 1.16 | 2.37 | 2.08 | 2.03 | 2.88 | 98.1 | 5 × 10 - 4 | 87.7 | 8 × 10 - 5 |\n| Asymmetric Gaussian | 7 | 61.50 | 64 | 0.961 | 2.53 | 2.25 | 2.23 | 3.06 | 91.3 | 1 × 10 - 2 | 77.3 | 1 × 10 - 2 |\n| BL06 | 7 | 53.09 | 64 | 0.830 | 2.45 | 2.18 | 2.16 | 3.00 | 82.9 | 1 | 68.9 | 1 |\n| Implied Errors | Implied Errors | Implied Errors | Implied Errors | Implied Errors | Implied Errors | Implied Errors | Implied Errors | Implied Errors | Implied Errors | Implied Errors | Implied Errors | Implied Errors |\n| Symmetric Gaussian | 5 | 92.51 | 66 | 1.40 | 2.37 | 2.08 | 2.03 | 2.88 | 114 | 4 × 10 - | 5 103 | 9 × 10 - 6 |\n| Asymmetric Gaussian | 7 | 74.10 | 64 | 1.16 | 2.53 | 2.25 | 2.23 | 3.06 | 104 | 6 × 10 - 3 | 89.9 | 6 × 10 - 3 |\n| BL06 | 7 | 63.97 | 64 | 1.00 | 2.45 | 2.18 | 2.16 | 3.00 | 93.8 | 1 | 79.7 | 1 | \nright panels of the top row of Figure 3.', '5.3. Asymmetric Gaussian and Image Anisotropy': "During the 2007 epoch the mm-VLBI visibilities are concentrated nearly exclusively along a single line, oriented nearly east-west (see the top-left panel of Figure 1). However, upon including the 2009 epochs, the portion of the u -v plane sampled covers a bow-tie shaped region with opening angle 26 · (see the bottom-center panel of Figure 1). While this is insufficient to generate an image directly, the coverage is sufficient to address the gross angular structure of Sgr A*'s image. Where the symmetric gaussian provides a phenomenological way in which to estimate the typical size of Sgr A*'s emitting region, an asymmetric gaussian can begin to probe its symmetry. \nFigure 5 shows the minimum χ 2 (or equivalently, the maximum likelihood) as a function of the average size, σ , and anisotropy parameter, A . The four left panels show this for the individual epochs, while the large right panel shows this for the combined data set. Unlike the symmetric gaussian model, the likely regions have somewhat different morphologies. This is due to the different coverage of the u -v plane during the various epochs (for example, the likely regions for epochs 2007 and 2009.95 are similar because the u -v coverage during these epochs is similar). Nevertheless, the region preferred by the combined data sets is present in all cases, implying that as with the symmetric gaussian all epochs are consistent with a single underlying image morphology. During this time the flux normalization of the compact component varied from 2 . 23Jy (2009.96) to 3 . 06Jy (2009.97). \nFor all epochs the reducedχ 2 is nearly unity, ranging from 0 . 45 (2007) to 1 . 34 (2009.95). The most likely configuration is highly asymmetric, with σ = 20 . 5 + 0 . 3 + 0 . 5 -0 . 8 -1 . 3 µ as, A = 0 . 70 + 0 . 03 + 0 . 05 -0 . 1 -0 . 18 , and ξ = -19 · + 3 · + 6 · -1 · -2 · , corresponding to a major-minor axis ratio of more than 2 . 4 + 0 . 2 + 0 . 3 -0 . 4 -0 . 6 , with symmetric models highly disfavored. The resulting FWHMs of the minor and major axes are then 37 ± 1 µ as and 88 ± 9 µ as, though these are significantly correlated due to the substantially larger fractional error on A in comparison to that on σ . The intrinsic image, scatterbroadened image and associated visibilities of the most likely configuration is shown in the middle row of Figure 3. \nThe χ 2 for the combined data set is 61 . 5, and much lower than that found for the symmetric case. As described in Section 4, a decrease in χ 2 is expected given the two additional addition of two parameters. However, the various ICs, given in Table 1, provide a means for identifying significantly lower χ 2 . The best fit asymmetric model has a BIC that is 6.8 lower than the best fit symmetric model, and an AIC that is 10.4 lower than \nthe best fit symmetric model. These provide 'strong' evidence against symmetric models for the image of Sgr A*, ruling these out at 2 . 6 σ (BIC) and 3 . 2 σ (AIC) levels, in terms of the relative significance. That is, despite the limited visibility coverage in the u -v plane, the existing mm-VLBI observations can conclusively detect asymmetric structure in Sgr A*.", '5.4. Accretion Flow and Implications of Physics': "The images of the BL06 model, described in Section 3.1, are characterized by asymmetric crescents. These arise due to the combination of gravitational lensing, the relativistic orbital motion and the opacity of the underlying accretion flow. The size and extent of the crescent depends upon the spin and inclination of the system, with nearly face-on disks (small θ ) appearing annular. \nBecause we have only have access to the visibility magnitudes, configurations rotated by 180 · are indistinguishable, imposing an unavoidable ambiguity upon any results. In addition, despite opacity, the images exhibit nearly exact symmetry between configurations viewed from above the equatorial plane (i.e., θ < 90 · ) and those viewed from below the equatorial plane (i.e., θ > 90 · ) at equal inclinations. As a consequence, there is also an ambiguity in the line-of-sight component of the spin vector. For this reason, here we discuss only 0 · ≤ θ ≤ 90 · . However we have performed the analysis for 90 · ≤ θ ≤ 180 · as well, finding no statistically significant differences in the parameter estimates. \nFigure 6 shows the minimum χ 2 (maximum likelihood) as a function of spin, a , and viewing angle, θ . As with Figure 5, the four panels on the left show this for the individual epochs, while the large panel on the right shows χ 2 for the combined data set. The morphology is similar in all cases, with the different u -v coverage during the different epochs manifesting itself primarily in the size of the likely region. As a result, we conclude that again the four epochs are consistent with a single underlying image morphology. Over all of the observations the flux normalization varied from 2 . 16Jy (2009.96) to 3 . 00Jy (2009.97), and is at all times sufficiently close to the value of 2 . 5Jy used to compute the images of the accretion flow. The intrinsic image, scatter-broadened image and corresponding visibilities of the most probable configuration (not necessarily the lowest χ 2 , see Section 6) is shown in the bottom row of Figure 3. \nAt 53.09, the χ 2 for the BL06 model (corresponding to a reducedχ 2 of 0.830) is the smallest of any model we consider. Again we may assess the significance of the this by appealing to the ICs described in Section 4. In this case the BIC and AIC are given by 82 . 9 and 68 . 9, respectively. These are much \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 5. Asymmetric gaussian χ 2 's as a function of σ and A , at the most likely ξ and V00 , for the 2007, 2009.95, 2009.96, and 2009.97 epochs, as well as when all epochs are combined. The minimum χ 2 , number of visibility observations ( N ) and fit parameters ( k ) are listed in each plot. In all cases the color map ranges from min( χ 2 ) (blue) to min( χ 2 ) + 6 (red). \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 6. BL06 accretion flow χ 2 's as a function of a and θ , at the most likely ξ and V00 , for the 2007, 2009.95, 2009.96, and 2009.97 epochs, as well as when all epochs are combined. The minimum χ 2 , number of visibility observations ( N ) and fit parameters ( k ) are listed in each plot. In all cases the color map ranges from min( χ 2 ) (blue) to min( χ 2 ) + 6 (red). \n<!-- image --> \nlower than those from the symmetric gaussian, providing 'decisive' evidence against symmetric configurations. Both are also below those of the asymmetric gaussian by 8 . 4 (since the number of parameters is the same in both models), implying 'strong' evidence for the physically motivated accretion flow model in contrast to the phenomenological asymmetric gaussian. This corresponds to a 2 . 9 σ confidence level in terms of the relative significance of the two models; i.e., the physically motivated accretion models are more than 67 times as likely as the phenomenological asymmetric gaussian models, and more than 2 × 10 3 times as likely as symmetric gaussian models. However, for two reasons this actually understates the case. \nFirst, the reducedχ 2 of the BL06 models is significantly less than unity. Indeed, with 64 degrees of freedom we expect \na χ 2 lower than that obtained (53.09) only 17% of the time. This suggests that the errors on the mm-VLBI visibilities have been over-estimated by a moderate amount. On the other hand, we may measure the 'true' errors by assuming that the BL06 model gives a sufficiently close approximation to the true image flux by renormalizing them until the reducedχ 2 is unity. This requires a roughly 10% reduction in the errors quoted in Section 2. This in turn alters the χ 2 , BIC and AIC for the other models as well. These values are listed under the 'Implied Errors' section of Table 1. The net effect is to increase the significance with which the asymmetric gaussian is ruled out in favor of the BL06 accretion model to 3 . 2 σ . \nSecond, the accretion flow models should generally be preferred on the basis of their motivation alone. That is, there is a strong prior in favor of physically motivated models by virtue \nFigure 7. Posterior probability of a given spin vector, marginalized over the V00 , obtained using the 2007, 2009.95, 2009.96, and 2009.97 epochs. Each sub-panel shows the posterior probability as a function of a and θ at fixed ξ , set to the value shown in the lower-left corner of each. In all cases the center panel shows the most probable position angle when all epochs are combined, and the color scheme spans identical ranges. Contours demark the 1 σ (solid), 2 σ (dashed), and 3 σ (dotted) regions, defined by cumulative probability. \n<!-- image --> \nof their design and connection to an existing body of knowledge. Furthermore, in the case of Sgr A* this includes the fact that these models were constrained to fit the pre-existing spectral data as well, something that is only possible because the physics governing the accretion flow provides a means to relate the properties of images at different wavelengths. \nThe fact that the BL06 model provides a significantly better fit to the mm-VLBI data implies that we can now distinguish phenomenological and physically motivated models on the basis of mm-VLBI observations alone. Since the most prominent features of the accretion flow image are due to the generic properties of black hole accretion flows, namely, the spacetime and orbital motion, this is likely to be robust among all radiatively inefficient accretion flow models for Sgr A*. That is, even with the extremely sparse u -v coverage presently available it is already possible to probe signatures of general relativity and accretion physics in the image itself.", '6. ESTIMATING BLACK HOLE SPIN': "Following Broderick et al. (2009) we produce posterior probability distributions for the parameters defining the BL06 \nmodel, assuming a flat prior on a and isotropic priors upon the spin direction, and marginalizing over the V00 . For the 3-dimensional parameter space defining the vector black hole spin, ( a , θ, ξ ), this is done for each epoch individually (Figure 7) as well as for the combined data set (Figure 8). For each epoch we show the probability distribution for the same position angle slices, defined such that the most probable values from the combined data set is exhibited in the central panel. In all cases we define 1 σ , 2 σ and 3 σ contours in terms of the cumulative probability, and these are shown by the solid, dashed and dotted lines, respectively. The probabilities are normalized to their average value, i.e., if all points were equally probable the probability density would be unity. \nNote that the way in which we have chosen the slices in ξ in Figure 7 does not capture the most likely configuration based upon the 2007 epoch alone (shown in Broderick et al. 2009). Nevertheless, the regions shown are well within the 1 σ region from that epoch. Here we explicitly see that all epochs produce consistent estimates for the spin, with varying degrees of statistical strength. \nThe combined data set dramatically restricts the parame- \nFigure 8. Posterior probability of a given spin vector, marginalized over the V00 when all epochs are combined. Each sub-panel shows the posterior probability as a function of a and θ at fixed ξ , set to the value shown in the lower-left corner of each. In all cases the center panel shows the most probable position angle when all epochs are combined, and the color scheme spans identical ranges. Contours demark the 1 σ (solid), 2 σ (dashed), and 3 σ (dotted) regions, defined by cumulative probability. \n<!-- image --> \nter estimates to a narrow sliver in the 3-dimensional spin parameter space. The most probable values for the spin are a = 0 . 0 + 0 . 64 + 0 . 86 , θ = 68 · + 5 · + 9 · -20 · -28 · , ξ = -52 · + 17 · + 33 · -15 · -24 · , where the errors quoted are the 1 σ and 2 σ errors. At this point the probability density is roughly 350 times that of the average value. In practice, these quantities are much more tightly correlated with \nθ /similarequal 68 · -42 · a ± 3 · ± 5 · (15) \nNote that these are degenerate with solutions for which ξ differs by 180 · , i.e., ξ = 128 · + 17 · + 33 · -15 · -24 · , and for which the line-of-sight component of the spin is reversed, i.e., θ = 112 · + 20 · + 28 · -5 · -9 · . \n9 We do not attempt to determine the systematic uncertainties associated with selecting a particular accretion model. However, we note that a number of efforts to fit alternative accretion flow models to the 2007 epoch have reached consistent results despite differences in the models, suggesting that these results are robust. Furthermore, the quality of the fits to the mm-VLBI visibility and spectral data, concurrently, suggests that the features of the BL06 model responsible for determining the spectral and image properties are generic, and are there- \nfore insensitive to the particulars of the accretion flow models. However, full studies of the systematic errors associated with the particular choices made for the accretion flow properties and the underlying spacetimes are now justified. While beyond the purview of the present paper, we will report upon such efforts elsewhere. \nProbability distributions for each of the spin parameters, marginalized over all others, are shown in Figure 9. In addition to the present case, these are also shown for the analysis of the 2007 epoch for comparison. In these the 1 σ and 2 σ ranges, defined by the cumulative probability, are also shown. In all cases the marginalized probability distributions from the combined data set are much more narrowly peaked than their 2007 epoch counterparts. Nevertheless, they are all consistent at the 1 σ level with those obtained from the 2007 epoch alone. \nIt is now possible to exclude a > 0 . 62 at the 2 σ level, with a = 0 + 0 . 32 + 0 . 62 , substantially preferring non-spinning models. Thus the high-spin island seen in Figure 7 of Broderick et al. (2009) is now eliminated. Similarly, the position angle is now very clearly constrained, choosing the solution less favored by the 2007 epoch data (though still consistent at the 1 σ level). In \n<!-- image --> \n<!-- image --> \nFigure 9. Posterior probability of a given a (left), θ (middle), and ξ (right), marginalized over all other parameters, obtained using all epochs. The filled regions show the 1 σ (dark) and 2 σ (light) regions, defined by cumulative probability. \n<!-- image --> \nthis case we have ξ = -60 · + 14 · + 29 · -5 · -12 · . Finally, the most probable viewing angle is θ = 61 · + 7 · + 24 · -9 · -15 · . This is somewhat higher than the most likely value from the 2007 epoch alone, though well within the 1 σ uncertainty. \nThese estimates for the orientation of the spin vector are in good agreement with a number of other efforts to estimate the properties of Sgr A*'s accretion flow. Estimates based upon fitting longer wavelength observations with numerical models of radiatively inefficient accretion flows produce position angles and inclination estimates with large uncertainties, though these are nevertheless consistent with the results obtained here (Huang et al. 2007). It is also in excellent agreement with more recent attempts to probe the spin orientation using the mmVLBI data from the 2007 epoch (Huang et al. 2009; Dexter et al. 2010). As before it is not possible to assess consistency with models that employ qualitatively different plasma distributions near the black hole (e.g., Markoff et al. 2007), though they tend to imply similarly large viewing angles. \nWe find similar spin orientations to those inferred from modelling of infrared polarization observations of Sgr A*'s flaring emission, though in this case the uncertainties are considerable (e.g., Meyer et al. 2007). Similarly, we find consistency with the spin directions obtained from modeling the spectrum and polarization using general relativistic MHD simulations, despite preferring significantly smaller spin magnitudes (Shcherbakov et al. 2010). \nUnlike the estimates in Broderick et al. (2009), there is no longer any allowed solution for the spin vector that aligns with either of the reported stellar disks in the inner 0 . 2pc of the Galactic center (Genzel et al. 2003). However, our revised position angle estimates are consistent with being aligned with the X-ray feature reported in Muno et al. (2008), bolstering the interpretation of this as related to a possible jet. Note, however, this interpretation may be inconsistent with the low spin magnitudes we prefer.", '7. OPTIMIZING FUTURE OBSERVATIONS': 'The constraints upon the accretion model parameters obtained in the previous sections have implications for future mm-VLBI experiments. With these it is possible both to make predictions for the expected visibilities on the various possible baselines, as well as identify which baselines are most likely to provide substantial improvements to the BL06 model parame- \nter estimation. To estimate these, here we compute the average visibility amplitudes as well as the variance associated with the uncertainty in the model parameters, weighted by the posterior probability distributions we have obtained using the combined mm-VLBI data set, following the method of Fish et al. (2009). \nThe probability-weighted mean visibility profile of the scatter-broadened 230GHz emission from Sgr A* is elongated in the ( + u , + v ) direction (Figure 10). For the moderately high values of θ favored by the mm-VLBI data, the intensity profile is dominated by Doppler-boosted emission on the approaching side of the accretion flow (e.g., in the northeast of lower left panel of Figure 3). This portion of the emission is elongated parallel to the projected direction of the black hole spin vector. Since the effective size of the emission is larger along the projected spin axis, the correlated flux density falls off faster with baseline length for baselines that are sensitive to structure in this direction than in the perpendicular direction. Our estimates of RIAF parameters suggest that a mm-VLBI baseline oriented southwest-northeast will detect more correlated flux density than an equal-length baseline oriented southeastnorthwest. \nThe standard deviation of the visibility amplitudes provides an estimate of which baselines would provide maximal additional constraints on RIAF model parameters (assuming equal sensitivity at all sites). Previous computations based on the 2007 epoch of data indicated that the largest scatter occurred at baseline lengths of approximately 3G λ and with orientations perpendicular to the Hawaii-SMT baseline (Fish et al. 2009). These findings still hold in light of the 2009 data. \nAmong possible observing baselines in the next few years, Chile-SPT and Chile-LMT probe the region of highest standard deviation, followed by baselines between the LMT and the continental US. Our model suggests that the LMT-SMT and LMTCARMA baselines will detect well over 1Jy. The increased sensitivity provided by phased ALMA may be important on the baselines to Chile, as the probability-weighted mean visibility amplitudes are /greaterorsimilar 0 . 1 Jy on the Chile-SMT baseline, /lessorsimilar 0 . 1Jy over most of the ( u , v ) track of the Chile-CARMA baseline, and smaller still on the longer baselines to Chile. However, the standard deviation of the predicted model visibility amplitudes is several × 10mJy on these baselines, leading to uncertainties of tens of percent in model amplitude predictions. Further mmVLBI data, either in the form of higher sensitivity on existing \nFigure 10. Left: Mean predicted visibility amplitudes in Jy. Right: Standard deviation of predicted visibility amplitudes. In both cases the model image was weighted by its posterior probability. \n<!-- image --> \nbaselines or detections on new baselines, will both reduce these uncertainties and test the RIAF model with increasing rigor.', '8. CONCLUSIONS': "The significantly increased number of long-baseline visibilities, significantly larger signal-to-noise, and improved northsouth coverage of the 2009 mm-VLBI observations have already paid substantial dividends in the estimation of the properties of Sgr A*. This is despite the fact that only three independent mm-VLBI stations (JCMT,CARMA,SMT) were used, and therefore the u -v plane remains extremely sparsely populated, with long baselines primarily in the east-west direction. Constraints upon the black hole spin have improved dramatically in all cases, with the spin and viewing angle becoming tightly correlated, with the most probable configuration, a =0 . 0 + 0 . 64 + 0 . 86 , θ =68 · + 5 · + 9 · -20 · -28 · , ξ = -52 · + 17 · + 33 · -15 · -24 · being roughly 350 times as likely as the average probability density, and 25 times as likely as the most probable configuration reported in Broderick et al. (2009). \nDespite the limited north-south coverage, the 2009 mmVLBI data conclusively excludes symmetric gaussian models for the source. The preferred asymmetric gaussian has a majorminor axis ratio of 2 . 4 + 0 . 2 + 0 . 3 -0 . 4 -0 . 6 , oriented with the major axis oriented -19 · + 3 · + 6 · -1 · -2 · east of north. This implies major and minor axis FWHMs of 37 ± 1 µ as and 88 ± 9 µ as, with the symmetric case excluded at 3 . 9 σ significance. Note that this orientation is not aligned with any particular feature in Sgr A*'s vicinity or the properties of the intervening interstellar electron scattering screen. \nIt is natural to give physically motivated models a prior bias over phenomenological models. Nevertheless, even when physically motivated accretion models are compared with asymmetric gaussian models are weighted equally, the accretion models provide a significantly better fits to the mm-VLBI data. Based upon both the BIC and AIC we find strong evidence in favor of the accretion model, corresponding to a posterior probability 67 to 160 times larger than that of the most likely asymmetric gaussian model. This is particularly strik- \ning given the simplicity of the accretion model we consider, suggesting that the image depends primarily upon the gross dynamical and geometric properties of the system: the orbital motion of the accreting material and the strong gravitational lensing by the black hole. In any case, it is clear that we have now entered the era of studying accretion and black hole physics with mm-VLBI. \nOur best fit accretion model requires a black hole spin of a = 0 . 0 + 0 . 64 + 0 . 86 viewed at an angle of θ = 68 · + 5 · + 9 · -20 · -28 · , oriented at a position angle of ξ = -52 · + 17 · + 33 · -15 · -24 · . Ambiguities due to the fact that only amplitudes of the visibilities were measured and the near symmetry of the accretion model images produce degeneracies corresponding to θ ↔ 180 · -θ and ξ ↔ ξ + 180 · , independently. The detection of closure phases will eliminate the ambiguity in position angle. \nDuring all of the mm-VLBI epochs presently available, three or fewer VLBI stations were employed, providing at most three baselines. In practice the two long baselines are nearly collinear, aligned predominantly east-west. The resulting sparse coverage within u -v plane is the primary factor limiting the estimation of black hole and accretion flow parameters. Therefore, despite the success attained thus far, there is a considerable opportunity to improve the constraints upon Sgr A* and its accretion flow dramatically in the near future by including additional VLBI stations. The Event Horizon Telescope is a mm and sub-mm wavelength VLBI network whose goal is to observe, image, and time resolve structures near the black hole event horizon (Doeleman et al. 2009a). Over the next few years, new sites will join the current array, and sensitivities of critical baselines (e.g., those found in Section 7) will be enhanced through technical developments to widen bandwidths and phase together multiple dishes at mm array sites (e.g., CARMA, SMA, ALMA). An important result is that within the next few years, interferometric phase information on long VLBI baselines will become available through robust measurement of the closure phase (Doeleman et al. 2009b), which will provide important new constraints on models of Sgr A*. \nThis work was supported in part by NSF grants AST0907890, AST-0807843, and AST-0905844, and NASA grants NNX08AL43G and NNA09DB30A.", 'REFERENCES': "Agol, E. 2000, ApJ, 538, L121 Aitken, D. K., Greaves, J., Chrysostomou, A., Jenness, T., Holland, W., Hough, J. H., Pierce-Price, D., & Richer, J. 2000, ApJ, 534, L173 Beckwith, K., Hawley, J. F., & Krolik, J. H. 2008, ApJ, 678, 1180 \n- Blandford, R. D., & Begelman, M. C. 1999, MNRAS, 303, L1\n- Bower, G. C., Goss, W. M., Falcke, H., Backer, D. C., & Lithwick, Y. 2006, ApJ, 648, L127\n- Bower, G. C., Wright, M. C. H., Falcke, H., & Backer, D. C. 2001, ApJ, 555, L103\n- -. 2003, ApJ, 588, 331\n- Broderick, A., & Blandford, R. 2004, MNRAS, 349, 994\n- Broderick, A. 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2010MNRAS.405L...1B
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Dark matter haloes determine the masses of supermassive black holes
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2010-01-01
| 14 | 0.48 | 160 |
['hydrodynamics', 'galaxies active', 'galaxies evolution', 'galaxies formation', 'galaxies quasars', 'cosmology theory', '-']
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[]
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The energy and momentum deposited by the radiation from accretion flows on to the supermassive black holes (BHs) that reside at the centres of virtually all galaxies can halt or even reverse gas inflow, providing a natural mechanism for supermassive BHs to regulate their growth and to couple their properties to those of their host galaxies. However, it remains unclear whether this self-regulation occurs on the scale at which the BH is gravitationally dominant, on that of the stellar bulge, the galaxy or that of the entire dark matter halo. To answer this question, we use self-consistent simulations of the co-evolution of the BH and galaxy populations that reproduce the observed correlations between the masses of the BHs and the properties of their host galaxies. We first confirm unambiguously that the BHs regulate their growth: the amount of energy that the BHs inject into their surroundings remains unchanged when the fraction of the accreted rest mass energy that is injected is varied by four orders of magnitude. The BHs simply adjust their masses so as to inject the same amount of energy. We then use simulations with artificially reduced star formation rates to demonstrate explicitly that BH mass is not set by the stellar mass. Instead, we find that it is determined by the mass of the dark matter halo with a secondary dependence on the halo concentration, of the form that would be expected if the halo binding energy were the fundamental property that controls the mass of the BH. We predict that the BH mass, m<SUB>BH</SUB>, scales with halo mass as m<SUB>BH</SUB> ~ m<SUP>α</SUP><SUB>halo</SUB>, with α ~ 1.55 +/- 0.05, and that the scatter around the mean relation in part reflects the scatter in the halo concentration-mass relation.
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https://arxiv.org/pdf/0911.0935.pdf
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{'C. M. Booth 1 /star and Joop Schaye 1': '1 Leiden Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, the Netherlands \n24 October 2018', 'ABSTRACT': 'The energy and momentum deposited by the radiation from accretion flows onto the supermassive black holes (BHs) that reside at the centres of virtually all galaxies can halt or even reverse gas inflow, providing a natural mechanism for supermassive BHs to regulate their growth and to couple their properties to those of their host galaxies. However, it remains unclear whether this self-regulation occurs on the scale at which the BH is gravitationally dominant, on that of the stellar bulge, the galaxy, or that of the entire dark matter halo. To answer this question, we use self-consistent simulations of the co-evolution of the BH and galaxy populations that reproduce the observed correlations between the masses of the BHs and the properties of their host galaxies. We first confirm unambiguously that the BHs regulate their growth: the amount of energy that the BHs inject into their surroundings remains unchanged when the fraction of the accreted rest mass energy that is injected, is varied by four orders of magnitude. The BHs simply adjust their masses so as to inject the same amount of energy. We then use simulations with artificially reduced star formation rates to demonstrate explicitly that BH mass is not set by the stellar mass. Instead, we find that it is determined by the mass of the dark matter halo with a secondary dependence on the halo concentration, of the form that would be expected if the halo binding energy were the fundamental property that controls the mass of the BH. We predict that the black hole mass, m BH , scales with halo mass as m BH ∝ m α halo , with α ≈ 1 . 55 ± 0 . 05 and that the scatter around the mean relation in part reflects the scatter in the halo concentration-mass relation. \nKey words: Cosmology: Theory - Galaxies: Active - Galaxies: Evolution - Galaxies: Formation - Hydrodynamics - Galaxies: Quasars: General', '1 INTRODUCTION': 'Almost all massive galaxies are thought to contain a central supermassive black hole (BH) and the properties of these BHs are tightly correlated with those of the galaxies in which they reside (e.g. Magorrian et al. 1998; Ferrarese & Merritt 2000; Gebhardt et al. 2000; Tremaine et al. 2002; Haring & Rix 2004; Hopkins et al. 2007b; Ho 2008). It is known that most of the mass of the BHs is assembled via luminous accretion of matter (Soltan 1982). The energy emitted by this process provides a natural mechanism by which BHs can couple their properties to those of their host galaxies. Analytic (e.g. Silk & Rees 1998; Haehnelt et al. 1998; Fabian 1999; Adams et al. 2001; King 2003; Wyithe & Loeb 2003; Murray et al. 2005; Merloni & Heinz 2008), semianalytic (e.g. Kauffmann & Haehnelt 2000; Cattaneo 2001; \n/star \nE-mail: [email protected] (CMB) \nGranato et al. 2004; Bower et al. 2006) and hydrodynamical (e.g. Springel et al. 2005; Di Matteo et al. 2005; Robertson et al. 2006; Sijacki et al. 2007; Hopkins et al. 2007a; Di Matteo et al. 2008; Okamoto et al. 2008; Booth & Schaye 2009) studies have used this coupling between the energy emitted by luminous accretion and the gas local to the BH to investigate the origin of the observed correlation between BH and galaxy properties, and the buildup of the supermassive BH population. \nBHs are expected to regulate the rate at which they accrete gas down to the scale on which they are gravitationally dominant. For example, gas flowing in through an accretion disk can become so hot that its thermal emission becomes energetically important. Scattering of the photons emitted by the accreting matter by free electrons gives rise to the so-called Eddington limit. If the accretion rate exceeds this limit, which is inversely proportional to the assumed radiative efficiency of the accretion disk, then the radiative force exceeds the gravitational attraction of the BH and the in- \nflow is quenched, at least within the region that is optically thin to the radiation. \nHowever, observations indicate that the time-averaged accretion rate is far below Eddington (Kollmeier et al. 2006), suggesting the presence of processes acting on larger scales. Indeed, the existence of tight correlations between the mass of the BH and the properties of the stellar bulge indicates that self-regulation may happen on the scale of the bulge ( ∼ 1 kpc; Adams et al. 2001; Hopkins et al. 2007a), far exceeding the radius within which the BH is gravitationally dominant. However, since galaxy-wide processes such as galaxy mergers can trigger gas flows into the bulge (Sanders et al. 1988; Mihos & Hernquist 1994), it is conceivable that BHs could regulate their growth on the scale of the entire galaxy ( ∼ 10 kpc; Haehnelt et al. 1998; Fabian 1999; Wyithe & Loeb 2003) or even on that of the DM haloes hosting the galaxies ( ∼ 10 2 kpc; Silk & Rees 1998; Ferrarese 2002). Finally, it is possible, perhaps even likely, that selfregulation takes place simultaneously on multiple scales. For example, frequent, short, Eddington-limited outbursts may be able to regulate the inflow of gas on the scale of the bulge averaged over much longer time scales. \nIn this paper we investigate, using self-consistent simulations of the co-evolution of the BH and galaxy populations, on what scale the self-regulation of BHs takes place. In Sec. 2 we describe the numerical techniques and simulation set employed in this study. In Sec. 3 we demonstrate that BH self-regulation takes place on the scale of the DM halo, and that the BH mass is determined by the binding energy of the DM halo rather than by the stellar mass of the host galaxy. Throughout we assume a flat ΛCDM cosmology with the cosmological parameters: { Ω m , Ω b , Ω Λ , σ 8 , n s , h } = { 0 . 238 , 0 . 0418 , 0 . 762 , 0 . 74 , 0 . 951 , 0 . 73 } , as determined from the WMAP 3-year data (Spergel et al. 2007).', '2 NUMERICAL METHODS': 'We have carried out a set of cosmological simulations using Smoothed Particle Hydrodynamics (SPH). We employ a significantly extended version of the parallel PMTree-SPH code gadget iii (last described in Springel 2005), a Lagrangian code used to calculate gravitational and hydrodynamic forces on a particle by particle basis. The initial particle positions and velocities are set at z = 127 using the Zeldovich approximation to linearly evolve positions from an initially glass-like state. \nIn addition to hydrodynamic forces, we treat star formation, supernova feedback, radiative cooling, chemodynamics and black hole accretion and feedback, as described in Schaye & Dalla Vecchia (2008), Dalla Vecchia & Schaye (2008), Wiersma et al. (2009a), Wiersma et al. (2009b) and Booth & Schaye (2009) (hereafter BS09) respectively. For clarity we summarize here the essential features of the BH model, which is itself a substantially modified version of that introduced by Springel et al. (2005).', '2.1 The black hole model': "Seed BHs of mass m seed = 10 -3 m g - where m g is the simulation gas particle mass - are placed into every DM halo that contains more than 100 DM particles and does not already \ncontain a BH particle. Haloes are identified by regularly running a friends-of-friends group finder on-the-fly during the simulation. After forming, BHs grow by two processes: accretion of ambient gas and mergers. Gas accretion occurs at the minimum of the Eddington rate, ˙ m Edd = 4 πGm BH m p //epsilon1 r σ T c and ˙ m accr = α 4 πG 2 m 2 BH ρ/ ( c 2 s + v 2 ) 3 / 2 , where m p is the proton mass, σ T is the Thomson cross-section, c is the speed of light, c s and ρ are the sound speed and density of the local medium, v is the velocity of the BH relative to the ambient medium, and α is a dimensionless efficiency parameter. The parameter α , which was set to 100 by Springel et al. (2005), accounts for the fact that our simulations possess neither the necessary resolution nor the physics to accurately model accretion onto a BH on small scales. Note that for α = 1 this accretion rate reduces to the so called BondiHoyle (Bondi & Hoyle 1944) rate. \nAs long as we resolve the scales and physics relevant to Bondi-Hoyle accretion, we could set α = 1. If a simulation resolves the Jeans scales in the accreting gas then it will also resolve the scales relevant for Bondi-Hoyle accretion onto any BH larger than the simulation mass resolution (BS09). We therefore generally set α equal to unity. However, this argument breaks down in the presence of a multiphase interstellar medium, because our simulations do not resolve the properties of the cold, molecular phase, and as such the accretion rate may be orders of magnitude higher than the Bondi-Hoyle rate predicted by our simulations for star-forming gas. We therefore use a power-law scaling of the accretion efficiency such that α = ( n H /n ∗ H ) β in star-forming gas, where n ∗ H = 0 . 1 cm -3 is the critical density for the formation of a cold, star-forming gas phase. The parameter β is a free parameter in our simulations. We set β = 2, but note that the results shown here are insensitive to changes in this parameter when β /greaterorsimilar 2 (see BS09), because in that case the growth of the BHs is limited by feedback. \nEnergy feedback is implemented by allowing BHs to inject a fixed fraction of the rest mass energy of the gas they accrete into the surrounding medium. The energy deposition rate is given by \n˙ E = /epsilon1 f /epsilon1 r ˙ m accr c 2 = /epsilon1 f /epsilon1 r 1 -/epsilon1 r ˙ m BH c 2 , (1) \nwhere /epsilon1 r is the radiative efficiency of the BH, ˙ m accr is the rate at which the BH is accreting gas, and ˙ m BH is the rate of BH mass growth. \nWe set /epsilon1 r to be 0.1, the mean value for radiatively efficient accretion onto a Schwarzschild BH (Shakura & Sunyaev 1973). We vary /epsilon1 f but use /epsilon1 f = 0 . 15 as our fiducial value. It was shown in BS09 that, for /epsilon1 f = 0 . 15, simulations identical to these reproduce the observed redshift zero m BH -m ∗ and m BH -σ relations, where σ is the one-dimensional velocity dispersion of the stars and m ∗ is the galaxy stellar mass. Energy is returned to the surroundings of the BH 'thermally', that is, by increasing the temperature of N heat of the BH's neighbouring SPH particles by at least ∆ T min . A BH performs no heating until it has built up enough of an energy reservoir to heat by this amount. The use of an energy reservoir is necessary in these simulations as otherwise gas will be able to radiate away the energy every timestep. Imposing a minimum temperature increase ensures that the radiative cooling time is sufficiently long for the feedback to be effective. In our fiducial model we set \nTable 1. Numerical parameters of the simulations. From left to right: Simulation identifier, comoving box size (Mpc/ h ), number of both gas and DM particles, final redshift, gas particle mass (10 7 M /circledot /h ), DM particle mass (10 7 M /circledot /h ), maximum physical gravitational softening (kpc/ h ). Each simulation was run multiple times using different values of /epsilon1 f . \n| Name | L | n part | z f | m g | m DM | /epsilon1 max , phys |\n|----------|------|----------|-------|--------|---------|------------------------|\n| L050N256 | 50 | 256 3 | 0 | 8 . 7 | 41 . 0 | 2 |\n| L050N128 | 50 | 128 3 | 0 | 69 . 6 | 328 . 0 | 4 |\n| L012N256 | 12.5 | 256 3 | 2 | 0 . 1 | 0 . 6 | 0.5 | \nN heat = 1 and ∆ T min = 10 8 K but the results are insensitive to the exact values of these parameters (see BS09).", '2.2 The simulation set': 'The simulations employed in the current work use cubic boxes of size 12.5 and 50 comoving Mpc/ h and assume periodic boundary conditions. Each simulation contains either 128 3 or 256 3 particles of both gas and collisionless cold DM. Comoving gravitational softenings are set to 1 / 25 of the mean interparticle separation down to z = 2 . 91, below which we switch to a fixed proper scale. The 12.5 Mpc/ h (50 Mpc/ h ) boxes are evolved as far as redshift two (zero). The numerical parameters of the simulations used in this study are summarized in Table 1. All results presented in this letter are derived from the 50.0 Mpc/ h , 256 3 particle simulations, with the other box sizes and particle numbers employed to demonstrate numerical convergence.', '3 RESULTS AND DISCUSSION': "It is instructive to first consider under what conditions BHs can regulate their growth. To regulate its growth on a mass scale M sr , a BH of mass m BH must be able to inject energy (or momentum) at a rate that is sufficient to counteract the force of gravity on the scale M sr , averaged over the dynamical time associated with this scale. The mass M sr could, for example, correspond to that of the BH, the stellar bulge, or the dark matter (DM) halo. If the BH cannot inject energy sufficiently rapidly, then gravity will win and its mass will increase. Provided that the maximum rate at which it can inject energy increases with m BH (as is for example the case for Bondi-Hoyle and Eddington-limited accretion with a constant radiative efficiency) and provided that this rate increases sufficiently rapidly to counteract the growth of M sr , the BH will ultimately reach the critical mass m BH , crit ( M sr ) required to halt the inflow on the scale M sr . If, on the other hand, m BH /greatermuch m BH , crit ( M sr ), then the BH will quickly quench the accretion flow and its mass will consequently remain nearly unchanged. The BH will in that case return to the equilibrium value m BH , crit ( M sr ) on the time scale which characterises the growth of M sr . \nIf the BH regulates its growth on the mass scale M sr and if m BH /lessmuch M sr , then the critical rate of energy injection required for self-regulation is independent of the mass of the BH. It then follows from Eq. 1 that ˙ m BH ∝ /epsilon1 -1 f , which implies \n( m BH -m seed ) ∝ /epsilon1 -1 f , (2) \nFigure 1. Predicted redshift zero global BH mass density (black diamonds) and normalization of the m BH -σ relation (black plus signs) as a function of the assumed efficiency of BH feedback, /epsilon1 f . Both quantities are normalized to their values in the simulation with /epsilon1 f = 0 . 15, which reproduces the observed relations between the mass of the BH and properties of the stellar bulge. Each point represents a different simulation. For 10 -4 < /epsilon1 f < 1 all data points track the dotted black line, which is a power-law with index minus one. This implies that in this regime BH mass is inversely proportional to m BH , and thus that the BHs inject energy into their surroundings at a rate that is independent of /epsilon1 f , as expected for self-regulated growth on scales that are sufficiently large for the gravity of the BH to be unimportant. The red data points show results from simulations with a mass resolution that is 8 times worse than the fiducial simulation. The blue data points correspond to simulations with 64 times better resolution than our fiducial resolution, but show results for redshift 2 rather than zero. The agreement between the black, red and blue points confirms numerical convergence and demonstrates that the BHs are already self-regulating at redshift 2. \n<!-- image --> \nwhere m seed is the initial mass of the BH. Hence, if the selfgravity of BHs is negligible on the maximum scale on which they regulate their growth and if m BH /greatermuch m seed , then we expect m BH ∝ /epsilon1 -1 f . \nThe black diamonds plotted in Fig. 1 show the predicted global mass density in BHs at redshift zero as a function of /epsilon1 f , the efficiency with which BHs couple energy into the ISM, normalised to the density obtained for /epsilon1 f = 0 . 15. Similarly, the black plus signs indicate the normalisation of the m BH -σ relation divided by that for the /epsilon1 f = 0 . 15 run. The feedback efficiency, /epsilon1 f , is varied, in factors of 4, from /epsilon1 f = 9 . 2 × 10 -6 to /epsilon1 f = 9 . 6, which implies that the fraction of the accreted rest mass energy that is injected (i.e. /epsilon1 r /epsilon1 f ) varies from 9 . 2 × 10 -7 to 0.96. BH mass is clearly inversely proportional to the assumed feedback efficiency for 10 -4 < /epsilon1 f < 1. For /epsilon1 f > 1 the trend breaks down because the BH masses remain similar to the assumed seed mass, in accord with Eq. 2. If we had used a lower seed mass, then the trend would have extended to greater values of /epsilon1 f . The deviation from inverse proportionality that sets in below /epsilon1 f = 10 -4 is more interesting. Such low values yield BH masses that are more than 0 . 15 / 10 -4 ∼ 10 3 times greater than observed, in which case they are no longer negligible compared to the masses of their host galaxies. In that case the critical rate of energy deposition will no longer be independent of m BH and we do not expect Eq. 2 to hold. \nWe have thus confirmed that feedback enables BHs to \nFigure 2. Median m BH -m halo ( left panel ) and m BH -m ∗ ( right panel ) relations for all BHs more massive than 10 m seed . The black curves correspond to a simulation using our fiducial star formation law and the red, dashed curves show the result for a run in which the star formation efficiency was decreased by a factor of 100. In order to isolate the effect of stellar mass, we turned off supernova feedback in both runs. The BH scaling relations therefore differ somewhat from those predicted by our fiducial model, which does include supernova feedback. Baryons dominate the gravitational potential in the central regions of the galaxy when we use our fiducial star formation law, but DM dominates everywhere in the run with the reduced star formation efficiency. While the m BH -m ∗ relation is strongly affected by the change in the star formation efficiency, the relation between BH and halo mass remains invariant. This demonstrates that the BH mass is insensitive to the mass distribution on scales where the stellar mass dominates, and must instead be determined by the mass distribution on larger ( /greatermuch 10 kpc) scales. \n<!-- image --> \nregulate their growth. Moreover, we demonstrated that this self-regulation takes places on scales over which the gravitational influence of the BHs is negligible, provided that the fraction of the accreted rest mass energy that is coupled back into the interstellar medium is /greaterorsimilar 10 -5 . \nTo test whether it is the stellar or the dark matter distribution that determines the mass of BHs, we compare the BH masses in two simulations that are identical except for the assumed efficiency of star formation. One uses our fiducial star formation law, but in the other simulation we reduced its amplitude by a factor of 100, making the gas consumption time scale much longer than the age of the Universe. Because changing the amount of stars would imply changing the rate of injection of supernova energy, which could affect the efficiency of BH feedback, we neglected feedback from star formation in both runs. In the simulation with 'normal' star formation the central regions of the galaxies are dominated gravitationally by the baryonic component of the galaxy, whereas in the simulation with reduced star formation the DM dominates everywhere. Fig. 2 shows the m BH -m halo and m BH -m ∗ relations at redshift 0. While the two runs produce nearly identical BH masses for a fixed halo mass, the m BH -m ∗ relation is shifted to lower stellar masses by more than an order of magnitude in the model with reduced star formation. The insensitivity of the relation between m BH and m halo to the assumed star formation efficiency demonstrates that the BH mass is not set by the gravitational potential on the scale of the galaxy. We have verified that the same result holds at redshift two for the simulations with 64 times better mass resolution. Clearly, stellar mass does not significantly influence the relation between the mass of the BH and that of its host halo. This implies that BH self-regulation occurs on the scale of DM haloes. \nIf the rate by which the BHs inject energy is independent of the assumed feedback efficiency, then we expect the \nFigure 3. The relation between BH mass and DM halo mass for all BHs that belong to central galaxies and have masses greater than 10 m seed . The DM halo mass, m 200 , is defined as the mass enclosed within a sphere, centred on the potential minimum of the DMhalo, that has a mean internal density of 200 times the critical density of the Universe. The grey pixels show the results from our fiducial simulation ( /epsilon1 f = 0 . 15), with the colour of each pixel set by the logarithm of the number of BHs in that pixel. The solid, red line shows the observational determination of the m BH -m halo relation (Bandara et al. 2009) and has a slope of 1.55. The dotted, red lines show the 1 σ errors on the observations. The simulation agrees very well with the observed relation. The value of the slope and the scatter (which correlates with the concentration of the DMhalo) suggest that the halo binding energy, rather than mass, determines the masses of BHs. \n<!-- image --> \nsame to be true for the factor by which BH feedback suppresses star formation. This is confirmed by comparison of the global SFRs in runs with different values of /epsilon1 f (see Fig. 6 of BS09). \nFig. 3 compares the predicted log 10 m BH -log 10 m halo relation with observation (Bandara et al. 2009). The agreement is striking. The slope and normalization of the ob- \nserved log 10 ( m BH / M /circledot ) -log 10 ( m halo / 10 13 M /circledot ) relation are 1 . 55 ± 0 . 31 and 8 . 18 ± 0 . 11 respectively, whereas the simulation predicts 1 . 55 ± 0 . 05 and 8 . 01 ± 0 . 04. Note that the simulation was only tuned to match the normalization of the relations between m BH and the galaxy stellar properties. \nIf the energy injected by a BH is proportional to the halo gravitational binding energy, then, for isothermal models (Silk & Rees 1998), m BH ∝ m 5 / 3 halo . Here we extend these models to the more realistic universal halo density profile (Navarro et al. 1997), whose shape is specified by a concentration parameter, c (we assumed c ∝ v 2 max /v 2 v , where v max and v v are the maximum halo circular velocity and the circular velocity at the virial radius respectively). It is known that concentration decreases with increasing halo mass, c ∝ m -0 . 1 halo (Bullock et al. 2001; Duffy et al. 2008), which then affects BH mass through the dependence of halo binding energy on concentration. If the total energy injected by a BH of a given mass is proportional to the energy required to unbind gas from a DM halo (/suppressLokas & Mamon 2001) out to some fraction of the virial radius, r ej /r v then \nm BH ∝ ( c ( ln(1 + c ) -c/ (1 + c ) ) 2 ) × ( 1 -1 (1 + c r ej r v ) 2 -2 ln(1 + c r ej r v ) 1 + c r ej r v ) m 5 / 3 v . (3) \nInserting c ∝ m -0 . 1 v and computing the logarithmic derivative with respect to m v in the mass range 10 10 M /circledot < m v < 10 14 M /circledot , we find that the slope is a weak function of r ej /r v that varies from 1.50 at r ej = 10 -1 r v to 1.61 at r ej = r v . The close match between theory, simulation and observation suggests that the halo binding energy, rather than halo mass, determines the mass of the BH. \nThe residuals from the m BH -m halo relation (∆log 10 m BH ) are correlated with halo concentration (Spearman rank correlation coefficient ρ = 0 . 29, probability of significance P = 0 . 9998) as would be expected if m BH is sensitive to the halo binding energy. The residuals are also correlated with galaxy stellar mass, though much less strongly ( ρ = 0 . 09; P = 0 . 96). Taken together, these correlations tell us that, at a given halo mass, galaxies with BHs more massive than the average will also contain a larger than average amount of stars, and are hosted by more concentrated haloes. This suggests that the galaxy stellar mass is also determined by the halo binding energy. Thus, outliers in the m BH -m halo relation may still lie close to the mean m BH -m ∗ relation. Furthermore, higher concentrations imply earlier formation times and spheroidal components do indeed typically host old stellar populations. \nIn addition to the 'quasar mode' of feedback discussed in this work, it has recently become clear that a second 'radio mode' may be required to quench cooling flows in galaxy groups and clusters (see e.g. Cattaneo et al. 2009, for a review). Although we do not explicitly include a 'radio mode' in the current work, the AGN feedback prescription explored here is capable of suppressing cooling flows, at least on group scales, providing excellent matches to observed group density and temperature profiles as well as galaxy stellar masses and age distributions (McCarthy et al. 2009). It is known that BHs obtain most of their mass in the 'quasar mode' (Soltan 1982) so any discussion of what detemines the masses of BHs must focus primarily on this mode of accre- \ntion. Finally, the ability of a BH to quench cooling flows in the 'radio mode' is expected to be closely related the virial properties of the hot halo (Cattaneo et al. 2009) and would therefore provide an additional link between BHs and DM haloes over and above what we discuss here and so serve to make any fundamental connection between BH mass and the properties of the DM halo even stronger. \nWe conclude that our simulation results suggest that in order to effectively halt BH (and galaxy) growth, gas must not return to the galaxy on a short timescale. This requires that the BH injects enough energy to eject gas out to scales where the DM halo potential is dominant. The mass of the BH is therefore determined primarily by the mass of the DM halo with a secondary dependence on halo concentration, of the form that would be expected if the BH mass were controlled by the halo binding energy. The tight correlation between m BH and m ∗ is then a consequence of the more fundamental relations between halo binding energy and both m BH and m ∗ .", 'ACKNOWLEDGMENTS': 'The authors thank Marijn Franx and the referee, Andrea Cattaneo, for useful discussions and suggestions. The simulations presented here were run on the Cosmology Machine at the Institute for Computational Cosmology in Durham as part of the Virgo Consortium research programme. This work was supported by an NWO Vidi grant.', 'REFERENCES': 'Adams F. C., Graff D. S., Richstone D. O., 2001, ApJL, 551, L31 \nBandara K., Crampton D., Simard L., 2009, ApJ, 704, 1135 Bondi H., Hoyle F., 1944, MNRAS, 104, 273 Booth C. M., Schaye J., 2009, MNRAS, 398, 53 (BS09) Bower R. G., Benson A. J., Malbon R., Helly J. C., Frenk C. S., Baugh C. M., Cole S., Lacey C. G., 2006, MNRAS, 370, 645 \nBullock J. S., Kolatt T. S., Sigad Y., et al., 2001, MNRAS, 321, 559 \nCattaneo A., 2001, MNRAS, 324, 128 Cattaneo A., Faber S. 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1999PhRvD..59l4014J
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Primordial black hole formation during first-order phase transitions
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1999-01-01
| 7 | 0.45 | 160 |
['-', '-', '-', '-', '-', 'methods numerical', 'black hole physics', 'particles', 'astrophysics', '-', '-']
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[]
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Primordial black holes (PBHs) may form in the early universe when pre-existing adiabatic density fluctuations enter into the cosmological horizon and recollapse. It has been suggested that PBH formation may be facilitated when fluctuations enter into the horizon during a strongly first-order phase transition which proceeds in approximate equilibrium. We employ general-relativistic hydrodynamics numerical simulations in order to follow the collapse of density fluctuations during first-order phase transitions. We find that during late stages of the collapse fluctuations separate into two regimes, an inner part existing exclusively in the high-energy density phase with energy density ɛ<SUB>h</SUB>, surrounded by an outer part which exists exclusively in the low-energy density phase with energy density ɛ<SUB>h</SUB>-L, where L is the latent heat of the transition. We confirm that the fluctuation density threshold δɛ/ɛ required for the formation of PBHs during first-order transitions decreases with increasing L and falls below that for PBH formation during ordinary radiation dominated epochs. Our results imply that, in case PBHs form at all in the early universe, their mass spectrum is most likely dominated by the approximate horizon masses during epochs when the universe undergoes phase transitions.
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https://arxiv.org/pdf/astro-ph/9901293.pdf
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{'Primordial Black Hole Formation during First-Order Phase Transitions': 'K. Jedamzik \nMax-Planck-Institut fur Astrophysik, Karl-Schwarzschild-Str. 1, 85740 Garching, Germany', 'J.C. Niemeyer': 'University of Chicago, Department of Astronomy and Astrophysics, 5640 S. Ellis Avenue, Chicago, IL 60637, USA \nPrimordial black holes (PBHs) may form in the early universe when pre-existing adiabatic density fluctuations enter into the cosmological horizon and recollapse. It has been suggested that PBH formation may be facilitated when fluctuations enter into the horizon during a strongly firstorder phase transition which proceeds in approximate equilibrium. We employ general-relativistic hydrodynamics numerical simulations in order to follow the collapse of density fluctuations during first-order phase transitions. We find that during late stages of the collapse fluctuations separate into two regimes, an inner part existing exclusively in the high-energy density phase with energy density /epsilon1 h , surrounded by an outer part which exists exclusively in the low-energy density phase with energy density /epsilon1 h -L , where L is the latent heat of the transition. We confirm that the fluctuation density threshold δ/epsilon1//epsilon1 required for the formation of PBHs during first-order transitions decreases with increasing L and falls below that for PBH formation during ordinary radiation dominated epochs. Our results imply that, in case PBHs form at all in the early universe, their mass spectrum is likely dominated by the approximate horizon masses during epochs when the universe undergoes phase transitions. \nPACS numbers: 04.70.Bw, 04.25.Dm, 97.60.Lf, 98.80.Cq', 'I. INTRODUCTION': "It is long known that only moderate deviations from homogeneity in the early universe may lead to abundant production of PBHs from radiation [1]. Slightly non-linear fluctuations with excess density contrast δ/epsilon1//epsilon1 ≈ 1, if horizon-size, are already very close to their own Schwarzschild radius. Therefore, already moderate collapse of such fluctuations may lead to the formation of a black hole. Under such conditions primordial angular momentum of fluctuations does generally not halt the collapse. The ultimate fate of an initially super-horizon density fluctuation, upon horizon crossing, is mainly determined by a competition between dispersing pressure forces and the fluctuation's self-gravity. For an ordinary radiation dominated equation of state (i.e. p = /epsilon1/ 3, where p is pressure) there is approximate equality between the Jeans-, M RD J , and horizon-, M h , masses. For fluctuation overdensities exceeding a critical threshold at horizon crossing ( δ/epsilon1//epsilon1 ) hc ≥ δ RD c ≈ 0 . 7 [2] gravity dominates and the formation of a PBH with mass M pbh M h results. Fluctuations with ( δ/epsilon1//epsilon1 ) hc < δ RD c disperse by pressure forces. \nSince PBH formation from pre-existing adiabatic density fluctuations is a fine competition between self-gravity and pressure forces any decrease of the pressure response ( ∂p/∂/epsilon1 ) S = v 2 s of the radiation fluid, or equivalently, decrease of the Jeans mass M J ≈ (4 π/ 3) /epsilon1 ( v 2 s / 4 πG/epsilon1 ) 3 / 2 , may yield a reduction of the threshold density contrast δ c for PBH formation. Here v s is the adiabatic speed of sound, with v s = 1 / √ 3 during ordinary radiation dominated epochs. A decrease in the pressure response of the radiation fluid is, in fact, anticipated to occur during cosmological first-order phase transitions. In essence, during a first-order transition between a high-energy density phase with energy density, /epsilon1 h , and a low-energy density phase with energy density /epsilon1 l = /epsilon1 h -L both phases may coexist in pressure equilibrium, p h = p l at a coexistence temperature T c . In a state where a fluid element is permeated by both phases, compression leads to an increase of energy density, since some low-energy density phase is converted into high-energy density phase. Nevertheless, there is no increase in pressure such that v eff s ≈ 0 [3,4] for a fluid element substantially larger than the mean separation between high-energy- and low-energy density phases. (see [3] for a more detailed discussion). These considerations are, of course, only valid under the assumption of approximate maintenance of thermodynamic equilibrium, in particular, negligible super-cooling and -heating. Less dramatic reductions of v eff s may also occur during higher-order phase transitions or particle annihilation periods in the early universe. \n∼ \nA reduction of the PBH formation threshold for fluctuations which enter the cosmological horizon during firstorder phase transitions (i.e. δ FPT c < δ RD c ) may have cosmological implications even if it is only modest. Conversion of cosmic radiation into PBHs at early epochs must be an extremely inefficient process if the contribution of PBH mass density to the present closure density, Ω pbh , is not too exceed unity. For Gaussian fluctuations this implies that PBH formation results only from those overdense fluctuations well within the exponentially declining tail of the density distribution function. PBH number density is dominated for fluctuations with ( δ/epsilon1//epsilon1 ) hc in a very narrow range \nbetween δ c and δ c + σ 2 /δ c , where σ is the variance of the Gaussian distribution. Typically, σ 2 /δ c < ∼ 10 -2 for Ω pbh < ∼ 1. For approximately scale-invariant Harrison-Zel'dovich spectra of the primordial density fluctuations, such as resulting from a multitude of proposed inflationary scenarios, fluctuations re-enter into the horizon with equal amplitudes on all mass scales. In this case, the slightest reduction of δ FPT c as compared to δ RD c may result in the formation of PBHs on essentially only the horizon scale during the first-order phase transition, yielding a highly peaked PBH mass function. We note here that there are other proposed scenarios for PBH formation during first-order phase transitions [6]. These usually involve production of seed fluctuations during the transition which collapse to PBHs. \nThe above considerations have led one of us to propose PBHs formed during the QCD color deconfinement transition at temperature T c ≈ 100MeV and with typical masses M pbh ∼ M QCD h ≈ 2 M /circledot ( T/ 100MeV) -2 as a candidate for halo dark matter [3,5]. Compact halo dark matter with approximate masses in the range 0 . 1 M /circledot < ∼ M < ∼ 1 M /circledot may have been detected by the MACHO [7] and EROS [8] collaborations. They monitored the light curves of millions of stars within the Large- and Small- Magellanic Clouds (LMC and SMC), searching for gravitational microlensing of LMC (SMC) stars by foreground compact objects. The simplest interpretation of their findings is that a significant fraction of halo dark matter is within compact objects of unknown nature. A dynamically significant population of halo redor white- dwarfs is subject to stringent constraints from either direct observations [9], or considerations of chemical evolution and contribution of MACHO baryon density to the critical density [10]. Nevertheless, recent observations of microlensing of a star in the SMC by a binary lens [11] may favor a cosmologically less interesting interpretation of the MACHO/EROS observations. It has been shown that the binary lens responsible for the microlensing event 98-SMC-1 most likely resides within the SMC itself, and not within the galactic halo. It may well be that lenses responsible for other microlensing events are not within the galactic halo but represent so far unknown stellar populations within the LMC and SMC. Clearly, further observations are needed to reveal the nature and location of the lens population. Note that a population of halo PBHs could be revealed by detection of gravitational waves emitted during PBH-PBH mergers [12]. \nIn this paper we employ a general-relativistic hydrodynamics code to follow the evolution of density fluctuations which enter the cosmological horizon during a first-order phase transition. In Section 2. we briefly summarize the adopted numerical technique which has been discussed in detail by [13] and [2] and introduce the equation of state describing the phase transition. Results and discussion of the PBH formation process are presented in Section 3.", 'II. THE PHASE TRANSITION AND NUMERICAL SIMULATION TECHNIQUE': "The algorithm employed to follow the evolution of spherically symmetric density fluctuations in an expanding Friedman-Robertson-Walker (FRW) universe is the same as the one adopted and described in detail in [13,2]. The method, based on an algorithm developed by Baumgarte et al. [13] and modified for use in a FRW background, is particularly well suited not only to simulate the formation of a black hole, but also to follow the late-time evolution of PBHs by virtue of the chosen Hernandez-Misner coordinates [14]. For details concerning the algorithm, zoning, size of time steps and computational domain, etc. the reader is referred to [2]. Initial conditions for the metric perturbations are chosen to be pure energy density fluctuations with Mexican-hat profile \n/epsilon1 ( R ) = /epsilon1 0 [ 1 + A ( 1 -R 2 R 2 h ) exp ( -3 R 2 2 R 2 h )] , (1) \nin unperturbed uniform Hubble expansion (essentially synchronous gauge with uniform Hubble expansion condition). Here, R is circumferential radius (i.e. R appears in the angular piece of the metric R 2 d Ω 2 , where d Ω is the solid angle element), which in the absence of perturbations, corresponds to what is commonly referred to as proper distance in cosmology, r p = ar c , where a is the scale factor of the universe and r c is the comoving cosmic distance. The distance R h is chosen to be the cosmological horizon distance, R h ( t 0 ), at the beginning of the simulation t 0 , i.e. \nR h ( t 0 ) = a ( t 0 ) ∫ t 0 0 dt a ( t ) , (2) \nThe increase of horizon distance with cosmic time in the presence of a phase transition is computed numerically. The amplitude A is adjusted such that fluctuation amplitudes are characterized by a certain average overdensity at horizon crossing \n( δ/epsilon1 /epsilon1 ) hc ≡ ( V h /epsilon1 0 ) -1 4 π ∫ R h ( t 0 ) 0 ( /epsilon1 ( R ) -/epsilon1 0 ) R 2 dR -1 , (3) \nwhere /epsilon1 0 = /epsilon1 0 ( t 0 ) is the unperturbed FRW energy density at t 0 and V h = (4 π/ 3) R 3 h ( t 0 ) is the initial horizon volume. \nThe properties of the equilibrium first-order phase transition are fully described by a choice for the equation of states of the low- and high-energy density phases, respectively. We adopt a phenomenological bag model for the highenergy density phase, where the energy density is given by the contributions of a quasi-free, extremely relativistic gas with statistical weight g h plus a temperature-independent self-interaction correction, the bag constant B . We approximate the low-energy density phase as a non-interacting, extremely relativistic gas with statistical weight g l . Thermodynamic quantities of the individual phases are [15] \n/epsilon1 l ( T ) = g l π 2 30 T 4 ; /epsilon1 h ( T ) = g h π 2 30 T 4 + B ; (4) \np l ( T ) = 1 3 g l π 2 30 T 4 ; p h ( T ) = 1 3 g h π 2 30 T 4 -B ; (5) \ns l ( T ) = 4 3 g l π 2 30 T 3 ; s h ( T ) = 4 3 g h π 2 30 T 3 , (6) \nwhere s denotes entropy density. Requiring the existence of a first-order phase transition at temperature T c , i.e. p l ( T c ) = p h ( T c ), we find /epsilon1 h ( T c ) -/epsilon1 l ( T c ) = 4 B , which together with the definition of latent heat, L = T c ( ∂/∂T )( p h -p l ) = T c ( s h -s l ), implies L = /epsilon1 h ( T c ) -/epsilon1 l ( T c ). We define the useful quantity \nη = L /epsilon1 l ( T c ) = /epsilon1 h ( T c ) -/epsilon1 l ( T c ) /epsilon1 l ( T c ) , (7) \ndescribing the strength of the phase transition. Since we assume the transition to proceed in close-to-equilibrium (i.e. negligible super -cooling and -heating) we may use conservation of entropy to relate the ratio of scale factors at the beginning of the transition, a 1 , and the end of the transition, a 2 , to η \na 2 a 1 = ( g h g l ) 1 / 3 = ( 1 + 3 4 η ) 1 / 3 . (8) \nDuring ordinary radiation dominated epochs the dynamics of fluctuations is self-similar for fluctuations on small length scales collapsing at early times and fluctuations on large length scales collapsing at late times, only dependent on the shape and density contrast of the fluctuation. Similarly, fluctuation dynamics in the presence of a phase transition should be independent of the temperature, energy density, and horizon mass at which the transition occurs. It is dependent, however, on the strength of the transition, η , as well as on the exact time at which the fluctuation crosses into the horizon, in particular, if shortly before onset, during, or shortly after completion of the transition. This 'time' may be parameterized by the ratio of cosmic average energy density at fluctuation horizon crossing and some typical energy density at the transition, τ hc ≡ /epsilon1 0 ( t 0 ) //epsilon1 h ( T c ). The threshold for PBH production is thus a function of τ hc and η only \nδ FPT c = ( δ/epsilon1 /epsilon1 ) c , hc ( τ hc , η ) . (9) \nIn contrast, the resulting PBH masses for collapsing fluctuations are approximately determined by the horizon mass, M h ≈ (4 π/ 3) /epsilon1 h R 3 h ( /epsilon1 h , η ), at which the transition occurs, and are only weakly dependent on η and τ hc . For a more accurate determination of the PBH mass spectrum it is necessary to convolve the distribution function for the preexisting density fluctuations, P ( δ ) dδ , with a scaling relation for the resulting PBH masses, M pbh = kM h ( δ -δ FPT c ) γ as shown in [16]. Here k is a dimensionless constant. This mass spectrum may be somewhat dependent on η and τ hc . \nIn order to follow the hydrodynamic evolution of fluid elements at T c , on length scales much larger than the mean separation between high- and low- energy density phase, it is necessary to specify an effective equation of state describing the mixture of phases in thermodynamic equilibrium. Note that the assumption of a near-to-equilibrium, first-order phase transition, implies negligible super -cooling and -heating and a typical mean separation between phases much shorter than the cosmological horizon. These assumptions are met if nucleation of new phase is efficient. Since a fluctuation scale length is of the order of the cosmological horizon the details of the distribution of phases on small scales should have vanishing influence on the all-over fluctuation dynamics. The fluid is in phase mixture, for energy densities between /epsilon1 h ( T c ), where the volume fraction occupied by high-energy density phase, f h = 1, and, /epsilon1 l ( T c ), where f h = 0. The effective equation of state may thus be written \np ( /epsilon1 ) = 1 3 ( /epsilon1 -L ) ; v 2 s = 1 3 ; /epsilon1 ≥ /epsilon1 h ( T c ) ; (10) \np ( /epsilon1 ) = 1 3 /epsilon1 l ( T c ) ; v 2 s = 0 ; /epsilon1 l ( T c ) < /epsilon1 < /epsilon1 h ( T c ) ; (11) \np ( /epsilon1 ) = 1 3 /epsilon1 ; v 2 s = 1 3 ; /epsilon1 ≤ /epsilon1 l ( T c ) , (12) \nwhere /epsilon1 l ( T c ) and /epsilon1 h ( T c ) are constants. Note again that the pressure response, v s , of mixed phase is exactly zero in thermodynamic equilibrium, whereas pressure itself is substantial.", 'III. RESULTS AND DISCUSSION': "We followed the evolution of density fluctuations upon horizon crossing during a cosmological phase transition with scaled latent heat η = 2 (in particular, we chose /epsilon1 h ( T c ) = 1 and /epsilon1 l ( T c ) = 0 . 5). Figures 1-3 show results for a fluctuation with overdensity ( δ/epsilon1//epsilon1 ) hc = 0 . 535 entering the cosmological horizon at time τ hc = 1, i.e. when the surrounding universe is at average energy density /epsilon1 h ( T c ) at the onset of the phase transition. In Figure 1 we show the evolution of the radial energy density profile of the fluctuation from the initial horizon crossing time t 0 to 20 . 1 t 0 . We choose constant proper time slicing, i.e. an individual curve in Figure 1 represents the energy density of all fluid elements evaluated at identical proper time. Energy density is shown as a function of scaled circumferential radius R sc = ( R/R h ( t 0 ))( a 0 /a ) such that R sc = constant for a fluid element in simple FRW expansion. The horizon at t 0 is located at R sc = 1. The two horizontal dotted lines in Figure 1 indicate the regime of energy densities in which fluid elements exist within mixed phases. \nFIG. 1. Energy density, /epsilon1 , as a function of scaled circumferential radius, R sc = ( R/R h ( t 0 ))( a 0 /a ), for a fluctuation with initial density contrast, ( δ/epsilon1//epsilon1 ) hc = 0 . 535, at horizon crossing. The initial horizon at t 0 is located at R sc = 1. From top to bottom, solid lines show the fluctuation at 1., 1.22, 1.49, 1.82, 2.23, 2.72, 3.32, 4.06, 4.95, 6.05, 7.39, 9.03, 11.0, 13.5, 16.4, and 20.1 times the initial time t 0 . Constant proper time slicing was used. The horizontal dashed lines indicate the energy densities at onset and completion of the phase transition. The formation of a PBH with M pbh ≈ 0 . 34 M h ( t 0 ) results. \n<!-- image --> \nThe formation of a PBH with final mass M pbh ≈ 0 . 34 M h ( t 0 ) results from the evolution of the fluctuation shown in Figure 1-3. Here we define the horizon mass at the initial time M h ( t 0 ) = /epsilon1 0 V h ( t 0 ). The fluctuation's selfgravity exceeds pressure forces such that the fluctuation separates from Hubble flow and recollapses to high-energy densities \nat the center until an event horizon forms. Subsequent accretion of material on the young PBH continues until the immense pressure gradients close to the event horizon launch an outgoing pressure wave which significantly dilutes the PBH environment. Accretion thereafter is negligible. The existence of a phase transition facilitates the PBH formation process as is evident from Figure 2. Figure 2 is a zoom into the core of the fluctuation shown in Figure 1. For comparison, this figure also shows the evolution of a fluctuation with the same initial conditions, but entering the cosmological horizon during an ordinary radiation dominated epoch, by the dotted line. The strong pressure gradients experienced by the fluctuation entering the horizon during an epoch with equation of state p = /epsilon1/ 3 prevent the formation of a PBH. \nFIG. 2. A zoom into the central region of Figure 1. From top to bottom, solid lines show the fluctuation at 1.0, 1.22, 1.49, 1.82, 2.22, 2.72, 3.32, 4.06, 4.95, and 5.47 times the initial time t 0 . The horizontal dashed lines indicate the energy densities at onset and completion of the phase transition. The dotted lines shows, for comparison, the evolution of a fluctuation with the same initial fluctuation parameters, but entering the cosmological horizon during an epoch with equation of state p = /epsilon1/ 3. \n<!-- image --> \nFIG. 3. Coordinate velocity, ∂R/∂τ , with τ proper time of fluid elements, as a function of scaled circumferential radius, R sc = ( R/R h ( t 0 ))( a 0 /a ). The solid lines show, from top to bottom, the coordinate velocities at times 1.0, 1.22, 1.49, 1.82, 2.22, 2.72, 3.32, 4.06, and 4.95 times t 0 for the same fluctuation and equation of state as shown in Figures 1 and 2 by the solid lines. \n<!-- image --> \nThe evolution of the fluctuation in the presence of a phase transition proceeds as follows. The initial phase between times t 0 and ∼ 2 t 0 is characterized by a core in high-energy density phase surrounded by an envelope in mixed phase. Cosmological expansion results in the decrease of energy densities in core and envelope. Nevertheless, the fluctuation's overdensity decelerates the material with respect to FRW expansion. This is evident from Figure 3, which shows the coordinate velocity ∂R/∂τ for the fluctuation shown in Figures 1 and 2, with τ proper time. In unperturbed Hubble flow, coordinate velocities would be straight lines with decreasing slope for increasing cosmic time. The deceleration of the expanding fluid is stronger for those fluid elements existing in mixed phase due to the absence of pressure gradients. Material in the core remains in high-energy density phase until t ≈ 2 . 2. Its deceleration is first stronger than FRW due to selfgravity until at t ≈ 2 t 0 , the increasing pressure gradient begins to counteract the collapse. The pressure gradient would give rise to the launching of a pressure wave and the dispersion of the fluctuation if core and envelope wouldn't join at t ≈ 2 . 2 t 0 and exist in mixed phase. In the subsequent phase between times ∼ 2 . 2 t 0 and ∼ 4 t 0 , a distinctively separated core develops. It is distinct through discontinuities in energy density and coordinate velocity. The core first undergoes homologous, but sub-Hubble, expansion which turns into homologous contraction at t ≈ 3 . 3 t 0 . The core energy density is almost uniform and increases with time. Initially the core mass-energy still reduces somewhat. Shells on the boundary of the core are expelled when they experience strong pressure gradients as their energy density falls below /epsilon1 l . In the final phase of collapse of the core region between ≈ 4 t 0 and ≈ 5 t 0 the fluid exists exclusively in high-energy density phase surrounded by fluid in low-energy density phase. The collapse proceeds \nno further homologously, rather, inner parts of the core contract faster than outer parts. At t ≈ 5 t 0 an event horizon forms. The resulting young PBH rapidly increases its mass up to M pbh ≈ 0 . 06 M h ( t 0 ) at t ≈ 5 . 5 t 0 . Subsequent slow long-term accretion onto the PBH increases the PBH mass further to M pbh 0 . 34 M h ( t 0 ). \nThe dynamics of PBH formation during a first-order phase transition is different from the one experienced during ordinary radiation dominated epochs. In the latter there is no obvious segregation between collapsing fluctuation and expanding universe, in particular, there are no discontinuities. Further, the collapse of fluctuations during ordinary radiation dominated epochs does not proceed in a homologous fashion. Figures 2 and 3 reveal the existence of instabilities in the evolutionary calculations of the PBH formation process. It may be observed that material within mixed phase develops a small-scale perturbation of growing amplitude. This is not surprising, since with v s = 0 in mixed phase the fluid is Jeans-unstable to collapse on all scales. The perturbation extends over many zones and its amplitude and size are unchanged with an increase of numerical viscosity and/or number of zones. We note that there may be other physical instabilities for v s = 0 which we, however, do not observe due to our choice of initial conditions. It has been argued that an instability of non-gravitational origin exists for sound waves which experience a sudden drop in the speed of sound [4,17]. Finally, we also observe a numerical instability of the adopted algorithm which occurs when the energy density of fluid shells falls slightly below /epsilon1 l ( T c ). Such shells experience immense acceleration when leaving the regime of mixed phase with large energy density gradients. The amplitude of the perturbations may be regulated by our choice of numerical viscosity, nevertheless, the final black hole mass is virtually unaffected by an increase/decrease of numerical viscosity. \n≈ \nFIG. 4. Energy overdensity threshold for PBH formation, δ FPT c (solid line), for fluctuations entering the cosmological horizon during, or close to a first-order phase transition, as a function of horizon crossing time, τ hc = /epsilon1 0 ( t 0 ) //epsilon1 h ( T c ). The energy densities at the onset and completion of the transition are chosen /epsilon1 h ( T c ) = 1 and /epsilon1 l ( T c ) = 0 . 5, respectively. The crosses are points determined from numerical simulation. The solid line is an interpolation between crosses. The dotted line shows δ FPT c / (1 + w ), with w the cosmic average p//epsilon1 at horizon crossing of the fluctuation. (See text for further explanations). \n<!-- image --> \nThe main result of our study is displayed in Figure 4. This figure shows the threshold for PBH formation, δ FPT c , for a phase transition with scaled latent heat, η = 2, as a function of the horizon crossing time of the fluctuation, τ hc , by the solid lines. Crosses represent the lowest ( δ/epsilon1//epsilon1 ) hc for which PBH formation results. The relative accuracy in δ FPT c is estimated to be on the 1% level. It is evident that over a range of horizon crossing times the energy overdensity required for PBH formation is below that for PBH formation during ordinary radiation dominated epochs. It should be clear from the introductory remarks that the slightest bias in favor of forming PBH during a phase transition may imply that essentially all PBH are formed on the scale associated with minimum δ FPT c . However, assuming a Harrison-Zel'dovich, exactly scale-invariant spectrum for the underlying density perturbations, the amplitudes of energy density perturbations in uniform Hubble constant gauge upon horizon crossing are not exactly constant during epochs with varying equation of state. Rather, modes with varying length scale cross into the horizon with equal ξ = (1+ w ) -1 ( δ/epsilon1//epsilon1 ) hc [18]. Here w = p//epsilon1 , such that for w smaller than 1 / 3, as in Eq. (10) - Eq. (12) for /epsilon1 > /epsilon1 l ( T c ), the horizon crossing amplitude ( δ/epsilon1//epsilon1 ) hc of perturbations is reduced. This partially decreases the bias to form PBH during \nphase transitions. In Figure 4 the dotted line shows a ˜ δ FPT c which corrects for this effect. Remarkably, the combined effects of reducing the threshold for PBH formation by the dynamics of fluctuations during a first-order transition and the decrease of the horizon crossing amplitude of energy density fluctuations due to the changed equation of state, result in a double dip structure. \nWe have also computed test runs for fluctuations entering the horizon during a phase transition with η = 10. Our results indicate that with increased strength of the phase transition δ FPT c is further decreased. With the discovery of critical phenomena in general relativity it has been recognized that the resulting black hole masses in nearly critically collapsing space-times obey scaling relations [19], such as M pbh = K ( δ -δ c ) γ . For a given matter model, γ is independent of initial conditions, while K depends on the specific choice of the control parameter δ . Specifically, a collapsing radiation fluid has γ ≈ 0 . 36 [20]. In our case, δ may be associated with ( δ/epsilon1//epsilon1 ) hc and K with kM h . We have explicitly verified that γ ≈ 0 . 36 also holds for the cosmological PBH formation process during ordinary radiation dominated epochs [2]. We have attempted to derive a scaling exponent γ for the PBH formation process during epochs with equation of state given by Eq. (10) - Eq. (12). Our numerical simulations do not clearly indicate a simple scaling relation as above. Preferred values of γ were rather large (i.e. γ > 2), even though somewhat dependent on the range of black hole masses to which we fit the scaling relation. In contrast to our work in [2], convincing verification of a scaling relation and a value for the exponent γ would require the use of adaptive mesh techniques. This would allow to simulate the formation of PBH with masses far smaller than 0 . 1 M h . We note that convolving a Gaussian density perturbation probability function with the preferred scaling relation of our numerical simulations would predict average masses M pbh far below 0 . 1 M h . Clearly, resolution of this issue would be desirable.", 'IV. CONCLUSIONS': "We have performed hydrodynamic general-relativistic simulations of the PBH formation process when initially super-horizon energy density fluctuations enter the cosmological horizon during a first-order phase transition. We have verified that the significantly diminished pressure forces for fluid undergoing the phase transition facilitate the PBH formation process. We have shown that the threshold of the amplitude of energy density fluctuations above which PBH formation results is smaller for fluctuations crossing into the horizon during a first-order phase transition than for fluctuations crossing into the horizon when the equation of state is ordinary (i.e. p = /epsilon1/ 3). PBH formation from pre-existing adiabatic fluctuations is more probable during first-order transitions, even when the reduction of fluctuation amplitudes of Harrison-Zel'dovich type which cross into the horizon for fluid pressures smaller than /epsilon1/ 3, is taken into account. Our simulations could not clarify if simple scaling relationships with scaling exponent γ ≈ 0 . 36 for the resulting PBH masses, as found in many studies of critical phenomena in general relativity, also apply to the PBH formation process during cosmological first-order phase transitions. Our simulations favor much larger scaling exponents, possibly yielding small typical PBH masses (in units of the horizon mass). Further study with adaptive mesh techniques is required to resolve this issue. \nOur results may have cosmological implications for the PBH mass spectrum. It was so far believed that PBH formation from pre-existing adiabatic perturbations in the early universe is equally likely on all scales, only dependent on the statistics of the pre-existing density perturbations. The dynamics of fluctuations during cosmological phase transitions may easily introduce a strong enough bias to form PBH only on the approximate horizon mass scales during epochs with phase transitions. This may yield a highly peaked mass function for PBHs. Here the term cosmological phase transition may be loosely interpreted, since a reduction of pressure forces is not only anticipated during first-order phase transitions, but may also occur during higher-order transitions, cross-overs, and particle annihilation periods. A prime candidate for such a transition is QCD color-confinement due to the large number of degrees of freedom participating in the transition. Whether, or not, PBHs have been produced in the early universe in cosmologically interesting numbers remains an open question. \nWe wish to thank T. Baumgarte for providing the original version of the hydrodynamical code, and T. Abel, E. Muller, and A. Olinto for helpful discussions. JCN acknowledges the support of an Enrico-Fermi-Fellowship. \n- [4] C. Schmid, D. J. Schwarz, and P. Widerin, Phys. Rev. Lett. 78 (1997) 791.\n- [5] K. Jedamzik, Phys. Rep, in press (astro-ph/9805147).\n- [6] M. Crawford and D. N. Schramm, Nature 298 (1982) 538; H. Kodama, M. Sasaki, and K. Sato, Prog. Theor. Phys. 68 (1982) 1979; L. J. Hall and S. D. Hsu, Phys. Rev. Lett. 64 (1990) 2848: P. Widerin and C. Schmid, submitted (astro-ph/9808142); V. Katalini'c and A. Olinto, in preparation.\n- [7] C.F. Alcock et. al. , ApJ 486 (1997) 697; K. Cook et. al. , AAS 191 (1997) 8301.\n- [8] C. Renault et al. (EROS Collaboration), Astron. Astrophys. 324 , L69 (1997).\n- [9] C. Flynn, A. Gould, and J. N. Bahcall, ApJL 466 (1996) L55; S. Charlot and J. Silk, ApJ 445 (1995) 124; D.S. Graff and K. Freese, ApJL 456 (1996) L49.\n- [10] B. D. Fields, K. Freese, and D. S. Graff, New Astronomy 3 (1998) 347.\n- [11] Afonso, C., et al. 1998, Astron. & Astroph., submitted (astro-ph/9806380); Alcock, C., et al. 1998, ApJ, submitted (astro-ph/9807163).\n- [12] T. Nakamura, M. Sasaki, T. Tanaka, and K. S. Thorne ApJL 487 (1997) L139; K. Ioka, T. Tanaka, and T. Nakamura, Phys. Rev. D, submitted (astro-ph/9809395).\n- [13] T. W. Baumgarte, S. L. Shapiro, and S. A. Teukolsky, Astrophys. J. 443 , 717 (1995).\n- [14] W. C. Hernandez and C. W. Misner, Astrophys. J., 143 , 452 (1966).\n- [15] G. M. Fuller, G. J. Mathews, and C. R. Alcock, Phys. Rev. D 37 (1988) 1380.\n- [16] J. C. Niemeyer and K. Jedamzik, Phys. Rev. Lett. 80 (1998) 5481.\n- [17] C. Schmid , D. J. Schwarz, and P. Widerin, Phys. Rev. D, in press (astro-ph/9807257).\n- [18] E. W. Kolb and M. S. Turner, The Early Universe (Addison-Wesley, New York, 1990).\n- [19] M. W. Choptuik, Phys. Rev. Lett. 70 (1993) 9; for recent reviews see: C. Gundlach, Adv. Theor. Math. Phys. 2 (1998) 1; M. W. Choptuik, (1998) gr-qc/9803075.\n- [20] C. R. Evans and J. S. Coleman, Phys. Rev. Lett. 72 (1994) 1782; T. Koike, T. Hara, and S. Adachi, Phys. Rev. Lett. 74 (1995) 5170."}
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2002ApJ...579..639B
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Gravitational Microlensing Events Due to Stellar-Mass Black Holes
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2002-01-01
| 10 | 0.47 | 160 |
['galaxy bulge', 'gravitational lensing', 'astrophysics']
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[]
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We present an analysis of the longest timescale microlensing events discovered by the MACHO Collaboration during a 7 year survey of the Galactic bulge. We find six events that exhibit very strong microlensing parallax signals due, in part, to accurate photometric data from the GMAN and MPS collaborations. The microlensing parallax fit parameters are used in a likelihood analysis, which is able to estimate the distances and masses of the lens objects based on a standard model of the Galactic velocity distribution. This analysis indicates that the most likely masses of five of the six lenses are greater than 1 M<SUB>solar</SUB>, which suggests that a substantial fraction of the Galactic lenses may be massive stellar remnants. This could explain the observed excess of long-timescale microlensing events. The lenses for events MACHO-96-BLG-5 and MACHO-98-BLG-6 are the most massive, with mass estimates of M/M<SUB>solar</SUB>=6<SUP>+10</SUP><SUB>-3</SUB> and M/M<SUB>solar</SUB>=6<SUP>+7</SUP><SUB>-3</SUB>, respectively. The observed upper limits on the absolute brightness of main-sequence stars for these lenses are less than 1 L<SUB>solar</SUB>, so both lenses are black hole candidates. The black hole interpretation is also favored by a likelihood analysis with a Bayesian prior using a conventional model for the lens mass function. We consider the possibility that the source stars for some of these six events may lie in the foreground Galactic disk or in the Sagittarius (Sgr) dwarf galaxy behind the bulge, but we find that bulge sources are likely to dominate our microlensing parallax event sample. Future Hubble Space Telescope observations of these events can either confirm the black hole lens hypothesis or detect the lens stars and provide a direct measurement of their masses. Future observations of similar events by the Space Interferometry Mission or the Keck or VLT interferometers, as explained by Delplancke, Górski, & Richichi, will allow direct measurements of the lens masses for stellar remnant lenses as well. Based in part on observations from NASA's Hubble Space Telescope.
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[]
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https://arxiv.org/pdf/astro-ph/0109467.pdf
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{'Gravitational Microlensing Events Due to Stellar Mass Black Holes 1': 'D.P. Bennett 2 , 3 , A.C. Becker 4 , J.L. Quinn 2 , A.B. Tomaney 5 , C. Alcock 3 , 6 , 7 , R.A. Allsman 8 , D.R. Alves 9 , T.S. Axelrod 10 , J.J. Calitz 11 , K.H. Cook 3 , 7 , A.J. Drake 7 , P.C. Fragile 2 , K.C. Freeman 10 , M. Geha 12 , K. Griest 13 , B.R. Johnson 14 , S.C. Keller 7 , C. Laws 5 , M.J. Lehner 6 , S.L. Marshall 7 , D. Minniti 15 , C.A. Nelson 7 , 16 , B.A. Peterson 10 , P. Popowski 17 , M.R. Pratt 5 , P.J. Quinn 18 , S.H. Rhie 2 , C.W. Stubbs 3 , 5 , W. Sutherland 19 , T. Vandehei 12 , D. Welch 20 \n(The MACHO and MPS Collaborations)', 'ABSTRACT': "We present an analysis of the longest timescale microlensing events discovered by the MACHO Collaboration during a seven year survey of the Galactic bulge. We find six events that exhibit very strong microlensing parallax signals due, in part, to accurate photometric data from the GMAN and MPS collaborations. The microlensing parallax fit parameters are used in a likelihood analysis, which is able to estimate the distance and masses of the lens objects based upon a standard model of the Galactic velocity distribution. This analysis indicates that the most likely masses of five of the six lenses are > 1M /circledot , which suggests that a substantial fraction of the Galactic lenses may be massive stellar remnants. This could explain the observed excess of long timescale microlensing events. The lenses for events MACHO-96-BLG-5 and MACHO98-BLG-6 are the most massive, with mass estimates of M/ M /circledot = 6 +10 -3 and M/ M /circledot = 6 +7 -3 , respectively. The observed upper limits on the absolute brightness of main sequence stars for these lenses are < 1L /circledot , so both lenses are black hole candidates. The black hole interpretation is also favored by a likelihood analysis with a Bayesian prior using a conventional model for \nthe lens mass function. We consider the possibility that the source stars for some of these six events may lie in the foreground Galactic disk or in the Sagittarius (SGR) Dwarf Galaxy behind the bulge, but we find that bulge sources are likely to dominate our microlensing parallax event sample. Future HST observations of these events can either confirm the black hole lens hypothesis or detect the lens stars and provide a direct measurement of their masses. Future observations of similar events by SIM or the Keck or VLTI interferometers (Delplancke, G'orski & Richichi 2001) will allow direct measurements of the lens masses for stellar remnant lenses as well.", '1. Introduction': "The abundance of old stellar remnants in our Galaxy is largely unknown because they emit little radiation unless they happen to be accreting material from a companion star, or for neutron stars, if they happen to emit pulsar radiation in our direction. Gravitational microlensing surveys (Liebes 1964; Paczy'nski 1986; Alcock et al. 1993; Aubourg et al. 1993; Udalski et al. 1993; Bond et al. 2001) have the potential to detect completely dark stellar remnants, but for most microlensing events, the mass can only be estimated very crudely based upon the observed Einstein ring diameter crossing time, ̂ t . For an individual microlensing event, the mass can only be estimated so crudely that a 7M /circledot black hole cannot be distinguished from a 0 . 5M /circledot star. However, for some microlensing events, it is possible to measure other parameters besides ̂ t that allow tighter constraints on the lens mass (Refsdal 1966; Gould 1992; Nemiroff & Wickramasinghe 1994; Alcock et al. 1995; Bennett et al. 1996; Han & Gould 1997; Alcock et al. 1997c; Afonso et al. 2000; Alcock et al. 2001a). For long timescale microlensing events, which are often due to massive lenses, it is frequently possible to measure the microlensing parallax effect (Refsdal 1966; Gould 1992; Alcock et al. 1995) which is an observable deviation in the microlensing light curve due to the orbital motion of the Earth. In this paper, we present an analysis of the microlensing events discovered by the MACHO Project which give a very strong microlensing parallax signal, and we show that some of these events are best explained as microlensing by black holes. \nThis paper is organized as follows: in Sections 2 and 3, we discuss the microlensing event data set and the long timescale sub-sample. The microlensing parallax fits are presented in Section 4, and in Section 5 we present our main analysis to determine the distances and masses of the lenses. This includes a discussion of the projected lens velocity distributions, the source star color magnitude diagrams, and a likelihood analysis of the distances and masses of the microlenses. In Section 6, we discuss possible follow-up observations with high resolution telescopes and interferometers that can directly determine the microlensing parallax event lens masses, and we conclude in Section 7.", '2. The Data Set': "The MACHO Project (Alcock et al. 1993) has monitored ∼ 10 -20 million stars in the Galactic bulge for 6-7 months per year during each of the 1993-1999 Galactic bulge seasons. During the last half of 1994, real-time microlensing discovery with the MACHO Alert system became possible (Alcock et al. 1996). (The OGLE collaboration developed this capability the same year (Udalski et al. 1994b).) This development allowed much more accurate photometry of the microlensing events, which were discovered in progress, from the CTIO 0.9m telescope where the MACHO/GMAN Project was allocated about 1 hour every night (Becker 2000). In 1997, the Microlensing Planet Search (MPS) Project (Rhie et al. 1999) began microlensing follow-up observations from the Mt. Stromlo 1.9m telescope. \nThe data set used for this analysis consists of the MACHO survey data from the Mt. Stromlo 1.3m 'Great Melbourne' telescope for all seven years, CTIO 0.9m data of selected alert events from 1995-1999, and Mt. Stromlo 1.9m data of alert events from the MPS 1997-1999 data sets. The initial selection of events consists of 42 events from 1993 (Alcock et al. 1997b), 252 events discovered by the MACHO Alert system from 1995-1999 (available from http://darkstar.astro.washington.edu/ ), and an additional 27 events discovered during the testing of the alert system, for a grand total of 321 events. There are ∼ 200 additional Galactic bulge events that have been discovered via other analyses that we have not considered here. This paper will focus on the six events from this list which give a strong microlensing parallax signal. The coordinates of these events are given in Table 1. A microlensing parallax study of a larger number of MACHOAlert events is presented in Becker (2000). \nThe MACHO and MPS data were reduced with slightly different versions of the SoDOPHOT photometry code (Bennett et al. 1993; Alcock et al. 1999). SoDOPHOT is quite similar to the DOPHOT photometry code (Schechter, Mateo, & Saha 1993) that it was derived from, but SoDOPHOT photometry generally exhibits smaller photometric scatter than DOPHOT photometry. This is due, in part, to SoDOPHOT's error flags which allow the removal of suspect data points (Alcock et al. 1999), but the scatter in DOPHOT photometry is often increased by the user's choice of PSF fitting parameters. Contrary to expectations, allowing the PSF fit box size to scale with the seeing generally causes increased photometric scatter (Bennett et al. 1993). The photometric errors reported by SoDOPHOT are modified by adding of 1.4% and 1.0% in quadrature to the MACHO and MPS data, respectively, to account for normalization and flat fielding errors. The CTIO data were reduced with the ALLFRAME package (Stetson 1994), with the error estimates multiplied by a factor of 1.5 to account for systematic errors. \nTable 2 gives the number of observations in each pass band for each event, and the data used for this paper are presented in Table 3. The complete set of macho survey data is available at http://wwwmacho.mcmaster.ca/ and http://wwwmacho.anu.edu.au . Only the MACHO survey data has been calibrated and transformed to standard pass bands (Alcock et al. 1999). The other data is given in instrumental magnitudes which have only been calibrated relative to other measurements with the same telescope and passband. The transformation between raw MACHO magnitudes given in Table 3 ( B MACHO and R MACHO ) and the Kron-Cousins V and R system is given by: \nR = 2 . 412 + c B MACHO + dR MACHO , (1) \nwhere the coefficients, a , b , c , and d are slightly different for each event as shown in Table 4.", '3. Long Timescale Events': "The timescale of a gravitational microlensing event is described by the Einstein diameter crossing time, ̂ t , which depends on the lens mass ( M ), distance ( D /lscript ), and transverse velocity ( v ⊥ ). It is given by \nwhere D s refers to the distance to the source (typically 8 kpc for a bulge source), and R E is the radius of the Einstein Ring. Eq. (2) indicates that long ̂ t events can be caused by large M , small v ⊥ or both. Fig. 1 shows the long timescale tail of the ̂ t distribution for our sample of 321 Galactic bulge microlensing events. In their analysis of the timescale distribution of the 1993 MACHO and OGLE bulge data sets, Han & Gould (1996) noted a surprisingly large fraction of the events with ̂ t ≥ 140 days: 4/51 or 8%. Such a large fraction of long timescale events would be expected less than 2% of the time with any of the stellar mass functions that they considered. With our data set of 321 events, we find 28, or 9%, with ̂ t > 140 days. The MACHO alert system is likely to be somewhat less sensitive to long timescale events than the 1993 analysis because the alert trigger is based upon the single most significant observation, so we would expect a slightly smaller fraction of long timescale events, but the fraction reported here is somewhat higher. The formal Poisson probability of 28/321 long events when 2% or less are expected is < 10 -10 , so the excess of long timescale events over the Han & Gould (1996) models is highly significant. This disagreement may be due to a population of massive stellar remnants, including black holes, that was not included in the Han &Gould (1996) models, but there are other possibilities as well. Other explanations include a set of sourcelens systems that have a low relative velocity from our vantage point in the Galactic disk or a more distant population of source stars. \n̂ t = 2 R E v ⊥ = 4 v ⊥ c √ GMD /lscript ( D s -D /lscript ) D s , (2) \nThe microlensing parallax effect refers to the effect of the orbital motion of the Earth on the observed microlensing light curve. The photometric variation for most microlensing events lasts only a month or two. For these events, the change in the Earth's velocity vector during the event is too small to generate a detectable deviation from the symmetric light curve, which is predicted for a constant velocity between the lens and the Earth-source star line of sight. For long timescale events, however, it is possible to see the effect of the Earth's motion in the microlensing light curve, and this is called the microlensing parallax effect.", '4. Microlensing Parallax Fits': "The magnification for a normal microlensing event with no detectable microlensing parallax is given by \n̂ \nA ( t ) = u 2 +2 u √ u 2 +4 ; u ( t ) ≡ √ u 2 0 +[2( t -t 0 ) / t )] 2 , (3) \nwhere t 0 is the time of closest approach between the angular positions of the source and lens, and u 0 = b/R E where b is the distance of the closest approach of the lens to the observer-source line. Eq. (3) can be generalized to the microlensing parallax case (Alcock et al. 1995) by assuming the perspective of an observer located at the Sun. We can then replace the expression for u ( t ) with \nu 2 ( t ) = u 2 0 + ω 2 ( t -t 0 ) 2 + α 2 sin 2 [Ω( t -t c )] +2 α sin[Ω( t -t c )] [ ω ( t -t 0 ) sin θ + u 0 cos θ ] + α 2 sin 2 β cos 2 [Ω( t -t c )] + 2 α sin β cos[Ω( t -t c )] [ ω ( t -t 0 ) cos θ -u 0 sin θ ] (4) \nwhere λ and β are the ecliptic longitude and latitude, respectively, θ is the angle between v ⊥ and the North ecliptic axis, ω = 2 / ̂ t , and t c is the time when the Earth is closest to the Sun-source line. The parameters α and Ω are given by \nα = ω (1AU) ˜ v (1 -/epsilon1 cos[Ω 0 ( t -t p )]) , (5) \nand \nΩ( t -t c ) = Ω 0 ( t -t c ) + 2 /epsilon1 sin[Ω 0 ( t -t p )] , (6) \nwhere t p is the time of perihelion, Ω 0 = 2 π yr -1 , /epsilon1 = 0 . 017 is the Earth's orbital eccentricity, and ˜ v is the lens star's transverse speed projected to the Solar position, given by \n˜ v = v ⊥ D s / ( D s -D /lscript ) , (7) \nThe 28 events shown in Fig. 1 have been fit with the microlensing parallax model described by eqs. (3)(5) which has 5 independent parameters: t 0 , u 0 , ̂ t , ˜ v , and θ . In the crowded fields that are searched for microlensing, it is also necessary to include two parameters for each independent photometric pass band (or telescope) to describe the flux of the source star and the total flux of any unlensed stars that are not resolved from the lensed source. Thus, a microlensing parallax fit to the dual-color MACHO data alone will have 9 fit parameters, and a fit that includes the CTIO and MPS follow-up data will have 13 fit parameters. \nThe microlensing parallax fits were performed with the MINUIT routine from the CERN Library, and the results for the 6 events that we discuss in this paper are summarized in Table 5. The best fit light curves and data are shown in Figs. 2-7. The significance of the microlensing parallax signal is represented by the parameter ∆ χ 2 shown in Table 5 which is the difference between the fit χ 2 for a standard microlensing fit with no parallax ( i.e. ˜ v = ∞ ) and the best fit presented here. All 28 events with standard microlensing fits (including blending) which indicated ̂ t std > 140 days where fit with a microlensing parallax model as well, and the 10 events with a microlensing parallax detection with a significance of ∆ χ 2 ≥ 50 are \nindicated with color in Fig. 1. The four events with 50 ≤ ∆ χ 2 < 200 are MACHO-101-B, MACHO95-BLG-27, MACHO-98-BLG-1, and MACHO-99-BLG-22, and the six strongest events with ∆ χ 2 ≥ 200 are MACHO-104-C, MACHO-96-BLG-5, MACHO-96-BLG-12, MACHO-98-BLG-6, MACHO-99-BLG1, and MACHO-99-BLG-8. These 6 events are the primary focus of this paper. \nNote that most of the events with ̂ t > 200 days and all the events with ̂ t > 300 days have a significant parallax signal. Microlensing parallax is more easily detected in such long events because the Earth's velocity changes significantly during the event, and because long events are likely to have low v ⊥ values. There are also a number of events with much shorter timescales that appear to have microlensing parallax signals significant at the ∆ χ 2 ≥ 50 level, but many of these have rather implausible parameters. This is likely to be due to the fact that other effects besides microlensing parallax can perturb the microlensing light curves in ways that can mimic the parallax effect. Examples of this include binary microlensing (see the discussion of MACHO-98-BLG-14 in Alcock et al. 2000a), and the reverse of the parallax effect, the orbital motion of a binary source star (Derue et al. 1999; Alcock et al. 2001a; Griest & Hu 1993; Han & Gould 1997), sometimes called the 'xallarap' effect. The xallarap effect can be particularly difficult to distinguish from microlensing parallax because a xallarap light curve can be identical to a parallax light curve if the period, inclination, eccentricity, and phase mimic that of the Earth. In practical terms, this is a difficulty only when the xallarap or parallax signal-to-noise is weak so that the fit parameters are poorly determined. \nIn order to avoid contamination of our microlensing parallax sample with non-parallax microlensing events, we have set a higher threshold for the events that we study in detail in this paper: ∆ χ 2 ≥ 200 . The 6 events that pass this threshold are listed in Tables 1 and 5. One of these events, MACHO-104-C, was the first microlensing parallax event ever discovered (Alcock et al. 1995), and the other five events were discovered by the MACHO Alert system. Because of this, they had the benefit of follow-up observations by the MACHO/GMAN Collaboration on the CTIO 0.9m telescope or by the MPS Collaboration on the Mt. Stromlo 1.9m telescope. Four of these five events would have passed the ∆ χ 2 ≥ 200 cut without the follow-up data, but event MACHO-98-BLG-6 only passes the cut because of the MPS follow-up data. This is probably due to a CCD failure that prevented the imaging of this event in the MACHO-Red band during most of the 1998 bulge season. \nWe have also compared our microlensing parallax fits to binary lens fits for each of these events. The parallax fits are preferred in every case with χ 2 improvements of 70.3, 81.6, 1625.8, 227.2, 1957.2, and 1601.3 for events MACHO-104-C, MACHO-96-BLG-5, MACHO-96-BLG-12, MACHO-98-BLG-6, MACHO-99-BLG-1, and MACHO-99-BLG-8, respectively.", '4.1. HST Observations of MACHO-96-BLG-5': "Event MACHO-96-BLG-5 is both the longest event in our sample, 1 and the event with the faintest source star. In fact, the microlensing parallax fit does not constrain the source star brightness very well. This is due to the faintness of the source, and due to a potential systematic error. The MACHO camera had a CCD upgrade in early 1999 which put a new CCD in the location that views MACHO-96-BLG-5 in the MACHO-Red passband. The new CCD probably had the effect of shifting the effective bandpass to a slightly different central wavelength, and so a slight systematic shift in the photometry of all the stars might be expected to occur with this upgrade. Because the MACHO-96-BLG-5 source is strongly blended with unlensed neighbors, the effect of this slight shift on the microlensing fit parameters can be relatively large because the fitting routine tries to explain all flux variation as resulting from microlensing. The best fit, with this suspect data removed, indicates that only about 12 ± 3 %of the flux associated with the 'star' seen in our ground-based images has been microlensed, which would imply that the remaining 88% of the flux must come from unlensed neighboring stars which are within ∼ 1 . 5 ' of the lensed source. Fortunately, we have a set of images from the Hubble Space Telescope's WFPC2 Camera that can be used to constrain the brightness of the source star more accurately than the fit does. \nWe had one orbit of HST data taken in the V and I (F555W & F814W) passbands of the WFPC2 Camera through Director's Discretionary Proposal # 8490, and this can be used to identify the microlensed source star. The first step in this identification process is to determine the centroid of the star that was lensed. This can be accomplished by subtracting two images which have substantially different microlensing magnifications (Alcock et al. 2001b). Since it is only the lensed source star that will appear to vary in brightness, this procedure will yield a point source centered on the location of the lens and source. Of course, the subtraction procedure must take into account the differences in the observing conditions of the two frames, including differences in seeing, pointing, sky brightness and air mass. We have accomplished this with the use of the DIFIMPHOT package of Tomaney & Crotts (1996). \nAset of 18 of our best CTIO images were selected to use for this source location task because the CTIO images generally have better seeing than the MACHO images, and because the highest magnification of the source was only observed from CTIO. These 18 images were combined to construct a master reference image which was then subtracted from each individual frame to construct a set of 18 difference images. The difference frames which had a negative flux at the location of our target were inverted, and then all the difference images were combined to make the master difference image shown in Figure 8. The centroid of the excess flux in this master difference image can be determined to better than 0.01.' \nIn order to identify the lensed source star on the HST images, we must find the correct coordinate transformation to match the ground and HST frames, but this is complicated by the fact that most of the 'stars' in the ground-based images actually consist of flux from several different stars that are blended together in the ground-based frames. We have dealt with this in two different ways: first, we used the HST \nimages to select a list of stars that were much brighter than their near neighbors, so that their positions should not be greatly affected by blending in the ground-based images. Then, we convolved the HST data with a 1.2' FWHM Gaussian PSF to simulate the resolution of the ground-based CTIO data. We then analyzed the convolved HST image with the same data reduction software used for the ground-based data. This gave an additional star list from the HST image. Two independent coordinate transformations between the groundbased and HST data were obtained by matching these two stars list to the star list for the ground-based data. \nThe HST images were dithered, and we combined them with the Drizzle routine (Fruchter & Hook 1997) prior to the comparison with the ground-based data. The CTIO R-band data were compared to the HST I and V band images as well as sum of the I and V band images. Coordinate transformation were determined to match the CTIO image coordinates to each of these HST images using the bright, isolated stars in the HST images and with the HST images convolved to ground-based seeing. This resulted in a total of 6 different comparisons between the location of the lensed star in the CTIO image and the HST images. All these comparisons yielded the same lens star location on the HST frames to better than 0.02', and this location coincides with the centroid of the star indicated in Fig. 8. This star was examined carefully in both the V and I images to determine if it could be a blend of more than one star. Model and DAOPHOT generated PSFs were subtracted at the centroid location of the lensed source, but no hint of any additional star was found. This star is very likely to be the source star for the MACHO-96-BLG-5 microlensing event. \nThe next step in the comparison of the HST and ground-based data is to determine what fraction of the flux of the object identified as a star in the ground-based frames is contributed by the source star identified in the HST images. This task is complicated by the fact that there is no close correspondence between the passbands of the ground based images and those used for the HST data. (This is due to the limitations imposed upon an HST Director's Discretionary time proposal. We requested prompt images in V and I to confirm the photometric variation implied by the microlensing parallax model, but prompt imaging in R could not be justified. Imaging in R was obtained in a subsequent GO program, and the analysis of these data will appear in a future publication.) Presumably, some combination of the V and I band images would provide a good representation of the R band ground based image. \nThe determination of the lensed flux fraction was made as follows: Photometry of the V+I combined HST frame was obtained using the IRAF implementation of DAOPHOT (Stetson 1987) and also using SoDOPHOT. Both of these packages were also used to reduce the HST images which had been convolved to mimic ground based seeing. The total stellar flux of isolated, bright stars was not conserved in these convolved images, so we found it necessary to renormalize the stellar flux in the convolved images to the ratio found for these isolated bright stars. This comparison yielded a flux fraction of 36% for the lensed component of the stellar blend identified as a single star in the ground-based images. We also followed this same procedure for the separate I and V images, and the results for the lensed flux fraction were quite similar as might be expected from the fit results shown in Table 5, which indicate no color dependence for the blending fit parameter. This is likely to be due to the fact that the stars contributing the blended light and the lensed source star are all main sequence stars of similar color which are just below the bulge turn-off. \nWe must also make a correction for the fact that the source star was still being magnified by the lens when the HST frames were taken. Because the event timescale depends upon the amount of blending that we determine from the HST analysis, it requires an iteration or two to find a fit that predicts the observed brightness of the lensed star in the HST frames. The best fit result is that the lensed source provides 33% of the total flux of the blended object that would be seen in the ground based frames in the absence of any microlensing magnification. At the time of the HST images, the lensing magnification was 1.063 according to this fit. \nFinally, we should mention the possibility that the star identified with the lensed source centroid is not, in fact, a single star. The HST images reveal no evidence of a chance superposition of unrelated stars, so this is unlikely. However, it could be that the superposition is not due to chance. Suppose, for example, that the star we've identified as the MACHO-96-BLG-5 source is actually the superimposed images of the lens and source. While this is a logical possibility, we will show below that there is no plausible scenario for this to occur because the implied lens mass cannot be made compatible with the observed brightness of the lens plus source.", '5. Lens Mass and Distance Estimates': 'The measurement of the projected speed of the lens, ˜ v , allows us to relate the lens mass to the lens and source distances \nIt is often assumed that the distance to the source, D s , is already known, at least approximately, so this relation can be considered to give the lens mass as a function of distance. Given the lens distance, one can also work out the lens velocity with respect to the line-of-sight to the source, v ⊥ . But, for some distances, the implied v ⊥ value can be unreasonably small or large. Thus, with some knowledge of the Galactic velocity distribution, we can work out an estimate for the distance and mass of the lens. This has been done for the MACHO-104-C event using a likelihood method in Alcock et al. (1995). This analysis assumes that the source star resides in the Galactic bulge, which is true for the vast majority of microlensing events seen towards the Galactic bulge. The result of similar analyses for the events presented in this paper are summarized in Table 6 and in Figs. 11-13. However, the microlensing parallax events are selected from a sample of unusually long microlensing events, so it may be that their source star locations are atypical as well. With the data currently available to us, we have two ways to investigate the location of the source stars for our microlensing parallax events. The first is to make use of the direction of projected velocity as determined by the microlensing parallax fit, and the second is to examine the location of the source star in a color-magnitude diagram of nearby stars. Another, perhaps more effective, discriminant between different source populations is radial velocity measurements. Radial velocities for some of the source stars have been measured by Cook et al. (2002), and they have provided us with some preliminary results. \nM = ˜ v 2 ̂ t 2 c 2 16 G D s -D /lscript D /lscript D s = ˜ v 2 ̂ t 2 c 2 16 G 1 -x xD s . (8)', '5.1. Source Star Locations': "The line-of-sight toward a Galactic bulge microlensing event passes through the Galactic disk, the bulge, and through the Sagittarius (SGR) Dwarf Galaxy behind the bulge. So, all of these are possible locations for the source stars. The variation in the source population/location can affect the inferred properties of the lens in several different ways: \n- 1. Microlensing rate: The microlensing rate per source star is very much lower for foreground Galactic disk stars and very much higher for SGR Dwarf stars than for Galactic bulge stars. Thus, foreground disk stars and SGR Dwarf stars will be under-represented and over-represented, respectively, in samples of microlensed stars when compared to stars in the Galactic bulge.\n- 2. Microlensing parallax detectability: some source star populations such as the foreground Galactic disk and the Sagittarius Dwarf Galaxy give rise to a larger fraction of events with microlensing parallax parameters that can be measured.\n- 3. Source distance: A source at a greater distance than the nominal Galactic bulge distance will usually imply a lower lens mass since M is a decreasing function of D s in eq. (8) (for fixed x ). Similarly, a smaller D s implies a larger mass.\n- 4. Source velocity: From eq. (7), we see that, for a fixed ˜ v value, a smaller v ⊥ value implies a smaller D s which, in turn, implies a smaller lens mass (for fixed D /lscript ). Smaller v ⊥ values are expected for lensing of foreground disk sources since the source and lens would both share the Galactic rotation velocity of the Sun. \nSeveral authors who have modeled microlensing parallax events (Mao 1999; Soszy'nski et al. 2001; Smith, Mao, & Wo'zniak 2001) have suggested that the source stars must be predominantly in the foreground Galactic disk because this makes a small v ⊥ more likely. A disk source is the only possibility for the OGLE1999-CAR-1 event since this star is located far from the bulge, but for events towards the Galactic bulge there are several factors that make a foreground disk source star less likely, including a much lower microlensing optical depth and a lower density of source stars. These are discussed in section 5.3, where we find that disk sources that are definitely in the foreground of the bulge at D s ≤ 5 kpc are quite unlikely.", '5.2. Projected Velocity Distributions': "One distinguishing characteristic of microlensing parallax distributions for different source populations is the distribution of the projected velocity, ˜ v including both the amplitude, ˜ v , and the direction θ . We use a Galactic model in which the stars around us are moving with a velocity dispersion of about 30 km/sec in both directions normal to the line of sight to the bulge. The Sun rotates at a speed of +16 km/sec faster than the kinematic Local Standard of Rest (LSR) and is moving towards Galactic North at 7 km/sec (Dehnen & Binney 1998). The Galactic disk rotates with an approximately flat rotation curve at v /similarequal 200 km/sec, while \nthe Galactic bulge probably has little rotation (Minniti 1996) and has a velocity dispersion of 80100 km/sec (Spaenhauer, Jones, & Whitford 1992; Minniti 1996; Zoccali et al. 2001). The Sagittarius Dwarf Galaxy is moving at 250 ± 90 km/sec in a direction that is only a few degrees away from Galactic North (Ibata et al. 1997). \nThe different velocity distributions of these source and lens populations lead to different expectations for the measured ˜ v distributions for events from different source star populations. However, the observed ˜ v distribution is strongly affected by selection effects since only a small fraction of microlensing events have detectable parallax signals. These selection effects can be difficult to precisely quantify because of the fact that much of the data taken for these events comes from follow-up programs with observing strategies that can be subjective and difficult to model. Therefore, instead of attempting a detailed simulation of the actual observing conditions, we investigate the ˜ v distribution using a 'toy model' of a microlensing survey and follow-up program. ((Buchalter & Kamionkowski 1997) also performed simulations of microlensing parallax events in a somewhat different context.) We assume a disk velocity dispersion of 30km / s in each direction, with a flat rotation curve of 200km / s and a bulge velocity dispersion of 80km / s with no bulge rotation, and the density profiles are a standard double-exponential disk and a barred bulge as in Han & Gould (1996). We assume that events are observed for 7 months per year by a microlensing survey system that makes photometric observations with 5% accuracy every 3 days. Once an event is magnified by at least 0.5 magnitudes, daily follow-up observations start with an accuracy of 1% for each day. This simulated data are then fit with a standard, no-parallax microlensing model, and the ∆ χ 2 is determined. (Since we have not added noise to the light curves, the fit χ 2 = 0 when there is no microlensing parallax signal.) Events with ∆ χ 2 ≥ 200 are considered microlensing parallax detections, and the ˜ v values for these simulated detected events are shown in Figure 9. This figure uses Galactic coordinates in which the y -axis is the direction of Galactic disk rotation, and the z -axis is Galactic North. \nAstriking feature of Figure 9 is that all six of our strong microlensing parallax events have ˜ v in the same quadrant with positive ˜ v y and negative ˜ v z . This is the region that is preferred for both bulge and SGR source stars, but not for foreground disk sources. In our simulations, 65% of the detectable SGR source events, and 50% of the detectable bulge sources, but only 29% of foreground disk sources lie in this quadrant. \nOne selection effect that affects each plot is that events with ˜ v roughly parallel to the ecliptic plane are easier to detect than events where ˜ v is approximately perpendicular to the ecliptic plane. This effect favors the positive ˜ v y -negative ˜ v z and negative ˜ v y -positive ˜ v z quadrants. The reason for this is that the Earth's orbital motion only affects u ( t ) near peak magnification when ˜ v is perpendicular to the ecliptic plane, but the orbital motion affects u ( t ) for a longer period of time when it is parallel to ˜ v . \nFor the bulge sources, there is a preference for positive ˜ v y motion because disk lens stars are passing inside of us at a higher angular velocity. If the source stars are rotating with us, as would be the case for disk sources in the foreground of the bulge, then the rotation is common to the source, lens and observer, and it has no effect. A smaller systematic effect occurs in the disk source case because the Sun is moving about 16 km/sec faster than the mean stellar motion around us. Thus, there is a slight enhancement of the abundance of negative ˜ v y events. \nFor SGR source stars, signal of the SGR proper motion toward the Galactic North can clearly be seen in the strong concentration of events at negative ˜ v z and positive ˜ v y . (Since ˜ v is a lens -source velocity, the ˜ v signal is in the opposite direction of the SGR motion.) For bulge lenses, the 250 km/sec velocity is reduced to 90 km/sec by the projection effect, and for the disk lenses that make up the bulk of microlensing parallax sample for SGR sources, the typical ˜ v z is -50 km/sec or so. For SGR sources, and disk lenses, the combination of SGR proper motion and disk rotation put the majority of ˜ v values in the positive ˜ v y -negative ˜ v z quadrant where the alignment with the ecliptic plane makes the parallax effect easy to detect.", '5.3. Microlensing Parallax Selection Effects': "For comparison between the different source populations, it is necessary to consider several different selection effects. First, the microlensing rate for SGR sources behind our fields is a factor of ∼ 6 larger than for bulge source stars (Cseresnjes & Alard 2001), and the fraction of SGR events with detectable microlensing parallax signals is a factor of 3 larger for sources in SGR than for bulge sources. Thus, it would appear that the probability of detecting a microlensing parallax event for a SGR source is a factor of ∼ 20 higher than for a bulge source (assuming that the sources are bright enough for reasonably accurate photometry). Of course, Galactic bulge sources are much more numerous, so microlensing parallax events with Galactic bulge source stars are likely to be more numerous than events with SGR source stars by an amount that is difficult to estimate. We consider this in detail in Section 5.4 when we present the source star color-magnitude diagrams. \nIt is quite difficult to distinguish Galactic bulge stars from stars in the inner Galactic disk because they are at similar distances and their velocity distributions overlap. In fact, this distinction is likely to be somewhat artificial because the two components are likely to have merged due to their mutual gravitational interactions. Therefore, we will limit our consideration of foreground disk sources to stars with a distance < 5 kpc. For stars at 5 kpc distance at a Galactic latitude of b = -3 · , the microlensing rate is a factor of about ∼ 40 lower than for Galactic bulge stars. (The optical depth is only a factor of ∼ 20 lower because of the longer time scales of disk-disk lensing events.) The physical density of disk stars is about an order of magnitude lower than the density of bulge stars, but there is also a volume factor that reduces the number of disk stars per unit distance modulus and solid angle by a factor of 4 at 5 kpc. The product of these factors yields a net suppression factor of 1 / 1600 for disk star lensing events for a fixed source star absolute magnitude. \nThis suppression factor must be multiplied by two enhancement factors. First, our simulations indicate that the chances of detecting a microlensing parallax signal are about a factor of 5 larger for disk sources than for bulge source stars. This increases the suppression factor to ∼ 1 / 320 . There is an additional enhancement factor due to the fact that the foreground disk stars are intrinsically fainter and the stellar luminosity function rises for fainter stars, but the difference between the disk stars at 5 kpc and the bulge stars at 8 kpc is only 1 magnitude. From Holtzman et al. (1998) we see that this factor is at most ≈ 10 if we select a source magnitude such that is 1-2 magnitudes above the bulge main sequence turnoff. For \nmagnitudes that correspond to bulge main sequence stars, it is less than a factor of two. Thus, we expect disk stars (with D < 5 kpc) to contribute less than 1% of the total number of detectable microlensing parallax events, except for source stars that are 1-2 magnitudes above the bulge main sequence turn-off where they might account for as many as 3% of the microlensing parallax events with bulge source stars. \nInner disk stars at D > 5 kpc will be accounted for by allowing their velocities to contribute to the assumed bulge velocity distribution. In fact, such inner disk stars are generally not excluded from star samples that are used to measure the bulge proper motion (Spaenhauer, Jones, & Whitford 1992; Zoccali et al. 2001). We will, therefore, classify all stars in the vicinity of the bulge ( 5kpc < D s < 11kpc ) as bulge stars. Instead of trying to distinguish different, but overlapping, populations of source stars, we consider a single model including all these stars. \nStars on the far side of the disk have velocities that make it very unlikely to see the microlensing parallax effect, while foreground disk stars are unlikely to be microlensed at all. Therefore, the SGR dwarf provides the only 'non-bulge' population of potential source stars that we will consider in the remainder of this paper.", '5.4. Color Magnitude Diagrams': 'Fig. 10 shows color-magnitude diagrams for all the stars within 2 arc minutes around each of our microlensing parallax source stars, with the lensed source indicated by a red circle. It is necessary to use different color magnitude diagrams for each event because of the large variation in reddening between different fields. By plotting only the stars within 2 arc minutes of our targets, we have minimized the variation in reddening. \nThese CM diagrams indicate that the MACHO-104-C and MACHO-96-BLG-12 source stars are located in the bulge red clump region, which means that they are likely to reside in the Galactic bulge. The MACHO-99-BLG-8 source star is more luminous than the red clump and is likely to be a bulge giant. Cook et al. (2002) find a radial velocity of v r = 195 ± 2km / s which confirms the bulge interpretation for this event. The MACHO-96-BLG-5 source star appears to be fainter than virtually all of the other stars in its CM diagram. This is a consequence of the extreme crowding of these Galactic bulge fields. The density of bright main sequence stars is ∼ 2 per square arc second, so main sequence stars are not individually resolved in these crowded Galactic fields. Instead, it is groups of unresolved main sequence stars that are identified as single stars, and it is these unresolved blends of multiple stars that make up the majority of the fainter objects identified as stars in these images. The majority of microlensed source stars in the Galactic bulge are blended main sequence stars like the MACHO-96-BLG-5 source, but the microlensing parallax signal is easier to detect for brighter source stars. \nThe source stars for events MACHO-98-BLG-6 and MACHO-99-BLG-1 appear to be on the bulge subgiant branch of the color magnitude diagram. They have a similar color to bulge red clump stars, but they are about 2 magnitudes fainter. This suggests that they could be red clump stars ∼ 14 kpc behind the bulge in Sagittarius (SGR) Dwarf Galaxy. This is about the only location on the color magnitude diagram were \nwe might expect to see microlensing of SGR source stars, because SGR red clump stars are probably the only abundant type of SGR stars that are brighter than the bulge main sequence stars that set the confusion limit. This SGR source interpretation appears to gain support from the location of these events in Fig. 9 which indicates that their parallax velocities are among the ones most consistent with Sagittarius Dwarf kinematics. \nA rough estimate of the probability of detecting microlensed SGR source stars can be made by noting that SGR Dwarf RR Lyrae stars are about 2.6% as numerous as bulge RR Lyrae in the MACHO fields (Alcock et al. 1997a). In a microlensing parallax sample, we should expect SGR source stars to be enhanced by a factor of ∼ 20 , but we must also include both bulge sub-giants and giants in the comparison with the SGR red clump giants. This would reduce the fraction of SGR events by a factor two or so. This would suggest that we might expect that for every 4 microlensing parallax events with bulge giant or sub-giant source we could expect one SGR giant source star event. 2 On the other hand, the ratio of red clump stars to RR Lyrae is likely to be higher for SGR stars than for Galactic bulge stars because of the lower metalicity of SGR, so we might expect fewer SGR events than this RR Lyrae comparison would suggest. \nThese considerations suggest that we should take the SGR source star hypothesis seriously for these events. However, Cook et al. (2002) used the Keck HIRES spectrograph to obtain spectra of the source stars for these events, and they find radial velocities of v r = -65 ± 2km / s and v r = 64 ± 2km / s for MACHO98-BLG-6 and MACHO-99-BLG-1, respectively. This is not consistent with the SGR radial velocity (Ibata et al. 1997) of v r = 140 ± 10km / s , and they are about 2 σ away from the expectation for a disk source star (Wielen 1982). Thus, these events are most likely to have bulge sub-giant source stars.', '5.5. Likelihood Distance and Mass Estimates': "Another, somewhat more general, constraint on x and M can be obtained if we make use of our knowledge of the velocity distributions of the source and lensing objects, since the likelihood of obtaining the observed value of ˜ v is a strong function of the distance to the lens. Note that this assumes that stellar remnant lenses have a velocity and density distribution that is similar to that of observed stellar populations. For neutron stars, this might be a questionable assumption because many neutron stars are apparently born with a large 'kick' velocity. However, for black holes, the evidence indicates that significant kick velocities are rare (Nelemans, Tauris, & van den Heuvel 1999). As an example of such an analysis, let us suppose that the disk and bulge velocity dispersions were negligible relative to the Galactic rotation velocity. Then, for disk lenses we would obtain the relation ˜ v = 200 D /lscript / ( D s -D /lscript ) km / s implying a lens distance of D /lscript = D s ˜ v/ (˜ v + 200km / s) . In reality, the random motions of both disk and bulge stars broaden this relationship somewhat, but we can still obtain a useful constraint. \nGiven the observed ˜ v , we obtain a likelihood function \nL ( x ; ˜ v ) ∝ √ x (1 -x ) ρ L ( x ) ˜ v (1 -x ) 3 ∫ f S ( v S ) f L ((1 -x )( v /circledot + ˜ v ) + x v S ) d v S , (9) \nwhere ρ L is the density of lenses at distance x = D /lscript /D s , and the integral is over combinations of source and lens velocities giving the observed ˜ v . v S and v L = (1 -x )( v /circledot + ˜ v ) + x v S are the 2-D source and lens velocity distribution functions (normalized to unity). We assume the same Galactic parameters as in our ˜ v simulations above: a disk velocity dispersion of 30km / s in each direction, a flat disk rotation curve of 200km / s , and a bulge velocity dispersion of 80km / s with no bulge rotation. The density profiles are a standard double-exponential disk and a Han & Gould (1996) barred bulge. For all events, the source is assumed to reside in the bulge, while the lens may be in the disk or the bulge. But for events MACHO-98BLG-6 and MACHO-99-BLG-1, we also consider the possibility of a SGR Dwarf source star with the lens in the disk or bulge, although this now appears to be ruled out (Cook et al. 2002). \nThe resulting likelihood functions for D /lscript is shown as the long-dashed curves in Figs. 11-13, and these are insensitive to specific parameter choices. These likelihood functions also provide a means for estimating the lens masses via the relation (8), which is also plotted in Figs. 11-13. Fig. 14 shows how the mass estimates correlate with the best fit event timescale for the six high signal-to-noise microlensing parallax events as well as four other events of lower signal-to-noise. (The lower signal-to-noise event with the highest mass is MACHO-99-BLG-22/OGLE-1999-BUL-32 which has been presented as a black hole candidate by Mao et al. 2001.) \nOne common way to interpret likelihood functions is the Bayesian method, in which the lens mass (or distance) probability distribution is given by the likelihood function times a prior distribution, which represents our prior knowledge of the probability distribution. In our case, the likelihood function represents all of our knowledge about the lens mass and location, so we select a uniform prior. With a uniform prior, the likelihood function becomes the probability distribution and we are able to calculate the lens mass confidence levels listed in Table 6. This table also includes lens mass confidence levels for models that differ from the preferred model in order to show how the mass estimates depend upon the amount of blending (for MACHO-96-BLG-5) and on whether the source star resides in the Galactic bulge or the SGR Dwarf. Note that the uncertainty in the mass estimates is smaller for SGR Dwarf sources due to the small velocity dispersion of the SGR Dwarf and the smaller range of likely lens distances.", '5.6. Constraints on Main Sequence Lenses': 'If we assume that the lens stars are main sequence stars, then we can obtain an additional constraint on their distances and masses by comparing the brightness of a main sequence star, of the implied mass, to the upper limit on the brightness of the lens star. We have assigned a conservative upper limit on the V-band brightness of each lens star based upon the available photometry and microlensing parallax fits listed in Table 5. In the case of MACHO-96-BLG-5, the upper limit is particularly stringent because it is based upon HST observations. Note that if we assign some of the flux of the star identified in the HST images to the \nlens star instead of the source, the best fit ̂ t will increase almost linearly with the inverse of the source star flux. This causes the lens mass estimate to increase as ∼ ̂ t 2 . Since stellar luminosity varies as a high power of the mass, a main sequence lens will be more strongly ruled out. \nIn order to apply these constraints to the likelihood functions for the mass and distances of the lens stars, we have multiplied the likelihood function by the Gaussian probability that the lens brightness exceeds the upper limit on the brightness of lens star. If a main sequence lens star would be fainter than the observed maximum brightness, there is no modification of the likelihood function. This gives the shortdashed likelihood curves shown in Figs. 11-13. These results are insensitive to our assumed L ∝ M 4 mass-luminosity relation. The assumed maximum lens brightnesses are V = 19 . 88 , 20 . 57 , and 16 . 92 for events MACHO-96-BLG-12, MACHO-98-BLG-6, and MACHO-99-BLG-8, respectively. These are based upon the amount of blending allowed by the fit, and each of these has an assumed 25% uncertainty which is also based upon the fit. For MACHO-96-BLG-5, the maximum lens brightness is V = 23 . 63 , with an assumed 50% uncertainty. For MACHO-104-C, and MACHO-99-BLG-8, the best fit has very little blended flux: V = 22 . 33 , and 23 . 08 , respectively. But, in both cases, the uncertainty in the blended flux is five times the best fit value. \nThe properties of the most likely main sequence lens models are given in Table 7, which is discussed in more detail in section 6.2. An important parameter in this table is the predicted lens-source separation in June, 2003, when they might plausibly be observed by HST. This can be calculated from the lens-source proper motion which is related to the projected velocity by µ = ˜ v ( D s -D /lscript ) / ( D s D /lscript ) .', '5.7. Stellar Remnant Lenses and Black Hole Candidates': 'The mean mass estimate for the six microlensing parallax events is 2 . 7M /circledot . Five of the six have best fit masses > 1M /circledot , and two of the events, MACHO-96-BLG-5 and MACHO-98-BLG-6, have best fit masses > 3M /circledot . This makes them black hole candidates because the maximum neutron star mass is thought to be ∼ 2M /circledot (Akmal, Pandharipande, & Ravenhall 1998). The 95% confidence level lower limits on the masses of these lenses are 1 . 64M /circledot and 0 . 94M /circledot , respectively, while the 90% confidence level lower limits are 2 . 3M /circledot and 1 . 9M /circledot . A main sequence star lens at the lower limit mass is strongly excluded in the case of MACHO96-BLG-5 because of the constraint on the lens brightness from HST images. However, a main sequence lens with a mass at the 95% confidence limit is not quite excluded for MACHO-98-BLG-6. The masses that have been measured for neutron stars are close to the Chandrasekhar mass, M NS = 1 . 35 ± 0 . 04M /circledot (Thorsett & Chakrabarty 1999), which is excluded at better than 95% confidence for MACHO-96-BLG-5 and better than 90% confidence for MACHO-98-BLG-6. Thus, both MACHO-96-BLG-5 and MACHO98-BLG-6 are both black hole candidates, but there is a small chance that MACHO-98-BLG-6 could be a neutron star or even a main sequence star. \nIn addition to these black hole candidates, three of the remaining four microlensing parallax events have best fit masses > 1M /circledot . For MACHO-104-C and MACHO-96-BLG-12, main sequence lens are disfavored, but not ruled out. MACHO-99-BLG-8 appears to be blended with a relatively bright source, so \na main sequence lens of M > ∼ 1M /circledot is a possibility. As we explain below, with HST imaging it will be straightforward to detect the lenses if they are main sequence stars. If HST images fail to detect the lens stars, then we can show that the lenses are almost certainly stellar remnants.', '5.8. Likelihood Analysis with a Mass Function Prior': "The Likelihood analysis presented in Section 5.5 attempts to estimate the distance to the lens based upon the measured value of the projected velocity, ˜ v , and then the lens mass is determined from eq. 8. If the lens mass function, dn/dM = φ ( M ) , is known, then it is possible to use the measured ̂ t value to make a more accurate estimate of the lens mass as advocated by Agol et al. (2002). The likelihood function, eq. 9, can be modified by multiplying by δ ( ̂ t -̂ t m ) M 1 / 2 φ ( M ) dM and integrating over M , where ̂ t m is the measured value of ̂ t . The factor of M 1 / 2 is the contribution of the lens mass to the lensing cross section, which is proportional to R E . The integral over δ ( ̂ t -̂ t m ) dM gives an additional factor of M . Thus, the likelihood analysis presented in Section 5.5 is equivalent to assuming a mass function of φ ( M ) ∝ M -1 . 5 . \nA more conventional mass function for the Galactic bulge is a broken power law initial mass function (Kroupa 2002) with φ ( M ) ∝ M -1 . 3 for 0 . 03M /circledot ≤ M ≤ 0 . 8M /circledot , and φ ( M ) ∝ M -2 . 35 for 0 . 8M /circledot ≤ M ≤ 100M /circledot . However, the stars with M > 1 . 0M /circledot will generally have ended their main sequence lifetimes and have become stellar remnants after significant mass loss. Following Fryer & Kalogera (2001), we can assume that all stars with an initial mass greater than a particular cutoff mass, M i > M BH become black holes. We take M BH = 20M /circledot (Fryer 1999; Fryer & Kalogera 2001). Similarly, we assume that all stars with 8M /circledot ≤ M i < M BH become neutron stars, and all stars with 1 . 0M /circledot < M i < 8M /circledot become white dwarfs. The mass functions of the stellar remnants are assumed to be Gaussians with mean masses of 0 . 6M /circledot , 1 . 35M /circledot , and 8M /circledot for white dwarfs, neutron stars, and black holes respectively. The Gaussian sigmas are 0 . 15M /circledot , 0 . 04M /circledot , and 2 . 5M /circledot , respectively. These are consistent with the measured mass functions (Bergeron, Saffer, & Liebert 1992; Bergeron, Leggett, & Ruiz 2001; Thorsett & Chakrabarty 1999; Bailyn, Jain, Coppi, & Orosz 1998), although the difficulty of directly observing old stellar remnants assures that the observed samples are incomplete. With this mass function, black holes would account for 3.7% of the Galaxy's stellar mass. \nA Bayesian analysis based upon this mass function gives a probability of 93% that the MACHO-96BLG-5 lens is a black hole and a probability of 69% that the MACHO-98-BLG-6 lens is a black hole. The probability of at least one black lens is 98%. This analysis may underestimate the black hole probability because the assumed mass function cannot account for the large number of long timescale microlensing events. An initial IMF that is slightly shallower than the Salpeter slope, φ ( M ) ∝ M -2 . 0 , might be appropriate if most of the stars in the Galaxy were formed in denser or more metal poor regions than is typical for present day star forming regions (Figer et al. 1999; Smith & Gallagher 2001). With this mass function and with M BH = 20M /circledot , black holes would account for 12% of the Galaxy's stellar mass. When we repeat the likelihood analysis with this mass function, we find black hole probabilities of 97% for MACHO-96-BLG-5 and 88% for MACHO-98-BLG-6. The probability of at least one black hole lens with this mass function is \n99.7%. If we retain the Salpeter IMF slope, and increase M BH to 40M /circledot , then the black hole probabilities for MACHO-96-BLG-5 and MACHO-98-BLG-6 drop to 82% and 43%, respectively. However, such a mass function probably cannot explain the excess of long timescale events. \nWe should note that these probabilities are substantially larger than those reported in a similar analysis in a preprint by Agol et al. (2002). This was due to a likelihood function calculation error by Agol et al. (2002). When this error is corrected, their results are quite similar to those presented here (Agol, private communication).", '6. Follow-up Observations': 'The detection of the microlensing parallax effect allows us to make a lens mass estimate that is accurate to about a factor of two, and to identify the black hole candidates. However, these estimates are not accurate enough to determine the black hole mass function, and they do not allow the unambiguous identification of neutron star or white dwarf lenses. However, follow-up observations with higher resolution instruments hold the promise of much more precise determinations of the lens masses.', '6.1. Interferometric Follow-up': "The most ambitious of microlensing event follow-up plans involve interferometric instruments such as the Keck and VLT interferometers (Delplancke, G'orski & Richichi 2001) and the Space Interferometry Mission (SIM) (Boden, Shao, & van Buren 1998). The most spectacular confirmation of a black hole event would be to measure the image splitting which is given by \nφ sep = 2 θ E √ 1 + u 2 / 4 , (10) \nM = ˜ v tθ E c 2 8 G . (11) \nwhere θ E is the image separation and u is given by eq. (4). For MACHO-96-BLG-5, we have θ E = 9 . 8 mas if the lens is at the distance preferred by the likelihood analysis. This compares to the 5 mas diffraction limit of an interferometer with a 100 m baseline operating at a wavelength of 2 µ m, such as the Keck or VLT Interferometers. In fact, these instruments are expected to be able to measure image splittings as small as ∼ 30 µ as (Delplancke, G'orski & Richichi 2001). Such measurements would allow a direct measurement of the lens mass: \nThe most challenging aspect of such measurements is the faintness of source stars such as the MACHO96-BLG-5 source, which is close to the (rather uncertain) magnitude limit of the VLT Interferometer (Delplancke, G'orski & Richichi 2001). \n̂ \nEven if the images cannot be resolved, it may be possible to measure the deflection of the image \ncentroid (Hog, Novikov, & Polnarev 1995; Miyamoto & Yoshi 1995; Walker 1995) which is given by \n∆ φ = ( u 2 +3 √ u 2 +4 -u ) θ E . (12) \nThis can be measured by a very accurate astrometry mission such as SIM (Boden, Shao, & van Buren 1998; Paczy'nski 1998; Gould 2000). Once again, however, the MACHO-96-BLG-5 source is a rather faint target for SIM, but in this case, the measurement is not so difficult because the amplitude of the centroid motion is very much larger than SIM's sensitivity limit. \nIf it should turn out that some of the more massive lenses are located very close to us, then it might be possible to directly observe the lensed images with HST. This is a realistic possibility for the MACHO99-BLG-22/OGLE-1999-BUL-32 event (Mao et al. 2001) because its ˜ v value is in the opposite quadrant from the events studied in this paper. This gives a likelihood function with two peaks: one at a distance of ∼ 500 pc for a lens in the disk and one at a distance of ∼ 6 kpc for a bulge lens (Bennett et al. 2002). The bulge lens solution predicts a mass of a few M /circledot , but the disk lens solution predicts a mass of > 100M /circledot and a lensed image separation of ∼ 0 . 1 . '", '6.2. Lens Detection and Source Proper Motion': "Another method can be used to make a direct determination of the lens mass for a bright lens star. If the lens can be detected and the relative proper motion of the lens with respect to the source is measured, then it is also possible to determine the lens mass from the proper motion and microlensing parallax parameters with the following formula: \nM = ˜ v t 2 µc 2 16 G , (13) \nwhere µ is the relative lens -source proper motion. This technique has the advantage that the proper motion measurements can be made many years after the peak magnification of the microlensing event. The lenssource separation can reach the 50-100 mas range within 5-10 years. Table 7 shows the predicted separations and lens brightness contrasts for our six strong microlensing parallax events. The columns are (1) the MACHO event name, (2) the lens mass with 1 σ errors, (3) a likely lens mass, M rmMS , if the lens is on the main sequence, (4) the lens distance, D /lscript -MS for a lens of mass M rmMS , (5) the predicted lens-source separation in June, 2003, (6) the apparent V magnitude of the lenses, and (7-11) the predicted contrast between the lens and source brightness in the UBVI bands. Positive ∆ -mags. imply that the source is brighter than the lens, so lens detection is easiest for events that have small or negative ∆ -mag. values. With the exceptions of MACHO-96-BLG-5, which doesn't have a viable main sequence lens model, all of the other lens stars should be detectable if they are not stellar remnants. \n̂ \nWhen the lens can be detected, it should also be possible to constrain the unlensed brightness of the source star, which will reduce the error bars on ̂ t . Also, it should be possible to get very accurate measures of the relative proper motion, µ , as the lens moves further from the source. Thus, the ultimate limits on the masses of the lenses may come from the uncertainties in the ˜ v values, which range from 2 -10 %. \nWhen the lenses are undetectable, it should still be possible to measure the proper motion of the source star with HST images separated by ∼ 5 years. The proper motion can only be measured with respect to the average of other, nearby stars because extra-galactic reference sources are not easily identified in these crowded Galactic bulge fields (Spaenhauer, Jones, & Whitford 1992; Zoccali et al. 2001). Proper motion measurements of the microlensed source stars would allow us to remove one degree of freedom from our likelihood analysis and reduce the uncertainty in the implied lens distances and masses. The proper motion distribution of the stars in the same field will also allow us to test the Galactic models that are used for the likelihood analysis, and so this should reduce the systematic uncertainties in the lens distance and mass estimates.", '7. Discussion and Conclusions': "We have performed microlensing parallax fits on the Galactic bulge events detected by the MACHO Collaboration with timescales of ̂ t ≥ 140 days, and found six events with highly significant detections of the microlensing parallax effect. Our analysis of the velocity distributions expected for parallax microlensing events from different source star populations suggests that source stars in the SGR Dwarf Galaxy might contribute to the detectable microlensing parallax events, and inspection of the source star color-magnitude diagrams indicates that two of our microlensing parallax events have source stars which could be SGR Dwarf red clump stars. However, radial velocity measurements (Cook et al. 2002) indicate that they are probably bulge sub-giant stars. \nA likelihood analysis has been employed to estimate the distance and masses of the lenses, and this indicates an average mass for our six lenses of 2 . 7M /circledot . Two of the lenses have masses large enough to imply that they are probably massive stellar remnants: The mass estimates for the MACHO-96-BLG-5 and MACHO-98-BLG-6 lenses are M/ M /circledot = 6 +10 -3 and M/ M /circledot = 6 +7 -3 , respectively, which implies that both are likely to be black holes. Together with MACHO-99-BLG-22/OGLE-1999-BUL-32 (Mao et al. 2001), these are the first black hole candidates that are truly black since we have not seen any radiation from matter that is gravitationally bound to the black hole. \nOur likelihood analysis differs from that of Agol et al. (2002) in that we compute the likelihood for the measured ˜ v value whereas Agol et al. (2002) attempt to compute the likelihood of the measured values of ̂ t as well as ˜ v . However, this requires that we input the mass function of the lenses, and this has never been measured for a complete sample of stellar remnants. Thus, the method of Agol et al. (2002) can give misleading results if the input mass function is not correct. Nevertheless, the results of such an analysis are consistent with the results that we have presented here. (Note that the preprint version of Agol et al. (2002) claimed an inconsistency with our results, but this was due to an error in the computation of the likelihood function (Agol, private communication).) For the MACHO-99-BLG-22/OGLE-1999-BUL-32 event, the method of Agol et al. (2002) does give potentially misleading results, however, because the shape of the Likelihood function for this event makes the results quite sensitive to the assumed black hole mass function (Bennett et al. 2002), which is, of course, unknown. \nSimilar events detected in the next few years may yield lens masses that are measured much more precisely due to follow-up observations from ground-based (Delplancke, G'orski & Richichi 2001) and spacebased (Gould 2000) interferometers. This will allow an unambiguous determination of the abundance and mass function of black hole and neutron star stellar remnants, although it may be difficult to determine if ∼ 2M /circledot objects are black holes or neutron stars. At present, there are three black hole microlens candidates in the sample of 321 microlensing events that was the starting point for this paper (although MACHO-99BLG-22 is only identified as a strong black hole candidate when OGLE data are included in the analysis (Mao et al. 2001; Bennett et al. 2002)). This is about 1% of the events, but far more than 1% of the total contribution to the microlensing optical depth. This suggests that the fraction of our Galaxy's stellar mass that is in the form of black holes may be significantly larger than 1%, which might help to explain the observed excess of long timescale microlensing events. However, we have not made an accurate determination of our microlensing event detection efficiency for this data set, and the detection efficiency is certainly larger for long timescale microlensing events than for short events. It is also possible that one of these three lenses may not be a black hole, and so these microlensing results may still be consistent with models which predict that of order 1% of the Milky Way's stellar mass should be in the form of black holes (Brown & Bethe 1994; Fryer & Kalogera 2001; Gould 2000). If all three of these events are truly due to black hole lenses, then a black hole mass fraction as high as ∼ 10 % might be preferred. These results appear to indicate that most stellar mass black holes do not reside in the X-ray binary systems where they are most easily observed (Bailyn, Jain, Coppi, & Orosz 1998). \nWe thank Eric Agol for discussions regarding the use of a mass prior in the likelihood analysis. This work was supported, in part, by NASA through the Space Telescope Science Institute (GO 8490) and through the NASA Origins Program (NAG5-4573). It was also supported by the National Science Foundation grants program (AST96-19575), and through the Office of Science and Technology Centers (AST-8809616). DM is supported by FONDAP Center for Astrophysic. CWS thanks the Packard Foundation for the generous support. WJS is supported by a PPARC Advanced Fellowship. CAN is supported in part by a NPSC Graduate Fellowship. TV and KG were supported in part by the DOE.", 'REFERENCES': "Afonso, C., et al. 2000, ApJ, 532, 340 \nAgol, E., et al. 2002, ApJ, submitted (astro-ph/0203257) \nAkmal, A., Pandharipande, V. R., & Ravenhall, D. G. 1998, Phys.Rev. 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W. 2000, ApJ, 530, 418 \nZoccali, M., Renzini, A., Ortolani, S., Bica, E., & Barbuy, B. 2001, AJ, 121, 2638 \nFig. 1.- The distribution of event timescales, ̂ t , for the 28 events with ̂ t > 140 days. The colored bars indicate the events with formally significant detections of microlensing parallax, but parallax signal for the events indicated in yellow is weak enough that the detection is not considered to be definitive. \n<!-- image --> \nFig. 2.- MACHO-104-C light curves normalized to the unlensed flux of the lensed star. The MACHO red and blue data are plotted in magenta and blue, respectively. The black curve is the parallax fit while the cyan curve is the best fit standard microlensing lightcurve. An additional 5 years of data showing no photometric variation are not shown. \n<!-- image --> \nFig. 3.- MACHO-96-BLG-12 lightcurve closeup with lightcurves normalized to the unlensed flux of the lensed star. The MACHO red and blue data are plotted in magenta and blue, respectively, and the CTIO data are shown in red. The black curve is the parallax fit while the cyan curve is the best fit standard microlensing lightcurve. An additional 5 years of data showing no photometric variation are not shown. \n<!-- image --> \nFig. 4.- MACHO-98-BLG-6 lightcurve closeup with lightcurves normalized to the unlensed flux of the lensed star. The MACHO red and blue data are plotted in magenta and blue, respectively, the CTIO data are shown in red, and the MPS data are shown in green. The black curve is the parallax fit while the cyan curve is the best fit standard microlensing lightcurve. The gap in the MACHO red data during the day 2280-2650 interval is due to a CCD failure. An additional year of data showing no photometric variation is not shown. \n<!-- image --> \nFig. 5.- MACHO-99-BLG-1 light curves normalized to the unlensed flux of the lensed star. The MACHO red and blue data are plotted in magenta and blue, respectively, the CTIO data are shown in red, and the MPS data are shown in green. The black curve is the parallax fit while the cyan curve is the best fit standard microlensing lightcurve. An additional 4 years of data showing very little photometric variation are not shown. \n<!-- image --> \nFig. 6.- MACHO-99-BLG-8 light curves normalized to the unlensed flux of the lensed star. The MACHO red and blue data are plotted in magenta and blue, respectively, the CTIO data are shown in red, and the MPS data are shown in green. The black curve is the parallax fit while the cyan curve is the best fit standard microlensing lightcurve. An additional 3 years of data showing very little photometric variation are not shown. \n<!-- image --> \nFig. 7.- MACHO-96-BLG-5 lightcurves normalized to the unlensed flux of the lensed star. The MACHO red and blue data are plotted in magenta and blue, respectively, and the CTIO data are shown in red. The black curve is the parallax fit while the cyan curve is the best fit standard microlensing lightcurve. An additional 4 years of data showing very little photometric variation are not shown. \n<!-- image --> \n<!-- image --> \nFig. 8.- The image on the left is the master difference image as described in the text. It has been registered to the same coordinate system as the F814W HST/WFPC2 image shown on the right. The red marks show the centroid of the variable flux in the master difference image and the location of this centroid when transformed to the coordinate system of the HST data. A single, main sequence bulge star is clearly identified as the lensed source star. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 9.- The distribution of ˜ v values in Galactic coordinates is shown for simulated microlensing parallax events towards the Galactic bulge for three different source star populations: the Galactic bulge, the foreground Galactic disk, and the Sagittarius Dwarf Galaxy. The large colored dots show the locations of our detected microlensing parallax events. The red circular spot is our best black hole candidate, MACHO-96BLG-5, and the green circular disk is the other black hole candidate: MACHO-98-BLG-6. The two green spots are the events with source stars that appear to be bulge sub-giants or Sagittarius Dwarf red clump stars. (MACHO-99-BLG-1 is the other). The blue squares are the bulge red clump source star events, and the blue triangle is MACHO-98-BLG-8 which has a red giant source and is probably also in the bulge. \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 10.- Color-Magnitude diagrams from MACHO data are shown for all the detected stars within a 2 arc minute circle around each of our microlensing parallax events. The red circles indicate the location of the lensed source star, assuming the blending implied by the microlensing parallax fits. \n<!-- image --> \n<!-- image --> \nFig. 11.- The mass vs. distance relations (solid curves) for our candidate black hole lenses are shown along with the likelihood functions (long dashed curves) computed assuming a standard model for the Galactic phase space distribution. The source star is assumed to reside in the bulge for both events. The implied best fit masses are M = 6 +10 -3 M /circledot for the MACHO-96-BLG-5 lens and M = 6 +7 -3 M /circledot for the MACHO-98-BLG6. The 95% confidence level lower limits on the masses are 1 . 6M /circledot and 0 . 94M /circledot respectively. The short dashed curves delineate the portion of the likelihood functions that is allowed when the lens is assumed to be a main sequence star. The ratio of the area below this portion to the entire area below the likelihood curve gives a probability that a lens is a main sequence star. For MACHO-96-BLG-5, the upper limit on the lens brightness is very stringent because of the HST images, and a main sequence lens is ruled out. \n<!-- image --> \n<!-- image --> \nFig. 12.- The mass vs. distance relations (solid curves) for our two bulge clump giant source events are shown along with the likelihood functions (long dashed curves) computed assuming a standard model for the Galactic phase space distribution. The implied best fit masses are M = 1 . 1 +1 . 1 -0 . 5 M /circledot for the MACHO104-C lens and M = 1 . 3 +1 . 8 -0 . 7 M /circledot for the MACHO-96-BLG-12. The 95% confidence level lower limits on the masses are 0 . 35M /circledot and 0 . 33M /circledot respectively. The short dashed curves delineate the portion of the likelihood functions that is allowed when the lens is assumed to be a main sequence star, and they indicate that main sequence lenses are disfavored but not ruled out. \n<!-- image --> \n<!-- image --> \nFig. 13.- The mass vs. distance relations (solid curves) for the two 1999 microlensing parallax events are shown along with the likelihood functions (long dashed curves) computed assuming a standard model for the Galactic phase space distribution. For both events the source star is assumed to reside in the Galactic bulge. The implied best fit masses are M = 0 . 7 +1 . 2 -0 . 4 M /circledot for the MACHO-99-BLG-1 lens and M = 1 . 2 +1 . 6 -0 . 6 M /circledot for the MACHO-99-BLG-8. The 95% confidence level lower limits on the masses are 0 . 14M /circledot and 0 . 3M /circledot respectively. The short dashed curves delineate the portion of the likelihood functions that is allowed when the lens is assumed to be a main sequence star, and these indicate that the lens brightness constraints are consistent with main sequence lens stars. For MACHO-99-BLG-1, a main sequence lens is disfavored, however. \n<!-- image --> \nFig. 14.- This plot shows M vs. ̂ t for the 10 events with ̂ t > 140 days with 1 σ error bars for the mass estimates. All events with formally significant detections of microlensing parallax are shown, but parallax signal for the events indicated in yellow is weak enough that the detection is not considered to be definitive. The green open symbols indicate the predicted lens masses for MACHO-98-BLG-6 and MACHO-99-BLG1 if their source stars were in the SGR Dwarf Galaxy, a possibility that appears to be contradicted by their spectra. \n<!-- image --> \nTable 1. Microlensing Parallax Event CoordinatesTable 2. Number of Observations \n| | | | | Galactic | Galactic | Ecliptic | Ecliptic |\n|-----------------|----------------|------------|-------------|------------|------------|------------|------------|\n| Event Name | MACHO Star ID | RA(J2000) | DEC(J2000) | l | b | λ | β |\n| MACHO-104-C | 104.20251.50 | 18:03:34.0 | - 28 :00:19 | 2.797 | - 2 . 933 | 270.790 | - 4 . 568 |\n| MACHO-96-BLG-5 | 104.20906.3973 | 18:05:02.5 | - 27 :42:17 | 3.219 | - 3 . 071 | 271.119 | - 4 . 270 |\n| MACHO-96-BLG-12 | 104.20382.803 | 18:03:53.2 | - 27 :57:36 | 2.871 | - 2 . 973 | 270.861 | - 4 . 524 |\n| MACHO-98-BLG-6 | 402.48103.1719 | 17:57:32.8 | - 28 :42:45 | 1.526 | - 2 . 132 | 268.762 | - 5 . 267 |\n| MACHO-99-BLG-1 | 121.22423.1032 | 18:08:50.0 | - 30 :31:56 | 1.138 | - 5 . 162 | 271.917 | - 7 . 106 |\n| MACHO-99-BLG-8 | 403.47849.756 | 17:56:25.2 | - 29 :40:31 | 0.569 | - 2 . 401 | 269.218 | - 6 . 237 | \n| Event | MACHO-Red | MACHO-Blue | CTIO | MPS |\n|-----------|-------------|--------------|--------|-------|\n| 104-C | 534 | 308 | 0 | 0 |\n| 96-BLG-5 | 558 | 1542 | 179 | 0 |\n| 96-BLG-12 | 584 | 466 | 103 | 0 |\n| 98-BLG-6 | 952 | 1083 | 29 | 212 |\n| 99-BLG-1 | 343 | 260 | 11 | 153 |\n| 99-BLG-8 | 386 | 310 | 213 | 155 | \nTable 3. Photometric Measurements \n| Event Name | Pass Band | time (MJD) | Magnitude | uncertainty |\n|----------------|-------------|--------------|-------------|---------------|\n| MACHO-104-C | MACHO-Red | 430.79500 | 14.0310 | 0.0221 |\n| | | 438.78620 | 13.9090 | 0.0188 |\n| | | 441.73940 | 13.8060 | 0.0163 |\n| | | 442.74640 | 13.8120 | 0.0163 |\n| | | 443.71500 | 13.7760 | 0.0172 |\n| | | 446.72880 | 13.7400 | 0.0155 |\n| | | 453.79520 | 13.5360 | 0.0182 |\n| | | 459.71700 | 13.3230 | 0.0301 |\n| | | 463.67350 | 13.2520 | 0.0172 |\n| | | 463.67660 | 13.2570 | 0.0200 |\n| | | ... | ... | ... |\n| | MACHO-Blue | 430.79500 | 13.0240 | 0.0167 |\n| | | 442.74640 | 12.7870 | 0.0157 |\n| | | 443.71500 | 12.7820 | 0.0157 |\n| | | 452.74020 | 12.5650 | 0.0157 |\n| | | 453.79520 | 12.5320 | 0.0160 |\n| | | 455.75470 | 12.4810 | 0.0157 |\n| | | 457.78640 | 12.4300 | 0.0157 |\n| | | 459.71700 | 12.3510 | 0.0200 |\n| | | 463.67350 | 12.2290 | 0.0157 |\n| | | 463.67660 | 12.2500 | 0.0167 |\n| | | ... | ... | ... |\n| MACHO-96-BLG-5 | MACHO-Red | 430.79500 | 16.2340 | 0.1299 |\n| | | 441.73940 | 16.3030 | 0.0726 |\n| | | 442.74640 | 16.4470 | 0.0814 |\n| | | 443.71500 | 16.2560 | 0.0942 |\n| | | 455.75470 | 16.2170 | 0.0952 |\n| | | 457.78640 | 16.3910 | 0.1110 |\n| | | 459.71700 | 15.8800 | 0.2295 |\n| | | 463.67350 | 16.3050 | 0.1538 |\n| | | 463.67660 | 15.9980 | 0.1796 |\n| | | 465.65960 | 16.4010 | 0.2574 |\n| | | ... | ... | ... |\n| | MACHO-Blue | 430.79500 | 17.0900 | 0.2155 |\n| | | 438.78620 | 17.2230 | 0.2066 |\n| | | 441.73940 | 17.1900 | 0.1031 |\n| | | 442.74640 | 17.3520 | 0.1140 |\n| | | 443.71500 | 17.2020 | 0.1388 |\n| | | 446.72880 | 17.4260 | 0.0580 |\n| | | 452.74020 | 17.1390 | 0.1577 |\n| | | 453.79520 | 17.0510 | 0.1747 |\n| | | 455.75470 | 17.2530 | 0.1547 |\n| | | 457.78640 | 17.3390 | 0.1647 |\n| | | ... | ... | ... |\n| | CTIO | 1560.39200 | 15.5760 | 0.0550 |\n| | | 1560.39600 | 15.5580 | 0.0550 |\n| | | 1561.40400 | 15.5380 | 0.0493 |\n| | | 1561.40800 | 15.5300 | 0.0507 |\n| | | 1564.40100 | 15.5250 | 0.0465 | \nTable 3-Continued \n| Event Name | Pass Band | time (MJD) | Magnitude | uncertainty |\n|-----------------|-------------|--------------|-------------|---------------|\n| | | 1564.40500 | 15.5670 | 0.0409 |\n| | | 1565.40300 | 15.4540 | 0.0437 |\n| | | 1565.40800 | 15.6850 | 0.0479 |\n| | | 1566.27800 | 15.5500 | 0.0409 |\n| | | 1566.28200 | 15.5280 | 0.0423 |\n| | | ... | ... | ... |\n| MACHO-96-BLG-12 | MACHO-Red | 441.73940 | 14.2730 | 0.0319 |\n| | | 455.75470 | 14.2670 | 0.0382 |\n| | | 457.78640 | 14.2860 | 0.0382 |\n| | | 463.67660 | 14.3630 | 0.0668 |\n| | | 465.65960 | 14.2380 | 0.0542 |\n| | | 468.75260 | 14.3240 | 0.0736 |\n| | | 471.69700 | 14.2670 | 0.0259 |\n| | | 474.70090 | 14.3630 | 0.0336 |\n| | | 476.62860 | 14.2410 | 0.0336 |\n| | | 480.61710 | 14.2430 | 0.0301 |\n| | | ... | ... | ... |\n| | MACHO-Blue | 441.73940 | 15.2830 | 0.0382 |\n| | | 446.72880 | 15.5230 | 0.0200 |\n| | | 457.78640 | 15.2740 | 0.0513 |\n| | | 471.69700 | 15.3200 | 0.0336 |\n| | | 476.62860 | 15.3130 | 0.0447 |\n| | | 480.61710 | 15.2760 | 0.0409 |\n| | | 485.60160 | 15.3200 | 0.0513 |\n| | | 489.62250 | 15.2720 | 0.0590 |\n| | | 500.60370 | 15.2830 | 0.0428 |\n| | | 501.61230 | 15.2660 | 0.0372 |\n| | | ... | ... | ... |\n| | CTIO | 1634.40000 | 13.3920 | 0.0194 |\n| | | 1634.40400 | 13.3740 | 0.0212 |\n| | | 1639.20200 | 13.3890 | 0.0243 |\n| | | 1640.14800 | 13.3640 | 0.0203 |\n| | | 1640.15200 | 13.3720 | 0.0222 |\n| | | 1653.21000 | 13.3330 | 0.0255 |\n| | | 1653.21400 | 13.3350 | 0.0255 |\n| | | 1661.15700 | 13.3090 | 0.0203 |\n| | | 1661.16100 | 13.3150 | 0.0194 |\n| | | 1668.01200 | 13.2860 | 0.0232 |\n| | | ... | ... | ... |\n| MACHO-98-BLG-6 | MACHO-Red | 1164.77190 | 15.8270 | 0.1180 |\n| | | 1168.75560 | 15.9450 | 0.1339 |\n| | | 1318.40870 | 15.8870 | 0.0590 |\n| | | 1319.40610 | 15.8430 | 0.0475 |\n| | | 1321.46010 | 16.1810 | 0.1448 |\n| | | 1323.46890 | 15.9430 | 0.1806 |\n| | | 1324.42410 | 16.2700 | 0.2814 |\n| | | 1325.41330 | 16.0320 | 0.1448 |\n| | | 1325.42910 | 15.9010 | 0.1220 |\n| | | 1326.46850 | 16.2210 | 0.1587 | \nTable 3-Continued \n| Event Name | Pass Band | time (MJD) | Magnitude | uncertainty |\n|----------------|-------------|--------------|-------------|---------------|\n| | | ... | ... | ... |\n| | MACHO-Blue | 1164.77190 | 17.1970 | 0.1856 |\n| | | 1168.75560 | 16.9490 | 0.1966 |\n| | | 1318.40870 | 17.1240 | 0.1170 |\n| | | 1319.40610 | 17.1600 | 0.0814 |\n| | | 1321.46010 | 17.1580 | 0.1210 |\n| | | 1323.46890 | 17.3430 | 0.2824 |\n| | | 1325.41330 | 17.1750 | 0.1796 |\n| | | 1325.42910 | 17.1360 | 0.1677 |\n| | | 1326.46850 | 17.2840 | 0.1468 |\n| | | 1327.43050 | 17.1130 | 0.0687 |\n| | | ... | ... | ... |\n| | CTIO | 2303.38300 | 13.6610 | 0.0303 |\n| | | 2305.36700 | 13.6520 | 0.0243 |\n| | | 2308.40000 | 13.6610 | 0.0243 |\n| | | 2318.39500 | 13.5660 | 0.0290 |\n| | | 2323.41900 | 13.5270 | 0.0266 |\n| | | 2362.26600 | 13.2020 | 0.0437 |\n| | | 2438.03400 | 13.2050 | 0.0202 |\n| | | 2439.03100 | 13.2370 | 0.0278 |\n| | | 2440.03800 | 13.2320 | 0.0243 |\n| | | 2441.04900 | 13.2510 | 0.0266 |\n| | | ... | ... | ... |\n| | MPS | 2306.71030 | 12.9630 | 0.0450 |\n| | | 2306.81750 | 13.0110 | 0.0395 |\n| | | 2308.55870 | 12.9080 | 0.0665 |\n| | | 2308.65670 | 13.1460 | 0.1479 |\n| | | 2308.71270 | 12.9020 | 0.0959 |\n| | | 2308.78110 | 12.9300 | 0.0395 |\n| | | 2316.68150 | 12.8670 | 0.0636 |\n| | | 2316.68370 | 12.9150 | 0.0535 |\n| | | 2328.59630 | 12.7660 | 0.0782 |\n| | | 2328.77350 | 12.7720 | 0.0593 |\n| | | ... | ... | ... |\n| MACHO-99-BLG-1 | MACHO-Red | 441.78840 | 15.3500 | 0.0301 |\n| | | 442.78270 | 15.3320 | 0.0251 |\n| | | 443.75950 | 15.3800 | 0.0428 |\n| | | 452.77860 | 15.3780 | 0.0504 |\n| | | 455.77750 | 15.3460 | 0.0400 |\n| | | 459.75610 | 15.3650 | 0.0419 |\n| | | 463.75080 | 15.3870 | 0.0629 |\n| | | 463.75430 | 15.3580 | 0.0600 |\n| | | 465.73690 | 15.6230 | 0.1637 |\n| | | 466.70240 | 15.3400 | 0.0687 |\n| | | ... | ... | ... |\n| | MACHO-Blue | 441.78840 | 16.2880 | 0.0428 |\n| | | 442.78270 | 16.2840 | 0.0327 |\n| | | 443.75950 | 16.2980 | 0.0629 |\n| | | 452.77860 | 16.3700 | 0.0795 | \nTable 3-Continued \n| Event Name | Pass Band | time (MJD) | Magnitude | uncertainty |\n|----------------|-------------|----------------|-------------|---------------|\n| | | 455.77750 | 16.3130 | 0.0648 |\n| | | 459.75610 | 16.2700 | 0.0854 |\n| | | 463.75080 | 16.1810 | 0.0972 |\n| | | 463.75430 | 16.1740 | 0.1041 |\n| | | 465.73690 | 16.7720 | 0.3813 |\n| | | 466.70240 | 16.4000 | 0.1349 |\n| | | ... | ... | ... |\n| | CTIO | 2732.31900 | 13.3080 | 0.0232 |\n| | | 2733.20300 | 13.3090 | 0.0266 |\n| | | 2769.22400 | 13.4460 | 0.0194 |\n| | | 2778.22500 | 13.5420 | 0.0187 |\n| | | 2778.99300 | 13.5280 | 0.0222 |\n| | | 2784.13200 | 13.5770 | 0.0222 |\n| | | 2789.15200 | 13.6390 | 0.0522 |\n| | | 2794.96300 | 13.7050 | 0.0232 |\n| | | 2819.99200 | 13.9730 | 0.0255 |\n| | | 2820.99400 | 14.0080 | 0.0232 |\n| | | ... | ... | ... |\n| | MPS | 2688.61020 | 12.6050 | 0.0328 |\n| | | 2688.71570 | 12.5720 | 0.1494 |\n| | | 2689.47020 | 12.6120 | 0.0841 |\n| | | 2689.55590 | 12.6540 | 0.0465 |\n| | | 2689.61810 | 12.6300 | 0.0507 |\n| | | 2689.71450 | 12.8030 | 0.1658 |\n| | | 2689.82380 | 12.6190 | 0.0382 |\n| | | 2690.56900 | 12.6450 | 0.0493 |\n| | | 2690.63600 | 12.6540 | 0.0221 |\n| | | 2690.66750 | 12.6410 | 0.0395 |\n| MACHO-99-BLG-8 | MACHO-Red | ... 1165.75610 | ... 11.7260 | ... 0.0194 |\n| | | | 11.7830 | 0.0155 |\n| | | 1168.75920 | | |\n| | | 1318.41170 | 11.7830 | 0.0153 |\n| | | 1319.40950 | 11.7760 | 0.0153 |\n| | | 1323.47160 | 11.7490 | 0.0155 |\n| | | 1324.42880 | 11.7530 | 0.0157 |\n| | | 1325.41700 | 11.7610 | 0.0155 |\n| | | 1325.43250 | 11.7630 | 0.0155 |\n| | | 1326.47170 | 11.7480 | 0.0155 |\n| | | ... | ... | ... |\n| | MACHO-Blue | 1168.75920 | 13.4070 | 0.0172 |\n| | | 1323.47160 | 13.3740 | 0.0177 |\n| | | 1324.42880 | 13.3880 | 0.0188 |\n| | | 1325.43250 | 13.3850 | 0.0172 |\n| | | 1326.47170 | 13.4110 | 0.0167 |\n| | | 1327.43530 | 13.4010 | 0.0157 |\n| | | 1329.46640 | 13.3660 | 0.0163 |\n| | | 1330.42570 | 13.3340 | 0.0163 | \nTable 3-Continued \n| Event Name | Pass Band | time (MJD) | Magnitude | uncertainty |\n|--------------|-------------|--------------|-------------|---------------|\n| | CTIO | 1342.45200 | 13.3370 | 0.0345 |\n| | | ... | ... | ... |\n| | | 2630.39700 | 11.3820 | 0.0212 |\n| | | 2630.40000 | 11.3760 | 0.0194 |\n| | | 2630.40300 | 11.3710 | 0.0255 |\n| | | 2630.40600 | 11.3960 | 0.0243 |\n| | | 2632.39400 | 11.3020 | 0.0290 |\n| | | 2632.39600 | 11.3220 | 0.0355 |\n| | | 2632.40000 | 11.3240 | 0.0355 |\n| | | 2632.40200 | 11.3840 | 0.0342 |\n| | | 2639.37200 | 11.3020 | 0.0243 |\n| | | 2639.37500 | 11.2510 | 0.0203 |\n| | | ... | ... | ... |\n| | MPS | 2688.60420 | 12.4140 | 0.0179 |\n| | | 2688.60530 | 12.4340 | 0.0186 |\n| | | 2688.60680 | 12.4310 | 0.0186 |\n| | | 2688.71320 | 12.3960 | 0.0493 |\n| | | 2689.46830 | 12.4360 | 0.0232 |\n| | | 2689.55290 | 12.3980 | 0.0221 |\n| | | 2689.61620 | 12.4230 | 0.0232 |\n| | | 2689.71320 | 12.4310 | 0.0202 |\n| | | 2690.47590 | 12.3160 | 0.0564 |\n| | | 2690.56730 | 12.4250 | 0.0202 |\n| | | ... | ... | ... | \nNote. - Complete data set available in electronic version. MJD = JD -248623 . 5 days. \nTable 4. Photometric Calibration Coefficients \n| Event Name | a | b | c | d |\n|-----------------|----------|----------|----------|----------|\n| MACHO-104-C | 0 . 8176 | 0 . 1824 | 0 . 1828 | 0 . 8172 |\n| MACHO-96-BLG-5 | 0 . 8076 | 0 . 1924 | 0 . 1804 | 0 . 8196 |\n| MACHO-96-BLG-12 | 0 . 8176 | 0 . 1824 | 0 . 1828 | 0 . 8172 |\n| MACHO-98-BLG-6 | 0 . 8191 | 0 . 1809 | 0 . 1829 | 0 . 8171 |\n| MACHO-99-BLG-1 | 0 . 8169 | 0 . 1831 | 0 . 1826 | 0 . 8174 |\n| MACHO-99-BLG-8 | 0 . 8188 | 0 . 1812 | 0 . 1829 | 0 . 8171 | \nTable 5. Microlensing Parallax Fit Parameters \n| Event | f MR | f MB | f CTIO | f MPS | t 0 (MJD) | u min | ̂ t (days) | ˜ v (km/sec) | θ | χ 2 (dof) | ∆ χ 2 |\n|-----------|------------|------------|------------|------------|-------------|-------------|-------------|----------------|-------------|-------------|---------|\n| 104-C | 1 . 00(1) | 0 . 99(2) | | | 508 . 3(6) | 0 . 15(1) | 220(2) | 77(4) | - 1 . 08(7) | 1 . 47 | 1051 |\n| 96-BLG-5 | 0 . 12(3) | 0 . 12(3) | 0 . 13(3) | | 1763(1) | 0 . 018(6) | 2000(500) | 30 . 9(1 . 3) | - 0 . 84(6) | 1 . 58 | 2395 |\n| (HST) | 0 . 28(1) | 0 . 30(1) | 0 . 33 | | 1767(1) | 0 . 048(6) | 970(20) | 30 . 9(1 . 3) | - 0 . 87(7) | 1 . 59 | 2371 |\n| | 0 . 31(1) | 0 . 33(1) | 0 . 37 | | 1768(1) | 0 . 054(7) | 900(20) | 31 . 0(1 . 3) | - 0 . 88(8) | 1 . 59 | 2363 |\n| 96-BLG-12 | 0 . 87(2) | 0 . 89(3) | 0 . 90(2) | | 1743 . 4(3) | - 0 . 11(2) | 294(5) | 47 . 5(1 . 3) | - 1 . 23(9) | 2 . 11 | 5914 |\n| 98-BLG-6 | 0 . 65(14) | 0 . 60(13) | 0 . 68(15) | 0 . 66(13) | 2388(3) | 0 . 16(4) | 490(50) | 79(5) | - 1 . 7(2) | 1 . 20 | 802 |\n| 99-BLG-1 | 0 . 96(9) | 0 . 98(10) | 1 . 0(1) | 0 . 97(7) | 2712(1) | 0 . 23(4) | 231(13) | 43 . 9(9) | - 1 . 85(2) | 1 . 54 | 1706 |\n| 99-BLG-8 | 0 . 75(12) | 0 . 73(12) | 0 . 76(12) | 0 . 79(13) | 2732 . 1(4) | 0 . 17(1) | 240(20) | 62(5) | - 1 . 53(3) | 2 . 34 | 2280 | \nNote. MJD = JD -248623 . 5 days. \nTable 6. Microlensing Parallax Likelihood Mass Estimates \n| | | | | Confidence Levels P ( M/ M /circledot < N) | Confidence Levels P ( M/ M /circledot < N) | Confidence Levels P ( M/ M /circledot < N) | Confidence Levels P ( M/ M /circledot < N) | Confidence Levels P ( M/ M /circledot < N) | Confidence Levels P ( M/ M /circledot < N) |\n|-----------|----------|------------|-----------|----------------------------------------------|----------------------------------------------|----------------------------------------------|----------------------------------------------|----------------------------------------------|----------------------------------------------|\n| Event | location | f MR | ̂ t (days) | ̂ v (km/sec) | P = 5% | P = 16% | P = 50% | P = 84% | P = 95% |\n| 104-C | bulge | 1 . 00(1) | 220(2) | 77(4) | 0.35 | 0.62 | 1.15 | 2.2 | 3.94 |\n| 96-BLG-5 | bulge | 0 . 12(3) | 2000(500) | 30 . 9(1 . 3) | 7 . 2 | 12 . 8 | 27 | 69 | 160 |\n| (HST) | bulge | 0 . 28(1) | 970(20) | 30 . 9(1 . 3) | 1.64 | 2.93 | 6.3 | 15.8 | 37 |\n| (HST) | bulge | 0 . 31(1) | 900(20) | 31 . 0(1 . 3) | 1 . 41 | 2 . 53 | 5 . 4 | 13 . 6 | 31 |\n| 96-BLG-12 | bulge | 0 . 87(2) | 294(5) | 47 . 5(1 . 4) | 0.33 | 0.62 | 1.29 | 3.1 | 6.7 |\n| 98-BLG-6 | bulge | 0 . 65(14) | 490(50) | 79(6) | 0.94 | 2.6 | 5.7 | 12.5 | 24 |\n| 98-BLG-6 | SGR | 0 . 65(14) | 490(50) | 79(6) | 1.23 | 1.61 | 2.52 | 4.2 | 6.7 |\n| 99-BLG-1 | bulge | 0 . 96(9) | 219(9) | 42 . 9(9) | 0.14 | 0.29 | 0.68 | 1.86 | 4.6 |\n| 99-BLG-1 | SGR | 0 . 96(9) | 219(9) | 42 . 9(9) | 0.10 | 0.16 | 0.31 | 0.70 | 1.51 |\n| 99-BLG-8 | bulge | 0 . 75(12) | 240(20) | 62(5) | 0.27 | 0.56 | 1.19 | 2.78 | 6.0 | \nNote. - Bold-faced type indicates the parameters that are considered to be most likely. Event 96-BLG-5 has parameters for three different fits listed. The first fit is the fit with no constraint on the source brightness, while the second and third fits have the lensed flux fixed to a value based upon our HST observations. The fit labeled HST is the best fit, while the third fit is provided to indicate the effect of the source flux uncertainty on the mass limits. For events 98-BLG-6 and 99-BLG-1, mass estimates based upon bulge and SGR sources are presented. In all cases, it is most likely that the source star is in the bulge. \nTable 7. Mass & Magnitude Estimates for the MACHO Microlensing Parallax Events \n| Event | M/ M /circledot | M MS / M /circledot | D /lscript - MS | sep-MS | V s | ∆ I /lscripts | ∆ V /lscripts | ∆ B /lscripts | ∆ U /lscripts |\n|-----------|----------------------|-----------------------|-------------------|----------|--------|-----------------|-----------------|-----------------|-----------------|\n| 104-C | 1 . 1 +1 . 1 - 0 . 5 | 0 . 74 | 2 . 7 kpc | 40 mas | 17 . 3 | 3 . 5 | 3 . 5 | 3 . 5 | 3 . 2 |\n| 96-BLG-5 | 6 +10 - 3 | - | - | - | - | - | - | - | - |\n| 96-BLG-12 | 1 . 3 +1 . 8 - 0 . 7 | 0 . 75 | 2 . 0 kpc | 28 mas | 18 . 0 | 2 . 1 | 2 . 2 | 2 . 2 | 2 . 3 |\n| 98-BLG-6 | 2 . 5 +1 . 7 - 0 . 9 | 0 . 88 | 5 . 7 kpc | 5 mas | 20 . 1 | 2 . 2 | 1 . 9 | 1 . 6 | 1 . 1 |\n| 99-BLG-1 | 0 . 7 +1 . 2 - 0 . 4 | 0 . 40 | 1 . 7 kpc | 17 mas | 18 . 9 | 1 . 8 | 3 . 2 | 3 . 6 | 3 . 9 |\n| 99-BLG-8 | 1 . 2 +1 . 6 - 0 . 6 | 1 . 2 | 1 . 6 kpc | 25 mas | 16 . 3 | 1 . 3 | 0 . 7 | - 0 . 3 | - 1 . 1 | \nNote. - These are the parameters of the 'most likely' main sequence star lenses for our best microlensing parallax events. For MACHO-96-BLG-5, a main sequence lens is ruled out."}
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2005PhRvD..71b7502G
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Log correction to the black hole area law
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2005-01-01
| 7 | 0.44 | 160 |
['-', '-', '-', '-', '-', '-']
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[]
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Various approaches to black hole entropy yield the area law with logarithmic corrections, many involving a coefficient 1/2, and some involving 3/2. It is pointed out here that the standard quantum geometry formalism with spin one-half at each puncture is not consistent with 3/2.
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[]
| 2 |
https://arxiv.org/pdf/gr-qc/0401070.pdf
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{'On the log correction to the black hole area law': "Amit Ghosh ∗ and P. Mitra † Saha Institute of Nuclear Physics Block AF, Bidhannagar Calcutta 700 064, INDIA \nVarious approaches to black hole entropy yield the area law with logarithmic corrections, many involving a coefficient 1/2, and some involving 3/2. It is pointed out here that the standard quantum geometry formalism is not consistent with 3/2 and favours 1/2. \nThere has long been an association of the area of the horizon of a black hole with an entropy [1]. This was not initially understood according to the Boltzmann definition of entropy as a measure of the number of quantum states of a black hole, because of the absence of a proper quantum theory of gravity. As a first step, however, considering gravity to be a statistical system, the na¨ıve Lagrangian path integral was seen quite early to lead to a partition function from which the area law of entropy was obtained [2] in the leading semiclassical approximation ignoring all quantum fluctuations. Subsequent support was obtained from considerations of quantum fields in black hole backgrounds [3,4]. The entropy calculated for the fields may be regarded as an additional contribution to the entropy of the black hole - matter system, and the gravitational entropy of the black hole itself may be imagined to get modified in this way. In these field theory calculations the leading term has a divergent multiplicative factor with the area of the horizon. This divergence may be thought of as a contribution to the bare or classical gravitational constant G , which is to be renormalized to a finite G R in the presence of quantized matter fields. \nRecently some statistical derivations of the area law have appeared in more elaborate models of quantum gravity - in string theory [5] as well as in quantum geometry [6]. Even though a complete and universally accepted quantum theory of gravity is not quite at hand, both of these approaches can accommodate the expected number of quantum micro-states of a black hole. \nWith the area law so well established for the entropy of large black holes, it is not surprising that even corrections to the area formula have been studied. The area of the horizon of an extremal dilatonic black hole vanishes, and in this case the matter field approach was seen to lead to a logarithm of the mass of the black hole [7] in the expression for the entropy. For black holes with non-vanishing area, the logarithm of the area appears as a sub-leading term after the dominant term proportional to the area. The coefficient of the logarithm depends on the black hole and is 1/90 in the Schwarzschild case. These coefficients are expected to be renormalized, as indicated above. Logarithmic corrections to the gravitational entropy, with coefficients which are negative, appeared later in many models. One approach [8] was related to the quantum geometry formulation but eventually mapped the counting problem to conformal blocks, leading to a negative coefficient of magnitude 3/2. Another [9] started from ideas about conformal symmetry in the near-horizon degrees of freedom and considered corrections to the Cardy formula, reaching the same coefficient. There were variations on these themes [10,11]. \nOn the other hand, there has been a conflicting set of calculations leading to a negative coefficient with the smaller magnitude 1/2. Among these, [12] has followed the same conformal symmetry approach as [9], but has dropped the assumption made there that the central charge is independent of the black hole area: it has in fact turned out to be proportional to the area. String theorists too [13,14] have obtained the value 1/2, which has also appeared in applications of statistical mechanics [15]. In view of this disagreement, it becomes necessary to examine the derivations of 3/2 more critically. Surprisingly, there has been no direct calculation in the quantum geometry approach [6], in spite of the continuing progress in this field: the derivation [8] used indirect conformal methods. As the leading expression for the entropy has been calculated directly in the quantum geometry approach [6], it is not difficult to look at the subleading contribution in the same manner. A bound which is readily derived is consistent with the value 1/2 but not with the value 3/2. It is argued that 1/2 is in fact the actual value. \nThe calculation of black hole entropy from quantum geometry is a simple counting problem described in detail in [6], whose notation we more or less follow. We consider a section of a spherically symmetric isolated horizon. There are non-zero spins J a ( a = 1 , ..., N ) associated with punctures on this sphere. We work in units such that 4 πγ/lscript 2 P = 1, where γ is a 'free' parameter (the Immirzi parameter) and /lscript P is the Planck-length. The spins are said to be permissible if the quantity | J | ≡ 2 ∑ N 1 [ J a ( J a +1)] 1 / 2 lies in the range \n| \nK -/epsilon1 ≤ | J | ≤ K + /epsilon1, (1) \nwhere K is an integer representing the horizon area in the above unit and /epsilon1 /lessmuch K compensates for the failure of | J | to be an integer. Roughly, J | /similarequal K N , all of these being large quantities. \n| /similarequal For a permissible set of spins, the a th puncture carries a vector space of dimensionality (2 J a + 1), so the net dimensionality of the representation is \n∼ \nd ( J ) = ∏ a (2 J a +1) . (2) \nThere is a further restriction to be imposed: boundary conditions require that \n∑ a 2 m a = 0 mod K (3) \nfor each allowed configuration. So the physical 'degeneracy' is \nd = ∑ Permissible J d phys ( J ) , where d phys ( J ) = ∫ π -π dθ 2 π ∏ a J a ∑ m a = -J a exp(2 im a θ ) . (4) \nIt may be noted here that N < K , so mod K does not contribute to the counting. Since d phys ( J ) ≤ d ( J ) for each permissible configuration, clearly d obeys a bound \nd ≤ ∑ Permissible J d ( J ) . (5) \n- (5) has been used to put an upper bound on d [6]: \nS = ln d ≤ K √ 3 ln 2 + O ( K ) , lim K →∞ O ( K ) K = 0 . (6) \nHowever, we shall concentrate on a lower bound, which, as in [6], can be obtained by considering all spins to be J a = 1 / 2. Then, | J | = N √ 3. Clearly, for /epsilon1 ≥ √ 3 it is always possible to find an even N obeying (1). The number of physical states or the degeneracy can be easily calculated from (4) \nd phys ( 1 / 2 ) = ( N N/ 2 ) for even N , d phys ( 1 / 2 ) = 0 for odd N . \nThus, the entropy \nln d ≥ ln ( N N/ 2 ) , N ∈ [ K √ 3 -1 , K √ 3 +1] . (7) \nAn estimate of the right hand side (7) can be made (cf [6]) with the Stirling approximation \nN ! = N N (2 πN ) 1 / 2 e -N ( 1 + O ( 1 N ) + · · · ) . (8) \nOne obtains \nln d ≥ N ln 2 -1 2 ln N + O (1) . (9) \nNow the two bounds on N can be exploited to obtain a bound on ln d . \nN ≥ K √ 3 -1 = ⇒ N ln 2 ≥ K √ 3 ln 2 + O (1) N ≤ K √ 3 +1 = ⇒-1 2 ln N ≥ -1 2 ln K + O (1 /K ) . (10) \nCombining the two inequalities, one gets \nS = ln d ≥ K √ 3 ln 2 -1 2 ln K + O (1) . (11) \n(11) clearly shows, after conversion of the integer K into the area A in appropriate units, that there is a lower bound on the entropy of a spherically symmetric isolated horizon: \nS ≥ A constant -1 2 ln A + O (1) . (12) \nIn the above derivation of the bound, only punctures with spin J = 1 / 2 have been considered, the reason being the dominance of spin 1 / 2 in the number of physical micro-states \nd = ∑ J d phys ( J ) χ /epsilon1 ( K -| J | ) (13) \nwhere χ /epsilon1 ( x ) is the characteristic function for the interval [ -/epsilon1, /epsilon1 ]. This constraint (1) reveals that the number of punctures N decreases as the spin J a at each puncture increases. An extreme case is one large J at a single puncture. That reduces the degeneracy to one, d phys = 1, since only the m = 0 state contributes. However, the analysis of other intermediate configurations J is involved. We present now some estimates of the contribution of other configurations to the entropy. \nln d phys ( j ) ≤ K ln(2 j +1) 2 √ j ( j +1) . (14) \nIf every puncture is associated with a common spin j , then d phys ( j ) ≤ (2 j +1) N j where N j ∼ K/ [2 √ j ( j +1)]. So \nIf the constraint (3) is implemented, one gets \nln d phys ( j ) = K ln(2 j +1) 2 √ j ( j +1) -j ln K + O (1) . (15) \nA logarithmic correction with the same coefficient was obtained earlier in a different context [16]. But the dominant term falls off as j increases: the factor \nln(2 j +1) √ j ( j +1) (16) \nhas a maximum at j = 1 / 2 (if j = 0, for which it is undefined, is excluded). Consequently, the contribution of a configuration with j > 1 / 2 to the entropy falls off at least like exp( -cK ) where c > 0, in comparison to a configuration with j = 1 / 2. E.g., j = 1 produces a correction that falls off at least like exp[ -0 . 02 K ] compared to the dominant term and vanishes rapidly for large K . \nFor mixed configurations, the falloffs are not exponential. It is not difficult to see that if a 1 / 2 in the 1 / 2 configuration is replaced by j > 1 / 2, the decrease of the expression (16) implies that the contribution to the entropy is reduced. Different contributions like this produce a O (1) factor which does not affect the dominant or logarithmic terms. We plan to present soon [17] detailed evidence in favour of an equality for the entropy with 1/2 as the coefficient for the logarithmic correction. \nIn conclusion, we have derived here a lower bound on the entropy of a black hole strictly following the quantum geometry formalism [6]. The popular value 1/2 for the logarithmic coefficient, which we will support more fully elsewhere, is consistent with this bound (12), but the older value 3/2 is not. It is of interest to understand why this bound is inconsistent with the value obtained in [8], which was also motivated by quantum geometry, though calculated through conformal methods. The condition (3), which originates from boundary conditions imposed on the isolated horizon [6], requires only a projection of the spins to add up to zero. In the approach of [8], weaker boundary conditions get imposed, corresponding to an enhancement of the 'gauge' symmetry from U(1) to SU(2) and a decrease in the number of 'physical' states. This is best seen in the counting in the second paper in [8], requiring not just a projection of the total spin, but also its other components to vanish. This reduces the number of states slightly, leaving the dominant term unaltered, but changing the coefficient of the negative logarithm to 3/2 instead of 1/2. \nWe thank Abhay Ashtekar for many helpful suggestions on improving the presentation. \n- [1] J. Bekenstein, Phys. Rev. D7 , 2333 (1973); Phys. Rev. D9 , 3292 (1974)\n- [2] G. Gibbons and S. Hawking, Phys. Rev. D15 , 2752 (1977)\n- [3] G. 't Hooft, Nucl. Phys. B256 , 727 (1985)\n- [4] L. Susskind and J. Uglum, Phys. Rev. D50 , 2700 (1994)\n- [5] A. Strominger and C. Vafa, Phys. Letters B379 , 99 (1996)\n- [6] A. Ashtekar, J. Baez and K. Krasnov, Adv. Theor. Math. Phys. 4 , 1 (2000) [gr-qc/0005126]\n- [7] A. Ghosh and P. Mitra, Phys. Rev. Letters 73 , 2521 (1994); D. V. Fursaev, Phys. Rev. D51 , 5352 (1995)\n- [8] R. Kaul and P. Majumdar, Phys. Rev. Letters 84 , 5255 (2000); S. Das, R. Kaul and P. Majumdar, Phys. Rev. D63 , 044019 (2001)\n- [9] S. Carlip, Class. Quant. Grav. 17 , 4175 (2000)\n- [10] G. Gour, Phys. Rev. D66 , 104022 (2002)\n- [11] R. Kaul and S. Kalyana Rama, Phys. Rev. D68 , 024001 (2003)\n- [12] J. L. Jing and M. L. Yan, Phys. Rev. D63 , 024003 (2001)\n- [13] S. Mukherjee and S. S. Pal, JHEP 0205 , 026 (2002)\n- [14] A. Krause, Class. Quant. Grav. 20 , S533 (2003)\n- [15] R. K. Bhaduri, M. N. Tran and S. Das, gr-qc/0312023\n- [16] S. Das and V. Husain, Class. Quant. Grav. 20 , 4387 (2003)\n- [17] A. Ghosh and P. Mitra, to appear"}
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2020JCAP...09..030I
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Gravitational lensing by black holes in the 4D Einstein-Gauss-Bonnet gravity
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2020-01-01
| 21 | 0.46 | 160 |
['-']
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[]
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Recently, a non-trivial 4D Einstein-Gauss-Bonnet (EGB) theory of gravity, by rescaling the GB coupling parameter as α/(D-4), was formulated in [1], which bypasses Lovelock's theorem and avoids Ostrogradsky instability. The theory admits a static spherically symmetric black hole, unlike 5D EGB or general relativity counterpart, which can have both Cauchy and event horizons. We generalize previous work, on gravitational lensing by a Schwarzschild black hole, in the strong and weak deflection limits to the 4D EGB black holes to calculate the deflection coefficients ā and bar b, while former increases and later decrease with increasing α. We also find that the deflection angle α<SUB>D</SUB>, angular position θ<SUB>∞</SUB> and u<SUB>m</SUB> decreases, but angular separation s increases with α. The effect of the GB coupling parameter α on positions and magnification of the source relativistic images is discussed in the context of SgrA* and M87* black holes. A brief description of the weak gravitational lensing using the Gauss-Bonnet theorem is presented.
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[]
| 3 |
https://arxiv.org/pdf/2004.01038.pdf
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{'Gravitational lensing by black holes in the 4 D Einstein-Gauss-Bonnet gravity': "Shafqat Ul Islam a , ∗ Rahul Kumar a , † and Sushant G. Ghosh a, b ‡ a Centre for Theoretical Physics, Jamia Millia Islamia, New Delhi 110025, India and b Astrophysics and Cosmology Research Unit, School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag 54001, Durban 4000, South Africa (Dated: September 17, 2020) \nRecently, a non-trivial 4 D Einstein-Gauss-Bonnet (EGB) theory of gravity, by rescaling the GB coupling parameter as α/ ( D -4), was formulated in [10], which bypasses Lovelock's theorem and avoids Ostrogradsky instability. The theory admits a static spherically symmetric black hole, unlike 5 D EGB or general relativity counterpart, which can have both Cauchy and event horizons. We generalize previous work, on gravitational lensing by a Schwarzschild black hole, in the strong and weak deflection limits to the 4 D EGB black holes to calculate the deflection coefficients ¯ a and ¯ b , while former increases and later decrease with increasing α . We also find that the deflection angle α D , angular position θ ∞ and u m decreases, but angular separation s increases with α . The effect of the GB coupling parameter α on positions and magnification of the source relativistic images is discussed in the context of SgrA* and M87* black holes. A brief description of the weak gravitational lensing using the Gauss-Bonnet theorem is presented. \nPACS numbers:", 'I. INTRODUCTION': "String theories, in the low-energy limits, give rise to the effective field theories of gravity such that the Lagrangian for these theories contains terms of quadratic and higher orders in the curvature in addition to the usual scalar curvature term [1-4]. Further, the gravitational action may be modified to include the quadratic curvature correction terms while keeping the equations of motion to second order, provided the quadratic terms appear in specific combinations corresponding to the Gauss-Bonnet (GB) invariants defined by [5] \nG = R 2 -4 R µν R µν + R µνρσ R µνρσ , (1) \nand such a theory is termed as D ≥ 5 dimensional Einstein Gauss-Bonnet (EGB) gravity theory with the D -4 extra dimensions. It turns out that the four-dimensional GB term arises as the next to leading order correction to the gravitational effective action in string theory [1-4]. The EGB gravity is a special case of the Lovelock theory of gravitation [6]. Since its equations of motion have no more than two derivatives of the metric, it has been proven to be free of ghosts [7]. Boulware and Deser [7], independently by Wheeler [8], found exact black hole solutions in EGB gravity theories, which are generalizations of the Schwarzschild-Tangherlini black holes [9]. However, the GB term (1) is a topological invariant in 4 D as its contribution to all components of Einstein's equations are in fact proportional to ( D -4), i.e, it does not contributes to the equations of motion and one requires D ≥ 5 for non-trivial gravitational dynamics. \nHowever, it was shown by Glavan and Lin [10] that by re-scaling the GB coupling constant, in the EGB gravity action, as α → α/ ( D -4), the GB invariant makes a non-trivial contribution to the gravitational dynamics even in the D = 4. This is evident by considering maximally symmetric spacetimes with curvature scale K \ng µσ √ -g δ G δg νσ = α ( D -2)( D -3) 2( D -1) K 2 δ ν µ , (2) \nobviously the variation of the GB action does not vanish in D = 4 because of the re-scaled coupling constant [10]. This 4 D EGB gravity has already attracted much attentions and being extensively studied [11-16]. The Boulware and Deser [7] version of the spherically symmetric static black hole to the 4 D EGB gravity was obtained in [10], which was also extended to the charged case [17]; this kind of solution for the Lovelock gravity has been obtained in \nRef. [18, 19]. The corresponding black holes in a string cloud model was considered in [20]. Also, nonstatic Vaidya-like spherical radiating black hole solutions have been obtained in 4 D EGB gravity [21, 22]. More black hole solutions can be found in Refs. [23-29]. Study of photon geodesics and the effects of GB coupling parameter on the shadow of 4 D EGB black hole is presented in Ref. [27, 28, 30, 31]. \nThe idea of 4 D regularization of EGB gravity was originally initiated by Tomozawa [32], and later Cognola et al. [33] reformulated it by accounting quantum corrections due to a GB invariant within a classical Lagrangian approach. Though the Glavan and Lin [10] 4 D regularization procedure is currently a subject of dispute [34-38], however, several alternate regularization procedures have been proposed [15, 19, 34, 39-42]. Interestingly, neither of the critics [34, 36, 38] disprove dimensional regularization procedure of [10] at least for the case of spherically symmetric spacetimes. Furthermore, the static spherically symmetric 4 D black hole solution found in [10, 33] remains valid for these alternate regularized theories [15, 19, 34, 41]. Furthermore, the semi-classical Einstein equations with conformal anomaly proportional to Euler density G [43, 44] and the 4 D non-relativistic Horava-Lifshitz theory of gravity [45] also admit the identical spherically symmetric black hole solution. \nGravitational lensing is one of the most powerful astrophysical tools for investigations of the strong-field features of gravity. Motivated by the pioneering work of Darwin [46], strong gravitational lensing by compact astrophysical objects with a photon sphere, such as black holes, has been extensively studied. The deflection of electromagnetic radiation in a gravitational field is commonly referred to as the gravitational lensing and an object causing a deflection is known as a gravitational lens. Virbhadra [47] studied the strong-field situation to obtain the gravitational lens equation using an asymptotically flat background metric and also analyzed the gravitational lensing by a Schwarzschild black hole. The results have been applied to the supermassive black hole Sgr A* using numerical techniques [47]. Later on, using a different formulation, an exact lens equation and integral expressions for its solutions were obtained [48]. However, it was Bozza et al. [49] who first defined the strong-field deflection limit to analytically investigate the Schwarzschild black hole lensing. This technique has been applied to static, spherically symmetric metrics which includes Reissner -Nordstrom black holes [50], braneworld black holes [51-54], charged black hole of heterotic string theory [55], and was also generalized to an arbitrary static, spherically symmetric metric by Bozza [56]. On the other hand, lensing in the strong gravitational field is a powerful tool to test the nature of compact objects, therefore, it continues to receive significant attention, and more recent works include lensing from other black holes [57-60]. However, the qualitative features in lensing by these black holes, in the presence of a photon sphere, is similar to the Schwarzschild case. Also, strong gravitational lensing from various modifications of Schwarzschild geometry in modeling the galactic center has been studied, e.g., lensing from regular black holes was studied in [61, 62], massive gravity black holes [63] and also lensing by wormholes [64, 65]. \nThe aim of this paper is to apply the prescription of Bozza et al. [56] to investigate the gravitational lensing properties of the 4 D EGB black hole. In particular, we calculate the strong lensing coefficients of the static spherically symmetric EGB black hole, from which we calculate the positions and magnification of the source's images and numerically compute them for the Schwarzschild black hole. The effect of the GB coupling parameter on the weak gravitational lensing is also investigated. The paper is organized as follows: we begin in Sect. II with a discussion of the static spherically symmetric black hole of the 4 D EGB gravity. Gravitational deflection of light in strong-field limits of these black holes is investigated in Sect. III. Corresponding lensing observables and numerical estimations of deflection angle, image positions, and magnifications in the context of supermassive black holes, namely Sgr A* and M87* are presented in Sect. IV. Section V is devoted to the weak gravitational lensing. Finally, paper ends in Sect. VI by summarizing our main findings.", 'II. THE 4 D EGB BLACK HOLE': "The EGB gravity action, with rescaled GB coupling α/ ( D -4) to restore the dimensional regularization, in the D -dimension spacetime yields [10] \nS = 1 16 πG ∫ d D x √ -g [ R + α D -4 G ] , (3) \nwhere g is the determinant of metric tensor g µν , R is the Ricci scalar, and α is the GB coupling constant considered to be positive. The 4 D EGB theory is defined in the limit of D → 4 at the level of equations of motion rather than in action, thereby the GB term makes a non-trivial contribution in gravitational dynamic [10], which admit static spherically symmetric black hole [10] given by \nds 2 = -f ( r ) dt 2 + 1 f ( r ) dr 2 + r 2 ( dθ 2 +sin 2 θ dφ 2 ) , (4) \n<!-- image --> \nFigure 1: (Left panel) Plot showing the horizons for various values of GB coupling parameter. The black solid line corresponds to the extremal value of ˜ α . (Right panel) The behavior of event horizon radii x + (solid black line), Cauchy horizon radii x -(dashed red line), and photon sphere radii x m (dashed blue line) with GB coupling parameter ˜ α . \n<!-- image --> \nwith \nf ± ( r ) = 1 + r 2 32 παG ( 1 ± √ 1 + 128 παMG 2 r 3 ) . (5) \nHere, M is the black hole mass parameter and G is the Newton's gravitational constant which is set to unity hereafter. Since various gravity theories [18, 19, 32-34, 41, 43-45] admit the spherically symmetric black hole solution identical to Eq. (4) with (5), therefore the presented study of gravitational lensing is also valid for these gravity black holes. It is clear that Eq. (5) leads to two different branches of solutions corresponding to the ' ± ' sign. However, at large distances, metric function (5) reduces to \nf -( r ) = 1 -2 M r , f + ( r ) = 1 + 2 M r + r 2 16 πα , (6) \nand only f -( r ) correctly identify with the Schwarzschild solution, even in the limit of vanishing GB coupling constant, α → 0, the metric (4) only with f -( r ) in Eq. (5) retains to the Schwarzschild black hole metric [10]. Henceforth, we shall restrict our discussion to ve branch. \nTo initiate discussion on gravitation lensing, we note that the static and spherically symmetric metric (4) is asymptotically flat and rewrite it using the dimensionless variables as follow [56] \n- \nd ˜ s 2 = (2 M ) -2 ds 2 = -A ( x ) dT 2 + 1 A ( x ) dx 2 + C ( x )( dθ 2 +sin 2 θ dφ 2 ) , (7) \nwhere \nx = r 2 M , T = t 2 M , ˜ α = α M 2 , (8) \nand accordingly from the metric (4), we have \nA ( x ) = 1 + x 2 8 π ˜ α ( 1 -√ 1 + 16 π ˜ α x 3 ) , C ( x ) = x 2 . (9) \nThe event horizon is the largest positive root of g rr = 0, i.e., horizon radii can be determined by solving \nA ( x ) = 0 , (10) \nwhich admits solutions \nx ± = 1 2 ( 1 ± √ 1 -16 π ˜ α ) . (11) \nDepending on the values of ˜ α and M , Eq. (10) has upto two real positive roots corresponding to the inner Cauchy horizon ( x -) and outer event horizon ( x + ). It turns out that A ( x ) = 0 has no solution if ˜ α > ˜ α E i.e., no black hole exists. Whereas, it has one double zero if ˜ α = ˜ α E , and two simple zeros if ˜ α < ˜ α E , respectively, corresponding to the 4 D EGB black hole with degenerate horizon ( x -= x + ≡ x E ), and a non-extremal black hole with two horizons ( x -/negationslash = x + ) (cf. Fig. 1). The event horizon radii decrease whereas Cauchy horizon radii increase with increasing ˜ α (cf. Fig. 1). It is evident that the event horizon radii for the EGB black holes is smaller than the Schwarzschild black hole value.", 'III. STRONG GRAVITATIONAL LENSING': "In this section, we focus on the gravitational deflection of light in the static spherically symmetric 4 D EGB black hole spacetime (7) and consider the propagation of light on the equatorial plane ( θ = π/ 2), as due to spherical symmetry, the same results can be applied to all θ . Then the metric (7) reduces to \nd ˜ s 2 = -A ( x ) dT 2 + A ( x ) -1 dx 2 + C ( x ) dφ 2 . (12) \nSince the spacetime is static and spherically symmetric, the projection of photon four-momentum along the Killing vectors of isometries are conserved quantities, namely the energy E = -p µ ξ µ ( t ) and angular momentum L = p µ ξ µ ( φ ) are constant along the geodesics, where ξ µ ( t ) and ξ µ ( φ ) are, respectively, the Killing vectors due to time-translational and rotational invariance [66]. This yields \ndt dτ = -E A ( x ) , dφ dτ = L C ( x ) , (13) \nwhere τ is the afine parameter along the geodesics. Using Eq. (13) for the null geodesics equation ds 2 = 0, we obtain \n( dx dτ ) 2 ≡ ˙ x 2 = E 2 -L 2 A ( x ) C ( x ) . (14) \nThe radial effective potential V eff ( x ) = L 2 A ( x ) /C ( x ), reads as \nV eff ( x ) = L 2 x 2 ( 1 + x 2 8 π ˜ α ( 1 -√ 1 + 16 π ˜ α x 3 )) , (15) \nand describes the different kinds of possible orbits. In particular, photons simply move on circular orbits of constant radius x m , at points where the potential is flat \ndV eff ( x ) dx = 0 ⇒ A ' ( x ) A ( x ) = C ' ( x ) C ( x ) , (16) \nEq. (16) reduces to \n4 x 3 -9 x +64 π ˜ α = 0 , (17) \nand solving which, gives the radius of photon circular orbits x m ; at x = x m potential has a unique maximum. These orbits are unstable against small radial perturbations, which would finally drive photons into the black hole or toward spatial infinity [66]. Due to spherical symmetry, these orbits are planer and generate a photon sphere around the black hole. Photons, coming from the far distance source, approach the black hole with some impact parameter and get deflected symmetrically to infinity, meanwhile reaching a minimum distance near the black hole. The impact parameter u is related to the closest approach distance x 0 , which follows from intersecting the effective potential V eff ( x ) with the energy of photon E 2 , this reads as \nV eff ( x ) = E 2 ⇒ u ≡ L E = √ C ( x 0 ) A ( x 0 ) . (18) \nFor x 0 = x m , the corresponding impact parameter is u m , which satisfy Eq. (16). The gravitational deflection angle of light is described as the angle between the asymptotic incoming and outgoing trajectories, and reads as [47, 67] \nα D ( x 0 ) = I ( x 0 ) -π, (19) \nthe integral (19) can be re-written as \nwith \nwhere \nI ( x 0 ) = 2 ∫ ∞ x 0 dφ dx dx, dφ dx = 1 √ A ( x ) C ( x ) √ C ( x ) A ( x 0 ) C ( x 0 ) A ( x ) -1 . (20) \nIt is worthwhile to note that the impact parameter coincides with the distance of closest approach only in the limit of vanishing deflection angle. Due to spacetime symmetries, the total change in φ as x decreases from ∞ to its minimum value x 0 and then increases again to ∞ is just twice its change from ∞ to x 0 . In the absence of a black hole, photons will follow the straight line trajectory and this change in φ is simply π and therefore by Eq. (19) the deflection angle is identically zero. The deflection angle α D ( x 0 ) monotonically increases as the distance of minimum approach x 0 decreases, and becomes higher than 2 π resulting in the complete loops of the light ray around the black hole before escaping to the observer for x 0 /similarequal x m (or u /similarequal u m ) and leads to the set of infinite or relativistic source's images [47, 67]. Only for the x 0 = x m (or u = u m ), deflection angle diverges logarithmically, whereas, photons with impact parameters u < u m get captured by the black hole and fall into the horizon. We consider the sign convention such that for α D ( x 0 ) > 0, light bend toward the black hole, whereas for α D ( x 0 ) < 0, light bend away from it. We are interested in the deflection of light in the strong-field limit, viz., light rays passing close to the photon sphere. In the strong deflection limit, we can expand the deflection angle near the photon sphere, where it diverges [56]. For this purpose, we define a new variable [56] \nz = A ( x ) -A ( x 0 ) 1 -A ( x 0 ) , (21) \nI ( x 0 ) = ∫ 1 0 R ( z, x 0 ) f ( z, x 0 ) dz, (22) \nR ( z, x 0 ) = 2 √ C ( x 0 )(1 -A ( x 0 )) C ( x ) A ' ( x ) , (23) \nwhere functions without subscript '0' are evaluated at x = A -1 [(1 -A ( x 0 )) z + A ( x 0 )] [56]. Equation (24), on performing the Taylor series expansion of the function within square root, reduces to \nwhere \nAnd \nf ( z, x 0 ) = 1 √ A ( x 0 ) -A ( x ) C ( x ) C ( x 0 ) , (24) \nf 0 ( z, x 0 ) = 1 √ ζ ( x 0 ) z + β ( x 0 ) z 2 , (25) \nζ ( x 0 ) = 1 -A ( x 0 ) A ' ( x 0 ) C ( x 0 ) [( C ' ( x 0 ) A ( x 0 ) -A ' ( x 0 ) C ( x 0 ))] , (26) \nβ ( x 0 ) = (1 -A ( x 0 )) 2 2 A ' ( x 0 ) 3 C ( x 0 ) 2 [ 2 C ( x 0 ) C ' ( x 0 ) A ' ( x 0 ) 2 + A ( x 0 ) A ' ( x 0 ) C ( x 0 ) C '' ( x 0 ) -C ( x 0 ) C ' ( x 0 ) A ( x 0 ) A '' ( x 0 ) -2 C ' ( x 0 ) 2 A ( x 0 ) A ' ( x 0 ) ] . (27) \nR ( z, x 0 ) = P + Qz x 3 0 c 1 2 ( -4 π ˜ α + x 3 0 ( -1 + c 1)) 4 (28) \n<!-- image --> \nFigure 2: Plot showing the strong lensing coefficients ¯ a and ¯ b as function of ˜ α . \n<!-- image --> \nP = 2 x 3 0 c 1( -1 + c 1) [ 2( -1 + c 1) x 12 0 + π ˜ α ( -68 + 52 c 1) x 9 0 + π 2 ˜ α 2 ( -696 + 344 c 1) x 6 0 + π 3 ˜ α 3 ( -1952 + 384 c 1) x 3 0 -512 π 4 ˜ α 4 ] , (29) \nQ \n= 2 x 3 0 c 1( -1 + c 1) \n[ +32 π 3 ˜ α 3 ( -183 + 39 c 1) x 3 0 -1536 π 4 ˜ α 4 , (30) \nc 1 = √ 1 + 16 π ˜ α x 3 m . (31) \n3( -1 + c 1) x 12 0 +2 π ˜ α ( -63 + 51 c 1) x 9 0 +8 π 2 ˜ α 2 ( -201 + 111 c 1) x 6 0 ] \nIn Eq. (22), R ( z, x 0 ) is regular for all values of z and x 0 , however, f ( z, x 0 ) diverges for x = x 0 or z → 0. With the above given definitions, the integral in Eq. (19) can be split into two, diverging and regular, parts as follows [56] \nI ( x 0 ) = I D ( x 0 ) + I R ( x 0 ) , (32) \nsuch that \nwith \nI D ( x 0 ) = ∫ 1 0 R (0 , x m ) f 0 ( z, x 0 ) dz, (33) \nThe integral I D ( x 0 ) converges for x 0 /negationslash = x m , as f 0 ( x 0 ) = 1 / √ z . But when x 0 = x m the integral I D ( x 0 ) has logarithmic divergences as φ ( x 0 ) = 0 and f 0 ( x 0 ) = 1 /z . Integral I R ( x 0 ) is the regularized term with the divergence subtracted. Solving the integrals in Eqs. (33) and (34), the deflection angle can be simplified [56] to \nwhere \nand \nI R ( x 0 ) = ∫ 1 0 ( R ( z, x 0 ) f ( z, x 0 ) -R (0 , x m ) f 0 ( z, x 0 ) ) dz. (34) \nα D ( u ) = -¯ a log ( u u m -1 ) + ¯ b + O ( u -u m ) , (35) \n¯ a = R (0 , x m ) 2 β ( x m ) = 8 c 1 4 ( 8 π ˜ α + x 2 m (1 -c 1) ) 2 ( -c 1 + 3) ( 1 + c 1) 2 ( x 4 m c 1 + 4 π ˜ α ( -9 + 4 x m c 1)) , (36) \n--¯ b = -π + I R ( x m ) + ¯ a log ( 2 β ( x m ) A ( x m ) ) , (37) \n- \nβ ( x m ) = ( -1 + c 1) 2 ( x 4 m c 1 + 4 π ˜ α ( -9 + 4 x m c 1) ) 4 c 1 3 (8 π ˜ α + x 2 m (1 -c 1)) 2 . (38) \n<!-- image --> \nFigure 3: Left Panel: plot showing the behavior of deflection angle α D ( u ) for strong-gravitational lensing with impact parameter u for different values of ˜ α . The colored points on the horizontal axis correspond to the impact parameter u = u m , for which deflection angle diverges. Right Panel: deflection angle α D ( u ) variation with ˜ α for u = 2 . 7. \n<!-- image --> \nIn this case I R ( x m ) can not be calculated analytically, so it has been calculated numerically. ¯ a and ¯ b are called the strong deflection limit coefficients, which depend on the metric functions evaluated at the x m . It is evident that ¯ a increases whereas ¯ b decreases with the ˜ α (cf. Fig. 2), and in the limit ˜ α → 0, they smoothly retain the values for the Schwarzschild black hole, viz., ¯ a = 1 and ¯ b = -0 . 4002. The deflection angle α D ( u ) in the strong deflection limits for the static spherically symmetric EGB black holes (4) are depicted in Fig. (3). The deflection angle α D ( u ) diverges for u = u m and steeply falls with u (cf. Fig. 3). It is evident that for fixed values of u , deflection angle decrease with increasing GB coupling parameter ˜ α ; Schwarzschild black hole cause larger deflection angle than the EGB black hole. It is worthwhile to note that the presented results are valid only in the strong deflection limit u /greaterorsimilar u m , whereas for u >> u m , the strong deflection limit is not a valid approximation. There exist a value of x 0 = x z or u = u z for which deflection angle becomes zero.", 'IV. LENSING OBSERVABLES': "Once we have known the deflection angle due to strong gravitational lensing Eq. (35), we can easily calculate the image positions using the lens equation. Lens equation, establishing a relation between the observational setup geometry, namely the positions of source S , observer O and the black hole L in a given coordinate system, and the position of the lensed images in the observer's sky, is given by [49] \nD OS tan β = D OL sin θ -D LS sin( α -θ ) cos( α -θ ) , (39) \nwhere β and θ are the angular separations, respectively, of the source and the image from the black hole. The distance between the source and black hole is D LS , whereas distance from the observer to the source and black hole is D OS and D OL respectively; all distances are expressed in terms of the Schwarzschild radius x s = R s / 2 M . Photon emitted with a impact parameter u from the source approaches the black hole and is received by an observer. The deflection angle α D is identified as the angle between the tangents to the emission and detection directions. Since it diverges for u = u m and the light rays perform several loops around the black hole before escaping to the observer, therefore, we can replace α D by 2 nπ +∆ α n , where n is the positive integer number corresponding to the winding number of loops around black hole and ∆ α n is the offset of the deflection angle. For the case of a far distant observer and source, and their nearly perfect alignment with the black hole, the lens equation (39) can be simplified as [49, 68] \nβ = θ -D LS D OS ∆ α n . (40) \nHowever, the lens equation has also been defined in more general setup [48, 50, 52-54, 69-72]. One can notice that in Eq. (40), only the offset angle ∆ α n comes into the lens equation rather than the complete deflection angle. To get \nwhere \nthe offset deflection angle for n th relativistic image, ∆ α n , we first solve the α D ( θ 0 n ) = 2 nπ , where θ 0 n is the image position for the α D = 2 nπ , this yields \nθ 0 n = u m D OL (1 + e n ) , (41) \nwith \ne n = e ¯ b -2 nπ/ ¯ a . (42) \nNow, making a Taylor series expansion of the deflection angle about ( θ 0 n ) to the first order, gives \nα D ( θ ) = α D ( θ 0 n ) + ∂α D ( θ ) ∂θ ∣ ∣ ∣ ∣ θ 0 n ( θ -θ 0 n ) + O ( θ -θ 0 n ) , (43) \n∣ \nusing ∆ θ n = θ -θ 0 n and the deflection angle Eq. (35), the Eq. (43) becomes \n∆ α n = -¯ aD OL u m e n ∆ θ n . (44) \nNeglecting the higher-order terms, the lens equation finally gives the position of n th image [56] \nθ n /similarequal θ 0 n + D OS D LS u m e n D OL ¯ a ( β -θ 0 n ) , (45) \nfor β = θ 0 n , viz., the image position coincides with the source position, the correction to the n th image position identically vanishes. Though Eq. (45) gives source's images on the same side of source ( θ > 0), we can replace β by -β in order to get the images on the other side. The brightness of the source's images will be magnified due to the gravitational lensing, the magnification of n th loop image is defined as [56] \nµ n = β θ dβ dθ ∣ ∣ ∣ ∣ ∣ θ 0 n -1 . (46) \n∣ The brightness magnification decreases steeply with the order n of the images, such that unless the source is almost perfectly aligned with the black hole and the observer, these images will be very faint as a result of high demagnification. Also Eq. (46) infers that for the perfect alignment of the source and the black hole, β → 0, the brightness magnification diverges. Considering only the outermost one-loop image θ 1 as distinguishable as a single image from the remaining inner packed images θ ∞ , we can have three characteristic observables [56] \nθ ∞ = u m D OL , (47) \ns = θ 1 -θ ∞ = θ ∞ e ¯ b -2 π ¯ a , (48) \nr = µ 1 ∑ ∞ n =2 µ n = e 2 π ¯ a , r mag = 2 . 5 log( r ) = 5 π ¯ a ln 10 . (49) \nu m = 2 √ 2 π ˜ α √ 1 + 8 π ˜ α -c 1 . (50) \nHere, θ ∞ is the angular radius of the photon sphere, i.e., the position of the innermost packed images, s is the angular separation between the one-loop image and the images at θ ∞ , and r is the ratio of the brightness flux from the outermost relativistic image at θ 1 to those from the remaining relativistic images at θ ∞ . It must be noted that, in contrary to θ ∞ and s , r mag is independent of the black hole distance from the observer. \nNow, we consider the supermassive black hole candidates at the galactic center of the Milky Way and the nearby galaxy Messier 87, respectively, Sgr A* and M87*, as the static spherically symmetric EGB black hole described by metric (4). Based on the latest observational data, we have taken their masses and distances from the Earth as, M = 3 . 98 × 10 6 M /circledot and D OL = 7 . 97 kpc for Sgr A* [73], and M = (6 . 5 ± 0 . 7) × 10 9 M /circledot and D OL = (16 . 8 ± 0 . 8) Mpc for M87* [74]. It is interesting to estimate the correction due to the GB coupling parameter ˜ α by comparing the results \nFigure 4: Plot showing the behavior of lensing observables, θ ∞ , r , and s as a function of ˜ α for Sgr A* (left panel) and M87* (right panel) black holes. \n<!-- image --> \nof EGB black hole with those for the Schwarzschild black hole (cf. Table I). The lensing observables for the Sgr A* and M87* black holes are shown in Fig. (4) as a function of ˜ α . Figure (4) and Table I clearly infer that the relativistic images have largest angular separation for small values of ˜ α . By measuring the lensing observables, namely θ ∞ , s , and r , for the EGB black hole and inverting the Eqs. 48 and (49), we can calculate the strong deflection coefficients ¯ a and ¯ b . Then the theoretically predicted values can be compared with the values inferred from the observational data to find the black hole parameters. \nTable I: Strong-lensing observables for the black hole Sgr A* and M87*, and lensing coefficients for various values of ˜ α . \n| | Sgr A* | Sgr A* | Sgr A* | M87* | M87* | M87* | Lensing Coefficients | Lensing Coefficients | Lensing Coefficients |\n|---------|-------------|-----------|----------|-------------|-----------|---------|------------------------|------------------------|------------------------|\n| ˜ α | θ ∞ ( µ as) | s ( µ as) | r m | θ ∞ ( µ as) | s ( µ as) | r m | ¯ a | ¯ b | u m /R s |\n| 0 . 0 | 25 . 530 | 0 . 031 | 6 . 825 | 19 . 780 | 0 . 024 | 6 . 825 | 1 . 0 | - 0 . 401 | 2 . 597 |\n| 0 . 005 | 25 . 032 | 0 . 043 | 6 . 412 | 19 . 395 | 0 . 034 | 6 . 412 | 1 . 063 | - 0 . 469 | 2 . 547 |\n| 0 . 01 | 24 . 473 | 0 . 063 | 5 . 931 | 18 . 961 | 0 . 048 | 5 . 931 | 1 . 150 | - 0 . 576 | 2 . 490 |\n| 0 . 015 | 23 . 829 | 0 . 096 | 5 . 330 | 18 . 462 | 0 . 074 | 5 . 330 | 1 . 279 | - 0 . 767 | 2 . 424 |\n| 0 . 019 | 23 . 218 | 0 . 146 | 4 . 691 | 17 . 988 | 0 . 113 | 4 . 691 | 1 . 454 | - 1 . 083 | 2 . 362 |", 'V. WEAK GRAVITATIONAL LENSING': 'Gibbon and Werner [75] invoked the Gauss-Bonnet theorem, in the context of optical geometry to calculate the deflection angle of light in the weak-field limits of the spherically symmetric black hole spacetime [76]. Later, considering the source and observers at finite distances from the black hole, corrections in the deflection angle due to static spherically symmetric black hole were calculated by Ishihara et al. [77, 78], which is further generalized by Ono et al. [79] for the stationary and axisymmetric black holes. Since then, their method have been extensively used for varieties of black hole spacetimes [62, 80-89]. We follow their approach to calculate the light deflection angle in the weak-field limit caused by the static and spherically symmetric EGB black hole. \nConsidering a coordinate system centered at the black hole ( L ), and assuming the observer ( O ) and the source ( S ) at the finite distances from the black hole (cf. Fig. 5), we can define the deflection angle at the equatorial plane as follow [77, 79] \nα D = Ψ O -Ψ S +Φ OS , (51) \nwhere, Φ OS is the angular separation between the observer and the source, Ψ S and Ψ O , respectively, are the angle made by light rays at the source and observer. The quadrilateral ∞ O /square ∞ S , which is made up of spatial light ray curves from source to the observer, a circular arc segment C r of coordinate radius r C ( r C → ∞ ), and two outgoing radial lines from O and S , is embedded in the 3-dimensional Riemannian manifold (3) M defined by optical metric γ ij (cf. Fig. 5). The surface integral of the Gaussian curvature of the two-dimensional surface of light propagation in this manifold gives the light deflection angle [77, 79] \nα D = -∫ ∫ ∞ O /square ∞ S KdS. (52) \nSolving Eq. (4) for the null geodesics ds 2 = 0, we get \nwith \ndt = ± √ γ ij dx i dx j , (53) \nγ ij dx i dx j = 1 f ( r ) 2 dr 2 + r 2 f ( r ) ( dθ 2 +sin 2 θ dφ 2 ) . (54) \nGaussian curvature of the surface of light propagation is defined as [90] \nK = 3 R rφrφ γ , = 1 √ γ ( ∂ ∂φ ( √ γ γ rr (3) Γ φ rr ) -∂ ∂r ( √ γ γ rr (3) Γ φ rφ )) , (55) \nwhere γ = det( γ ij ). For EGB black hole metric (4), in the weak-field limits, Eq. (55) simplifies to \nK = -2 M r 3 + 3 M 2 r 4 + 640 M 2 πα r 6 -1152 M 3 πα r 7 + O ( M 3 α 2 r 9 , M 4 α 2 r 10 ) . (56) \nThe surface integral of Gaussian curvature over the closed quadrilateral ∞ O /square ∞ S reads [79] \n∫ ∫ ∞ O /square ∞ S KdS = ∫ φ O φ S ∫ r 0 ∞ K √ γdrdφ, (57) \nFigure 5: Schematic figure for the quadrilateral ∞ O /square ∞ S embedded in the curved space. \n<!-- image --> \nwhere r 0 is the distance of closest approach to the black hole. In the weak-field approximation, the light orbits equation can be considered as [91] \nb = sin φ u + M (1 -cos φ ) 2 u 2 -M 2 (60 φ cos φ +3sin3 φ -5 sin φ ) 16 u 3 + O ( M 2 α u 5 ) , (58) \nwhere b = 1 /r , and u is the impact parameter. The integral Eq. (57) can be recast as \n∫ ∫ ∞ O /square ∞ S KdS = ∫ φ O φ S ∫ b 0 -K √ γ b 2 dbdφ, (59) \nwhich for the metric Eq. (54) reads as \n∫ ∫ KdS = ( cos -1 ub o +cos -1 ub s ) ( 15 M 2 4 u 2 -60 M 2 πα u 4 ) + ( √ 1 -u 2 b 2 o + √ 1 -u 2 b 2 s )( 2 M u + 128 M 3 6 u 3 -14852 M 3 πα 25 u 5 ) + ( b o √ 1 -u 2 b 2 o + b s √ 1 -u 2 b 2 s )( -M 2 4 u -60 M 2 πα u 3 ) + ( b 2 o √ 1 -u 2 b 2 o + b 2 s √ 1 -u 2 b 2 s )( M 3 6 u -7426 M 3 πα 25 u 3 ) + O ( M 4 u 4 , M 4 α u 6 ) , (60) \nwhere, b o and b s , respectively, are the reciprocal of the distances of observer and source from the black hole, i.e., b o = 1 /r o and b s = 1 /r s . We have used the approximation cos φ o = -√ 1 -u 2 b 2 o , cos φ s = √ 1 -u 2 b 2 s , the relative negative sign is because the source and the observer are at the opposite sides to the black hole. In the limits of far distant observer and source, b o → 0 and b s → 0, the deflection angle for the EGB black hole in the weak-field limits reads as follows \nα D = 4 M u + 15 πM 2 4 u 2 -60 M 2 π 2 α u 4 + 128 M 3 3 u 3 -29704 M 3 πα 25 u 5 + O ( M 4 u 4 , M 4 α u 6 ) . (61) \nFurther, in the limiting case of α = 0, it reduces as \nα D | Schw = 4 M u + 15 πM 2 4 u 2 + 128 M 3 3 u 3 + O ( M 4 u 4 ) , (62) \nwhich corresponds to the value for the Schwarzschild black hole [47, 91]. In terms of the normalized impact parameter and GB coupling parameter, u → u/M and α → ˜ α = α/M 2 , the deflection angle Eq. (61), reads as \nα D = 4 u + 15 π 4 u 2 -60 π 2 ˜ α u 4 + 128 3 u 3 -29704 π ˜ α 25 u 5 + O ( 1 u 4 , ˜ α u 6 ) . (63) \nTable II: The corrections in the deflection angle δα D = α D | Schw -α D for weak-gravitational lensing around the EGB black hole with source at b s = 10 -4 and observer at b o = 0; δα D is in units of µ as. \n| | ˜ α = 0 . 001 | ˜ α = 0 . 003 | ˜ α = 0 . 005 | ˜ α = 0 . 01 | ˜ α = 0 . 019 |\n|--------------|--------------------------------------------------------|-----------------|-----------------|----------------|-----------------|\n| u = 1 ∗ 10 3 | 0.122428 | 0.367283 | 0.612139 | 1.22428 | 2.32613 |\n| u = 2 ∗ 10 3 | 0.00762716 | 0.0228815 | 0.0381359 | 0.0762718 | 0.144916 |\n| u = 3 ∗ 10 3 | 0.00150446 | 0.0045134 | 0.0075223 | 0.0150446 | 0.0285848 |\n| u = 4 ∗ 10 3 | 0.000475089 0.0014253 0.00237546 0.00475093 0.00902681 | | | | | \nFigure 6: The corrections in the deflection angle δα D = α D | Schw -α D for weak-gravitational lensing around EGB black hole with b s = 10 -4 , b o = 0 and varying u ; δα D is in units of µ as. \n<!-- image --> \nIt is clear from Eq. (63), that the GB coupling parameter ˜ α reduces the deflection angle, i.e., in the weak-field limits the EGB black hole leads to smaller deflection angle than the Schwarzschild black hole. We presented the correction in the deflection angle δα D = α D -α D | Schw due to ˜ α in Table II and Fig. 6 for various values of impact parameter u . Table II infers that, for fixed value of impact parameter u , the correction δα D increases with the ˜ α and is of the order of micro-arc-second. However, for a fixed value of ˜ α , δα D decreases with u .', 'VI. CONCLUSIONS': "The GB correction to the Einstein-Hilbert action provides a natural extension to the Einstein's theory of general relativity in D > 4 dimensional spacetime, however, it is a topological invariant quantity in D → 4, and therefore does not contribute to the gravitational field equations. Recently proposed 4 D regularized EGB gravity theories, in which GB term in the gravitational action makes a non-trivial contribution to the field equations in 4 D and contrary to the Schwarzschild black hole solution of GR, a black hole in this theory can possess up to two horizons. \nGravitational lensing is unequivocally a potentially powerful tool for the analysis of strong fields and for testing general relativity. The strong-field limit provides a useful framework for comparing lensing by different gravities, the 4 D EGB gravity is a very interesting model to consider and discuss the observational signatures this quadratic curvature corrected gravity has than the Schwarzschild black hole of general relativity. In view of this, we investigated the gravitational lensing of light around the 4 D EGB black hole in the strong deflection limits. Depending on the value of impact parameter u , photons get deflected from their straight path and leads to the multiple images of a source, and for the particular value of u = u m , photons follow the circular orbits around the black hole and deflection angle diverges. It is found that the static spherically symmetric EGB black holes lead to smaller deflection angle as compared to the Schwarzschild black hole value, and the deflection angle decreases with the GB coupling parameter ˜ α . The effect of ˜ α on the deflection angle immediately reflect on the relativistic images. Considering the observer and the light source at far distances from the black hole, we calculate the observables for the strong lensing, namely the angular separation between the relativistic source's images and the relative brightness magnitude of the first image. It is noted that, with the increasing GB coupling parameter, the photon circular orbits radii decrease, the angular \nseparation between the set of source images increase, whereas the brightness magnitude decrease. \nWe modeled the supermassive black holes Sgr A* and M87,* respectively, in the Galactic Center and at the center of galaxy M87 as the 4 D EGB black hole (lens), and estimate the lensing observables. The corrections in the angular separation of the images, due to the GB coupling parameter α , is of the order of µ as, which is within the limit of current observational outreaches. The weak gravitational lensing around EGB black holes is also discussed and corrections in the deflections angle due to ˜ α are calculated, which increase with ˜ α . \nInvestigation of the gravitational lensing around the rotating EGB black holes in the strong and weak field limits is the topic of our future investigation.", 'VII. 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2013Natur.503..500L
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Puzzling accretion onto a black hole in the ultraluminous X-ray source M 101 ULX-1
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2013-01-01
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There are two proposed explanations for ultraluminous X-ray sources (ULXs) with luminosities in excess of 10<SUP>39</SUP> erg s<SUP>-1</SUP>. They could be intermediate-mass black holes (more than 100-1,000 solar masses, ) radiating at sub-maximal (sub-Eddington) rates, as in Galactic black-hole X-ray binaries but with larger, cooler accretion disks. Alternatively, they could be stellar-mass black holes radiating at Eddington or super-Eddington rates. On its discovery, M 101 ULX-1 had a luminosity of 3 × 10<SUP>39</SUP> erg s<SUP>-1</SUP> and a supersoft thermal disk spectrum with an exceptionally low temperature--uncomplicated by photons energized by a corona of hot electrons--more consistent with the expected appearance of an accreting intermediate-mass black hole. Here we report optical spectroscopic monitoring of M 101 ULX-1. We confirm the previous suggestion that the system contains a Wolf-Rayet star, and reveal that the orbital period is 8.2 days. The black hole has a minimum mass of 5, and more probably a mass of 20-30, but we argue that it is very unlikely to be an intermediate-mass black hole. Therefore, its exceptionally soft spectra at high Eddington ratios violate the expectations for accretion onto stellar-mass black holes. Accretion must occur from captured stellar wind, which has hitherto been thought to be so inefficient that it could not power an ultraluminous source.
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https://arxiv.org/pdf/1312.0337.pdf
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{'Puzzling accretion onto a black hole in the ultraluminous X-ray source M101 ULX-1': "Jifeng Liu 1 , Joel N. Bregman 2 , Yu Bai 1 , Stephen Justham 1 , Paul Crowther 3 , \n- 1 KeyLaboratoryofOpticalAstronomy,NationalAstronomicalObservatories,ChineseAcademy ofSciences,20ADatunRd,ChaoyangDistrict,Beijing,China100012 \n2 DepartmentofAstronomy,UniversityofMichigan,500ChurchSt.,AnnArbor,MI40185,US \n3 DepartmentofPhysics&Astronomy,UniversityofSheffield,HounsfieldRd,SheffieldS37RH, \nUK \nThere are two proposed explanations for ultraluminous X-ray sources 1, 2 (ULXs) with luminosities in excess of 10 39 erg s -1 . They could be intermediate-mass black holes (more than 100-1,000, solar masses, M /circledot ) radiating at sub-maximal (sub-Eddington) rates, as in Galactic black-hole X-ray binaries but with larger, cooler accretion disks 3-5 . Alternatively, they could be stellar-mass black holes radiating at Eddington or super-Eddington rates 2, 6 . On its discovery, M101 ULX-1 4, 7 had a luminosity of 3 × 10 39 erg s -1 and a supersoft thermal disk spectrum with an exceptionally low temperature ł- uncomplicated by photons energized by a corona of hot electrons - more consistent with the expected appearance of an accreting intermediate-mass black hole 3, 4 . Here we report optical spectroscopic monitoring of M101 ULX-1. We confirm the previous suggestion 8 that the system contains a Wolf-Rayet star, and and reveal that the orbital period is 8.2 days. The black hole has a minimum mass of 5 M /circledot , and more probably a mass of 20 -30 M /circledot , but we argue that it is very unlikely to be \nan intermediate-mass black hole. Therefore its exceptionally soft spectra at high Eddinton ratios violate the expectations for accretion onto stellar-mass black holes 9-11 . Accretion must occur from captured stellar wind, which has hitherto been thought to be so inefficient that it could not power an ultraluminous source 12, 13 . \nWhile it is desirable to obtain the primary mass for ultraluminous X-ray sources (ULXs) through measuring the motion of its companion, this is only possible in the X-ray low state because the X-ray irradiated accretion disk will dominate the optical light in the high state 14, 15 . A spectroscopic monitoring campaign for M101 ULX-1 was carried out from February to May 2010 during its expected X-ray low states. The optical spectrum (Figure 1) is characterized by broad helium emission lines, including the He II 4686 ˚ Aline. Given the absence of broad hydrogen emission lines, which are detected in some ULXs from their X-ray irradiated accretion disk at very high luminosities 14, 15 , the donor cannot be hydrogen rich, and thus must be a Wolf-Rayet (WR) star or a helium white dwarf. The latter can be excluded because a white dwarf is roughly a million times dimmer than the observed optical counterpart even during the low states. Indeed, the optical spectrum is unique to WR stars, and the intensities of the helium emission lines can be reproduced well by an atmospheric model 16 of a WR star, the mass of which is estimated to be 19 M /circledot based on the empirical mass-luminosity relation 17, 18 . Given the relatively low luminosities in the X-ray low state, the helium emission lines are expected to originate mainly from the WR secondary with little contribution from the accretion disk. Such emission lines have been used to measure the black hole (BH) mass in both IC10 X-1 ( 21 -35 M /circledot ) 19, 20 and NGC300 X-1 ( 12 -24 M /circledot ) 21, 22 , systems which exhibit luminosities an order of magnitude lower than the peak luminosity of M101 ULX-1. \nSince the centroid of the He II 4686 ˚ A emission line varied by ± 60 km/s over three months of our monitoring campaign, we have been able to obtain the orbital period of P = 8 . 2 ± 0 . 1 days and the mass function f ( M ∗ , M · , i ) = 0 . 18 ± 0 . 03 M /circledot for M101 ULX-1 (Figure 2). Because we already know the mass of the donor star we are able to infer the mass of the accretor to be M · ≥ 4 . 6 ± 0 . 3 M /circledot (for inclination angle i ≤ 90 · ), where the error is computed from the uncertainties in the secondary mass and in the mass function. Even for the minimum mass, obtained when the system is aligned perfectly edge-on to the line of sight (for which i = 90 · ), such a compact primary can only be a BH. Higher BH masses are easily obtained for lower inclination angles. For example, a stellar mass BH of 20 M /circledot corresponds to i = 19 · , and an intermediate mass black hole (IMBH) of 1000 M /circledot ( 300 M /circledot ) corresponds to i = 3 · ( i = 5 · ). The probability of discovering a pole-on binary with i < 3 · ( i = 5 · ) by mere chance is lower than 0.1% (0.3%). This makes it very unlikely that this system contains an IMBH of 1000 M /circledot ( 300 M /circledot ). If the peak luminosity of M101 ULX-1 corresponds to less than 30% of the Eddington level - which is commonly-assumed to be required to produce the thermally-dominated spectral state 9, 23 - then the BH mass would exceed 50 -80 M /circledot . The true BH mass seems likely to be ∼ 20 -30 M /circledot (see the Supplementary Information for details). \nThe confirmation of a WR star in the system, independent of the dynamical mass measurement, also suggests that M101 ULX-1 is unlikely to be an IMBH. IMBHs cannot form directly through the collapse of massive stars, but it is suggested that they can form through mergers in dense stellar environments 24, 25 . However, any IMBH formed would not be seen as a ULX unless they capture a companion as a reservoir from which to accrete matter. Such a capture is a rare \nevent even in dense stellar environments such as globular clusters or galactic bulges, to which M101 ULX-1 apparently does not belong, and captures that can provide high-enough accretion rates to power a ULX are even more unusual 26, 27 . Given the rarity of WR stars, with roughly 2000 WRstars out of 200 billions of stars in a typical spiral galaxy like the Milky Way 18 , it is extremely unlikely that M101 ULX-1 is such a revived IMBH. Alternatively a huge population of IMBHs would somehow remain undetected, both with and without companions. \nM101 ULX-1 is thus a stellar black hole, although it is a member of the class of supersoft ULXs which have been considered to be outstanding IMBH candidates 4, 5 . Its combination of high luminosities and low disk temperatures (Figure 3) strains our current understanding of accretion by stellar-mass BHs 9-11 . Studies of Galactic black hole X-ray binaries suggest that radiation at less than roughly 30% of the Eddington luminosity is dominated by the thermal emission from a hot disk ( ∼ 1 keV). A hard power-law component due to Comptonization by the disk corona becomes more and more significant when the luminosity increases to near-Eddington levels. When the luminosity increases further to Eddington or super-Eddington levels, the Comptonized component begins to dominate over the disk component, as observed for ULXs in the ultraluminous state 2, 6 . For example, the ultraluminous microquasar in M31 with a stellar-mass black hole ( ∼ 10 M /circledot ) and a luminosity of 10 39 erg s -1 exhibited hard X-ray spectra 28 . If it were the same phenomenon, a hard X-ray spectra would be expected for a stellar-mass BH in M101 ULX-1, whether it is radiating at sub-, near- or super-Eddington luminosities. The observed supersoft X-ray spectra lack hard photons above 1.5 keV, and can be described purely by cool accretion disks, uncomplicated by Comptonization, with exceptionally low temperatures of 90-180 eV 4, 7 . Including extra pho- \nectric absorption by the surrounding WR wind into spectral analysis would further lower the underlying disk temperatures and increase the luminosities 4 , which would drive M101 ULX-1 to deviate even farther from the expected hard spectra. This unambiguously demonstrates that stellar mass BHs can have very cool accretion disks uncomplicated by the Comptonized component, contrary to standard expectations 3, 9, 11 . \nM101 ULX-1 is the third known WR/BH binary but is distinctly different from NGC 300 X-1 and IC 10 X-1. While M101 ULX-1 is a recurrent transient with supersoft spectra and low disk temperatures, both IC 10 X-1 and NGC 300 X-1 show constant X-ray output (despite apparent variations due to orbital modulation), hard spectra with a minor disk component, and disk temperatures above 1 keV 19, 21, 29 (Figure 3). Hence the compact object in M101 ULX-1 was considered to be an excellent IMBH candidate, while IC 10 X-1 and NGC 300 X-1 were expected to host stellar mass black holes (as was later confirmed). The 8.2-day orbital period shows that M101 ULX-1 is a wide binary, with components which would be separated by 50 R /circledot for M · = 5 M /circledot ( 75 R /circledot for M · = 60 M /circledot ). The Roche lobe radius for the secondary is always greater than 22 R /circledot , twice as large as the WR star itself. Mass transfer by Roche lobe overflow is thus impossible, and the black hole must be accreting matter by capturing the thick stellar wind. Given the geometry of the system, the disk is very large, and thus there will be a helium partial ionization zone. Such a disk is prone to instability, causing the observed X-ray transient behaviors for M101 ULX-1. In contrast, IC 10 X-1 and NGC 300 X-1 have shorter orbital periods (34.9 hr and 32.3 hr respectively) and smaller separations ( ∼ 20 R /circledot ). Since those WR stars fill their Roche lobes, the BHs accrete via Roche-lobe overflow. These systems also have much smaller and hotter accretion disks without \nhelium partial ionization zones, which explains why IC 10 X-1 and NGC 300 X-1 do not display disk-instability outbursts (see also the Supplementary Information). \nMass transfer through wind-accretion usually has a very low efficiency, as in the case of many low-luminosity high mass X-ray binaries, and is typically not considered for populations that require high accretion rates. However, M101 ULX-1 demonstrates that this expectation is not always correct. In particular, transient outbursts of such wind-accreting system have generally not been included in theoretical ULX populations 12, 13 , but M101 ULX-1 does attain ULX luminosities. 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Mass Transfer in Mira-type Binaries. Baltic Astronomy , 21 , 88-96 (2011) \nAcknowledgements We thank Drs. Jeffery McClintock, Rosanne Di Stefano, Qingzhong Liu, Xiangdong \nLi, Feng Yuan, and ShuangNan Zhang for helpful discussions. J.F. Liu acknowledges support for this work provided by NASA through the Chandra Fellowship Program (grant PF6-70043) and support by National Science Foundation of China through grant NSFC-11273028. The paper is based on observations obtained at the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Minist'erio da Ciˆencia, Tecnologia e Inovac¸ ˜ao (Brazil) and Ministerio de Ciencia, Tecnolog'ıa e Innovaci'on Productiva (Argenti na). \nAuthor contributions J.F. Liu and J.N. Bregman proposed the observations, J.F. Liu and Y. Bai reduced the data and carried out the analysis, J.F.Liu, J.N. Bregman and S. Justham discussed the results and wrote the paper, P. Crowther helped to confirm the properties of the Wolf-Rayet star. All commented on the manuscript and contributed to revise the manuscript. \nAuthor information Reprints and permissions information is available at npg.nature.com/reprints. The authors declare that they have no competing financial interests. Readers are welcome to comment on the online version of the paper. Correspondence and requests for materials should be addressed to J.F. Liu (email: [email protected]).", 'Figure 1: The secondary of M101 ULX-1 is confirmed to be a Wolf-Rayet star based on the': "optical spectrum , combined from 10 Gemini/GMOS observations with a total exposure time of 16 hours. The spectrum shows narrow nebular lines with FWHM of ∼ 4 ˚ A at the instrumental spectral resolution, including hydrogen Balmer lines and forbidden lines such as [OIII] 4960/5006 (the latter is mostly in the CCD gap and only partly shown), [NII] 6548/6583, and [SII] 6716/6731, all at a constant radial velocity over observations consistent with that of M101. Also present are broad emission lines with FWHM up to 20 ˚ A, including strong HeII 4686, HeI 5876, HeI 6679, weaker HeI 4471, HeI 4922, and HeII 5411, and NIII 4640 lines. The observed HeI 5876/He II 5411 equivalent width ratio suggests a WR star of WN8 sub-type, consistent with the absence of carbon emission lines for WC stars (such as CIII 5696 and CIV 5812). The intensities of the helium emission lines can be best reproduced by an atmospheric model 16 of a Wolf-Rayet star with R ∗ = 10 . 7 R /circledot , M ∗ = 17 . 5 M /circledot , L ∗ = 5 . 4 × 10 5 L /circledot , T ∗ = 48kK , ˙ M ∗ = 2 ± 0 . 5 × 10 -5 M /circledot /yr, v ∞ = 1300 ± 100 km s -1 (with 68.3% uncertainties for the two continuously variable parameters), consistent with those for a WN8 star. The mass-luminosity relation 17, 18 for WR stars gives a more reliable mass estimate of 19 M /circledot , which we use in the main text, with an estimated formal error of 1 M /circledot . \nFigure 2: An orbital period of ∼ 8 . 2 days is revealed from the radial velocity measurements over three months for M101 ULX-1. The upper panel (a) shows the radial velocities of the HeII 4686 ˚ A emission line (with 68.3% uncertainties computed mainly from the dispersion of the wavelength calibration) from nine observations over three months. The lower right panel (b) shows the χ 2 computed for a sine fit (under the assumption of a circular orbit) to the radial velocity curve as a function of trial periods. The trial periods range from a minimum of 3 days, when the WR secondary fills its Roche lobe, to a maximum of 10 days as suggested by the last five measurements. The best fit is achieved at minimal χ 2 ∼ 1 . 6 for P = 8 . 2 days and K = 61 km/s, for which the folded radial velocity curve is shown in the lower left panel (c). The 68.3% uncertainties for the best fit are estimated to be ∆ P = 0 . 1 days and ∆ K = 5 km/s using χ 2 -χ 2 best = 1 . All other trial periods (such as those at P ∼ 6 . 4 days) are worse by ∆ χ 2 > 4 . The successful fit with a sine curve suggests that the orbital eccentricity is small. This leads to a mass function f ( M ∗ , M · , i ) = PK 3 2 πG = 0 . 18 ± 0 . 03 M /circledot , where the error accounts for the 68.3% uncertainties in P and K . \nFigure 3: The prototype ultraluminous supersoft X-ray source M101 ULX-1 exhibited distinct spectral characteristics. M101 ULX-1 is compared to Galactic black hole X-ray binaries (GBHXRBs), WR/BH binaries IC 10 X-1 and NGC 300 X-1, and other ULXs on the disk X-ray luminosity ( L X ) - temperature ( T d ) plane, all plotted with the 68.3% uncertainties from the Xray spectral fitting. Except for M101 ULX-1, which can be fitted with a disk blackbody model with temperatures of 90-180 eV 4, 7 , all other X-ray sources are complicated by the presence of a hard power-law component due to Comptonization by a corona, and can be best fitted with a disk blackbody plus power-law composite model 3, 29 . While GBHXRBs 3 and the other two WR/BH binaries 29 with stellar black holes cluster in the same region, M101 ULX-1 lies within a distinct region that has been expected to contain IMBH candidates, the same region as for some ULXs 3 . The dotted lines describe the expected disk luminosity ( L d ) for different disk temperatures for a fixed disk inner radius based on the relation L d ∝ R 2 in T 4 d . The two lines are offset by 4 orders of magnitudes in luminosity, implying a factor of 100 difference in the disk inner radii, and a factor of 100 difference in the black hole masses if the disk radius is tied to the innermost stable orbit of the black hole. Fitting ULX spectra with alternative Comptonization models can yield high disk temperatures consistent with those of stellar mass black holes 6 . However, the location of M101 ULX-1 on the L X -T d plane does not change because its spectra are not complicated by Comptonization at all. \nExtended Data Figure 1. M101 ULX-1 as observed in the optical. (a) M101 ULX-1 is located on a spiral arm of the face-on grand-design spiral galaxy M101, as indicated by the arrow. The color image of M101 is composed of GALEX NUV, SDSS g, and 2MASS J images. (b) ULX-1 is identified as a blue object with V=23.5 mag at the center of the 1 '' circle on the HST image. The color image is composed of ACS/WFC F435W, F555W and F814W images. \nExtended Data Figure 2. Physical properties of the WR secondary from spectral line modeling. Distributions of computed ∆ 2 as a function of (a) stellar masses, (b) stellar mass loss rate, (c) stellar radii, and (d) terminal velocity. Here ∆ 2 = ∑ i (EW -EW i ) 2 computes the difference between observed and synthetic equivalent widths for six broad helium lines present in the Gemini/GMOS spectrum. We have computed synthetic spectra for a group of 5000 real stars from the evolution tracks (as shown by the thick stripes in the mass plot and the radius plot) and for another group of 'fake' stars with continuous distributions in mass, radius and luminosity. The best model is labeled by a filled pentagon in all panels. \nExtended Data Figure 3. Properties for the Wolf-Rayet/black hole binary for different black hole masses. Shown are the binary separation (solid), the Roche lobe sizes for the Wolf-Rayet star (dotted) and for the black hole (dashed), the capture radius for the black hole when using the terminal velocity (dash-dotted) or when using a simplified velocity law v ( r ) = v ∞ (1 -R ∗ /r ) (long-short dashed). \nExtended Data Figure 4. The black hole accretion rate for different black hole mass. The accretion rates are computed adopting the terminal velocity (dotted) and a simplified velocity law \nv ( r ) = v ∞ (1 -R ∗ /r ) (solid). To power the observed average luminosity of 3 × 10 38 erg/s, the black hole mass must exceed 13 M /circledot ( 8 M /circledot ) using the terminal velocity (the velocity law) for a Kerr black hole, and exceed 46 M /circledot ( 28 M /circledot ) for a Schwarzschild black hole. \nExtended Data Figure 5. Disk temperature structures for M101 ULX-1. (a) The disk temperature profiles for M101 ULX-1 (for P = 8 . 24days , M ∗ = 19 M /circledot , R ∗ = 10 . 7 R /circledot , M · = 10 / 100 M /circledot ) and NGC300 X-1 (for P = 32 . 4hr , M ∗ = 26 M ∗ , R ∗ = 7 . 2 R /circledot , M · = 16 . 9 M /circledot ; Crowther et al. 2010). (b) The disk temperature at the outer edge for different black hole mass in M101 ULX-1. \nExtended Data Table 1. Gemini/GMOS spectroscopic observations of M101 ULX-1. The columns are: (1) Observation date, (2) Modified Julian Date, (3) exposure time in seconds, (4) barycentric correction computed with rvsao , and (5) the corrected radial velocity as measured with HeII 4686, with an error of 15 km/s as mainly from the uncertainties in the wavelength calibration. \nExtended Data Table 2. Properties of emission lines. The columns are: (1) emission line ID, (2) FWHM as obtained from Gaussian fit, which equals to 2 . 35 σ , (3) equivalent width, (4) line luminosity in unit of 10 34 erg/s, and (5) equivalent width from the best WR synthetic model. \n<!-- image --> \n<!-- image --> \n<!-- image -->", 'Online Methods': 'This Online Methods part provides details about background information for M101 ULX-1, data reduction and analysis of the Gemini/GMOS spectroscopic observations, search for the orbital periodicity, and properties of the Wolf-Rayet/black hole binary. It contains 5 figures and 2 tables, and additional references.', '1 M101 ULX-1 is an outstanding IMBH candidate': "M 101 is a nearby face-on grand design spiral galaxy, a frequent target of various observations. These include the optical monitoring observations in search of Cepheids with the Hubble Space Telescope, yielding a distance of 6.855 Mpc 31 . M101 ULX-1 (CXO J140332.3+542103) is located near a spiral arm (Extended Data Figure 1), and identified with a unique optical counterpart of V = 23 . 5 mag 32 . At this location, the metallicity is 0 . 4 × solar according to the M101 gas-phase oxygen abundance gradient 33 . \nThis ULX has been observed intensively by X-ray missions including ROSAT, XMM and Chandra since early 1990's, which exhibited spectral state transitions between the low-hard state and the high-soft state reminiscent of Galactic black hole X-ray binaries. This ULX was once the brightest X-ray point source in M101 with a Chandra/ACIS count rate of 0.10 count/second 34 , observed during the 2000 March observation (ObsID 934). The Chandra/XMM-Newton spectra during its outbursts 4, 35 were very soft and can be generally fitted with an absorbed blackbody \nmodel with n H = 1 -4 × 10 21 cm -2 and temperatures of 50-100 eV, and the peak 0.3-7 keV luminosity reached 3 × 10 40 erg/s, with a bolometric luminosity of about 10 41 erg/s, suggesting an intermediate mass black holes of a few thousand solar masses. It was argued that it is unphysical to adopt a high neutral absorber column density of ≥ 10 21 cm -2 , and fitting the spectra as blackbody plus a diskline component centered at 0.5 keV with N H fixed at the Galactic value of 4 × 10 20 cm -2 yielded the maximum outburst bolometric luminosity of 3 × 10 39 erg s -1 , consistent with the Eddington luminosity of a black hole of 20-40 M /circledot 7 . \nEven at the lowered luminosities of 3 × 10 39 erg s -1 , the combination of the disk luminosities and disk temperatures makes M101 ULX-1 an outstanding IMBH candidate. It is believed that the accretion disks for IMBHs should have larger inner radii and consequently lower disk temperatures 3-5 , occupying the upper left portion in the T disk ∼ L X plane as shown in Figure 3. The position of M101 ULX-1 on this plane suggests that it is distinctly different from the Galactic BH X-ray binaries in the lower right portion, but belongs to the league of IMBH candidates along with some extreme ULXs above 10 40 erg/s. The practice of placing these ULXs on this plane was questioned because decomposing ULX spectra into DISKBB+PL is unphysical given the dominance of the hard power-law component. However, in the case of M101 ULX-1 the spectra are supersoft without any hard power-law component, so its location on the plane should reflect the accretion disk uncomplicated by Comptonization. For comparison, we also put on this plane the other two known WR/BH binaries 29 IC 10 X-1 and NGC 300 X-1, which apparently belong to the league of stellar mass black holes, and dynamical mass measurements have yielded mass estimates of 20 -30 M /circledot . \nCombined analysis of 26 HST observations and 33 X-ray observations over 16 years 8 revealed two optical outbursts in addition to 5 X-ray outbursts. While there is no 'exact' period for the recurring outbursts, the outbursts occur once roughly every six months. Such outbursts last 10-30 days, suggesting a outburst duty cycle of 10%-15%. Outside outbursts, ULX-1 stays in a low-hard state with an X-ray luminosity of 2 × 10 37 erg/s 4, 7, 8, 35 . Such behaviors is reminiscent of those of soft X-ray transients in low-mass X-ray binaries, albeit with higher luminosities and lower disk temperatures, but are different from the recently discovered high mass fast transients due to clumping winds at much lower X-ray luminosities ( ∼ 10 34 erg/s). Detailed studies of the optical spectral energy distribution, after removal of optical emission from the X-ray irradiated accretion disk in the outbursts, suggest that the secondary is a Wolf-Rayet star of initially 40-60 M /circledot , currently 18-20 M /circledot , 9-12 R /circledot and about 5 × 10 4 Kelvin 8 . This claim of a WR companion is supported by the presence of the He II λ 4686 emission line in the Gemini/GMOS-N spectrum taken in 2005 36 .", '2 Gemini/GMOS data reduction': "M101 ULX-1 was monitored spectroscopically from February to May in 2010 during its expected low states under the Gemini/GMOS-N program GN-2010A-Q49 (PI: Jifeng Liu). Extended Data Table 1 lists the observations taken in ten nights distributed from February to May, with a total exposure of 15.6 hours. All exposures were taken with the 0 . '' 75 slit and the B600 grating tuned for a wavelength coverage from 4000 ˚ A to 6900 ˚ A; such a slit/grating combination will yield a spectral resolution of about 4.5 ˚ A. We followed standard procedures to reduce the observations and \nextract 1-D spectra using the gmos package in IRAF . All consecutive sub-exposures during one night were combined into one spectrum to increase the signal-to-noise ratio, and we obtained ten spectra with exposure times ranging from 3200 seconds to 9600 seconds (Extended Data Table 1). \nFor each spectrum, the wavelength solution was obtained using the Copper-Argon arc lamp spectra taken with the same slit/grating setting right before and after the science exposures during the same night or occasionally the night after. We verified the wavelength solution by comparing thus obtained wavelengths to the intrinsic wavelengths for a dozen of strong night sky emission lines identified in the spectra before sky subtraction, and revealed wavelength differences with a dispersion of about 0.25 ˚ A, or ∼ 15 km/s. The extracted spectra were converted to flux spectra using the standard star HZ44 taken during the night of February 15, and we scaled the spectra to have f λ = 1 . 5 × 10 -18 erg/s/cm 2 / ˚ A at 5500 ˚ A corresponding to F555W = 23.5 mag based on previous HST/WFPC2 observations 8 . \nFigure 1 shows the flux-calibrated sky-subtracted spectrum combined from the ten spectra. The combined spectrum is free of absorption lines but abundant in emission lines as identified and listed in Extended Data Table 2. For each emission line, we fit a Gaussian profile to derive its line width and compute its line flux and luminosity. Two categories of lines are present in the spectrum. The first category is the broad helium emission lines with FWHM of up to 20 ˚ A, five times broader than the instrumental spectral resolution, and includes strong HeII 4686, HeI 5876, HeI 6679, and weaker HeI 4471, HeI 4922, and HeII 5411 lines. The broad NIII 4634 emission line is also present. The second category is the the narrow emission lines with line widths consistent \nwith the instrumental spectral resolution, and includes the Balmer lines and forbidden lines such as [OIII] 4960/5006 (the latter is mostly in the CCD gap and not listed), [NII] 6548/6583, and [SII] 6716/6731. \nThe emission line properties are derived from the Gaussian line profile fitting. The average line properties including FWHM, Equivalent Width, and line luminosities are measured from the combined spectrum (Extended Data Table 2). The shifts of the line centers were also measured for individual spectra, with the barycentric correction computed using the rvsao package in IRAF as listed in Extended Data Table 1 for each spectrum. It was found that the line shifts, after barycentric correction, are consistent with being constant for narrow emission lines over all observations at 230 ± 15 km/s, consistent with the radial velocity of 241 ± 2 km/s for the face-on M101. However, the broad helium emission lines, as measured with the strongest He II 4686 line, shifted from observation to observation between 210 km/s and 330 km/s as listed in Extended Data Table 1, with an average of 270 km/s that is significantly different from that for nebular lines. \nThe properties of the nebular lines help to determine the environmental metallicity and the neutral hydrogen column density. N 2 ≡ [NII] λ 6583 /H α can be used as an abundance indicator 37 with 12 + log(O/H) = 8.90 + 0.57 × N 2 , albeit with a large dispersion in log(O/H) of ± 0 . 41 . Given the equivalent width of these two lines (Extended Data Table 2), we find 12 + log(O/H) = 8.70, close to solar metallicity (8.66). This is higher than but marginally consistent with the value of 0 . 4 × solar according to the M101 gas-phase oxygen abundance gradient 33 given the location of ULX-1. The observed Balmer line flux ratios can be used to infer the dust extinction between \nthe nebula and the observer. In the nebular emission around ULX-1, the intrinsic ratio H α /H β is 2.74 in case B for a thermal temperature of T=20,000K 38 . Assuming E(B-V) = 0.1 mag, then A 6564 = 0 . 250 mag, A 4863 = 0 . 360 mag, ∆ A = 0.11 mag, ∆ H α / ∆ H β = 1 . 1 , and reddened H α /H β ∼ 3 . The observed H α /H β is 2.85, suggesting that the extinction is low, and using the Galactic value is reasonable.", '3 ULX-1 is a Wolf-Rayet binary of the WN subclass': 'The broad helium emission lines in the newly obtained Gemini/GMOS spectrum are typical of an extremely hot, hydrogen depleted Wolf-Rayet star. Accretion disks around a compact object can also give rise to broad helium emission lines, but broad Balmer line is expected to be present and much stronger than the helium lines. Indeed, broad H β emission lines are present in two ULXs with optical spectra (4000-5400 ˚ A), NGC1313 X-2 14 and NGC 5408 X-1 15 , and are stronger than the He II 4686 emission line. In the ULX-1 spectrum (Figure 1), while the Balmer emission lines are present, they are narrow emission lines like forbidden lines, and should come from the surrounding nebulae, as evidenced by their nearly constant line shifts from observation to observation, in distinct contrast to helium lines with line shift difference of ± 60 km/s. \nThe sub-type of this Wolf-Rayet star can be determined from the presence or absence of line species in the spectrum 18 . There are two main types of Wolf-Rayet stars, WN stars with R ∼ 5 -12 R /circledot revealing H-burning products and subsequently more compact WC stars with R ∼ 2 -3 R /circledot revealing He-burning products. WC stars are dominant by carbon lines (such as \nCIII 4650, CIII 5696 and CIV 5812) that are stronger than helium lines, but none of the carbon lines are present in the ULX-1 spectrum. WN stars from WN4 to WN8 show 39 increasing absolute magnitudes M V from -3.5 mag to -6 mag, increasing mass loss rates from 10 -5 to 10 -4 M /circledot /yr , decreasing effective temperatures from 80 kK to 45 kK, hence increasing fraction of HeI atoms relative to HeII ions. Comparing the observed spectrum to the spectral atlas of WN stars 18, 40 , we estimate a late-type WN8 star. A WN8 subtype is also inferred based on the HeI 5876/He I 5411 equivalent width ratio 41 . Such a subtype is roughly consistent with its absolute magnitude of M V = - 5.9 mag (after extinction correction using Galactic E(B-V) = 0.1 mag and R V = 3.1), and the effective temperature of about 50 kK derived from its broad-band spectral energy distribution 8 .', '4 Physical parameters for the Wolf-Rayet star': "As for the case of NGC 300 X-1 22 , we have calculated synthetic models using the line-blanketed, non-local thermodynamic equilibrium model atmosphere code 16 . To select the best physical parameters of the WR star, we compare the model equivalent width with observed values for the six helium emission lines and minimize the quantity ∆ 2 = ∑ i (EW -EW i ) 2 . In all model calculations, elemental abundances are set to 40 percent of the solar value for the metallicity of 0 . 4 Z /circledot at the location of ULX-1. We vary the stellar radius R ∗ between 4 to 20 R /circledot , stellar mass M ∗ between 5 -35 M /circledot , stellar luminosity L ∗ between 5 -100 × 10 4 L /circledot , the outer radius for line-forming region R MAX up to 40 R /circledot , the terminal velocity v ∞ between 400 -2000 km/s, and the stellar wind mass loss rate ˙ M ∗ between 5 -100 × 10 -6 M /circledot /yr . \nWe have run ∼ 5000 models with the combination of stellar mass, radius and luminosity determined by the stellar evolution tracks 42 of Z = 0 . 4 Z /circledot for all possible WN stars, and another ∼ 5000 models with 'fake' stars whose mass, radius and luminosity are completely independent of each other. After a total of ∼ 10000 model evaluations, a best fitting model is found with R ∗ = 10 . 7 R /circledot , M ∗ = 17 . 5 M /circledot , L ∗ = 5 . 4 × 10 5 L /circledot , v ∞ = 1300 km/s, R MAX = 22 R /circledot , and ˙ M ∗ = 2 . 0 × 10 -5 M /circledot /yr . The model reproduces the helium emission lines extremely well (Extended Data Table 2), with an average difference of | ∆ | = 0.6 ˚ A. In comparison, the majority of models and all models with 'fake' stellar parameters are much worse-fitting with ∆ 2 >> 10 (Extended Data Figure 2). Based on the ∆ 2 distribution, our model evaluations picked up the stellar parameters effectively, and we estimate, with equivalently ∆ χ 2 = 1 , the errors to be ˙ M ∗ = 2 ± 0 . 5 × 10 -5 M /circledot /yr , and v ∞ = 1300 ± 100 km/s. Note that, if we adopt a solar metalicity, as allowed by the abundance indicator N 2 ≡ [NII] λ 6583 /H α , the best model will change to R ∗ = 11 . 1 R /circledot , M ∗ = 17 . 5 M /circledot , L ∗ = 4 . 9 × 10 5 L /circledot , v ∞ = 1700 km/s, and ˙ M ∗ = 2 . 4 × 10 -5 M /circledot /yr . This is consistent with the 0 . 4 Z /circledot results within the errors except for a significantly higher terminal velocity. \nThe stellar parameters of this best model belong to a 'real' WN star from the stellar evolution tracks, with an effective temperature of 48 kK, an initial mass of 42 M /circledot , an age of about 5 Myrs, and a remaining lifetime of about 0.3 Myrs before it loses another ∼ 6 M /circledot and collapses into a black hole of ∼ 12 M /circledot . This model is actually one of the best models derived from studies of the optical spectral energy distribution 8 . Comparing to the physical properties of WR stars in the Milky Way 18 , we find that T ∗ , L ∗ , ˙ M ∗ , and v ∞ are consistent with those for a WN7/WN8 star. The \nabsolute magnitude M V for ULX-1 ( M V = -5 . 9 mag after extinction correction) is brighter by 0.5 mag, fully within the spread of absolute magnitudes for WN subtypes. \nThe mass of the WR star can be more reliably estimated with the the empirical massluminosity relation 17, 18 as done for NGC 300 X-1 22 . In our case, L ∗ = 5 . 4 × 10 5 L /circledot , this corresponds to a WR mass of 19 M /circledot , quite consistent with the mass for the best model. The luminosity derived for solar metalicity will correspond to a WR mass of 18 M /circledot . Hereafter we will use 19 M /circledot for the WR mass, with an estimated formal error of 1 M /circledot to roughly reflect the difference between the model value and the empirical value. Given the stellar mass and radius of 10 . 7 R /circledot , we can obtain the orbital period 43 as P = √ ρ/ 110hr /similarequal 72hr if the WR star is filling its Roche lobe. The true orbital period will be longer than 72 hrs if the WR star is only filling part of its Roche lobe.", '5 Search for orbital periodicity': 'The radial velocity changes between 210 km/s and 330 km/s as measured by the HeII 4686 emission line should reflect the orbital motion of the WR star. While broad HeII 4686 emission line can be produced from the X-ray heated accretion disk in some ULXs with rather high X-ray luminosities (e.g., in NGC 1313 X-2 with ∼ 10 40 erg/s 14 ), this should not be the case for M101 ULX-1 because its X-ray luminosities during the Gemini/GMOS observations were three orders of magnitude lower, and the disk heating effects are insignificant even in its outburst based on the optical studies 8 . In addition, the line ratios for the heated accretion disk are different from the line ratios for the WR star because the emission line forming regions and temperature structures are \nquite different, yet the observed line ratios can be well reproduced by the WR star. \nIn order to search for the orbital periodicity, we assume a circular orbit and fit a since curve v r = v 0 + K sin[2 π ( t -t 1 ) /P + φ ] to nine barycenter-corrected radial velocities; the radial velocity for March 17th was dropped from the analysis because the spectrum had a very low signal-tonoise ratio. The four parameters are the radial velocity of the binary mass center v 0 , the radial velocity semi-amplitude K , the orbital period P , and phase φ at the first observation. The search is carried out by minimizing χ 2 defined as χ 2 = ∑ 10 i =1 [ v r ( t i ) -v r,i ] 2 /σ 2 v r i . The radial velocity errors σ v r i are taken as the wavelength calibration error of 0.25 ˚ A, or 15 km/s. The five radial velocity measurements from May 13th to May 19th suggest a period no longer than 10 days (Figure 2). The amoeba technique is used for χ 2 minimization, using initial guesses taken from the parameter grids with P from 3 to 10 days in step of 0.01 days, K from 20 to 150 km/s in step of 5 km/s, and φ from 0 · to 360 · in step of 10 · . The best solution is found at the minimum χ 2 = 1 . 6 , for which the best period P = 8 . 24 ± 0 . 1 days and the best radial velocity semi-amplitude K = 61 ± 5 km/s, with the 68.3% error determined with ∆ χ 2 = 1 . The fact that the radial velocity curve can be fitted with a sine curve suggests that the orbital eccentricity is small. \nGiven P and K , the mass function for M101 ULX-1 can be computed as f ( M ∗ , M · , i ) = PK 3 2 πG = M 3 · ( M · + M ∗ ) 2 sin 3 i = 0 . 178 M /circledot . This sets an absolute lower limit for the mass of the primary. In the case of ULX-1, more information can be extracted because we already know M ∗ = 19 M /circledot . Given the equation M 3 · ( M · + M ∗ ) 2 sin 3 i = 0 . 178 M /circledot , the primary mass will increase monotonically when the inclination angle decreases, i.e., changing from edge-on ( i = 90 · ) toward face-on ( i = \n0 · ). Thus the minimum mass for the primary can be obtained when i = 90 · , which is M · = 4 . 6 M /circledot after solving the equation M 3 · ( M · + M ∗ ) 2 = 0 . 178 M /circledot . The minimum mass will be M · = 4 . 4 M /circledot if we use M ∗ = 17 . 5 M /circledot . Such a compact primary can only be a black hole. This is thus the dynamical evidence for a black hole in a ULX.', '6 The Wolf-Rayet/black hole binary properties': "This section duplicates some text from the main article, but with additional technical details. \nM101 ULX-1 is thus a Wolf-Rayet/black hole binary, only the third discovered so far after IC 10 X-1 and NGC300 X-1. The binary separation can be computed with Kepler's Law a 3 = G ( M ∗ + M · ) 4 π 2 P 2 , which increases monotonically for increasing black hole mass, starting from a = 50 R /circledot for M · = 4 . 6 M /circledot to a = 75 R /circledot for M · = 60 M /circledot (Extended Data Figure 3). The Roche lobe size for the secondary can be computed with R cr = a · f ( q ) = a · 0 . 49 q 2 / 3 / [0 . 6 q 2 / 3 + ln (1 + q 1 / 3 )] with q = M ∗ /M · , and the Roche lobe size for the black hole can be computed with the same formula but with different q = M · /M ∗ . As shown in Extended Data Figure 3, the Roche lobe size for the black hole increases with the increasing black hole mass, but the Roche lobe size for the secondary does not change much, from R cr, ∗ = 25 R /circledot for M · = 4 . 6 M /circledot to R cr, ∗ = 23 R /circledot for M · = 10 M /circledot , and to R cr, ∗ = 22 R /circledot for M · = 20 M /circledot . \nRegardless of the black hole mass, the secondary is filling only half of its Roche lobe by radius, and the black hole must be accreting from the Wolf-Rayet star winds. Since the black hole is at least 50 R /circledot away from the WR star, the stellar wind must have reached close to its \nterminal velocity. The capture radius for the wind accretion can be computed as r acc = 2 GM · v 2 ∞ , and the accretion rate can be computed as ˙ M · = πr 2 acc 4 πa 2 ˙ M ∗ . Given that the average luminosity for M101 ULX-1 is about 3 × 10 38 erg/s, the required accretion rate is ˙ M · = L/ηc 2 /similarequal L 38 η 2 × 10 -9 M /circledot /yr = 1 η 6 × 10 -9 M /circledot /yr . To capture this much stellar wind matter, as shown in Extended Data Figure 4, the black hole mass must be greater than 46 M /circledot for η = 0 . 06 in the case of a nonspinning Schwarzschild black hole, and greater than 13 M /circledot for η = 0 . 42 in the case of a maximally spinning Kerr black hole. If we use the velocity law v ( r ) = v ∞ (1 -R ∗ /r ) β with β = 1 for the inner wind 16 , then the black hole mass must be greater than 28 M /circledot for η = 0 . 06 in the case of a non-spinning Schwarzschild black hole, and greater than 8 M /circledot for η = 0 . 42 in the case of a maximally spinning Kerr black hole. If we adopt a typical η value of 0.1, the required accretion rate corresponds to M · > 24 M /circledot (and i < 17 · ) for a wind velocity of v /similarequal 1100 km s -1 , and corresponds to M · > 32 M /circledot (and i < 14 · ) for the terminal velocity. The accretion rate argument thus requires a black hole of > 8 -46 M /circledot , likely a black hole of 20 -30 M /circledot similar to IC 10 X-1 and NGC 300 X-1. \nThe recurring X-ray/optical outbursts dictates the presence of an accretion disk prone to instability, and the disk formation under stellar wind accretion places stringent constraints on the binary system. To explore why the number of Galactic X-ray stars is so small, it has been shown 44 that in the case of accretion of stellar wind matter in a detached binary system the specific angular momentum of the matter captured by the compact object is typically small. Therefore, usually no accretion disk is formed around the compact object. Consequently, very special conditions are required for a black hole in a detached binary system to be a strong X-ray source. A disk may form \nif the specific angular momentum of accreting matter, Q acc = 1 4 2 π P r 2 acc , exceeds the specific angular momentum of the particle at the innermost stable circular orbit, Q ISCO = √ 3 r g c = √ 3 2 GM · c 2 c . This is usually expressed as P < 4 . 8 M · /M /circledot v 4 1000 δ 2 hr, where δ ∼ 1 is a dimensionless parameter 19, 45 . Given P = 8 . 24 ± 0 . 1 days and v ∞ = 1300 ± 100 km s -1 for M101 ULX-1, the black hole mass is required to be M · > 80 M /circledot , corresponding to i = 9 · (i.e., nearly face-on). If the wind velocity from the velocity model of the inner wind 16 is adopted, then the black hole mass is required to be M · > 48 M /circledot , corresponding to i = 11 · . \nTo investigate the possible presence of partial ionization zone, we need to compute the temperature structure T d ( r ) for the accretion disk, especially for the outer disk. Following the procedures designed for an X-ray irradiated black hole binary model for ULXs 46 , we compute the disk temperature structure for a standard accretion disk with the α prescription 47 plus X-ray irradiation 43 . As shown in Extended Data Figure 5, regardless of the black hole mass for M101 ULX-1, its outer disk temperature is as low as 4000K in the low-hard state due to its large separation and large disk, and the helium partial ionization zone at about 15000K is bound to exists unless the black hole mass is lower than 5.5 M /circledot . In comparison, the disk temperature for NGC 300 X-1, with an orbital period of 32.8 hrs and its WN5 star ( M ∗ = 26 M /circledot , R ∗ = 7 . 2 R /circledot ) filling its Roche lobe 22 , never drops below 20000K due to its small separation and small disk, and there is no helium partial ionization zone in the disk. This explains naturally why NGC 300 X-1 and similar IC 10 X-1 exhibit steady X-ray radiation despite the apparent variations due to orbital modulation under the edge-on viewing geometry. \nThe existence of an accretion disk in M101 ULX-1 is also supported by the observed spectral state changes, which resemble those for Galactic black hole binaries 9, 11 that are believed to reflect changes in the properties of their accretion disks 10 . During its outbursts, M101 ULX-1 exhibits an X-ray spectrum 4, 7 that can be classified as a thermal dominant state (albeit with exceptionally low disk temperatures), a well-defined spectral state that corresponds to a standard thin accretion disk at about 10% of its Eddington luminosity. Quantitative studies 23 show that when the luminosity exceeds 30% of the Eddington limit, the emission changes such that the X-ray spectrum includes a steep power-law with a significant hard component above 2 keV. The presence of such a hard component is not seen in the X-ray spectra of M101 ULX-1. Given its bolometric luminosity of 3 × 10 39 erg s -1 in the thermal dominant state at less than 30% of its Eddington limit, we infer that the black hole mass is above 80 M /circledot . If this is true, the inferred black hole mass of M101 ULX-1 may challenge the expectations of current black hole formation theories. The most massive black holes that can be produced for solar metallicity are about 15 M /circledot , and about 20 M /circledot ( 25 M /circledot , 30 M /circledot ) for 0 . 6 × ( 0 . 4 × , 0 . 3 × ) solar metallicity due to reduced stellar winds and hence reduced mass loss in the final stages before stellar collapse 48 . \n- 31. Freedman, W. et al. Final results from the Hubble Space Telescope key project to measure the Hubble Constant, Astrophys. J. 553 , 47-72 (2001)\n- 32. Kong, A. K. H., Rupen, M. P., Sjouwerman, L. O. & Di Stefano, R. Proceedings Papers of the 22nd Texas Symposium on Relativistic Astrophysics at Stanford. (Stanford Univ. 2005)\n- 33. Bresolin, F. The oxygen abundance in the inner H II regions of M101: implications for the \ncalibration of strong-line metallicity indicators. Astrophys. J. 656 , 186-197 (2007) \n- 34. Liu, J. F. Chandra ACIS survey of X-ray point sources in 383 nearby galaxies. I. The source catalog. Astrophys. J. 192 (suppl.). 10-64 (2011)\n- 35. Kong, A. K. H. & Di Stefano, R. An unusual spectral state of an ultraluminous very soft X-ray source during outburst. Astrophys. J. 632 , L107-L110 (2005)\n- 36. Kuntz, K. D. et al. The optical counterpart of M101 ULX-1. Astrophys. J. 620 , L31-L34 (2005)\n- 37. Pettini, M. & Pagel, B. E. J. [OIII]/[NII] as an abundance indicator at high redshift. Mon. Not. R. Astron. Soc. 348 , L59-L63 (2004)\n- 38. Osterbrock, D. Astrophysics of Gaseous Nebulae and Active Galactic Nuclei. (University Science Books 1989)\n- 39. Hamann, W. R., Koesterke, L. & Wessolowski, U. Spectra analysis of the Galactic Wolf-Rayet stars - a comprehensive study of the WN class. Astron. Astrophys. 274 , 397-414 (1993)\n- 40. Crowther, P. A. & Hadfield, L. J. Reduced Wolf-Rayet line luminosities at low metallicity. Astron. Astrophys. 449 , 711-722 (2006)\n- 41. Smith, L. F., Shara, M. M. & Moffat, A. F. J. A three-dimensional classification for WN stars. Mon. Not. R. Astron. Soc. 281 , 163-191 (1996)\n- 42. Girardi, L. et al. Theoretical isochrones in several photometric systems. I. Johnson-CousinsGlass, HST/WFPC2, HST/NICMOS, Washington, and ESO imaging survey filter sets. Astron. Astrophys. 391 , 195-212 (2002) \n- 43. Frank, J., King, A. & Raine, D. Accretion Power in Astrophysics. (Cambridge Univ. Press, 2002)\n- 44. Illarionov, A. F. & Sunyaev, R. A. Why the number of Galactic X-ray stars is so small? Astron. Astrophys. 39 , 185-195 (1975)\n- 45. Ergma, E. & Yungelson, L. R. CYG X-3: can the compact object be a black hole? Astron. Astrophys. 333 , 151-158 (1998)\n- 46. Liu J. F., Orosz, J. & Bregman, J. N. Dynamical mass constraints on the ultraluminous X-ray source NGC 1313 X-2. Astrophys. J. 745 , 89-110 (2012)\n- 47. Shakura, N. I. & Sunyaev, R. A. Black holes in binary systems. Observational appearance. Astron. Astrophys. 24 , 337-355 (1973)\n- 48. Belczynski, K. et al. On the maximum mass of stellar black holes. Astrophys. J. 714 , 12171226 (2010) \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nTable 1: Gemini/GMOS spectroscopic observations of M101 ULX-1 \n| OBSDATE | MJD | exposure (second) | bary. (km/s) | velocity (km) |\n|------------|---------|---------------------|----------------|-----------------|\n| 2010-02-15 | 55242.6 | 3200 | 7.4 | 212 |\n| 2010-02-16 | 55243.5 | 3200 | 7.3 | 236 |\n| 2010-03-16 | 55271.5 | 3200 | 0.1 | 301 |\n| 2010-03-17 | 55272.5 | 3200 | -0.2 | - |\n| 2010-04-17 | 55303.5 | 4800 | -7.7 | 326 |\n| 2010-05-13 | 55329.3 | 6400 | -12.2 | 302 |\n| 2010-05-14 | 55330.4 | 6400 | -12.4 | 256 |\n| 2010-05-15 | 55331.4 | 6400 | -12.5 | 227 |\n| 2010-05-18 | 55334.4 | 9600 | -13 | 244 |\n| 2010-05-19 | 55335.4 | 9600 | -13.1 | 305 | \nThe columns are: (1) Observation date, (2) Modified Julian Date, (3) exposure time in seconds, (4) barycentric correction computed with rvsao , and (5) the corrected radial velocity as measured with HeII 4686, with an error of 15 km/s as mainly from the uncertainties in the wavelength calibration. \nTable 2: Properties of emission lines \n| Line | FWHM E.W. | | Lum. | model |\n|---------------|-------------|--------------|-------------|---------|\n| ID | ( ˚ A) | ( ˚ A) | 10 34 erg/s | ( ˚ A) |\n| HeII 4686 | 19.3 | 21.83 ± 0.20 | 43 | 21.75 |\n| HeI 5876 | 19.0 | 34.78 ± 0.29 | 49 | 34.21 |\n| HeI 6679 | 18.8 | 25.74 ± 0.37 | 24 | 26.56 |\n| HeII 5411 | 20.5 | 5.46 ± 0.13 | 8.3 | 6.10 |\n| HeI 4922 | 13.4 | 5.80 ± 0.64 | 8.4 | 3.91 |\n| HeI 4471 | 12.1 | 3.86 ± 0.65 | 7.0 | 5.18 |\n| H γ | 3.6 | 1.35 ± 0.22 | 2.7 | |\n| H β | 4.5 | 7.51 ± 0.06 | 12 | |\n| H α | 4.7 | 26.54 ± 0.46 | 34 | |\n| [ OIII ] 4960 | 4.4 | 23.70 ± 0.49 | 40 | |\n| [ NII ] 6548 | 3.8 | 3.85 ± 0.39 | 4.7 | |\n| [ NII ] 6583 | 4.7 | 16.66 ± 0.08 | 18 | |\n| [ SII ] 6716 | 4.0 | 4.58 ± 0.07 | 4.0 | |\n| [ SII ] 6731 | 4.6 | 3.81 ± 0.06 | 3.1 | | \nThe columns are: (1) emission line ID, (2) FWHM as obtained from Gaussian fit, which equals to 2 . 35 σ , (3) equivalent width, (4) line luminosity in unit of 10 34 erg/s, and (5) equivalent width from the best WR synthetic model. \nFigure 1 M101 ULX-1 as observed in the optical. (a) M101 ULX-1 is located on a spiral arm of the face-on grand-design spiral galaxy M101, as indicated by the arrow. The color image of M101 is composed of GALEX NUV, SDSS g, and 2MASS J images. (b) ULX-1 is identified as a blue object with V=23.5 mag at the center of the 1 '' circle on the HST image. The color image is composed of ACS/WFC F435W, F555W and F814W images. \nFigure 2 Physical properties of the WR secondary from spectral line modeling. Distributions of computed ∆ 2 as a function of (a) stellar masses, (b) stellar mass loss rate, (c) stellar radii, and (d) terminal velocity. Here ∆ 2 = ∑ i (EW -EW i ) 2 computes the difference between observed and synthetic equivalent widths for six broad helium lines present in the Gemini/GMOS spectrum. We have computed synthetic spectra for a group of 5000 real stars from the evolution tracks (as shown by the thick stripes in the mass plot and the radius plot) and for another group of 'fake' stars with continuous distributions in mass, radius and luminosity. The best model is labeled by a filled pentagon in all panels.", 'Figure 3 Properties for the Wolf-Rayet/black hole binary for different black hole masses.': 'Properties for the Wolf-Rayet/black hole binary for different black hole masses. Shown are the binary separation (solid), the Roche lobe sizes for the Wolf-Rayet star (dotted) and for the black hole (dashed), the capture radius for the black hole when using the terminal velocity (dash-dotted) or when using a simplified velocity law v ( r ) = v ∞ (1 -R ∗ /r ) (long-short dashed). \nFigure 4 The black hole accretion rate for different black hole mass. The black hole accretion rate for different black hole mass if adopting the terminal velocity (dotted) or a simplified velocity law v ( r ) = v ∞ (1 -R ∗ /r ) (solid). To power the observed average luminosity of 3 × 10 38 erg/s, the black hole mass must exceed 13 M /circledot ( 8 M /circledot ) using the terminal velocity (the velocity law) for a Kerr black hole, and exceed 46 M /circledot ( 28 M /circledot ) for a Schwarzschild black hole. \nFigure 5 Disk temperature structures for M101 ULX-1. (a) The disk temperature profiles for M101 ULX-1 (for P = 8 . 24days , M ∗ = 19 M /circledot , R ∗ = 10 . 7 R /circledot , M · = 10 / 100 M /circledot ) and NGC300 X-1 (for P = 32 . 4hr , M ∗ = 26 M ∗ , R ∗ = 7 . 2 R /circledot , M · = 16 . 9 M /circledot ; Crowther et al. 2010). (b) The disk temperature at the outer edge for different black hole mass in M101 ULX-1.'}
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2018ApJ...863....1C
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A Population of Bona Fide Intermediate-mass Black Holes Identified as Low-luminosity Active Galactic Nuclei
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2018-01-01
| 34 | 0.55 | 160 |
['cosmology observations', 'cosmology early universe', 'galaxies active', 'galaxies nuclei', 'galaxies seyfert', 'galaxies quasars', '-', '-', '-']
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[]
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Nearly every massive galaxy harbors a supermassive black hole (SMBH) in its nucleus. SMBH masses are millions to billions of solar mass, and they correlate with properties of spheroids of their host galaxies. While the SMBH growth channels, mergers, and gas accretion are well established, their origin remains uncertain: they could have emerged either from massive “seeds” (10<SUP>5</SUP>-10<SUP>6</SUP> M<SUB>⊙</SUB>) formed by direct collapse of gas clouds in the early universe or from smaller (100 M<SUB>⊙</SUB>) BHs, end products of first stars. The latter channel would leave behind numerous intermediate-mass BHs (IMBHs, 10<SUP>2</SUP>-10<SUP>5</SUP> M<SUB>⊙</SUB>). Although many IMBH candidates have been identified, none are accepted as definitive; thus, their very existence is still debated. Using data mining in wide-field sky surveys and applying dedicated analysis to archival and follow-up optical spectra, we identified a sample of 305 IMBH candidates having masses 3× {10}<SUP>4</SUP> {M}<SUB>⊙ </SUB>< {M}<SUB>BH</SUB>}< 2× {10}<SUP>5</SUP> {M}<SUB>⊙ </SUB>, which reside in galaxy centers and are accreting gas that creates characteristic signatures of a type I active galactic nucleus (AGN). We confirmed the AGN nature of 10 sources (including five previously known objects that validate our method) by detecting the X-ray emission from their accretion disks, thus defining the first bona fide sample of IMBHs in galactic nuclei. All IMBH host galaxies possess small bulges and sit on the low-mass extension of the {M}<SUB>BH</SUB>}{--}{M}<SUB>bulge</SUB>} scaling relation, suggesting that they must have experienced very few if any major mergers over their lifetime. The very existence of nuclear IMBHs supports the stellar-mass seed scenario of the massive BH formation.
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https://arxiv.org/pdf/1805.01467.pdf
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{'No Header': "Draft version May 7, 2018 \nTypeset using L A T E X twocolumn style in AASTeX62 \nA Population of Bona Fide Intermediate Mass Black Holes Identified as Low Luminosity Active Galactic Nuclei \nIgor V. Chilingarian, 1, 2 Ivan Yu. Katkov, 2 Ivan Yu. Zolotukhin, 2, 3, 4 Kirill A. Grishin, 2, 5 Yuri Beletsky, 6 Konstantina Boutsia, 6 and David J. Osip 6 \n1 Smithsonian Astrophysical Observatory, 60 Garden St. MS09, Cambridge, MA 02138, USA 2 Sternberg Astronomical Institute, M.V.Lomonosov Moscow State University, Universitetsky prospect 13, Moscow, 119992, Russia 3 Special Astrophysical Observatory of the Russian Academy of Sciences, Nizhnij Arkhyz 369167, Russia 4 Universit'e de Toulouse; UPS-OMP, IRAP, 9 avenue du Colonel Roche, BP 44346, F-31028 Toulouse Cedex 4, France 5 Department of Physics, M.V. Lomonosov Moscow State University, 1 Vorobyovy Gory, Moscow, 119991, Russia 6 Las Campanas Observatory, Carnegie Institution of Washington, Colina el Pino, Casilla 601 La Serena, Chile \n(Received April 26, 2018; Revised -; Accepted -) \nSubmitted to ApJ", 'ABSTRACT': "Nearly every massive galaxy harbors a supermassive black hole (SMBH) in its nucleus. SMBH masses are millions to billions M glyph[circledot] , and they correlate with properties of spheroids of their host galaxies. While the SMBH growth channels, mergers and gas accretion, are well established, their origin remains uncertain: they could have either emerged from massive 'seeds' (10 5 -10 6 M glyph[circledot] ) formed by direct collapse of gas clouds in the early Universe or from smaller (100 M glyph[circledot] ) black holes, end-products of first stars. The latter channel would leave behind numerous intermediate mass black holes (IMBHs, 10 2 -10 5 M glyph[circledot] ). Although many IMBH candidates have been identified, none is accepted as definitive, thus their very existence is still debated. Using data mining in wide-field sky surveys and applying dedicated analysis to archival and follow-up optical spectra, we identified a sample of 305 IMBH candidates having masses 3 × 10 4 < M BH < 2 × 10 5 M glyph[circledot] , which reside in galaxy centers and are accreting gas that creates characteristic signatures of a type I active galactic nucleus (AGN). We confirmed the AGN nature of ten sources (including five previously known objects which validate our method) by detecting the X-ray emission from their accretion discs, thus defining the first bona fide sample of IMBHs in galactic nuclei. All IMBH host galaxies possess small bulges and sit on the lowmass extension of the M BH -M bulge scaling relation suggesting that they must have experienced very few if any major mergers over their lifetime. The very existence of nuclear IMBHs supports the stellar mass seed scenario of the massive black hole formation. \nKeywords: cosmology: observations - early universe - galaxies: active - galaxies: nuclei - galaxies: Seyfert - quasars: supermassive black holes", '1. INTRODUCTION AND MOTIVATION': "The existence of stellar mass black holes (Abbott et al. 2016) and giant black holes millions to billions times more massive than the Sun is observationally established (Schodel et al. 2002; Miyoshi et al. 1995). Massive black holes in galaxy centers are believed to co-evolve with spheroids of their hosts (Kormendy & Ho 2013), grow via coalescences during galaxy mergers (Merritt \nCorresponding author: Igor Chilingarian [email protected], [email protected] \n& Milosavljevi'c 2005) and by accreting gas during the AGN/quasar phase (Volonteri 2012), however, their origin still remains unclear: they could have either emerged from massive 'seeds' (10 5 -10 6 M glyph[circledot] ) formed by direct collapse of large gas clouds in the early Universe (Loeb & Rasio 1994) or from smaller (100 M glyph[circledot] ) black holes, end-products of first stars (Madau & Rees 2001), which must have also created a rich, yet undetected population of intermediate mass black holes (IMBHs, 10 2 10 5 M glyph[circledot] ). \nTwo observational phenomena allow us to detect and estimate masses of central black holes in large samples of galaxies: (i) dynamic signatures observed as high rota- \n- \ntional velocities or velocity dispersions of stars and gas in circumnuclear regions of galaxies (Miyoshi et al. 1995; Kormendy & Richstone 1995; Seth et al. 2014); (ii) AGN and quasars, which appear when a massive black hole is caught while accreting gas (Elvis 2000) and, hence, growing. The discovery of quasars in the early Universe ( z > 6 . 3, that is 750-900 Myr after the Big Bang) hosting super-massive black holes (SMBHs) as heavy as 10 10 M glyph[circledot] (Mortlock et al. 2011; Wu et al. 2015) cannot be explained by gas accretion on stellar mass black hole seeds ( ∼ < 100 M glyph[circledot] ) alone. Even if formed right after the Big Bang by the first generation of Population iii stars, it would take over 1 Gyr to foster an SMBH because the accretion rate cannot significantly exceed the Eddington limit during extended periods of time. Population iii stars might have formed in dense clusters in primordial density fluctuations, which could then evolve into more massive SMBH seeds by collisions and/or core collapse (Portegies Zwart et al. 2004). Alternatively, the rapid inflow and subsequent direct collapse of gas clouds (Loeb & Rasio 1994; Begelman et al. 2006) can form massive seeds ( M > 10 5 -10 6 M glyph[circledot] ). The latter scenario solves the SMBH early formation puzzle but leads to a gap in the present-day black hole mass function in the IMBH regime (100 ∼ < M BH ∼ < 1 × 10 5 M glyph[circledot] ), whereas stellar mass seeds should leave behind a large number of IMBHs. Therefore, the elusive IMBH population holds a clue to the understanding of SMBH formation. \nThe first evidence for the existence of IMBHs came in late 80s, when two dwarf galaxies with stellar masses of about 10 9 M glyph[circledot] hosting AGN were identified: Pox 52, a dwarf elliptical galaxy (Kunth et al. 1987; Barth et al. 2004) and NGC 4395, a low-luminosity spiral (Filippenko & Sargent 1989; Wrobel & Ho 2006). They both host central black holes with the mass estimates around 3 × 10 5 M glyph[circledot] (Filippenko & Ho 2003; Peterson et al. 2005; Thornton et al. 2008) and are nowadays considered too massive to be called IMBH, however, they ignited the interest towards search for less massive examples. \nDespite substantial observing time investments over the past two decades, only a few IMBH candidates were identified with reliable mass estimates: (i) the serendipitously discovered hyperluminous X-ray source HLX1 in a nearby galaxy (Webb et al. 2012) having the mass between 10 4 and 10 5 M glyph[circledot] estimated from the Xray flux and radio emission of the relativistic jet, which, however, may be a stellar mass black hole accreting in the super-critical regime with a beamed X-ray radiation along the line-of-sight (King & Lasota 2014); (ii) a 4,000 M glyph[circledot] IMBH in the globular cluster 47 Tuc detected using stellar dynamics and pulsar timing (Kızıltan et al. 2017), although one has to keep in mind that several \npast claims of IMBHs in globular clusters (Noyola et al. 2010) were refuted by subsequent analysis (Zocchi et al. 2017); (iii) several low luminosity AGN in dwarf galaxies found using optical spectra and later confirmed in X-ray (Dong et al. 2007; Reines et al. 2013; Baldassare et al. 2015) (5 × 10 4 < M BH < 3 × 10 5 M glyph[circledot] ) but the search approach excluded luminous galaxies and involved substantial amount of manual data analysis applied on a per object basis. Here we call 'reliable' the black hole mass estimates techniques, which have calibration uncertainties of at most a factor of 2-3, such as reverberation mapping, stellar dynamics, pulsar timing, X-ray variability, broad H α scaling. We do not consider IMBH candidates relying on some average Eddington ratios in AGN, M BH -σ bulge relation or the fundamental plane of the black hole activity, which have intrinsic uncertainties of 0.8-1.5 dex. \nIn this paper we present the results of the first systematic search for IMBHs in AGN without applying a priori pre-selection filtering criteria to the input galaxy sample. We developed an automated workflow that analyzed 1 million galaxies spectra from the SDSS Data Release 7 (DR7) spectroscopic catalog and measured central BH masses for those objects that demonstrate broad H α line and narrow-line photoionization signatures of accreting BHs. Throughout this work we define an IMBH as an object in the mass range between 10 2 and 10 5 M glyph[circledot] and we look for IMBH candidates having a mass < 2 × 10 5 M glyph[circledot] because of the internal precision of the virial mass estimate of about ∼ 0 . 3 dex.", '2.1. An automated IMBH search workflow': 'An AGN creates specific signatures in an optical spectrum of a galaxy (Elvis 2000): X-ray photons from the corona of an accretion disc around the black hole ionize gas out to a few kiloparsec away, which then produces easily detectable emission lines (Fig. 1). The width and the flux of an allowed recombination line (e.g. hydrogen H α ) emitted from the broad line region in the immediate vicinity of a central black hole provide a virial estimate of its mass (Greene & Ho 2005; Reines et al. 2013). We have designed an automated workflow that uses data mining in optical and X-ray astronomical data archives publicly available in the international Virtual Observatory to search for AGN signatures of IMBHs. \nThe workflow automatically analyzed (Fig. 1) about 1,000,000 optical spectra of galaxies and quasars from the legacy sample of the Sloan Digital Sky Survey Data Release 7 (SDSS DR7, Abazajian et al. 2009) without any pre-selection on host galaxy luminosity or redshift. We used a non-parametric representation of a narrow \n× \nemission line profile (Chilingarian et al. 2017), which produced lower fitting residuals in Balmer lines and allowed us to detect fainter broad line components, thus boosting the sensitivity of our analysis technique. Then we took the resulting sample of galaxies with broad line detections, computed emission line flux ratios [O iii ]/H β and [N ii ]/H α from narrow-line components, and used the Baldwin-Phillips-Terlevich (BPT) (Baldwin et al. 1981) diagnostics to reject objects where the ionization was induced by star formation, because such objects often have broad Balmer lines originating from transient stellar events (Baldassare et al. 2016) rather than from AGN. After that, we eliminated statistically insignificant measurements by filtering the sample on relative strengths and widths of narrow and broad line components, signal-to-noise ratios, relative radial velocity offsets, and assembled a list of 305 candidates in the IMBH mass range ( M BH < 2 10 5 M glyph[circledot] ). \nThis list (further referred as the parent sample ) included all known nuclear IMBH candidates (Dong et al. 2007; Reines et al. 2013; Baldassare et al. 2015) except those which fell on the star forming sequence in the BPT diagram; our black hole mass estimates agreed within uncertainties with those obtained from dedicated deep spectroscopic observations (Reines et al. 2013; Baldassare et al. 2015). Then we searched in data archives of Chandra , XMM-Newton , and Swift orbital X-ray observatories and detected 10 X-ray counterparts for candidates with virial masses as low as 4 . 3 × 10 4 M glyph[circledot] . Five of them were mentioned in the literature (Reines et al. 2013; Baldassare et al. 2015) and one object had a spatially extended X-ray emission probably related to star formation which we excluded from further analysis. All 4 remaining X-ray point sources were observed serendipitously.', '2.2. Non-parametric emission line analysis': 'We developed a dedicated technique to analyze optical spectra, which allows us to estimate a central black hole mass by measuring a broad component in allowed emission lines (Greene & Ho 2005). As an input dataset for our study we use galaxy spectra from SDSS DR7, which we re-analyzed and presented in the value-added catalog RCSED (Chilingarian et al. 2017). We extended RCSED by adding AGN spectra classified as quasars in SDSS. This input sample contains 938,487 galaxy spectra of 878,138 unique objects. We analyzed the follow-up optical spectra obtained with the 6.5-m Magellan telescope in the same fashion. We fitted and subtracted stellar continuum in SDSS spectra using the NBursts full spectrum fitting technique (Chilingarian et al. 2007b,a) and then measured emission lines. \nThe core of our analysis method is a simultaneous fitting of all strong emission lines (H β , [O iii ], [N ii ], H α , [S II ]) by a linear combination of a narrow line component having a non-parametric shape and a broad Gaussian component in the Balmer lines. The broad-line component parameters, velocity dispersion ( σ BLR ) and the central radial velocity of the BLR component are fitted in a non-linear minimization loop. All other parameters are fitted linearly within it at every evaluation of the function using an iterative procedure that includes the following two steps: (i) we determine fluxes of all emission line components solving a linear problem with the non-negative constrain; (ii) then we recover the shape of the narrow line component in a non-parametric way by solving a linear convolution problem with the regularization, which requires a smoothness of a solution. By using a non-parametric NLR component shape, we can successfully model complex gas kinematics and avoid the degeneracy, which affects the traditionally used multiple Gaussian profile decomposition (Greene & Ho 2005; Reines et al. 2013), because Gaussian functions are not orthogonal and, therefore, do not form a basis. We then repeated our analysis by excluding a BLR component from the model in order to compare the χ 2 values between the two approaches and conclude whether adding a BLR component improved the fitting quality in a statistically significant way.', '2.3. Constructing the sample: IMBH selection criteria': "Having obtained the flux and the width of the H α broad component in all 938,487 spectra of the input sample, we use the conservative empirical calibration to estimate a virial black hole mass (Reines et al. 2013): \nM BH = 3 . 72 × 10 6 (FWHM H α / 10 3 kms -1 ) 2 . 06 × ( L H α / 10 42 erg s -1 ) 0 . 47 M glyph[circledot] (1) \nBy comparing a broad H α based virial estimates with other black hole mass measurement techniques (e.g. reverberation, stellar dynamics) it was demonstrated (Dong et al. 2012) that they agree within 0.3 dex, i.e. a factor of 2. One, however, has to keep in mind that this uncertainty also includes statistical and systematic errors of black hole mass measurements used in the calibration, which might significantly contribute to the 0.3 dex error budget. We use this as a rough estimate of the systematic uncertainty of our method, which also defines the mass range of our search: M BH < 2 × 10 5 M glyph[circledot] . \n× We then apply multiple selection criteria in order to filter reliable IMBH candidates from the input sample and eliminate spurious broad line detections: \nFigure 1. A black hole mass determination in AGN from optical spectra. Top row: A black hole with an accretion disc ionizes the interstellar medium in its host galaxy. Dense gas clouds in the immediate vicinity of the black hole (0.001-0.1 pc; broad line region or BLR) are virialized and move at velocities up-to thousands km/s, thus broadening recombination lines originating from allowed transitions. Rarefied gas clouds further away from the black hole ( ∼ < 1 kpc; narrow line region or NLR) move much slower (up-to hundreds km/s) and emit also in forbidden transitions, however, the narrow line shape depends on the exact NLR morphology. Middle and bottom rows: We model the stellar content of a galaxy by fitting its observed spectrum against a grid of stellar population models; then fit emission line residuals, first by using the same non-parametric shape for all detected lines and then by adding Gaussian broad line components in the hydrogen Balmer lines. If the fitting results differ significantly, we estimate the virial black hole mass from the broad line component width and luminosity using the calibration by Reines et al. (2013). \n<!-- image --> \n- · M BH < 2 × 10 5 M glyph[circledot] in order to select objects in the IMBH mass range given the assumed 0.3 dex systematic uncertainty.\n- · No night sky airglow lines falling in the regions around H α +[N ii ], [O iii ] 5007 ˚ A, and H β 4861 ˚ A, which we use for the spectral line profile fitting and decomposition.\n- · Empirical constraint that the width of the broad line component is at least √ 5 times larger than that of the narrow line component.\n- · Signal-to-noise ratio exceeding 3 in every emission line used in the BPT (Baldwin et al. 1981) classi- \ncation (H β , [O iii ], [N ii ], H α ), which ensures its reliability. \n- · The BPT classification is 'AGN' or 'composite' (Kewley et al. 2006), that discards star-forming galaxies because broad line components in them are often transient (Baldassare et al. 2016).\n- · The H α /H β Balmer decrement for both narrow and broad line components < 4.\n- · Statistical error on M BH better than 33%\n- · | v BLR -v NLR | < 3 σ NLR to reject strongly asymmetric BLR profiles. \nThese criteria joined together with boolean and form our main selection filter. It leaves a sample of 305 IMBH \ncandidates out of nearly 1 million input spectra. We call it the parent sample .", '3. FOLLOW-UP OBSERVATIONS AND ANALYSIS OF NEW AND ARCHIVAL DATA': 'In order to exclude possible transient phenomena such as core collapse supernova or tidal disruption events, we followed up 3 galaxies with X-ray counterparts, 4 targets selected for our X-ray observations, and 5 additional IMBH candidates (12 targets in total) using the intermediate resolution Magellan Echellette Spectrograph (MagE) at the 6.5-m Magellan Baade telescope (see Table 1). We processed MagE spectra through our data analysis technique and obtained independent second-epoch IMBH mass estimates consistent within uncertainties with SDSS (see Table 2) for 8 galaxies. We did not detect a broad H α component in 3 objects. \nThe observed flux in the forbidden oxygen line [O iii ] ( λ = 5007 ˚ A) in AGN correlates with the X-ray luminosity L X (Heckman et al. 2005a), because the NLR is ionized by energetic photons originating from the active nucleus. Using this correlation, we selected 4 IMBH candidates with estimated X-ray fluxes > 5 × 10 -15 erg cm -2 s -1 which can be detected in a 10,000 s exposure for follow-up X-ray observations. We obtained a solid confirmation for one source using Chandra ( M BH = 1 . 2 × 10 5 M glyph[circledot] , MagE) and a low-confidence detection for another source using XMM-Newton ( M BH = 7 . 5 × 10 4 M glyph[circledot] , SDSS, no broad H α component detected with MagE). The two remaining objects were not detected in X-ray suggesting that either we observed them in a low phase of activity and they fell below the [O iii ]-L X correlation or that the broad lines were due to transient phenomena. Finally, we ended up with a sample of 10 bona fide broad-line AGN with virial black hole masses between 43,000 and 202,000 M glyph[circledot] estimated from SDSS spectra having point source X-ray counterparts positioned at galaxy centers (Fig. 2). \nOne object from the final sample, SDSS J171409.04+584906.2, has archival Hubble Space Telescope images. We observed 4 new confirmed IMBH host galaxies and 6 additional candidates with the Magellan Baade telescope using the FourStar near-infrared imaging camera in the K s photometric band ( λ = 2 . 2 µ m). We performed a light profile decomposition of IMBH host galaxy images, computed the luminosities of the spheroidal components and converted them into stellar masses using published ages and metallicities from RCSED.', '3.1. Optical and NIR observations': "We carried out follow-up imaging and spectroscopic observations of several IMBH candidates and their host \ngalaxies in the optical and near-infrared domains using the 6.5-m Magellan Baade telescope, Las Campanas Observatory, Chile. \nOur primary goal was to obtain quasi-simultaneous optical spectroscopy of the IMBH galaxies selected for follow-up X-ray observations using Chandra and XMMNewton within a period of 2-6 weeks between observations. Our secondary goal was to obtain the second spectroscopic epoch for several prominent X-ray confirmed IMBHs and get an independent black hole mass estimates. Finally, we aimed to take advantage of superior seeing conditions at the Magellan telescope to obtain near-infrared images of several IMBH host galaxies with the spatial resolution 0.5-0.7 arcsec crucial for the analysis of structural properties, that is 2-3 times better than the resolution of SDSS images. The complete log of our follow-up observations for confirmed IMBHs is provided in Table 1. \nFor our spectroscopic observations, we used the Magellan Echellette spectrograph (Marshall et al. 2008) with the 10-arcsec long 0.7 arcsec wide slit that provides a cross-dispersion spectroscopy with the spectral resolving power λ/ ∆ λ = 6500 or σ inst = 20 km s -1 in 14 spectral orders covering the wavelength range 0 . 3 < λ < 1 . 0 µ m. Each object was integrated for 40 min to 1 h 20 min in individual 20 min long exposures either along the major or minor axis of its host galaxy. Objects which were small enough to fit in the slit ( < 5 arcsec) were observed along the minor axis and the sky model was constructed from the 'empty' part of the slit. Larger galaxies which did not fit in a 10 arcsec slit were observed along the major axis; then we used offset sky observations of 5 min re-normalized in flux to match science observations. \nWe reduced the data using a dedicated MagE data reduction pipeline, which we developed. The pipeline builds a wavelength solution with uncertainties as small as 2 km s -1 ; merges echelle orders and creates a flux calibrated sky subtracted merged 2D spectrum, which is then fed to the NBursts spectral fitting procedure to subtract the stellar continuum and then to the emission line analysis procedure. \nThe pipeline uses standard stars observed shortly before or after a science source to perform flux calibration and telluric correction. However, in order to perform an extra check and eliminate possible systematic flux calibration errors, we compare our reduced spectra to SDSS and use stellar continuum to perform independent flux calibration. We first extract a spectrum in a 3 × 0.7 arcsec box and use a published light profile of a each galaxy (Simard et al. 2011) in order to calculate the expected flux difference between the circular 3 arcsec SDSS fiber aperture and the box extraction. Then, we calibrate a \nTable 1. The log of follow-up observations of confirmed IMBHs and their host galaxies. \n| Object | Instrument | Date | Exp. time (s) | Seeing (arcsec) |\n|-----------------------|--------------|------------|-----------------|-------------------|\n| J122732.18+075747.7 | MagE | 10/07/2017 | 3600 | 1.2 |\n| | MagE | 06/07/2017 | 2400 | 1.3 |\n| J110731.23+134712.8 | FourStar | 10/07/2017 | 466 | 0.77 |\n| | Chandra | 17/07/2017 | 9960 | n/a |\n| J134244.41+053056.1 | MagE | 30/05/2017 | 4800 | 1.5 |\n| J134244.41+053056.1 | FourStar | 09/07/2017 | 384 | 0.53 |\n| J022849.51 - 090153.8 | MagE | 01/01/2018 | 3600 | 0.9 |\n| J022849.51 - 090153.8 | FourStar | 01/01/2018 | 384 | 0.7 | \nreduced 2D spectrum using that flux ratio, and perform an optimal extraction of a nuclear point source using the value of the image quality reported by the guider in order to estimate an extraction profile, because the BLR in an AGN is supposed to stay unresolved. At the end, we apply a flux correction computed for a point source observed through a 0.7 arcsec wide slit to the extracted spectrum. This approach yields a flux calibrated spectrum of the galaxy nucleus that can be directly compared to SDSS. \nFor our imaging observations we used the FourStar camera (Persson et al. 2013) that covers a field of view of 11 × 11 arcmin with a mosaic containing 4 Hawaii2RG detectors. We observed each IMBH host galaxy from a subsample selected for imaging in the K s band with the total on-source integration of 8 min. We used the Poisson random dithering pattern with a box size of 52 arcsec in order to provide enough background sampling for flat fielding and background subtraction. We reduced FourStar images using the fsred data reduction pipeline that performs pre-processing of raw NIR images obtained in the fowler2 mode, that is 2 readouts in the beginning and 2 at the end of each exposure; flat fielding; background subtraction; and flux calibration using 2MASS (Skrutskie et al. 2006) sources inside the field of view. The final result of the pipeline is a sky subtracted flux calibrated image and its flux uncertainties. \nFor one galaxy, SDSS J1714+5849, we used archival Hubble Space Telescope images in the F814W photometric band downloaded from the Hubble Legacy Archive (http://hla.stsci.edu/; dataset JA2S0M010). For SDSS J1227+0757, which we did not observe with FourStar, we used Pan-STARRS archival images (Chambers et al. 2016) with sub-arcsec seeing quality. \nIn order to check the position of all our candidates on the M BH -M bulge scaling relation, we analyzed imaging data for the candidate IMBH host galaxies. For all of them but one we have used a two-dimensional decomposition using the galfit v.3 software (Peng et al. 2010). For one galaxy, SDSS J1714+5849, which harbors a strong stellar bar, we used instead a one-dimensional decomposition of a light profile extracted using the ellipse task in noao iraf . For SDSS J1342+0530, SDSS J1107+1347, and SDSS J0228 -0901 the followup imaging data taken on FourStar has been used. We used psfextractor (Bertin 2011) to extract the point spread function convolution kernel for every image. Usually we found the best-fitting solution with the simple photometric model 'bulge+disk'. In case of compact bulges being limited by the atmospheric seeing quality we had to model a bulge using a point source. Then, apparent magnitudes of bulges were translated into luminosities using the WMAP9 cosmology (Hinshaw et al. 2013) and later converted into stellar masses using stellar population properties provided in the RCSED (Chilingarian et al. 2017).", '3.2. X-ray data and observations analysis': "We performed X-ray observations of 2 objects with Chandra (observations 20114 and 20115), and 2 objects with XMM-Newton (observations 0795711301 and 0795711401) using director's discretionary time quasisimultaneously with optical observations. \nBoth Chandra observations were carried out with the Advanced CCD Imaging Spectrometer (ACIS) detector in the faint data mode with 10,000 s long exposures. Target galaxies were always placed on-axis of the backilluminated S3 chip of ACIS-S. The data were reduced and analyzed with ciao 4.9 package following standard recipes. \nIn Chandra observation 20114 a single bright pointlike X-ray source was detected at the position of the optical center of SDSS J1107+1347 galaxy. We performed its aperture photometry using srcflux task and detected 518 net counts in 0.5-7 keV band which corresponds to the observed flux (4 . 5 ± 0 . 4) × 10 -13 erg s -1 cm -2 . \nNo source was detected at the position of SDSS J135750.71+223100.8 in the Chandra observation 20115. We estimate a 3 σ detection limit of this observation as 8 . 0 10 -15 erg s -1 cm -2 . \nTwo XMM-Newton observations 10,000 s long each were performed with EPIC detector in FullFrame mode with Thin filter. The data were reduced and analyzed with the common XMM-Newton analysis threads with the sas 16.1.0 software package running in a virtual machine. \n× \nWe were not able to detect any source at a position of SDSS J161251.77+110621.6 in the XMM-Newton observation 0795711301 up-to the limiting flux level of 5 . 0 × 10 -15 erg s -1 cm -2 . \n× However, we marginally detected a faint source coincident within positional errors with the nucleus of SDSS J1440+1155 galaxy in the XMM-Newton observation 0795711401. We estimate its flux in the standard XMM-Newton 0.2-10 keV band as (5 ± 2) × 10 -15 erg s -1 cm -2 . \nFor other sources in this study (including previously known objects) we used X-ray data from XMMNewton 's 3XMM-DR7 catalog (Rosen et al. 2016) accessible through the catalog website (Zolotukhin et al. 2017, http://xmm-catalog.irap.omp.eu), Chandra Source Catalog Release 1.1 (Evans et al. 2010) and Swift 1SXPS catalog (Evans et al. 2014). We performed a cross-match with point non-spurious subset of sources in those catalogs using their 3 σ X-ray positional uncertainties.", '4.1. Detected IMBH candidates and their properties': "Using data mining in wide-field sky surveys and applying dedicated analysis to archival and follow-up optical spectra, we identified a sample of 305 IMBH candidates having masses 3 × 10 4 < M BH < 2 × 10 5 M glyph[circledot] , which reside in galaxy centers and are accreting gas that creates characteristic signatures of a type-I AGN. We confirmed the AGN nature of ten sources (including five previously known (Dong et al. 2007; Reines et al. 2013; Baldassare et al. 2015)) by detecting the X-ray emission from their accretion discs, thus defining the first bona fide sample of IMBHs in galactic nuclei. The very existence of nuclear IMBHs supports the stellar mass seed scenario of the massive black hole formation. \nIn Table 2 we present main properties of 10 IMBHs confirmed as AGN by the X-ray identification and their host galaxies. Every object is identified by the IAU designation, which includes its J2000 coordinates. For every source we present a central BH virial mass estimate, flux and width of the broad H α component, redshift, X-ray flux, and an estimated stellar mass of a spheroidal component. For host galaxies of 5 newly detected sources presented in the top part of the table, we also provide estimates of the absolute magnitude of the bulge or spheroid obtained from the photometric decomposition of their direct images. For the confirmed sources from the literature (bottom part of the table) we also provide published BH mass estimates. In Fig. 36 we present SDSS and MagE (when available) spectra and line profile decomposition results for these 10 objects. \nHere we briefly describe properties of bona fide IMBHs detected in X-ray for the first time: \n- · J122732.18+075747.7: The least massive IMBH ( M BH = 3 . 6 × 10 4 M glyph[circledot] ) detected by our workflow hosted in a barred spiral galaxy with a starforming ring; the X-ray counterpart is very faint. The BPT diagnostics places a galaxy in the composite region because the AGN emission is heavily contaminated by star formation in the inner ring, that becomes less of a problem in MagE data where it is spatially resolved.\n- · J134244.41+053056.1: A particularly interesting source, which we matched with the Swift source 1SXPS J134244.6+053052. Dou et al. (2016) classified it as a tidal disruption event (TDE) candidate based on the variability of highly ionized iron lines in its optical spectrum. The claim is that the TDE must have happened close to the SDSS spectrum epoch (2002-04-09) which is, however, in clear contradiction with the hypothesis that its Xray emission with the luminosity 1 . 3 × 10 41 erg s -1 observed by Swift 7 years later on 2009-05-15 is connected to the TDE. Therefore we attribute the detected X-ray source to the AGN activity in SDSS J1342+0530.\n- · J171409.04+584906.2: Hosted in a barred spiral with a compact bulge well resolved in archival HST images, this IMBH is another example of a source falling into the composite region in the BPT diagram. Because of high declination we were unable to obtain the second epoch spectroscopy.\n- · J111552.01 -000436.1: Located in a nearly edgeon spiral galaxy with a compact bulge, this is an- \nFigure 2. Optical images of ten IMBH host galaxies. Sloan Digital Sky Survey images of galaxies hosting IMBHs detected by our automated data analysis workflow demonstrate low luminosity spheroidal stellar systems or spiral galaxies with small bulges. The locations of X-ray counterparts with the corresponding 3 σ positional uncertainties is shown by red circles. The bottom row displays objects mentioned in the literature previously, which our workflow has successfully recovered. A virial mass estimate in M glyph[circledot] from the analysis of SDSS spectra is shown in the bottom left corner of every panel followed by an estimate from MagE when available, the physical scale in the host galaxy plane is in the bottom right. \n<!-- image --> \nFigure 3. Spectral decomposition of MagE and SDSS data for IMBHs detected in this work. Top row panel: The observed optical SDSS spectrum is shown in blue, the orange line is the best-fitting stellar population template without emission lines. Middle row: A close-up view of several emission lines in the SDSS spectrum. The emission line profile (observed data are shown in black) is constructed first by subtracting the best-fitting stellar population template, then in allowed lines it is decomposed into narrow line (gray) and broad line (red) components. The total emission line model is shown in magenta and its residuals are displayed in gray shifted downward for clarity. A non-parametric narrow line model (rightmost panel, blue histogram) is computed simultaneously for all forbidden emission lines in the spectrum that reduces fitting residuals and allows us to detect even a very faint broad line component (rightmost panel, red line), a proxy for the BH mass. Bottom row: Same as the middle row but for MagE data. \n<!-- image --> \nTable 2. IMBHs identified as AGN and some of their properties. \n| Object | M BH (10 3 M glyph[circledot] ) | Lit. M BH (10 3 M glyph[circledot] ) | σ BLR (km s - 1 ) | L BLR H α (10 39 erg s - 1 ) | z | M sph abs (mag) | M ∗ sph (10 9 M glyph[circledot] ) | L X (10 40 erg s - 1 ) |\n|-----------------------|-----------------------------------|----------------------------------------|---------------------|--------------------------------|------------------|-------------------|--------------------------------------|-----------------------------|\n| This work | This work | This work | This work | This work | This work | This work | This work | This work |\n| J122732.18+075747.7 | 43 ± 10 1 36 ± 7 2 | | 214 ± 20 200 ± 10 | 1 . 5 ± 0 . 4 1 . 4 ± 0 . 4 | 0.033 | - 17.8 (r) | 0.9 | 0.55 ( XMM ) |\n| J134244.41+053056.1 | 65 ± 7 1 96 ± 13 2 | | 216 ± 10 286 ± 13 | 3 . 5 ± 0 . 4 2 . 4 ± 0 . 5 | 0.037 | - 20.7 (r) | 3.5 | 13.5 ( Swift ) |\n| J171409.04+584906.2 | 115 ± 24 1 | | 373 ± 31 | 1 . 1 ± 0 . 3 | 0.030 | - 17.4 (F814W) | 0.7 | 2.5 ( XMM ) |\n| J111552.01 - 000436.1 | 115 ± 38 1 | | 315 ± 41 | 2 . 3 ± 0 . 9 | 0.039 | - 16.8 (r) | 0.4 | 4.9 ( XMM ) |\n| J110731.23+134712.8 | 122 ± 18 1 71 ± 10 2 | | 269 ± 17 244 ± 10 | 5 . 1 ± 0 . 8 2 . 5 ± 0 . 6 | 0.045 | - 18.0 (K) | 0.3 | 190 ( Chandra ) glyph[star] |\n| Previously known | Previously known | Previously known | Previously known | Previously known | Previously known | Previously known | Previously known | Previously known |\n| J152304.97+114553.6 a | 70 ± 20 1 | 50 | 350 ± 30 | 0 . 5 ± 0 . 2 | 0.024 | | 0.7 | 0.4 ( Chandra ) a |\n| J153425.58+040806.7 b | 111 ± 7 1 | 130 | 246 ± 6 | 6 . 2 ± 0 . 3 | 0.039 | | 1.3 | 85 ( Chandra ) d |\n| J160531.84+174826.1 b | 116 ± 11 1 | 160 | 316 ± 12 | 2 . 3 ± 0 . 2 | 0.032 | | 1.7 | 12.7 ( XMM ) |\n| J112333.56+671109.9 c | 157 ± 36 1 | 200 | 341 ± 34 | 3 . 1 ± 0 . 6 | 0.055 | | 2.4 | 53.5 ( XMM ) |\n| J022849.51-090153.8 c | 202 ± 13 1 367 ± 27 2 | 316 | 250 ± 7 340 ± 9 | 21 ± 1 19 ± 2 | 0.072 | | 0.7 | 275 ( Chandra ) d | \n- 1 Spectrum from SDSS\n- 2 Spectrum from Magellan/MagE \na Baldassare et al. (2015) \nb Reines et al. (2013) \nc Greene & Ho (2007) \n- d Dong et al. (2012) \nNote -The upper part of the table lists objects found in this work, while 5 objects in the bottom part were known previously (see references near object names) but had their properties re-measured using our data analysis approach. The X-ray luminosity is computed in this work from the flux reported in a corresponding X-ray catalog unless a reference is given. An asterisk marks dedicated Chandra X-ray observations from this study. The columns are: SDSS IAU name, black hole mass (derived in this study from SDSS data and from Magellan/MagE data where available), black hole mass from the literature (where applicable; see reference near object name), redshift, H α BLR velocity dispersion (from SDSS and from MagE data where available), H α BLR luminosity (from SDSS and from MagE data where available), absolute magnitude and mass of spheroidal component (bulge) of a host galaxy, X-ray luminosity. \nTable 3. A list of 305 candidate IMBHs identified as active galactic nuclei based on the SDSS archival data. \n| Object | z | M BH (10 3 M glyph[circledot] ) | σ BLR (km s - 1 ) | L BLR H α (10 39 erg s - 1 ) | M sph abs (10 9 M glyph[circledot] ) |\n|-----------------------|-------|-----------------------------------|---------------------|--------------------------------|----------------------------------------|\n| J111835.82+002511.2 | 0.025 | 138 ± 20 | 230 ± 13 | 13.24 ± 2.07 | 1.87 ± 0.04 |\n| J112209.97+010114.8 | 0.075 | 99 ± 19 | 216 ± 14 | 86.3 ± 2.51 | 1.94 ± 0.3 |\n| J141215.60 - 003759.0 | 0.025 | 62 ± 17 | 269 ± 30 | 1.24 ± 0.39 | 0.46 ± 0.05 |\n| J094733.06+001302.9 | 0.063 | 181 ± 39 | 322 ± 30 | 5.47 ± 1.15 | 10.65 ± 2.50 |\n| J003826.70+000536.8 | 0.071 | 100 ± 22 | 257 ± 25 | 4.06 ± 0.87 | n/a | \nNote -This table is published in its entirety in the machine-readable format. A random subset is shown here for guidance regarding its form and content. \nother example of a weak AGN whose signature is contaminated by star formation in its host galaxy. \n- · J110731.23+134712.8: This IMBH located in a low luminosity disk galaxy with a very compact bulge is the only one of five falling in the AGN region of the BPT diagram despite its contamination by star formation. This object has a very bright X-ray counterpart detected in our Chandra dataset, that corresponds to the X-ray luminosity alone over 10% of the Eddington limit for a 70,000 M glyph[circledot] black hole, which suggests that the bolometric luminosity should be close to the Eddington limit. \nWe also mention one object previously described in the literature, J022849.51 -090153.8 (Greene & Ho 2007). Its black hole mass estimate from the follow-up spectroscopy with MagE, 3.7 ± 0.3 · 10 5 M glyph[circledot] puts it above the IMBH mass threshold adopted in this work, however, similarly to J110731.23+134712.8 it also exhibits very bright X-ray emission, that corresponds to the bolometric luminosity close to the Eddington limit for its mass. \nTable 3 contains properties of all 305 sources selected as IMBH candidates ( the parent sample ) regardless of the availability of X-ray data. The columns are similar to Table 2 with the exception of X-ray identification and literature data. We used bulge luminosities in the r band reported in the photometric catalog by Simard et al. (2011) in the 'Sersic+disk' decomposition table, which we converted into stellar masses using mass-tolight ratios of SDSS galaxies presented in Saulder et al. (2016). \n4.2. The IMBH mass detection limit and reliability of M BH estimates \nWe studied the behavior of our non-parametric emission line analysis algorithm with Monte-Carlo simulations. For each object in our final sample of 10 X-ray confirmed IMBHs we generated a set of 90,000 mock emission spectra, which included all strong optical emission lines (H β , [O iii ], [N ii ], H α , [S II ]). To generate synthetic forbidden lines in these spectra we took their model profiles derived from the narrow line LOSVD, which was recovered at the first pass of the algorithm, and added random noise with the distribution derived from the observed spectrum noise in the vicinity of a line. For allowed lines we also added broad Gaussian components with a random Balmer decrement in the range 2.8-3.2, whose widths σ and luminosities L H α were distributed in a grid to cover the region of interest in the parameter space. At each point of this grid we generated 100 spectra with random noise realizations which were fed to the non-parametric emission line analysis algorithm. M BH recovered by the algorithm was then compared to the true input value used. We considered an individual trial successful in case the recovered black hole mass fell within 0.3 dex from the input value, and computed recovery rate at each point of ( σ , L H α ) grid as a fraction of successful to total number of trials. \nThe maps of recovery rate for MagE and SDSS spectra of objects from the final sample of 10 IMBHs are presented in Fig. 7. In almost all cases the derived black holes masses lie in the region with reliable recovery rate. This modeling also shows that under favorable circumstances our non-parametric emission line analysis algorithm could recover from SDSS spectra black hole masses as low as 10 4 M glyph[circledot] .", '4.3. Contamination estimate of the parent sample of IMBH candidates': 'We estimate the contamination of the parent sample of IMBH candidates produced by our method using sev- \nFigure 3. contd. \n<!-- image --> \neral approaches. By contamination here we mean the fraction of sources in the sample that are not actual IMBHs, that is they do not exhibit all the required observed IMBH properties: persistent broad H α emission, X-ray emission from an accretion disk and AGN or composite emission lines ratio in the BPT diagram. By construction, our parent sample contains sources with single-epoch spectroscopic mass estimate without X-ray confirmation for most of them. Hence, it inevitably contains sources that e.g. showed IMBH features at some point in time and then changed their appearance. The \ncontaminating sources likely have diverse origin. These could be supernovae, transient stellar processes (Reines et al. 2013; Baldassare et al. 2016) or spurious detections caused by the imperfection of our spectral analysis. \nIt is generally very hard to precisely estimate the contamination, so here for simplicity we derive an upper limit of the contamination, i.e. the most pessimistic estimate of the quality of our IMBH search method. It requires that bona fide IMBH candidates satisfy the most stringent observing constraints: they must have an Xray detection and consistent multi-epoch spectroscopic \nFigure 4. Spectral decomposition of SDSS data for IMBHs detected in this work. Panels are the same as top and middle rows in Figure 3. \n<!-- image --> \nmass estimates (more precisely, we require that broad H α emission is detected at different epochs possibly with different instruments and all its detections satisfy our quality criteria, the mass estimates at different epochs are consistent within 0.3 dex, and the BPT classification does not change). \nFirst we checked our parent sample against the 3XMM-DR5 upper limit server (http://www.ledas.ac. uk/flix/flix.html) and found 14 objects that serendipitously fell in the footprint of archival XMM-Newton observations but were not detected in them. We compared detection limits of these observations with the fluxes expected from the 14 IMBH candidates given existing L X -L [OIII] correlation (Heckman et al. 2005b). None of these observations were deep enough to exclude X-ray emission at the level of L X -L [OIII] correlation minus its 1 σ scatter. We, therefore, cannot reject the accreting IMBH hypothesis for these objects. Given that the objects of our interest are all low-mass AGN with small luminosities, we expect that other X-ray archives are unlikely to contain many deep enough observations of our IMBH candidates. \nHence, out of 305 IMBH candidates in our parent sample only 18 possess sufficiently deep X-ray observations to confirm or rule out the accreting IMBH hypothesis: 14 from X-ray archives and 4 IMBH candidates observed with dedicated X-ray observations by Chandra and XMM-Newton in this work. Out of those 18, two sources (SDSS J161251.77+110621.6 observed by XMMNewton and SDSS J135750.71+223100.8 observed by Chandra ) were not detected in X-ray at the level below of that expected from L X -L [OIII] correlation minus its 1 σ scatter which secures their non-IMBH nature. One source, SDSS J144005.82+115508.7, while detected in X-ray with XMM-Newton in our observation, did not display broad H α emission at the second spectroscopic epoch when observed with Magellan/MagE. In addition to this, we discarded five sources without performing the second epoch spectroscopic follow-up observations considering them spurious detections. This happens, for example, when our automated emission line decomposition algorithm underestimates the broad H α emission flux and after more careful emission line decomposition with manually adjusted constraints, a black hole mass \nFigure 5. Spectral decomposition of MagE and SDSS data for a previously known IMBH re-measured in this work. Panels are the same as in Figure 3. \n<!-- image --> \nestimate exceeds 2 × 10 5 M glyph[circledot] . Thus, out of 18 sources with deep X-ray data we discard eight sources in total. 10 remaining sources are those presented in Table 2. For six sources from Table 2 we have both X-ray and second epoch spectroscopic confirmation (objects found in this work: SDSS J1227+0757, SDSS J1342+0530, SDSS J1115 -0004, SDSS J1107+1347; previously known objects: SDSS J1523+1145, SDSS J0228 -0901). Four remaining sources from Table 2 (object found in this work: SDSS J1714+5849; previously known objects: SDSS J1534+0408, SDSS J1605+1748, SDSS J1123+6711) are detected in X-ray and have reliable single epoch detections of broad H α emission but do not possess second epoch spectroscopic observations and therefore cannot be used for the contamination estimate. \nThis leaves us with a sample of 14 sources that have enough evidence to tell if they pass all required tests (multi-epoch spectroscopy and deep enough X-ray observations) or not: 6 sources successfully pass them all, and 8 sources fail in at least one test. The upper limit on the contamination of our parent sample can be estimated as 8 / 14 = 57%. A more realistic estimate should lower this value. In particular, it was shown (Baldassare et al. 2016) that 100% of objects classified as AGN on the BPT diagram which at the same time show the broad H α emission, exhibit the same properties in second spectroscopic epoch. Out of 4 objects without the second spectroscopic epoch, one (SDSS J1605+1748, Dong et al. (2007)) is classified as AGN on the BPT diagram. It is then natural to anticipate that it is true \nIMBH which would lower the contaminating fraction in our sample to 53%. \nTherefore, we have all evidence to expect that at least 0 . 43 × 305 = 131 sources in our parent sample of IMBH candidates are real IMBHs in a sense that they will satisfy the most stringent observing criteria once the necessary follow-up observations have been performed. If we assume no strong dependence of the contamination level on the black hole mass, we find that at least 42 of our IMBH candidates from Table 3 with M BH < 10 5 M glyph[circledot] must be actual IMBHs.', '4.4. Implications for co-evolution of central black holes and their host galaxies': "In Fig. 8 we compare IMBH masses and host galaxy properties to the recent compilations of bulge/spheroid masses of host galaxies of massive black holes (Graham & Scott 2015; Graham et al. 2015). All IMBH hosts have stellar masses of their bulges between 4 × 10 8 and 4 × 10 9 M glyph[circledot] and reside on the low-mass extension of the M BH -M bulge scaling relation established by more massive black holes and their host galaxies, thus filling a sparsely populated region of the parameter space. This argues for the validity of the search approach that looks for AGN signatures created by IMBHs and also supports the connection between the black hole mass growth and the growth of their host galaxy bulges via mergers down to the IMBH regime. \nGalaxy mergers were frequent when the Universe was younger (redshifts z > 1, Conselice et al. 2003; Bell et al. \nFigure 6. Spectral decomposition of SDSS data for previously known IMBHs re-measured in this work. Panels are the same as in Figure 4. \n<!-- image --> \n2006; Lotz et al. 2011). They are thought to be responsible for the growth of bulges (Aguerri et al. 2001; BoylanKolchin et al. 2006), hence suggesting the co-evolution of central black holes and their hosts (Kormendy & Ho 2013), and explaining the observed correlations between central black hole masses and bulge properties, stellar velocity dispersion (Ferrarese & Merritt 2000; Gebhardt et al. 2000; van den Bosch 2016) and stellar mass (Marconi & Hunt 2003; Haring & Rix 2004; Gultekin et al. \n2009). Therefore, IMBH host galaxies must have experienced very few if any major mergers over their lifetime. \nFrom the multiple epoch optical spectroscopy and Xray observations, we estimate that our IMBH candidate sample may include up-to 57% of transient broad lines or spurious detections (see a detailed discussion above). Even though, keeping in mind that virial masses are uncertain to a factor of 2, it should contain at least 42 objects with masses smaller than 10 5 M glyph[circledot] . These ob- \nFigure 6. contd. \n<!-- image --> \nFigure 7. Monte-Carlo simulations of the emission line analysis. Black lines show equal black hole masses of 10 4 , 3 × 10 4 , 6 × 10 4 , 9 × 10 4 , 1 . 2 × 10 5 , 1 . 5 × 10 5 , 2 × 10 5 solar masses. Color indicates the recovery fraction of the black hole mass determination in every point of the grid derived from 100 random noise realizations. The source of the spectral data is MagE if indicated on top of the panel or SDSS otherwise. The panels are in the same order as in Fig. 2. \n<!-- image --> \njects are the relics of the early SMBH formation survived through the cosmic time almost intact and their host galaxies must have had very poor merger histories. Their existence suggests that at least some SMBHs did not originate from massive ( > 10 5 M glyph[circledot] ) seeds but from stellar mass black holes. The efficiency of mass growth via super-Eddington accretion is questionable because of radiation driven outflows (King & Muldrew 2016). Therefore, stellar mass black hole mergers must have played an important role in the SMBH assembly. \nOur sample likely represents the tip of the iceberg of the IMBH population. The sphere of influence of an IMBH is too small to significantly affect the stellar dynamics of its host galaxy and cannot be detected beyond the Local Group with currently available instruments, \ntherefore we can find only IMBHs in the actively accreting phase, which requires the gas supply onto the galaxy centre. On the other hand, most non-starforming galaxies with small bulges reside in galaxy clusters, which does not favour AGN, because they lost their gas completely due to environmental effects. Therefore, while the exact fraction of actively accreting IMBHs is unknown, it is likely smaller than that of more massive black holes. \nThe main limitations of our technique are the relatively shallow flux limit of the SDSS spectroscopy and the lack of wide field X-ray surveys reaching the flux limit (5 × 10 -15 erg cm -2 s -1 ) of a snapshot Chandra or XMM-Newton observation: 4 serendipitously detected sources reside in glyph[similarequal] 2 % of the sky observed by both \nFigure 8. A scaling relation between the central black hole mass M BH and the mass of a host galaxy bulge/spheroid M bulge . Masses of quiescent and active black holes and stellar masses of bulges of their host galaxies (Graham & Scott 2015; Graham et al. 2015) show a strong correlation for galaxies having different morphological types, which supports the scenario of their co-evolution (Kormendy & Ho 2013). The gray dashed line is a power-law fit of the relation for host galaxies with S'ersic light profiles (Graham & Scott 2015). The X-ray confirmed IMBHs and their hosts (see Fig. 2 for their images) are shown as large green ( HST and FourStar observations) and red (literature) star symbols; IMBH candidates without X-ray confirmation ( FourStar observations) as small black stars. They extend the correlation to lower masses: its continuity suggests that the nuclear IMBHs represent the low mass extension of the mass function of central black holes in galaxies. \n<!-- image --> \nSDSS and X-ray satellites. The future deep eRosita X-ray survey may confirm several dozen IMBH candidates from our current sample of 305. A targeted optical spectroscopic probe of nearby galaxies with small bulges deeper than SDSS will likely bring new IMBH identifications at even lower masses. \nIC, IK, IZ, and KG acknowledge the support by the Russian Scientific Foundation grant 17-72-20119 that was used in the final stages of this work and the manuscript preparation. IZ acknowledges the Russian Scientific Foundation grant 14-50-00043 for the development of the pipeline system used for input catalog assembly. The authors acknowledge partial support from the M.V.Lomonosov Moscow State University Program of Development, and a Russian-French PICS International Laboratory program (no. 6590) co-funded by the RFBR (project 15-52-15050), entitled 'Galaxy evolution mechanisms in the Local Universe and at intermediate redshifts'. The authors are grateful to citizen scientist A. Zolotov for his help with the figure \n1 of the manuscript. Support for this work was provided by the National Aeronautics and Space Administration through Chandra Award Number 18708581 issued by the Chandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for and on behalf of the National Aeronautics Space Administration under contract NAS8-03060. The scientific results reported in this article are based in part on observations made by the Chandra X-ray Observatory (observations 20114, 20115). Based on observations obtained with XMM-Newton , an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA (observations 0795711301, 0795711401). This paper includes data gathered with the 6.5 meter Magellan Telescopes located at Las Campanas Observatory, Chile. Based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the Data Archive at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with the HST program 11142. This research \nhas made use of TOPCAT, developed by Mark Taylor at the University of Bristol; Aladin developed by the Centre de Donn'ees Astronomiques de Strasbourg (CDS); the VizieR catalogue access tool (CDS); Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration, 2013); Atlassian JIRA issue tracking system and Bitbucket source code hosting service. Funding for the SDSS and SDSS -II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/. The Pan-STARRS1 Surveys (PS1) and the PS1 public science archive have been made possible through contributions by the In- \ntitute for Astronomy, the University of Hawaii, the Pan-STARRS Project Office, the Max-Planck Society and its participating institutes, the Max Planck Institute for Astronomy, Heidelberg and the Max Planck Institute for Extraterrestrial Physics, Garching, The Johns Hopkins University, Durham University, the University of Edinburgh, the Queen's University Belfast, the Harvard-Smithsonian Center for Astrophysics, the Las Cumbres Observatory Global Telescope Network Incorporated, the National Central University of Taiwan, the Space Telescope Science Institute, the National Aeronautics and Space Administration under Grant No. NNX08AR22G issued through the Planetary Science Division of the NASA Science Mission Directorate, the National Science Foundation Grant No. AST-1238877, the University of Maryland, Eotvos Lorand University (ELTE), the Los Alamos National Laboratory, and the Gordon and Betty Moore Foundation.", 'REFERENCES': "Chilingarian, I. 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2018PhRvD..98h4011B
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Radial perturbations of the scalarized Einstein-Gauss-Bonnet black holes
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2018-01-01
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Recently a new class of scalarized black holes in Einstein-Gauss-Bonnet (EGB) theories was discovered. What is special for these black hole solutions is that the scalarization is not due to the presence of matter, but it is induced by the curvature of spacetime itself. Moreover, more than one branch of scalarized solutions can bifurcate from the Schwarzschild branch, and these scalarized branches are characterized by the number of nodes of the scalar field. The next step is to consider the linear stability of these solutions, which is particularly important due to the fact that the Schwarzschild black holes lose stability at the first point of bifurcation. Therefore we here study in detail the radial perturbations of the scalarized EGB black holes. The results show that all branches with a nontrivial scalar field with one or more nodes are unstable. The stability of the solutions on the fundamental branch, whose scalar field has no radial nodes, depends on the particular choice of the coupling function between the scalar field and the Gauss-Bonnet invariant. We consider two particular cases based on the previous studies of the background solutions. If this coupling has the form used in [D. D. Doneva and S. S. Yazadjiev, Phys. Rev. Lett. 120, 131103 (2018)] the fundamental branch of solutions is stable, except for very small masses. In the case of a coupling function quadratic in the scalar field [H. O. Silva, J. Sakstein, L. Gualtieri, T. P. Sotiriou, and E. Berti, Phys. Rev. Lett. 120, 131104 (2018)], though, the whole fundamental branch is unstable.
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https://arxiv.org/pdf/1805.05755.pdf
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{'Radial perturbations of the scalarized EGB black holes': "Jose Luis Bl'azquez-Salcedo ∗ 1 , Daniela D. Doneva † 2,3 , Jutta Kunz ‡ 1 , and Stoytcho S. Yazadjiev § 2,4,5 \n1 Institut fur Physik, Universitat Oldenburg, Postfach 2503, D-26111 Oldenburg, Germany 2 Theoretical Astrophysics, Eberhard Karls University of Tubingen, Tubingen 72076, Germany 3 INRNE - Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria 4 Department of Theoretical Physics, Faculty of Physics, Sofia University, Sofia 1164, Bulgaria \n5 Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Street 8, Sofia 1113, Bulgaria", 'Abstract': 'Recently a new class of scalarized black holes in Einstein-Gauss-Bonnet (EGB) theories was discovered. What is special for these black hole solutions is that the scalarization is not due to the presence of matter, but it is induced by the curvature of spacetime itself. Moreover, more than one branch of scalarized solutions can bifurcate from the Schwarzschild branch, and these scalarized branches are characterized by the number of nodes of the scalar field. The next step is to consider the linear stability of these solutions, which is particularly important due to the fact that the Schwarzschild black holes lose stability at the first point of bifurcation. Therefore we here study in detail the radial perturbations of the scalarized EGB black holes. The results show that all branches with a nontrivial scalar field with one or more nodes are unstable. The stability of the solutions on the fundamental branch, whose scalar field has no radial nodes, depends on the particular choice of the coupling function between the scalar field and the Gauss-Bonnet invariant. We consider two particular cases based on the previous studies of the background solutions. If this coupling has the form used in [1] the fundamental branch of solutions is stable, except for very small masses. In the case of a coupling function quadratic in the scalar field [2], though, the whole fundamental branch is unstable.', '1 Introduction': 'Very recently new black holes with a nontrivial scalar field have been constructed in the extended scalar-tensor-Gauss-Bonnet (ESTGB) theories [1, 2, 3, 4]. What is interesting in these results is the presence of non-uniqueness of the solutions - in addition to the pure general relativistic solution, that exists in the whole domain of the parameter space, for a certain range of parameters new branches of solutions with a nontrivial scalar field are present. These branches can be characterized by the number of nodes of the scalar field. In fact, besides the fundamental branch which possesses no nodes of the scalar field, there arises a whole sequence of radially excited branches 1 . Moreover, the Schwarzschild solution loses stability at the point, where the first nontrivial branch bifurcates from it. Then the fundamental branch of scalarized black holes could represent the stable one. This would represent a direct analogy with the spontaneous scalarization of neutron stars in the standard scalar-tensor theories considered in [8], and also with the scalarized black holes in scalartensor theories in the presence of nonlinear matter sources [6, 7, 9, 10]. The main difference with respect to those results, though, is that in the ESTGB case the scalar field is not sourced by matter, but instead by the curvature of spacetime through the Gauss-Bonnet scalar. In fact such spontaneous scalarization in ESTGB theories is observed also for neutron stars [11, 2]. \nThe ESTGB theories are very interesting on their own because of the following reasons. Their theoretical motivation comes from attempts to quantize gravity and the fact that pure general relativity is not a renormalizable theory. A way to cure this problem is to supplement the Einstein-Hilbert action with all possible algebraic curvature invariants of second order [12]. A serious problem that appears, though, is that the resulting field equations are of order higher than two, which leads to Ostrogradski instability and to the appearance of ghosts. However, this can be avoided in the special case when the additional dynamical scalar field is non-minimally coupled to a special combination of the second order invariants, namely the Gauss-Bonnet invariant, since the resulting field equations in this case are of second order [13]. These are exactly the ESTGB theories. \nOne of the most studied class of ESTGB theories are the dilatonic EGB theories, where the coupling function between the scalar field and the Gauss-Bonnet invariant has the form αe 2 γϕ , where α and γ are arbitrary constants, and the scalar field potential is set to zero. Black holes in dilatonic EGB theories have been extensively studied, both in the perturbative and non-perturbative regime and also including rapid rotation [14]-[24]. In contrast with the dilatonic EGB theories, the considered class of ESTGB theories in [1, 2, 3, 4] is characterized by a coupling function that can lead to non-uniqueness of the solutions and scalarization/descalarization. \nStability of black holes in dilatonic EGB theories was examined in [25, 26, 18, 27, 28, 29] and it was shown that the primary branch of black holes is stable, while the secondary branch, that appears for sufficiently strong dilaton coupling, is unstable. The linear stability of the scalarized black holes obtained in [1, 2, 3, 4] has not been studied yet. It was already \nproved, though, in [1, 2, 30] that the Schwarzschild solution loses stability at the point of bifurcation, where the first nontrivial branch of solutions appears. Examining the stability of the nontrivial branches of solutions is much more involved and represents the goal of the present paper. Based on thermodynamical analysis, it has been argued in [1] that, for the particular coupling function considered there, the fundamental ESTGB black hole branch should be the stable one, whereas all radially excited branches should be unstable. Of course in order to check this hypothesis more rigorously, one has to examine the linear stability of the branches of scalarized black holes, as done below. \nIn Section II the field equations used to obtain the background solutions are presented. The radial perturbations are examined in Section III, while the lengthy formulae are given in a separate Appendix. The results for the stability of the scalarized black holes are presented in Section IV. The paper ends with Conclusions.', '2 Field equations': 'The action of ESTGB theories in vacuum, in its general form, can be written as \nS = 1 16 π ∫ d 4 x √ -g [ R -2 ∇ µ ϕ ∇ µ ϕ -V ( ϕ ) + λ 2 f ( ϕ ) R 2 GB ] , (1) \nwhere R is the Ricci scalar with respect to the spacetime metric g µν , ϕ is the scalar field, V ( ϕ ) and f ( ϕ ) are the potential and the coupling function that depend on ϕ only, λ is the Gauss-Bonnet coupling constant that has dimension of length . The Gauss-Bonnet invariant R 2 GB is defined as R 2 GB = R 2 -4 R µν R µν + R µναβ R µναβ where R is the Ricci scalar, R µν is the Ricci tensor and R µναβ is the Riemann tensor. \nWe will consider static and spherically symmetric spacetimes as well as static and spherically symmetric scalar field configurations. In addition, we will concentrate on the case of zero scalar field potential V ( ϕ ) = 0. Thus, the spacetime metric can be written in the following form \nds 2 = -e 2Φ( r ) dt 2 + e 2Λ( r ) dr 2 + r 2 ( dθ 2 +sin 2 θdφ 2 ) . (2) \nAfter varying the action and performing a dimensional reduction of the resulting field equations one can obtain the following system of ordinary differential equations (more details can \nbe found in [1]) \n2 r [ 1 + 2 r (1 -3 e -2Λ )Ψ r ] d Λ dr + ( e 2Λ -1) r 2 -4 r 2 (1 -e -2Λ ) d Ψ r dr -( dϕ dr ) 2 = 0 , (3) \n2 r [ 1 + 2 r (1 -3 e -2Λ )Ψ r ] d Φ dr -( e 2Λ -1) r 2 -( dϕ dr ) 2 = 0 , (4) \nd 2 Φ dr 2 + ( d Φ dr + 1 r )( d Φ dr -d Λ dr ) + 4 e -2Λ r [ 3 d Φ dr d Λ dr -d 2 Φ dr 2 -( d Φ dr ) 2 ] Ψ r -4 e -2Λ r d Φ dr d Ψ r dr + ( dϕ dr ) 2 = 0 , (5) \nd 2 ϕ dr 2 + ( d Φ dr -d Λ dr + 2 r ) dϕ dr -2 λ 2 r 2 df ( ϕ ) dϕ { (1 -e -2Λ ) [ d 2 Φ dr 2 + d Φ dr ( d Φ dr -d Λ dr )] +2 e -2Λ d Φ dr d Λ dr } = 0 , (6) \nwhere \nΨ r = λ 2 df ( ϕ ) dϕ dϕ dr . (7) \nFurthermore, we assume zero cosmological value of the scalar field ϕ | r →∞ ≡ ϕ ∞ = 0, and the coupling function f ( ϕ ) is chosen such that it satisfies the conditions df dϕ (0) = 0 and b 2 = d 2 f dϕ 2 (0) > 0. Without loss of generality we can set b = 1, which can be achieved after rescaling of the coupling parameter λ → bλ and redefinition of the function f → b -2 f . Another observation one can make is that the field equations do not depend on f ( ϕ ) itself, but only on its derivative with respect to ϕ which leaves the freedom to choose f (0) = 0. \nAfter performing an expansion of the reduced field equations at the horizon and requiring regularity of the metric functions and the scalar field, one finds [1] that black hole solutions with a real scalar field exist only when the following condition is fulfilled \nr 4 H > 24 λ 4 ( df dϕ ( ϕ H ) ) 2 , (8) \nwhere r H is the radius of the black hole horizon and ϕ H is the value of the scalar field at the horizon. \nThe boundary conditions are derived by the requirement for asymptotic flatness \nΦ | r →∞ → 0 , Λ | r →∞ → 0 , ϕ | r →∞ → 0 . (9) \nOn the other hand, the very existence of a black hole horizon requires \ne 2Φ | r → r H → 0 , e -2Λ | r → r H → 0 . (10) \nIn the present paper we will concentrate mainly on the following coupling function \nf ( ϕ ) = 1 12 ( 1 -e -6 ϕ 2 ) , (11) \nsince it was shown in [1] that it can produce non-negligible deviations from pure GR. In addition, eq. (11) is quite similar to the coupling function employed in the studies of spontaneous scalarization of neutron stars [8]. \nIn the last part of the paper we will also present results for the case of the quadratic potential previously considered in [2] \nf ( ϕ ) = 1 2 ϕ 2 . (12) \nIt is worth noting that in the case of small scalar field, the coupling (12) is the leading term of the coupling (11), f ( ϕ ) = 1 12 ( 1 -e -6 ϕ 2 ) ≈ 1 2 ϕ 2 + O ( ϕ 4 ), and both couplings will share many features in the small ϕ domain.', '3.1 Ansatz and equations': 'We consider time dependent radial perturbations over the spherically symmetric and static background black holes obtained after solving the reduced system of equations (3)-(6) \nds 2 = -e 2Φ( r )+ /epsilon1F t ( r,t ) dt 2 + e 2Λ( r )+ /epsilon1F r ( r,t ) dr 2 + r 2 ( dθ 2 +sin 2 θdφ 2 ) , ϕ = ϕ 0 ( r ) + /epsilon1ϕ 1 ( r, t ) , (13) \nwith /epsilon1 being the control parameter of the perturbations. The field equations result in three second order differential equations and two algebraic constraints on the first derivatives [26]. However, the system can be simplified into a single second order differential equation \ng 2 ( r ) ∂ 2 ϕ 1 ∂t 2 -∂ 2 ϕ 1 ∂r 2 + C 1 ( r ) ∂ϕ 1 ∂r + U ( r ) ϕ 1 = 0 , (14) \nwhere the functions U ( r ), g ( r ) and C 1 ( r ) depend only on the background metric and scalar field. Their expressions are given in the appendix A. \nIn order to study the mode stability of the background configuration, we decompose the perturbation function ϕ 1 as \nϕ 1 ( r, t ) = ϕ 1 ( r ) e iωt (15) \nand we obtain the master equation for the eigenvalue problem, namely \nd 2 ϕ 1 dr 2 = C 1 ( r ) dϕ 1 dr + [ U ( r ) -ω 2 g 2 ( r ) ] ϕ 1 ( r ) . (16) \nThe master equation (16) can be cast into the standard Schrodinger form by defining the function Z ( r ): \nϕ 1 ( r ) = C 0 ( r ) Z ( r ) , (17) \nwhere C 0 ( r ) is the solution of the following differential equation \n1 C 0 dC 0 dr = C 1 -1 g dg dr . (18) \nThus we obtain \nd 2 Z dR 2 = V ( R ) -ω 2 ] Z, (19) \n[ \n] where we have defined the tortoise coordinate R and the effective potential as \ndR dr = g, (20) \nV ( R ) = 1 g 2 ( U + C 1 C 0 dC 0 dr -1 C 0 d 2 C 0 dr 2 ) . (21) \nSince we are interested in the stability analysis of the background solutions, we will focus on perturbations with purely imaginary eigenfrequencies: ω = iω I .', '3.2 Boundary conditions and numerical method': 'We want the perturbation to be outgoing at infinity and ingoing at the horizon: \nZ ---→ r →∞ e iω ( t -R ) = e -ω I ( t -R ) , (22) \nZ ---→ r → r H e iω ( t + R ) = e -ω I ( t + R ) . (23) \nThese boundary conditions simplify a lot for unstable modes possessing ω I < 0, and it is straightforward to show that Z r = ∞ = Z | r = r H = 0 (see e.g. [7]). \n| \n| In order to obtain the unstable modes, we implement the following numerical procedure. The first step is to generate a background solution for a set of fixed values for r H , λ and the number of nodes of the scalar field. We make use of Colsys [31] in order to integrate the equations. Typically the background solutions have a relative precision of 10 -10 with around 1000 points on a grid in the compactified coordinate x = 1 -r H /r . With these solutions we calculate numerically the coefficients of eq. (16), which is the equation we solve, since it is slightly simpler than eq. (19). \nOnce the coefficients are calculated, we follow a scheme similar to the one described in [32] to obtain the bifurcation points. We define the quantity ω 2 ≡ E , and promote it to an auxiliary function E ( r ). This function satisfies a trivial differential equation, dE dr = 0, which is added to eq. (16) to form a system of differential equations. The three boundary conditions that we impose on this system are ϕ 1 | r = r H = 0, ϕ 1 | r = r 0 = 1 and ϕ 1 | r = ∞ = 0, where r H < r 0 < ∞ . The procedure is automatized, allowing us to rapidly calculate the eigenfrequency for several thousands of solutions in the parameter space. \nFigure 1: Domain of existence of black holes parametrized by r H and ϕ H for λ = 1. \n<!-- image -->', '4 Stability analysis': 'Using the procedure described above we calculate the potential and the unstable modes of the Schwarzschild branch as well as the first few branches of scalarized black holes for the two coupling functions (11) and (12).', '4.1 Exponential coupling function': 'In the following we will focus on the exponential coupling function (11), employed in [1], discussing first the nonperturbative background solutions and subsequently the unstable modes.', '4.1.1 Background solutions': 'The domain of existence of these black holes is summarized in Fig. 1, where we show the space of solutions in the ( r H , ϕ H ) plane for λ = 1 2 . The yellow area represents the region where condition (8) is not fulfilled, with the red line representing the saturation of this inequality (i.e., the singular limit). The area in cyan is filled by solutions that in general do not satisfy the condition ϕ ∞ = 0. This condition is only satisfied for the solutions shown with black curves. Thus the ϕ ∞ = 0 solutions form a system of branches bifurcating from the Schwarzschild solution (shown by the vertical line in Fig. 1). In the following we will only consider these branches of ϕ ∞ = 0 solutions. \nEach branch of scalarized black holes can be characterized by the number of nodes of the scalar field as it extends along the radial coordinate. The fundamental branch possesses \nFigure 2 also indicates the radial stability of each branch, discussed in detail below. Solid curves correspond to radially unstable solutions (i.e., on solid curves the solutions possess at least one unstable radial mode). The dotted curves correspond to solutions that are radially stable (i.e., the effective potential is everywhere positive). The dashed curves correspond to solutions that do not seem to possess unstable modes although the potential is not strictly positive. \n<!-- image --> \nFigure 2: (left) The scalar charge D vs the mass M , both quantities scaled with λ . In red we show the Schwarzschild solution, in orange the fundamental branch and in blue, green, violet, brown and black the branches n = 1 , 2 , 3 , 4 and 5, respectively. Solid curves represent radially unstable solutions, dashed curves solutions without unstable modes, and dotted curves solutions with a strictly positive potential. (right) A zoom of the n = 1 ... 5 branches. All the solutions shown in this region are unstable. \n<!-- image --> \nsolutions without nodes ( n = 0), the first excited branch has solutions with one node ( n = 1), etc. Here we present results for the first six branches ( n = 0 ... 5), although branches with higher number of nodes exist, presenting similar features. \nThe first few branches can also be seen in Fig. 2, where we exhibit the scalar charge D/λ vs the mass M/λ . The Schwarzschild solution exists for arbitrary values of M/λ , while the scalarized branches exist only in certain intervals. The fundamental branch (shown in orange) exists between M/λ = 0 and the upper bound M/λ = 0 . 587, where the scalar hair disappears and the branch merges with the Schwarzschild branch. \nThe radially excited branches exist only in very small intervals of M/λ since the condition (8) is quickly violated. All these branches have the same structure, as seen in Figure 2. The n ≥ 1 branches bifurcate from the Schwarzschild branch at certain values of M/λ , which decrease with increasing node number. Along these scalarized branches, the scalar charge increases with increasing M . The branches then end at some critical value of M/λ , where the scalarized solution becomes singular [1]. The branches rapidly decrease in size as the node number increases. \nFig. 3 shows that the branch of Schwarzschild solutions possesses sets of unstable modes, indicated by the red curves. All these modes behave like ( M 2 /λ ) -1 when M/λ → 0 (see the scaling). The first set of unstable modes extends from M/λ = 0 up to M/λ = 0 . 587. This set of modes corresponds to the instability previously investigated in [1] by analyzing the potential. At M/λ = 0 . 587 the curve reaches a zero mode (i.e., ω I = 0), which appears because at that point there is the bifurcation point of the fundamental branch. This set of unstable modes exists in the same range of parameters as the fundamental branch, which coexists with the Schwarzschild solution in this region of the parameter space (see Fig. 2 (left), the orange curve). \n<!-- image --> \nFigure 3: (left) Eigenfrequency ω I scaled with M 2 /λ vs the scaled mass M/λ . The unstable modes in red correspond to the Schwarzschild solution. Note, that there are no unstable Schwarzschild modes beyond M/λ = 0 . 587. In orange we show the mode corresponding to the fundamental branch. The vertical dashed line in orange marks the value M/λ = 0 . 171 at which the unstable mode of the fundamental branch disappears. In blue, green, pink, brown and black we show the modes corresponding to the n = 1 ... 5 branches, respectively. (right) A zoom focused on the modes of the n = 1 ... 5 branches. \n<!-- image -->', '4.1.2 Unstable Schwarzschild modes': 'Let us now turn to a detailed discussion of the radial stability of the Schwarzschild branch and the scalarized branches, starting with a summary of our findings as exhibited in Fig. 3. Here we exhibit the value of ω I multiplied by M 2 /λ vs the (scaled) mass M/λ for all considered branches of solutions. In red we show the modes of the Schwarzschild solution, and with various other colours the modes corresponding to the scalarized branches. The right panel of Fig. 3 shows a zoom focusing on the region, where the n = 1 ... 5 scalarized branches reside. \nConsequently the Schwarzschild black hole in this theory is unstable under radial perturbations in the full intervall where the fundamental branch of scalarized black holes exists. Moreover, the black hole solutions are not unique in this interval. Nevertheless, as soon as the fundamental branch ceases to exist (i.e., for M/λ > 0 . 587), the instability of the Schwarzschild solutions disappears. The potential, however, is only strictly positive for \nM/λ > √ 3 / 4. Hence in Fig. 2 (left) a solid red line indicates the unstable Schwarzschild solutions in the interval 0 < M/λ < 0 . 587, a dashed line indicates the solutions between 0 . 587 < M/λ < √ 3 / 4, and a dotted line marks the stable Schwarzschild solutions for 0 . 587 < M/λ . \nClearly, this is not the only set of unstable modes that the Schwarzschild solution possesses. In Fig. 3 we see additional sets of unstable modes extending from M/λ = 0 up to certain values of M/λ , where further zero modes are reached. Again, these zero modes appear precisely at the bifurcation points of the scalarized branches, but now with node number n > 0 (shown by the short blue to black curves in Fig. 2). We note, that the different sets of unstable modes of the Schwarzschild solutions can be characterized by the number of nodes of the perturbation function ϕ 1 : the set of unstable Schwarzschild modes connected to the zero mode associated with the emergence of the n -th scalarized branch always possesses n nodes in the function ϕ 1 . \nTo summarize the above analysis, we conclude that the number of unstable modes of the Schwarzschild solution depends on the value of M/λ : if the Schwarzschild solution resides in between bifurcation points of the n -th and the ( n +1) -th branch of scalarized solutions, then it has n +1 unstable modes.', '4.1.3 Unstable n > 0 modes': 'Let us next address the unstable modes of the n > 0 branches before discussing the stability of the fundamental branch. In Fig. 3 these unstable modes are marked in blue, green, pink, brown and black for the branches with n = 1 , 2 , 3 , 4 and 5, respectively. We see that these scalarized black holes also possess several sets of radially unstable modes. One of these sets is always related to the respective zero mode of the Schwarzschild solution at the bifurcation point. For instance, in Fig. 3 we see that at each zero mode (for n ≥ 1), a set of unstable modes of the corresponding n -th branch appears and extends up to the maximum value of M/λ , where the background solutions become singular. \nThe additional sets of unstable modes are not directly connected to the zero modes. However, since the n -th excited branch bifurcates from the Schwarzschild solution, its unstable modes bifurcate from the unstable modes of the Schwarzschild solution as well as from the zero mode. This occurs at the bifurcation point as dictated by continuity (although for zero modes there could also arise a mode that is not purely imaginary). Therefore the n excited branch should have a total of n + 1 unstable modes, as indeed seen in Fig. 3. In Fig. 4 (left) we focus on the subset of unstable modes that bifurcate from the set of unstable Schwarzschild modes connecting with the zero mode of the fundamental branch, while in Fig. 4 (right) we zoom in on these unstable modes of the n = 1 branch. The behaviour is analogous for the sets of unstable modes bifurcating from the other branches of unstable Schwarzschild modes. We also note, that each set of unstable modes can be characterized by the nodes in the ϕ 1 function, possessing the same number of nodes as the modes of the corresponding Schwarzschild family they are connected with. \nLet us again summarize the above analysis and conclude that the n -th excited branch of scalarized black holes (where n > 0 ) possesses n +1 distinct unstable modes. \n<!-- image --> \n<!-- image --> \n2 \nFigure 4: (left) Subset of eigenfrequencies ω I scaled with M 2 /λ vs the scaled mass M/λ . The unstable modes in red correspond to those of the Schwarzschild solution and connect with the zero mode associated with the bifurcation point of the fundamental branch. In further colors shown are a subset of unstable modes of the n > 0 branches. (right) A zoom focusing on a set of unstable modes of the n = 1 branch (blue). \n<!-- image --> \nFigure 5: (left) The scaled potential V vs 1 -r H /r for several black holes on the excited branches with n = 1, 2 and 3 for λ = 1 and several values of r H . (right) A similar figure for black holes on the fundamental branch. \n<!-- image --> \nThis conclusion is supported further by an inspection of the potential for the excited branches. In Fig. 5 (left) we show the potential V vs the function of the radial coordinate 1 -r H r for black holes of the n = 1, 2 and 3 branches. Here we clearly see that the potential is dominantly negative. In fact, the integral of the potential is negative \nn > 0 ⇒ ∫ ∞ -∞ V ( R ) dR < 0 , (24) \nin accordance with the existence of the unstable modes. Hence in Fig. 2 (left), the n > 1 branches are shown by solid lines, since they are always radially unstable.', '4.1.4 Stability analysis of the fundamental branch': 'Let us now turn to the analysis of the radial stability of the fundamental branch of scalarized black holes. In Fig. 5 (right) we show the potential V vs the function of the radial coordinate 1 -r H r for several black holes along this branch (with λ = 1 and different values of r H ). What we observe is that the potential is always positive for solutions in the range 0 . 285 /lessorsimilar M/λ /lessorsimilar 0 . 542. Hence we conclude that no unstable modes exist on the fundamental branch for black holes as long as the solution belongs to this region. In Fig. 2 (left) this region of the fundamental branch (orange) is marked with a dotted line. \nFor black holes on the fundamental branch in the interval 0 . 542 /lessorsimilar M/λ /lessorsimilar 0 . 587 the potential becomes slightly negative in a region close to the horizon, while for black holes with 0 . 171 /lessorsimilar M/λ /lessorsimilar 0 . 288 the potential becomes slightly negative in some intermediate region of r , as seen, for instance, in Fig. 5 (left) for the potentials corresponding to λ = 1, r H = 1 . 16 (in red) and r H = 0 . 3 (in purple). However for both intervals the integral of the potential is always positive, meaning that \nn = 0 , M/λ /greaterorsimilar 0 . 171 ⇒ ∫ ∞ -∞ V ( R ) dR > 0 . (25) \nAlthough this does not exclude the possibility of unstable modes, we have not been able to generate any solution to the perturbation equation (16) describing an unstable mode and satisfying the boundary and regularity conditions. We interpret this as a strong indication that solutions along this branch are mode stable as long as M/λ /greaterorsimilar 0 . 171. In Fig. 2 (left) the two intervals of the fundamental branch are marked with a dashed line, where stability can thus not be decided with certainty. \nThe black holes on the fundamental branch in the interval 0 < M/λ /lessorsimilar 0 . 171 need a more involved investigation. Let us first consider the problem on the level of the Schrodinger equation (19). For the interval under consideration the tortoise coordinate becomes illdefined. In order to show this, we plot in Fig. 6 the function (1 -r H r ) 2 g 2 vs 1 -r H r , for black holes belonging to the fundamental branch with λ = 1 and for different values of r H . This function should be positive in order to have a well defined tortoise coordinate R (see eq. (20)). As r H is decreased, the function deviates more and more from the Schwarzschild case, which corresponds to g = (1 -r H r ) -1 . For small enough values of r H , the function \nFigure 6: The function (1 -r H r ) 2 g 2 vs 1 -r H /r for black holes on the fundamental branch for λ = 1 and several values of r H . \n<!-- image --> \nbecomes negative in the range r H ≤ r ≤ r ∗ where g ( r ∗ ) = 0. As a consequence, the potential becomes singular for M/λ /lessorsimilar 0 . 171. \nThe more fundamental reason for this strange behavior is that for M/λ /lessorsimilar 0 . 171 equation (14) is not hyperbolic for r H ≤ r ≤ r ∗ . This is an interesting and highly nontrivial phenomenon and we leave its investigation to future work. What is immediately clear from this fact is that the study of the linear stability as a Cauchy problem is ill-posed. Nevertheless, we can formally study the stability on the level of the eigenvalue problem. It is possible to analyze the mode stability by using the standard coordinate r and integrating eq. (16). Then it is possible to find that indeed, black holes of the fundamental branch with M/λ /lessorsimilar 0 . 171 possess an unstable mode. In Fig. 3, this unstable mode is shown in orange. The mode extends from M/λ = 0 up to M/λ ≈ 0 . 171, where the mode diverges. (This limit is marked by a vertical dashed line in orange in Fig. 3 (left).) Interestingly, this unstable mode does not bifurcate from any Schwarzschild mode (although in the Figure it crosses the sets of unstable Schwarzschild modes for small values of M/λ , it is not connected with them). The perturbation function ϕ 1 of this family of modes always presents zero nodes. \nMarking consequently this part of the fundamental branch extending from M/λ = 0 to M/λ = 0 . 171 with a solid orange line in Fig. 2 we note, that black holes belonging to the fundamental branch are not always radially stable , i.e., for arbitrary values of M/λ (despite them having always the larger value of the entropy, as it shown in [1]), and even worse, for small enough black holes (as compared to the coupling parameter) the theory is not hyperbolic. Stability is only found for sufficiently large values of M/λ . \nWe remark that the structure found is to some extent reminiscent (although much more complicated) to the radial instability observed in the dilatonic EGB black holes in [26, 29]. This theory corresponds to f ( ϕ ) = e 2 γϕ and λ 2 = α 4 (following the conventions of [28, 29]). For certain values of the coupling constant γ , one secondary branch of solutions is present in a small region of the parameter space, in addition to the main branch of dilatonic black \n<!-- image --> \nFigure 7: (left) Domain of existence r H vs ϕ H of the solutions with the quadratic coupling (12). For comparison, we include the singular limit for the coupling (11) as a dashed blue curve. (right) The scalar charge D vs the mass M for both quantities scaled with λ . \n<!-- image --> \nholes, resulting in non-uniqueness of the solutions. The secondary branch appears close to the minimum value of the black hole mass allowed by the theory, and it was found to be radially unstable [26, 29]. \nIt is also interesting to note that in dilatonic EGB black holes the existence of a minimum mass is caused by the existence of a limiting value of the normalized coupling constant (i.e., the maximum value of ζ = α/M 2 [29], when condition (8) is no longer satisfied). No regular black hole solutions can be found for smaller values of this mass. This is different from the scalarized EGB black holes on the fundamental branch considered here, which exist for arbitrarily small values of the mass. However, this branch also possesses an effective minimum mass ( M/λ ≈ 0 . 171), below which the theory is no longer hyperbolic (i.e., where no longer stable configurations can be found).', '4.2 Quadratic coupling function': 'Finally we turn to the case of the quadratic coupling (12). In Fig. 7 (left) we show the domain of existence r H vs ϕ H . As expected, we find that the space of solutions is very similar to the one of the previous coupling in the small ϕ H and r H region (compare Fig. 1). In particular, since the quadratic coupling (12) is obtained in the small ϕ limit of the exponential coupling (11), the bifurcation points of the Schwarzschild solution coincide. Even more, the structure of the zero modes and sets of instabilities of the Schwarzschild solution is exactly the same for both couplings. This is seen in Fig. 7 (right), where we show D/λ vs M/λ for solutions with the quadratic coupling. \nThe most important difference appears for the fundamental branch n = 0, which in this case extends from the bifurcation point M/λ = 0 . 587 to larger values of M/λ and reaches the singular limit at a finite value of ϕ . This gives the n = 0 branch a similar structure to the excited n > 0 branches. In fact, the fundamental branch turns out to be unstable like \nFigure 8: The scaled eigenfrequencies ω I vs the scaled mass M/λ focused on the n = 0, 1 branches for the quadratic coupling (12). For comparison, we include the unstable modes of the n = 1 branch of the exponential coupling (11) with a cyan curve. \n<!-- image --> \nthe excited branches. In Fig. 8 we exhibit the scaled unstable modes ω I vs M/λ focusing on the n = 0, 1 branches for the quadratic coupling. Again in red we show the first two sets of unstable Schwarzschild modes that connect with the n = 0 and n = 1 zero modes. In the present case at the zero mode of the n = 0 branch, a new set of unstable modes appears and remains present on the whole fundamental branch (shown by the orange curve). The branch and with it its set of unstable modes end when the singular configuration is reached. The n = 1 modes are also shown (in blue). Moreover, for comparison the respective unstable modes corresponding to the exponential coupling (11) are shown (in cyan). \nHence in this case, all the scalarized branches are unstable, and the n -th branch possesses n +1 unstable modes, including the fundamental branch, n = 0 . This means there are no stable configurations below the first branching point at M/λ = 0 . 587. As a matter of fact this conclusion coincides with the thermodynamical results - after performing a calculation of the black hole entropy similar to [1] but for the particular coupling function (12), it turns out that all of the branches with a nontrivial scalar field, including the fundamental one, have lower entropy than the Schwarzschild solution and therefore they are thermodynamically less stable.', '5 Conclusions': "In the present paper we have studied the stability with respect to radial perturbations of scalarized black holes in EGB theories. The two particular cases of the coupling functions between the scalar field and the Gauss-Bonnet invariant considered have been motivated by the previous studies of black holes with a nontrivial scalar field. The first one is the case of f ( ϕ ) = 1 12 ( 1 -e -6 ϕ 2 ) considered in [1] which leads to a well manifested fundamental branch (i.e., the branch characterized by a scalar field which has no nodes) that deviates \nsignificantly from the Schwarzschild black holes. Second, the coupling function f ( ϕ ) = 1 2 ϕ 2 , considered in [2], has been examined. \nIt was previously shown [1, 2] that the Schwarzschild solution is stable only up to the first point of bifurcation where the fundamental branch appears. Therefore a natural question to ask has been whether another solution with a nontrivial scalar field is stable for masses smaller than the critical mass of the bifurcation. The results from the linear stability analysis performed show that all solutions characterized by a scalar field with n > 0 nodes possess n +1 unstable modes. Thus all radially excited branches of solutions are radially unstable. \nThe picture is more complicated for the fundamental branch. In the case of the coupling function f ( ϕ ) = 1 12 ( 1 -e -6 ϕ 2 ) , the scalarized solutions are stable from the bifurcation point until some small critical mass M ∗ , where they formally lose stability as well. The presence of this instability is formally different in nature from the instability of the other scalarized branches - the unstable modes are not connected with any zero mode of the Schwarzschild solution and are due most probably to the fact that for small masses the tortoise coordinate is ill defined and leads to singularities of the potential of the perturbations equations. In fact the situation is even worse because below the critical mass M ∗ the theory loses its hyperbolicity, and the stability has only been investigated formally on the basis of a formal eigenvalue problem. \nIn constrast, in the case of the coupling function f ( ϕ ) = 1 2 ϕ 2 the whole fundamental branch, which in this case is short and terminates at some nonzero mass because of violation of condition (8), is unstable. This instability is a more 'classical' one since it is connected with a zero mode of the Schwarzschild solution. \nWe should note that these conclusions are in agreement with the thermodynamical studies of the stability performed in [1]. More precisely, the entropy of all branches of solutions possessing a scalar field which has one or more nodes is smaller than the entropy of the Schwarzschild black holes. On the other hand, the results show that the fundamental branch for f ( ϕ ) = 1 12 ( 1 -e -6 ϕ 2 ) is thermodynamically more stable as compared to the Schwarzschild solution, while for f ( ϕ ) = 1 2 ϕ 2 the entropy of the fundamental branch is always smaller than the Schwarzschild one. \nIt is interesting to make a comparison of these solutions with the black holes in the dilatonic EGB theory, where the coupling function has the form f ( ϕ ) = e 2 γϕ . In this case solutions can only exist if they are larger than a certain minimum value of the mass. For some values of the coupling constant γ , a secondary branch of black holes appears close to this limit. The solutions on this secondary branch were shown to be always radially unstable [26, 29]. \nThus, the stability and the existence of solutions is highly controlled by the coupling function. A general conclusion, though, is that for all of the considered cases there exists a threshold mass below which there are no stable black hole solutions (including the Schwarzschild one). As a matter of fact one might be able to cure this problem by a better choice of the coupling function (or even by varying the numerical constants in the coupling functions (11) or (12)). Answering this question and studying the loss of hyperbolicity for small black holes require, however, a much more thorough investigation that will be the \nsubject of a future study.", 'Acknowledgements': 'JLBS would like to acknowledge support from the DFG project BL 1553. JLBS and JK would like to acknowledge support by the DFG Research Training Group 1620 Models of Gravity and the COST Action CA16104. SY and DD would like to thank for support by the COST Action CA16214. SY would like to thank for support by the COST Action CA16104 and the Sofia University Research Grant under No 3258. DD would like to thank the European Social Fund, the Ministry of Science, Research and the Arts Baden-Wurttemberg for the support. DD is indebted to the Baden-Wurttemberg Stiftung for the financial support of this research project by the Eliteprogramme for Postdocs.', 'A Additional equations': "The functions that appear in the master equation (16) depend only on the background configurations. The function g that characterizes the tortoise coordinate can be written like \ng 2 = A/B , (26) \nwith the functions A and B : \nA = -8 e 6Λ ϕ ' 0 λ 2 r 3 df dϕ 0 + e 8Λ r 4 +16 ϕ ' 0 2 λ 6 ( e 2Λ -1 ) 2 e 2Λ ( df dϕ 0 ) 2 d 2 f dϕ 2 0 -16 λ 6 ( e 2Λ -1 ) ( e 2Λ Λ ' ϕ ' 0 -e 2Λ ϕ '' 0 +3Λ ' ϕ ' 0 + ϕ '' 0 ) e 2Λ ( df dϕ 0 ) 3 -4 e 4Λ λ 4 ( e 4Λ -4 e 2Λ Λ ' r -4 ϕ ' 0 2 r 2 -2 e 2Λ +4Λ ' r +1 ) ( df dϕ 0 ) 2 , (27) \nB = -8 e 4Λ+2Φ ϕ ' 0 λ 2 r 3 df dϕ 0 + e 6Λ+2Φ r 4 +16 e 2Φ Φ ' ϕ ' 0 λ 6 ( e 4Λ +2 e 2Λ -3 ) ( df dϕ 0 ) 3 -4 e 2Φ+2Λ λ 4 ( -4 ϕ ' 0 2 r 2 +4 e 2Λ Φ ' r + e 4Λ -4Φ ' r -2 e 2Λ +1 ) ( df dϕ 0 ) 2 . (28) \nThe coefficient C 1 can be written like \nC 1 = C 2 /C 4 , (29) \nwith the functions C 2 and C 4 being: \nC 4 = [ 2 df dϕ 0 ϕ ' 0 ( e 2Λ -3 ) λ 2 + e 2Λ r ] × [ -8 df dϕ 0 e 4Λ ϕ ' 0 λ 2 r 3 + e 6Λ r 4 +16Φ ' ϕ ' 0 λ 6 ( e 2Λ -1 ) ( e 2Λ +3 ) ( df dϕ 0 ) 3 -4 e 2Λ λ 4 ( 4Φ ' e 2Λ r + e 4Λ -4 ϕ ' 0 2 r 2 -4Φ ' r -2 e 2Λ +1 ) ( df dϕ 0 ) 2 ] , (30) \nC 2 \n= \nd 2 f dϕ 2 0 [ 4 ϕ ' 0 2 λ 2 r 4 ( e 8Λ -e 6Λ ) +32Φ ' ϕ ' 0 3 λ 8 ( 15 e 4Λ -7 e 2Λ -5 e 6Λ -3 ) ( df dϕ 0 ) 3 -8 ϕ ' 0 2 λ 6 e 2Λ ( 8 e 4Λ Φ ' r +8 e 2Λ Φ ' r -16Φ ' r +9 e 4Λ -9 e 2Λ -3 e 6Λ +3 ) ( df dϕ 0 ) 2 +4 ϕ ' 0 λ 4 re 4Λ ( e 4Λ ϕ ' 0 2 r 2 -6 e 2Λ ϕ ' 0 2 r 2 +5 ϕ ' 0 2 r 2 +8 e 2Λ Φ ' r +2 e 4Λ -8Φ ' r -4 e 2Λ +2 ) df dϕ 0 ] +32 ϕ ' 0 λ 8 ( df dϕ 0 ) 4 [ -7 e 4Λ Φ ' 2 ϕ ' 0 -11 e 4Λ Φ ' Λ ' ϕ ' 0 +15 e 2Λ Φ ' 2 ϕ ' 0 -9 e 2Λ Φ ' Λ ' ϕ ' 0 +Φ ' 2 ϕ ' 0 e 6Λ +9Φ ' Λ ' ϕ ' 0 e 6Λ + e 4Λ Φ '' ϕ ' 0 +13 e 4Λ Φ ' ϕ '' 0 +9 e 2Λ Φ '' ϕ ' 0 -25 e 2Λ Φ ' ϕ '' 0 -Φ '' ϕ ' 0 e 6Λ -9Φ ' 2 ϕ ' 0 +27Φ ' Λ ' ϕ ' 0 -3Φ ' ϕ '' 0 e 6Λ -9Φ '' ϕ ' 0 +15Φ ' ϕ '' 0 ] \n+8 λ 6 e 2Λ ( df dϕ 0 ) 3 [ 6Λ ' ϕ ' 0 +18Φ '' ϕ ' 0 r +6Φ ' ϕ ' 0 3 r 2 -6Λ ' ϕ ' 0 3 r 2 +12Φ ' 2 ϕ ' 0 r +40 e 2Λ Φ ' Λ ' ϕ ' 0 r +3 ϕ '' 0 + e 4Λ ϕ '' 0 + ϕ '' 0 e 6Λ -5 e 2Λ ϕ '' 0 +24 ϕ ' 0 3 r -6 e 4Λ Φ ' ϕ ' 0 3 r 2 -2 e 4Λ Λ ' ϕ ' 0 3 r 2 +8 e 2Λ Φ ' ϕ ' 0 3 r 2 +16 e 2Λ Λ ' ϕ ' 0 3 r 2 +4 e 4Λ Φ ' 2 ϕ ' 0 r -16 e 2Λ Φ ' 2 ϕ ' 0 r +2 e 4Λ Φ '' ϕ ' 0 r -12 e 4Λ Φ ' ϕ '' 0 r -20 e 2Λ Φ '' ϕ ' 0 r +32 e 2Λ Φ ' ϕ '' 0 r -6Φ ' ϕ ' 0 -8 e 2Λ ϕ ' 0 3 r -2 e 4Λ Φ ' ϕ ' 0 +6 e 2Λ Φ ' ϕ ' 0 +2Φ ' ϕ ' 0 e 6Λ -20Φ ' ϕ '' 0 r -72Φ ' Λ ' ϕ ' 0 r -6Λ ' ϕ ' 0 e 6Λ +10 e 4Λ Λ ' ϕ ' 0 -10 e 2Λ Λ ' ϕ ' 0 ] \n(31) \n+4 λ 4 e 4Λ ( df dϕ 0 ) 2 [ -1 -4Λ ' r -34 ϕ ' 0 2 r 2 +6Φ ' r +26Φ ' Λ ' r 2 -13Φ ' ϕ ' 0 2 r 3 -2Φ ' 2 r 2 -4Φ '' r 2 + e 4Λ Φ ' ϕ ' 0 2 r 3 -e 4Λ Λ ' ϕ ' 0 2 r 3 + e 4Λ ϕ ' 0 ϕ '' 0 r 3 -4 e 2Λ Λ ' ϕ ' 0 2 r 3 -6 e 2Λ ϕ ' 0 ϕ '' 0 r 3 +5 ϕ ' 0 ϕ '' 0 r 3 +Λ ' ϕ ' 0 2 r 3 + e 4Λ + e 2Λ +8 e 2Λ ϕ ' 0 2 r 2 +4 e 2Λ Λ ' r +2 e 4Λ ϕ ' 0 2 r 2 +2 e 2Λ Φ ' 2 r 2 -18 e 2Λ Φ ' Λ ' r 2 -e 6Λ +4 e 2Λ Φ '' r 2 +2 e 4Λ Φ ' r -8 e 2Λ Φ ' r ] -e 8Λ r 4 [ -Λ ' r +Φ ' r +2 ] -2 λ 2 r 3 e 6Λ df dϕ 0 [ e 4Λ ϕ ' 0 -2 e 2Λ ϕ '' 0 r -6Φ ' ϕ ' 0 r +2Λ ' ϕ ' 0 r +2 e 2Λ ϕ ' 0 +2 ϕ '' 0 r -15 ϕ ' 0 ] . \nFinally, the function U , related with the effective potential, can be written like \nU = C 3 /C 4 , (32) \nwith \nand \nD 0 = -ϕ ' 0 r 5 e 8Λ ( -2Φ ' ϕ ' 0 r +2Λ ' ϕ ' 0 r + e 2Λ ϕ ' 0 -3 ϕ '' 0 r -5 ϕ ' 0 ) , (34) \nD 2 = -2 r 3 e 6Λ [ e 2Λ ϕ ' 0 4 r 2 -3 ϕ ' 0 4 r 2 -2 e 2Λ Φ ' ϕ ' 0 2 r +2 e 2Λ Λ ' ϕ ' 0 2 r -3 e 2Λ ϕ ' 0 ϕ '' 0 r +4Φ ' ϕ ' 0 2 r -4Λ ' ϕ ' 0 2 r + e 4Λ ϕ ' 0 2 -e 2Λ Φ ' 2 + e 2Λ Φ ' Λ ' -4 e 2Λ ϕ ' 0 2 +3 ϕ ' 0 ϕ '' 0 r -e 2Λ Φ '' +Φ ' 2 -3Φ ' Λ ' +3 ϕ ' 0 2 +Φ '' ]( d 2 f dϕ 2 0 ) +2 r 4 e 6Λ ϕ ' 0 3 ( e 2Λ -1 ) ( d 3 f dϕ 3 0 ) +2 r 2 e 6Λ [ 10 e 2Λ ϕ ' 0 3 r 2 +4Λ ' ϕ ' 0 3 r 3 -4Φ ' ϕ ' 0 3 r 3 -12 ϕ ' 0 2 ϕ '' 0 r 3 +4Φ ' ϕ '' 0 r 2 +3Φ ' ϕ ' 0 r +2Λ ' ϕ ' 0 r -30 ϕ ' 0 3 r 2 -e 4Λ ϕ ' 0 +2 e 2Λ ϕ ' 0 -ϕ '' 0 r + e 2Λ ϕ '' 0 r +4Φ ' 2 ϕ ' 0 r 2 +3Φ '' ϕ ' 0 r 2 +2 e 2Λ Φ ' Λ ' ϕ ' 0 r 2 -ϕ ' 0 -2 e 2Λ Φ ' ϕ '' 0 r 2 -2 e 2Λ Φ ' ϕ ' 0 r -14Φ ' Λ ' ϕ ' 0 r 2 + e 4Λ Φ ' ϕ ' 0 r +2 e 2Λ Φ ' ϕ ' 0 3 r 3 -2 e 2Λ Λ ' ϕ ' 0 3 r 3 +2 e 2Λ ϕ ' 0 2 ϕ '' 0 r 3 -2 e 2Λ Φ ' 2 ϕ ' 0 r 2 -e 2Λ Φ '' ϕ ' 0 r 2 ]( df dϕ 0 ) , (35) \nC 3 = D 0 + D 2 λ 2 + D 4 λ 4 + D 6 λ 6 + D 8 λ 8 , (33) \nD 4 \n= \n-4 ϕ ' 0 4 r 3 ( e 8Λ -4 e 6Λ +3 e 4Λ ) ( d 2 f dϕ 2 0 ) 2 + [ -8 e 8Λ Φ ' Λ ' ϕ ' 0 r 2 -16 e 6Λ Φ ' Λ ' ϕ ' 0 r 2 +8 e 4Λ Φ ' Λ ' ϕ ' 0 r 2 -8 ϕ '' 0 e 6Λ r +4 e 8Λ ϕ '' 0 r -24 e 4Λ ϕ ' 0 5 r 4 -4 e 4Λ ϕ ' 0 +4 e 8Λ ϕ ' 0 +4 ϕ ' 0 e 6Λ -4 ϕ ' 0 e 10Λ +4 e 4Λ ϕ '' 0 r +36 e 8Λ ϕ ' 0 3 r 2 -104 e 6Λ ϕ ' 0 3 r 2 +68 e 4Λ ϕ ' 0 3 r 2 +8 e 6Λ ϕ ' 0 5 r 4 -56 e 4Λ Λ ' ϕ ' 0 3 r 3 +8 e 8Λ ϕ ' 0 2 ϕ '' 0 r 3 -56 e 6Λ ϕ ' 0 2 ϕ '' 0 r 3 +48 e 4Λ ϕ ' 0 2 ϕ '' 0 r 3 +8 e 8Λ Φ ' 2 ϕ ' 0 r 2 -32 e 6Λ Φ ' 2 ϕ ' 0 r 2 +24 e 4Λ Φ ' 2 ϕ ' 0 r 2 +8 e 8Λ Φ '' ϕ ' 0 r 2 -24 e 6Λ Φ '' ϕ ' 0 r 2 +16 e 6Λ Φ ' ϕ '' 0 r 2 +16 e 4Λ Φ '' ϕ ' 0 r 2 -16 e 4Λ Φ ' ϕ '' 0 r 2 +8 e 8Λ Φ ' ϕ ' 0 r -16 e 4Λ Λ ' ϕ ' 0 r -8 e 8Λ Λ ' ϕ ' 0 3 r 3 -88 e 6Λ Φ ' ϕ ' 0 3 r 3 +32 e 6Λ Λ ' ϕ ' 0 3 r 3 +124 e 4Λ Φ ' ϕ ' 0 3 r 3 -32Φ ' ϕ ' 0 e 6Λ r +16Λ ' ϕ ' 0 e 6Λ r +12 e 8Λ Φ ' ϕ ' 0 3 r 3 +24 e 4Λ Φ ' ϕ ' 0 r ]( df dϕ 0 )( d 2 f dϕ 2 0 ) +4 ϕ ' 0 2 r [ -6 ϕ ' 0 2 e 6Λ r 2 + e 8Λ ϕ ' 0 2 r 2 +5 e 4Λ ϕ ' 0 2 r 2 +4Φ ' e 6Λ r -4 e 4Λ Φ ' r -2 e 6Λ + e 8Λ + e 4Λ ]( df dϕ 0 )( d 3 f dϕ 3 0 ) + [ 8 e 8Λ Φ ' Λ ' ϕ ' 0 2 r 3 +8 e 6Λ Φ ' Λ ' ϕ ' 0 2 r 3 +80 e 4Λ Φ ' Λ ' ϕ ' 0 2 r 3 -4 e 8Λ Φ ' ϕ ' 0 ϕ '' 0 r 3 -16 e 6Λ Φ ' ϕ ' 0 ϕ '' 0 r 3 +4 e 4Λ Φ ' ϕ ' 0 ϕ '' 0 r 3 +4 e 4Λ Φ ' +24 e 4Λ ϕ ' 0 2 r +24 e 6Λ Φ ' 2 r +168 e 4Λ ϕ ' 0 4 r 3 +4 e 10Λ Φ ' -4 e 8Λ Φ ' -4 e 6Λ Φ ' +8 e 8Λ ϕ ' 0 4 r 3 -32 e 6Λ ϕ ' 0 2 r -8 e 8Λ Φ ' 2 r +8 e 6Λ Φ '' r -4 e 8Λ Φ '' r -8 e 6Λ Φ ' 3 r 2 -16 e 4Λ Φ ' 2 r +8 e 4Λ Φ ' 3 r 2 -4 e 4Λ Φ '' r -80 e 6Λ ϕ ' 0 4 r 3 +8 e 8Λ ϕ ' 0 2 r -16 e 6Λ Φ ' ϕ ' 0 4 r 4 +16 e 6Λ Λ ' ϕ ' 0 4 r 4 -16 e 6Λ ϕ ' 0 3 ϕ '' 0 r 4 +48 e 4Λ ϕ ' 0 3 ϕ '' 0 r 4 -8 e 8Λ Φ ' 2 ϕ ' 0 2 r 3 +16 e 6Λ Φ ' 2 ϕ ' 0 2 r 3 -40 e 4Λ Φ ' 2 ϕ ' 0 2 r 3 -4 e 8Λ Φ '' ϕ ' 0 2 r 3 +16 e 6Λ Φ '' ϕ ' 0 2 r 3 -44 e 4Λ Φ '' ϕ ' 0 2 r 3 -24 e 8Λ Φ ' ϕ ' 0 2 r 2 -24 e 8Λ Λ ' ϕ ' 0 2 r 2 +40 e 6Λ Φ ' 2 Λ ' r 2 +104 e 6Λ Φ ' ϕ ' 0 2 r 2 +8 e 6Λ Λ ' ϕ ' 0 2 r 2 -56 e 4Λ Φ ' 2 Λ ' r 2 -128 e 4Λ Φ ' ϕ ' 0 2 r 2 +8 e 8Λ ϕ ' 0 ϕ '' 0 r 2 \n(36) \n-16 e 6Λ Φ '' Φ ' r 2 -8 e 6Λ ϕ ' 0 ϕ '' 0 r 2 +16 e 4Λ Φ '' Φ ' r 2 -16 e 6Λ Φ ' Λ ' r +16 e 4Λ Φ ' Λ ' r ]( df dϕ 0 ) 2 , \nD 6 \n= 8 e 2Λ ϕ ' 0 3 [ 2 e 4Λ ϕ ' 0 2 r 2 -8 e 2Λ ϕ ' 0 2 r 2 +6 ϕ ' 0 2 r 2 -12 e 4Λ Φ ' r +32 e 2Λ Φ ' r -20Φ ' r \n-6 ϕ ' 0 2 r 2 +2 e 4Λ Φ ' r -20 e 2Λ Φ ' r +18Φ ' r + e 6Λ -5 e 4Λ +7 e 2Λ -3 ]( df dϕ 0 ) 2 ( d 3 f dϕ 3 0 ) -8 e 2Λ [ 3Φ ' ϕ '' 0 + e 4Λ Φ ' ϕ '' 0 +7 e 2Λ Φ '' ϕ ' 0 -2 e 2Λ Φ ' 2 ϕ ' 0 -5 e 2Λ Φ ' ϕ '' 0 +Φ ' ϕ '' 0 e 6Λ +2Φ ' 2 ϕ ' 0 e 6Λ -5 e 4Λ Φ '' ϕ ' 0 +6Φ ' Λ ' ϕ ' 0 -16 e 4Λ Φ ' Λ ' ϕ ' 0 3 r 2 +4 e 2Λ Φ ' Λ ' ϕ ' 0 3 r 2 +6 e 4Λ Φ ' ϕ ' 0 2 ϕ '' 0 r 2 -12 e 2Λ Φ ' ϕ ' 0 2 ϕ '' 0 r 2 +10 e 4Λ Φ ' 2 Λ ' ϕ ' 0 r -8 e 2Λ Φ ' 2 Λ ' ϕ ' 0 r -4 e 4Λ Φ '' Φ ' ϕ ' 0 r -8 e 2Λ Φ '' Φ ' ϕ ' 0 r -6Φ '' ϕ ' 0 3 r 2 -14Φ ' 2 ϕ '' 0 r +6Φ ' 3 ϕ ' 0 r -3Φ '' ϕ ' 0 -10 e 4Λ Φ ' ϕ ' 0 3 r -4 e 2Λ Φ ' 3 ϕ ' 0 r +22 e 2Λ Φ ' ϕ ' 0 3 r -10 e 4Λ Φ ' 2 ϕ '' 0 r +24 e 2Λ Φ ' 2 ϕ '' 0 r -8 e 2Λ Φ ' 2 ϕ ' 0 3 r 2 +12Φ ' Λ ' ϕ ' 0 3 r 2 +6Φ ' ϕ ' 0 2 ϕ '' 0 r 2 -18Φ ' 2 Λ ' ϕ ' 0 r +12Φ '' Φ ' ϕ ' 0 r +2 e 4Λ Φ '' ϕ ' 0 3 r 2 -4 e 2Λ Φ '' ϕ ' 0 3 r 2 +2 e 6Λ Φ ' ϕ ' 0 3 r 2 e 4Λ Φ ' 3 ϕ ' 0 r 6Φ ' Λ ' ϕ ' 0 e 6Λ +10 e 4Λ Φ ' Λ ' ϕ ' 0 \n+ e 6Λ + e 4Λ -5 e 2Λ +3 ]( df dϕ 0 )( d 2 f dϕ 2 0 ) 2 +8 e 2Λ ϕ ' 0 [ -9 ϕ ' 0 Φ ' +2 e 2Λ ϕ '' 0 -4 e 4Λ ϕ '' 0 +10 e 4Λ Λ ' ϕ ' 0 + ϕ ' 0 Φ ' e 6Λ -6Λ ' ϕ ' 0 e 6Λ -e 6Λ Φ ' Λ ' ϕ ' 0 r +13 e 4Λ Φ ' Λ ' ϕ ' 0 r -11 e 2Λ Φ ' Λ ' ϕ ' 0 r +12Λ ' ϕ ' 0 3 r 2 -2Φ ' ϕ '' 0 r -18 e 4Λ ϕ ' 0 3 r +2 e 6Λ ϕ ' 0 3 r -30 ϕ ' 0 3 r +6Λ ' ϕ ' 0 -4 e 4Λ ϕ ' 0 2 ϕ '' 0 r 2 +16 e 2Λ ϕ ' 0 2 ϕ '' 0 r 2 -10 e 4Λ Φ ' ϕ ' 0 3 r 2 +4 e 4Λ Λ ' ϕ ' 0 3 r 2 +24 e 2Λ Φ ' ϕ ' 0 3 r 2 -8 e 2Λ Λ ' ϕ ' 0 3 r 2 +3 e 4Λ Φ ' 2 ϕ ' 0 r -9 e 2Λ Φ ' 2 ϕ ' 0 r +Φ ' 2 ϕ ' 0 e 6Λ r -9 e 4Λ Φ '' ϕ ' 0 r -10 e 4Λ Φ ' ϕ '' 0 r +11 e 2Λ Φ '' ϕ ' 0 r +12 e 2Λ Φ ' ϕ '' 0 r +Φ '' ϕ ' 0 e 6Λ r -9Φ ' Λ ' ϕ ' 0 r +11 e 2Λ Φ ' ϕ ' 0 -3 e 4Λ Φ ' ϕ ' 0 +2 ϕ '' 0 e 6Λ -3Φ '' ϕ ' 0 r -12 ϕ ' 0 2 ϕ '' 0 r 2 -10 e 2Λ Λ ' ϕ ' 0 +46 e 2Λ ϕ ' 0 3 r -30Φ ' ϕ ' 0 3 r 2 +5Φ ' 2 ϕ ' 0 r ]( df dϕ 0 ) 2 ( d 2 f dϕ 2 0 ) +8 e 2Λ ϕ ' 0 3 [ -2 e 4Λ ϕ ' 0 2 r 2 +8 e 2Λ ϕ ' 0 2 r 2 \n---10 e 2Λ Φ ' Λ ' ϕ ' 0 +Φ '' ϕ ' 0 e 6Λ -30Φ ' ϕ ' 0 3 r ]( df dϕ 0 ) 3 , (37) \nD 8 = -32 ϕ ' 0 4 Φ ' [ 3 e 6Λ -13 e 4Λ +25 e 2Λ -15 ]( df dϕ 0 ) 2 ( d 2 f dϕ 2 0 ) 2 -32 ϕ ' 0 4 Φ ' ( e 6Λ -e 4Λ -9 e 2Λ +9 ) ( df dϕ 0 ) 3 ( d 3 f dϕ 3 0 ) -64 ϕ ' 0 2 [ -Φ ' 2 ϕ ' 0 e 6Λ -5Φ ' Λ ' ϕ ' 0 e 6Λ +3 e 4Λ Φ ' 2 ϕ ' 0 +10 e 4Λ Φ ' Λ ' ϕ ' 0 -5 e 2Λ Φ ' 2 ϕ ' 0 -9 e 2Λ Φ ' Λ ' ϕ ' 0 +Φ '' ϕ ' 0 e 6Λ +2Φ ' ϕ '' 0 e 6Λ -4 e 4Λ Φ '' ϕ ' 0 -7 e 4Λ Φ ' ϕ '' 0 +3 e 2Λ Φ '' ϕ ' 0 +8 e 2Λ Φ ' ϕ '' 0 +3Φ ' 2 ϕ ' 0 -3Φ ' ϕ '' 0 ]( df dϕ 0 ) 3 ( d 2 f dϕ 2 0 ) +64 ϕ ' 0 Φ ' [ Φ ' 2 ϕ ' 0 e 6Λ -5Φ ' Λ ' ϕ ' 0 e 6Λ -4 e 4Λ Φ ' 2 ϕ ' 0 +10 e 4Λ Φ ' Λ ' ϕ ' 0 +3 e 2Λ Φ ' 2 ϕ ' 0 -9 e 2Λ Φ ' Λ ' ϕ ' 0 +2Φ '' ϕ ' 0 e 6Λ +Φ ' ϕ '' 0 e 6Λ -8 e 4Λ Φ '' ϕ ' 0 -3 e 4Λ Φ ' ϕ '' 0 +6 e 2Λ Φ '' ϕ ' 0 +5 e 2Λ Φ ' ϕ '' 0 -3Φ ' ϕ '' 0 ]( df dϕ 0 ) 4 . 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2015ApJ...813...84G
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X-Ray Reflection Spectroscopy of the Black Hole GX 339--4: Exploring the Hard State with Unprecedented Sensitivity
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2015-01-01
| 32 | 0.52 | 160 |
['accretion', 'accretion disks', 'atoms', 'black hole physics', 'line formation', 'astronomy x rays', '-']
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[]
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We analyze simultaneously six composite RXTE spectra of GX 339-4 in the hard state comprising 77 million counts collected over 196 ks. The source spectra are ordered by luminosity and span the range 1.6%-17% of the Eddington luminosity. Crucially, using our new tool pcacorr, we re-calibrate the data to a precision of 0.1%, an order of magnitude improvement over all earlier work. Using our advanced reflection model relxill, we target the strong features in the component of emission reflected from the disk, namely, the relativistically broadened Fe K emission line, the Fe K edge, and the Compton hump. We report results for two joint fits to the six spectra: For the first fit, we fix the spin parameter to its maximal value (a<SUB>*</SUB> = 0.998) and allow the inner disk radius R<SUB>in</SUB> to vary. Results include (i) precise measurements of R<SUB>in</SUB>, with evidence that the disk becomes slightly truncated at a few percent of Eddington and (ii) an order-of-magnitude swing with luminosity in the high energy cutoff, which reaches >890 keV at our lowest luminosity. For the second fit, we make the standard assumption in estimating spin that the inner edge of the accretion disk is located at the innermost stable circular orbit (R<SUB>in</SUB> = R<SUB>ISCO</SUB>) and find {a}<SUB>*</SUB>={0.95}<SUB>-0.05</SUB><SUP>+0.03</SUP> (90% confidence, statistical). For both fits, and at the same level of statistical confidence, we estimate that the disk inclination is i = 48° ± 1° and that the Fe abundance is super-solar, A<SUB>Fe</SUB> = 5 ± 1.
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https://arxiv.org/pdf/1505.03607.pdf
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{'X-RAY REFLECTION SPECTROSCOPY OF THE BLACK HOLE GX 339-4: EXPLORING THE HARD STATE WITH UNPRECEDENTED SENSITIVITY': "Javier A. Garc'ıa 1 , James F. Steiner 1 , Jeffrey E. McClintock 1 , Ronald A. Remillard 2 , Victoria Grinberg 2 , Thomas Dauser 3 \nDraft version October 6, 2018", 'ABSTRACT': 'We analyze simultaneously six composite RXTE spectra of GX 339-4 in the hard state comprising 77 million counts collected over 196 ks. The source spectra are ordered by luminosity and span the range 1.6% to 17% of the Eddington luminosity. Crucially, using our new tool pcacorr , we re-calibrate the data to a precision of 0.1%, an order of magnitude improvement over all earlier work. Using our advanced reflection model relxill , we target the strong features in the component of emission reflected from the disk, namely, the relativistically-broadened Fe K emission line, the Fe K edge and the Compton hump. We report results for two joint fits to the six spectra: For the first fit, we fix the spin parameter to its maximal value ( a ∗ = 0 . 998) and allow the inner disk radius R in to vary. Results include (i) precise measurements of R in , with evidence that the disk becomes slightly truncated at a few percent of Eddington; and (ii) an order-of-magnitude swing with luminosity in the high energy cutoff, which reaches > 890 keV at our lowest luminosity. For the second fit, we make the standard assumption in estimating spin that the inner edge of the accretion disk is located at the innermost stable circular orbit ( R in = R ISCO ) and find a ∗ = 0 . 95 +0 . 03 -0 . 05 (90% confidence, statistical). For both fits, and at the same level of statistical confidence, we estimate that the disk inclination is i = 48 ± 1 deg and that the Fe abundance is super-solar, A Fe = 5 ± 1.', '1. INTRODUCTION': 'GX 339-4 is one of the most thoroughly studied of the roughly 50 known black-hole X-ray binaries. Its orbital period is around 1.7 days, and for the best candidate period of 1.7557 days the mass function is 5 . 8 ± 0 . 5 M glyph[circledot] (Hynes et al. 2003). Like nearly all black hole binaries, the X-ray source is transient, having undergone more than a dozen outburst cycles since its discovery in the early 1970s by Markert et al. (1973). During a cycle, GX339-4 often exhibits all known X-ray states, which unfold in the canonical pattern (Remillard & McClintock 2006; Dunn et al. 2010). During the rising phase, the source can reach exceptional luminosities in the hard state, which is the focus of this paper. \nThe hard state is strongly dominated by a hard powerlaw component (Γ ∼ 1 . 6). The thermal component, which contributes glyph[lessorsimilar] 20% of the 2-20 keV flux, is faint and cool ( kT glyph[lessorsimilar] 0 . 2 keV) compared to the thermal state (Remillard & McClintock 2006). The Fe K line is a ubiquitous spectral feature. Strong variability is a hallmark of the hard state (rms power > 10% in the band 0.1-10 Hz), while QPOs may be either present or absent. The state is associated with the presence of an AU-scale steady jet, and clear correlations between the radio and X-ray intensities are observed (Corbel et al. 2013). A major question for the hard state is the geometry of the corona: While there is significant evidence that the corona in the hard \n1 Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138 USA; [email protected], [email protected], [email protected] \n2 MIT Kavli Institute for Astrophysics and Space Research, MIT, 70 Vassar Street, Cambridge, MA 02139, USA; [email protected], [email protected] \n3 Dr. Karl Remeis-Observatory and Erlangen Centre for Astroparticle Physics, Sternwartstr. 7, 96049 Bamberg, Germany; [email protected] \nstate is compact, it is quite unclear whether, e.g., it is ADAF-like and quasi-spherical, hugs the disk, or originates at the base of a jet (Corbel et al. 2000; Shidatsu et al. 2011).', "1.1. Controversy over the Location of the Disk's Inner Edge": "In the thermal state, there is abundant evidence that the accretion disk is truncated near the innermost stable circular orbit (ISCO) (e.g. Gierli'nski & Done 2004; Steiner et al. 2010; Penna et al. 2010; Zhu et al. 2012). The standard paradigm for the faint hard state is that as the luminosity decreases the inner edge of the disk recedes from the ISCO, leaving a hot advection-dominated accretion flow (ADAF) or other coronal flow (Narayan & Yi 1994; Narayan & McClintock 2008). While there is good evidence that at very low luminosities the disk is grossly truncated (for a review, see Narayan & McClintock 2008), the location of the inner edge relative to the ISCO for luminosities in the range ∼ 0 . 1 -10% of Eddington is a hotly-debated topic. With GX 339-4 as a principal test bed, two methods have been widely used to estimate the radius R in of the inner edge of the disk in the hard state: (1) modeling the component of emission reflected from the disk, principally the Fe K line; and (2) fitting the continuum spectrum of the accretion disk. The former method, which is addressed in the following section, is the central topic of this paper. \nEfforts to estimate the inner edge of the accretion disk in the low/hard state via disk reflection go back farther, but the first strong indication that disks may remain close to the ISCO in bright phases of the low/hard state was made by Miller et al. (2006b). Based on fits to the thermal component, a number of papers claim that there is an optically thick disk that extends inward to the ISCO in the hard state (Miller et al. 2006b,a; Rykoff \net al. 2007; Reis et al. 2009, 2010; Reynolds et al. 2010). This claim is strongly contested by Done et al. (2007) and Done & Diaz Trigo (2010); the claim is all the more questionable when one considers that self-consistent disk coronal models (e.g., Steiner et al. 2009) return larger values of the inner-disk radius. More recently, Miller and coworkers have invoked extreme values of the spectral hardening factor in making the case for an untruncated hard-state disk (Reynolds & Miller 2013; Salvesen et al. 2013). This evidence for the presence of such a disk does not appear to us compelling given the difficulties of obtaining accurate estimates of R in by modeling a faint, cool ( kT glyph[lessorsimilar] 0 . 2 keV) thermal component that is strongly Comptonized and cut off by interstellar absorption.", '1.2. Reflection Spectroscopy': "The reflection spectrum results from the reprocessing of high-energy coronal photons in the optically-thick accretion disk. The result is a rich spectrum of radiative recombination continua, absorption edges and fluorescent lines, most notably the Fe K complex in the 6-8 keV energy range. This reflected radiation leaves the disk carrying information on the physical composition and condition of the matter in the strong fields near the black hole. The Fe K emission line (and other fluorescent lines) are broadened and shaped by Doppler effects, light bending and gravitational redshift. By modeling the reflection spectrum, one can estimate both the disk inclination and the dimensionless spin parameter a ∗ = cJ/GM 2 ( -1 ≤ a ∗ ≤ 1). In measuring a ∗ , one estimates the radius of the inner edge of the accretion disk and identifies it with the radius of the innermost stable circular orbit, R ISCO , which simply and monotonically maps to a ∗ (Bardeen et al. 1972). For the three canonical values of the spin parameter, a ∗ = +1, 0 and -1, R ISCO = 1 M , 6 M and 9 M (for c = G = 1), respectively. \nThe reflection model most widely used in the past for both general application and measuring black hole spin is reflionx (Ross & Fabian 2005). Recently, an improved reflection model has been developed, relxill 4 , which is based on the reflection code xillver (Garc'ıa & Kallman 2010; Garc'ıa et al. 2011, 2013, 2014a), and the relativistic line-emission code relline (Dauser et al. 2010, 2013, 2014). Compared to reflionx , relxill incorporates a superior treatment of radiative transfer and Compton redistribution, and it allows for the angular dependence of the reflected spectrum. Furthermore, by implementing the routines of the photoionization code xstar (Kallman & Bautista 2001), relxill provides an improved calculation of the ionization balance. At the same time, limitations of the model include assuming that the density of the disk is independent of vertical height, that the illuminating radiation strikes the disk at a fixed angle of 45 deg, and that apart from Fe all the elemental abundances are assumed to be solar. The results presented in this paper were derived using relxill to model the relativistically-blurred reflection component from the inner disk and xillver to model a distant reflector. \nIt is important to appreciate the faintness of the reflected features that are crucial for probing effects in the regime of strong gravity, the features that one relies on for estimating R in and constraining black hole spin. For \nexample, in the spectrum of GX 339-4, even the most prominent feature, the Fe K line, has a typical equivalent width of ∼ 0 . 1 keV, and the peak intensity of the line is only about 10% of the local continuum (Section 3). Sensitivity to such faint features requires both high-count spectra and a well-calibrated detector.", '1.3. The Special Quality of This Study': "The principal detector aboard the Rossi X-ray Timing Explorer ( RXTE ) was the Proportional Counter Array (PCA), which was comprised of five nearly identical Proportional Counter Units (PCUs), each with an effective area of 1600 cm 2 and with sensitivity from 2-60 keV. Despite the limited spectral resolution of the instrument ( ≈ 17% at 6 keV) the archive of PCA data amassed during the RXTE mission (1995-2012) continues to be preeminent for the synoptic study of stellar-mass black holes. A few-dozen bright black holes were observed daily during their outburst cycles with typical exposure times of a few ks. Some 15,000 individual spectra were obtained with a net total exposure time of 30 Ms (1 year). In this paper, we report the results of our analysis of six hard-state spectra of GX 339-4, each a summation of dozens of individual exposures (Section 3). For the spectrum obtained at maximum luminosity ( L/L Edd = 17%) with an exposure time of 46 ks, the total number of counts is 40 million and the counts-per-keV in the continuum at 6.4 keV is 4.4 million, while the total number of counts in the Fe K line region (3-10 keV) is 28 million. \nA limitation of the PCA, which has not allowed the implied statistical precision to be realized in modeling data, has been the appreciable ∼ 1% uncertainties in the detector response (Jahoda et al. 2006; Shaposhnikov et al. 2012). We have overcome this limitation by developing a calibration tool, called pcacorr , that increases the sensitivity of the RXTE PCA detector to faint spectral features - such as the Fe K line/edge - by up to an order of magnitude (Garc'ıa et al. 2014b). By applying pcacorr to a large number of spectra for three black holes, we found that the tool improved the quality of all the fits, and that the improvement was dramatic for spectra with glyph[greaterorsimilar] 10 7 counts. The tool allows one to achieve a precision of ∼ 0 . 1% rather than ∼ 1%, thereby making full use of spectra of bright sources with ∼ 10 6 counts per channel. \nConsequently, our study of the reflection spectrum of GX 339-4 greatly improves on earlier studies using the PCA, such as that by Plant et al. (2015). A limitation of PCAdata is its modest resolution, while its major advantage is its freedom from the problematic effects of pileup, which is commonly a serious problem in analyzing and interpreting data for bright sources obtained using CCD detectors (see Section 6.1.3). Another advantage of the PCA, which has only recently been matched by NuSTAR , is its high-energy coverage, which allows observations of both the Fe K region and the Compton hump using a single detector. \nThis paper is organized as follows: Section 2 describes the observations and data reduction, and Section 3 outlines our procedure for combining the individual spectra into six composite spectra. The luminosities of these spectra, which we refer to throughout as Spectra A-F, range over an order of magnitude. Fitting the spectra individually, while emphasizing the importance of correct- \nFig. 1.Hardness-intensity diagram for all PCU-2 RXTE observations of GX 339-4. The vertical axis shows the raw PCU-2 count rate (for reference, 1 Crab ≈ 2600 counts s -1 ), a proxy for the X-ray intensity and luminosity. Plotted on the horizontal axis is the hardness ratio HR defined as the ratio of source counts at 8.6-18 keV to the counts at 5-8.6 keV. Following further the conventions of Remillard & McClintock (2006), the hard-state data considered exclusively in this paper are defined to have HR > 0 . 75. The six boxes labeled A-F define the data sets we sum to create Spectra A-F, which are used in our analysis throughout the paper. \n<!-- image --> \ning the data using the pcacorr tool, is the subject of Section 4. Our key results appear in Section 5. Therein, we describe how we fit Spectra A-F simultaneously, first fixing the spin parameter and letting the inner-disk radius vary, and then allowing the spin parameter to vary while fixing the inner radius at the ISCO. We discuss our results in Section 6 and offer our conclusions in Section 7.", '2. OBSERVATIONS AND DATA REDUCTION': "Our reduction and analysis of the RXTE PCA data are detailed in Garc'ıa et al. (2014b), which follows the procedures of McClintock et al. (2006). The data were obtained in 'Standard 2' mode and segmented into contiguous intervals with exposure times ranging from 300 s to 5000 s. Background spectra, which were derived using pcabackest and the model pca bkgd cmvle eMv20111129.mdl , were subtracted from the data. Response files were generated using pcarmf (version 11.7) and the energy-to-channel conversion table (version e05v04 ) described in Shaposhnikov et al. (2012). Throughout, we analyze just the data collected using the best-calibrated detector, PCU2, which also provides the richest data set. \nAs a crucial final step, we apply the tool pcacorr (Garc'ıa et al. 2014b) to the data and thereby calibrate the detector to a precision of ∼ 0 . 1%; we include a systematic error of this magnitude in all our data analysis. This step greatly enhances the sensitivity of the detector (Section 4.3) to the reflection features that are our focus. \nFigure 1 presents a hardness-intensity diagram, PCA count rate versus PCA hardness ratio HR , for all 1471 RXTE PCU-2 observations of GX 339-4 obtained be- \ntween 1996 July 26 and 2011 April 5. The hard-state data considered exclusively in this paper are defined to have HR > 0 . 75. In order to boost the signal-to-noise, we define the six boxes A-F shown in Figure 1. Each box contains a number of spectra, all of them corresponding to roughly the same source intensity. We combine all the spectra within a box using the procedures described in Section 3; importantly, we do not combine spectra obtained during different outburst cycles. Except for Box A, which is comprised of observations taken during the 2002 outburst, all the other boxes contain observations taken during the 2010 outburst. Ultimately, we produce six master spectra (A-F), one for each box.", '3. COMBINING SPECTRA': 'We now outline our procedure for combining the individual spectra in a box to create Spectra A-F in such a way as to eliminate small variations in the power-law index under the assumption that the reflection features are unaffected by small changes in the continuum. For each box separately, we first fitted the individual spectra to a simple absorbed power-law ( Tbabs*powerlaw ) using a fixed hydrogen column density of N H = 3 × 10 21 cm -2 , which is similar to the expected column in the direction of GX 339-4 (Kalberla et al. 2005). No evidence for a thermal component was found in any of the spectra. The fits were performed in the 3-45 keV band where the Fe K features are most pronounced 5 . We then created individual residual spectra (data counts minus model) and summed them, thereby greatly enhancing the residual features present in these spectra. \nFigure 2 shows the residual spectra for the six boxes. The striking features in each spectrum are the Fe K line and the K edge, which are revealed with precision in these high signal-to-noise spectra. Surprisingly, the overall structure of the residuals are in all cases quite similar, despite the factor of ∼ 10 spread in luminosity (Figure 1). Upon closer examination, however, one sees that the line width, the position of the edge, and the shape of the Compton hump differ to some degree among the boxes. This point is discussed in more detail in Section 5.1. \nWe now use these residuals to create Spectra A-F, which constitute our prime data set. To be specific, consider the creation of Spectrum A: For box A we generate a single template continuum spectrum using the average values of the fit parameters Γ and the normalization N . This spectrum is generated synthetically using the fakeit task in xspec ; its net exposure time is the sum of the exposure times of all the 23 individual spectra in box A. Finally, we add this continuum spectrum to its corresponding summed residual spectrum to complete the generation of Spectrum A. \nThe resulting spectrum is superior to that obtained by simply summing the individual spectra directly because it seamlessly eliminates the effects of small differences in the power-law index and normalization among the spectra. The mean power-law parameters and other information describing these six spectra, which are hereafter our focus, are summarized in Table 1. We compute the luminosity for a spectrum using our model fluxes in the 1-100 keV band and assuming a distance of D = 8 kpc and black hole mass of M = 10 M glyph[circledot] (for details, see Footnote a to Table 1).', '4. FITTING SPECTRA A-F INDIVIDUALLY': "The residual plots in Figure 2 unambiguously demonstrate that a strong reflected component is present, which is widely attributed to the illumination of the disk by a hot corona. Invoking this paradigm, we proceed to fit each of the six spectra using our physically-motivated reflection code relxill v0.2g (Garc'ıa et al. 2014a). As before, Galactic absorption is modeled using Tbabs with fixed column density ( N H = 3 × 10 21 cm -2 ). For the Tbabs model (Wilms et al. 2000), we used the Anders & Grevesse (1989) set of solar abundances and the Verner et al. (1996) photoelectric cross sections. \nWe fit Spectra A-F in turn to a succession of four models; the final adopted model in each case yields a good fit with χ 2 /ν ∼ 1. Table 2 provides detailed information on the quality of the fit for each spectrum and each model. In Figure 3, we show for Spectrum A with 4 × 10 7 counts - the most challenging case - residual plots for the progression of the four models, which we now describe. \nModel 0: Tbabs*powerlaw . An absorbed power-law model, which is clearly deficient, prominently displays the principal reflection features, the Fe K line/edge and Compton hump, in the residuals (Figure 3). \nModel 1: Tbabs*relxill . A greatly improved fit to all six spectra is achieved by replacing the power-law with our fully relativistic reflection model. For simplicity and to achieve definiteness, we fix the spin to its extreme value of a ∗ = 0 . 998 and assume the canonical dependence of disk emissivity with radius, namely ∝ r -3 . This model already delivers fits of reasonable quality (Table 2). Some pronounced residual features remain, which are most evi- \nfor the most luminous case, Spectrum A (Figure 3). Specifically, two apparent absorption features flank the Fe K line at ∼ 5 . 6 and ∼ 7 . 2 keV. Other features are also present at higher energies in the region of the Compton hump ( ∼ 20 -45 keV). \nModel 2: Tbabs*(relxill+xillver) . The residuals are significantly reduced by including an unblurred reflection component via xillver . Physically, this reflector could be cold material in a wind or in the outer region of a flared disk (see below). The xillver parameters are linked to those of relxill with two exceptions: The ionization parameter was fixed at its minimum value, log ξ = 0, and the Fe abundance was fixed to solar (i.e., A Fe = 1). Linking the Fe abundance results in a significantly worse fit and a compromise value of abundance that is midway between the low value required by the unblurred component and the super-solar value required by the blurred component (see Section 6.1.4). This result, and the uncertain origin of the unblurred component, motivate our choice of solar Fe abundance for the distant reflector. We have no good explanation for the different Fe abundances required in fitting the blurred and unblurred reflection components. Further discussion on the Fe abundance is presented in Section 6.1.4. \nInclusion of the xillver component, which introduces only one new free parameter, namely its normalization, quite significantly improves the fit to all the spectra except Spectrum F, which has the fewest counts. While the xillver component improves the fit at low energies and in the region of the Compton hump, a strong residual feature remains at ∼ 7 . 2 keV (Figure 3). \nModel 3: Tbabs*(relxill+xillver)*gabs . We model the remaining residual feature near 7.2 keV phenomenologically as absorption using a single Gaussian. The addition of this component improves the fits substantially for Spectra A-C, i.e., those with many counts, while it has only a marginal effect for Spectrum D and a negligible effect for Spectra E and F (Table 2). As expected, its importance is greatest for Spectrum A where it completely eliminates the strong 7.2 keV residual feature (Figure 3) and produces a very good fit to this spectrum, despite its extreme statistical precision (4 × 10 7 total counts), with an allowance for systematic error of only 0.1% (Section 2). \nIt is important to note that the inclusion of the 7.2 keV feature has a significant effect on some important model parameters. In particular, we find that including the Gaussian component (Model 3) increases the inclination and decreases the inner-disk radius by about 4 degrees and 20%, respectively, compared to excluding the component. The changes in the other fit parameters are relatively much smaller. We adopt Model 3 as our fiducial model for all six spectra, thereby assuming that the 7.2 keV absorption feature has a physical origin. While it is beyond the scope of this work to establish a definite physical interpretation of the feature, we now briefly consider some plausible explanations.", '4.1. The 7.2 keV absorption feature and the efficacy of the xillver component': 'We first consider the likely possibility that the absorption feature is largely an artifact related to the uncertain energy resolution of the PCA. We then discuss the one plausible physical explanation for the feature known to \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFig. 2.Residuals (data-minus-model) are computed for each box (defined in Figure 1) by subtracting an absorbed power-law fit to the individual spectra in the box. The extreme statistical precision results from summing millions of counts, ranging from ∼ 28 million for Box A to ∼ 3 million for Box F (3-10 keV). Note that the spectra are all scaled differently; e.g., the peak signal for the highest-luminosity Box A is ∼ 15 times greater than for the lowest-luminosity Box F. Remarkably, the appearance of the spectra is quite insensitive to luminosity. \n<!-- image --> \nTABLE 1 Properties of Spectra A-F and the boxes defined in Figure 1 \n| Spectrum | L/L Edd (%) a | Count Rate | # Spectra | < Γ > | < N > | Exp. (ks) |\n|------------|-----------------|--------------|-------------|-----------------|-----------------|-------------|\n| A | 17.3 | 1000 - 1100 | 23 | 1 . 72 ± 0 . 01 | 2 . 12 ± 0 . 11 | 45.7 |\n| B | 14.2 | 800 - 900 | 7 | 1 . 75 ± 0 . 03 | 1 . 81 ± 0 . 10 | 10.3 |\n| C | 11.9 | 600 - 700 | 11 | 1 . 69 ± 0 . 01 | 1 . 22 ± 0 . 06 | 27 |\n| D | 7.9 | 350 - 400 | 7 | 1 . 61 ± 0 . 60 | 0 . 60 ± 0 . 03 | 15.7 |\n| E | 3.9 | 150 - 200 | 18 | 1 . 52 ± 0 . 01 | 0 . 23 ± 0 . 02 | 24.6 |\n| F | 1.6 | 50 - 100 | 43 | 1 . 59 ± 0 . 08 | 0 . 11 ± 0 . 03 | 72.7 | \na Note. - Eddington-scaled luminosities assuming D = 8 kpc and M = 10 M glyph[circledot] (corresponding to L Edd = 1 . 25 × 10 39 erg s -1 ), and based on the fluxes computed over the 1-100 keV band using the model and fit parameters summarized in Table 3. \nus, namely that it is produced by absorption in a highlyionized wind. Finally, we consider the role of the xillver component not only in modeling the residual features near the Fe K line, but also its role in improving the fit quite generally.', '4.1.1. On the accuracy of the PCA energy resolution': 'The presence of residual absorption features bracketing the Fe line at ∼ 5 . 6 keV and ∼ 7 . 2 keV suggests the possibility that the PCA resolution may be better than assumed in generating the PCU-2 response. We have explored this possibility for Spectrum A. We test the effects of slight changes in the value assumed for the detector resolution by moderately smoothing the data, with the results shown in Figure 4. The smoothing is accomplished using a Gaussian kernel operating over the detector channels; the parameter f specifies the width of the Gaussian as a percent of the channel width. Accordingly, the curve in Figure 4 labeled f = 0 is unsmoothed, while the curves labeled f = 40 and f = 50 correspond to degrading the resolution of the data by 0.9% and 1.5% at 6.4 keV. \nAlthough this approximate approach to artificially tuning the detector resolution does not eliminate the residu- \nals flanking the Fe line, it does significantly reduce their strength. The test demonstrates that at this extreme level of statistical precision the fit to a line feature is very sensitive to the value assumed for the detector resolution. Specifically, if one assumes that the nominal value of resolution for the unsmoothed case ( f = 0) at 6 keV is 17.0%, then the net resolution for f = 50 is 17.3% (i.e., the additional blurring has a width of one-half channel, equivalent to ∼ 0 . 2 keV at 6.4 keV, which is combined in quadrature with the nominal resolution width). Meantime, the resolution of the PCA is not known to sufficient accuracy to discriminate such fine differences (N. Shaposhnikov, private communication). This suggests that the residuals near the Fe line may result from the PCA resolution being slightly better than assumed in modeling the detector response. However, this test is inconclusive. To properly assess the importance of tuning the resolution, one must carry out a systematic analysis using the PCA calibration software, which is beyond the scope of this paper. \n4.1.2. On the possibility that the feature originates in a highly ionized wind \n<!-- image --> \nFig. 3.Data-to-model ratio ( left ) and contributions to χ 2 ( right ) for Spectrum A resulting from fitting a sequence of four models. From top to bottom, the models increase in completeness and competence, starting with an absorbed power-law model to which, incrementally, is added a blurred reflection component ( relxill ); unblurred reflection ( xillver ); and a Gaussian absorption line gabs . The comparable residual plots for the other five spectra are qualitatively similar. \n<!-- image --> \nTABLE 2 Statistics of the individual fits to Spectra A-F \n| Spectrum | Model | χ 2 | ν | χ 2 ν | ∆ χ 2 / ∆ ν |\n|------------|---------|----------|-----|----------|---------------|\n| A | 0 | 25094.9 | 69 | 363.694 | |\n| | 1 | 299.07 | 63 | 4.747 | 4132.64 |\n| | 2 | 151.07 | 62 | 2.437 | 148.00 |\n| | 3 | 67.82 | 60 | 1.13 | 41.63 |\n| B | 0 | 7653.27 | 69 | 110.917 | |\n| | 1 | 117 | 63 | 1.857 | 1256.05 |\n| | 2 | 81.16 | 62 | 1.309 | 35.84 |\n| | 3 | 54.14 | 60 | 0.902 | 13.51 |\n| C | 0 | 11849.3 | 70 | 169.276 | |\n| | 1 | 142.44 | 64 | 2.226 | 1951.14 |\n| | 2 | 93.21 | 63 | 1.48 | 49.23 |\n| | 3 | 68.68 | 61 | 1.126 | 12.27 |\n| D | 0 | 4880.44 | 70 | 69.721 | |\n| | 1 | 89.91 | 64 | 1.405 | 798.42 |\n| | 2 | 41.36 | 63 | 0.657 | 48.55 |\n| | 3 | 35.69 | 61 | 0.585 | 2.84 |\n| E | 0 | 2552.13 | 70 | 36.459 | |\n| | 1 | 116.4 | 65 | 1.791 | 487.15 |\n| | 2 | 65.97 | 64 | 1.021 | 50.43 |\n| F | 0 | 2311.81 | 70 | 33.026 | |\n| | 1 | 63.17 | 65 | 0.9719 | 449.73 |\n| | 2 | 63.17 | 64 | 0.987 | 0.0 | \nIf the 7.2 keV feature is not an instrumental artifact, a potentially plausible explanation is that it originates in a highly ionized wind that envelops the primary source. Disk winds have been observed in many black hole binary systems, particularly at high accretion rates (e.g., Ponti et al. 2012; Neilsen et al. 2012; Miller et al. 2006c). We investigated this possibility by replacing the gabs component by the photoionized warm absorber model \n( warmabs ). We forced the Fe Ly α line at ∼ 6 . 9 keV to be the dominant feature by setting the ionization parameter to its maximum value (log ξ = 5). We linked the Fe abundances of warmabs and relxill while the abundances of all the other elements remain at solar. The fitted blueshift of the Fe Ly α required to model the 7.2 keV feature is z = 0 . 0576 ± 0 . 0101, which corresponds to an outflow velocity of v = 1 . 7 × 10 4 km s -1 . The model provides a good fit ( χ 2 ν = 1 . 17), which is very comparable to that achieved using Model 3 (see the top-left panel of Figure 5 for details and a comparison of the residuals). However, this interpretation seems unlikely on physical grounds due to the extreme column density required by the warm absorber, namely, N abs H = (7 . 7 ± 0 . 2) × 10 23 cm -2 . If one links the warmabs Fe abundance to that of the xillver component (i.e., A Fe = 1), the fit pegs at the hard limit of the warmabs model (10 24 cm -2 ).', '4.1.3. On the inclusion of the unblurred reflection component': 'Our initial motivation for including the unblurred xillver component of reflection was the presence of the ∼ 5 . 6 keV and ∼ 7 . 2 keV residual features flanking the Fe line. However, as Figure 3 makes clear, while the xillver component effectively eliminated the low-energy feature, it actually enhanced the 7.2 keV feature. The simplest ad hoc phenomenological approach to eliminating both features is to include a pair of Gaussian absorption lines in our model, which we did, fixing the widths of the Gaussians to 0.01 keV and allowing the energy and strength of each line to vary. While this model ( Tbabs*relxill*gabs*gabs ) does clean up the ∼ 5 . 6 keV and ∼ 7 . 2 keV features, the quality of the fit, χ 2 ν = 2 . 36 ( χ 2 = 139 . 25 for 59 d.o.f.), is much poorer \nthan that achieved with Model 3, χ 2 ν = 1 . 13 ( χ 2 = 67 . 82 for 60 d.o.f.), which uses xillver and a single Gaussian. As the top-right panel of Figure 5 makes clear for the stringent case of Spectrum A, the latter model not only does a better job cleaning up the pair of targeted residual features, it is also more effective at improving the fit at most other energies as well. We consider this strong evidence for the presence in GX 339-4 of a distant reflector.', '4.2. Comparing combined versus summed spectra': 'As discussed in Section 3, we combined the individual spectra in order to increase our sensitivity to the reflection features while minimizing the effects of jitter in the power-law index. To verify the procedures we used in combining spectra, we compare the results of fitting our Spectrum A to those obtained by fitting a spectrum created by summing directly all the spectra in Box A 6 . Applying pcacorr and fitting both spectra with our adopted model (i.e., Tbabs*(relxill+xillver)*gabs ), we find that the model parameters are all consistent. However, the fit to the summed spectrum is of significantly lower quality (∆ χ 2 = 62 . 97) than the fit to Spectrum A created using the procedures described in Section 3. Furthermore, as shown in the bottom-left panel of Figure 5, the residuals for the summed spectrum are larger in almost every energy channel. These results demonstrate that our method of combining the individual spectra significantly improves the quality of the fit.', '4.3. The importance of applying the pcacorr tool': "As fully described in Garc'ıa et al. (2014b) and discussed in Section 1.3, the pcacorr tool greatly reduces the effects of instrumental features in PCA spectra, thereby making it possible to achieve good fits to high-count spectra at the 0.1% level of statistical precision. The bottom-right panel in Figure 5 demonstrates the importance of applying the tool to Spectrum A, with its 4 × 10 7 counts. The figure compares residuals for a fit to uncorrected data to one using pcacorr -corrected data. To most clearly illustrate the power of pcacorr , we use Model 1 ( Tbabs*relxill ) and set the systematic errors to zero. Concerning the ∼ 5 . 6 keV and ∼ 7 . 2 keV features flanking the Fe K line, we note that they are present prior to the application of pcacorr , which confirms that they are not introduced by the correction. The key message of the bottom-right panel in Figure 5 is the degree to which pcacorr diminishes these features and others, especially those below 10 keV and the one near 30 keV, a feature that is likely related to the detector Xe K-edge (Shaposhnikov et al. 2012; Garc'ıa et al. 2014b).", '5. FITTING SPECTRA A-F SIMULTANEOUSLY': "In order to achieve the strongest possible constraints on the key model parameters, we fitted Spectra A-F simultaneously. This composite data set is an assemblage of 106 individual PCA/PCU-2 spectra of GX 339-4 in the hard state. The total number of counts is 77 million, 34 million of which are in the 3-10 keV Fe K band. \nAll fits are performed using Model 3 ( const*Tbabs*(relxill+xillver)*gabs ). To accommodate the order-of-magnitude range of luminosity, \nFig. 4.Residual plots (data-model) illustrating the extreme sensitivity in fitting the Fe K line to the value assumed for the energy resolution of PCU-2. The residuals are for fits to Spectrum A using Model 1 ( Tbabs*relxill ). The resolution of the data have been slightly degraded by convolving them with a Gaussian whose smoothing width is characterized by the parameter f : The cases f = 40 (green) and f = 50 (blue) correspond to decreases in the resolution of 0.9% and 1.5%, respectively, and f = 0 (red) is the unsmoothed case. \n<!-- image --> \nwe included a normalization constant that is unity for Spectrum A and floats for Spectra B-F. Where sensible, key physical parameters are tied: the spin a ∗ of the black hole; inclination i of the system; line-of-sight column density N H ; Fe abundance A Fe ; and the normalization of the relativistic reflection component N r (see Table 3). Given the uncertain origin of the absorption feature near 7.2 keV, which we model as a Gaussian, we also tie the central energy of this component while allowing its normalization to vary from spectrum to spectrum. The width of the Gaussian was fixed at 0.01 keV in all the spectra. Like the Gaussian normalization, all other model parameters are allowed to vary independently. \nDespite the extreme signal-to-noise of the composite spectrum, we must impose some additional assumptions in order to simultaneously constrain both the spin parameter a ∗ and the radius of the inner edge of the accretion disk R in . This is because these two quantities, which manifest almost indistinguishably in the red wing of the Fe K line profile, are extremely degenerate (Dauser et al. 2013). Therefore, we have conducted two complementary analyses, which we will refer to as jf-i and jf-ii , where jf signifies that these are joint fits (i.e., simultaneous) to Spectra A-F (rather than the fits to Spectrum A alone that are featured in earlier sections). For jf-i , our aim is to constrain R in , and we therefore keep the spin fixed at its maximum value of a = 0 . 998. For jf-ii , we tie the spin parameter for the six spectra and fit for it, while fixing the inner edge of the disk at the ISCO; i.e., R in = R ISCO . We follow the guidelines of Fabian et al. (2012) and fix the emissivity index to its canonical value of 3; for both jf-i and jf-ii , we do not attempt to fit for this parameter because of the PCA's limited spectral resolution. This choice is motivated by several jf-i and jf-ii tests we performed that returned values of the emissivity index that were always glyph[lessorsimilar] 4. \nThe number of free fitting parameters is large, 52 for jf-i and 47 for jf-ii . The complexity of the analysis dictated our approach: We performed Markov Chain Monte-Carlo (MCMC) runs using the emcee-hammer Python package (Foreman-Mackey et al. 2013), which \nFig. 5.Comparative plot showing contributions to the total χ 2 (data-model) for each channel for fits to Spectrum A: ( top-left ) The red curve was computed for our adopted Model 3 ( Tbabs*(relxill+xillver)*gabs ) and is identical to the plot shown in the lower-right panel in Figure 3. The blue curve is for an alternative model that substitutes the warm absorber model wabs for gabs in our adopted Model 3. The alternative model provides a good fit to the data: χ 2 = 69 . 06 for 59 d.o.f. ( χ 2 ν = 1 . 130); compare χ 2 values for Model 3 in Tables 3 and 4. ( top-right ) The red curve was computed for our adopted Model 3 and is identical to the plot shown in the lower-right panel in Figure 3. The blue curve is for an alternative model that replaces the unblurred ( xillver ) reflection component by a second Gaussian absorption line at ∼ 5 . 6 keV (blue), which results in a distinctly inferior fit. As this residual plot shows, Model 3 performs better at almost all energies. ( bottom-left ) Fits of our adopted Model 3 for two cases: (1) A direct sum of the 23 spectra in Box A (blue), and (2) a fit to Spectrum A (red), which was prepared by combining the spectra according to the procedures described in Section 3. While the model parameters are consistent for the two cases, Spectrum A provides a superior fit, as this comparison of the residuals makes clear. ( bottom-right ) Fits using Model 1 ( Tbabs*relxill ) for two cases: (1) The 23 individual spectra that comprise Spectrum A are corrected using the pcacorr tool (blue) and (2) they are left uncorrected (red). In this instance only, all systematic errors have been zeroed to most clearly illustrate the effect of applying pcacorr to the data. \n<!-- image --> \nimplements affine-invariant sampling. MCMC methods are powerful for high-dimensional analysis. Specifically, they enable an efficient exploration of parameter space and determine a posterior probability structure for the model of interest.", '5.1. Joint Fit I: Fixed spin and variable inner radius': 'A principal goal of our study is to track the radius of the inner edge of the disk R in as the luminosity varies by an order of magnitude (i.e., over the range 1.6-17% of Eddington; Table 1). As discussed in Section 1.1, a question of great interest is whether the inner disk is truncated in the hard state at low luminosities and, if so, to what extent. In order to be maximally sensitive to a disk that is only slightly truncated, we fix the spin to its maximum allowed value, namely a ∗ = 0 . 998. In so doing, our focus is on determining how R in trends with luminosity rather than obtaining accurate estimates of this parameter. We note that most spin determinations in the literature (which assume R in = R ISCO ) suggest that the spin is high (see Section 6.1.3). \nFigure 6 shows the fit residuals for jf-i for two cases: \n(1) The top panel shows a data-to-model ratio for a fit to Model 0, i.e., the simple absorbed power-law model used to produce the residual plot for a fit to Spectrum A only (which is shown in the left-top panel of Figure 3). This simple fit prominently displays the reflection features, which are strong for all six spectra. The profile of the Fe K line shows moderate variations among the spectra. As the luminosity increases, so does the intensity of the line (cf. Figure 2). Additionally, the blue wing of the line extends to higher energies for the high luminosity spectra, which could be evidence for a shift in the Fe K edge caused by an increase in the ionization of the gas. There are obvious changes at high energies among the spectra, which are likely due to the evolution of the highenergy cutoff with luminosity. (2) Of chief interest, the middle and bottom panels of Figure 6 show the residuals for our adopted Model 3 (for a ∗ = 0 . 998). \nModel 3 performs remarkably well for all spectra as indicated by the goodness of fit, χ 2 ν = 1 . 06, and also by the uniformly ergodic appearance of the residuals across the energy band. The fit results are summarized in Table 3. A key result, which is discussed in detail in the \nTABLE 3 \nResults for jf-i: Fit parameters for Model 3, const*Tbabs*(relxill+xillver)*gabs , with fixed maximum spin a ∗ and free R in . \nTABLE 4 \n| Model | Parameter | Spectrum A | Spectrum B | Spectrum C | Spectrum D | Spectrum E | Spectrum F |\n|-----------------|---------------|----------------------------------|----------------------------|-------------------------------|--------------------------------------|------------------------------|--------------------------------|\n| Tbabs | N H (cm - 2 ) | (7 . 0 ± 1 . 0) × 10 21 | | | | | |\n| relxill | a ∗ | 0 . 998 | | | | | |\n| relxill | i (deg) | | | 48 . 4 ± | 1 . 1 | | |\n| relxill | A Fe | | | 4 . 6 +0 - 0 | . 5 . 3 | | |\n| relxill | N r | | | 1 . 48 +0 - 0 | . 05 . 03 | | |\n| gabs | E (keV) | | | 7 . 19 +0 0 | . 04 | | |\n| | | | | - | . 06 | | |\n| Constant | R in ( R ) | 1 +0 . 2 | 0 . 91 ± 0 . 04 +0 . 4 | 0 . 71 ± 0 . 02 1 . 8 +0 . 1 | 0 . 36 ± 0 . 01 2 . 1 +0 . 3 - 0 . 5 | 0 . 16 ± 0 . 01 2 . 7 +0 . 5 | 0 . 08 ± 0 . 01 3 . 7 +0 . 7 0 |\n| relxill | ISCO | 1 . 7 - 0 . 3 | 1 . 5 - 0 . 2 | - 0 . 2 | 1 . 628 ± 0 . 015 | - 1 . 5 | - 1 . |\n| relxill | Γ | 1 . 620 ± 0 . 013 3 . 31 +0 . 03 | 1 . 682 ± 0 . 016 3 07 | 1 . 672 ± 0 . 013 07 | 3 . 031 +0 . 020 | 1 . 588 ± 0 . 010 | 1 . 648 +0 . 007 - 0 . 012 |\n| relxill relxill | log ξ E cut | - 0 . 07 97 +3 - 5 | . 24 ± 0 . 129 ± 10 | 3 . 12 +0 . - 0 . 03 179 ± 14 | - 0 . 013 660 +130 - 170 | 2 . 02 ± 0 . 17 > 840 | 2 . 05 +0 . 26 - 0 . 10 > 890 |\n| relxill | R f | 0 . 21 ± 0 . 02 | 0 . 22 ± 0 . 03 | 0 . 21 ± 0 . 03 | 0 . 34 +0 . 04 - 0 . | 0 . 31 ± 0 . 03 | 0 . 31 +0 . 02 - 0 . 03 |\n| xillver | N x | 0 . 27 +0 . 02 - 0 . 03 | 0 . 25 ± 0 . 04 | 0 . 25 ± 0 . 04 | 05 0 . 34 +0 . 04 | < 0 . 05 | < 0 . 03 |\n| gabs | Strength | 0 . 021 +0 . 009 - 0 . 007 | 0 . 029 +0 . 017 - 0 . 012 | 0 . 036 ± 0 . 016 | - 0 . 05 0 . 05 +0 . 07 - 0 . 03 | 0 . 08 +0 . 16 - 0 . 05 | 0 . 14 +0 . 10 - 0 . 05 |\n| | L/L Edd (%) | 17.3 | 14.2 | 11.9 | 7.9 | 3.9 | 1.6 |\n| | χ 2 | 402.49 | 402.49 | 402.49 | 402.49 | 402.49 | 402.49 |\n| | ν 2 | | | 379 | | | | \nNote . - For the given model components, the parameters from top to bottom are: hydrogen column density ( N H ); dimensionless spin parameter ( a ∗ = cJ/GM 2 , where J is the angular momentum of the black hole); inclination of the inner disk ( i ); iron abundance with respect to its solar value ( A Fe ); normalization of the blurred reflection component plus power-law continuum ( N r ); energy of the absorption Gaussian centroid ( E ); constant multiplicative factor between spectra; inner-disk radius ( R in ), with R ISCO = 1 . 237 R g for a ∗ = 0 . 998 ( R g = GM/c 2 ); power-law photon index (Γ); log of the ionization parameter ( ξ = 4 πF x /n , where F x is the ionizing flux and n is the gas density); high-energy cutoff ( E cut ); reflection fraction ( R f , ratio of the reflected flux to that in the power-law, in the 20-40 keV band); normalization of the distant (unblurred) reflection ( N x ); strength of the absorption Gaussian; X-ray luminosity in terms of Eddington (see notes in Table 1); goodness of the fit ( χ 2 ); number of degrees of freedom ( ν ); goodness of the fit per degree of freedom ( χ 2 ν = χ 2 /ν ). Uncertainties are based on a 90% confidence level. \nResults for jf-ii: Fit parameters for Model 3, const*Tbabs*(relxill+xillver)*gabs , with a ∗ free and R in = R ISCO . \n| Model | Parameter | Spectrum A | Spectrum B | Spectrum C | Spectrum D | Spectrum E | Spectrum F |\n|--------------|-----------------|---------------------------------|---------------------------------|---------------------------------|---------------------------------|---------------------------------|---------------------------------|\n| Tbabs | N H (cm - 2 ) | (5 . 9 +0 . 6 - 1 . 9 ) × 10 21 | (5 . 9 +0 . 6 - 1 . 9 ) × 10 21 | (5 . 9 +0 . 6 - 1 . 9 ) × 10 21 | (5 . 9 +0 . 6 - 1 . 9 ) × 10 21 | (5 . 9 +0 . 6 - 1 . 9 ) × 10 21 | (5 . 9 +0 . 6 - 1 . 9 ) × 10 21 |\n| relxill | a ∗ | 0 . 95 +0 - 0 . 05 | | | . 03 | | |\n| relxill | i (deg) | | | 47 . 8 | +0 . 9 - 1 . 4 | | |\n| relxill | A Fe | 5 . 4 +1 - 0 . 5 | | | . 9 | | |\n| | N r | 1 . 44 +0 . 04 | 1 . 44 +0 . 04 | 1 . 44 +0 . 04 | 1 . 44 +0 . 04 | 1 . 44 +0 . 04 | 1 . 44 +0 . 04 |\n| relxill gabs | E (keV) | - 0 . 08 | - 0 . 08 | 7 . 23 ± | 0 . 08 | - 0 . 08 | - 0 . 08 |\n| Constant | | 1 | 0 . 90 +0 . 03 - 0 . 04 | 0 . 71 +0 . 02 - 0 . 03 | 0 . 37 ± 0 . 02 | 0 . 17 ± 0 . 01 | 0 . 08 ± 0 . 01 |\n| relxill | R in ( R ISCO ) | | | | 1 | | |\n| relxill | Γ | 1 . 604 +0 . 010 - 0 . 027 | 1 . 658 ± 0 . 018 | 1 . 651 +0 . 015 - 0 . 022 | 1 . 62 +0 . 02 - 0 . 04 | 1 . 578 +0 . 009 - 0 . 013 | 1 . 637 +0 . 009 - 0 . 013 |\n| relxill | log ξ | 3 . 33 ± 0 . 03 | 3 . 35 ± 0 . 04 | 3 . 16 +0 . 10 - 0 . 05 | 3 . 05 +0 . 04 - 0 . 02 | 1 . 96 +0 . 12 - 0 . 21 | 2 . 0 +0 . 2 - 0 . 2 |\n| relxill | E cut | 92 +2 - 6 | 118 ± 8 | 160 +12 - 16 | 440 +230 - 110 | > 830 | > 940 |\n| relxill | R f | 0 . 20 ± 0 . 01 | 0 . 20 +0 . 02 - 0 . 01 | 0 . 20 ± 0 . 02 | 0 . 27 +0 . 03 - 0 . 05 | 0 . 25 +0 . 05 - 0 . 03 | 0 . 28 +0 . 02 - 0 . 04 |\n| xillver | N x | 0 . 23 +0 . 05 - 0 . 02 | 0 . 26 ± 0 . 04 | 0 . 24 ± 0 . 05 | 0 . 13 ± 0 . 06 | 0 . 08 ± 0 . 06 | < 0 . 09 |\n| gabs | Strength | 0 . 024 +0 . 009 - 0 . 007 | 0 . 028 +0 . 018 - 0 . 011 | 0 . 037 +0 . 016 - 0 . 012 | 0 . 04 +0 . 03 - 0 . 02 | 0 . 025 ± 0 . 014 | 0 . 020 +0 . 017 - 0 . 009 |\n| | L/L Edd (%) | 17.3 | 14.2 | 11.9 | 7.9 | 3.9 | 1.6 |\n| | χ 2 | 418.66 | 418.66 | 418.66 | 418.66 | 418.66 | 418.66 |\n| | ν 2 | 384 | 384 | 384 | 384 | 384 | 384 | \nNote . - For the given model components, the parameters from top to bottom are: hydrogen column density ( N H ); dimensionless spin parameter ( a ∗ = cJ/GM 2 , where J is the angular momentum of the black hole); inclination of the inner disk ( i ); iron abundance with respect to its solar value ( A Fe ); normalization of the blurred reflection component plus power-law continuum ( N r ); energy of the absorption Gaussian centroid ( E ); constant multiplicative factor between spectra; inner-disk radius ( R in ); power-law photon index (Γ); log of the ionization parameter ( ξ = 4 πF x /n , where F x is the ionizing flux and n is the gas density); high-energy cutoff ( E cut ); reflection fraction ( R f , ratio of the reflected flux to that in the power-law, in the 20-40 keV band); normalization of the distant (unblurred) reflection ( N x ); strength of the absorption Gaussian; X-ray luminosity in terms of Eddington (see notes in Table 1); goodness of the fit ( χ 2 ); number of degrees of freedom ( ν ); goodness of the fit per degree of freedom ( χ 2 ν = χ 2 /ν ). Uncertainties are based on a 90% confidence level. \nFig. 6.A simultaneous fit to Spectra A-F comprising a total of 106 individual spectra and 77 million counts. ( top ) Ratio plots for a joint fit ( jf-i ) using Model 0, the simple absorbed power-law model, which strikingly reveal the principal signatures of reflection. ( middle ) Ratio plots for these same data obtained by fitting our canonical Model 3 ( Tbabs*(relxill+xillver)*gabs ). This model produces an excellent fit with χ 2 ν = 1 . 06. ( bottom ) Contributions to the total χ 2 (data-model), again for Model 3 and the same data. following section, is evidence that the disk is moderately - but significantly - truncated at the lowest luminosities that can be effectively explored using RXTE (i.e., at a few percent of Eddington). \n<!-- image -->', '5.2. Joint Fit II: Constraining the spin of the black hole': 'In order to obtain constraints on black hole spin using either leading method, reflection spectroscopy or continuum fitting, one must assume that R in = R ISCO (e.g., Reynolds 2014; McClintock et al. 2014). In performing jf-ii , we make this assumption for all six spectra in order to constrain the spin of the black hole. Doing so allows us to obtain a precise estimate of spin: a ∗ = 0 . 95 +0 . 03 -0 . 05 at 90% confidence. As the summary of results in Table 4 shows, the other parameters are quite close to those obtained in jf-i (Table 3), and the goodness of fit is of very comparable quality: χ 2 ν = 1 . 09. Given the extreme statistical precision, the residual spectra (data/model and χ 2 ) for jf-ii (not shown) are essentially indistinguishable by eye from the spectra for jf-i , which are shown in Figure 6. \nConcerning our spin estimate and the fundamental assumption that R in = R ISCO , we again note that jf-i provides evidence for disk truncation at our lowest luminosities 7 . Meantime, these low-luminosity spectra are included in computing our single, tied jf-ii estimate of spin. The incorporation of disk truncation effects (which we have ignored) would imply an even higher spin value than is quoted above.', '6. DISCUSSION': "The MCMC runs utilized 120 ( jf-i ) and 100 ( jf-ii ) 'walkers,' each navigating a chain with a length of \n100,000 elements, after having been initialized in a cluster distributed about the best fit. The first 50,000 elements of each walker were discarded in the 'burn-in' phase during which the chain reaches its stationary state. The typical autocorrelation length, which is the interval over which the chain forgets its previous location, was several thousand elements; the corresponding net number of independent samples of the parameter space was ∼ 10 4 . From the full distribution, we trivially obtain a probability distribution for any given set of parameters of interest by marginalizing over all the parameters that were outside that set. Flat priors were adopted for all model parameters.", '6.1. The four intrinsic parameters of the system': 'We first discuss four parameters that are global for GX 339-4 (i.e., the same for all six spectra), namely, the Galactic hydrogen column density N H , the spin parameter a ∗ of the black hole, the inclination i , and the Fe abundance A Fe . The entries for these (and all other) parameters estimated in the MCMC analysis given in Tables 3 and 4 are 90% minimum-width confidence intervals about the posterior maxima.', '6.1.1. Hydrogen column density': "Despite the limited low-energy coverage of the PCA, the hydrogen column density is well-constrained in both jf-i and jf-ii to N H = (6 . 5 +0 . 8 -1 . 5 ) × 10 21 cm -2 (Tables 3 and 4), which is consistent with other estimates in the literature, including (in units of 10 21 cm -2 ) 4-6 (Kong et al. 2000); 6 (Zdziarski et al. 2004); 5.4 (Shidatsu et al. 2011); and 5-8 (M'endez & Klis 1997).", '6.1.2. Inclination of the inner disk': "Previous estimates of inclination, which have been obtained by modeling the reflected component, run the gamut (see Table 5). Three papers using the same XMMNewton and RXTE data uniformly report low values: Miller et al. (2006b) and Reis et al. (2008) obtained i = 20 +5 -15 deg and i glyph[lessorsimilar] 20 deg, respectively, while Done & Diaz Trigo (2010), using a different strategy for reducing the data, found i ∼ 20 -27 deg. A much larger inclination, i = 46 ± 8 degrees, was determined by Shidatsu et al. (2011) using Suzaku data. All of these results were obtained using the reflionx models (Ross & Fabian 2005). Recently, Plant et al. (2015) fitted simultaneously XMM-Newton and Suzaku data sets using both reflionx and xillver and reported two estimates of inclination: i = 36 +3 -6 deg and i = 42 +11 -6 deg. In a different work, Plant et al. (2014b) analyzed three low/hard state observations of GX 339-4 using a recent version of the relxill model and found i = 30 +5 -4 deg, a result that may be biased because it relies solely on low-energy XMM-Newton data. \nIn jf-i , we obtained a tight constraint on inclination, i = 48 . 4 ± 1 . 1 deg (Table 3), a result that is consistent with that obtained in jf-ii (Table 4). Note that the relxill and xillver models used here properly treat the angular distribution of the reflected radiation, unlike earlier reflection models, which only provided an angleaveraged solution (Garc'ıa et al. 2014a). \nAs an aside, if one makes the usual assumption that the spin of the black hole is aligned with the orbital an- \nFig. 7.A probability density map for the strength of the Gaussian absorption component ( gabs ) and the inclination angle from our MCMC analysis for jf-ii . While the inclination is tied between the six spectra A-F, the strength of the gabs component is free to vary. The distribution for each spectrum is coded by color. Only weak correlation is observed, with the gabs component becoming slightly stronger as the inclination increases and as the luminosity decreases. \n<!-- image --> \ngular momentum vector (Fragos et al. 2010; Steiner & McClintock 2012), then the lower estimates of inclination discussed above imply implausibly large values of black hole mass based on the Hynes et al. (2003) estimate of the mass function: e.g., M ∼ 5 . 8 / sin 3 i ∼ 100 M glyph[circledot] for i ∼ 25 deg. Meanwhile, our inclination implies M ∼ 15 M glyph[circledot] , consistent with the range of values observed for stellar-mass black holes ( Ozel et al. 2010; Farr et al. 2011). \nThe inclination angle is largely determined by the shape and position of the blue wing of the Fe K line, which in our fits is somewhat affected by the inclusion or exclusion - of the Gaussian absorption feature. The energy of this feature was linked in all six spectra and constrained to 7 . 23 ± 0 . 08 keV (Table 4), while the normalization was free to vary. Figure 7, based on our MCMC analysis, shows that while the strength of the gabs component does increase by a factor ∼ 2 with decreasing luminosity, it only weakly interplays with inclination. This implies that this component, whose origin is uncertain (Section 4.1), has at most a modest affect on our estimate of inclination.", '6.1.3. Black hole spin': "The spin of the black hole has been estimated via reflection modeling using three independent data sets: a ∗ = 0 . 939 ± 0 . 004 using XMM-Newton /EPIC-MOS plus RXTE spectral hard-state data (Miller et al. 2006b; Reis et al. 2008); a ∗ = 0 . 93 ± 0 . 02 using XMM-Newton /EPICpn plus RXTE spectral data in the very high (or steep power-law) state (Miller et al. 2004; Reis et al. 2008); and a ∗ = 0 . 93 ± 0 . 01 (statistical) ± 0 . 04 (systematic) using Suzaku data (Miller et al. 2008). The corresponding estimates of inclination were all low ( i ∼ 10 -20 deg) implying implausibly high estimates of black hole mass assuming spin-orbit alignment (Section 6.1.2). \nThere is considerable uncertainty associated with these estimates of spin and inclination because of the effects of pileup, i.e., the arrival of two or more photons in the same or adjacent CCD pixel within a single frame time. For example, Done & Diaz Trigo (2010), analyzing precisely \nthe same hard-state XMM-Newton /EPIC-MOS data as Miller et al. (2006b) and Reis et al. (2008), conclude that the high spin reported by Miller et al. and Reis et al. is the result of severe pileup effects; using PN Timing-Mode data (presumably unaffected by pileup), Done & Diaz Trigo (2010) report evidence for a narrow Fe line and a truncated disk. Meantime, Miller et al. (2010) rebut the conclusions of Done and D'ıaz-Trigo. As a second example, the high spin reported by (Miller et al. 2008) based on their analysis of Suzaku data (see above), is challenged by Yamada et al. (2009) who find - using the same data set - evidence for a truncated disk and no need to invoke a rapidly spinning black hole. \nKolehmainen & Done (2010) applied the alternative continuum-fitting method to disk-dominated RXTE data collected during three different outbursts of GX 3394. This method relies on accurate knowledge of the mass, distance, and inclination of the system, all of which are highly uncertain for GX 339-4. Using approximate bounds on these parameters, Kolehmainen & Done obtained 'a strict upper limit' on the spin of a ∗ < 0 . 9, which they claim is inconsistent with the spin estimates obtained by modeling the reflection spectrum. \nA chief virtue of the PCA data upon which we rely is its freedom from the confusing effects of pileup. A further virtue is the abundance of data, which allows us to track the behavior of GX 339-4 over a range of luminosity, as well as to reach extreme ( ∼ 0 . 1%) levels of statistical precision. By assuming that the inner radius of the disk always remains at the ISCO ( jf-ii ; Section 5.2), we established a firm constraint on the spin at a ∗ = 0 . 95 +0 . 03 -0 . 05 (90% confidence) while obtaining a precise estimate of the inclination, i = 47 . 8 +0 . 9 -1 . 4 degrees. Our spin result is in accord with the earlier Fe-line estimates, but our inclination estimate is distinctly different, and more in line with expectation (Section 6.1.2). \nThe results of our MCMC analysis allow us to search for possible degeneracies of the spin parameter with other fit parameters. Figure 8 shows for jf-ii spin probability distributions for three key parameters: (1) the inclination angle, which affects the blue wing of the line; (2) the Fe abundance, which affects the strength of both the line and edge; and (3) the strength of the Gaussian absorption, which also could affect the blue wing of the line. While there is no evidence for a substantial correlation between the spin and the inclination or the strength of the Gaussian, there is indication of a moderate positive correlation with the Fe abundance. From this positive correlation, it follows that an increase in Fe abundance will produce a diminished inner radius for fits performed with a fixed spin and variable R in (Section 6.3).", '6.1.4. Fe abundance': 'The Fe abundance for the blurred reflection component ( relxill ) is surprisingly high: 4 . 6 +0 . 5 -0 . 3 and 5 . 4 +1 . 9 -0 . 5 in solar units for jf-i and jf-ii , respectively (Tables 3 and 4). Most studies have merely assumed that the abundance is solar, while Allured et al. (2013) fitted Swift / RXTE hard-state data and also found a super-solar abundance: A Fe = 2 . 4 +1 . 47 -0 . 62 . The high abundance results directly from the remarkable strength of the Fe K line/edge relative to the Compton hump (top panel of Figure 6), which is usually the highest-amplitude feature in the re- \n<!-- image --> \n<!-- image --> \nFig. 8.Probability contours from our MCMC analysis of jf-ii for the spin parameter and three other parameters of interest: inclination angle ( left ), Fe abundance ( middle ), and the strength of the Gaussian absorption component ( gabs ) for Spectrum A ( right ). A modest positive correlation is observed between the Fe abundance and the spin parameter. \n<!-- image --> \nflection spectrum. A lower Fe abundance underpredicts the strength of the line/edge required to fit the Compton hump. Thus, our ability to constrain the Fe abundance is likely a consequence of our broad bandpass that provides high-sensitivity coverage of all the principal reflection features, from the Fe K line on through complete coverage of the Compton hump. This quality of coverage is not provided by XMM-Newton data (even when RXTE data with a floating normalization are included) which may explain why others have not reported a supersolar Fe abundance. \nQuantitatively, forcing the Fe abundance to the solar value ( A Fe = 1) results in a grossly unacceptable fit with χ 2 ν = 9 . 9 and large residuals across the PCA band (Figure 9), while the inner-disk radius grows by about a factor of 10 compared to the fit with variable Fe abundance. We tried several alternative models (e.g., varying the emissivity index) in an unsuccessful attempt to find an acceptable model with lower Fe abundance. We note that fits to NuSTAR data for the most recent outburst of GX 339-4 likewise require a large Fe abundance (Fuerst et al. 2015). \nFig. 9.Spectral fitting results, similar to jf-i , but here fixing the Fe abundance to the solar value ( A Fe = 1). The top and bottom panels show respectively a ratio plot and contributions to χ 2 . This model fails to fit jointly the three principal reflection features (Fe K line and edge and the Compton hump), most noticeably for the higher-luminosity spectra, and the fit is unacceptable ( χ 2 ν = 9 . 9). A vertical dashed line marks the 6.4 keV rest-frame energy of the Fe K line. \n<!-- image --> \nThe Fe abundance has an influence on the shape of the Fe K line, and it may therefore in turn affect such parameters of interest as the inclination or inner-disk radius. Moreover, the abundance also alters the continuum pho- \ntoelectric opacity at higher energies, which modifies the depth of the Fe K edge and the red side of the Compton hump. These effects, which are subtle, are driving our fits because of the unprecedented signal-to-noise we have achieved. We shall return to this point in Section 6.3. \nWe conclude that super-solar Fe abundance is a strong and inescapable requirement of these data. At the same time, the data also require the unblurred (distant) reflection to have moderate - near-solar - Fe abundance. In our tests, fitting the spectrum with the most counts (Spectrum A), we found that adopting a single abundance for both blurred and unblurred components (i.e., relxill and xillver ) results in A Fe ∼ 3 . 6, an intermediate value between relxill ( A Fe ∼ 5), and the low value required for the xillver component. Though this approach may be intuitively more satisfying, it is strongly rejected by the data, with an increase in χ 2 of ∼ 55. The effect of linking the abundances is to decrease the importance of the xillver component by reducing its normalization parameter from ∼ 0 . 2 to ∼ 0 . 07. An inspection of the residual contributions to χ 2 (lower panel of Figure 9) reveals that the quality of the fit is degraded not only in the Fe K region but over the entire energy band. In summary, we find strong empirical evidence for the presence of an unblurred reflection component whose Fe abundance is much less than that of the blurred component. \nAt the same time, there is no obvious reason why the inner-disk abundances should be so high. We note that similar physical processes may be occurring in AGN, since large Fe abundances are likewise found in many cases when fitting relativistic reflection models (Fabian 2006), with 1H 0707-495 being a prime example (Dauser et al. 2012; Kara et al. 2015). Possibly, an unknown physical effect is being overlooked in current models that is artificially driving the Fe abundance to high values. For example, Reynolds et al. (2012) proposed that radiative levitation of Fe ions in the accretion disk atmosphere could cause an apparent enhancement of their abundance.', '6.2. Parameters that evolve systematically with luminosity': 'Setting aside the Gaussian absorption component (already discussed in Section 4), for jf-i there are six important parameters that are fitted separately for Spectra A-F: the inner-disk radius R in ; the photon index Γ; \nthe ionization parameter ξ ; the high-energy cutoff E cut ; the reflection fraction R f ; and the normalization of the unblurred reflection component ( xillver ). In this section, we show how these parameters depend on luminosity, and we discuss the causes of these dependencies. \nFigure 10 illustrates our MCMC results for jf-i (Table 3), the case of fixed spin. The probability distribution for each parameter is shown plotted versus the floating constant factor, which can be regarded as a proxy for the luminosity. The luminosity ranges over somewhat more than an order of magnitude. Each Spectrum is colorcoded (see legend in top-left panel). The breadth of a distribution is a measure of uncertainty, while its shape indicates the degree of correlation of that particular parameter with luminosity. We now discuss in turn the behavior of each parameter.', '6.3. Inner edge of the disk': "The evolution of the inner-disk radius R in with luminosity is shown in the top-left panel of Figure 10. Each spectrum delivers a good constraint on R in , allowing us to conclude that the inner edge of the disk moves outward by a factor of a few as the luminosity decreases by an order of magnitude, from a nominal value of 17% of Eddington to 1.6% of Eddington (Table 1). \nThis is a principal result of our paper because R in and its dependence on luminosity is a matter of central importance for the study of black hole binaries in the hard state (Section 1.1). In Table 5 we summarize estimates of R in in the literature for GX 339-4 in the hard state, while considering only those results obtained via reflection spectroscopy. The compilation includes results obtained using a wide variety of data and over a large range in luminosity ( ∼ 0 . 1 -20% of Eddington). At a glance, one notes the extreme range of values reported for R in . The most notable conflict are two grossly disparate values reported for the same XMM-Newton observation: Reis et al. (2008) analyzed MOS and RXTE PCA data and reported R in = 2 . 04 +0 . 07 -0 . 02 R g , while Plant et al. (2015) analyzed EPIC-pn timing-mode data and reported R in = 318 +165 -74 R g . \n74 Figure 11 shows all the values of R in that appear in Table 5 plotted as a function of the Eddington-scaled luminosity. Several studies report results for multiple observations over a range of luminosity; in these cases, the individual data points are highlighted in the left panel of Figure 11 using colored tracks. Meanwhile, individual measurements are shown in the right panel. Our results are shown in both panels with a solid red track connecting the data points. As noted above, we find that R in increases modestly with decreasing luminosity: Best-fit values trend upward from 2 . 1 R g to 4 . 6 R g as the luminosity decreases from 17% to 1.6% of Eddington. (Note that the values of R in in Table 3 are in units of R ISCO = 1 . 237 R g .) \nThis trend is consistent with that found in previous studies except for that of Allured et al. (2013) (yellow track), who fitted Swift and RXTE data using the reflionx model. Despite the general agreement that the inner radius shrinks with increasing luminosity, our estimates of R in at comparable values of luminosity are much smaller than those reported by others. \nFor example, Plant et al. (2015) found the disk to \nbe extremely truncated based on fits to a Suzaku and three XMM-NewtonRXTE spectra using xillver and reflionx (light and dark blue tracks). Kolehmainen et al. (2014) reported a similar trend but smaller values of radius (green track) by analyzing the same three XMM-Newton spectra (excluding the RXTE data) using the rfxconv model (based on the reflionx tables; Kolehmainen et al. 2011). For their highest-luminosity data, they report two values of inner radius: One is a lower limit that is consistent with our results, R in > 2 . 2 R g , while the other (which includes a ∼ 9 keV instrumental feature in the fit; orange track) is reasonably consistent with the results of Plant et al. (2015), R in ≈ 47 R g . Overall, the results of Kolehmainen et al. (2014) and Plant et al. (2015) are similar, with the former authors reporting somewhat smaller values of R in . Interestingly, our results appear to be in reasonable agreement with an extrapolation of the low-luminosity values of R in reported by Petrucci et al. (2014) (purple track), which is not necessarily expected since these authors assumed solar Fe abundance. \nThough the gross disparities in the reported values of R in may be partially due to differences in the models, this should be a secondary effect since, e.g., tests show that the models xillver and reflionx perform similarly (Garc'ıa et al. 2013). The more likely reason for the inconsistent results is limitations of the data. One of the most severe of these is the effects of pileup, especially for the crucial XMM-Newton data (see Section 6.1.3). Another effect leading to major differences in results is whether or not high-energy data were used. For example, Plant et al. (2015) and Kolehmainen et al. (2014) used the same EPIC-pn data in timing mode, but while Plant et al. (2015) used simultaneous RXTE data to extend the energy coverage, Kolehmainen et al. (2014) eschewed its use because of their concern over the cross-calibration of the two detectors. As a consequence of employing XMMNewton data only, the results of Kolehmainen et al. (2014) are highly sensitive to calibration issues associated with the rapidly-falling and uncertain response of the EPIC-pn detector at energies glyph[greaterorsimilar] 9 keV. \nThe PCA has important advantages despite its limited energy resolution and lack of coverage below 3 keV. Most notably, the PCA data are free from the contentious effects of pileup that are inherent to CCD observations of bright sources. Meanwhile, the use of a single detector eliminates problems associated with cross-calibrating a pair of detectors. The much higher effective area of the PCA around the Fe line and Compton hump - and the many dozens of observations - yields spectra with ordersof-magnitude more counts than CCD spectra (Section 5). Moreover, one can now fully utilize these many millions of counts per spectrum to detect subtle effects in reflection features because the response of the PCA has been successfully calibrated to ∼ 0 . 1% precision (Shaposhnikov et al. 2012; Garc'ıa et al. 2014b). These virtues of the PCA data are attested to by our success in fitting our reflection models ( χ 2 ν ∼ 1) to six extremely high signal-to-noise spectra, which individually contain between 3 and 28 million total counts in the 3-10 keV band. Finally, the great abundance of data makes the PCA database unrivaled for synoptic studies of Galactic black holes. \nWe now return to the question of the grossly discrepant \nFig. 10.Variation of key model parameters with X-ray luminosity. The clouds of points in each panel (color-coded to correspond to a particular one of the six spectra) show the posterior density of the MCMC results for: the inner radius R in in units of the ISCO radius; the photon index Γ of the power law; the ionization parameter ξ ; the high-energy cutoff E cut ; the reflection fraction R f ; and the normalization N x of the unblurred reflection component xillver . The Constant Factor on the x -axis, which is proportional to the Eddington-scaled luminosity, is normalized to unity (corresponding to L/L Edd = 17 %) for Spectrum A (Table 1). \n<!-- image --> \nresults reported for R in (Table 5; Figure 11) while reminding the reader that the Fe abundance affects the Fe K line profile and other reflection features at a detectable level given our signal to noise (Section 6.1.4). In turn, the Fe abundance affects other parameters, notably the inner-disk radius and spin parameter, which correlates positively with Fe abundance (Section 6.1.3). We now show that values of R in found by others are significantly biased by either the low signal-to-noise of their data or inadequate high-energy coverage. Such data make it difficult to distinguish between small R in with large A Fe and large R in with solar abundances, as we illustrate in Figure 12, which compares fits to two relxill models, one with A Fe = 1 and the other with A Fe = 5. The model with solar abundance (black curve) can only fit the data when the disk is strongly truncated, which serves to minimize the relativistic effects that blur the line profile. Note, however, that this model then fails to repro- \nduce the depth of the Fe K edge and underpredicts the continuum above ∼ 30 keV. Figure 12 should be compared directly with Figure 9, where the limitations of the A Fe = 1 model are apparent from the fit residuals. Unlike most other data sets, the extreme signal in our data clearly discriminates between the two models. \nThe truncation of the inner disk and the decrease in R in with increasing L/L Edd , which we find, is a prediction of the advection-dominated accretion flow (ADAF) model (see Section 1.1). In this paradigm, the inner disk evaporates becoming a very hot and optically thin accretion flow that fills the inner region (see, e.g., MeyerHofmeister et al. 2009). Our results are in line with this model, although our observations do not extend to the lower luminosities at which extreme truncation likely occurs. \nHowever, we note that these results are apparently at odds with our non detection of a thermal disk component \n<!-- image --> \nFig. 11.A comparison for GX 339-4 of our estimates with those in the literature (see Table 5) of the inner-disk radius versus luminosity obtained by reflection modeling of hard-state spectra. The solid red track shown in both panels links our six measurements of R in (Table 3). ( left ) The various colored tracks show the evolution of R in with Eddington-scaled luminosity (see footnote to Table 1) reported by us and others (Table 5). Plant et al. (2015) and Kolehmainen et al. (2014) each present a pair of tracks, which are labeled (a) and (b) ; the latter track in the case of Kolehmainen et al. (2014) includes a notch feature at ∼ 9 keV. ( right ) Single measurements of R in reported in a variety of studies (Table 5). The track labeled Done+Diaz10 corresponds to results reported by Done & Diaz Trigo (2010) for fits to Epic-pn data. Data points with identical values of luminosity are slightly offset for clarity (precise values are given in Table 5). \n<!-- image --> \nFig. 12.A comparison of two reflection models calculated using our relxill code: The red curve is for the parameters of jf-i (see Table 3), in particular for A Fe = 5 and R in = R ISCO . The black curve is for a model with solar abundances ( A Fe = 1) that has been doctored by increasing R ISCO by 100-fold to best match the structure in the Fe K line. Meantime, the models are seen to differ greatly in the depth and structure of the Fe K edge and in the shape of the Compton hump. \n<!-- image --> \nof emission. Making the usual assumption that all the observed power-law photons are generated by Compton up-scattering of disk photons, we would have expected to detect a thermal component for a disk that extends to such small radii (see Section 7).", '6.4. Parameters of the continuum: Γ and E cut': 'We find that the power-law photon index Γ is relatively constant despite the order of magnitude increase in luminosity (top-right panel of Figure 10), a result that has been previously reported for GX 339-4 (e.g. Wilms et al. 1999; Zdziarski et al. 2004; Plant et al. 2014a). Its average value is 1 . 640 ± 0 . 035 for jf-i and 1 . 625 ± 0 . 030 for jf-ii (std. dev., N = 6; Table 3 and 4), firmly in the range for the hard state (1 . 4 < Γ < 2 . 1; Remillard & McClintock 2006). \nIn contrast to the constancy of the power-law index, the cutoff energy E cut systematically decreases with increasing luminosity from > 890 keV for Spectrum F down \nto 97 ± 4 keV for Spectrum A (right-middle panel of Figure 10). This lower value of E cut for our highest luminosity spectrum is of the same order of magnitude as the 58 . 5 ± 2 . 2 keV value reported by Droulans et al. (2010), which is based on their analysis of simultaneous RXTE and INTEGRAL data obtained during another bright hard state of GX 339-4. \nOur model achieves good constraints for all six spectra. Remarkably, this is true for even the lowest-luminosity data (Spectrum F) for which the cutoff energy of > 890 keV is far beyond the 45 keV limit of the PCA bandpass. This surprising result is a consequence of the detectable effects that are imprinted on the reflected component in the 3-45 keV band by photons with energies of hundreds of keV. We discuss the capability of the relxill model to probe the spectrum at extreme energies in Garcia et al. (2015). \nIn a Comptonized and isothermal corona, the highenergy cutoff is set by the electron temperature: E cut ∼ (2 -3) kT e . In such a plasma, thermal disk photons are Compton up-scattered, thereby cooling the coronal electrons while producing the observed power-law continuum. The slope of the power law depends on the interplay between the electron temperature and the optical depth τ e , \nΓ = -1 2 + √ 9 4 + 1 θ e τ e (1 + τ e / 3) , (1) \n(Lightman & Zdziarski 1987), where θ e = kT e /m e c 2 and m e c 2 = 511 keV is the electron rest mass. Values of these parameters for Spectra A-F, which are consistent with previous determinations (e.g. Wilms et al. 1999), are summarized in Table 6 for our nominal value of the photon index (Γ = 1 . 6). We find, as predicted by Equation 1, that the coronal temperature decreases with increasing luminosity, while the optical depth increases.', '6.5. The reflection fraction': "The reflection fraction R f is a third parameter (in addition to Γ and E cut ) that provides information on the \nCompilation of literature estimates of R in obtained by fitting reflection models to hard-state spectra of GX 339-4. \nTABLE 5 \n| Satellite | Instrument | L/L Edd (%) | R in ( R g ) | i (deg) | q | High-energy? | Ref. |\n|-------------|----------------|---------------|----------------------------|------------|----------------|----------------|--------|\n| XMM-Newton | EPIC-pn (TM) a | 1.42 | 684 +301 - 378 972 +28 643 | 42 +11 - 6 | 3 | Yes | [1] |\n| | | | - | 36 +3 - 6 | 3 | Yes | [2] |\n| | | | 110 +80 - 40 | 60 | 3 | No | [3] |\n| | | | 150 ∗ - 50 | 60 | 3 | No | [4] |\n| | EPIC-MOS | 3.25 | 5 ± 0 . 5 | 20 +5 - 10 | 3 | Yes | [5] |\n| | | | 2 . 04 +0 . 07 - 0 . 02 | 20 - 1 . 3 | 3 . 16 ± 0 . 5 | Yes | [6] |\n| | EPIC-pn | | 10 ± 2 | 27 ± 3 | 3 | Yes | [7] |\n| | | | 60 +40 - 20 | 60 | 3 | Yes | [8] |\n| | EPIC-pn (TM) a | | 318 +165 - 74 | 42 +11 - 6 | 3 | Yes | [1] |\n| | | | 125 +21 - 51 | 36 +3 - 6 | 3 | Yes | [2] |\n| | | | 89 +55 - 23 | 60 | 3 | No | [3] |\n| | | | 128 +73 - 43 | 60 | 3 | No | [4] |\n| | | 10.2 | 155 +139 - 53 | 42 +11 - | 3 | Yes | [1] |\n| | | | 72 +42 | 6 +3 | 3 | | [2] |\n| | | | - 21 2 . 2 ∗ | 36 - 6 | 3 | Yes | [3] |\n| | | | +10 | 60 | | No | [4] |\n| | | | 47 - 7 | 60 | 3 | No | |\n| | | < 0 . 05 | 21 +17 - 9 | 30 +5 - 4 | 3 | No | [14] |\n| | | < 0 . 05 | 27 +6 - 6 | 30 +5 - 4 | 3 | No | [14] |\n| | | < 0 . 05 | 16 +7 - 4 | 30 +5 - 4 | 3 | No | [14] |\n| Suzaku | XIS0,1,3/PIN | 0.14 | > 65 | 18 | 2-3 | Yes | [9] |\n| | | | > 798 | 42 +11 - 6 | 3 | Yes | [1] |\n| | | | > 745 | 36 +3 - 6 | 3 | Yes | [2] |\n| | | | 190 +170 - 90 | 50 | 2.3 | Yes | [10] |\n| | | | > 180 | 20 | 3 | Yes | [11] |\n| | | 0.13 | > 30 | 20 | 3 | Yes | [11] |\n| | | 0.19 | > 10 | 20 | 3 | Yes | [11] |\n| | | 0.25 | > 70 | 20 | 3 | Yes | [11] |\n| | | 0.91 | - 19 7 . 0 +1 . 1 - 1 . 3 | 20 | 3 | Yes | [11] |\n| | | 2.0 | 13 . 3 +6 . 4 - 6 . 0 | 46 ± 8 | 2 . 3 ± 0 . 1 | Yes | [10] |\n| Swift | XRT | 1.33 | 3 . 6 +1 . 4 - 1 . 0 | 20 | 3 . 2 ± 0 . 6 | Yes | [12] |\n| | | | 6 . 7 +9 . 1 - 2 . 3 | 20 | 3 | Yes | [13] |\n| | | 0.46 | < 10 | 20 | 3 . 1 ± 0 . 4 | Yes | [12] |\n| | | | 3 . 6 +1 . 9 - 0 . 9 | 20 | 3 | Yes | [13] |\n| | | 1.08 | 12 . 8 +19 . 8 - 9 . 0 | 20 | 3 | Yes | [13] |\n| | | 1.12 | 6 . 9 +9 . 1 - 3 . 7 | 20 | 3 | Yes | [13] |\n| | | 1.68 | 19 . 5 +25 - 8 . 5 | 20 | 3 | Yes | [13] |\n| | | 2.23 | 16 . 3 +11 . 7 - 5 . 2 | 20 | 3 | Yes | [13] |\n| | | | 19 . 7 +12 . 1 | 20 | 3 | | |\n| | | 5.21 | - 6 . 5 | | | Yes | [13] | \nNote . - [1] Plant et al. (2015) implementing xillver ; [2] Plant et al. (2015) implementing reflionx ; [3] Kolehmainen et al. (2014); [4] Kolehmainen et al. (2014) including a notch feature at ∼ 9 keV; [5] Miller et al. (2006b); [6] Reis et al. (2008); [7] Done & Diaz Trigo (2010); [8] Done & Diaz Trigo (2010) with fixed inclination; [9] Tomsick et al. (2009); [10] Shidatsu et al. (2011); [11] Petrucci et al. (2014); [12] Tomsick et al. (2008); [13] Allured et al. (2013); [14] Plant et al. (2014b). \nTABLE 6 Coronal properties a \n| Box | L/L Edd (%) | E cut (keV) | T e (10 9 K) | τ e |\n|-------|---------------|---------------|----------------|-------|\n| A | 17.3 | 97 | 0.45 | 3.03 |\n| B | 14.2 | 129 | 0.6 | 2.5 |\n| C | 11.9 | 179 | 0.83 | 1.99 |\n| D | 7.9 | 660 | 3.06 | 0.72 |\n| E | 3.9 | 840 | 3.9 | 0.59 |\n| F | 1.6 | 890 | 4.13 | 0.56 | \nstructure of the corona. In relxill , the parameter is empirically defined as the ratio of the reflected flux to the power-law flux in the 20-40 keV band. The results of jf-i show that the reflection fraction of the relativistically blurred component ( relxill ) ranges from 0 . 2 glyph[lessorsimilar] R f glyph[lessorsimilar] 0 . 3, decreasing modestly with increasing luminosity (bottom-left panel, Figure 10). This trend is surprising given that, at the same time, R in is decreasing so that the area of the reflector should increase. As a further wrinkle, one expects R f glyph[greaterorsimilar] 1 based on simple arguments (Dauser et al. 2014). \nThere are several scenarios that can plausibly account for values of R f < 1. We mention four and then discuss a new, alternative explanation. (1) An obvious expla- \nnation is a severely truncated disk (more specifically, a disk with truncation radius large compared to the size of the corona). We discard this possibility as inconsistent with the small values we find for R in (Section 6.3). (2) Another option is for the corona to be continuously outflowing at relativistic speeds, beaming the bulk of its emission away from the disk (e.g. Miller et al. 2014; Keck et al. 2015). While this is a possible explanation for R f < 1, one outcome of this scenario is a resultant low value of the emissivity index, which is not obviously required by our data. (3) The value of R f may be depressed by our assumption of a constant-density disk atmosphere, as Ballantyne et al. (2001) have shown in their studies of hydrostatic atmospheres. The hotter gas layer at the surface of a hydrostatic atmosphere additionally scatters and blurs reflection features (see also Nayakshin & Kallman 2001) thereby diluting the reflection signal relative to a constant density model. (4) The apparent strength of reflection features may also be reduced by the Comptonization of these features in an extended corona, as Wilkins & Gallo (2014) recently proposed. However, for such a corona to be effective in reducing R f appreciably, it must have a large covering fraction which may interfere with detection of blurred reflection features from the inner disk. \nWe propose an alternative explanation for R f < 1 based on the strong dependence of the reflected spectrum on the angle at which an illuminating photon strikes the disk. This angle crucially determines the characteristic depth in the disk at which the photon interacts; this in turn affects the limb-darkening/brightening of the disk (Svoboda et al. 2009; Garc'ıa et al. 2014a). A deficiency of reflionx , relxill , xillver and other widely-used reflection models is the simplifying assumption of a fixed incidence angle of 45 deg. However, a larger angle of incidence (measured with respect to the normal to the disk plane), for example, results in a hotter surface layer and therefore a weaker reflection signature (see Figure 5 in Dauser et al. 2013). \nTo test whether the assumption of near-grazing illumination substantially increases fitted values of R f , we produced a new table of xillver reflection models with a fixed incidence angle of 85 deg and merged them with relline to create a new high-incidence-angle version of relxill (Section 1.2). Fitting Spectra A-F as in Section 5.1 (i.e., jf-i ), the fit is slightly worse (∆ χ 2 = 9 . 71) but statistically comparable and still quite reasonable ( χ 2 ν = 1 . 09). Notably, for the 85-deg model we find that R f increases with luminosity and that R f > 1 for the three most luminous spectra (A-C). Meanwhile, all the other parameters are consistent with those for jf-i (Table 3). Importantly, the Fe abundance remains unchanged. \nFigure 13 compares the reflection factors computed for the two models. The large-angle model is more in accord with expectation, namely, the reflection fraction trends upward with luminosity and the values at the higher luminosities (with R in near the ISCO and with correspondingly large reflector area) are sensibly glyph[greaterorsimilar] 1. Thus, our results qualitatively suggest that the accretion disk in GX 339-4 is illuminated at near-grazing angles with respect to the surface of the disk. \nWithin a few gravitational radii of the horizon, the \nFig. 13.Comparison of the reflection fraction versus luminosity for two fixed value of the incident angle of the illuminating radiation: the 45 deg value (blue points), which is widely assumed in reflection modeling, and 85 deg (red points), the value we have assumed for this test. \n<!-- image --> \nextreme bending of light rays causes photons to strike the disk over a wide range of angles (Dauser et al. 2013; see in particular the middle panel of their Figure 5). Given the strong dependence of R f on the angle of incidence, reaching firm conclusions concerning the reflection factor will require building a new generation of models, a task beyond the scope of this paper that will be addressed in future work.", '6.6. Ionization parameter and geometry': "As expected, both the normalization N r and the ionization parameter ξ of the blurred reflection component ( relxill ) increase with luminosity (Table 3; Figure 10, middle-left panel). In particular, the ionization parameter changes from ξ = 112 . 2 to ξ = 2041 . 7, which traces very well the ten-fold increase in luminosity, from 1.6% L Edd to 17% L Edd . For the strongly-illuminated portion of the disk, the variations in ξ and L deviate mildly from the simple relation ξ = L/nD 2 , where n is the density of the gas (fixed at n = 10 15 cm -3 for the relxill and xillver models used here 8 , and D is the distance from the coronal source to the strongly heated portion of the disk. Thus, D increases only modestly as the luminosity decreases by an order of magnitude in passing from Spectrum A to Spectrum F: \nD F D A = √ L F ξ A L A ξ F ∼ 1 . 3 . (2) \nThis small change is reasonable given the correspondingly mild increase in the inner radius obtained for jf-i : \nR F R A = 2 . 2 . (3) \nLikewise, the normalization N x of the unblurred reflection component ( xillver ) increases with luminosity (Figure 10, bottom-right panel); although presumably the ionization parameter of this component also increases with luminosity, we approximate the state of the gas in the distant reflector as cold and neutral. \n8 The choice of gas density is relatively unimportant; it is the ionization parameter that largely determines the properties of the reflected spectrum (see Garc'ıa & Kallman 2010; Garc'ıa et al. 2013).", '7. SUMMARY AND CONCLUSIONS': 'We have presented an analysis of six composite RXTE PCA spectra of the X-ray binary black hole GX 339-4. All these spectra were taken when the source was in the hard state. The spectra correspond to luminosities ranging from 1.6% to 17% of the Eddington luminosity. The six spectra, each spanning the energy range 3-45 keV, comprise in total 77 million counts and a total exposure time of 196 ks. A unique feature of this work is our use of the tool pcacorr , which allows us to calibrate the PCA data to a precision of 0.1%. \nThe spectra individually, and jointly, are well fitted by a model with three principal components: relxill , our model of relativistic ionized reflection; xillver , a minor component that models the effects of a cold, distant reflector; and Tbabs , a standard model of Galactic absorption. We include an ad hoc Gaussian component ( gabs ) to model an absorption feature near 7 keV. The origin of this feature is unclear, but it is likely an artifact resulting from a misestimate of the PCA energy resolution. \nWe performed two joint fits of the six spectra. In the first of these, we fixed the spin to its maximal value, which allows the inner-disk radius R in to approach the ISCO radius, and we derived precise estimates for the evolution of R in with luminosity. We find that the disk becomes increasingly truncated with decreasing luminosity. Specifically, as the luminosity ranges from 17% to 1.6% of Eddington, R in increases from 2 . 1 R g to 4 . 6 R g . While this trend has been previously reported (e.g., Petrucci et al. 2014; Kolehmainen et al. 2014; Plant et al. 2015), our values of R in for comparable values of luminosity are much smaller than those found by others. The grossest discrepancy is the hundredfold larger values reported by Plant et al. (2015). \nThat we find such small values of the inner-disk radius and no evidence for a thermal disk component is at odds with the current models. This is particularly true for Spectrum A with R in = 2 . 1 R g . One expects such a modestly-truncated disk to be sufficiently hot (particularly because it is heated by the corona; e.g., Haardt & Maraschi 1993) that we should have detected it with the PCA. This implies that our model somewhat underestimates the true value of R in and that our model is incomplete. To address this problem, we are in the process of exploring an extended model that self-consistently treats the thermal, power-law and reflected components. This is a challenging problem whose solution is beyond the scope of this paper. \nOur analysis indicates that the factor of ∼ 100 range in the values of R in at fixed luminosity, which have been reported in the literature, is unlikely to result from the use of different reflection models; the shifts in R in attributable to this cause appear to be relatively minor. Instead, the large disparity appears to be attributable to limitations of the data, one of which is the well-known effects of pileup. In this paper, we highlight a particularly important effect, namely, the modest statistical quality of most data, which has resulted in observers fitting the blurred reflection component assuming that the Fe abundance is solar, whereas we demonstrate that super-solar Fe abundance is required for fits to data with extreme statistical precision. Specifically, we strongly constrain \nthe Fe abundance (in solar units) to be A Fe = 5 . 0 +1 . 2 -0 . 4 , which is the average value for our two joint fits. This strict requirement of the data is a promising and likely explanation for why, at luminosities ∼ 1% of Eddington, we find evidence for relatively mild disk truncation compared to earlier studies. \nWe acknowledge that the accuracy of our results are limited, systematically, by the presence of an absorption feature near 7.2 keV (Section 4.1) whose origin is unknown. However, our principal conclusions regarding the inner radius of the disk are sound, being subject to a minor uncertainty of about 20% arising from whether or not this feature is included in the model. \nAs the source luminosity and the radiation field bathing the disk grow, the disk becomes increasingly ionized and its structure changes as R in shrinks. At the same time, the large and steady decrease in the high-energy cutoff indicates that the illuminating coronal source is likewise evolving, as its temperature drops and its optical depth increases. \nIn the second of our two joint fits to the six spectra, we made the standard assumption used in estimating black hole spin, namely, we fixed R in to the radius of the ISCO. Doing so, we constrained the spin of the black hole to be a ∗ = 0 . 95 +0 . 03 -0 . 05 . We were able to achieve this statistical precision despite the limited spectral resolution of the PCA because of the quality of the data and its calibration. \nIf there is some truncation of the inner disk (i.e., R in > R ISCO ), then the spin is greater than the estimate given above. Our estimate of spin agrees well with previous determinations made using the Fe line method (Miller et al. 2004, 2006b, 2008; Reis et al. 2008). It is, however, inconsistent with the upper limit of a ∗ < 0 . 9 derived using the continuum-fitting method (Kolehmainen & Done 2010), a result that is uncertain because the accurate values of black hole mass, disk inclination and distance that are required for successfully applying the continuum-fitting method are unknown for GX 339-4. Our result is also formally incompatible with the value of spin predicted for GX 339-4 by Steiner et al. (2013) based on the relationship between spin and jet power proposed by Narayan & McClintock (2012), which has been challenged by Russell et al. (2013). \nWe also obtain a precise estimate for the inclination of the inner disk of i = 48 . 1 +1 . 0 -1 . 3 deg. This value is further subject to an estimated systematic uncertainty of about 4 deg arising from whether or not one chooses to include the 7.2 keV absorption feature in the model. Our value is inconsistent with the low values found earlier using the Fe-line method (Miller et al. 2004, 2006b, 2008; Reis et al. 2008), while it is more in line with reasonable expectations for the mass of the black hole based on the value of the mass function. \nWe thank Felix Furst, Mike Nowak, Tim Kallman, Rubens Reis, Francesco Tombesi, and Andrzej Zdziarski for useful and valuable discussions. JG and JEM acknowledge the support of NASA grant NNX11AD08G. JFS has been supported by NASA Hubble Fellowship grant HST-HF-51315.01. 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2013MNRAS.436.3856S
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Energy, momentum and mass outflows and feedback from thick accretion discs around rotating black holes
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2013-01-01
| 32 | 0.52 | 160 |
['accretion', 'accretion disks', 'black hole physics', 'relativity', 'methods numerical', 'galaxies jets', '-']
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[]
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Using long-duration general relativistic magnetohydrodynamic simulations of radiatively inefficient accretion discs, the energy, momentum and mass outflow rates from such systems are estimated. Outflows occur via two fairly distinct modes: a relativistic jet and a subrelativistic wind. The jet power depends strongly on the black hole spin and on the magnetic flux at the horizon. Unless these are very small, the energy output in the jet dominates over that in the wind. For a rapidly spinning black hole accreting in the magnetically arrested limit, it is confirmed that jet power exceeds the total rate of accretion of rest mass energy. However, because of strong collimation, the jet probably does not have a significant feedback effect on its immediate surroundings. The power in the wind is more modest and shows a weaker dependence on black hole spin and magnetic flux. Nevertheless, because the wind subtends a large solid angle, it is expected to provide efficient feedback on a wide range of scales inside the host galaxy. Empirical formulae are obtained for the energy and momentum outflow rates in the jet and the wind.
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https://arxiv.org/pdf/1307.1143.pdf
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{'Energy, momentum and mass outflows and feedback from thick accretion discs around rotating black holes': 'Aleksander S˛adowski 1 /star , Ramesh Narayan 1 /star , Robert Penna 1 /star , Yucong Zhu 1 /star \n1 Harvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02134, USA \n30 April 2018', 'ABSTRACT': 'Using long-duration general relativistic magnetohydrodynamic simulations of radiatively ine ffi cient accretion discs, the energy, momentum and mass outflow rates from such systems are estimated. Outflows occur via two fairly distinct modes: a relativistic jet and a subrelativistic wind. The jet power depends strongly on the black hole spin and on the magnetic flux at the horizon. Unless these are very small, the energy output in the jet dominates over that in the wind. For a rapidly spinning black hole accreting in the magnetically arrested limit, it is confirmed that jet power exceeds the total rate of accretion of rest mass energy. However, because of strong collimation, the jet probably does not have a significant feedback e ff ect on its immediate surroundings. The power in the wind is more modest and shows a weaker dependence on black hole spin and magnetic flux. Nevertheless, because the wind subtends a large solid angle, it is expected to provide e ffi cient feedback on a wide range of scales inside the host galaxy. Empirical formulae are obtained for the energy and momentum outflow rates in the jet and the wind. \nKey words: accretion, accretion discs - black hole physics - relativity - methods: numerical - galaxies: jets \ntween the mass M BH of the SMBH and the velocity dispersion σ bulge of its host galaxy bulge (Ferrarese & Merritt 2000; Gebhardt et al. 2000; Gültekin et al. 2009; Kormendy & Ho 2013), and the earlier Magorrian et al. (1998) relation between M BH and the bulge mass M bulge. Given the very large size ratio ∼ 10 8 and mass ratio ∼ 10 3 between the galaxy bulge and the SMBH, it is reasonable to assume that the coupling occurs via feedback of energy or momentum (eg., Silk & Rees 1998; King 2003, 2005, 2010; Hopkins, Murray & Thompson 2009). \nA second line of evidence is the observed exponential cuto ff in the number density of galaxies at the high mass / luminosity end (Schechter 1976), even though there is no cuto ff at the equivalent mass scale in the distribution of dark matter halos. One of the mechanisms invoked to explain the galaxy cuto ff is a reduction in the star formation rate in massive systems due to ine ffi cient cooling (White & Rees 1978; White & Frenk 1991). However, cooling effects alone are insu ffi cient (Thoul & Weinberg 1995), and attempts to fit observed luminosity functions need to include additional feedback processes (e.g., Benson et al. 2003; Croton et al. 2006). The most viable process is expulsion of gas from young galaxies in superwinds as a result of feedback from supernovae and / or agtive galactic nuclei (AGN). Observations of starburst galaxies (e.g., McNamara et al. 2006), and midly-relativistic winds in AGN (Tombesi et al. 2010a,b) confirm this picture. \nYet another puzzling aspect of the galaxy population is the fact that the most massive galaxies, typically ellipticals in clus-', '1.1 Feedback': "Black hole (BH) accretion discs are some of the most energetic objects in the Universe (Frank, King & Raine 2002; Kato, Fukue & Mineshige 2008). Geometrically thin discs (Shakura & Sunyaev 1973; Novikov & Thorne 1973) are radiatively e ffi cient and convert about 10% of the rest mass energy of the accreting gas into radiation. Geometrically thick discs, on the other hand, are advection dominated accretion flows (ADAFs, Narayan & Yi 1994, 1995; Abramowicz et al. 1996; Narayan & McClintock 2008) and produce little radiation relative to their mass accretion rates. Instead, they produce outflows in the form of jets and winds which carry huge amounts of energy, mass and momentum. This is the topic of the present paper. \nEnergy and momentum outflow, and to a lesser extent mass outflow, can a ff ect the BH's surroundings. This e ff ect is most profound in the case of supermassive BHs (SMBHs) in the centers of galaxies, where a number of observations suggest the existence of strong'feedback' e ff ects from the SMBH accretion disc on the evolution of the entire host galaxy. The best-known evidence for such coupling is the celebrated M BHσ bulge relation be- \ners, are made of the oldest stars (Bender & Saglia 1999). This 'downsizing' is counterintuitive, since it seems to conflict with hierarchical growth of structure in a CDM cosmogony, where massive dark halos assemble at lower redshift than lower mass halos (e.g., Lacey & Cole 1993). Again, feedback provides a plausible mechanism; it prevents significant accretion in massive galaxies, thus suppressing star formation at late times (Benson et al. 2003; Springel, Di Matteo & Hernquist 2005). \nFinally, the observed X-ray emission in the centers of galaxy clusters implies a cooling time much shorter than the age of the system, suggesting that gas at the centers of these clusters must condense and turn into stars; however, there is no observational evidence for star formation at the required level (Fabian et al. 2001; Peterson et al. 2003; Kaastra et al. 2004; Croton et al. 2006). While thermal conduction may be part of the explanation (e.g., Narayan & Medvedev 2001), an important clue comes from the observation (Burns, Gregory & Holman 1981) that every cluster with a strong cooling flow also contains an active SMBH in a central radio galaxy. This suggests that energy feedback from the SMBH keeps the cluster gas hot (Ciotti & Ostriker 2001; Brüggen & Kaiser 2002; Ruszkowski & Begelman 2002; Churazov et al. 2004). \nThe evidence summarized above indicates that feedback from accreting SMBHs plays a crucial role not only on galaxy scales but even on the scale of galaxy clusters. Two kinds of SMBH feedback are discussed in the literature (see Fabian 2012, Kormendy & Ho 2013 for reviews). One occurs in the 'quasar mode' when the SMBH accretes at a good fraction ( ∼ 0 . 1 to > 1) of the Eddington rate and deposits energy in its surroundings either directly through radiation or via radiatively driven winds. The second kind of feedback takes place in the 'radio mode' (Croton et al. 2006) or 'maintenance mode' (Hopkins 2010). Here, accretion occurs via an ADAF and the radiative luminosity is low. Hence, feedback is almost entirely in the form of mechanical energy and momentum. This ADAF-specific form of feedback is the topic we wish to investigate. \nMaintenance mode feedback has been (i) invoked for the 'cooling flow problem' in galaxy clusters (Ciotti & Ostriker 2001; Brüggen & Kaiser 2002; Ruszkowski & Begelman 2002; Churazov et al. 2004; Gaspari et al. 2011), (ii) included in semi-empirical models of galaxy formation (Croton et al. 2006; Best et al. 2005; Hopkins et al. 2006; Somerville et al. 2008), and (iii) modeled via simple prescriptions in gas dynamical computer simulations of galaxy / cluster formation in the universe (Di Matteo, Springel & Hernquist 2005; Springel, Di Matteo & Hernquist 2005; Cox et al. 2006; Ciotti, Ostriker & Proga 2010; Novak, Ostriker & Ciotti 2011; Scannapieco et al. 2012). However, all these e ff orts are highly empirical since nobody knows exactly how much mechanical energy or momentum flows out from an accreting SMBH. The general practice is to employ the Bondi model (Springel, Di Matteo & Hernquist 2005), or some variant of it (Debuhr et al. 2010), to relate the energy or momentum output from a SMBH to boundary conditions in the surrounding ISM. However, whether or not the Bondi model is a reasonable description of accretion from an external medium is still very much in debate (see, e.g., Igumenshchev & Narayan 2002; Narayan & Fabian 2011). Recently, Gaspari, Ruszkowski & Oh (2013) carried out an in-depth investigation and showed that cold gas condenses out of the hot phase via nonlinear thermal instabilities. As a result the cold filaments / clouds collide inelastically and boost the accretion rate, making the Bondi model very unrealistic. \nA final important question is the following: Which is more important, energy feedback or momentum feedback? The former is traditionally considered in cosmological simulations (e.g., Springel, Di Matteo & Hernquist 2005; Di Matteo, Springel & Hernquist 2005), but the latter may also be important (King 2003, 2010; Debuhr et al. 2010; Ostriker et al. 2010). For a given energy flux, the momentum flux is smallest in the case of a relativistic jet and largest for a non-relativistic wind. Thus the dependence of momentum feedback e ffi ciency on parameters such as the mass accretion rate or the BH spin could be quite di ff erent compared to energy feedback e ffi ciency. In this work we estimate both e ffi ciency factors for ADAFs using numerical simulations.", '1.2 Blandford-Znajek': "It is a remarkable prediction of general relativity that magnetic field lines threading a black hole can extract the hole's rotational energy (Ru ffi ni & Wilson 1975). Rapidly rotating black holes can drive powerful jets. In the standard black hole jet model, the jet power scales as \nP jet ∼ Ω 2 H Φ 2 BH , Ω H = a ∗ / 2 r H , r H = 1 + √ 1 -a 2 ∗ , (1) \nwhere Ω H is the angular velocity of the outer BH horizon with radius r H, and Φ BH is the magnetic flux threading the horizon (Blandford & Znajek 1977; MacDonald & Thorne 1982; Phinney 1982; Thorne, Price & MacDonald 1986). The main predictions of the Blandford-Znajek jet model are supported by GRMHD simulations (Komissarov 2001; Hirose et al. 2004; De Villiers et al. 2005; McKinney 2005; Hawley & Krolik 2006; Beckwith, Hawley & Krolik 2008; Tchekhovskoy, Narayan & McKinney 2010, 2011a; Tchekhovskoy, McKinney & Narayan 2012). The BlandfordZnajek (BZ) model is a close cousin of the Goldreich & Julian (1969) model for pulsar magnetospheres, a relationship which becomes particularly clear in the membrane formulation of the BZ model (MacDonald & Thorne 1982; Thorne, Price & MacDonald 1986). Recently it has been observed that the scaling of jet power with black hole spin in galactic X-ray binaries is consistent with the BZ model (Narayan & McClintock 2012; Steiner, McClintock & Narayan 2013). \nOne of the aims of the present paper is to check how well jets and winds in simulated ADAFs agree with the scaling shown in equation (1).", '1.3 Previous work': 'Outflows of mass and energy are multi-dimensional and are best studied with numerical simulations. Multi-dimensional numerical hydro- and magnetohydro-dynamical simulations of hot accretion discs have been performed for more than a decade. Already early works based on pseudo-Newtonian codes with purely hydrodynamic visosity showed that a significant fraction of the inflowing mass near the equatorial plane can flow out along the poles (Stone, Pringle & Begelman 1999; Igumenshchev & Abramowicz 1999, 2000). Li, Ostriker, & Sunyaev (2013, see also earlier work by Proga & Begelman 2003, and Janiuk et al. 2009) ran a set of hydrodynamical axisymmetric simulations of low-angular momentum gas. For their viscous models they observed conical outflows almost balancing inflow. \nPseudo-Newtonian MHD simulations have been then performed by a number of authors (Machida, Hayashi & Matsumoto 2000; Machida, Matsumoto & Mineshige 2001; \nStone & Pringle 2001; Hawley & Balbus 2002; Igumenshchev, Narayan & Abramowicz 2003). Outflows were observed and it was claimed that the initial configuration of the magnetic field may play an important role in determining the mass outflow rate. On the contrary, in a series of numerical MHD simulations, Pen, Matzner & Wong (2003) and Pang et al. (2011) found little evidence for either outflows or convection. Even though the entropy gradient was unstable the gas was apparently prevented from becoming convective by the magnetic field. Recently, Yuan, Wu & Bu (2012) and Yuan, Bu & Wu (2012) carried out 2D hydrodynamical and MHD simulations of ADAFs which cover a very large range of radius and show fairly strong outflows. \nBeginning with the work of De Villiers, Hawley & Krolik (2003), accretion flows have been studied using general relativistic magneto-hydrodynamic (GRMHD) codes. The authors observed two kinds of outflows: bipolar unbound jets and bound coronal flow. The coronal flow supplied gas and magnetic field to the coronal envelope, but apparently did not have su ffi -cient energy to escape to infinity. The jets on the other hand were relativistic and escaped easily, though carrying very little mass. Jets have been studied in detail by a number of authors (McKinney & Gammie 2004; De Villiers et al. 2005; McKinney 2006). Tchekhovskoy, Narayan & McKinney (2011a) simulated a strongly magnetized disc around a rapidly spinning BH, and obtained very powerful jets with energy e ffi ciency η > 100%, i.e., jet power greater than 100% of ˙ M BH c 2 , where ˙ M BH is the mass accretion rate on to the BH. Their work showed beyond doubt that at least some part of the jet power had to be extracted from the spin energy of the BH. \nRecently, McKinney, Tchekhovskoy & Blandford (2013) have studied a large set of thick accretion disc simulations both for rotating and non-rotating BHs. They found that models initiated with poloidal magnetic field showed mass loss both in the jet and a magnetized wind. The energy reaching large radii was dominated by the power produced via the magnetic flux penetrating the horizon as in the BZ mechanism.', '1.4 This work': 'The present paper attempts to obtain via global general relativistic magnetohydrodynamic (GRMHD) simulations quantitative estimates of the amount of energy, mass and momentum that flow out from an ADAF around a BH. In a previous paper (Narayan et al. 2012), we studied mass outflow from an ADAF around a nonrotating BH, and showed that a wind flows out at relatively large radii, r /greaterorsimilar 40 1 . Here we extend our analysis to accretion flows around rotating BHs. In the process we show that there are two kinds of outflows in such systems: (i) relatively slow winds at larger radii, similar to the winds studied in Narayan et al. (2012), and (ii) relativistic jets which flow out from close to the BH and are uniquely associated with spinning holes. We study in detail the energy, mass and momentum in these two kinds of outflow. In an \naccompanying paper (Penna, Narayan & Sa¸dowski 2013), we discuss the physics of relativistic jets and demonstrate that our numerical simulations strongly validate the Blandford & Znajek (1977) mechanism. \nThe paper is organized as follows. In Section 2 we introduce the numerical scheme we used to simulate the discs. In Section 3 we discuss outflows emerging from them. In particular, in Sections 4.5, 4.6 and 4.7 we present radial profiles of outflowing energy, mass and momentum, respectively, and in Section 4.8 we give approximate formulae for the corresponding fluxes. In Section 5 we compare the e ffi ciencies of generating outflows by thick and thin discs. We conclude with Section 6 discussing implications of our results.', '2 NUMERICALSETUP': "We performed seven simulations of radiatively ine ffi cient accretion flows around spinning and non-spinning BHs, as listed in Table 1. Following the methods described in Narayan et al. (2012), two distinct initial conditions were used for the seed magnetic field threading the initial gas torus: (i) In one set of simulations the seed field consisted of multiple poloidal loops of magnetic field with changing polarity (multi-loop or SANE, which stands for 'Standard And Normal Evolution'). (ii) In the other set of simulations the initial field was a single poloidal loop threading the entire torus (singleloop or MAD, 'Magnetically Arrested Disc'). SANE runs are designed such that relatively little magnetic flux accumulates around the BH. In contrast, MAD runs quickly saturate at the maximum allowed magnetic flux on the BH for the given mass accretion rate; the back-reaction of the saturated field causes the accretion flow to be magnetically arrested and to settle down to the MAD state (Narayan, Igumenshchev & Abramowicz 2003). \nAll the simulations were performed using the GRMHD code HARM (Gammie, McKinney & Tóth 2003). The coordinates, initial torus, and magnetic field were set up following Narayan et al. (2012). We used modified Kerr-Shield horizon penetrating coordinates which covered uniformly the full range of azimuthal and polar angles. The radial size of cells increased exponentially with radius, with the innermost radius chosen to fit six cells inside the BH horizon. The outermost radius was set to r = 10000. The adopted resolutions are given in Table 1. The initial torus of gas (set up following Penna, Kulkarni & Narayan 2013) was threaded either by multiple and counter-orientated, or single poloidal magnetic field loops for SANE and MAD models, respectively. \nTwo of the simulations discussed here have BH spin a / M ≡ a ∗ = 0, and are the same as the ones described in Narayan et al. (2012), except that the a ∗ = 0 MADmodel has been run up to a time t final = 200000 (to match the corresponding SANE run), which is twice as long as in the previous work. The runs with spinning BHs ( a = 0 . 7, 0.9, 0.98) are all new. \nTo validate if the magneto-rotational instability (MRI) is resolved we calculate the parameters, \nQ ˆ θ = 2 π Ω dx ˆ θ | b ˆ θ | √ 4 πρ , Q ˆ φ = 2 π Ω dx ˆ φ | b ˆ φ | 4 πρ , (2) \n√ \n∗ \n√ where dx ˆ i (grid cell size) and b ˆ i (the magnetic field) are evaluated in the orthonormal fluid frame, Ω is the angular velocity, and ρ is the gas density. For SANE runs, the gas inside r = 100 and within one density scale height of the midplane has Q ˆ θ and Q ˆ φ between 10 -20, for all of the time chunks except the first one (see below for the definition of time chunks). The MAD simulations have Q ˆ θ > \nTable 1. Disc modelsTable 2. Time chunks \n| Model | BH spin ( a ∗ ) | Initial magnetic field | Resolution ( r , θ , φ ) | t final |\n|-------------------|-------------------|--------------------------|----------------------------|-----------|\n| a ∗ = 0 . 0 SANE | 0 . 0 | multi-loop | 256x128x64 | 200000 |\n| a ∗ = 0 . 7 SANE | 0 . 7 | multi-loop | 256x128x64 | 100000 |\n| a ∗ = 0 . 9 SANE | 0 . 9 | multi-loop | 256x128x64 | 50000 |\n| a ∗ = 0 . 98 SANE | 0 . 98 | multi-loop | 256x128x64 | 25000 |\n| a ∗ = 0 . 0 MAD | 0 . 0 | single-loop | 264x126x60 | 200000 |\n| a ∗ = 0 . 7 MAD | 0 . 7 | single-loop | 264x126x60 | 100000 |\n| a ∗ = 0 . 9 MAD | 0 . 9 | single-loop | 264x126x60 | 50000 | \n| Chunk | Time Range | t chunk |\n|---------|---------------|-----------|\n| T1 | 3000-6000 | 3000 |\n| T2 | 6000-12000 | 6000 |\n| T3 | 12000-25000 | 13000 |\n| T4 | 25000-50000 | 25000 |\n| T5 | 50000-100000 | 50000 |\n| T6 | 100000-200000 | 100000 | \n100 and Q ˆ φ ∼ 50 in the same regions. Therefore, MRI is properly resolved in both cases (Hawley, Guan & Krolik 2011). \nFigure 1 shows snapshots of density and Lorentz factor u t for the a ∗ = 0 . 7 SANE and MAD runs. In both models, the disc is geometrically thick (root mean square h / r ≈ 0 . 4) and turbulent. However, gas escapes in two fairly distinct structures: a fast collimated laminar flow along the axis, which we refer to as the 'jet', and a slower outflow covering a wider range of angles, which we call the 'wind'. We focus on these two components in the rest of the paper. Compared to the SANE run, the outflowing jet velocity is much higher in the MAD simulation, sometimes exceeding u t = 5 ( v jet > 0 . 98 c ).", '3.1 Averaging': 'As in our previous work (Narayan et al. 2012), we used timeaveraged disc properties to extract radial profiles of quantities of interest. The time-averaging was done over logarithmically increasing chunks of time as listed in Table 2, with each successive time chunk being twice as long as the previous one. The two a ∗ = 0 simulations were run up to t final = 200000, allowing us to compute time averages for chunks T1-T6. The shorter a ∗ = 0 . 7 runs only went up to chunk T5, the a ∗ = 0 . 9 runs reached chunk T4, while the a ∗ = 0 . 98 run stopped at chunk T3. 2 Apart from averaging over time, we also averaged the data over azimuth, and symmetrized it with respect to the equatorial plane.', '3.2 Quantites of interest': "We are interested in estimating the amount of mass, energy and momentum flowing out of the accretion disc. The radial flux of rest mass is given by, \n˙ m = ρ u r , (3) \n2 The physical wall time of all the simulations was comparable because of the smaller horizon radius of spinning BHs, which required a correspondingly smaller time step. \nwhere a positive (negative) sign indicates that matter flows away from (towards) the BH. The total energy flux is \n˙ e tot = -T r t , (4) \nwhere T r t is the ( r , t ) component of the magnetohydrodynamical stress energy tensor (e.g., Misner, Thorne & Wheeler 1973), \nT r t = ( ρ + Γ u + b 2 ) u r ut -b r bt . (5) \nand b µ is the magnetic field four-vector (e.g., Gammie, McKinney & Tóth 2003). The negative sign in equation (4) is because ut is negative; thus, a positive value of ˙ e tot means that total energy flows outward, and vice versa. Note that T r t represents the total energy transported by the fluid and the magnetic field, including the rest mass energy of the gas. However, this is not very convenient when considering energy at 'infinity' since the rest mass energy plays no role in feedback. Therefore, we consider a di ff erent measure of energy flux in which we eliminate the rest mass energy, \n˙ e = ˙ e tot -˙ m = -T r t -ρ u r , (6) \nwhich we hereafter refer to as 'the energy flux'. Positive values of ˙ e correspond to energy lost from the system, i.e., energy flows out into the surrounding medium. \nIntegrating over θ and φ and normalizing by the net mass flow rate at r = 10 (to avoid numerical issues near the horizon) we obtain the normalized radial profiles of mass and energy flow. The mass accretion rate is thus \n˙ M ( r ) = 1 | ˙ M net | ∫ θ ∫ φ ˙ mdA θφ, (7) \nand the two energy loss rates are \n˙ E tot( r ) = 1 | ˙ M net | ∫ θ ∫ φ ˙ e tot dA θφ, (8) \n˙ E ( r ) = 1 | ˙ M net | ∫ θ ∫ φ ˙ e dA θφ, (9) \nwhere dA θφ = √ -gd θ d φ is an area element in the θ -φ plane, \n˙ M net = ∫ θ ∫ φ ˙ m ( r = 10) dA θφ, (10) \nand signs have been chosen so as to make the integrated fluxes positive for outflow. The integrands in the above integrals correspond to time-averages over the duration of the time chunk of interest. The φ integral is over the range 0 to 2 π , while the range of the θ integral depends on the quantity of interest. When we wish to calculate the net outflow of mass or energy, we integrate over the full range θ = 0 -π . When we are interested in outflow in the jet or the wind, we limit the θ range accordingly, as described in the next two subsections. From the integrated rate of outflow of mass and energy, we calculate the integrated momentum in the outflow using the relativistic formula, \n˙ P ( r ) = √ ( ˙ E tot( r )) 2 -( ˙ M ( r )) 2 . (11) \nFinally, we quantify the magnetic field strength at the BH horizon by means of the magnetic flux parameter (Tchekhovskoy, Narayan & McKinney 2011a), \nΦ BH( t ) = 1 2 √ ˙ M ∫ θ ∫ φ | B r ( r H , t ) | dA θφ, (12) \nwhere B r is the radial component of the magnetic field \nFigure 2 shows profiles of µ versus θ for SANE and MAD runs with BH spin a ∗ = 0 . 7. (This is just an example; other spins give similar results.) We see that gas with θ values within ∼ 1 rad of the two poles has a positive value of µ and can escape to infinity, while gas closer to the equatorial plane has negative µ and cannot escape. The figure also shows profiles of the radial mass flux ˙ m , which changes sign at practically the same values of θ as µ . In other words, gas that has a positive value of µ has an outward-pointed velocity (it is escaping from the BH), while gas with negative µ has an inward-pointed velocity (it is accreting on to the BH). This confirms that the Bernoulli parameter µ is an excellent diagnostic of outflows in our simulations. \n<!-- image --> \nFigure 1. Snapshots of a ∗ = 0 . 7 SANE (left) and MAD (right panel) models at t = 90000. Reddish and blueish colors show the density and the Lorentz gamma factor u t , respectively, in the poloidal ( r , θ ) plane for an arbitrary value of the azimuthal angle φ . Arrows show the direction of gas velocity in that plane. \n<!-- image --> \nand r H is the radius of the horizon. The integral is over the whole sphere, and the factor of 1 / 2 converts the result to one hemisphere. An accretion flow with geometrical thickness h / r ≈ 0 . 4 transitions to the MAD state once Φ BH reaches ∼ 50 (Tchekhovskoy, Narayan & McKinney 2011a; Tchekhovskoy, McKinney & Narayan 2012). As we show later, the three MAD runs reach this limit quickly and remain there, whereas the four SANE runs are for the most part well below this limit.", '3.3 The outflow criterion': 'Since the simulations extend over only a finite range of radius, we need a criterion to decide whether a particular parcel of fluid can escape to infinity. The quantity we use to determine this is the Bernoulli parameter µ (Narayan et al. 2012), \nµ = -T r t ρ u r √ grr + T θ t ρ u θ √ g θθ ( ρ u r ) 2 grr + ( ρ u θ ) 2 g θθ -1 . (13) \nTo understand this expression, note that the quantities ˙ e and ˙ m introduced earlier correspond to the radial components of the respective fluxes. Correspondingly, there are θ components of these fluxes, and the two together may be viewed as two-vectors /vector ˙ ep and /vector ˙ mp in the poloidal r θ plane. We see then that µ is equal to /vector ˙ ep · /vector ˙ mp / /vector ˙ mp · /vector ˙ mp , i.e., it is the flux of energy parallel to the flow streamline. This quantity has to be positive for gas to be able to escape to infinity. \nTo be safe, in this paper we consider gas to be outflowing to infinity only if the conditions µ > 0 and ˙ m > 0 are both satisfied. \nHowever, as Fig. 2 shows, the two conditions are practically degenerate, so we could equally well have used just one of them.', '3.4 Three regions: Disc, wind, jet': "We identify the region of the solution where the gas flows in ( u r < 0) as the 'disc'. In our radiatively ine ffi cient ADAF-like simulations, the disc region is geometrically thick and extends over a range of θ ∼ ± 0 . 6 rad around the mid-plane (see Fig. 2). Outside the disc zone we have outflowing gas, which we further subdivide into two components, a slowly-moving 'wind' and a rapidly-moving 'jet'. The distinction between wind and jet is motivated by the shapes of the µ and ˙ m profiles in Fig. 2. In both the SANE and MAD simulations we see that for θ values within about 0.4 rad of the poles, µ is large and ˙ m is small. The outflowing material here is clearly relativistic and has a large energy per unit mass; we call it a 'jet'. The rest of the outflowing region, which lies in between the jet and the disc, has a low value of µ and larger ˙ m . This is a slowly-moving outflow, which we call a 'wind'. \nHow do we determine the boundary between the jet and the wind? Unfortunately, there is no unique way of separating these regions since the transition is smooth and the regions of high mass flux (wind) and high energy flux (jet) overlap each other. One possibility is to follow streamlines of the poloidal energy flux T p t , and to identify all outflowing streamlines anchored on the horizon as the jet and the remaining outflowing streamlines that originate in the disc as the wind (compare the bottom-most right panel of Fig. 5 and see Penna, Narayan & Sa¸dowski 2013). This is similar to the approach used by Tchekhovskoy, Narayan & McKinney (2011a) and is a convenient way of distinguishing the part of the outflow that is powered by the BH from that which is powered by the disc. However, magnetic fields extract rotational energy from the BH at all latitudes (Penna, Narayan & Sa¸dowski 2013), even in the disc region where the inflowing rest mass flux overwhelms this e ff ect and causes the flux of total energy to be negative. \nAnother approach, which we follow in this paper, is to choose a critical value of µ , or equivalently a critical value of the 'velocity at infinity', and to identify all outflowing gas with µ larger than this \nFigure 2. Vertical profiles of µ and ˙ m for a ∗ = 0 . 7 SANE (top panel) and MAD (bottom) simulations. The profiles were calculated at r = 80 and r = 160, respectively. \n<!-- image --> \ncritical value as the jet and the rest as the wind. We choose \nµ crit = 0 . 05 , (14) \nwhich at infinity corresponds to β crit = v crit / c ≈ 0 . 3, a reasonable demarcation point between jet and wind. In Fig. 3 we plot five contours of µ on top of the outflowing energy fluxes for the a ∗ = 0 . 7 MAD simulation. Our particular choice of µ crit = 0 . 05 reasonably separates the region of high energy flux, which corresponds to the jet, and the region of low energy flux, which corresponds to the wind.", '3.5 Radius of inflow equilibrium': "Since each of our simulations has been run for only a finite duration, the solutions reach inflow equlibrium only within a limited volume. Gas outside this volume has not had enough time to be influenced by the accretion flow structure near the horizon. To quantify the range of radii over which the disc solution is reliable, we adopt the 'loose' equilibrium criterion from Narayan et al. (2012), \nFigure 3. Contours of µ in the poloidal plane for the a ∗ = 0 . 7 MAD model plotted on top of the magnitude of the radial flux of energy ˙ e . Five contours are plotted corresponding to µ = 0 . 3 (thinnest blue line), 0 . 1, 0 . 05, 0 . 02 (thickest blue line), and 0 . 0 (green line). \n<!-- image --> \nTable 3. Inflow / outflow equlibrium radii \n| Model | Disc | Wind | Jet |\n|-------------------|--------|--------|-------|\n| a ∗ = 0 . 0 SANE | 110 | 210 | - |\n| a ∗ = 0 . 7 SANE | 70 | 130 | 21000 |\n| a ∗ = 0 . 9 SANE | 50 | 110 | 16000 |\n| a ∗ = 0 . 98 SANE | 50 | 80 | 8000 |\n| a ∗ = 0 . 0 MAD | 340 | 720 | 65000 |\n| a ∗ = 0 . 7 MAD | 160 | 320 | 29000 |\n| a ∗ = 0 . 9 MAD | 140 | 260 | 13000 | \ni.e, we search for an inflow equilibrium radius r eq which satisfies \nr eq = | v r ( r eq) | t chunk , (15) \nwhere v r ( r eq) is the density-weighted average velocity at a given radius, and t chunk is the duration of the last time chunk for the particular simulation. We carry out this calculation separately for the disc, the wind, and the jet. In the case of the wind and jet v r ( r eq) is outward, so technically r eq is the outflow equilibrium radius, but the principle remains the same. \nTable 3 gives values of the limiting inflow / outflow equilibrium radii r eq for each of the runs for each of the three regions. These radii decrease with increasing BH spin, since the durations of the simulations become shorter. In all cases, the wind region reaches equilibrium to larger radii than the disc since the radial velocity of the gas is larger. Similarly, since the jet has a relativistic velocity, this region reaches equlibrium to very large radii (essentially the entire domain of the simulation). Another systematic e ff ect is that the MAD simulations, because of their larger radial velocities (see Narayan et al. 2012), are in inflow equlibrium over a significantly larger volume compared to the SANE simulations.", '4.1 Accretion rate and magnetic flux': 'All the simulations were initialized with an equilibrium gas torus threaded by a weak poloidal magnetic field. Once the magnetorotational instability (MRI) develops, gas accretes towards the BH and the inner regions of the torus are depleted of matter. The accretion rate on the BH thus decreases with time, the variation being determined by the density profile assumed in the initial torus \nFigure 4. Accretion rate ˙ M into the BH (top) and magnetic flux Φ BH threading the BH horizon (bottom) versus time for SANE (dotted lines) and MAD (solid lines) models with di ff erent BH spins. \n<!-- image --> \n(Narayan et al. 2012). The upper panel of Fig. 4 shows the mass accretion rate ˙ M on the BH versus time for all the radiatively ine ffi cient models studied here. Solid and dotted lines correspond to MAD and SANE models, respectively. For a given BH spin, the accretion rate evolution is roughly the same for SANE and MAD models. However, at any given time, the higher the spin, the lower is the accretion rate. This appears to be because of the increasing mass loss rate (discussed in Section. 4.6). \nBy monitoring the magnetic flux Φ BH threading the BH horizon (eq. 12) we can evaluate whether a particular simulation is in the SANE or MAD state. The bottom panel of Fig. 4 shows the evolution of Φ BH for each of the models. The magnetic flux at the horizon for the MAD runs (solid lines) remains always near Φ BH ∼ 50, showing that the flux has saturated at the maximum allowed value, as appropriate for the MAD state (Tchekhovskoy, Narayan & McKinney 2011a). In contrast, the SANE models are characterized by lower values of Φ BH. However, once in a while even these simulations show Φ BH approaching the MAD limit, though the flux subsequently falls as an oppositely polarized magnetic loop reaches the BH. For all the three SANE models with non-zero BH spin, Φ BH approaches the saturation value appropriate to the MAD state near the end of the simulation. Thus, despite our e ff orts to avoid the MAD state in our SANE simulations, the accretion flow apparently has a tendency to be pushed towards the MAD limit.', '4.2 Structure of the outflow regions': "In this section we discuss the general properties of the outflow regions. Fig. 5 shows the magnitude of mass and energy fluxes ( ˙ m , ˙ e and ˙ e tot) for a ∗ = 0 . 7 SANE and a ∗ = 0 . 7 MAD models together with corresponding streamlines in the poloidal plane. The blue contours denote the border between wind and jet regions (eq. 14) while the green contours separate the outflow and inflow regions. \nThe left column shows the rest mass flux. At large radii mass flows mostly inside the bulk of the disc. However, the streamlines clearly show that the inflowing accretion rate is not constant in all models - some rest mass is lost from the inflow region and forms the wind. Narayan et al. (2012) have shown that for a ∗ = 0 models \nsuch a magnetically-driven wind from the accretion disc itself does not extend all the way towards the BH but stops around r = 40. \nThe mass outflows are enhanced for rotating BHs. For the a ∗ = 0 . 7 SANE model the extra energy flux along the polar axis (middle panel) supresses accretion in the polar region. However, the accretion flow wants to deposit the same amount of gas as in the non-rotating case (locally the disc does not feel the impact of spin at large radii). The surplus of gas has to find its way out of the system and flows out in the wind. For the a ∗ = 0 . 7 MAD model the energy flux along the axis is so strong that it not only reduces the solid angle available for the inflow but also directly drives the outflow, converting its magnetic energy to kinetic energy of gas. It results in a strong, 'cocoon'-shaped, mass outflow originating in the inner region (10 < r < 20) which significantly decreases the inflowing accretion rate and contributes to the jet region. The amount of mass lost in this region may be comparable to the net mass accretion rate ˙ M BH on the BH. Note that the region of increased mass outflow close to the polar axis and near the BH is a result of numerical corrections (floors) imposed to retain reasonable ratios of energy densities and make the code stable, and therefore should be regarded as unphysical. Fortunately, the amplitudes of all the fluxes emerging from that region are negligible when compared to the outflows at r > 10. \nThe middle column of Fig. 5 shows the magnitude of the energy flux (˙ e ). The outflowing energy is highly concentrated in the jet region and is much higher for the MAD model. The magnetic jet is more collimated than the mass loaded jet - most of the rest mass in this simulation flows out along a conical surface with opening angle ∼ 25 deg. The locations of both jets are visualised in Fig. 6 showing the density distribution in the left part of the plot and outflows of rest mass and energy in the right part of the plot with magenta and blue, respectively. It is clear that the strong energy flux region is surrounded by the region where the mass loss is most e ffi cient. \nThe right column in Fig. 5 shows the magnitude and streamlines of the total energy flux ˙ e tot, i.e., the sum of the first two columns. Inside the disc the inflowing stream of rest mass dominates and the net energy flux points inward. In contrast, in the wind and jet the total energy flux is positive. \nThe solid angles of the jet regions in the a ∗ = 0 . 7 simulations are Ω jet /lessorsimilar 1 . 0 sr for the SANE and MAD models, respectively. The corresponding solid angles covered by the wind are Ω wind /greaterorsimilar 5 . 0 sr. The wind thus subtends a significantly larger solid angle.", '4.3 Angle-integrated mass and energy fluxes': 'Figure 7 shows radial profiles of various mass flow rates for the a ∗ = 0 . 7 SANE and MAD simulations. The red lines show the absolute value of the normalized net mass accretion rate (eq. 7 with θ integrated over the full range 0 to π ) while the black lines show the magnitude of the total inflowing mass flux (same integral but over inflowing gas only). The curves are terminated at the limiting inflow equilibrium radius of the disc. As expected, the net ˙ M is equal to unity (because of the normalization) and is independent of radius (indicative of steady state), though constancy is violated near the outer edge, where complete equilibrium has not been achieved. \nThe blue curves show the mass outflow rate in the jet; in this case the integral in equation (7) is limited to those values of θ where the Bernoulli parameter µ exceeds the critical value µ crit and u r > 0. Since the jet originates close to the BH, we see that the blue lines asymptote to a constant value at large radii. Thus the simulations provide a reliable estimate of the mass loss rate in the jet. The rate \nFigure 5. Rest mass flux ( ρ u r , left), energy flux ( -T r t -ρ u r , middle), and total energy flux ( -T r t , right column) in the poloidal plane for models a ∗ = 0 . 7 SANE and a ∗ = 0 . 7 MAD. Colors show the logarithm of magnitude of the radial flux while streamlines show the flux direction on the poloidal plane. Green contour shows µ = 0 and separates the outflow region from the inflowing disc region. Blue contour stands for µ = 0 . 05 and separates the jet and wind regions. The third and fourth rows zoom in on the innermost region. \n<!-- image --> \nFigure 6. 3D visualisation of a snapshot of the a ∗ = 0 . 7 MAD model. The yellow-green colors in the left part of the plot show the spatial distribution of mass density. The magenta colors in the right part correspond to the magnitude of the rest mass flux ρ u r while the blue colors show the magnitude of the energy flux T r t + ρ u r . \n<!-- image --> \nis about 10% of the accretion rate for the SANE model and roughly equal to the accretion rate for the MAD model. \nIn contrast to the jet, the mass outflow rate in the wind (orange lines) increases steadily with radius and there is no sign of convergence at large radii. This confirms that mass loss in the wind is dominated by large radii. Since the simulations reach inflow equilibrium at best out to a few hundred RG , whereas real flows in nature extend to outer radii R out ∼ 10 5 RG or even larger, this means that estimates of mass loss rates require considerable extrapolation. This is discussed below. \nFigure 8 shows corresponding results for the energy flow rate. Here the red lines show the e ffi ciency of the accretion flow, i.e., the energy that flows out to infinity normalized by the net mass accretion rate. We see that the a ∗ = 0 . 7 SANE run has an e ffi ciency of about 5% whereas the MAD run has a much larger e ffi ciency of about 70%. The stronger magnetic flux around the BH in the latter enables much more e ffi cient tapping of the BH spin energy. The energy outflow rates in the jet (blue curves) are well converged, just like the mass. Even the energy outflows rates in the wind (orange curves) seem reasonably well converged. This is consistent with the simple analysis we present next.', '4.4 Radial scalings of outflows': 'Let us define the mass inflow rate ˙ M in( r ) at a given radius as the sum of the net mass accretion rate, which we call ˙ M BH, and the mass outflow rate in the jet and wind ˙ M wind (e.g., Stone, Pringle & Begelman 1999; Yang et al. 2013). Thus, ˙ M in is the mass accretion rate we would calculate if we restricted the θ integral in equation 7 to the regions with u r < 0. It is reasonable to assume that ˙ M wind scales with radius as a power-law (Blandford & Begelman 1999). Hence let us assume that \n˙ M wind( r ) = ˙ M BH ( r r in ) s , (16) \n˙ M in( r ) = ˙ M BH [ 1 + ( r r in ) s ] , (17) \nwhere we expect the index s to lie in the range 0 -1, and r in is some characteristic radius, typically of order tens of RG . The black curves in Fig. 7 show the variation of ˙ M in vs r for the two simulations. Note the approximate power-law behavior, with s /greaterorsimilar 0 . 5. \nFrom equation (16) it is evident that the mass outflow rate is dominated by large radii. Therefore, unless we have a reliable estimate of the value of s , we cannot hope to obtain an accurate estimate of the mass loss rate in an ADAF. The situation is better in the case of energy outflow. The di ff erential mass loss at a given \nFigure 9 shows radial profiles of energy outflow in the jet and wind regions in the final three time chunks of our simulations: chunks T4, T5, T6 for the a ∗ = 0 SANE and MAD runs, T3, T4, T5 for the two a ∗ = 0 . 7 runs, and T2, T3, T4 for the two a ∗ = 0 . 9 runs. We see that the wind energy profiles (orange curves) are poorly converged, in agreement with Narayan et al. (2012) who found that outflows from discs around non-rotating BHs showed poor convergence with time. On average, the longer the duration of a given simulation, the further from BH the winds originate. There is non-monotonic behavior in the a ∗ = 0 . 7 SANE and a ∗ = 0 . 9 MAD models, but we believe this is simply because the magnetic flux around the BH increased in the last time chunk for these two simulations (compare Fig. 4). Note that the energy lost in the wind for the SANE models, and to a lesser extent for the MAD models, is independent of radius, i.e., the energy budget is dominated by the innermost region of the wind, in agreement with the scalings discussed in Section 4.4. \n<!-- image --> \nFigure 7. Radial profiles of the mass flux ˙ M for a ∗ = 0 . 7 SANE (top) and MAD (bottom panel) at time chunk T5. Red, orange and blue lines show the net mass accretion rate, mass outflow in the wind, and mass outflow in the jet, respectively. All fluxes are normalized by ˙ M at r = 10, and each line is terminated at the corresponding radius of inflow or outflow equlibrium (Table 3). The black lines show the total inflow flux, i.e., the integral over sphere of the inflowing rest mass (eq. 7). \n<!-- image --> \nradius is given by \nd ˙ M wind = d ˙ M in = s ˙ M BH r ( r r in ) s dr . (18) \nWe expect that any mass that flows out at radius r will carry with it an energy equal to some fraction ξ of the local potential energy. Thus, we estimate the local energy loss rate in the wind to be \nd ˙ E wind = d ˙ M wind ξ c 2 r = ξ s ˙ M BH c 2 r s -2 r s in dr , (19) \nwhich gives a cumulative energy loss rate of \n˙ E wind( r ) = ξ s 1 -s ˙ M BH c 2 r in [ 1 -( r in r ) 1 -s ] ≈ ξ s 1 -s ˙ M BH c 2 r in . (20) \nWe see that the energy loss in the wind is dominated by the innermost regions (as confirmed in Fig. 8), hence the simulations ought to provide reliable estimates of the energy feedback rate into the surroundings. The momentum outflow rate has a scaling intermediate between those of mass and energy outflow. \nGRMHD simulations of BH accretion discs are limited by computational power. Even in the best cases (e.g., the simulations discussed here), the inflow equilibrium region of a simulation extends only to radii of order a few hundred RG (Table 3). Because of this limitation, we can obtain meaningful estimates of the total \n<!-- image --> \nFigure 8. Similar to Fig. 7 but for the energy flux ˙ E . \n<!-- image --> \nmass loss rate in the wind only if the mass outflow behaves in a self-similar fashion and the range of inflow equilibrium of the simulation is far enough from the BH that we can obtain a reliable estimate of the index s . Having this caveat in mind, we discuss first the more reliable energy outflow.', '4.5 Energy outflow': 'The energy profiles ˙ E ( r ) of the simulated jets show very good spatial convergence, asymptoting to a constant value at large radii. This indicates that the jet criterion we adopted ( µ > µ crit) closely follows the energy flux streamlines at large radii. Variations among \nFigure 9. Fluxes of energy ( ˙ E ) for SANE (top) and MAD (bottom) simulations with given value of BH spin. Blue and orange lines correspond to the jet and wind regions, respectively. On each sub-panel three sets of lines are plotted corresponding to three most recent chunks of time for each simulation. \n<!-- image --> \ndi ff erent time chunks is due to fluctuations in the magnetic flux at the horizon, as discussed further below. \nBecause of the lack of convergence with time of the wind regions in the simulations, it is not obvious how one should compare di ff erent simulations to study the e ff ect of BH spin. In the following, we choose to compare simulations at the same physical time, viz., time chunk T4. Figure 10 shows the results. The power of both the jet and the wind increases with BH spin for both SANE and MADsimulations. The variation in the case of the wind is modest, whereas jet power shows a very strong dependence on BH spin. In fact, the jet power in the a ∗ = 0 . 9 MAD simulation exceeds ˙ Mc 2 , showing that the jet is powered by more than accretion energy. At least some part of the power must come directly from the BH (Tchekhovskoy, Narayan & McKinney 2011a). \nTheoretical jet models indicate that the power extracted from a spinning accreting BH scales as (Blandford & Znajek 1977; Tchekhovskoy & McKinney 2012; Penna, Narayan & Sa¸dowski 2013), \nη jet = P jet ˙ Mc 2 ∼ 0 . 05 4 π c Φ 2 BH Ω 2 H , (21) \nwhere Φ 2 BH is the magnetic flux threading the horizon (eq. 12), and Ω H is the angular velocity of the outer BH horizon (eq. 1). The jet powers in our simulations are generally consistent with the spin dependence in this formula; the a ∗ = 0 . 9 has the strongest jet and a ∗ = 0 has essentially no jet. Note that the non-zero jet power we find for the a ∗ = 0 . 0 MAD model is an artifact caused by the activation of numerical floors at the polar axis and the corresponding injection of mass and energy. \nIn Fig. 11 we test the scaling of jet power with Φ 2 BH . The horizontal axes show the magnetic flux 〈 Φ BH 〉 (see eq. 12) averaged over a given chunk of time. For each simulation, three points are shown, corresponding to the final three time chunks of that simulation (only two points in the case of the a ∗ = 0 . 98 model because of the very short duration of this run). The horizontal error bars reflect the standard deviation of the magnetic flux within the par- \n<!-- image --> \nFigure 10. Radial profiles of the energy flux ( ˙ E ) for SANE (top) and MAD (bottom panel) models at time chunk T4. Profiles for three values of BH spins are presented. All fluxes are normalized to ˙ M at r = 10 and the lines are terminated at the radius of outflow equilibrium of the corresponding a ∗ = 0 . 9 model. \n<!-- image --> \nFigure 11. Outflow rates of the energy flux ( ˙ E ) in the jet (top) and the wind (bottom panel) as a function of the magnetic flux at BH horizon averaged over the duration of given time chunk. For each model values corresponding to the three most recent time chunks are presented. Triangles and crosses are for SANE and MAD models, respectively. Colors denote BH spin. The outflow rate in the wind was measured at r = 80 while the jet power is averaged over r = 100 ÷ 1000 and the vertical errorbars show the minimum and maximum values in this range. \n<!-- image --> \ntime chunk. Colors denote BH spin. Triangles and crosses correspond to SANE and MAD simulations, respectively. \nThe top panel in Fig. 11 shows the energy loss rate in the jet as a function of magnetic flux at the horizon. Within each model and for a given spin, the jet power follows the Φ 2 BH scaling quite well, validating this prediction of theory. The bottom panel of the same figure shows the energy carried away by the wind measured at a common radius of r = 80. This quantity again increases with the magnetic flux Φ BH, but the dependence is much weaker than for the jet. This shows that the wind receives only some of its energy from the BH, the rest coming directly from the gravitational energy released by the accreting gas.', '4.6 Mass outflow': 'Figure 12 shows radial profiles of mass outflow for SANE and MAD models with BH spin a ∗ = 0, 0 . 7, and 0 . 9. For each model, the last three time chunks are shown. Lack of convergence of the mass outflow rate in the wind as a function of time is clearly visible \nFig. 13 shows profiles of mass outflow in the jet and wind for time chunk T4 and three BH spins, a ∗ = 0, 0 . 7, and 0 . 9. In the SANE simulations (top panels), where the jet is weak, mass loss is dominated by the wind at all radii. Mass loss is strongest and starts closest to the BH for the highest value of BH spin. In the case of the MAD solutions (bottom panels), mass outflow in the jet (cocoon) dominates at smaller radii (except for a ∗ = 0), and the wind takes over only at larger radii. Outflow rates initially overlap each other and then diverge showing the same dependence on BH spin. The jet mass loss rate in particular shows noticeable dependence on the BH spin. \n<!-- image --> \nFigure 13. Similar to Fig. 10 but for the mass flux ( ˙ M ). \n<!-- image --> \nfor most of the models - the orange lines move steadily outward with time. The mass outflow rate in the jet is well converged in space, i.e., it quickly saturates at a consant value. However, it varies with time (most profound for a ∗ = 0 . 9 SANE model) as a result of changing magnetic flux threading the horizon (Fig. 4). \nThe top panel of Fig. 14 shows the relation between the magnetic flux at the horizon Φ BH and the mass loss rate in the jet. There is a strong correlation, with a dependence approximately ∝ Φ 1 . 5 BH , which is similar to the scaling of the jet energy loss rate ( Φ 2 BH ) but a little shallower. The slightly di ff erent slope is because the terminal Lorentz factor of the jet ( u t ) scales with the magnetic flux roughly as Φ 0 . 5 BH , i.e., jets in MAD solutions are typically more relativistic than those in SANE solutions. \nThus the mass in the jet cocoon is driven essentially by energy outflow from the BH horizon. There is also a clear dependence on BH spin, with larger spins giving stronger mass loss in the jet. The mass loss in the jet for the a ∗ = 0 MAD model is an artifact, and \nFigure 12. Similar to Fig. 9 but for the mass flux ( ˙ M ). \n<!-- image --> \nFigure 14. Similar to Fig. 11 but for the rest mass flux ˙ M . \n<!-- image --> \nrepresents leakage of mass from the the disc wind into the jet region. \nThe bottom panel shows the mass loss rate in the wind as measured at r = 80 (which lies inside the wind outflow equilibrium region for all the simulations) as a function of Φ BH. It is clear that, for a given BH spin, there is essentially no correlation between the mass loss in the wind and the magnetic flux. This is reasonable. Winds flow out from relatively large radii in the disc, where the gas is not sensitive to the magnetic flux near the BH. Note, however, that there is a correlation between the wind mass loss rate and BH spin - on average higher spin leads to stronger mass outflow.', '4.7 Outflows of momentum': 'The relativistic momentum flux given by eq. (11) reduces for particles to, \n˙ P = ˙ M v √ 1 -v 2 , (22) \nand in the non-relativistic limit gives ˙ P = ˙ M v . Therefore, one could expect that the profiles of outflowing momentum are similar to the profiles of lost rest-mass but normalized by a coe ffi cient related to the total gas velocity v . \nFor the mildly-relativisitic wind region the dominant component of gas velocity is the azimuthal component ( v ≈ r -1 / 2 ). Fig. 15 shows the radial profiles of the integrated momentum flux ˙ P . Indeed, the momentum lost in the wind (orange lines) resembles profiles of the rest-mass lost in that region (Fig. 12). Its magnitude is noticeably smaller and follows the rescaling with characteristic gas velocity, e.g., at r = 100 is a factor of v = 0 . 1 lower. As a result of the radial dependence of the azimuthal velocity the profiles are less steep than for the rest-mass flux. \nThe gas in the jet region is highly relativistic (Fig. 1) and its velocity is dominated by the radial component. In a similar way, the radial profiles of momentum lost in the jet region should resemble the rest-mass loss rates with the scaling factor depending on the gas \nFigure 15. Similar to Figs. 9 and 12 but for fluxes of momentum ( ˙ P ) for SANE (top) and MAD (bottom) simulations. \n<!-- image --> \nvelocity. For the jet region with the characteristic Lorentz factor u t ≈ 2 this factor is close to unity. Fig. 15 shows that indeed the momentum lost in the jet region (blue lines) is quantitatively similar to the profiles of rest-mass lost in this region. The fact that both are constant in radius for r /greaterorsimilar 50 proves that the characteristic velocity in the jet region does not change.', '4.8 Approximate model of outflows': "Outflows in a typical strongly magnetized ADAF around a rotating BH can be divided into a slow wind at large radii and a relativistic jet along the poles. Such a structure is schematically shown in Fig. 16. Most of the energy is lost via the jet; for large enough BH spins, the power of the jet may even exceed the total energy budget ˙ Mc 2 of the accretion flow. Mass is lost through both the wind and the 'cocoon' surrounding the jet. At large radii, mass loss is dominated by the wind. \nIn the case of the jet, as discussed in previous sections, the energy and mass outflow are sensitive both to the BH spin as well as the magnetic flux at the horizon. Outflows in the wind also depend on these parameters, but less less sensitively. \nAccording to the BZ model (Section 1.2) the power extracted from the BH (eq. 21), P BH, is proportional to the square of the magnetic flux threading the horizon Φ BH and the square of the angular velocity of the BH horizon Ω H. Using this scaling as a guide and keeping in mind the discussion in Sections 4.5-4.7, we construct here simple empirical fits which reasonably reproduce the simulation results. Except for ˙ M wind, all the other outflow rates are good to within a factor of order unity (Fig. 17). This is true even for ˙ P wind so long as s is not very di ff erent from 1 / 2. \nAll the estimates given in this subsection are normalized by the mass accretion rate on the BH, ˙ M BH. Thus, energy outflow rates are given in units of ˙ M BH c 2 , momentum outflow rates in units of ˙ M BH c , and mass outflow rates in units of ˙ M BH. \nThe cumulative energy, mass and momentum outflow at radius r may be written as a sum of contributions from the jet and the \nwind, \n˙ E = ˙ E jet + ˙ E wind , (23) \n˙ M = ˙ M jet + ˙ M wind , (24) \n˙ P = ˙ P jet + ˙ P wind . (25) \nAs discussed in Section 4.4, our simulations provide reliable estimates of the two energy fluxes. The energy loss from the system is dominated by the outflow in the jet (Fig. 9) which is driven by energy extracted from the BH through the BZ process. We approximate the jet energy as \n˙ E jet ≈ 0 . 5 ˜ Φ 2 BH ˜ Ω 2 H , (26) \nwhere \n˜ Φ BH = Φ BH / 50 (27) \nis the magnetic flux threading the horizon, normalized by the characteristic value for a MAD disc, and \n˜ Ω H = Ω H / 0 . 2 (28) \nis the horizon angular velocity (eq. 1), normalized by the angular velocity for a spin a ∗ = 0 . 7 BH Ω H( a ∗ = 0 . 7) ≈ 0 . 2. As the top panel of Fig. 17 shows, the coe ffi cient 0.5 in eq. (26) provides a good fit to the results presented in Section 4.5. 3 \nThe energy loss in the wind is much weaker (Fig. 9) and we estimate it as, \n˙ E wind ≈ 0 . 005(1 + 3 ˜ Φ 2 BH ˜ Ω 2 H ) . (29) \nwhere the empirical factor (1 + 3 ˜ Φ 2 BH ˜ Ω 2 H ) accounts for the increase of the wind power with BH spin and magnetic flux (see the bottom panel of Fig. 11). The middle panel of Fig. 17 validates the choice of the coe ffi cient. \nFigure 16. Schematical picture of the disc structure near a rotating BH. \n<!-- image --> \nThe mass loss rates we measure are robust only in the jet region. Following the discussion in Section 4.5, we approximate it as, \n˙ M jet ≈ 0 . 7 ˜ Φ 1 . 5 BH ˜ Ω H , (30) \nwhere the coe ffi cient 0 . 7 is our best guess from the simulation results (see bottom panel of Fig. 17). \nThe radial profiles of the mass loss rate in the wind in the various simulations are not converged in time and are therefore less reliable. Even estimating the power-law index s and the characteristic radius r in (eq. 16) is di ffi cult because the region where the profiles show power-law behavior is close to the radius r conv defined in eq. (15). Roughly, it appears that 0 . 5 < s < 0 . 7 (in agreement with Yuan, Bu & Wu 2012). Let us write, \n˙ M wind ≈ ( r r in ) s , (31) \nwhere r in is the radius where the mass flux in the wind equals the net accretion rate on the BH: ˙ M wind( r in) = 1. From Fig. 12 it appears that r in /greaterorsimilar 100 and that it tends to be smaller (outflows stronger) for rotating BHs. Better estimation of r in is not possible due to the limitations of our simulations. \nFinally, the fluxes of momentum in the jet and wind are approximately related to the corresponding mass loss rates by, \n˙ P jet ˙ Ejet , (32) \n˙ P wind ≈ ˙ M wind r -0 . 5 , (33) \n≈ \nwhere we have assumed u t ≈ 2 in the jet, that the characteristic velocity of the wind originating at radius r is v ≈ v φ = r -0 . 5 , and that s > 0 . 5.", '5 COMPARISON WITH THIN DISCS': 'So far we have shown that outflows of mass and energy are common in thick accretion discs. This class of discs corresponds to radiatively ine ffi cient flows forming at very low ( L /lessorsimilar 0 . 01 L Edd, Narayan & McClintock 2008) or very high ( L /greaterorsimilar L Edd) accretion rates; the latter regime is known as the slim disc (Abramowicz et al. 1988). However, many BHs in microquasars and galactic nuclei are \nTable 4. Thin disc models \n| Model | BH spin ( a ∗ ) | Initial magnetic field | Resolution | t final |\n|------------------|-------------------|--------------------------|--------------|-----------|\n| a ∗ = 0 . 0 thin | 0 . 0 | multi-loop | 264x64x64 | 50000 |\n| a ∗ = 0 . 9 thin | 0 . 9 | multi-loop | 264x64x64 | 50000 | \nknown to accrete with moderate accretion rates which correspond to a radiatively e ffi cient, geometrically thin disc. Do such discs produce outflows similar to those found in the case of thick discs? \nJets in Galactic BHs are quenched around the time they change state from the low-hard, presumably geometrically thick, to the high-soft state corresponding to a geometrically thin disc (e.g., Remillard & McClintock 2006). This fact may suggest that relativistic outflows are not characteristic for such class of discs. \nNumerical studies of geometrically thin discs have been limited so far because of the requirement of including radiative transfer. All GRMHD simulations of thin discs performed so far (e.g., Shafee et al. 2008; Penna et al. 2010; Noble et al. 2011; Zhu et al. 2012) are based on an artificial cooling function which drives discs to an arbitrarily chosen entropy corresponding to a required given disc thickness. More sophisticated treatment of radiation in the context of thin accretion flows is being developed (e.g., Fragile et al. 2012; Sa¸dowski et al. 2013). \nTo compare the power of outflowing mass and energy between thick and thin discs we have performed two additional simulations which use the cooling function as decribed in Zhu et al. (2012) to drive the disc towards h / r ≈ 0 . 1 which corresponds to L ≈ 0 . 3 L Edd. We tested two values of BH spin a ∗ = 0 . 0 and a ∗ = 0 . 9. The initial magnetic field was set in a similar way to the SANE simulations descibed earlier, i.e., its poloidal component formed a set of counterorientated loops. Details of the simulations are given in Table 4. \nFig. 18 shows the history of the accretion rate and the magnetic flux through the BH horizon for the thin disc simulations. For both values of BH spin the accretion rate is roughly contant. The magnetic flux, however, decreases with time. For both simulations the non-dimensional flux parameter Φ BH (eq. 12) is well below the \nFig. 19 shows the magnitude of the local rest mass flux ( ˙ m , Eq. 3), the energy flux (˙ e , Eq. 6), and the total energy flux (˙ e tot, Eq. 4) in the poloidal plane, averaged over t = 25000 ÷ 50000, for thin disc simulations with a ∗ = 0 . 0 and 0 . 9, respectively. Because of the much lower radial velocity in thin discs, the range of inflow equilibrium in the disc region is limited to r /lessorsimilar 15. Even with this limitation, we believe the simulations should be su ffi cient to provide useful estimates of the outflow in a jet. It is thus significant that we see no sign of a jet in either of the two simulations, suggesting that thin discs are not conducive to producing relativistic jets. We also do not see any outflowing mass or energy in a wind. However, this result is less significant since, even in the geometri- \n<!-- image --> \n<!-- image --> \nFigure 17. Coe ffi cients in fitting formulae (top to bottom: eqs. (26), (29) and (30) for various models. The horizontal dashed lines show the values chosen for the fits. \n<!-- image --> \nMAD-characteristic value 4 Φ BH , MAD = 25 reflecting the fact that the magnetic pressure does not saturate and that the discs are in the SANE state. \nFigure 18. Accretion rate (top) and vertical magnetic flux threading the horizon (bottom) history for thin disc simulations with a ∗ = 0 . 0 and a ∗ = 0 . 9. \n<!-- image --> \ncally thick ADAF runs described earlier, the wind usually begins only at relatively large radii. Our thin disc simulations have not reached inflow equilibrium at such radii. \nThe thin disc simulations described here were set up with multiple poloidal magnetic field loops which, as explained earlier, results in unsaturated magnetic field around the BH and thus are destined to produce weaker jets. Initializing the magnetic field with a single loop, as in the MAD ADAF runs, should result in stronger magnetic flux at the horizon. This would be the thin disc equivalent of the MAD state. (Note that, historically, the original conceptual paper on the MAD solution by Narayan, Igumenshchev & Abramowicz (2003) considered thin discs, whereas the numerical simulations by Igumenshchev, Narayan & Abramowicz (2003) which motivated this paper dealt with ADAFs.) Such an experiment is worthwhile and is left for future work. However, comparing the flow structure of the two thin disc simulations discussed here (Fig. 19) with the SANE ADAF simulations discussed earlier (e.g., third row of Fig. 5) already suggests that geometrically thin discs are less e ffi -cient in generating relativistic outflows of rest mass and energy than equivalent thick discs. Because of the very limited region of inflow equilibrium in Fig. 19, we cannot address the e ffi ciency of generating wind-like outflows except in the innermost regions (where there is no wind).', '6 DISCUSSION AND SUMMARY': "In this paper we presented a number of GRMHD simulations of magnetized non-radiative geometrically-thick accretion discs around BHs, and used them to investigate the outflow of energy, momentum and mass from these systems. The simulations covered a range of BH spin parameters a ∗ = a / M or, equivalently, BH horizon angular frequencies Ω H, and thus probed the e ff ect of this important parameter on outflows. We also tested two initial configurations of the magnetic field, which led respectively to configurations with a weak magnetic field around the BH (SANE configuration) and a saturated magnetic field (MAD configuration). This enabled \nFigure 19. Similar to Fig. 5 but for thin disc models with a ∗ = 0 . 0 (top) and a ∗ = 0 . 9 (bottom panels). \n<!-- image --> \nus to study the dependence of outflow properties on the magnetic flux Φ BH around the BH. \nEach simulation was run for a long time in order to reach steady state over as wide a range of radius as possible (see Table 3 for estimates of the size of the inflow and outflow equilibrium region). In each simulation, we divided the flow into an inflowing 'disc', and an outflowing 'wind' and 'jet'. The latter two zones were distinguished on the basis of their velocity at infinity. The jet consists of all matter that flows out with v /greaterorsimilar 0 . 3 at infinity and the wind is the rest of the outflowing matter with asymptotic v /lessorsimilar 0 . 3. More precisely, the boundary is defined by a critical Bernoulli parameter, µ crit = 0 . 05 (eq. 13). Figure 16 shows the three zones schematically. \nOur results are summarized in Section 4.8 in the form of approximate fitting functions. In the case of the jet, we are able to provide fairly reliable estimates of its outflow properties. Specifically, we give fitting functions for the jet energy outflow ˙ E jet , momentum outflow ˙ P jet and mass outflow ˙ M jet. The energy and momentum outflow rates vary proportional to Φ 2 BH Ω 2 H , as predicted by the BZ model. \nThe situation is less clear in the case of the wind. We believe our estimates of the wind energy outflow ˙ E wind and, to a lesser extent, the wind momentum outflow, ˙ P wind, are fairly reliable. However, the mass outflow in the wind is highly uncertain. This is because mass outflow is dominated by large radii and our simulations do not go out to a large enough radius to enable us to extrapolate reliably to the putative outer edge of the accretion flow at (say) the Bondi radius. Specifically, in equation (31), we cannot determine the value of s with any certainty. The situation is exacerbated by the fact that, as the simulation progresses in time, the wind launch radius appears to track the limiting radius of inflow equilibrium in the disc. That is, the wind seems always to begin within a factor \nof about two of that radius, but this is where the results are least reliable. Modulo this serious uncertainty, it appears that the energy, momentum and mass outflow rates in the wind have some dependence on the BH angular velocity Ω H and the BH magnetic flux Φ BH. \nSMBH 'feedback' in galaxy clusters is believed to play an important role in keeping the cluster gas hot and preventing catastrophic star formation (Ciotti & Ostriker 2001; Brüggen & Kaiser 2002; Ruszkowski & Begelman 2002; Churazov et al. 2004). Observational evidence suggests that the feedback occurs via relativistic jets. The scaling relations given in this paper for ˙ E jet and ˙ P jet are likely to be useful in this context. \nSMBHfeedback also appears to occur inside AGN host galaxies (Silk & Rees 1998; King 2003; Hopkins, Murray & Thompson 2009), but in this case it is less obvious that the jet is important. Jets are much too collimated to have much of an e ff ect, e.g., our jets subtend less than 10% of the solid angle around the BH at a radius r ∼ 10 2 , and the solid angle at larger radii is much less because of continued collimation. On the other hand, the wind in our simulations covers nearly 50% of the solid angle around the BH (the other 50% being covered by the thick disc). Thus, the wind is likely to have a strong e ff ect on the host galaxy, especially since, unlike radiation which can escape through optically thin regions of the galaxy, a wind is certain to deposit all its energy and momentum in the interstellar medium of the galaxy. The scaling relations given in this paper for ˙ E wind and ˙ P wind (of which the energy scaling is more reliable) are thus relevant. \nWe also presented simulations of artifically cooled thin accretion discs ( h / r ≈ 0 . 1) for two BH spins: a ∗ = 0, 0.9. Neither simulation showed any evidence for either a jet or a wind. We note that both simulations were in the SANE regime and the radius of inflow equilibrium was only r ≈ 15. Nevertheless, our preliminary \nFigure 20. Qualitative description of outflows in BH accretion discs. \n<!-- image --> \nconclusion from these simulations is that thin discs are significantly less e ffi cient in producing relativisic jets. We cannot say anything about winds from thin discs. \nThe flowchart in Fig. 20 summarizes our qualitative conclusions from this study of outflows from accreting BHs. Geometrically thin discs appear not to produce relativisitic outflows, nor do they have magnetically-driven winds from small radii. Mass loss through magnetically- or line-driven winds at larger radii is certainly possible, but is beyond the scope of this work. Geometrically thick ADAFs exhibit two kinds of outflows: wind and jet. The wind originates from relatively large radii in the flow. It is responsible for most of the mass outflow, and carries a modest amount of energy and momentum. The properties of the wind depend relatively weakly on the BH spin and the magnetic flux at the horizon. The wind covers a large solid angle and is an excellent agency for feedback. The jet is a relativistic outflow which emerges from radii close to the BH horizon and is highly collimated along the poles. Its properties depend strongly on both the BH spin and the magnetic flux at the horizon. In favorable cases - rapid BH spin and large magnetic flux (MAD limit) - the flux of energy and momentum in the jet are very much larger than the corresponding fluxes in the wind. However, because of the strong collimation, jet feedback is probably important only on the largest scales, e.g., galaxy clusters. \nApart from the mass accretion rate ˙ M BH and the BH an- \nular velocity Ω H, the present study confirms the importance of a third parameter, the dimensionless magnetic flux Φ BH at the BH horizon. In principle, di ff erent systems might have di ff erent values of Φ BH, making it much more di ffi cult to come up with useful predictions for individual systems. One mitigating factor is that the numerical simulations described here as well as other recent work (Tchekhovskoy, Narayan & McKinney 2011a; Tchekhovskoy, McKinney & Narayan 2012; Narayan et al. 2012; McKinney, Tchekhovskoy & Blandford 2013) suggest that ADAFs easily reach the MAD state in which the magnetic flux saturates at its limiting value Φ BH ∼ 50 -60. Thus, it is conceivable that the majority of observed systems would be in the MAD limit and have a similar value of Φ BH, thus eliminating this uncertainty. Note that plenty of magnetic flux is available in the external medium, more than enough to saturate the field at the horizon (Narayan, Igumenshchev & Abramowicz 2003), and theoretical arguments suggest that a geometrically thick ADAF is likely to transport the magnetic field inward e ffi ciently (Guilet & Ogilvie 2012, 2013). \nWe conclude with two caveats. First, our formulae for the energy and momentum outflow in the jet and wind are expressed in terms of the net mass accretion rate on the BH, ˙ M BH. In practice, to use these relations, one needs to be able to estimate ˙ M BH given the mass supply rate at the feeding zone of the BH, say the Bondi ra- \nus R B /greaterorsimilar 10 5 RG . Unfortunately, there is considerable uncertainty on this issue. In the language of equation 16, if the Bondi accretion rate at R = R B is ˙ M B, then very roughly we expect \n˙ M B = ˙ M in( r B) = ˙ M BH [ 1 + ( r B r in ) s ] . (34) \nDepending on the values of s and r in, the ratio ˙ M BH / ˙ M B could vary over a wide range. What can be done about this uncertainty? There is hope that r in could be determined via simulations. However, it might be much harder to obtain a reliable estimate of s , at least through GRMHD simulations. Some authors have obtained fairly good estimates of s via large dynamic range hydro simulations (e.g., Yuan, Bu & Wu 2012, obtain s ∼ 0 . 5), but it is uncertain if their results will carry over to MHD. This problem needs to be resolved before accretion disc simulation results can be used for galaxy feedback studies. \nSecond, the simulations presented here correspond to nonradiative flows, whereas ADAFs in nature do produce at least some radiation. Dibi et al. (2012) included radiative cooling in their simulations and found that it introduces significant di ff erences in the structure of the flow near the black hole for accretion rates above 10 -7 Eddington. Their result, however, depends on assumptions regarding the electron temperature in the two-temperature plasma. More conservatively, one might expect noticeable e ff ects from radiation for accretion rates /greaterorsimilar 10 -4 Eddington. Radiation can also have a strong e ff ect at and beyond the Bondi radius. Specifically, X-rays produced by inverse Compton scattering near the black hole can heat gas near the Bondi radius and significantly modify its thermal properties (Sazonov, Ostriker & Sunyaev 2004; Yuan, Xie & Ostriker 2009). In turn, this can significantly modify the mass supply at the Bondi radius, thereby forming a feedback loop.", '7 ACKNOWLEDGEMENTS': 'A.S. and R.N. were supported in part by NASA grant NNX11AE16G. 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2019PhRvD.100b3011V
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New search pipeline for compact binary mergers: Results for binary black holes in the first observing run of Advanced LIGO
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2019-01-01
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In this paper, we report on the construction of a new and independent pipeline for analyzing the public data from the first observing run of Advanced LIGO for mergers of compact binary systems. The pipeline incorporates different techniques and makes independent implementation choices in all its stages including the search design, the method to construct template banks, the automatic routines to detect bad data segments ("glitches") and to insulate good data from them, the procedure to account for the nonstationary nature of the detector noise, the signal-quality vetoes at the single-detector level and the methods to combine results from multiple detectors. Our pipeline enabled us to identify a new binary black hole merger GW151216 in the public LIGO data. This paper serves as a bird's eye view of the pipeline's important stages. Full details and derivations underlying the various stages will appear in accompanying papers.
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https://arxiv.org/pdf/1902.10341.pdf
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{'A New Search Pipeline for Compact Binary Mergers: Results for Binary Black Holes in the First Observing Run of Advanced LIGO': "Tejaswi Venumadhav, 1, ∗ Barak Zackay, 1 Javier Roulet, 2 Liang Dai, 1 and Matias Zaldarriaga 1 1 School of Natural Sciences, Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA 2 Department of Physics, Princeton University, Princeton, NJ, 08540, USA (Dated: March 21, 2019) \nIn this paper, we report on the construction of a new and independent pipeline for analyzing the public data from the first observing run of advanced LIGO for mergers of compact binary systems. The pipeline incorporates different techniques and makes independent implementation choices in all its stages including the search design, the method to construct template banks, the automatic routines to detect bad data segments ('glitches') and to insulate good data from them, the procedure to account for the non-stationary nature of the detector noise, the signal-quality vetoes at the singledetector level and the methods to combine results from multiple detectors. Our pipeline enabled us to identify a new binary black-hole merger GW151216 in the public LIGO data. This paper serves as a bird's eye view of the pipeline's important stages. Full details and derivations underlying the various stages will appear in accompanying papers.", 'I. INTRODUCTION': "The LIGO and Virgo observatories reported the detection of several gravitational wave (GW) events from compact binary coalescence in their First and Second Observing Runs (O1 and O2 respectively) [1]. These detections required technically sophisticated analysis pipelines to reduce the strain data. This is because typical events are buried under the detector noise, and cannot be simply 'seen' in raw data at current sensitivities. Hence, any search for signals in the data needs to properly and precisely model the detector noise. \nThe simplest model is that the detector noise is stationary and Gaussian in nature. Under these assumptions, the best method to detect signals is matched-filtering, which involves creating a bank of possible signals, constructing optimal filters (or templates) for the signals given the noise model, and running the templates over the data. The resulting scores are distributed according to known (chi-squared) distributions in the presence or absence of real signals [2]. \nUnfortunately, both the assumptions underlying matched-filtering fail at some level: the noise statistics vary even on the timescales of the (putative) signals, and there are intermittent non-astrophysical artifacts which are clearly not produced by Gaussian random noise ('glitches') [3], examples of such disturbances can be found in Ref. [4]. These systematics pollute the distribution of the matched-filtering scores. Moreover, the templates describing different astrophysical signals have finite overlaps, and thus often trigger on the same underlying noise transients. Detectable real events lie in the tails of the score distribution, and hence it is crucial to properly correct for systematics in order to maximize the sensitivity to GW events, and to quote reliable falsealarm rates (FARs). \nThe official LVC catalog of GW events comprises candidates from two independent pipelines: PyCBC [5] and GstLAL [6]. Additional analysis of the data was presented in Ref. [7]. Each of these pipelines has developed solutions for the data complexities described above. In this paper, we describe a new and independent analysis pipeline that we have developed for analyzing the publicly available data from the first observing run of advanced LIGO [8]. Our solutions and implementation choices were guided by the desire to attain, as much as possible, the ideal of the distributions in the Gaussian case, which are easily understood and interpreted. \nFirst, we developed a method to construct template banks that enumerates not over physical waveforms, but over linear combinations of a complete set of basis functions for their phases. Correlations between templates have a uniform and isotropic metric in this space. \nSecond, when dealing with systematics, we use procedures with analytically tractable behavior in the case of Gaussian random noise, which enables us to set thresholds based on well-defined probabilities. We developed a simple method to empirically correct for the nonstationary nature of the detector noise (PSD drift). Under this procedure, segments of data with no apparent glitches produce trigger scores with perfect chi-squared distributions. At the first pass, we attempt to veto out residual 'glitches' using a collection of simple tests (either at the signal-processing level or after triggering), while still using the matched-filtering scores as the ranking statistics to leave the Gaussian 'floor' untouched. We also developed methods to condition masked data in a way that guarantees that the following matched filtering step would have zero response to the masked data segments. \nFinally, we estimate the background of coincident triggers between the two detectors using time slides (akin to PyCBC ). Our pipeline includes methods to use the information from background triggers to combine physical triggers from different detectors in a statistically optimal \nmanner for distinguishing astrophysical events from noise transients. \nOur paper is organized as follows: Section II provides an overview of the stages in the pipeline. Section III expands upon each of the stages while omitting derivations and precise details, which we present in accompanying papers [9-11]. In Section IV we present the results of our search for binary black hole mergers in O1.", 'II. PIPELINE STAGES': "We construct our pipeline in several stages, which are organized as follows: \n- 1. Construction of a template bank: We divide the mergers into banks with logarithmic spacing in the chirp mass, and analyze each bank separately. Section III A provides further details on the underlying method, and the properties of the resulting banks.\n- 2. Analysis of single detector data: We first analyze the data streams from the Hanford (H1) and Livingston (L1) detectors separately, as follows:\n- (a) We preprocess data from each detector in chunks of glyph[similarequal] 4096 s. Section III B details our initial signal processing.\n- (b) We iteratively whiten the data stream, perform several tests to detect and remove bad data segments ('glitches'), and condition the remaining data to preserve astrophysical signals. Sections III C and III D describe this procedure.\n- (c) We correct for the non-stationary nature of the noise (PSD drift), which if untreated, systematically pollutes the connection between the matched-filtering scores and probability. Section III F provides more details.\n- (d) We generate matched-filtering overlaps for the waveforms in our banks with the whitened data stream, apply the PSD drift correction, and record triggers whose matched-filtering scores are above a chosen threshold (Section III E).\n- 3. Coincidence analysis between detectors: We analyze triggers that are coincident in H1 and L1. In Section III G, we describe how we collect coincident triggers with combined incoherent score above a threshold, at both physical (candidates) and unphysical (background) time delays.\n- 4. Refining on a fine grid: We refine the parameters of the candidates and the background on a finer grid around the triggers in order to account for template bank inefficiency, and allow room for more stringent signal quality vetoes. \n- 5. Trigger quality vetoes: We apply vetoes on the triggers based on the signal quality, as well as the data quality. The vetoes have to be applied either at the single-detector level, to avoid biasing the calculation of the coincident background using time slides. Section III I lists the vetoes we applied to the triggers.\n- 6. Estimating the significance of candidates: We use the set of background triggers to estimate the FAR for the candidates at physical lags between H1 and L1. We do this in two stages:\n- (a) We first compute a ranking score that is purely a function of the incoherent scores of the triggers, under the assumption that the noise processes that produce the background are independent between detectors (Section III J).\n- (b) Section III K describes our coherent score, which adds all the information encapsulated in the phase, amplitude, relative sensitivity and arrival time differences between the detectors to create our final candidate ranking statistic.\n- (c) Section III L describes how we construct an estimate for the probability of a coincident event being of astrophysical origin given an astrophysical event rate.", 'A. Template bank': "We perform our search by matching the strain data to a discrete set of waveform templates that sufficiently closely resemble any gravitational wave signal within our target parameter space. We target our search at coalescing binary black holes (BBH), defined here as compact binary objects with individual masses between 3 M glyph[circledot] and 100 M glyph[circledot] and with aligned spins. We allow spin magnitudes up to | χ 1 , 2 | < 0 . 85. We restrict the mass ratios to be q -1 < 18. \nAs described in Ref. [9], we construct five BBH template banks ( BBH 0-4 ) that together span this target parameter space, and conduct a separate search within each of them. The banks are defined by regions in the plane of component masses, as shown in Fig. 1. We place the bounds between adjacent banks at M = { 5 , 10 , 20 , 40 } M glyph[circledot] , where M = ( m 1 m 2 ) 3 / 5 / ( m 1 + m 2 ) 1 / 5 is the chirp mass and m 1 , 2 are the individual masses. We find several motivations for dividing the search. The lowmass banks have many more templates than the heavier banks, and thus they inherently have a larger lookelsewhere penalty. Dividing the search prevents this from strongly affecting the sensitivity of the high-mass searches: in this way, on astrophysical grounds we might expect roughly comparable numbers of signals in each \nFIG. 1. Division of the BBH parameter space into five template banks ( BBH 0-4 ) by component masses. A separate search is conducted on each. The points represent the input waveforms used to construct the banks (not the templates themselves), and the colors encode the division of each bank into subbanks according to the shapes of the waveform amplitude. Approximate detector-frame masses are indicated for BBH detections reported to date (in O1 and O2) and for GW151216. \n<!-- image --> \nbank, regardless of the largely different number of templates they have. Moreover, this splitting enables us to discriminate between the different types of background events that each search is subject to. The different duration of the signals in each bank will require us to use different thresholds when masking bad data segments (see Section III C). The prevalence of non-Gaussian glitches will be different in each bank and thus the score we assign to events with the same signal-to-noise ratio (SNR) is different in each bank (see Section III J). Table I summarizes the template bank parameter ranges and sizes. \nThe template bank needs to be effectual, that is, to guarantee a sufficiently high match between a GW waveform and at least one template in the bank. We define the inner product between waveforms h i , h j \n( h i | h j ) := 4 ∫ ∞ 0 d f ˜ h i ( f ) ˜ h ∗ j ( f ) S n ( f ) , (1) \nwhere S n ( f ) is the one-sided noise power spectral density (PSD) of the detector and a tilde indicates a Fourier transform into the frequency domain. It is used to define the match \nm ij = max τ ∣ ∣ ( h i | h j e i 2 πfτ ) ∣ ∣ ; (2) \nthroughout this section we assume that all waveforms are normalized to ( h | h ) = 1. We assess the effectualness E of each bank by computing the best match with 10 4 random waveforms in its target parameter space. We apply the down-sampling and sinc-interpolation described \nTABLE I. Summary of template bank parameters. M is the chirp mass range that the bank is designed to cover. E 0 and E are the effectualnesses without and with refinement (Section III H) respectively, as quantified by the best match within the bank achieved by the top 99 . 9% of random astrophysical templates. N templates is the total number of templates in each bank. \n| Bank | M ( M glyph[circledot] ) | E 0 | | E N templates |\n|--------|----------------------------|----------|------|-----------------|\n| BBH 0 | < 5 | 0 . 90 0 | . 97 | 6465 |\n| BBH 1 | (5 , 10) | 0 . 92 0 | . 96 | 7919 |\n| BBH 2 | (10 , 20) | 0 . 94 0 | . 96 | 5855 |\n| BBH 3 | (20 , 40) | 0 . 95 0 | . 96 | 594 |\n| BBH 4 | > 40 | 0 . 97 0 | . 97 | 57 |\n| Total | Total | | | 20 890 | \nin Section III E and the waveform optimization described in Section III H to the test waveforms, to properly simulate the search procedure. We report the effectualness of the banks in Table I. When designing banks, we set the reference PSD to be the aLIGO MID LOW PSD [12], which is representative of O1. \nIn order to correct the PSD drift at manageable computational cost, our search pipeline requires that the frequency domain templates, of the form \n˜ h ( f ) = A ( f ) e iψ ( f ) , (3) \nshare a common amplitude profile A ( f ) (see Section III F) and differ only in the phase ψ ( f ). In order to avoid excessive loss of effectualness due to this approximation, we split each bank into several subbanks, each of which is assigned a different A ( f ) profile. We use the method of 'stochastic placement' to determine as many subbanks as needed to guarantee that every waveform within the target parameter range has an amplitude match, \n∫ d f A ( f ) A ( f ) S n ( f ) glyph[greaterorequalslant] 0 . 95 , (4) \nwith at least one subbank. The resultant divisions into subbanks are color-coded in Fig. 1. \nThe remaining task is to place templates in each subbank to efficiently capture the possible phase shapes ψ ( f ). We achieve that with a geometric approach, where we use the mismatch between templates to define a mismatch distance, which quantifies the similarity between any two waveforms. We abandon the physical parameters as a description of the templates in favor of a new basis of coordinates c , in which the mismatch distance induces an Euclidean metric. We then set up a regular grid in this space. Our templates take the form \nh ( f ; c ) = A ( f ) exp [ i ( ψ ( f ) + ∑ α c α ψ α ( f ) )] , (5) \nwhere ψ ( f ) is the average phase, and { ψ α ( f ) } are phase basis functions which are orthonormalized such that the \nmismatch distance satisfies \nd 2 c , c + δc := 1 -m ( h ( c ) , h ( c + δc )) = 1 2 ∑ α δc 2 α + O ( δc 3 ) . (6) \nAn input set of physical waveforms representing the target signals are used, first to define the subbanks and then to determine the appropriate phase basis functions. The input waveforms may be generated with any frequencydomain model; we use the IMRPhenomD approximant [13]. The phase basis functions are found from a singular value decomposition of the input waveforms which identifies the minimal set of linear independent components that need to be kept. A small number of basis functions are enough to approximate all possible phases to sufficient accuracy. All banks require five linearly independent bases or fewer, with about half of them having only three or fewer. While the coefficient for the lowest order bases may vary over a range of several hundred units, the coefficients for the highest order bases vary within narrow ranges, sometimes by less than one unit.", 'B. Loading and preprocessing the data': "The strain data is provided by LIGO in sets of files of length 4096 seconds for each detector (H1 and L1 in O1). The natural choice is to split the analysis along the same lines, i.e., file by file. We would like to preserve our sensitivity to events near the edges of files, and hence we pull in data from adjacent files if available. The length of data we pull in is set by the following considerations: (a) there should be no artifacts in the whitened strain at the edge of a file due to missing data at the right edge, (b) events that straddle files should be contained inside the padded and whitened data stream, and (c) relatively short segments of data ( < 1024 s) near file edges, with a large adjoining segment ( > 64 s) of missing data, are analyzed as part of the adjoining file instead of on their own. Even after padding, the boundary of the (expanded) data stream will still have artifacts from the whitening filter. To treat this, we further append 64 s of zeros to the padded strain data on either side, that we will later inpaint using the method of Section III D. \nAdditionally, we observe that long segments ( glyph[greaterorsimilar] 64 s) of bad data, as marked by LIGO's quality flags, can have a few unmarked extra seconds of bad data adjoining the marked segments (this can happen due to latency in the flagging system, for example). The procedure outlined in Section III C is designed to catch such segments, as well as other kinds of misbehaved data. However, we only reach this stage after some initial signal processing and sufficiently bad data segments might pollute good data segments through each step of the analysis. Therefore, we trim an additional 2 s of data when these segments occur at the right edges of files. \nThe next step after loading the data is to estimate its PSD. We use Welch's method [14], in which several \noverlapping chunks of data are windowed and their periodograms are averaged (we use the implementation in scipy.signal with a Hann window). We make our PSD estimation robust to bad data by (a) disregarding chunks that overlap with segments that were marked by LIGO's quality flags, and (b) averaging using the median instead of the mean (see Appendix B of Ref. [15]). \nAn important choice to make is the length of the individual chunks whose periodograms enter the averages ('chunksize' in what follows). In pure Gaussian random noise, the choice of chunksize is governed by the following (conflicting) considerations: (a) controlling the statistical uncertainty in the averages, which depends on the number of independent samples within a file, and (b) mitigating the loss in matched-filter sensitivity around underresolved spectral lines. As we discuss in Section III F, the advanced LIGO data is typically not described by purely Gaussian random noise (even in the absence of 'bad' segments with excess power) due to systematic drifts in the PSD within a file. We find that using 64 s chunks to measure the PSD yields an acceptable compromise between the above effects. This choice also affects the minimum length of the files that we choose to analyze: the first consideration above (the measurement noise in the PSD) implies that we take a 4% loss in sensitivity for files that are shorter than 16 times the chunksize. If a file is shorter than this limit (not including the segments marked by LIGO's quality flags), we try to analyze it using a chunksize of 16 s instead, while enforcing the same minimum number of chunks. \nWe restrict ourselves to analyzing frequencies f < 512 Hz by down-sampling the data to 1024 Hz. This is safe to do since all compact binary merger signals accumulate more than glyph[similarequal] 99% of their matched filtering SNR below 512 Hz at the O1 detector sensitivity, and since we already budget for glyph[greaterorsimilar] 1% losses in the template bank. This choice reduces the sizes of the template banks and saves us computational time during triggering, at the expense of a negligible loss in sensitivity. We also apply a high-pass filter to the data (implemented as a fourthorder Butterworth filter with f min = 15Hz, applied from the left and the right to preserve phases). This removes low-frequency artifacts in the data (that could later trigger our flagging procedure in Section III C), and is safe to do since we only use frequencies f > 20 Hz in building the template bank. \nFinally, we construct the whitening filter from the estimated PSD, and use it to whiten the data. The whitening filter typically has most of its power at small lags, but exhibits a long tail at large lags due to spectral lines in the data. Our procedure for inpainting bad data segments (described in Section III D) requires that the whitening filter has finite support, hence we zero the filter at large lags (while ensuring that we retain glyph[greaterorsimilar] 99 . 9% of its weight, typically the filter is left with an impulse response length of glyph[similarequal] 16 s). Zeroing the whitening filter in the time domain corresponds to convolution with a sinc function in the frequency domain, which fills in the lines; thus, the \nfilter does not reject spectral lines completely. Hence, we take care that our flagging procedure does not trigger on spectral lines in the data.", 'C. Identifying bad data segments': "Advanced LIGO data contains intermittent loud disturbances that are not marked by the provided data quality flags. We need to flag and remove these segments to prevent them from polluting our search, while taking care to preserve astrophysical signals of interest. This is the fourth analysis of the data, and hence we assume that any new signals we find will have an integrated matched filter SNR ρ < 30 in a single detector. This assumption allows us to bound the influence of a true signal on our procedure. \nWe devise several complementary tests to flag bad data segments. We design our tests to satisfy the following conditions: \n- 1. The test statistics have analytically known distributions for Gaussian random noise.\n- 2. The thresholds are set to values of the test statistics achieved by waveforms with single-detector ρ = 30 in noiseless data. Signals at this SNR have a probability of glyph[similarequal] 0 . 5 of triggering a single test in the presence of Gaussian random noise. We found empirically that signals satisfying ρ ≤ 20 are almost always retained.\n- 3. If the above thresholds are too low, they are adjusted so that a single test is triggered at most once per five files due to Gaussian random noise alone. This is important for template banks with long waveforms.\n- 4. The tests are safeguarded from being triggered by PSD drifts over long timescales ( t glyph[greaterorsimilar] 10 s), which can manifest as excess power over shorter timescales. \nThese conditions ensure that we are sensitive to gravitational waves while not over-flagging the data. It is important that the tests be done at the single-detector level in order to avoid biasing the calculation of the background using time slides. \nOur tests trigger on the following anomalies: (a) outliers in the whitened data-stream, (b) sine-Gaussian transients in particular bands, (c) excess power localized to particular bands and timescales, and (d) excess power (summed over frequencies) on particular timescales. We picked timescales and frequency bands for the tests based on inspecting the spectrograms of the bad segments; Table II details the choices. \nThe data has spectral lines at which the PSD is several orders of magnitude higher than in the continuum. The power in these lines often significantly varies in a non-Gaussian manner within a single file. The lines do \nTABLE II. Summary of tests for identifying bad data segments. For each test, we show the frequency band and timescale of the disturbance that it is sensitive to, and the length of the data we excise around the disturbance. \n| Test type | Frequency band | Excess duration (s) 3 | Hole duration (s) |\n|------------------|------------------|-------------------------|---------------------|\n| Whitened outlier | [20 , 512] | 10 - | 0 . 6 |\n| Excess power | [20 , 512] | 0 . 2 | 0 . 2 |\n| Excess power | [20 , 512] | 1 | 1 |\n| Excess power | [55 , 65] | 1 | 1 |\n| Excess power | [70 , 80] | 1 | 1 |\n| Excess power | [40 , 60] | 1 | 1 |\n| Excess power | [40 , 60] | 0 . 5 | 0 . 5 |\n| Excess power | [20 , 50] | 1 | 1 |\n| Excess power | [100 , 180] | 1 | 1 |\n| Excess power | [25 , 70] | 0 . 1 | 0 . 1 |\n| Excess power | [20 , 180] | 0 . 05 | 0 . 05 |\n| Excess power | [60 , 180] | 0 . 025 | 0 . 025 |\n| Excess power | [25 , 70] | 0 . 2 | 1 |\n| Sine-Gaussian a | [55 , 65] | - | 0 . 1 |\n| | [20 , 60] | - | 0 . 1 |\n| | [100 , 140] | - | 0 . 1 |\n| | [50 , 150] | - | 0 . 1 |\n| | [70 , 110] | - | 0 . 1 |\n| | [50 , 90] | - | 0 . 1 |\n| | [125 , 175] | - | 0 . 1 |\n| | [75 , 125] | - | 0 . 1 | \nnot contribute to the matched-filtering overlap, since the PSD is effectively infinite at their frequencies. Hence it is preferable that varying lines do not trigger our tests. \nWe detect sine-Gaussian artifacts in a given band by matched-filtering with a complex waveform that saturates the time-frequency uncertainty principle and contains most of its power in the band. We apply notch filters to the sine-Gaussian template to remove any overlap with spectral lines. We flag any outliers in the matched-filtering results above a threshold defined to satisfy the aforementioned conditions (see second paragraph of Sec. III C), which is a procedure safe to any relevant events) \nWe detect excess power using a spectrogram (computed using the spectrogram function in scipy.signal with its default Tukey window). We sum the power in the frequency ranges of interest, disregarding frequency bins that overlap with varying lines. For Gaussian random noise, this sum has a chi-squared distribution. This is not achieved in practice unless correcting for the effects of PSD changes. We make the excess power statistic robust to the drifting of the PSD by comparing the instantaneous excess power with with a local moving-average power baseline. \nThe simplest test is to look for outliers in the whitened strain, since individual samples should be independent and normally distributed with unit variance. We flag segments of whitened data, with a safety margin in time, \naround outliers above a chosen threshold. \nWhenever one or more of these tests fire, we excise the offending segments (which we refer to as 'holes') and inpaint the raw data within as described in Section III D. In practice, we observe that the outlier test often does not catch all of the 'bad' data, in which case the inpainted and whitened data contain further outliers. Hence, we iterate over the 'identify bad segments, inpaint, whiten' cycle multiple ( < 7) times, increasing the safety margin in time by successively larger multiples of 0.1 s, until the process converges. \nWe treat any part of the data that was marked with any of the LIGO quality flags as if it contained large disturbances. After all the data quality tests done in this section, we are left with roughly 46 days of coincident on-time between the detectors, with slight changes from bank to bank, as all the test thresholds are waveform dependent.", 'D. Inpainting bad data segments': "The matched-filtering score for a template h with data d with a noise covariance matrix C is: \nZ = h † C -1 d = 4 ∑ f h ∗ ( f ) d ( f ) S n ( f ) , (7) \nwhere f denotes the frequencies, and in the last equality we assumed that the noise is diagonal in Fourier space. The tests described in Section III C flag bad data segments that we would like to mask. The operator C -1 (the 'blueing filter') is not diagonal in the time domain; when viewed as a linear filter operating on the data, its impulse response length (typically glyph[similarequal] 32 s) is set by the PSD spectral lines and the chunktime used to estimate the PSD. Thus the scores evaluated using Eq. (7) can be significantly affected even tens of seconds away from a masked segment. \nTo deal with this problem, if we consider a fraction of the data of length N d in which we have masked N h samples, we filter the data with a filter F and define a new score by: \n˜ Z = h † C -1 F d. (8) \nThe filter F is given by \nF = 1 -WM -1 W T C -1 , (9) \nwhere the matrix W has one column of length N d for every sample that is masked with all the entries zero except for a one at the position of the masked sample and M is the N h × N h matrix M = W T C -1 W . The computationally expensive part of this filtering procedure is to invert the matrix M . \nThe filter F is such that the score ˜ Z is independent of the value of the template waveform h inside masked segments. That is to say, F can be obtained by demanding \nthat C -1 Fd is identically zero inside the masked regions. F is a projection operator ( F 2 = F ) that commutes with C -1 , i.e., C -1 F = F T C -1 , and depends only on the mask and the covariance matrix C . In particular, it is independent of the waveform h , and thus can be computed once and for all before performing matched filtering. Note also that for computing F , it is not important that C -1 be the exact noise covariance; it just needs to be consistently used to define the scores in the section of data. \nWe can also derive F as the solution of several related linear algebra problems. We can model the presence of the mask as if the data had an additional source of noise inside the masked region, and take the limit of zero additional noise outside the holes and infinite additional noise inside. The filtered data ˜ d = F d equals the original data outside the masked segments, and the best linear prediction for the data inside the hole based only on the data outside (Wiener filter). It can also be thought of as the ˜ d that minimizes \nχ 2 = 1 2 ˜ d † C -1 ˜ d (10) \nsubject to the constraint that ˜ d equals the original data outside the mask, but can take any value inside. The computation of F is explained in detail in Ref. [10]. \nFigure 2 shows an example of a small section of the data containing a 'glitch' artifact. We show the difference between 'gating' the bad data by applying a window function to it, and creating a hole and inpainting it with the algorithm we described. We can see that gating substantially changes the standard deviation of the samples in the hole and the few seconds surrounding it, which can potentially create spurious triggers, and can damage any real signals that happen to be in the data at the same time. In our method, the 'blued' data is set to be identically zero inside the hole.", 'E. Matched filtering': "Given the whitened, hole-filled data, we compute the overlaps with all templates in the template bank, and register the times and templates when the SNR 2 is above a triggering threshold. The choice of the threshold was driven by the requirement to produce a manageable number of triggers per file, and was generally in the range 20 < SNR 2 thresh < 25 for the various banks and subbanks. \nIn order for the statistics of the overlaps to have a standard complex normal distribution, we need to apply two corrections: one is for the PSD drift, and one for the existence of holes (masked data segments). As we show in Ref. [10], the PSD correction depends only on the amplitude of the waveform, and hence we pre-compute it for each representative A ( f ). The hole correction is waveform dependent: we evaluate it under the stationary phase approximation, which assumes that there are many waveform cycles inside the hole, and accounts for the \nFIG. 2. Effect of masking and inpainting glitches. Top panel: A segment of whitened strain data (in units of the noise standard deviation) that has an identified glitch. The orange line is the standard deviation σ over a running window of 100 samples, and is typically close to unity as expected for whitened data. Second panel: Gating the glitch with an inverse Tukey window (green) and then whitening generates artifacts in the whitened data, even outside the window. For example, σ remains above 1.1 for approximately 2 s to each side of the glitch. Third panel: The inpainted whitened data has unit variance outside the hole (shaded). Bottom panel: After inpainting, the 'blued' strain is identically zero inside the hole, so overlaps with templates do not depend on what is inside the hole. \n<!-- image --> \nchange in the variance due to the missing cycles in the hole. This approximation works only for long waveforms, and hence we use overlaps in the vicinity of holes only for waveforms that are longer than 10 s. We also ignore overlaps where more than half of the variance (and hence SNR 2 ) is inside holes as these are anyway a negligible part of the volume (and are also non-declarable even if they contain a genuine candidate). \nIn order to compute the overlaps and hole variance corrections efficiently, we first notice that the waveform is shorter than a typical data segment, so we can use the overlap-save method in order to reduce the FFT sizes. Because the maximum frequency of the whitened data is taken to be 512 Hz, all information about matching the template to the data is in the complex overlaps we compute. Looking at single overlaps and comparing to the triggering threshold is not sufficient since the SNR could be reduced by as much as 10% due to sub-sample shifts in the GW arrival time (we down-sampled the data to 1024 Hz). We recover this sensitivity by first setting \nFIG. 3. It is necessary to track the drifting PSD on time scales of seconds. In blue we show the power spectrum of the square of the absolute value of the overlaps with a template in the BBH 0 bank for a repesentative set of files. It reaches the level of Gaussian fluctuations only close to ∼ 0 . 1 Hz, and has a rednoise power spectrum fit by a power-law (red dashed curve). The orange curve shows the PSD drift correction we apply to the data, which correctly traces the actual fluctuations in the standard deviation of the overlaps up to the Gaussian floor. \n<!-- image --> \na lower SNR bar, and sinc-interpolating the overlaps (by a factor of 4) within each contiguous segment above this lower bar, before checking for overlaps above the (higher) triggering threshold.", 'F. Applying corrections due to the varying Power Spectral Density of the Noise': "The power spectral density of the LIGO detectors can slightly vary with time. These changes may be hard to track and would inevitably result in PSD mis-estimation. As Ref. [10] shows, if we mis-estimate the PSD by a factor (1 + glyph[epsilon1] ( f )), the information loss in matched filtering scales as O ( glyph[epsilon1] 2 ), but the overlap's standard deviation differs by O ( glyph[epsilon1] ). This means that O (100) segments of data are required in order to measure the PSD well enough to aim for discarding less than 1% sensitivity. In order to resolve the lines well enough to aim for the same loss, tens of seconds of data are required. Therefore, an order of a thousand seconds are needed for estimating the PSD. We choose to measure the PSD using the Welch method, in which the signal is cut into overlapping segments, and the PSD power at frequency f is the (scaled) median of all the power estimates at this frequency from all the segments. It turns out, though, that the PSD varies on time-scales as short as ∼ 10 s, as seen in Figure 3. \nWhile at first sight it may seem impossible to both capture the width of the lines and track the fast variation in the PSD, we accomplish it by correcting the first order effect of PSD mis-estimation on time-scales that are as short as the PSD changes, to precision of ∼ 1%. \nThis correction is basically a local estimate of the stan- \nFIG. 4. Estimated changes to the variance of the overlap measurements, measured over periods of ≈ 16 s defined to guarantee a 2% precision. Measurement errors are shown by shuffling the overlaps in time and calculating the local averages. Vertical lines are one standard deviation away from the mean for each distribution. It is evident that the variance changes we are tracking are not random measurement fluctuations and can lead to severe changes in the significance assessment of a particular event. \n<!-- image --> \ndard deviation of the overlaps, and is derived (along with some other nice properties it has) in Ref. [10]. In Figure 4, we present a histogram of the distribution of the local variance estimates. Notice the large deviations from unity in both directions. We note that the tail reaches values as high as 1 . 5; at such high values, there are visible disturbances in the spectrogram, sometimes referred to as glitches. However, at values in the range [0 . 85 , 1 . 2], the data mostly behaves in a regular fashion, and there is no apparent sign something bad is going on in the spectrogram of the data. These changes can cause substantial loss of sensitivity in binary coalescence analyses that neglect this effect 1 . \nTo illustrate why correcting for these variance estimates is crucial for determining the exact significance of a candidate event, we point out that the most economic way of creating a (spurious) ρ = 8 event is to wait for a lucky time where the PSD mis-estimation is large (say, 1.2), and then create a (genuine) ρ = 7 . 3 fluctuation. In Figure 5, we see the tail of the trigger distribution is substantially inflated if the PSD drift is not corrected. \nFIG. 5. Effect of the PSD drift correction on the trigger distribution. Trigger distributions of binary black hole merger waveforms in bank BBH 0 ( M∈ [2 . 6 , 5] M glyph[circledot] ) and a subbank from BBH 3 ( M∈ [20 , 40] M glyph[circledot] ), in the Hanford detector, before applying any vetoes. \n<!-- image -->", 'G. Coincidence Analysis of the two detectors': 'After all single detector triggers above a critical ρ 2 are collected, we need to find pairs of triggers that share the same template, and have a time-lag difference that is less than 10 ms. In order to generate background coincident triggers, we also need to collect trigger pairs with all other considered time slides (we choose integer jumps of 0 . 1 s in the range [ -1000 s , 1000 s]). We collect the background events and the physical events by the following process: First, we define that a real trigger has ρ 2 > 0 . 9 ρ 2 max -5 where ρ max is the maximum trigger in the segment of 0 . 01 s. The reason for this choice is that triggers that are too close to a major erratic event are not declarable and that if there is a glitch that slipped through our net, we do not want a large amount of accompanying triggers to coincide with random fluctuations in the other detector. This massively reduces the load of the subsequent stages. \nWe then take each remaining trigger, and insert it into a dictionary according to the template key. This would allow us to immediately find all the times at which this template triggered. Using queries to the dictionary, we find all the pairs of triggers that belong to either the background or the foreground group, and pass the threshold ρ collect . This threshold depends on the bank via computing the Gaussian noise threshold for obtaining one significant event per O1, and then multiplied by the bank effectualness, to guarantee that every trigger that can acquire the one-per-O1 significance after optimization is included. \nWe now view the H1 component of all pairs of triggers and group them to groups of 0.1s. We use the less stringent version of the veto to vet the trigger with the highest SNR in each group, and upon failure discard the entire group (the logic here is that similar triggers are all passing or failing the veto together). We do the same for the L1 component of all remaining trigger pairs. \nWe then optimize every trigger by computing the overlaps with the data of every template in the sub-grid c values (see Sec. III H). We further sinc interpolate with a long support to obtain further time resolution for the overlaps. We then choose the sub-grid template that maximizes the quadrature sum of the single detector SNRs. This trigger pair is now vetoed with the stringent veto. If a trigger pair passes all these, it is registered.', 'H. Refining triggers on a finer template grid': 'The template-bank is organized as a regular grid, which facilitates refinement in places of interest. This enables us to squeeze more sensitivity and imitate the strategy of a continuous template bank, which is more objective than an arbitrarily chosen grid. The effectualness achieved by the top 99.9% of injections with the template banks used for the search varies between 0.9 and 0.96. Refining the grid by a factor of two in each dimension would bring it to > 0 . 96 in all cases, but would also substantially increase the number of waveforms in the bank (which in turn increases the computational complexity and memory requirements of our search). We therefore take the approach of refining every candidate and background trigger pair. Since we know the maximum amount of SNR increase that is possible for a real event, we refine all candidates that have a score that is high enough to have a chance of reaching a FAR of 1/O1 after refinement. We greatly speed up the candidate refinement by calculating the likelihood using the relative binning method [16] (using the original grid-point trigger as the reference waveform). Table I reports the improvement in effectualness achieved by this procedure for our banks.', 'I. Vetoing triggers': "The matched-filtering score is the optimal statistic for detecting signals buried in Gaussian random noise. As emphasized in the previous sections, the LIGO strain data is not well-described by purely Gaussian random noise, and hence, the matched-filtering score may be triggered (i.e., pushed above the Gaussian-noise significance threshold) by either transient or prolonged disturbances in the detector. Our pipeline attempts to reject these candidates by identifying bad segments at the preprocessing level (Section III C), or downweighting the scores by their large (empirically measured) variance (Section III F). However, this is not enough to bring us down to the Gaussian detection limit, especially for the heavier black hole banks. Thus, we need additional vetoes at the final stage to reject glitches. We use vetoes that are based on either the quality of the neighboring data, as well as that of the signal. \nOur most selective vetoes are based on signal quality, and check that the matched-filtering SNR builds up the \nright way with frequency. We perform the following tests: \n- 1. We subtract the best-fit waveforms from the data and repeat the excess power tests of Section III C, but with lower thresholds computed using waveforms with ρ = 3 (and bounded to fire once per 10 files due to Gaussian noise). Moreover, when we see excess power in a particular band and at a particular time, we only reject candidates with power at the same time in their best-fit waveforms (in order to avoid vetoing candidates due to unrelated excess power).\n- 2. We split the best-fit waveform into disjoint chunks, and check for consistency between their individual matched-filtering scores. This test is similar in philosophy to the chi-squared veto described in Ref. [17], but improves upon it by accounting for the mis-estimation of the PSD (which is an inevitable consequence of PSD drift) and by projecting out the effects of small mismatch with the template bank grid.\n- 3. We empirically find triggers that systematically miss the low-frequency parts of the waveforms, or have large scores at intermediate frequencies. The check described above is agnostic to the way the matched-filtering scores in various chunks disagree, and hence is not the most selective test for these triggers. We reject these triggers by using 'splittests' that optimally contrast scores within two sets of chunks. \nThe final two tests are the most selective vetoes, and hence their thresholds must be set with care. Our method for constructing template banks enables us to set these thresholds in a rigorous and statistically well-defined manner to ensure a given worst-case false-positive probability, which, accounting for the inefficiency in the bank, is achieved with adversarial template mismatches. Hence we set the worst-case false-positive probability of 10 -2 for each of these tests. The details of the tests, and the methods to set thresholds, are described in Ref. [11]. We note that all hardware injections that triggered passed the single-detector signal-based veto. \nThe data-quality vetoes are relatively simple in nature, and motivated by segments with excess power (as observed in spectrograms) that slip through the combination of the flagging procedure (of Sec. III C) and PSD drift correction (of Sec. III F). The tests are as follows: \n- 1. Sometimes, our flagging procedure only partially marks the bad segments, in which case short templates (such as those of the heavier black hole banks) can trigger on the adjoining unflagged data. This is mitigated by our choice, described in Section III E, to discard candidates with short waveforms in the vicinity of holes in our data (in practice, we reject waveforms < 10 s long within 1 s of a hole). \nFIG. 6. The impact of signal and data quality vetoes on the distribution of Hanford detector triggers in the BBH 3 bank. GW151216 is deep in the Gaussian part of the distribution with ρ 2 H = 39 . 4, and is not shown in this plot. \n<!-- image --> \n- 2. There are rare bad segments on timescales of glyph[similarequal] 5 -10 s, which is too long for our flagging procedure but too short for the PSD drift correction. We flag segments of duration 25 s with a statistically significant number of loud triggers ( ρ 2 glyph[greaterorsimilar] 30) that are local maxima within subintervals of 0 . 1 s. We set a generous threshold that should be reached at most once per run (approximately accounting for correlations between templates) within Gaussian noise, and is robust to astrophysical events (due to the maximization over time).\n- 3. Finally, we account for rare cases with significant PSD drifts on finer timescales than the ones used while triggering (described in Section III F and Ref. [10]). When this PSD drift is statistically significant, we veto coincidence candidates (both at zero-lag and in timeslides) whose combined incoherent scores, after accounting for the finer PSD drift correction, are brought down below our collection threshold. \nFigure 6 shows the cumulative effect of our vetoes on the score distribution of the triggers in the BBH 3 bank, which contains short waveforms of heavy binary black hole mergers. Also shown are the hardware injections present in the data stream and GW150914 which belongs to this bank's chirp mass range. We note that the veto retained every hardware injection in this chirp mass domain that passed the flagging procedure of Section III C. It is interesting to note that GW150914 does not stand out from the single detector trigger distribution before the application of the veto, and is clearly detected even without resorting to coincidence after it. \nFIG. 7. Relation between our new rank-based score ˜ ρ and the SNR ρ , for the Hanford detector. The initial linear dependence reflects the Gaussian part of the trigger distribution, the curve saturates due to the non-Gaussian glitch tail. This effect is more prominent in the higher-mass banks, which are more sensitive to glitches. \n<!-- image -->", 'J. Incoherent Ranking': 'When constructing a statistic to rank events an important part is P ( ρ 2 H , ρ 2 L | H 0 ), the probability of obtaining a trigger with squared SNRs ( ρ 2 H , ρ 2 L ) in each detector under the null hypothesis H 0 . Under the assumption that the noise in both detectors is independent, \nP ( ρ 2 H , ρ 2 L | H 0 ) = P ( ρ 2 H | H 0 ) P ( ρ 2 L | H 0 ) . (11) \nIf the noise in each detector was Gaussian, \nlog P ( ρ | H 0 ) = -ρ 2 / 2 + const (12) \nand \nlog P ( ρ H , ρ L | H 0 ) = -( ρ 2 H + ρ 2 L ) / 2 + const . (13) \nUnder this assumption it is optimal to use ρ 2 H + ρ 2 L to rank candidate events. Unfortunately this is an invalid assumption for two reasons: firstly, even for Gaussian noise, at high SNR the maximization over templates, phase and arrival time leads to \nlog P ( ρ | H 0 ) = -ρ 2 / 2 + c log( ρ ) + const , (14) \nwhere the constant c depends on the bank dimension. However, in practice this is a minor correction, the more substantial problem is the non-Gaussian tail of the noise, the so-called glitches. In the high-SNR limit P ( ρ | H 0 ) is much larger than the Gaussian prediction. \nThe non-Gaussian tail in the ρ distribution has an important consequence when combining the scores of multiple detectors. If we were simply to use ρ 2 H + ρ 2 L as a score, we would be ranking coincidences in which the trigger in one of the detectors is coming from this non-Gaussian tail, as we would be misjudging its probability by many orders of magnitude. \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFIG. 8. Left panels: Two dimensional histogram of the SNR 2 = ρ 2 of the background for the BBH 2 (top) and BBH 3 (bottom) banks obtained by shifting the data in time so as to recreate 2 × 10 4 O1 observing runs. The non-Gaussian glitch tail is clearly visible at high SNR. Right panels: similar histogram but using the rank-based score ˜ ρ 2 . The lines of constant probability are straight (solid contours). We show the line corresponding to one event per O1 for this statistic for each bank. Our sub-threshold candidates in these banks are shown together with GW151012 and GW151216. GW150914 is too far to the upper right to be included in this histograms. \n<!-- image --> \nTo correct this problem we empirically determine log[ P ( ρ i | H 0 )] for each detector. We do so by taking our triggers and ranking them according to decreasing ρ i for each detector i . We then model \nP ( ρ 2 i | H 0 ) ∝ Rank( ρ 2 i ) , (15) \nwhich is a good approximation for distributions with ex- \nponential or polynomial tails. We denote \n˜ ρ 2 i = -2 log P ( ρ 2 i | H 0 ) . (16) \nAssuming independence, we can use \n˜ ρ 2 = -2 log P ( ρ 2 H , ρ 2 L | H 0 ) = ˜ ρ 2 H + ˜ ρ 2 L (17) \nas a robust approximation of the optimal score. In principle, a parametric model for the probability density might \nBBH 3 \noutperform the rank estimate, but practical reasons as too few surviving glitches made such estimates prone to fine tuning. Moreover, at the high SNR parts of the distribution, single-detector glitches find background in many timeslides, which makes it problematic to estimate the uncertainty in any such procedure. For this reason, and to maintain simplicity, we chose to use the rank function as a proxy for the single detector trigger probability distribution function. \nFigure 7 shows the relation between ρ and our new Rank-based score ˜ ρ for both LIGO detectors and triggers in bank BBH 2 . This mapping is dependent on the bank as the prevalence of non-Gaussian glitch triggers is very different as one changes the length of the templates, i.e., the target chirp mass of the bank. ˜ ρ and ρ agree at low values (only differ by a conventional additive constant), but as ρ increases, ˜ ρ saturates due to the tail in the distribution of triggers. \nIn Figure 8 we show the two-dimensional histogram of the background obtained by adding 20 000 unphysical time shifts between detectors to the O1 LIGO data (so as to recreate an equivalent of 20 000 O1 observing runs) for banks BBH 2 and BBH 3 . In the left panels we show the distribution of background triggers using ρ as the score. The tail of non-Gaussian glitches is clearly visible leading to an overproduction of triggers where the SNR in one detector is much larger than in the other. On the right panels we show the distribution of the same triggers but now using our rank score to bin them. The lines of constant probability are now straight. Our sub-threshold candidates in these banks are shown together with GW151012, which is a clear outlier, and with GW151216. \nFor reference, in Figure 8 we show the line corresponding to a false alarm rate of one event per O1 observing run based on this statistic. For example, for BBH 2 this corresponds to ˜ ρ 2 H ∼ ˜ ρ 2 L ∼ 37 if divided evenly among both detectors. Figure 7 shows that for this threshold SNR values the relation between ρ and ˜ ρ is still linear. This demonstrates that although very visible in the histograms, at the detection limit the background is still dominated by the Gaussian part of the noise. The presence of the non-Gaussian glitches does not significantly overproduce the background at the detection threshold. It is also important to note that when we demand that the parameters of the events in both detectors are consistent, according to our so-called coherent score described in the next section, many of these outlier events are heavily down-weighted.', 'K. Coherent Score': "In this section we further improve the statistic used to rank candidates by exploiting the information encapsulated in the relative phases, amplitudes and arrival times to the different detectors. We begin with the standard \nexpression: \nmax T P ( ρ 2 H , ρ 2 L , ∆ t, ∆ φ, t | H 1 ( T )) P ( ρ 2 H , ρ 2 L , ∆ t, ∆ φ, t | H 0 ) , (18) \nwhere T is a template in the continuous template bank. Because the maximization procedure on T is done incoherently, and prior to the application of all these terms, we will drop it from the notation. Note that in principle we should have maximized the full expression, but for practical reasons we decided to do the maximization prior to the coherent analysis. In favor of this approximation stands the fact that to linear order, the phase and time shifts are built to be orthogonal to the template identity [9], so the template's fine optimization is expected to preserve the φ and δt of a candidate to high accuracy. We further develop this expression using Bayes rule (and using some basic independence arguments): \nP ( ρ 2 H , ρ 2 L , ∆ t, ∆ φ, t | H 1 ) = P ( ρ 2 H , ρ 2 L , ∆ φ, ∆ t ∣ ∣ n H /n L , H 1 ) × P ( t ∣ ∣ H 1 , n 2 H ( t ) + n 2 L ( t ) ) P ( ρ 2 H , ρ 2 L , ∆ t, ∆ φ, t | H 0 ) = P ( ρ 2 H , ρ 2 L ∣ ∣ H 0 ) P ( ∆ φ, ∆ t ∣ ∣ H 0 ) , (19) \nwhere n i is the momentary response of detector i computed from the measured PSD, PSD drift correction and the ovelap of the waveform with holes using the data of detector i . ∆ φ is the difference between detectors in overlap phase of matched filtering the best-fit T with the data. ∆ t is the difference in arrival time of the maximum score between the detectors. P ( ρ 2 H , ρ 2 L | H 0 ) was computed using the ranking approximation detailed in Section III J. \nP (∆ φ, ∆ t | H 0 ) is taken to be the uniform distribution by symmetry. Here we note that in principle, P ( ρ i | t, H 0 ) can be non-uniform, if there are bad times where glitches conglomerate. Also, glitches could have a waveform model that prefers a particular phase for a particular template. We currently choose not to introduce these complications (other than the bad times veto applied in Sec. III I). \nP ( ρ 2 H , ρ 2 L , ∆ φ, ∆ t ∣ ∣ n H /n L , H 1 ) is measured by drawing samples that are uniformly distributed in volume out to a distance where the expected value of the SNR is four, calculating the detector response, and adding noise with the standard complex normal distribution. Out of these samples, we have created a binned histogram of the observed meaningful values ∆ t, ∆ φ, ρ 2 H , ρ 2 L ; the probability of an observed configuration given the signal hypothesis is proportional to the histogram's occupancy. The same number of samples is used for all values of n H /n L so that the pipeline's preference for detecting events with equal response between the detectors could be evaluated. This is very similar to the coherent score used in [18]. \nThe term \nP ( t ∣ ∣ H 1 , n 2 H ( t ) + n 2 L ( t ) ) ∝ ( n 2 H + n 2 L ) 3 / 2 (20) \nreflects the changes in sensitivity in the detector as a function of time. Including it allows to analyze different \nFIG. 9. Significance assessment of GW151012. In blue, the cumulative histogram of the coherent scores of background events in bank BBH 2 is presented. The flattening at low values is an artifact of the threshold used while collecting background triggers. GW151012 is clearly detected with high significance. We show that its FAR is smaller than 1 in 2 × 10 4 O1 observing runs. Extrapolation of the background distribution yields a FAR of roughly one in 5 × 10 5 O1. We note that at this low rate, many more time slides are required for exact assessment of the FAR \n<!-- image --> \nsegments of data with very different sensitivities, including multiple runs together (say O1 and O2) while maintaining a consistent detection bar, down-weighting the significance of spurious events from less sensitive detector times. One important note is that once we include this term, the FAR does not have units of inverse time, but units of inverse volume time.", 'L. Determination of FAR': 'We combine the two detectors in different time-slides with unphysical shifts between -1000 s and 1000 s in jumps of 0 . 1 s to obtain an empirical measurement of the inverse false alarm rate of up to 2 × 10 4 observing runs. To these unphysical shifts we apply all stages detailed above, exactly as we do the zero-lag data. Because the optimization and veto stages are computationally expensive, we cannot operate them on all trigger pairs for all time-slide shifts. We ensure that any trigger that has potential of entering the background distribution with an inverse FAR that is better than one per observing run is vetoed, optimized and ranked coherently.', 'M. Determination of the probability of a source being of astrophysical origin': "While the FAR is largely agnostic of the astrophysical rates (beyond the use of the model in constructing the detection statistic) and is objectively and accurately measurable through time-slides, it is hard to convert to an assessment of the astrophysical origin of a particular \nevent. Such an assessment depends both on the exact (potentially multidimensional) noise probability density at the event's location (contrast with the one dimensional cumulative probability density the FAR depends on) and the exact probability density given the astrophysical model, including the unknown rate (also as a function of physical parameters). Essentially, if all exact details in the model were known, the probability of an event being of astrophysical origin would be exactly computable, but in the presence of rate uncertainties, especially when considering the rate as a function of physical parameters, the determination of p astro may be dominated by rate uncertainties and astrophysical prejudice. Nevertheless, the objectivity of p astro to ranking functions and its immunity to the existence of the few last glitches that are left after our heavy vetoing are compelling, and we therefore proceed in computing it. \nTo do that, we strictly assume all templates inside a bank are equally probable (even though parameter dependant rate differences probably exist). We further assume that the background probability density is uniform in time and phase, an assumption we find is extremely good when the SNR value is in the region where the Gaussian noise is dominant. \nWe then compute the rate at which we observe such an event in coincidence between the two detectors: \nR (event | H 0 ) = R bg P (∆ t, ∆ φ, ρ 2 H , ρ 2 L | H 0 ) = R bg P ( ρ 2 H | H 0 ) P ( ρ 2 L | H 0 ) 2 πT , (21) \nwhere T is the allowed physical time shift between the detector, and P ( ρ 2 H | H 0 ) , P ( ρ 2 L | H 0 ) were fit using \nP ( ρ 2 i ∣ ∣ H 0 ) = ( α i + β i ρ 2 i ) e -ρ 2 i / 2 . (22) \nα i and β i are fit to the background computed from timeslides in the region close to the ( ρ 2 H , ρ 2 L ) combination of the event. We find this approximation robust in all cases where the event is close to the detection threshold and when the difference between ρ 2 H and ρ 2 L is not big. \nWe then compute the rate ratio \nW = R (event | H 1 ) R > 100 = P (∆ t, ∆ φ, ρ 2 H , ρ 2 L | H 1 ) P ( ρ 2 H + ρ 2 L > 100 | H 1 ) (23) \nusing the table constructed in Section III K. Here, R > 100 = R ( ρ 2 H + ρ 2 L > 100 ∣ ∣ H 1 , n H , n L ) is the astrophysical rate of detecting gravitational wave mergers in the event's bank , with the detector sensitivity at the time of the event. Because R > 100 can be easily estimated and updated using a list of known astrophysical events, it is assumed to be known. We then provide the estimate for the event's astrophysical origin to be: \np astro (event) = P (event | H 1 ) P (event | H 0 ) + P (event | H 1 ) = R > 100 W R (event | H 0 ) 1 + R > 100 W R (event | H 0 ) . (24) \nFor ease of future interpretation of the results, we report in Section IV both W/ R (event | H 0 ) and the computed p astro using our best knowledge of R > 100 at the time of writing.", 'IV. RESULTS OF THE BBH SEARCH': "Here we report all the signals and sub-threshold candidates found in the search. We report the FAR in units of 'O1' to reflect the fact that there was a volumetric correction factor in the coherent score. If we assume the sensitivity of the first observing run to be roughly constant, then the 'O1' unit can be converted to roughly 46 days, the effective coincident time we used in the analysis (that has some variation across banks due to differences in the data flagging thresholds). There was no background trigger with a better coherent score than GW150914, GW151012 and GW151226 in their respective banks, so we obtain only an upper limit on the FAR of 1 / (20 000 O1) for all of these events, with an effective p astro = 1 for all of them. We report their recovered squared SNR for each detector. We further found an additional event, GW151216, with a FAR of 1 / (52 O1), reported in greater detail in a companion paper [19]. These and two additional sub-threshold candidates with FAR of approximately 1/O1 are reported in Table III.", 'V. CONCLUSIONS AND DISCUSSION': "In this paper we presented an overview of a new and independent pipeline to analyze the publicly available data from the first observing run of Advanced LIGO. We used this pipeline to identify a new gravitational merger event in the O1 data. In companion papers we will provide additional details of our techniques and implementation \n- [1] B. P. Abbott et al. (LIGO Scientific, Virgo), (2018), arXiv:1811.12907 [astro-ph.HE].\n- [2] P. Jaranowski and A. Kr'olak, Living Reviews in Relativity 15 , 4 (2012).\n- [3] M. Cabero, A. Lundgren, A. H. Nitz, T. Dent, et al. , arXiv e-prints , arXiv:1901.05093 (2019), arXiv:1901.05093 [physics.ins-det].\n- [4] M. Zevin, S. Coughlin, S. Bahaadini, E. Besler, et al. , Classical and Quantum Gravity 34 , 064003 (2017), arXiv:1611.04596 [gr-qc].\n- [5] S. A. Usman, A. H. Nitz, I. W. Harry, C. M. Biwer, et al. , Classical and Quantum Gravity 33 , 215004 (2016), arXiv:1508.02357 [gr-qc].\n- [6] C. Messick, K. Blackburn, P. Brady, P. Brockill, et al. , Phys. Rev. D 95 , 042001 (2017).\n- [7] A. H. Nitz, C. Capano, A. B. Nielsen, S. Reyes, R. White, D. A. Brown, and B. Krishnan, arXiv e-prints , arXiv:1811.01921 (2018), arXiv:1811.01921 [gr-qc].\n- [8] M. Vallisneri, J. Kanner, R. Williams, A. Weinstein, and \nchoices and further characterize our search by providing simple estimates of the space-time volume searched as a function of parameters. \nThere are several areas for future development and improvements in this pipeline, including precise determination of the merger rate/sensitive volume, analysis of single detector triggers, and triggers with subthreshold candidates in the other detector. For future runs, it also remains to incorporate more than two detectors into the ranking of coincident triggers in our pipeline.", 'ACKNOWLEDGMENT': "We thank the participants of the JSI-GWPAW 2018 Workshop at the University of Maryland, and the Aspen GWPop conference (2019) for constructive discussions and comments. \nThis research has made use of data, software and/or web tools obtained from the Gravitational Wave Open Science Center (https://www.gw-openscience.org), a service of LIGO Laboratory, the LIGO Scientific Collaboration and the Virgo Collaboration. LIGO is funded by the U.S. National Science Foundation. Virgo is funded by the French Centre National de Recherche Scientifique (CNRS), the Italian Istituto Nazionale della Fisica Nucleare (INFN) and the Dutch Nikhef, with contributions by Polish and Hungarian institutes. \nTV acknowledges support by the Friends of the Institute for Advanced Study. BZ acknowledges the support of The Peter Svennilson Membership fund. LD acknowledges the support by the Raymond and Beverly Sackler Foundation Fund. MZ is supported by NSF grants AST1409709, PHY-1521097 and PHY-1820775 the Canadian Institute for Advanced Research (CIFAR) program on Gravity and the Extreme Universe and the Simons Foundation Modern Inflationary Cosmology initiative. \n- B. Stephens, in Journal of Physics Conference Series , Journal of Physics Conference Series, Vol. 610 (2015) p. 012021, arXiv:1410.4839 [gr-qc].\n- [9] J. Roulet et al. , in preparation.\n- [10] B. Zackay et al. , in preparation.\n- [11] T. Venumadhav et al. , in preparation.\n- [12] LIGO Scientific Collaboration, 'LIGO Algorithm Library - LALSuite,' free software (GPL) (2018).\n- [13] S. Khan, S. Husa, M. Hannam, F. Ohme, M. Purrer, X. J. Forteza, and A. Boh'e, Physical Review D 93 (2016), 10.1103/physrevd.93.044007.\n- [14] P. Welch, IEEE Transactions on audio and electroacoustics 15 , 70 (1967).\n- [15] B. Allen, W. G. Anderson, P. R. Brady, D. A. Brown, and J. D. E. Creighton, Physical Review D 85 (2012), 10.1103/physrevd.85.122006.\n- [16] B. Zackay, L. Dai, and T. Venumadhav, arXiv e-prints (2018), arXiv:1806.08792 [astro-ph.IM].\n- [17] B. Allen, Phys. Rev. D 71 , 062001 (2005), arXiv:gr- \nTABLE III. Events and subthreshold candidates in all of the binary black hole banks. \n| Name | Bank | M ( M glyph[circledot] ) | GPS time a ρ 2 H | ρ 2 L | FAR - 1 (O1) b | W R (event | H 0 ) (days) | R > 100 (days - 1 ) | p astro |\n|------------|--------|----------------------------|----------------------------|---------|------------------|-----------------------------|-----------------------|-----------|\n| GW151226 | BBH 1 | 9.74 | 1135136350.585 120.0 | 52.1 | > 20 000 | - c | - | 1 c |\n| GW151012 | BBH 2 | 18 | 1128678900.428 55.66 46.75 | | > 20 000 | 7 × 10 5 d | 0.01 | 0.9998 d |\n| GW150914 | BBH 3 | 28 | 1126259462.411 396.1 184.3 | | > 20 000 | - c | - | 1 c |\n| GW151216 e | BBH 3 | 29 | 1134293073.164 39.4 | 34.8 | 52 | 74 ± 2 | 0.033 | 0.71 |\n| 151231 | BBH 3 | 30 | 1135557647.145 37.5 | 25.2 | 0.98 | 5 . 4 ± 0 . 4 | 0.033 | 0.15 |\n| 151011 | BBH 4 | 58 | 1128626886.595 24.5 | 39.9 | 1 . 1 | 16 ± 1 | 0.01 | 0.14 | \n- a Times are given as the linear-free times, that is, the times corresponding to when the waveforms generated by the bank where orthogonal to the time shift component given the fiducial PSD.\n- b The false alarm rates (FAR) given are computed within each bank. The inverse false alarm rate is given in terms of 'O1' to reflect the volumetric weighting of events using the momentary detector sensitivity. Under the approximation of constant sensitivity of the detectors during the observing runs, the unit 'O1' corresponds to roughly 46 days.\n- c We found no credible way of computing the probability density of the background distribution at these high SNRs.\n- d Estimating p astro for GW151012 required some extrapolation of the background trigger distribution.\n- e A new event we are reporting in a companion paper [19]. \nqc/0405045 [gr-qc]. \n- [18] A. H. Nitz, T. Dent, T. Dal Canton, S. Fairhurst, and D. A. Brown, Astrophys. J. 849 , 118 (2017), arXiv:1705.01513 [gr-qc]. \n- [19] B. Zackay, T. Venumadhav, L. Dai, J. Roulet, and M. Zaldarriaga, arXiv e-prints , arXiv:1902.10331 (2019), arXiv:1902.10331 [astro-ph.HE]."}
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2012ApJ...746..168D
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The Relationship between Black Hole Growth and Star Formation in Seyfert Galaxies
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2012-01-01
| 34 | 0.49 | 159 |
['galaxies active', 'galaxies nuclei', 'galaxies seyfert', '-', '-']
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[]
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We present estimates of black hole accretion rates (BHARs) and nuclear, extended, and total star formation rates for a complete sample of Seyfert galaxies. Using data from the Spitzer Space Telescope, we measure the active galactic nucleus (AGN) luminosity using the [O IV] λ25.89 μm emission line and the star-forming luminosity using the 11.3 μm aromatic feature and extended 24 μm continuum emission. We find that black hole growth is strongly correlated with nuclear (r < 1 kpc) star formation, but only weakly correlated with extended (r > 1 kpc) star formation in the host galaxy. In particular, the nuclear star formation rate (SFR) traced by the 11.3 μm aromatic feature follows a relationship with the BHAR of the form SFR\propto \dot{M}_{BH}^{0.8}, with an observed scatter of 0.5 dex. This SFR-BHAR relationship persists when additional star formation in physically matched r = 1 kpc apertures is included, taking the form SFR\propto \dot{M}_{BH}^{0.6}. However, the relationship becomes almost indiscernible when total SFRs are considered. This suggests a physical connection between the gas on sub-kiloparsec and sub-parsec scales in local Seyfert galaxies that is not related to external processes in the host galaxy. It also suggests that the observed scaling between star formation and black hole growth for samples of AGNs will depend on whether the star formation is dominated by a nuclear or an extended component. We estimate the integrated black hole and bulge growth that occurs in these galaxies and find that an AGN duty cycle of 5%-10% would maintain the ratio between black hole and bulge masses seen in the local universe.
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[]
| 2 |
https://arxiv.org/pdf/1106.3565.pdf
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{'THE RELATIONSHIP BETWEEN BLACK HOLE GROWTH AND STAR FORMATION IN SEYFERT GALAXIES': 'Aleksandar M. Diamond-Stanic 1,2 , George H. Rieke 3 Accepted for publication in ApJ', 'ABSTRACT': 'We present estimates of black hole accretion rates and nuclear, extended, and total star-formation rates for a complete sample of Seyfert galaxies. Using data from the Spitzer Space Telescope, we measure the active galactic nucleus (AGN) luminosity using the [O iv ] λ 25 . 89 µ memission line and the star-forming luminosity using the 11.3 µ m aromatic feature and extended 24 µ m continuum emission. We find that black hole growth is strongly correlated with nuclear ( r < 1 kpc) star formation, but only weakly correlated with extended ( r > 1 kpc) star formation in the host galaxy. In particular, the nuclear star-formation rate (SFR) traced by the 11.3 µ m aromatic feature follows a relationship with the black hole accretion rate (BHAR) of the form SFR ∝ ˙ M 0 . 8 BH , with an observed scatter of 0.5 dex. This SFR-BHAR relationship persists when additional star formation in physically matched r = 1 kpc apertures is included, taking the form SFR ∝ ˙ M 0 . 6 BH . However, the relationship becomes almost indiscernible when total SFRs are considered. This suggests a physical connection between the gas on sub-kpc and sub-pc scales in local Seyfert galaxies that is not related to external processes in the host galaxy. It also suggests that the observed scaling between star formation and black hole growth for samples of AGNs will depend on whether the star formation is dominated by a nuclear or extended component. We estimate the integrated black hole and bulge growth that occurs in these galaxies and find that an AGN duty cycle of 5-10% would maintain the ratio between black hole and bulge masses seen in the local universe. galaxies: active, galaxies: nuclei, galaxies: Seyfert \nSubject headings: galaxies: active, galaxies: nuclei, galaxies: Seyfert', '1. INTRODUCTION': 'The discovery of correlations between the masses of supermassive black holes and the properties of galaxy bulges such as mass (Kormendy & Richstone 1995; Magorrian et al. 1998; Marconi & Hunt 2003; Haring & Rix 2004) and velocity dispersion (Ferrarese & Merritt 2000; Gebhardt et al. 2000; Tremaine et al. 2002) suggests a connection between the processes that regulate the growth of the central black hole and the galaxy bulge. Some models have explained this connection via energetic feedback from the accreting black hole (Silk & Rees 1998; Wyithe & Loeb 2003; Di Matteo et al. 2005; Somerville et al. 2008), which is often assumed to be triggered by galaxy mergers (e.g., Kauffmann & Haehnelt 2000; Hopkins et al. 2005). While such violent, merger-driven activity may be important for fueling luminous quasars, secular processes in the host galaxy (e.g., Hopkins & Hernquist 2006; Jogee 2006) may be sufficient to explain the black hole accretion rates (BHARs) of lower-luminosity active galactic nuclei (AGNs) such as Seyfert galaxies. \nImportant insights regarding the black hole-galaxy bulge connection and mechanisms for fueling AGNs can be obtained by studying the stellar populations of AGN host galaxies. There is evidence that circumnuclear stellar populations around AGNs include a relatively young component (e.g., Cid Fernandes et al. 2001; Davies et al. 2007; Riffel et al. 2009) and that \n- 1 Center for Astrophysics and Space Sciences, University of California, San Diego, La Jolla, CA, 92093\n- 2 Center for Galaxy Evolution Fellow\n- 3 Steward Observatory, University of Arizona, Tucson, AZ, 85721 \nhigher luminosity AGNs tend to be associated with younger stellar populations (e.g., Kauffmann et al. 2003; Wild et al. 2007). This connection may be driven by the fact that both black hole growth and star formation occur when large amounts of gas are injected into the central region of a galaxy (e.g., Sanders et al. 1988; Barnes & Hernquist 1991; Storchi-Bergmann et al. 2001), although a mechanism is necessary to transport gas from r ∼ 100 pc down to sub-pc scales, regardless of the external fuel supply (e.g., Shlosman et al. 1990; Wada 2004; Escala 2007; Hopkins & Quataert 2010). A more direct connection between nuclear star formation and AGN activity has also been proposed where mass loss from evolved stars (e.g., Norman & Scoville 1988; Ciotti & Ostriker 2007) or angular-momentum loss associated with supernova-generated turbulence (e.g., von Linden et al. 1993; Hobbs et al. 2011) supplies the necessary material to the black hole accretion disk. \nMeasurements of ongoing star formation in AGN host galaxies are useful for constraining the above models, but standard star-formation rate (SFR) diagnostics such as the ultraviolet (UV) continuum, the H α emission line, or the infrared (IR) continuum (e.g., Kennicutt 1998a) are often contaminated by the AGN itself. The mid-IR aromatic (also known as PAH) features offer a solution because they probe the strength of the UV radiation field in photo-dissociation regions near young, massive stars (e.g., Peeters et al. 2004; Tielens 2005; Smith et al. 2007a). While high-energy photons or shocks associated with the AGN could destroy the molecular carriers of the aromatic features (e.g., Voit 1992; Genzel et al. 1998), Diamond-Stanic & Rieke (2010) showed that the 11.3 µ m feature remains a valid measure of the SFR for \nthe nuclear environments of local Seyfert galaxies. \nPrevious studies that have used the aromatic features to assess the level of star formation in AGN host galaxies have found it to be correlated with the AGN luminosity (e.g., Imanishi & Wada 2004; Schweitzer et al. 2006; Shi et al. 2007; Netzer et al. 2007; Lutz et al. 2008; Shi et al. 2009). This is broadly consistent with models where star formation and AGN activity are both triggered by an external supply of cold gas or models where nuclear star formation fuels subsequent AGN activity. However, it is unclear how this empirical relationship behaves for a more complete sample of AGNs, how it varies as function of scale in the galaxy (e.g., nuclear v. extended star formation), or to what extent there are differences as a function of AGN luminosity or obscuration (e.g., Lutz et al. 2010). \nIn this paper, we extend the study of SFRs in AGN host galaxies to a complete sample of Seyfert galaxies drawn from the revised Shapley-Ames galaxy catalog (RSA, Sandage & Tammann 1987; Maiolino & Rieke 1995; Ho et al. 1997a). All galaxies have nuclear mid-IR spectra from the Infrared Spectrograph (IRS, Houck et al. 2004) and imaging from the Multiband Imaging Photometer for Spitzer (MIPS, Rieke et al. 2004) onboard the Spitzer Space Telescope (Werner et al. 2004), and we are able to treat nuclear ( < 1 kpc) and extended regions separately. This spatially resolved information allows us to assess the extent to which star formation and black hole accretion activity are physically connected. We present measurements of BHARs based on the [O iv ] λ 25 . 89 µ m emission line (e.g., Diamond-Stanic et al. 2009; Rigby et al. 2009), nuclear SFRs based on the 11.3 µ m aromatic feature (e.g., Diamond-Stanic & Rieke 2010), and extended SFRs based on 24 µ m flux (e.g., Rieke et al. 2009). We explore the relationships among these quantities and the constraints they place on models of black hole growth and galaxy evolution.', '2. SAMPLE, DATA, AND MEASUREMENTS': "We consider the RSA Seyfert sample analyzed by Diamond-Stanic et al. (2009), which includes the 89 Seyferts brighter than B T = 13 from Maiolino & Rieke (1995) and Ho et al. (1997a). This galaxy-magnitudelimited sample is unique in its sensitivity to lowluminosity and highly obscured AGNs (Maiolino & Rieke 1995; Ho et al. 1997a). The median distance of the sample is 22 Mpc, where the 3.7 '' slit width of the IRS ShortLow (SL) module subtends 390 pc. Distances and Seyfert types are compiled by Diamond-Stanic et al. (2009). \nFor the analysis of BHARs, we use [O iv ] λ 25 . 89 µ m emission-line fluxes from Pereira-Santaella et al. (2010), which are based on data from the Long-High IRS module. For the nine galaxies without [O iv ] measurements from Pereira-Santaella et al. (2010), we use [O iv ] upper limits Diamond-Stanic et al. (2009), which are based on data from the Long-Low IRS module. When considering [O iv ] / [Ne ii ] ratios, we use [Ne ii ] fluxes from Pereira-Santaella et al. (2010), which are based on data from the Short-High IRS module. Among the 84 sources with 11.3 µ m aromatic feature measurements (see below), we focus our analysis on the 74 galaxies with [O iv ] / [Ne ii ] > 0 . 15, including 12 sources with [O iv ] upper limits that are consistent with this ratio. For these \ngalaxies the contribution of star formation to [O iv ] is sub-dominant ( < 30%, Pereira-Santaella et al. 2010), so we are able to use their [O iv ] fluxes to robustly estimate BHARs. \nFor the analysis of nuclear SFRs (Section 4), we use measurements of the 11.3 µ m aromatic feature and the [Ne ii ] λ 12 . 81 µ m emission line based on data from the IRS SL module. We used the IRS SL module (rather than the SH module) for [Ne ii ] SFRs to enable comparisons with aromatic-based SFRs in the same aperture. Following Diamond-Stanic & Rieke (2010), one-dimensional spectra were extracted using CUBISM (Smith et al. 2007a) with small apertures (3 . 6 '' × 7 . 2 '' ) designed to isolate nuclear emission, and spectral fitting was performed with PAHFIT (Smith et al. 2007b). For three sources (NGC3031, NGC3783, NGC4151), we did not obtain an adequate continuum fit near 11.3 µ m, so we defined the local continuum using a power-law fit in the λ = 10 . 75-10.9 µ m and λ = 11 . 65-11.85 µ m regions. Our measurements are compiled in Table 1. We exclude two sources (CIRCINUS, NGC1068) with saturated SL2 data and three sources (NGC777, NGC4168, NGC4472) without 11.3 µ m aromatic feature or [O iv ] λ 25 . 89 µ m detections, leaving a sample of 84 galaxies. \nAperture corrections are necessary to determine total nuclear fluxes because the above measurements only cover the central portion (1 . 2 × 2 . 3 λ/D ) of the pointspread function function (PSF) at λ ≈ 12 µ m. Using the IRAC channel 4 (8.0 µ m) PSF, scaled to match the expected beam size at λ = 11 . 3 µ m ( λ = 12 . 8 µ m), we find that our extraction aperture contains only 51.9% (45.3%) of the total flux from a point source. When determining SFRs, we therefore apply an aperture correction of 1.93 (2.21) to the 11.3 µ m aromatic feature ([Ne ii ]) fluxes. \nUsing MIPS 24 µ m images, we also consider the star formation in regions of each galaxy that are outside the IRS SL 3 . 6 '' × 7 . 2 '' extraction aperture. We measure 24 µ m fluxes in circular apertures with radii r = 2 . 9 '' , r = 1 kpc, and r = r galaxy . For each galaxy we attribute all the 24 µ m flux measured in the 2 . 9 '' aperture (diameter ≈ λ/D ) 4 to a central point source (i.e., unresolved emission from both the AGN and nuclear star formation) and we use aperture corrections based on the MIPS 24 µ m PSF to determine the point-source contribution in the larger apertures. We exclude these contributions and convert the remaining fluxes into SFRs using the Rieke et al. (2009) calibration. The choice of r = 1 kpc matches the area subtended by the 3 . 6 '' × 7 . 2 '' aperture for the most distant galaxies and corresponds to the smallest physical aperture that can be used for the whole sample 5 .", '3. BLACK HOLE ACCRETION RATES': "The [O iv ] line has been shown to be an accurate tracer of AGN intrinsic luminosity (Mel'endez et al. 2008a; Rigby et al. 2009; Diamond-Stanic et al. 2009). The line is dominated by the AGN unless the IR luminosity associated with SF exceeds the AGN intrinsic luminosity by an order of magnitude (Pereira-Santaella et al. \n4 r = (3 . 6 '' × 7 . 2 '' /π ) 0 . 5 = 2 . 9 '' \nFig. 1.The demographics of AGN luminosity, Eddington ratio, and black hole mass for the RSA Seyfert sample. The sample includes low- to moderate-luminosity AGNs with masses generally below 10 8 M /circledot . The most rapidly accreting sources with L/L Edd > 0 . 1 all have M BH < 3 × 10 7 M /circledot . \n<!-- image --> \n2010). We use the calibration of Rigby et al. (2009) to convert between [O iv ] luminosity and AGN intrinsic luminosity, L AGN = L [O iv ] × 2550, which has an rms scatter of 0.4 dex. Assuming a radiative efficiency η = 0 . 1 ( L AGN = η ˙ M BH c 2 ), this is equivalent to the following: \n˙ M BH ( M /circledot yr -1 ) = 1 . 7 × 10 -9 L ([O iv ] , L /circledot ) (1) \nWe note that there are theoretical expectations that the radiative efficiency may drop at both high ( L/L Edd > 1, e.g., Abramowicz et al. 1988) and low ( L/L Edd < 0 . 01, e.g., Narayan & Yi 1995) accretion rates due to advection of matter onto the black hole. The latter regime is relevant for our sample, such that equation 1 may underestimate the true mass accretion rate for sources with small L/L Edd values. \nTo assess the demographics of the sample in terms of the ratio of the AGN intrinsic luminosity to the Eddington luminosity, L EDD = 1 . 3 × 10 46 ( M BH / 10 8 M /circledot ) erg s -1 , we gathered estimates of black hole mass from the literature based on highresolution gas, stellar, or maser dynamics (5 objects), reverberation mapping (10 objects), and bulge velocity dispersion (46 objects). The values of AGN intrinsic luminosity, black hole mass, and Eddington ratio for the \nsample are shown in Figure 1 and compiled in Table 1. Most objects fall in the range L AGN = 10 42 -10 45 erg s -1 , M BH = 10 6 -10 8 M /circledot , and L/L Edd = 10 -4 -1.", '4. NUCLEAR STAR-FORMATION RATES': "The mid-IR aromatic features (e.g., Peeters et al. 2004; Smith et al. 2007a; Calzetti et al. 2007), the [Ne ii ] line (e.g., Ho & Keto 2007; D'ıaz-Santos et al. 2010), and the 24 µ m continuum luminosity (e.g., Calzetti et al. 2007; Rieke et al. 2009) can all be used as tracers of the SFR for normal star-forming galaxies. However, when a galaxy contains a central AGN, dust heated by the AGN will likely dominate the 24 µ m continuum (e.g., Brand et al. 2006), ionizing photons from the AGN can contribute significantly to [Ne ii ] (e.g., Groves et al. 2006; Pereira-Santaella et al. 2010), and high-energy photons or shocks associated with the AGN may destroy or modify the molecules that produce the mid-IR aromatic features (e.g., Voit 1992; O'Dowd et al. 2009; Diamond-Stanic & Rieke 2010). Nonetheless, Diamond-Stanic & Rieke (2010) showed that the 11.3 µ m aromatic feature is robust to the effects of AGN- and shock-processing, and Mel'endez et al. (2008b) outlined a method to determine the starformation contribution to [Ne ii ] for AGNs. In this section, we evaluate the merit of the 11.3 µ m aromatic feature and the [Ne ii ] emission line for estimating nuclear SFRs of AGN host galaxies.", '4.1. SFR Calibrations': 'We used the Rieke et al. (2009) star-forming galaxy templates to convert the 11.3 µ m aromatic feature and [Ne ii ] emission-line strengths into SFRs. Based on spectral decompositions with PAHFIT, we determined the strength of these features for the templates in the L IR = 10 9 . 75 -10 10 . 75 L /circledot range, which are appropriate for the nuclear SFRs in the sample ( < 10 M /circledot yr -1 ). For these templates, the 11.3 µ m aromatic feature contributes 1 . 2% ± 0 . 1% of the IR luminosity, while [Ne ii ] contributes 0 . 13% ± 0 . 01% (above L IR = 10 11 L /circledot these fractions drop to ∼ 0 . 5% and ∼ 0 . 07%, respectively). Using the Rieke et al. (2009) calibration between L IR and SFR 6 , we find for this luminosity range: \nSFR ( M /circledot yr -1 ) = 9 . 6 × 10 -9 L (11 . 3 µm,L /circledot ) (2) \nSFR ( M /circledot yr -1 ) = 8 . 9 × 10 -8 L ([Ne ii ] , L /circledot ) (3) \nWe note that equation 2 of Ho & Keto (2007) implies that [Ne ii ] contributes, on average, only ∼ 0.05% of the IR luminosity (albeit with a large scatter of 0.51 dex). Such a small L [Ne ii ] /L IR ratio is more consistent with the Rieke et al. (2009) templates above L IR = 10 11 L /circledot . For lower luminosities, Treyer et al. (2010) note that the sample used by Ho & Keto (2007) includes a number of low-metallicity galaxies where [Ne iii ] is the dominant Ne species (e.g., Wu et al. 2006), resulting in smaller L [Ne ii ] /L IR ratios. Thus for the sources considered in this paper, our equation 3 is more appropriate for converting [Ne ii ] luminosities into SFRs. \nFig. 2.The relationship between [Ne ii ], 11.3 µ m aromatic feature, and [O iv ] luminosity. We show the [Ne ii ]-[O iv ] relationship for the 12 micron, Swift-BAT, and RSA samples, as well as the aromatic-[O iv ] relationship for the RSA sample. The [Ne ii ] and aromaticfeature axes are normalized based on their typical ratio in star-forming galaxies, and the handful of starburst-dominated Seyferts with [O iv ] / [Ne ii ] < 0 . 15 are not plotted. This figure illustrates that [Ne ii ]-[O iv ] relationships is tighter than the aromatic-[O iv ] relationship, with the discrepancy being driven by sources with larger [Ne ii ]-to-aromatic-feature ratios. \n<!-- image -->', '4.2. Comparing Aromatic and [Ne II] SFRs': "We show the relationship between [Ne ii ], [O iv ], and 11.3 µ m aromatic feature luminosities for RSA Seyferts in Figure 2. For reference, we also show the [Ne ii ]-[O iv ] relationship for Seyferts from the 12 µ m sample (e.g., Tommasin et al. 2010) and the Swift-BAT sample (e.g., Weaver et al. 2010) that are not starburst dominated (i.e., excluding the ≈ 10% of sources with [O iv ] / [Ne ii ] < 0 . 15, see Section 2). This figure illustrates that there is less scatter in the [Ne ii ]-[O iv ] relationship (0.41 dex) than in the aromatic-[O iv ] relationship (0.52 dex), implying a tighter connection between the two quantities. This figure also illustrates that if one attributes all of the [Ne ii ] flux to star formation, the [Ne ii ]-based SFR often exceeds the aromatic-based SFR. \nTo investigate this behavior in more detail, we compare aromatic and [Ne ii ] SFRs as a function of the equivalent width (EW) of the 11.3 µ m aromatic feature in the top panel of Figure 3. We find good agreement between aromatic and [Ne ii ] SFRs for sources with EW(11.3 µ m) > 0 . 3 µ m, with 0.2 dex of scatter around the median ra- \nSFR [Ne ii ] /SFR 11 . 3 µm = 1 . 14, consistent with previous results from Diamond-Stanic & Rieke (2010) for a subset of this sample. The largest outlier in this regime, NGC2639, shows evidence for suppressed aromatic features (Diamond-Stanic & Rieke 2010). However, for sources with EW(11.3 µ m) < 0 . 3 µ m, the [Ne ii ] SFR estimates are systematically larger than the aromatic SFRs. Given that the aromatic feature EW is a proxy for the ratio of star-forming to AGN luminosity (e.g., Genzel et al. 1998), this behavior illustrates that the SFR discrepancy is associated with AGN-dominated sources. \nIn the bottom panel of Figure 3, we show that the sources with discrepant SFRs also tend to have larger [O iv ]/[Ne ii ] ratios, suggesting that the radiation field is dominated by the AGN. The typical [O iv ]/[Ne ii ] ratio for AGNs is ∼ 3 (e.g., Sturm et al. 2002; Mel'endez et al. 2008b; Pereira-Santaella et al. 2010), so for sources with [O iv ] / [Ne ii ] > 3 and large [Ne ii ]-to-aromatic-feature ratios, it is likely that the observed [Ne ii ] is dominated by the AGN rather than star formation. For sources \nFig. 3.Top panel: The ratio of SFRs derived from the [Ne ii ] line to SFRs derived from the 11.3 µ m aromatic feature as a function of EW(11.3 µ m). Sources whose [Ne ii ] and aromatic SFRs differ by more than a factor of two are noted by filled symbols. This ratio scatters around unity (dashed line) for sources with EW(11.3 µ m) > 0 . 3 µ m (dotted line). However, sources with smaller EWs tend to have systematically larger [Ne ii ]-toaromatic-feature ratios. Bottom panel: The [O iv ]/[Ne ii ] ratio as a function of EW(11.3 µ m). Sources with discrepant [Ne ii ] SFRs are most commonly found in the AGN-dominated region (top-left) of this plot. The two sources with [O iv ] / [Ne ii ] < 0 . 3 and EW(11.3 µ m) < 0 . 1 µ m (NGC3031, NGC1275) are discussed in the text. \n<!-- image --> \nwith large [Ne ii ]-to-aromatic-feature ratios but smaller [O iv ]/[Ne ii ] ratios, the situation is less clear. However, the amount of [Ne ii ] produced by the AGN is strongly dependent on the ionization parameter (e.g., Groves et al. 2006), and it is not uncommon for the observed [O iv ]/[Ne ii ] ratio for AGNs to be as large as ∼ 10 or smaller than unity (e.g., Pereira-Santaella et al. 2010, Figure 3). The two sources with [O iv ] / [Ne ii ] < 0 . 3 and EW(11.3 µ m) < 0 . 1 µ m are the nearby spiral galaxy M81 (NGC3031) and the radio galaxy Perseus A (NGC1275) 7 , both of which may exhibit advectiondominated accretion flows with softer spectral energy distributions (e.g., Quataert et al. 1999; Balmaverde et al. 2008; Miller et al. 2010). \nFig. 4.The ratio of SFRs derived from the [Ne ii ] line, using the method of Mel'endez et al. (2008b) to subtract the AGN contribution to [Ne ii ], to SFRs derived from the 11.3 µ m aromatic feature as a function of the [O iv ]/[Ne ii ] ratio. We find that the Mel'endez et al. (2008b) method can overestimate SFRs for sources with [O iv ] / [Ne ii ] < 0 . 3 and underestimate SFRs for sources with [O iv ] / [Ne ii ] > 1. \n<!-- image --> \nSeveral authors have discussed the question of what fraction of [Ne ii ] emission is produced by the AGN (e.g., Sturm et al. 2002; Schweitzer et al. 2006; Mel'endez et al. 2008b; Weaver et al. 2010; Pereira-Santaella et al. 2010). This fraction can be estimated by assuming a fiducial relationship between [O iv ] and [Ne ii ] emission for pure AGNs (i.e., with no star formation contribution to [Ne ii ]), and then attributing excess [Ne ii ] emission to star formation. For example, (Sturm et al. 2002) adopted the ratio [O iv ] / [Ne ii ] = 2 . 7 for pure AGNs, and Mel'endez et al. (2008b) determined a luminositydependent relationship between [Ne ii ] and [O iv ] for Seyfert galaxies with undetected aromatic features, yielding a pure AGN ratio of [O iv ] / [Ne ii ] = 0 . 9 for sources with L [O iv ] = 10 39 erg s -1 and [O iv ] / [Ne ii ] = 3 . 1 for sources with L [O iv ] = 10 42 erg s -1 . We compare aromatic SFRs to [Ne ii ] SFRs estimated based on this latter method in Figure 4. There is rough agreement in the median SFR between the two methods, but there is significant disagreement on a source-by-source basis. In particular, the Mel'endez et al. (2008b) method assigns SFR=0 for a large number of RSA Seyferts with clearly detected aromatic features, and it gives larger SFRs for a number of sources with smaller [O iv ]/[Ne ii ] ratios (e.g., NGC3031 and NGC1275, as discussed above). \nThus, for any individual Seyfert galaxy, it is not straightforward to determine an accurate SFR based on its [Ne ii ] emission line, primarily because the AGN contribution to [Ne ii ] varies significantly from source to source (e.g., Groves et al. 2006; Pereira-Santaella et al. 2010). If one assumed that all of the [Ne ii ] emission were produced by star formation, one would overestimate SFRs for AGN-dominated sources and underestimate the scatter in the relationship between AGN and \nFig. 5.The relationship between nuclear SFR as traced by the 11.3 µ m aromatic feature and BHAR as traced by [O iv ]. Seyferts with high accretion rates also tend to have enhanced nuclear SFRs. The solid line is the best-fit relationship (equation 4), and the dotted lines show the 95% confidence interval on the regression line. \n<!-- image --> \nstar-forming luminosity (e.g., see Figure 2), thus concluding that this relationship is stronger than it actually is. The destruction of aromatic molecules by the AGN does not appear to have a significant effect on the 11.3 µ m aromatic feature in the circumnuclear environment of local Seyfert galaxies (Diamond-Stanic & Rieke 2010). That said, if such destruction were important, it would mean that the connection between star-formation rate and black hole accretion rate (a central result of this paper, see Section 5) is actually stronger than we've presented. We therefore adopt the 11.3 µ maromatic feature as the most robust tracer of the SFR for our sample. We adopt an uncertainty of 0.2 dex on conversions between the 11.3 µ maromatic feature strength and IR luminosity based on the scatter in this ratio for the SINGS sample (Smith et al. 2007a) and an additional uncertainty of 0.2 dex for conversions between IR luminosity and SFR (Rieke et al. 2009). Adding these in quadrature, the uncertainty on SFRs obtained from equation 2 is 0.28 dex.", '5. RESULTS': "5.1. Black Hole Accretion v. Nuclear Star Formation \nIn Figure 5, we show the relationship between BHAR, as traced by [O iv ], and nuclear SFR, as traced by the 11.3 µ m aromatic feature. A strong correlation is apparent: Seyferts with larger BHARs tend to have larger nuclear SFRs. We use the linear regression method 8 outlined by Kelly (2007) to quantify the relationship between nuclear SFR and BHAR: \nSFR (11 . 3 µm,M /circledot yr -1 ) = 7 . 6 +9 . 8 -3 . 9 ( ˙ M BH M /circledot yr -1 ) 0 . 80 +0 . 14 -0 . 12 (4) \nThe uncertainties on the regression parameters above correspond to the interval that includes 90% of the posterior distribution for each parameter (see Table 2). The best-fit regression line and 95% confidence interval, given the uncertainties in the regression parameters, are shown as solid and dashed lines in Figure 5. The observed scatter around this relationship is 0.52 dex (treating BHAR upper limits as detections), although the posterior median estimate of the intrinsic scatter is 0 . 37 dex (see Ta- \n8 code available from the IDL Astronomy User's Library (linmix err.pro), http://idlastro.gsfc.nasa.gov/ \nFig. 7.The nuclear SFR/BHAR ratio as a function of BHAR. The median ratio SFR/BHAR=23 is shown as a dotted line. There is mild anti-correlation such that sources with large accretion rates tend to have smaller SFR/BHAR ratios. Plot symbols are the same as in Figure 1. \n<!-- image --> \nFig. 6.The relationship between [O iv ] flux and 11.3 µ m aromatic feature flux. The correlation between these two observed quantities is statistically significant, illustrating that the connection between the derived physical quantities (nuclear SFR and BHAR) is real and not just driven by the distance-squared factor in luminosity-luminosity plots. \n<!-- image --> \nble 2), suggesting that much of the observed scatter may be driven by the measurement errors on SFR and BHAR. \nTo test whether the relationship between SFR and BHAR could be driven by the distance dependence inherent in luminosity-luminosity plots, in Figure 6 we show the relationship between the observed quantities, 11.3 µ m aromatic feature and [O iv ] flux. The correlation in this flux-flux plot is still statistically significant (Spearman's ρ =0.66, probability of no correlation p < 1 × 10 -6 ; Isobe et al. 1986; Lavalley et al. 1992), confirming the reality of this relationship. \nWe investigate the behavior of the SFR/BHAR ratio as a function of BHAR in Figure 7; the median ratio SFR/BHAR=23 is shown as a dotted line. A mild anticorrelation exists such that sources with large accretion rates tend to have smaller SFR/BHAR ratios (Spearman's ρ =-0.48, p = 8 × 10 -5 ). Thus, consistent with the sub-linear slope in equation 4, this indicates that the nuclear SFR does not keep pace with the BHAR towards high AGN luminosities. We find no significant difference in the nuclear SFRs or SFR/BHAR ratios between different Seyfert types (see Section 6.4).", '5.1.1. Nuclear Star Formation in Physically Matched Apertures': 'As described in Section 2, we also measured 24 µ m fluxes inside r = 1 kpc apertures to assess contributions from star formation that falls outside our IRS extraction aperture. In Figure 8, we show that this contribution can be significant for sources with D < 30 Mpc. Since the nearby sources tend to be have smaller BHARs, including this additional star-formation contribution leads to a shallower relationship between nuclear SFR and BHAR. We show this relationship in Figure 9, which is described \nby the following: \nSFR ( r = 1 kpc , M /circledot yr -1 ) = 4 . 7 +6 . 8 -2 . 3 ( ˙ M BH M /circledot yr -1 ) 0 . 61 +0 . 15 -0 . 11 (5) \nAgain, the uncertainties on the values above correspond to the interval that includes 90% of the posterior distribution for each parameter (see Table 2). The observed scatter around this relationship is 0.50 dex and the posterior median estimate of the intrinsic scatter is 0.41 dex. The bottom panel of Figure 9 illustrates how the SFR/BHAR ratio tends to decrease as a function of BHAR. \nIt is worth noting that our sample does not include galaxies with larger SFR/BHAR ratios by definition since this is a sample of AGNs and is limited to sources that have [O iv ] / [Ne ii ] > 0 . 15, where we can accurately estimate AGN luminosities (see Section 2). Based on equations 1 and 3, this cut corresponds to a SFR/BHAR ratio of ≈ 350, which is similar to the maximum ratio in Figure 9.', '5.2. Black Hole Accretion v. Extended Star Formation': 'The connection described above between black hole accretion and nuclear star formation motivates us to consider whether such a relationship also exists between black hole activity and star formation on larger scales in the host galaxy. We examine this relationship using extended ( r > 1 kpc) SFRs estimated from MIPS 24 µ m fluxes (see Section 2, Table 1, and Figure 10) and total SFRs estimated from the nuclear and extended components (see Figure 11). \nThere are correlations present in Figures 10-11, but the scatter is significantly larger than for nuclear SFRs (Figures 5 and 9). The posterior median estimate of the intrinsic scatter in the SFR-BHAR relationship is \nFig. 8.The ratio of the the SFR determined from the 11.3 µ m feature to the SFR in a r = 1 kpc aperture, where additional contributions from 24 µ m emission outside the spectroscopic aperture are included, as a function of galaxy distance. Such contributions are often significant for sources with D < 30 Mpc. \n<!-- image --> \n0.73 dex for r > 1 kpc SFRs and 0.86 dex for total SFRs. Furthermore, the correlation probabilities for the fluxflux versions of these extended and total SFR relationships are not highly significant (see Table 2), illustrating that the connection between the physical quantities is weak. In addition, the observed significance is enhanced by a Malmquist-type bias against more distant galaxies with lower SFRs, related to the galaxy apparent magnitude limit of the RSA Seyfert sample. \nTo illustrate the stronger correlation and smaller scatter associated with nuclear SFRs, in Figure 12 we show the posterior distributions for the correlation coefficient and intrinsic scatter of the SFR-BHAR relationship when considering (1) 11.3 µ m aromatic feature, (2) r = 1 kpc, (3) r > 1 kpc, and (4) total galaxy SFRs. This figure illustrates that while the BHAR correlates reasonably well with star formation on sub-kpc scales, it is only weakly related to extended and total star formation activity.', '6. DISCUSSION': 'Our results present a picture where the star formation on sub-kpc scales in AGN host galaxies traces the BHAR in a somewhat sub-linear fashion, while star formation on larger scales only weakly traces the BHAR. Recently, Lutz et al. (2010) argued that host galaxy star formation only shows a clear dependence on AGN luminosity for high-luminosity sources ( L AGN > 10 45 erg s -1 or ˙ M BH > 0 . 1 M /circledot yr -1 , see their Figure 6), but our results show that this relationship persists towards lower AGN luminosities if one considers only the nuclear component of the host galaxy. Given that estimates for samples of AGNs usually provide only total SFRs (see Section 6.1), the observed scaling between SFR and BHAR may depend on whether the star formation is dominated by a nuclear or extended component. \nFig. 9.The relationship between nuclear SFR (measured on r = 1 kpc scales) and the BHAR. A strong correlation exists, and the slope of the relationship is sub-linear as sources with larger BHARs tend to have smaller SFR/BHAR ratios (see equation 5). \n<!-- image -->', '6.1. Comparison with Previous SFR and BHAR Measurements': "Several authors have estimated SFRs for AGN host galaxies and explored the relationship with BHAR. Some find an approximately linear relationship ( α ≥ 0 . 8, where SFR ∝ ˙ M α BH , e.g., Satyapal et al. 2005; Netzer 2009; Shi et al. 2009) consistent with our results for nuclear SFRs traced by the 11.3 µ maromatic feature (equation 4, Figure 5), while others finding a much shallower relationship ( α ≤ 0 . 5, e.g., Hao et al. 2005; Silverman et al. 2009; Bonfield et al. 2011), more consistent with our results for extended and total SFRs (Figures 10-11). \nAmong studies that have used the aromatic features to estimate SFRs, Netzer (2009) compiled Spitzer measurements for 28 z ∼ 0 . 1 QSOs (Netzer et al. 2007) and 12 z ∼ 2 QSOs (Lutz et al. 2008) to complement their own study of a large sample of type 2 Seyferts and LINERs from the Sloan Digital Sky Survey (SDSS). The AGN luminosities for the SDSS sources were estimated using the [O iii ] λ 5007 and [O i ] λ 6300 lines, and SFRs were estimated using the method of Brinchmann et al. (2004). They found a relationship of the form SFR ∝ ˙ M 0 . 8 BH with a SFR/BHAR ratio ∼ 30 for ˙ M BH = 0 . 1 M /circledot yr -1 . Similarly, Shi et al. (2009) considered the relationship between 5-6 µ m continuum luminosity and aromaticfeature luminosity for a sample of 89 PG quasars at \nFig. 11.The relationship between a galaxy's total SFR and BHAR, which exhibits only a weak correlation. \n<!-- image --> \nFig. 10.The relationship between extended SFR (measured on r > 1 kpc scales) and the BHAR. Only a mild correlation exists between these quantities. \n<!-- image --> \nz < 0 . 5 (Shi et al. 2007) and 57 SDSS quasars at z ∼ 1 and found a relationship of the form SFR ∝ ˙ M 0 . 97 ± 0 . 08 BH with a SFR/BHAR ratio ∼ 10 for sources with ˙ M BH = 1 M /circledot yr -1 . These relationships are consistent with our results for aromatic-based SFRs (Section 5.1), although they extend to higher luminosities. \nA number of studies have also used measurements of far-IR luminosities of AGNs to estimate host galaxy SFRs. There are uncertainties regarding poorly sampled IR spectral energy distributions and contributions from dust heated by the AGN (e.g., Schweitzer et al. 2006; Shi et al. 2007), but these estimates are nonetheless instructive. For example, Satyapal et al. (2005) expanded on the work of Dudik et al. (2005) and compiled AGN and far-IR luminosities for a sample including 86 Seyferts and quasars, for which they found a relationship of the form SFR ∝ ˙ M 0 . 89 BH with an SFR/BHAR ratio ∼ 60 at ˙ M BH = 0 . 1 M /circledot yr -1 . This is similar to our Equation 4, but with a somewhat higher SFR normalization. On the other hand, a shallower relationship ( α = 0 . 29) with a much larger SFR normalization (SFR/BHAR ∼ 3000 at ˙ M BH = 0 . 1 M /circledot yr -1 ) was found by Hao et al. (2005) for a sample of 31 AGN-ULIRGs at z < 0 . 5, where excess emission at 60 µ m was attributed to star formation. More recently, such a shallow slope ( α ≤ 0 . 5) has also been found by several studies of optically selected quasars (e.g., Serjeant & Hatziminaoglou \n2009; Hatziminaoglou et al. 2010; Bonfield et al. 2011) in fields that have Spitzer or Herschel (Pilbratt et al. 2010) coverage. \nSeveral studies of X-ray selected AGNs have also found a shallow SFR-BHAR relationship. Silverman et al. (2009) used the [O ii ] λ 3727 line to estimate SFRs for a sample of COSMOS (Scoville et al. 2007) X-ray AGNs and found SFR ∝ ˙ M 0 . 28 ± 0 . 22 BH with a SFR/BHAR ratio ∼ 50 at ˙ M BH = 0 . 1 M /circledot yr -1 . Atlee et al. (2011) considered a sample of X-ray and IR-selected AGNs in galaxy clusters, with SFRs estimated from spectral decompositions of mid-IR data (Assef et al. 2010), and found SFR ∝ ˙ M 0 . 46 ± 0 . 06 BH . While these sources reside in denser environments, their AGN luminosities and total SFRs fall in the range of RSA Seyferts. Lutz et al. (2010) and Shao et al. (2010) measured 870 µ m and 100-160 µ m emission, respectively, for samples of Chandra X-ray-selected AGNs (Alexander et al. 2003; Lehmer et al. 2005; Tozzi et al. 2006), and compiled 60 µ mmeasurements for local AGNs detected by Swift-BAT (Cusumano et al. 2010). They also find a shallow SFR-BHAR slope ( α ∼ 0 . 4), but argue that star formation and black hole growth are more closely linked at higher AGN luminosities. Recently, Mullaney et al. (2011) found no correlation between Xray luminosity and far-IR luminosity for moderate luminosity X-ray sources ( L X = 10 42 -10 44 erg s -1 ) up to z ≈ 3, consistent with the weak relationship we find for \n<!-- image --> \nFig. 12.The posterior distributions for the correlation coefficient and intrinsic scatter in the SFR-BHAR relationships for different SFR values. This figure illustrates that the relationships with nuclear SFR exhibit a stronger correlation with smaller scatter than the relationships with extended and total star formation. \n<!-- image --> \ntotal SFRs.", '6.2. Comparison with Model Predictions': 'A number of authors have made theoretical predictions for the behavior of the SFR and the BHAR during the AGN phase. These models differ primarily in their predictions for the nuclear SFR/BHAR ratio, ranging from ∼ 1 (e.g., Kawakatu & Wada 2008) to ∼ 10 3 (e.g., Thompson et al. 2005) for AGNs with ˙ M BH ∼ 0 . 1 M /circledot yr -1 . \nFor example, in the galaxy merger models of Di Matteo et al. (2005) that produce a final black hole mass ∼ 4 × 10 7 M /circledot , which is appropriate for our sample (see Figure 1), the SFR/BHAR ratio reaches ∼ 200 at the peak of star-formation activity as the galaxies coalesce and drops to ∼ 5 at the end of the bright AGN phase ( ˙ M BH > 0 . 1 M /circledot yr -1 ). Although the RSA Seyferts show little evidence for merger activity, our results for nuclear SFRs are broadly consistent with these values. For lower accretion rates ( ˙ M BH = 10 -3 -10 -2 M /circledot yr -1 ) characteristic of earlier merger phases in the Di Matteo et al. (2005) model, the SFR/BHAR ratio falls in the 500-1000 range, consistent with our results for total SFRs (see Figure 11). \nThe starburst disk models of Thompson et al. (2005) \nthat produce ˙ M BH ∼ 0 . 1 M /circledot yr -1 suggest that most of the gas being supplied at an outer radius R out = 200 pc will be consumed by a starburst near that outer radius, resulting in SFR/BHAR ∼ 10 3 . While inconsistent with our measurements for local Seyfert galaxies, this model is perhaps consistent with the findings of Hao et al. (2005) for AGN-ULIRGs. Ballantyne (2008) used a scaleddown version of the Thompson et al. (2005) model to study less powerful starbursts around AGNs. They found SFR/BHAR ∼ 200 inside r = 100 pc for a fiducial model with ˙ M BH ∼ 0 . 3 M /circledot yr -1 , which is more star formation than we observe in r = 1 kpc apertures (see Figure 9). They also predict that the SFR/BHAR ratio should increase toward lower AGN luminosities due to competition for gas between between star formation and black hole accretion. This model assumes a constant gas supply, and could perhaps be reconciled with our observations if the gas supply were depleted in such a way that the SFR decreased more quickly than the BHAR. \nEscala (2007) studied the evolution of the central kpc of a massive nuclear disk with a central black hole and find that the SFR and BHAR trace each other during the primary growth phase ( SFR ∝ ˙ M BH ) with SFR/BHAR ratios in the 3-50 range. This is consistent with our results regarding nuclear SFRs for sources with ˙ M BH > 0 . 01 M /circledot yr -1 (see Figure 9). Focusing on smaller scales, Kawakatu & Wada (2008) studied a 100 pc circumnuclear disk with accretion being driven by turbulence from supernovae. They found a correlation between SFR and BHAR only for sources with ˙ M BH > 0 . 01 M /circledot yr -1 , and they predict that the SFR/BHAR ratio should increase with AGN luminosity, reaching a maximum ratio ∼ 2. This model predicts less circumnuclear star formation than we observe in r = 1 kpc apertures, and can only be reconciled with our observations if there is 10 × more star formation on 100 pc < r < 1 kpc scales than on r < 100 pc scales. \nHopkins & Quataert (2010) went a step further and made predictions for the relationship between SFR and BHAR as a function of radius. They used results from large-scale (100 kpc to 100 pc) simulations of galaxy mergers and barred galaxy disks, and then re-simulated the central kpc (1 kpc to 10 pc) and the central 10 pc (10 pc to 0.1 pc) at higher spatial resolution. They find SFR/BHAR ratios ranging from ∼ 10 for R < 100 pc to ∼ 30 for R < 1 kpc, and ∼ 300 for the whole galaxy, which is consistent with our results for nuclear and total SFRs. They also find SFR ∝ ˙ M BH on the smallest scales ( R < 10 pc) and SFR ∝ ˙ M 0 . 7 BH on larger scales, which is in general agreement with the scalings we find for nuclear star formation.', '6.3. Implications for Black Hole and Bulge Growth': "It is worthwhile to consider our measurements of nuclear SFRs and BHARs in terms of the observed scaling between galaxy bulge and black hole masses (e.g., Marconi & Hunt 2003; Haring & Rix 2004). Heckman et al. (2004) analyzed a sample of bulgedominated galaxies from the Sloan Digital Sky Survey and found that the integrated black hole growth (from galaxies hosting AGNs) and star formation (from all galaxies) corresponded to SFR/BHAR ∼ 10 3 , consistent with the observed bulge/black hole mass ratio. Fu et al. \n(2010) performed a similar exercise for a sample of luminous infrared galaxies and found similar results for the integrated population. For our sample, we find a median SFR/BHAR = 36 on scales of r = 1 kpc (the integrated total corresponds to SFR/BHAR = 23), which is factor of ∼ 20 below the median bulge/black hole mass ratio (Haring & Rix 2004). This implies an AGN duty cycle of ∼ 5%, which is within a factor of two of the Seyfert fraction in the RSA galaxy sample ( ∼ 10%, Ho et al. 1997a). \nThere are several caveats associated with the above estimate, including the fact we excluded the most starformation dominated Seyferts (see Section 2) and that the our nuclear apertures have not been customized to match the bulge of each galaxy. A systematic bulge/disk decomposition is beyond the scope of this work, but r = 1 kpc corresponds roughly to the effective radius of a 10 10 M /circledot bulge (Marconi & Hunt 2003; Shen et al. 2003; Graham & Worley 2008), meaning that our nuclear apertures encompass half the light of such a bulge (and less light for more massive bulges). Accounting for these effects, as well as potential bulge growth through dynamical process that relax the orbits of pre-existing stars, would tend to increase our estimate of the AGN duty cycle. We have also assumed that local Seyfert galaxies obey the black hole-bulge scaling relations defined primarily by early-type galaxies with classical bulges, which may or may not be the case (e.g., Graham 2008; Hu 2008; Greene et al. 2010; Kormendy et al. 2011). \nWe now consider our results in the context of AGN fueling mechanisms. A number of morphological studies of AGN host galaxies at z < 1 have argued that the fueling for most systems is not merger driven (e.g., Grogin et al. 2005; Pierce et al. 2007; Gabor et al. 2009; Cisternas et al. 2011). While large-scale bars can also provide the necessary gravitational torques to drive gas down to ∼ 100 pc, there is not strong evidence that Seyfert galaxies exhibit a larger bar fraction than do star-forming galaxies (e.g., Ho et al. 1997b; Mulchaey & Regan 1997; Hunt & Malkan 1999). However, both Seyfert and star-forming activity appear to be associated with a higher incidence of bars than found in quiescent galaxies (Laurikainen et al. 2004; Hao et al. 2009). We have demonstrated that the BHAR is correlated with the amount of gas on sub-kpc scales (adopting the SFR ∝ Σ gas assumption of the Schmidt-Kennicutt relationship, Schmidt 1959; Kennicutt 1998b), but such gas still needs to shed most of its angular momentum to reach the black hole. Future studies with ALMA of the spatial distribution and kinematics of molecular gas down to ∼ pc scales in local Seyfert galaxies will probe the nature of this fueling more directly. We note that Davies et al. (2007) and Wild et al. (2010) presented evidence for a time delay between the onset of star formation and AGN activity, which can be explained if the black hole is being fed by outflows from intermediate-age stars. Our observations are not sensitive to detecting such a ∼ 100 Myr time delay because the aromatic features will continue to be excited by UV photons from longer-lived B stars (e.g., Peeters et al. 2004; D'ıaz-Santos et al. 2010). The required loss of angular momentum could also be explained by dynamical instabilities in self-gravitating disks (e.g., Hopkins & Quataert 2010).", '6.4. Behavior as a Function of Seyfert Type': "There have been suggestions from theoretical (e.g., Wada & Norman 2002; Ballantyne et al. 2006) and observational work (e.g., Maiolino et al. 1995; Mouri & Taniguchi 2002; Buchanan et al. 2006; Deo et al. 2007; Mel'endez et al. 2008b) that star formation is enhanced in obscured (i.e., type 2) AGNs. This would be inconsistent with the standard unified model (e.g., Antonucci 1993; Urry & Padovani 1995) where differences between obscured and unobscured AGNs are attributed to our viewing angle towards a central obscuring torus, and would suggest that the obscuring material in type 2 AGNs is related to star-formation activity in the host galaxy. However, we find that the distributions of nuclear SFRs, extended SFRs, total SFRs, and SFR/BHAR ratios for our sample do not exhibit any statistically significant differences between type 1 and type 2 Seyferts. While this result does not definitively rule out any difference between the star-forming properties of type 1 and type 2 AGNs, it does imply that such differences are not dramatic. \nWe note that results on star-formation activity as a function of Seyfert type likely depend on sample selection and the method used to measure the SFR. A comprehensive analysis of these factors is beyond the scope of this work, but their effect is apparent in previous published results. For example, Maiolino et al. (1995) studied a sample including 51 Seyferts from the CfA sample (Huchra & Burg 1992), 59 from the RSA sample, and 84 from the 12 µ m sample (Rush et al. 1993). By comparing ground-based 10 µ m observations with IRAS 12-100 µ m fluxes, they found that the extended IR emission in type 2 Seyferts tends to have a much redder color than that in type 1s, consistent with enhanced star formation in the type 2 galaxies. Similarly, Mouri & Taniguchi (2002) studied a sample of 50 RSA and 37 CfA Seyferts and found that Seyfert 2s had larger infrared/B-band and larger 100 µ m/60 µ m flux ratios, implying that they were more often starburst or hostgalaxy dominated. On the other hand, Imanishi & Wada (2004) studied a sample of 24 CfA and 33 12 µ m Seyferts and found no significant difference in 3.3 µ m aromatic feature luminosities between type 1 and type 2 Seyferts. More recently, Pereira-Santaella et al. (2010) found a significant enhancement in the fraction of type 2 Seyferts that exhibit excess [Ne iii ] emission associated with star formation for a heterogeneous sample of 201 Seyferts, but found no significant difference when considering more complete samples of 70 RSA and 97 12 µ m Seyferts.", '7. CONCLUSIONS': "We have measured BHARs based on the [O iv ] emission line and nuclear, extended, and total SFRs based on the 11.3 µ maromatic feature and extended 24 µ mcontinuum emission for a complete sample of Seyfert galaxies. We find a strong correlation between the nuclear star formation and black hole accretion, where the nuclear star formation is traced by the 11.3 µ m aromatic feature and 24 µ m emission external to the central PSF but within r = 1 kpc (see Figures 5 and 9). On the other hand, the extended ( r > 1 kpc) and total SFRs are only weakly correlated with the BHAR (see Figure 10-12 and Table 2). This suggests a connection between gas on sub-kpc scales \nthat is forming stars and the gas on sub-pc scales that is accreting onto the black hole. While the physical processes that drive this relationship (e.g., mass loss from evolved stars, angular momentum loss from gravitational instabilities) are not clearly identified by these data, the connection is apparently unrelated to external processes on kpc scales in the host galaxy. \nWe thank the anonymous referee for helpful suggestions that have improved the paper. We acknowledge useful discussions with colleagues at all five Center for Galaxy Evolution campuses (UCI, UCLA, UCR, UCSB, \nUCSD), where this work was presented prior to submission. We also acknowledge constructive feedback from David Ballantyne, Jill Bechtold, Xiaohui Fan, Alister Graham, Richard Green, Marcio Mel'endez, John Moustakas, Greg Novak, Feryal Ozel, and Miguel PereiraSantaella. AMD acknowledges support from the Southern California Center for Galaxy Evolution, a multicampus research program funded by the University of California Office of Research. This work was also partially supported by contract 1255094 from Caltech/JPL to the University of Arizona. \nFacilities: Spitzer", 'REFERENCES': "- Abramowicz, M. A., Czerny, B., Lasota, J. P., & Szuszkiewicz, E. 1988, ApJ, 332, 646\n- Alexander, D. M., et al. 2003, AJ, 126, 539\n- Antonucci, R. 1993, ARA&A, 31, 473 Assef, R. J., et al. 2010, ApJ, 713, 970 \nAtlee, D. W., Martini, P., Assef, R. J., Kelson, D. D., & Mulchaey, J. 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B., & Loeb, A. 2003, ApJ, 595, 614 \n14 \nDiamond-Stanic et al. \nT AB L E 1 M e a s u r e m e n t s a n d D e r i v e d P a r a m e t e r s \n| d d | · · · - 0 2 · · 0 2 - 0 2 · · · · · · - 0 2 - 0 3 - 0 3 · · · | - 0 2 · · · · · · - 0 3 - 0 2 - 0 4 - 0 3 - 0 4 · · · - 0 5 | · · · - 0 1 - 0 4 - 0 2 - 0 2 | - 0 3 - 0 1 - 0 3 - 0 2 | - 0 2 - 0 2 - 0 4 - 0 4 | - 0 2 - 0 1 - 0 4 - 0 2 - 0 3 | e - 0 4 - 0 4 - 0 1 e - 0 3 e - 0 4 e - 0 4 e - 0 2 | - 0 4 · · · - 0 2 - 0 5 - 0 3 |\n|---------------|-----------------------------------------------------------------------------------------------|---------------------------------------------------------------------------|-----------------------------------------|---------------------------------|---------------------------------|-----------------------------------------|-------------------------------------------------------|-----------------------------------------|\n| / L E ( 9 ) | 7 . 2 e · 9 . 6 e - 4 . 8 e 1 . 1 e 2 . 9 e 8 . 1 e | 1 . 5 e 7 . 3 e 1 . 3 e 8 . 8 e 2 . 0 e 4 . 1 e 1 . 7 e | 1 . 4 e 5 . 0 e 2 . 7 e 1 . 6 e | 1 . 2 e 4 . 2 e 2 . 9 e 3 . 7 e | 4 . 5 e 4 . 1 e 3 . 0 e 2 . 2 e | 2 . 5 e 1 . 2 e 7 . 7 e 5 . 2 e 2 . 2 e | 3 . 0 4 . 6 e 1 . 2 e 9 . 6 1 . 0 2 . 3 7 . 5 | 3 . 2 e 5 . 5 e 4 . 0 e 1 . 1 e |\n| L | | < | < | < < | < | | < | < |\n| f . | | | | | | | | |\n| , r e | 2 | 1 1 3 3 3 4 | 5 | 2 | 2 | 6 | 7 2 | 0 |\n| o d | s , 1 v , s , 1 s , 1 s , 3 s , 1 s , 1 | s , s , s , s , s , s , | s , 1 s , 3 s , 3 v , | s , 3 s , 1 s , 3 v , | s , 3 v , s , 3 s , 3 | s , 1 v , 5 s , 3 v , s , 3 | a s , s , 3 s , 1 v , s , 3 s , 1 s , 1 | s , 3 v , 8 a , 1 s , 3 |\n| B H e t h 8 ) | d i r e d i d i d i d i i | d i d i d i d i d i g a | d i d i d i e | d i d i d i r e | d i r e d i d i | d i r e d i r e d i | , m , d i , d i , r e , d i d i d i | d i r e s t , d i |\n| M m ( | , , , , , , , d | , , , , , , | , , , , r | 7 , 7 , 6 , 7 , | 7 , 7 , 7 , 8 , | 6 , 6 , 7 , 7 , 7 , | 0 7 0 8 7 5 7 7 , 7 , | 7 , 6 , 8 , 6 |\n| , | + 0 6 + 0 8 + 0 7 + 0 7 + 0 8 + 0 7 + 0 7 | 0 7 0 7 0 7 0 7 0 6 0 7 | 0 7 0 8 0 6 0 7 | 0 0 0 0 | + 0 + 0 + 0 + 0 | + 0 + 0 + 0 + 0 + 0 | + 0 0 0 + 0 + 0 | + 0 0 0 0 |\n| a l u e | · · 8 e · · 4 e 2 e · · · · 9 e 8 e 6 e · · 7 e | · · · · 6 e + 0 e + 7 e + 5 e + 4 e + · · 0 e + | · · 3 e + 0 e + 3 e + 2 e + | 6 e + 9 e + 5 e + 3 e + | 3 e 0 e 6 e 1 e | 2 e 7 e 8 e 6 e 4 e | 8 e + 1 e 7 e + 6 e + 2 e + 2 e 8 e | 9 e · · 8 e + 7 e + 1 e + |\n| v | · 6 . · 1 . 3 . · · 2 . 3 . 7 . · | 1 . · · 7 . 2 . 8 . 5 . 6 . · 8 . | · 2 . 2 . 3 . 4 . | 1 . 1 . 1 . 4 . | 3 . 3 . 2 . 1 . | 1 . 1 . 1 . 4 . 4 . | 3 . 1 . 1 . 3 . 8 . 7 . 3 . | 2 . · 9 . 5 . 7 . |\n| | 2 2 2 1 2 5 3 2 2 2 3 | 4 3 2 3 3 3 5 2 | 2 3 3 2 | 4 1 5 2 | 2 4 4 | 4 3 4 2 3 | 4 3 2 5 4 4 2 | 4 |\n| H | - 0 - 0 - 0 - 0 - 0 - 0 · · · - 0 - 0 - 0 - 0 - | 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 5 | · · · - 0 - 0 - 0 - 0 | - 0 - 0 - 0 - 0 | - 0 2 - 0 - 0 - 0 | - 0 - 0 - 0 - 0 - 0 | - 0 - 0 - 0 - 0 - 0 - 0 - 0 | - 0 · · · - 0 2 - 0 4 - 0 4 |\n| ˙ B ( 7 ) | . 1 e . 1 e . 5 e . 2 e . 6 e . 3 e . 7 e . 6 e . 4 e . 8 e . 0 e | . 7 e . 7 e . 3 e . 2 e . 8 e . 6 e . 2 e . 4 e . 2 e | . 6 e . 3 e . 0 e . 6 e | . 5 e . 9 e . 9 e . 7 e | . 4 e . 9 e . 8 e . 7 e | 3 e 0 e 2 e 5 e 2 e | . 6 e . 2 e . 8 e . 1 e . 0 e . 0 e . 7 e | . 1 e 3 e 3 e 9 e |\n| M | 5 1 2 3 3 < 2 7 2 1 3 | 6 < 6 1 1 6 1 < 2 6 8 3 | 7 < 2 2 1 | < 4 1 < 9 3 | 3 2 < 1 5 | 7 . 5 . 3 . 5 . 2 . | 2 < 1 4 8 2 4 6 | 2 1 . 5 . 1 . |\n| | | | | | | | | < |\n| p c | 0 1 2 0 0 0 0 | 1 0 0 0 1 1 1 1 | 0 1 0 1 1 | 1 2 | 0 1 0 | 1 1 1 1 | 1 1 1 0 | 1 1 1 |\n| F R 1 k 6 ) | e + 0 · · · · · · e - 0 e - 0 e + 0 e + 0 · · e + 0 e + 0 · · | e - 0 e + 0 e + 0 · · e + 0 e - 0 e - 0 e - 0 e - 0 | e + 0 e - 0 e + 0 e - 0 e - 0 | e - 0 · · e - 0 · · | e + 0 · · e - 0 e + 0 | e - 0 e - 0 e - 0 · · e - 0 | e - 0 e - 0 e - 0 e - 0 2 e - 0 2 e + 0 · · | e - 0 e - 0 1 e - 0 e - 0 e - 0 1 |\n| S r > ( | 1 . 2 2 . 1 5 . 9 2 . 3 2 . 4 1 . 2 8 . 4 | 1 . 7 2 . 5 7 . 8 1 . 2 3 . 1 1 . 1 5 . 4 1 . 7 | 2 . 1 3 . 1 4 . 7 1 . 4 2 . 1 | 4 . 4 7 . 0 | 2 . 9 1 . 5 1 . 7 | 4 . 9 8 . 8 1 . 1 1 . 4 | 3 . 9 5 . 6 2 . 4 2 . 0 2 . 5 1 . 7 | 5 . 3 4 . 7 4 . 8 6 . 6 1 . 2 |\n| p c | 0 1 0 1 0 2 0 0 0 1 0 2 0 0 0 1 0 1 0 2 0 0 0 2 | 0 2 0 1 0 1 0 1 0 1 0 1 0 2 0 1 0 2 | 0 0 0 1 0 1 0 1 0 1 | 0 2 0 1 0 2 1 | 0 1 0 2 0 2 0 2 | 0 1 1 2 2 2 | 0 2 0 2 0 1 0 3 0 2 2 0 1 | 0 2 0 2 0 1 0 2 0 2 |\n| R 1 k 5 ) | 6 e - e - e - e + e - e - e + e - e - e - e + - | - e - e - e - e - e - e - e - e - | + - - - - | 0 | - - - - | - - 0 - 0 - 0 - 0 | - - - - - 0 - | - - - - - |\n| S F ( | 0 9 7 2 1 3 9 5 2 8 4 e | 9 e 8 4 6 0 2 0 7 7 | 4 e 5 e 5 e 7 e 8 e | 1 e - 7 e - 8 e - 0 e - | 1 e 0 e 0 e 3 e | 5 e 3 e 8 e 8 e 5 e | 6 e 9 e 1 e 3 e 2 e 8 e - 9 e | e 7 e 6 e 4 e 1 e |\n| r = | 3 . 7 . 8 . 4 . 1 . 2 . 2 . 5 . 6 . 8 . 4 . 5 . | 6 . 1 . 5 . 7 . 1 . 1 . 1 . 7 . 1 . | 3 . 1 . 1 . 1 . 4 . | 1 . 8 . 2 . 2 . | 5 . 5 . 1 . 9 . | 1 . 1 . 2 . 5 . 6 . | 4 . 5 . 2 . 1 . 2 . 6 . 9 . | 4 . 0 4 . 1 . 3 . 2 . |\n| r e | . 4 2 . 2 3 . 1 8 . 0 2 . 8 9 . 3 2 . 5 8 . 8 8 . 4 5 . 8 7 . 7 5 . 3 7 | . 4 6 . 6 8 . 1 4 . 9 9 . 4 9 . 8 5 . 5 7 . 1 9 . 1 3 | . 7 1 . 1 9 . 4 3 . 7 4 . 7 2 | . 8 2 . 5 6 . 2 6 . 3 6 | . 4 3 . 2 6 . 6 6 . 3 2 | . 5 9 . 5 9 . 5 9 . 7 1 . 2 3 | . 2 8 . 2 3 . 5 9 . 1 6 . 5 9 . 5 9 . 0 8 | . 3 4 . 5 9 . 4 4 . 7 0 . 5 9 |\n| r t u 4 ) | × 1 × 1 × 2 × 5 × 1 × 0 × 0 × 1 × 2 × 1 × 0 × 0 | × 0 × 0 × 2 × 0 × 1 × 0 × 0 × 1 × 0 | × 0 × 1 × 1 × 0 × 0 | × 0 × 1 × 0 × 1 | × 1 × 1 × 0 × 1 | × 0 0 0 0 1 | × 0 × 1 × 0 × 0 × 0 × 0 × 2 | × 0 × 0 × 1 × 0 × 0 |\n| p e ( | 7 1 6 2 0 9 5 1 9 4 1 6 2 9 9 4 2 2 9 4 3 8 1 9 | 2 3 3 4 0 7 5 0 7 4 4 3 2 8 6 0 0 6 | 3 6 6 0 7 1 3 7 3 6 | 4 1 7 8 1 3 6 8 | 7 2 6 3 3 3 6 6 | 0 0 × 0 × 5 × 1 × | 1 4 6 1 2 9 0 8 2 9 2 9 4 | 1 7 2 9 7 2 3 5 2 9 |\n| a | 0 . 0 . 1 . 2 . 0 . 0 . 0 . 0 . 1 . 0 . 0 . 0 . | 0 . 0 . 1 . 0 . 0 . 0 . 0 . 0 . 0 . | 0 . 0 . 0 . 0 . 0 . | 0 . 0 . 0 . 0 . | . 0 . 0 . 0 . | 3 . 3 0 . 3 0 . 3 0 . 6 | . 0 . 0 . 0 . 0 . 0 . 1 . 0 | 0 . 0 . 0 . . |\n| | | | | | 0 | 0 . 0 | 0 | 0 . 0 |\n| | 0 1 0 0 0 1 0 0 0 1 0 3 0 1 0 1 0 0 0 1 0 0 0 2 | 0 2 0 1 0 1 0 1 0 1 0 1 0 3 0 0 0 3 | 0 1 1 1 1 | 3 0 3 1 | 1 1 3 2 | 2 1 2 1 1 | 2 2 1 3 2 2 0 0 | 3 1 1 2 3 |\n| R | 7 e - 4 e + 4 e - 0 e + 6 e - 8 e - 5 e - 5 e - 8 e + 0 e - 0 e + 9 e - | 3 e - 0 e - 9 e - 6 e - 1 e - 6 e - 3 e - 4 e + 6 e - | 8 e + 0 2 e - 0 9 e - 0 3 e - 0 1 e - 0 | 0 e - 0 3 e + 0 2 e - 0 5 e - 0 | 6 e - 0 0 e - 0 3 e - 0 1 e - 0 | 0 e - 0 1 e - 0 8 e - 0 8 e - 0 4 e - 0 | 8 e - 0 7 e - 0 2 e - 0 1 e - 0 0 e - 0 1 e - 0 7 e + | 9 e - 0 1 e - 0 5 e - 0 8 e - 0 9 e - 0 |\n| i ] S F | 7 , 9 . 7 , 1 . 8 , 2 . 7 , 8 . 8 , 5 . 9 , 2 . 7 , 1 . 8 , 8 . 7 , 5 . 8 , 3 . 7 , 1 . 7 , 4 | . 8 , 3 . 8 , 1 . 8 , 6 . 7 , 9 . 8 , 5 . 8 , 1 . 9 , 6 . 7 , 1 . 7 , 8 . | , 1 . , 4 . , 1 . , 1 . , 8 . | , 6 . , 1 . , 1 . , 2 . | , 3 . , 7 . , 8 . , 7 . | 8 , 6 . , 1 . , 1 . , 3 . 1 . | 8 , 1 . 9 , 2 . 7 , 5 . 8 , 2 . 8 , 2 . 8 , 4 . , 3 . | 8 , 6 . 7 , 1 . 8 , 2 . 8 , 9 . 9 , 9 . |\n| e i t y , 3 ) | 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 | - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 | - 0 6 - 0 8 - 0 8 - 0 8 - 0 7 | 0 9 0 7 0 9 0 8 | - 0 8 0 7 0 9 0 8 | 0 0 7 0 8 0 7 0 8 , | - 0 - 0 - 0 - 0 - 0 - 0 0 7 | 0 - 0 - 0 - 0 - 0 |\n| [ N s i ( | 3 e - 3 e 2 e 2 e 7 e 0 e 3 e 7 e 3 e 1 e 6 e 4 e | 1 e 0 e 8 e 3 e 9 e 5 e 6 e 3 e 3 e | 1 e 0 e 0 e 6 e 5 e | e - e - e - e - | 7 e e - e - e - | - 6 e - 5 e - 7 e - 9 e - | e 3 e 5 e 4 e 7 e e 3 e - | - e 8 e 7 e e |\n| e n | 0 . 0 . 0 . 0 . 1 . 1 . 0 . 5 . 0 . 3 . 0 . 0 | . 2 . 2 . 2 . 0 . 0 . 3 . 5 . 0 0 . 0 | 0 . 1 . 3 . 3 . 0 | . 6 0 . 0 3 . 2 0 . 3 9 | 3 2 1 6 1 3 0 | 2 7 e 0 0 0 2 | 0 7 7 0 0 0 0 5 0 | 2 9 e 0 2 3 0 5 7 |\n| n t | 0 . 0 0 0 0 1 0 0 0 0 0 0 | 0 0 0 0 0 0 0 0 ± | 0 . 0 0 0 0 | 0 0 3 0 | 0 . 0 . 0 . 0 . | 0 . 0 . 0 . 0 . 0 . | 0 . 0 . 0 . 0 . 0 . 0 . 0 . | ± 0 . 0 . 0 . 0 . 0 . |\n| i | 5 8 ± 1 1 ± 6 7 ± 0 5 ± 1 6 ± 9 4 ± 4 4 ± 8 8 ± 1 7 ± 7 9 ± 0 7 ± 1 8 ± | 9 6 ± 4 6 ± 9 7 ± 2 0 ± 5 5 ± 1 0 ± 4 7 ± 2 6 ± 7 9 | 7 ± 0 ± 0 ± 6 ± 5 ± | 1 ± 7 ± 3 ± 4 ± | 4 ± 4 ± 5 ± 4 ± | 5 9 ± 0 4 ± 1 ± 6 ± 9 ± | 7 ± 6 ± 8 ± 4 ± 6 ± 4 ± 8 1 ± | 9 9 3 ± 2 ± 3 ± 6 ± |\n| | 1 . 3 . 1 . 1 . 5 . 8 . 1 . 7 . 3 . 2 . 6 . 1 . | 4 . 7 . 4 . 3 . 7 . 7 . 6 . 3 . 1 . | 1 . 1 9 . 6 3 . 1 7 . 5 5 . 1 | 2 . 9 1 . 7 5 . 8 4 . 5 | 5 . 8 1 . 4 6 . 2 1 . 3 | 5 . 1 . 1 . 7 2 . 4 2 . 9 | 7 . 4 5 . 9 4 . 9 2 . 6 1 . 9 3 . 9 2 . | 1 . 1 . 0 4 . 0 6 . 6 9 . 4 |\n| | 1 1 2 0 1 3 1 1 1 2 1 2 | 2 1 1 1 2 1 3 1 | | | 1 | 2 | 3 2 1 4 2 1 | 3 |\n| | - 0 - 0 - 0 + 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 | - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 3 | + 0 0 - 0 1 - 0 1 - 0 1 - 0 1 | - 0 3 - 0 1 - 0 3 - 0 1 | - 0 - 0 2 - 0 3 - 0 2 | - 0 - 0 1 - 0 2 - 0 2 - 0 2 | - 0 - 0 - 0 - 0 - 0 - 0 2 - 0 | - 0 - 0 2 - 0 1 - 0 3 - 0 2 |\n| R | . 0 e . 1 e . 8 e . 7 e . 2 e . 7 e . 4 e . 9 e . 5 e . 2 e . 3 e . 8 e | . 8 e . 4 e . 3 e . 6 e . 6 e . 0 e . 3 e . 0 e . 2 e | . 1 e . 5 e . 3 e . 6 e . 8 e | . 0 e . 7 e . 7 e . 2 e | 5 e 0 e 6 e 5 e | . 3 e 1 e 1 e 8 e 4 e | 7 e 6 e 6 e 1 e 5 e 0 e . 9 e | . 9 e 1 e 6 e 3 e 7 e |\n| S F | , 3 , 6 , 7 , 4 , 1 , 5 , 1 , 5 , 6 , 8 , 7 , 3 | 0 , 3 6 , 1 6 , 5 0 , 7 3 , 8 7 , 1 4 , 4 8 , 6 6 , 1 | , 3 6 , 1 4 , 1 0 , 1 , 4 | , 6 , 8 , 1 , 1 | , 4 . , 5 . , 1 . , 7 . | 3 , 1 . , 1 . , 5 . , 5 . | , 7 . , 5 . , 1 . , 2 . , 1 . , 3 . , 9 | 4 3 . , 1 . , 5 . , 1 . |\n| m , | . 1 0 . 2 0 . 0 2 . 1 0 . 0 4 . 6 8 . 0 7 . 8 3 . 0 3 . 2 7 . 3 4 . 1 3 | . 4 . 5 . 1 . 5 . 5 . 4 . 2 . 2 . 0 | 7 0 . 0 . 2 . 6 . 4 6 | 4 6 1 0 2 4 0 4 | . 3 1 . 0 1 . 0 5 . 7 1 | . 6 7 , . 1 2 . 1 8 . 0 2 . 1 6 | . 1 5 . 6 2 . 2 3 . 1 8 . 6 2 . 8 7 . 0 6 | . 3 7 , . 2 6 , . 0 4 . 0 6 . 6 7 |\n| µ E W 2 ) | , 0 , 0 , 0 , 0 , 0 , 1 , 1 , 0 , 0 , 0 , 0 , 0 | , 1 , 0 , 1 , 0 , 0 , 0 , 0 , 0 , 0 | , 9 . , 0 , 0 , 1 , 0 | , 0 . , 0 . , 1 . , 0 . | , 0 , 0 , 0 , 0 | , 0 , 0 , 0 , 0 , 0 | , 0 , 0 , 0 , 0 , 0 , 0 , 0 | 7 , 0 , 0 , 0 , 0 , 0 |\n| 1 . 3 t y , ( | 0 7 0 6 0 8 0 7 0 7 0 7 0 6 0 7 0 7 0 8 0 6 0 7 | 0 7 0 6 0 7 0 6 0 7 0 7 0 8 0 6 - 0 7 | 0 5 0 7 0 7 0 6 0 6 | 0 8 0 6 0 8 0 7 | 0 7 0 7 0 8 0 7 | 0 7 0 6 0 7 0 7 0 7 | 0 7 0 7 0 6 0 8 0 7 0 7 0 7 | 0 0 7 0 7 0 8 - 0 7 |\n| 1 n s i | 0 e - 5 e - 0 e - 4 e - 5 e - 6 e - 3 e - 1 e - 6 e - 1 e - 5 e - 4 - | e 5 e - 1 e - 8 e - 3 e - 5 e - 9 e - 9 e - 2 e - 6 e | 3 e - 2 e - 9 e - 4 e - e - | 8 e - 5 e - 0 e - 6 e - | 0 e - 0 e - 5 e - 5 e - | 9 e - 3 e - 3 e - 2 e - 5 e - | 3 e - 4 e - 3 e - 1 e - 4 e - 3 e - 8 e - | 9 e - 3 e - 9 e - 9 e - 3 e |\n| n t e | 0 . 2 0 . 0 1 . 4 0 . 1 0 . 1 0 . 0 0 . 0 1 . 0 0 . 1 1 . 8 0 . 0 | 0 . 3 0 . 1 0 . 0 0 . 3 0 . 0 0 . 0 0 . 1 0 . 2 0 . 0 0 . 1 | 0 . 0 0 . 1 0 . 1 0 . 0 0 . 0 3 | 0 . 3 0 . 2 1 . 7 0 . 2 | 0 . 3 0 . 2 0 . 3 0 . 2 | 0 . 1 0 . 0 0 . 0 1 . 7 0 . 1 | 0 . 0 0 . 0 0 . 0 0 . 3 0 . 0 0 . 0 0 . 1 | 0 . 1 0 . 1 0 . 2 1 . 0 0 . 0 |\n| i | 4 ± 9 ± 0 ± 0 ± 7 ± 1 ± 0 ± 3 ± 5 ± 7 ± 1 ± ± | 3 8 ± 6 ± 1 ± 9 ± 4 ± 1 ± 8 ± 7 ± 4 ± | 2 ± 6 ± 3 ± 0 ± 4 ± | 6 ± 4 ± 2 ± 9 ± | 9 ± 0 ± 9 ± 0 ± | 1 ± 2 ± 5 ± 3 ± 4 ± | 0 ± 9 ± 8 ± 2 ± 5 ± 6 ± 5 ± | 7 ± 3 ± 5 ± 7 ± 7 ± |\n| | 5 . 1 1 . 3 5 . 7 6 . 5 1 . 1 1 . 9 1 . 5 5 . 8 3 . 7 8 . 0 4 . 5 | 9 . 7 6 . 0 1 . 0 4 . 0 2 . 6 1 . 3 4 . 8 4 . 6 1 . 4 2 . 5 | 2 . 1 3 . 6 2 . 1 1 . 0 3 . 2 | 3 . 0 1 . 2 8 . 8 2 . 2 | 7 . 5 1 . 1 1 . 2 1 . 5 | 3 . 2 1 . 1 1 . 0 4 . 0 1 . 2 | 3 . 4 1 . 2 1 . 5 2 . 8 1 . 5 3 . 0 7 . 9 | 1 . 4 3 . 1 2 . 6 3 . 7 1 . 6 |\n| | 1 5 9 8 7 1 5 8 5 | 6 3 6 7 3 9 5 5 2 1 | 9 1 7 5 7 | 4 1 6 6 | 5 3 1 6 | 2 1 8 1 5 | 8 8 8 5 7 1 7 | 5 9 3 4 9 |\n| M E 1 ) | 6 0 3 9 S 1 1 2 K 5 0 7 8 8 1 0 5 1 0 9 1 2 4 1 2 7 1 3 5 1 3 6 3 8 | C 1 C 1 4 3 C 1 5 6 C 1 6 6 C 2 2 7 C 2 6 3 C 2 6 5 C 2 6 8 C 2 9 9 3 0 3 | C 3 0 7 C 3 0 8 C 3 1 4 C 3 1 8 C 3 2 2 | C 3 2 5 C 3 2 8 C 3 4 8 C 3 5 1 | 3 7 3 3 7 8 3 9 4 3 9 7 | 9 8 0 5 4 1 3 4 1 5 4 2 3 | 4 2 5 4 3 7 4 3 8 4 3 9 4 4 7 4 5 0 4 5 0 | 4 5 6 4 5 7 4 5 9 4 5 9 4 6 3 |\n| N A ( | 2 5 C 3 6 R A R G C G C G C G C G C G C G C | G G G G G G G G G G C | G G G G G | G G G G | G C G C G C G C | G C 3 G C 4 G C G C G C | G C G C G C G C G C G C G C | C G C G C G C G C |\n| | I C I I M N N N N N N N | N N N N N N N N N N | N N N N N | N N N N | N N N N | N N N N N | N N N N N N N | N G N N N N | \n< 3 . 3 e -0 4 2 . 0 e -0 4 · · · \n4 . 1 e + 0 7 , d i s , 3 3 . 2 e + 0 7 , d i s , 3 · · · \n< 3 . 1 e -0 4 1 . 5 e -0 4 7 . 7 e -0 2 \n7 . 4 e -0 2 3 . 4 e -0 1 2 . 4 e + 0 0 \n5 . 1 e -0 3 1 . 0 e -0 2 5 . 9 e -0 2 \n0 . 2 9 × 0 . 5 9 0 . 2 2 × 0 . 4 3 0 . 8 1 × 1 . 6 3 \n5 . 8 5 ± 0 . 5 0 e -0 9 , 6 . 1 e -0 3 6 . 0 1 ± 2 . 3 5 e -0 9 , 3 . 4 e -0 3 6 . 0 5 ± 0 . 1 6 e -0 8 , 4 . 9 e -0 1 \n9 . 7 5 ± 3 . 3 4 e -0 9 , 0 . 0 8 , 9 . 7 e -0 4 2 . 8 3 ± 1 . 6 8 e -0 8 , 0 . 0 6 , 1 . 5 e -0 3 7 . 7 8 ± 0 . 9 4 e -0 8 , 0 . 0 7 , 5 . 9 e -0 2 \nN G C 4 6 9 8 N G C 4 7 2 5 N G C 4 9 3 9 \n-C o n t i n u e d \nT AB L E 1 \nN n s ( \nBlack Hole Growth and Star Formation in Seyfert Galaxies \n15 \ne n s i t y i n s i d e m a s s e r e n c e e t a l . 6 ) . \nr e h y [ i c e s e y \nN o t e . -C o l . ( 1 ) : G a l a x y n a m e . C o l . ( 2 ) : 1 1 . 3 [ W m -2 s r -1 ] a n d d e r i v e d s t a r -f o r m a t i o n r a t e [ M /circledot r = 1 k p c [ M /circledot y r -1 ] . C o l . ( 6 ) : S t a r -f o r m a t i o n r a t [ M /circledot ] , m e t h o d u s e d t o d e t e r m i n e b l a c k h o l e m a s s f o r b l a c k h o l e m a s s ( s e e b e l o w ) . C o l . ( 9 ) : E d d i n g ( 2 0 0 9 ) . ( 6 ) B e n t z e t a l . ( 2 0 0 6 ) . ( 7 ) H e r r n s t e i n e t a \n| E d d ) e - 0 2 | · · · · · · e - 0 3 e - 0 4 e - 0 1 e - 0 3 e - 0 2 · · · · · · e + 0 0 e - 0 4 | · · · e - 0 2 · · · · · · · · · e - 0 2 e - 0 3 e - 0 2 e - 0 2 · · · | · · · · · · · · · e - 0 3 · · · | e - 0 2 e - 0 3 e - 0 3 | 3 ) : [ N e i i ] i r m a t i o n r a t 8 ) : B l a c k h | o i c s ) , a n d r . ( 5 ) D e n n o l d e t a l . ( 2 |\n|---------------------|-----------------------------------------------------------------------------------------------|---------------------------------------------------------------------------|-----------------------------------------|---------------------------|-------------------------------------------------------------|-----------------------------------------------------------|\n| L / L ( 5 . 2 | 1 . 7 8 . 4 2 . 5 3 . 5 2 . 5 1 . 1 5 . 5 | 2 . 8 2 . 2 5 . 0 5 . 0 2 . 8 | 2 . 5 | 4 . 5 5 . 6 7 . 6 | o l . ( a r - f o | o l . ( y n a m 0 0 3 ) W |\n| | < | | | | . C S t | C r d . ( 2 ( 1 2 |\n| e f . | | | | | - 1 ] 5 ) : 1 | ] . l l a a l ) . |\n| , r 1 | 1 1 0 1 1 1 1 3 | 1 1 1 3 1 1 | 5 3 | 1 2 1 3 | y r l . ( - | r s t e x e t 0 0 9 |\n| o d , | i s , a , i s , i s , i s , i s , i s , | i s , v , i s , i s , i s , | v , s , | a s , s , s , | /circledot C o | M /circledot y e : e u ( 2 |\n| B H e t h 8 ) d i s | d s t d d d d d | , d , r e , d , d , d | , r e , d i | g d i d i | M . | [ s t e r a l . |\n| M m ( 0 6 , | 0 7 , 0 7 , 0 7 , 0 6 , 0 6 , 0 6 , 0 7 , | 0 8 0 7 0 7 0 7 0 7 | 0 7 0 7 | 0 7 , 0 6 , 0 6 , | a t e [ k p c ] | t y i c s , D e v e t |\n| u e , + | · e + e + e + e + e + · · e + e + · | e + · · · e + e + e + e + · | · · e + e + · | e + e + e + | n r c × | o s i n a m ( 4 ) n t z |\n| l 4 e | · · · . 4 . 0 . 2 . 9 . 1 · · . 5 . 7 · · | 1 . 6 · · · · · · 1 . 8 2 . 2 3 . 3 4 . 7 · | · · · · 1 . 2 4 . 8 · | 5 6 . 2 5 . 3 | t i o p | i n y e |\n| v a 3 . | · · 4 7 2 8 2 · · 4 6 | · | · | 5 . | a s [ k | l u m d ) . ) B |\n| - 0 3 | · · · · · · - 0 3 - 0 3 - 0 1 - 0 4 - 0 3 · · · - 0 3 - 0 1 - 0 4 - 0 2 | - 0 1 - 0 2 · · · - 0 3 - 0 3 - 0 3 - 0 2 - 0 2 · · · | - 0 2 - 0 3 · · · - 0 3 · · · | 0 2 0 4 0 4 | f o r m u m n | i v ] g a s 0 0 9 ( 1 1 |\n| B H ( 7 ) . 1 e | . 8 e . 4 e . 3 e . 4 e . 2 e . 7 e . 2 e . 6 e . 4 e | . 1 e . 2 e . 4 e . 7 e . 6 e . 9 e . 1 e | . 6 e . 1 e . 8 e | . 8 e - . 1 e - . 4 e - | s t a r - s c o l | e [ O g a s : a l . ( 2 0 9 ) . |\n| ˙ M 4 | 1 1 1 7 1 3 1 < 8 1 | 1 2 3 9 2 3 3 | 1 4 2 | 5 8 9 | e d i o u | t h n g , o e t ( 2 0 |\n| | | | < | | e r i v r e v | r o m p p i H a l . |\n| p c 1 | 0 0 0 1 0 0 0 0 2 1 | 0 0 0 1 0 0 0 1 1 | 1 1 0 1 | 0 0 2 | d d o p | d f m a ( 3 ) i e t |\n| F R 1 k 6 ) - 0 | + 0 + 0 + 0 - 0 + 0 + 0 · · + 0 + 0 · · - 0 - 0 | + 0 + 0 + 0 - 0 + 0 + 0 + 0 - 0 - 0 | - 0 - 0 · · + 0 - 0 | + 0 + 0 - 0 | a n t w | v i e n ) . l a r |\n| S > ( . 2 e | . 1 e . 8 e . 8 e . 1 e . 6 e . 2 e · . 4 e . 9 e · . 1 e . 7 e | . 7 e . 4 e . 6 e . 3 e . 4 e . 0 e . 7 e . 8 e . 9 e | . 2 e . 0 e · . 7 e . 1 e | . 9 e . 1 e . 9 e | m ] , t h e | d e r a t i o 0 0 4 p e l |\n| r 1 | 1 1 1 7 3 2 2 2 3 8 | 1 2 3 9 1 2 6 6 3 | 6 3 1 5 | 1 1 4 | [ µ o r | t e b e r . ( 2 C a p |\n| c 2 | 1 1 1 1 0 1 2 2 1 1 2 1 | 0 1 0 1 2 1 0 1 2 | 2 2 0 1 1 | 1 2 | t h e d f | n r a v e r a l 0 ) |\n| R k p ) - 0 | - 0 - 0 - 0 - 0 + 0 - 0 - 0 - 0 - 0 - 0 - 0 - 0 | + 0 - 0 + 0 - 0 - 0 - 0 + 0 - 0 - 0 | - 0 - 0 + 0 - 0 - 0 | + 0 0 - 0 - 0 | i d s | i o r e e t ( 1 |\n| S F = 1 ( 5 . 3 e | . 6 e . 3 e . 3 e . 7 e . 1 e . 8 e . 7 e . 9 e . 3 e . 8 e . 3 e . 4 e | . 4 e . 3 e . 7 e . 1 e . 2 e . 7 e . 3 e . 9 e . 4 e | 6 e 6 e 5 e 2 e 7 e | 1 e 6 e 9 e | t w r e u | r e t e v : s o n ) . |\n| r | 5 6 3 3 6 2 5 9 2 5 1 2 | 1 4 1 2 9 8 4 7 4 | 6 . 4 . 8 . 3 . 2 . | 2 . 1 . 7 . | a l e n e r t u | a c c , r t e r 9 8 8 |\n| 1 | | 0 3 9 7 9 9 4 0 1 7 | 3 7 4 3 1 | 7 7 5 | u i v a p | l e s i o n P e ( 1 |\n| u r e 0 . 5 9 | 0 . 1 5 0 . 7 4 0 . 6 5 0 . 1 5 2 . 0 1 0 . 2 9 0 . 7 4 1 . 6 3 1 . 4 1 1 . 0 5 1 . 1 4 0 . 5 | 1 . 4 1 . 4 0 . 6 0 . 4 0 . 8 0 . 8 2 . 4 1 . 3 0 . 8 | 0 . 7 0 . 8 2 . 3 1 . 1 0 . 8 | 0 . 7 0 . 7 0 . 8 | , e q . 2 ' ' | c k h o s p e r ( 2 ) e n d y |\n| r t ( 4 ) × | × × × × × × × × × × × × | × × × × × × × × × | × × × × × | × × × | - 1 ] × 7 | B l a d i 7 ) . r m |\n| a p e . 2 9 | 8 3 7 3 3 0 8 0 1 1 5 3 7 8 2 7 1 5 2 5 7 5 | 7 2 7 5 3 4 2 4 4 5 4 2 2 0 6 6 4 3 | 3 6 4 3 1 7 5 7 4 0 | 3 8 3 8 4 3 | s r 6 ' ' | : i t y 0 0 o |\n| 0 | . 0 . . . . . . . . . . . 2 | 0 . 0 . 0 . 0 . 0 . 0 . 1 . 0 . . | 0 0 1 0 0 | 0 0 0 | m - e 3 . | 7 ) c 2 K |\n| 1 | 0 0 0 0 1 0 0 0 0 0 0 0 | 0 | . . . . . | . . . | 2 | l . ( e l o u ( ( 9 ) |\n| e - 0 | e - 0 1 e - 0 1 e - 0 1 e - 0 1 e + 0 1 e - 0 2 e - 0 2 e - 0 2 e - 0 1 e + 0 0 e - 0 2 e - 0 | 1 e + 0 0 e - 0 1 e + 0 0 e - 0 1 e - 0 1 e - 0 1 e + 0 0 e + 0 0 e - 0 1 | e - 0 1 e - 0 2 e + 0 1 e - 0 1 e - 0 1 | e - 0 1 e - 0 2 e - 0 1 | [ W o f t h | C o g e v & G 6 ) . |\n| F R 1 . 1 | 3 . 7 5 . 5 1 . 6 1 . 4 1 . 0 4 . 4 4 . 6 3 . 6 2 . 4 1 . 9 1 . 7 1 . 8 | 1 . 4 6 . 1 2 . 4 1 . 3 1 . 5 5 . 3 9 . 3 1 . 2 3 . 1 | 1 . 0 6 . 6 1 . 9 4 . 5 5 . 5 | 6 . 5 4 . 9 1 . 0 | n s i t y s i z e - 1 | r ] . : b u l B i a n . ( 2 0 0 | \ne e e e e e e e e e e e e e e e e e e e e e e e e e e e e e \n[ R i n t e 3 e -0 2 1 . 0 9 ± 0 . 0 1 . 5 e -0 1 5 . 4 7 ± 0 . 0 5 4 e -0 1 3 . 2 7 ± 0 . 0 4 5 e -0 1 1 . 2 7 ± 0 . 0 2 8 e -0 2 2 . 0 8 ± 0 . 0 2 8 e + 0 0 8 . 1 3 ± 0 . 0 8 7 e -0 2 1 . 6 8 ± 0 . 0 2 1 e -0 2 2 . 7 4 ± 0 . 3 0 4 e -0 2 4 . 4 4 ± 0 . 5 0 5 e -0 1 4 . 0 3 ± 0 . 0 6 8 e -0 1 5 . 6 0 ± 0 . 0 6 8 e -0 3 4 . 4 1 ± 0 . 5 5 4 e -0 1 2 . 2 9 ± 0 . 0 9 2 e + 0 0 2 . 1 8 ± 0 . 0 2 1 e -0 1 8 . 9 3 ± 0 . 3 5 5 e -0 1 1 . 7 7 ± 0 . 0 2 1 e -0 1 1 . 7 6 ± 0 . 0 2 4 e -0 2 6 . 1 5 ± 0 . 3 6 1 e -0 1 2 . 4 4 ± 0 . 0 4 3 e + 0 0 5 . 2 9 ± 0 . 0 8 3 e -0 1 2 . 2 2 ± 0 . 0 2 4 e -0 2 1 . 3 4 ± 0 . 0 3 9 e -0 2 6 . 4 0 ± 0 . 1 4 8 e -0 2 2 . 9 1 ± 0 . 1 9 5 e + 0 0 1 . 1 7 ± 0 . 0 1 2 e -0 1 1 . 1 6 ± 0 . 1 1 7 e -0 1 2 . 7 7 ± 0 . 0 4 0 e -0 1 3 . 6 0 ± 0 . 0 4 8 e -0 2 2 . 7 3 ± 0 . 3 4 2 e -0 2 4 . 5 2 ± 0 . 2 8 µ m a r o m a t i c f e a t u y r -1 ] . C o l . ( 4 ) : P e o u t s i d e r = 1 k p c ( m a s : m a s e r d y n a m t o n r a t i o . R e f e r e n c l . ( 2 0 0 5 ) . ( 8 ) D e n n \n1 1 . 3 µ m i n t e n s i t y , E W , S F ( 2 ) \nN A M E \n( 1 ) \n1 . 3 1 ± 0 . 0 9 e -0 7 , 0 . 0 8 , 1 . 2 . 3 6 ± 0 . 0 2 e -0 5 , 2 6 . 1 1 , 1 3 . 3 7 ± 0 . 0 5 e -0 6 , 2 . 5 9 , 5 . 1 . 2 3 ± 0 . 0 2 e -0 6 , 1 . 3 2 , 1 . 2 . 8 0 ± 0 . 2 8 e -0 6 , 0 . 0 6 , 1 . 4 . 9 3 ± 0 . 0 5 e -0 6 , 1 . 0 5 , 5 . 6 . 8 0 ± 0 . 1 5 e -0 7 , 0 . 8 8 , 1 . 3 . 2 0 ± 0 . 1 9 e -0 7 , 0 . 6 3 , 5 . 1 . 2 2 ± 0 . 0 5 e -0 7 , 1 . 3 0 , 9 . 2 . 6 9 ± 0 . 0 4 e -0 7 , 0 . 3 8 , 1 . 1 . 8 2 ± 0 . 0 6 e -0 6 , 0 . 1 0 , 5 . 2 . 6 1 ± 0 . 3 8 e -0 8 , 0 . 2 0 , 9 . 1 . 9 8 ± 0 . 0 9 e -0 6 , 0 . 4 0 , 1 . 2 . 1 0 ± 0 . 0 2 e -0 6 , 1 . 4 4 , 1 . 6 . 4 2 ± 0 . 3 2 e -0 7 , 0 . 9 2 , 4 . 6 . 5 0 ± 0 . 0 7 e -0 6 , 0 . 8 3 , 8 . 3 . 0 7 ± 0 . 0 3 e -0 6 , 0 . 3 4 , 2 . 2 . 3 3 ± 0 . 2 1 e -0 7 , 0 . 1 0 , 5 . 2 . 0 3 ± 0 . 0 6 e -0 6 , 1 . 5 9 , 4 . 2 . 6 1 ± 0 . 0 7 e -0 6 , 0 . 6 2 , 4 . 1 . 4 6 ± 0 . 0 6 e -0 6 , 1 . 7 0 , 7 . 2 . 0 4 ± 0 . 0 3 e -0 7 , 0 . 0 6 , 4 . 2 . 5 5 ± 0 . 1 1 e -0 7 , 0 . 1 2 , 3 . 1 . 7 4 ± 0 . 1 3 e -0 7 , 1 . 1 7 , 3 . 5 . 4 0 ± 0 . 0 5 e -0 6 , 0 . 3 8 , 8 . 8 . 7 0 ± 2 . 2 0 e -0 7 , 0 . 1 4 , 3 . 1 . 4 2 ± 0 . 0 3 e -0 6 , 0 . 4 9 , 2 . 4 . 1 0 ± 0 . 0 4 e -0 6 , 1 . 4 2 , 7 . 4 . 5 7 ± 0 . 8 1 e -0 7 , 1 . 5 4 , 7 . 3 . 4 2 ± 0 . 0 3 e -0 7 , 1 . 2 3 , 7 . \nN G C 4 9 4 1 N G C 4 9 4 5 N G C 5 0 0 5 N G C 5 0 3 3 N G C 5 1 2 8 N G C 5 1 3 5 N G C 5 1 9 4 N G C 5 2 7 3 N G C 5 3 9 5 N G C 5 4 2 7 N G C 5 5 0 6 N G C 5 6 3 1 N G C 5 6 4 3 N G C 5 7 2 8 N G C 5 8 9 9 N G C 6 2 2 1 N G C 6 3 0 0 N G C 6 8 1 4 N G C 6 9 5 1 N G C 7 1 3 0 N G C 7 1 7 2 N G C 7 2 1 3 N G C 7 3 1 4 N G C 7 4 1 0 N G C 7 4 6 9 N G C 7 4 7 9 N G C 7 4 9 6 N G C 7 5 8 2 N G C 7 5 9 0 N G C 7 7 4 3 \nTABLE 2 Correlation Analysis: SFR v. ˙ M BH \n| aperture radius | 11.3 µ PAH | m 24 µ m continuum | α | β | σ | luminosity correlation | flux correlation | flux probability |\n|-------------------|--------------|----------------------|-------------------------|-------------------------|-------------------------|--------------------------|-------------------------|--------------------|\n| < 300 pc > | Y | N | 0 . 88 +0 . 36 - 0 . 31 | 0 . 80 +0 . 14 - 0 . 12 | 0 . 37 +0 . 18 - 0 . 20 | 0 . 95 +0 . 04 - 0 . 07 | 0 . 69 +0 . 11 - 0 . 13 | < 1 × 10 - 6 |\n| 1 kpc | Y | Y | 0 . 67 +0 . 39 - 0 . 29 | 0 . 61 +0 . 15 - 0 . 11 | 0 . 41 +0 . 23 - 0 . 24 | 0 . 93 +0 . 06 - 0 . 10 | 0 . 49 +0 . 16 - 0 . 18 | 2 . 9 × 10 - 5 |\n| > 1 kpc | N | Y | 1 . 30 +0 . 76 - 0 . 45 | 0 . 57 +0 . 28 - 0 . 17 | 0 . 73 +0 . 13 - 0 . 20 | 0 . 78 +0 . 13 - 0 . 20 | 0 . 33 +0 . 21 - 0 . 22 | 0.02 |\n| r galaxy | Y | Y | 1 . 59 +1 . 15 - 0 . 57 | 0 . 71 +0 . 45 - 0 . 22 | 0 . 86 +0 . 22 - 0 . 20 | 0 . 65 +0 . 17 - 0 . 22 | 0 . 14 +0 . 20 - 0 . 21 | 0.39 | \nNote . -The quoted values in columns 4-8 correspond to the median of the posterior distribution, and the range that encompasses 90% of that distribution (see Kelly 2007). The values in the last column correspond to the probability that no correlation exists in the flux-flux version of the relationship, based on generalized Spearman's ρ (Isobe et al. 1986; Lavalley et al. 1992)."}
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1992PhRvD..46.2445V
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Dirty black holes: Thermodynamics and horizon structure
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1992-01-01
| 4 | 0.45 | 159 |
['-', '-', '-', '-', 'black hole physics', '-']
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Considerable interest has recently been expressed in (static spherically symmetric) black holes in interaction with various classical matter fields (such as electromagnetic fields, dilaton fields, axion fields, Abelian Higgs fields, non-Abelian gauge fields, etc.). A common feature of these investigations that has not previously been remarked upon is that the Hawking temperature of such systems appears to be suppressed relative to that of a vacuum black hole of equal horizon area. That is, kT<SUB>H</SUB><=ħ/(4πr<SUB>H</SUB>)==ħ/ √4πA<SUB>H</SUB> . This paper will argue that this suppression is generic. Specifically, it will be shown that kT<SUB>H</SUB>=(ħ/4πr<SUB>H</SUB>)e<SUP>-φ(r</SUP><SUB>H</SUB>) (1-8πGρ<SUB>H</SUB>r<SUP>2</SUP><SUB>H</SUB>). Here φ(r<SUB>H</SUB>) is an integral quantity, depending on the distribution of matter, that is guaranteed to be positive if the weak energy condition is satisfied. Several examples of this behavior will be discussed. Generalizations of this behavior to nonsymmetric nonstatic black holes are conjectured.
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https://arxiv.org/pdf/hep-th/9203057.pdf
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{'Matt Visser ∗': 'Physics Department, Washington University, St. Louis, Missouri 63130-4899 \n(Received 20 March 1992) \nConsiderable interest has recently been expressed in (static spherically symmetric) blackholes in interaction with various classical matter fields (such as electromagnetic fields, dilaton fields, axion fields, Abelian Higgs fields, non-Abelian gauge fields, etc ). A common feature of these investigations that has not previously been remarked upon is that the Hawking temperature of such systems appears to be suppressed relative to that of a vacuum blackhole of equal horizon area. That is: kT H ≤ ¯ h/ (4 πr H ) ≡ ¯ h/ √ 4 πA H . This paper will argue that this suppression is generic. Specifically, it will be shown that \nkT H = ¯ h 4 πr H e -φ ( r H ) ( 1 -8 πG ρ H r 2 H ) . \nHere φ ( r H ) is an integral quantity, depending on the distribution of matter, that is guaranteed to be positive if the Weak Energy Condition is satisfied. Several examples of this behaviour will be discussed. Generalizations of this behaviour to non-symmetric non-static blackholes are conjectured. \n04.20.-q, 04.20.Cv, 04.60.+n; hepth/9203057', 'I. INTRODUCTION': "For a variety of reasons, considerable attention has recently been focussed on static spherically symmetric blackholes in interaction with various static spherically symmetric classical fields. For example, the system (gravity + electromagnetism + dilaton) has been discussed by Gibbons and Maeda [1], by Ichinose and Yamazaki [2,3], and in an elegant paper by Garfinkle, Horowitz and Strominger [4], this particular system currently being deemed to be of interest due to its tentative connection with low energy string theory. The resulting charged dilatonic blackholes were rapidly generalized by Shapere, Trivedi, and Wilczek [5] to the dyonic dilatonic blackholes appropriate to the system (gravity + electromagnetism + dilaton + axion). The system (gravity + electromagnetism + axion) has been considered by Allen, Bowick, and Lahiri [6], by Campbell, Kaloper, and Olive [7], and by Lee and Weinberg [8]. The considerably simpler system of (gravity + axion) and the associated axionic blackholes had previously been discussed by Bowick, Giddings, Harvey, Horowitz, and Strominger [9]. The system (gravity + electromagnetism + Abelian Higgs field) has been discussed by Dowker, Gregory, and Traschen [10] using Euclidean signature formalism. Coloured blackholes, arising in the system (gravity + non-Abelian gauge field), have been discussed by Galtsov and Ershov [11], by Straumann and Zhou [12], by Bizon [13], and by Bizon and Wald [14]. A variation on these themes: the system (gravity + axion + nonAbelian gauge field), has recently been considered by Lahiri [15]. For brevity, any blackhole in interaction with nonzero classical matter fields will be refereed to as 'dirty'. \nA common feature of these various investigations is that whenever the Hawking temperature of the resulting dirty blackhole can be computed, the Hawking temperature (equivalently, the surface gravity) appears to be suppressed relative to that of a clean vacuum Schwarzschild blackhole of equal horizon area (equivalently, of equal entropy). Specifically, the inequality \nkT H ≤ ¯ h 4 πr H ≡ ¯ h √ 4 πA H (1.1) \nappears to be satisfied. \nI claim that this inequality is not an accident, but rather that this inequality is related to the classical nature of the fields interacting with the blackhole. Indeed it shall be shown that, for a general spherically symmetric distribution of matter with a blackhole at the center, the Hawking temperature is given by \nkT H = ¯ h 4 πr H e -φ ( r H ) ( 1 -8 πG ρ H r 2 H ) . (1.2) \nNow r H and ρ H , the radius and matter density at the horizon, clearly depend only on conditions local to the horizon itself. In contrast, φ ( r H ) is an integral quantity that depends on the distribution of matter all the way from r = r H to r = ∞ . The remarkable feature of the analysis is that, if the matter surrounding the blackhole satisfies the Weak Energy Condition (WEC), which is certainly the case for classical matter, then the Einstein field equations imply that φ ( r H ) is non-negative. The inequality kT H ≤ ¯ h/ (4 πr H ) follows immediately. \n(Warning: Since semiclassical quantum effects are capable of violating the WEC, it follows that quantum physics may allow a violation of this inequality. On the other hand, violations of the WEC in the vicinity of the event horizon are quite likely to destabilize the horizon, disrupt the blackhole, and lead to a traversable wormhole, thereby rendering moot the question of the Hawking temperature [16].) \nA side effect of the investigation is the discovery of a particularly pleasant functional parameterization of the static spherically symmetric metric that permits a simple (formal) integration of the Einstein field equations in a form suitable for the direct application of the WEC. \nAlso of note is the fact that the matter fields at the horizon (as measured by a fiducial observer - a FIDO) are constrained to satisfy the boundary condition ρ H = τ H if the horizon is to be 'canonical' in a sense to be described below. This boundary condition is in fact equivalent to demanding that the energy density measured by a freely falling observer (FFO) remain integrable as the observer crosses the horizon. \nSeveral examples are discussed in detail: The Reissner-Nordstrom geometry and a 'thin shell' example are particularly instructive elementary examples. The dyonic dilatonic black- \nholes and their ilk are decidedly nontrivial examples. \nFinally a conjecture is formulated as to a possible generalization of these results to spherically asymmetric non-static dirty blackholes. \nUnits: Adopt units where c ≡ 1, but all other quantities retain their usual dimensionalities, so that in particular G ≡ /lscript P /m P ≡ ¯ h/m 2 P ≡ /lscript 2 P / ¯ h .", 'A. Functional form': "The spacetime metric generated by any static spherically symmetric distribution of matter may (without loss of generality) be cast into the form \nds 2 = -g tt dt 2 + g rr dr 2 + r 2 ( dθ 2 +sin 2 θ dϕ 2 ) . (2.1) \nThis form corresponds to the adoption of Schwarzschild coordinates. While one can relatively easily adopt the brute force approach of inserting this metric into the curvature computation formalism and 'turning the crank', the resulting expression for the Einstein tensor is not as illuminating as it might otherwise be. \nThere is an art to further specifying the functional form of g tt and g rr in such a manner as to keep computations (and their interpretations) simple. For instance, to discuss traversable wormholes Morris and Thorne found the choices g tt = exp(2 φ ( r )); g rr = (1 -b ( r ) /r ) -1 to be particularly advantageous [16]. For the discussion currently at hand I propose \ng tt = e -2 φ ( r ) ( 1 -b ( r ) r ) , g rr = ( 1 -b ( r ) r ) -1 . (2.2) \nThat is: \nds 2 = -e -2 φ ( r ) (1 -b ( r ) /r ) dt 2 + dr 2 (1 -b ( r ) /r ) + r 2 ( dθ 2 +sin 2 θ dϕ 2 ) . (2.3) \nFollowing Morris and Thorne, the function b ( r ) will be referred to as the 'shape function'. The shape function may be thought of as specifying the shape of the spatial slices. On the other hand, φ ( r ) might best be interpreted as a sort of 'anomalous redshift' that describes how far the total gravitational redshift deviates from that implied by the shape function. As will subsequently be seen the Einstein field equations have a particularly nice form when written in terms of these functions.", 'B. Putative horizons': "For now, explore the meaning of the metric in the form (2.3) without yet applying the field equations. Firstly, applying boundary conditions at spatial infinity permits one to set φ ( ∞ ) = 0 without loss of generality. Once this normalization of the asymptotic time coordinate is adopted one may interpret b ( ∞ ) in terms of the asymptotic mass b ( ∞ ) = 2 GM . (Naturally one is assuming an asymptotically flat geometry). \nThe metric (2.3) has putative horizons at values of r satisfying b ( r H ) = r H . Only the outermost horizon is of immediate interest and comments will be restricted to that case. Now for the outermost horizon one has ∀ r > r H that b ( r ) < r , consequently b ' ( r H ) ≤ 1. The case b ' ( r H ) = 1 is anomalous and will be discussed separately. Assuming then that b ' ( r H ) < 1 the behaviour of the metric near the putative horizon is \nds 2 ≈ -e -2 φ ( r H ) ( r -r H r H ) (1 -b ' ( r H )) dt 2 + 1 (1 -b ' ( r H )) ( r H r -r H ) dr 2 + r 2 H ( dθ 2 +sin 2 θ dϕ 2 ) . (2.4) \nThus the putative horizon is seen to possess all the usual properties of a Schwarzschild horizon provided that e -2 φ ( r ) is positive and of finite slope at r = r H , corresponding to | φ ( r H ) | and | φ ' ( r H ) | being finite. \nThe putative horizon at r H = b ( r H ) will be said to be of canonical type if \nb ' ( r H ) < 1; | φ ( r H ) | < ∞ ; | φ ' ( r H ) | < ∞ . (2.5) \nNoncanonical horizons are of interest in their own right. On the one hand, if b ' ( r H ) = 1 one may Taylor expand \nb ( r ) = b ( r H ) + b ' ( r H )( r -r H ) + b '' ( r H ) 2 ( r -r H ) 2 + ... = r + γ 2 2 r H ( r -r H ) 2 + ... (2.6) \nThis allows the simple expansion (1 -b/r ) = 1 2 γ 2 ( r -r H ) 2 /r 2 H + ... , thus indicating that in this case g rr does not change sign at the horizon (provided that γ 2 = 0). This behaviour is \n/negationslash \nan indication of the merging of an inner and an outer horizon. In fact, the horizon of an extreme Q = M Reissner-Nordstrom blackhole is precisely of this type (with φ ( r ) ≡ 0). If γ 2 = 0 then one must go to higher order in the Taylor series expansion. If the first nonzero term is of oder n , that is if b ( r ) -r = 1 n ! γ n ( r -r H ) n /r n H + ... , then one may easily convince oneself that one is dealing with a n -fold merging of n degenerate horizons. \nOn the other hand, even if b ' ( r H ) < 1, one may still obtain noncanonical horizon structure due to the behaviour of φ ( r ) near the putative horizon. For instance, take φ ( r ) = + 1 2 ln( r -r H r H ) + f ( r ), where f ( r ) is smooth and finite at the putative horizon. In this case the behaviour of the metric near the putative horizon is \nds 2 ≈ -e -2 f ( r H ) (1 -b ' ( r H )) dt 2 + 1 (1 -b ' ( r H )) ( r H r -r H ) dr 2 + r 2 H ( dθ 2 +sin 2 θ dϕ 2 ) . (2.7) \nThus g tt remains nonzero on the putative horizon, so that the putative horizon is not in fact a horizon at all, but rather is the throat of a traversable wormhole [16]. \nFinally, one should consider the possibility that the 'anomalous redshift' might diverge in a region where the 'shape function' is still well behaved. Specifically, consider the possibility that φ ( r ) → + ∞ as r → r H , while b ( r ) → r 0 ≡ 2 Gm 0 < r H . Such a horizon is certainly noncanonical. Analysis of the Einstein field equations (see below) indicates that this case corresponds to a divergence in the stress-energy density as the horizon is approached. \nFurther discussion of noncanonical horizons will be postponed, and henceforth all horizons are taken to be of canonical type.", 'A. Surface gravity': "The Hawking temperature of a blackhole is given in terms of its surface gravity by kT H = (¯ h/ 2 π ) κ . Now in general for a spherically symmetric system the surface gravity can be computed via \nκ = lim r → r H { 1 2 ∂ r g tt √ g tt g rr } . (3.1) \n(This result holds independently of whether or not one chooses to normalize the g θθ and g ϕϕ components of the metric by adopting Schwarzschild coordinates.) For the choice of functional form described in (2.3) this implies \nκ = lim r → r H { 1 2 e φ ∂ ∂r [ e -2 φ ( 1 -b ( r ) r )]} = lim r → r H { 1 2 e -φ [ -2 φ ' ( r ) ( 1 -b ( r ) r ) + b ( r ) r 2 -b ' ( r ) r ] . } (3.2) \nNow for a canonical horizon | φ ( r H ) | and | φ ' ( r H ) | are both finite so that \nκ = 1 2 r H e -φ ( r H ) (1 -b ' ( r H )) . (3.3) \nAt this stage of course, this formula is largely definition . This formula receives its physical significance only after b ' ( r H ) and φ ( r H ) are related to the distribution of matter by imposing the Einstein field equations. Note that the derivation of the formula for the surface gravity continues to make perfectly good sense for degenerate horizons ( ie b ' ( r H ) = 1), merely asserting in this case that κ = 0.", 'B. Euclidean signature techniques': "Another way of calculating the Hawking temperature is via the periodicity of the Euclidean signature analytic continuation of the manifold [17]. Proceed by making the formal substitution t →-it to yield \nds 2 E = + e -2 φ ( r ) (1 -b ( r ) /r ) dt 2 + dr 2 (1 -b ( r ) /r ) + r 2 ( dθ 2 +sin 2 θdϕ 2 ) . (3.4) \nAs is usual, discard the entire r < r H region, retaining only the (analytic continuation of) that region that was outside the outermost horizon ( ie: r ≥ r H ). Taylor series expand the metric in the region r ≈ r H . Provided that the horizon is canonical one may write (1 -b/r ) ≡ ( r -b ) /r ≈ ( r -r H ) r -1 H (1 -b ' ( r H )) to give \nds 2 E ≈ -e -2 φ ( r H ) (1 -b ' ( r H )) ( r -r H r H ) dt 2 + 1 (1 -b ' ( r H )) ( r H r -r H ) dr 2 + r 2 H ( dθ 2 +sin 2 θ dϕ 2 ) . \n(3.5) \nConstruct a new radial variable /rho1 by taking \nd/rho1 = 1 √ 1 -b ' ( r H ) √ r H r -r H dr = 2 √ 1 -b ' ( r H ) d ( √ r H ( r -r H )) . (3.6) \nThen r H ( r -r H ) = 1 4 (1 -b ' ( r H )) /rho1 2 , and the Euclidean signature metric may be written as \nds 2 E ≈ -e -2 φ ( r H ) (1 -b ' ( r H )) 2 1 4 r 2 H ( /rho1 2 dt 2 ) + d/rho1 2 + r 2 H ( dθ 2 +sin 2 θ dϕ 2 ) . (3.7) \nNow the ( /rho1, t ) plane is a smooth two dimensional manifold if and only if t is interpreted as an angular variable with period \nβ = 2 π 2 r H e φ ( r H ) (1 -b ' ( r H )) -1 . (3.8) \nInvoking the usual incantations [17], this periodicity in imaginary (Euclidean) time is interpreted as evidence of a thermal bath of temperature kT = ¯ h/β , so that the Hawking temperature is identified as \nkT H = ¯ h 4 πr H e -φ ( r H ) (1 -b ' ( r H )) . (3.9) \nThis is the same result as was obtained by direct calculation of the surface gravity, though this formulation has the advantage of (1) shedding further illumination on the subtleties associated with noncanonical horizons, and (2) verifying the relationship between Hawking temperature and surface gravity.", 'A. Formal solution': "The Einstein tensor corresponding to (2.3) can be obtained by the standard simple but tedious computation. Choose an orthonormal basis attached to the ( t, r, θ, ϕ ) coordinate system ( ie , choose a fiducial observer basis - a FIDO basis) \nG ˆ t ˆ t = b ' r 2 (4.1) \nG ˆ r ˆ r = -2 r ( 1 -b r ) φ ' -b ' r 2 (4.2) \nWhereas the forms of G ˆ t ˆ t and G ˆ r ˆ r are quite pleasing, the form of G ˆ θ ˆ θ ≡ G ˆ ϕ ˆ ϕ is quite horrible. \nFortunately one will not need to use G ˆ θ ˆ θ or G ˆ ϕ ˆ ϕ explicitly. For completeness note: \nG ˆ θ ˆ θ = G ˆ ϕ ˆ ϕ = ( 1 -b r ) ( -φ '' + φ ' ( φ ' -1 r )) -3 2 φ ' ( b r 2 -b ' r ) -1 2 b '' r . (4.3) \nAll other components of the Einstein tensor are zero. To minimize computation use the results of Morris and Thorne [16] with the substitution φ Morris -Thorne = -φ here + 1 2 ln(1 -b/r ). \nThe Einstein field equations are \nG αβ = 8 πG T αβ = 8 π /lscript 2 P ¯ h T αβ . (4.4) \nIn the FIDO orthonormal basis used above, the nonzero components of the stress-energy tensor are \nT ˆ t ˆ t = ρ ; T ˆ r ˆ r = -τ ; T ˆ θ ˆ θ = T ˆ ϕ ˆ ϕ = p. (4.5) \nThe first two Einstein equations are then simply rearranged to give \nb ' = 8 πG ρ r 2 , (4.6) \nφ ' = -8 πG 2 ( ρ -τ ) r (1 -b/r ) . (4.7) \nInstead of imposing the third Einstein equation G ˆ θ ˆ θ = G ˆ ϕ ˆ ϕ = 8 π G p , observe that (as is usual) this equation is redundant with the imposition of the conservation of stress-energy. Thus one may take the third equation to be \nτ ' = ( ρ -τ )[ -φ ' + 1 2 { ln(1 -b/r ) } ' ] -2( p + τ ) /r. (4.8) \nTaking ρ and τ to be primary, one may formally integrate the Einstein equations, and then substitute this into the conservation of stress-energy to determine p . Specifically: \nb ( r ) = r H +8 πG ∫ r r H ρ ˜ r 2 d ˜ r, (4.9) \n- \nφ ( r ) = 8 πG 2 ∫ ∞ r ( ρ -τ )˜ r (1 b/ ˜ r ) d ˜ r, (4.10) \np ( r ) = r 2 [ ( ρ -τ ) 2(1 -b/r ) { b -8 πG τ r 3 r 2 } -τ ' ] -τ. (4.11) \nInserting these results into the formula for the Hawking temperature now yields the promised result \nkT H = ¯ h 4 πr H exp ( -8 πG 2 ∫ ∞ r H ( ρ -τ ) r (1 -b/r ) dr ) ( 1 -8 πG ρ H r 2 H ) . (4.12) \nThe Hawking temperature is seen to depend both on data local to the event horizon ( r H , ρ H ) and on a 'redshift' factor whose computation requires knowledge of ρ ( r ) and τ ( r ) all the way from the horizon to spatial infinity. \nOnce the problem has been cast in this form the role of the Weak Energy Condition is manifest. WEC implies that ρ -τ ≥ 0 and that ρ ≥ 0. Consequently ∀ r , φ ( r ) ≥ 0. Also b ' ( r H ) ≥ 0. Thus adopting WEC allows one to assert the promised inequality \nkT H ≤ ¯ h 4 πr H ≡ ¯ h √ 4 πA H . (4.13)", 'B. Convergence issues': "Several points regarding these formulae are worth mentioning. Firstly, the condition b ' ( r H ) ≤ 1 which is automatically satisfied by the outermost putative horizon (regardless of \nwhether or not it be canonical) implies, via the Einstein field equations, a constraint on ρ H , viz ρ H < 1 / (8 πGr 2 H ) ≡ ¯ h/ (8 π/lscript 2 P r 2 H ). This constraint has the nice feature of guaranteeing that the Hawking temperature is non-negative. Turning to questions of convergence of the various integrals encountered, note that \n2 GM = r H +2 G ∫ ∞ r H 4 πρr 2 dr, (4.14) \nso that this integral is guaranteed to converge by the assumed asymptotic flatness of the spacetime. The only questionable integral is that for φ ( r H ). Specifically, its convergence properties near the putative horizon are somewhat subtle. Assuming b ' ( r H ) < 1 one may write this integral as \nφ ( r H ) ≡ 8 πG 2 ∫ ∞ r H ( ρ -τ ) r (1 -b/r ) dr, ≈ (finite) + 8 πGr 2 H 2(1 -b ' ( r H )) ∫ (1+ /epsilon1 ) r H r H ( ρ -τ ) ( r -r H ) dr. (4.15) \nThis integral converges provided that ( ρ -τ ) ≤ k ( r -r H ) α as r → r H for some arbitrary constant k and some constant α > 0. In particular this implies that ρ H = τ H is a necessary condition for the existence of a canonical horizon. It should come as no great surprise then to observe that all 'reasonable' classical field solutions satisfy this boundary condition. Indeed, this boundary condition is equivalent to requiring the energy density measured by a freely falling observer (FFO) to remain integrable as one crosses the horizon. \nTo see this, consider a freely falling observer who starts falling from spatial infinity with initial velocity zero. Let V µ denote the four-velocity of the FFO, and let K µ denote the timelike Killing vector. That is, K µ = (1 , 0 , 0 , 0); K µ = ( -g tt , 0 , 0 , 0). Then the inner product K µ V µ is conserved along geodesics, so that V t = 1, V t = -g tt = -1 /g tt . Since the four-velocity must be normalized ( ‖ V ‖ = -1), one may solve for the radial component to find (outside the outermost horizon): \nV µ = ( 1 g tt , √ 1 g rr √ 1 g tt -1 , 0 , 0) . (4.16) \nIn the FIDO basis \nV ˆ µ = ( √ 1 g tt , √ 1 g tt -1 , 0 , 0) . (4.17) \nSo the energy density measured by a FFO is ρ FFO ≡ T ˆ µ ˆ ν V ˆ µ V ˆ ν = ρ/g tt +( -τ )( g -1 tt -1) = τ +( ρ -τ ) /g tt . Finally, inserting the functional form for g tt one sees \nρ FFO = τ + ( ρ -τ ) e -2 φ (1 -b/r ) ≈ e +2 φ ( ρ -τ ) r H (1 -b ' )( r -r H ) . (4.18) \nSo that the boundary condition ( ρ -τ ) ≤ k ( r -r H ) α , α > 0, required to keep φ ( r H ) finite, implies the integrability of ρ FFO . Conversely, the integrability of ρ FFO implies either (1) the finiteness of φ ( r H ) (canonical horizon), or (2) φ ( r ) → -∞ (corresponding to a traversable wormhole).", 'A. Reissner-Nordstrom': 'For the Reissner-Nordstrom geometry the symmetries of the situation together with the form of the electromagnetic stress-energy tensor implies \nρ = τ = p = E 2 / 8 π. (5.1) \nThis automatically gives φ ( r ) = 0, ∀ r . The electromagnetic field equations imply E = Q/r 2 , so that \nkT RN H = ¯ h 4 πr H ( 1 -GQ 2 r 2 H ) . (5.2) \nThis is an unusual, though correct formula for the Hawking temperature of a ReissnerNordstrom blackhole. To see this note that explicit solution of the Einstein-Maxwell field equations gives g tt = ( g rr ) -1 = 1 -(2 GM/r ) + ( GQ 2 /r 2 ), whence κ = 1 2 lim r → r H ∂ r g tt = 1 2 ( { 2 GM/r 2 H } - { 2 GQ 2 /r 2 H } ) = (1 / 2 r H )( { 2 GM/r H } - { GQ 2 /r 2 H } ) = (1 / 2 r H )(1 -{ GQ 2 /r 2 H } ), which is the above result.', 'B. Thin shell geometry': "Consider a thin spherical shell of matter of density ρ S , radius r S , and thickness ( δr ) S , which surrounds a vacuum blackhole of Schwarzschild radius r H . The mass of this thin shell is m S = 4 πρ S r 2 S ( δr ) S , and the asymptotic total mass satisfies 2 GM = r H + 2 Gm S . The shape function exhibits a step function discontinuity: b ( r ) = r H +Θ( r -r S )2 Gm S . Direct integration of φ ' ( r ) is not an appropriate way of calculating φ ( r H ) due to the discontinuity in b ( r ). Rather it is more appropriate to solve for φ ( r H ) by using the continuity of g tt to develop matching conditions. Everywhere except at the shell itself both ρ and τ are zero, so φ ( r ) is piecewise constant. Applying boundary conditions at the horizon and at spatial infinity gives φ ( r ) = φ ( r H )Θ( r S -r ). The matching conditions are thus \ng tt ( r + S ) = 1 -2 GM/r S , (5.3) \ng tt ( r -S ) = e -2 φ ( r H ) (1 -r H /r S ) . (5.4) \nOne immediately obtains \ne -2 φ ( r H ) = 1 -2 GM/r S 1 -r H /r S = 1 -2 Gm S /r S 1 -r H /r S . (5.5) \nFinally, noting that ρ = 0 on the horizon, one sees that the Hawking temperature is suppressed by \nkT H = ¯ h 4 πr H √ √ √ √ 1 -2 Gm S /r S (1 -r H /r S ) . (5.6) \nPhysically, this suppression of the Hawking temperature may be attributed to the fact that the shell introduces an extra gravitational redshift that decreases the energy of the Hawking photons on their way out to spatial infinity.", 'C. Charged dilatonic blackhole': 'As a decidedly nontrivial example consider geometry and fields surrounding a charged dilatonic blackhole [1,4]. The calculation about to be exhibited is a rather obtuse way of calculating the Hawking temperature, depending as it does on delicate cancellations amoung r H , ρ H , and φ H . The only virtue of this computation is that it illustrates general features of the formalism. (Units: For this section only set G ≡ 1.) \nConsider then a solution to the combined (gravity + electromagnetism + dilaton) equations of motion. The Lagrangian is \nL = √ -g { -R/ 8 π +2( ∇ Φ) 2 + F 2 / 4 π } . (5.7) \n(Warning: Φ /negationslash = φ !) In Schwarzschild coordinates the solution corresponding to an electric monopole is \nds 2 = -( 1 -2 M a + √ r 2 + a 2 ) dt 2 + ( 1 -2 M a + √ r 2 + a 2 ) -1 r 2 r 2 + a 2 dr 2 \n+ r 2 ( dθ 2 +sin 2 θ dϕ 2 ) , (5.8) \nF ˆ t ˆ r = Q/r 2 , (5.9) \ne 2Φ = 1 -Q 2 M ( a + √ r 2 + a 2 ) (5.10) \nHere one has used the freedom to make an overall shift in Φ to set Φ( ∞ ) = 0. The parameter a is defined by a ≡ Q 2 / 2 M . In terms of the formalism developed in this paper \n1 -b r = ( 1 + a 2 r 2 )( 1 -2 M a + √ r 2 + a 2 ) , (5.11) \ne -2 φ ( r ) = ( 1 + a 2 r 2 ) -1 = r 2 r 2 + a 2 . (5.12) \nThe horizon occurs at 2 M = a + √ r 2 H + a 2 , that is, r 2 H + a 2 = (2 M -a ) 2 , so that the surface gravity is \nκ = 1 2 r H r H √ r 2 H + a 2 ( 1 -8 π ρ H r 2 H ) = 1 2(2 M -a ) ( 1 -8 π ρ H r 2 H ) . (5.13) \nTo calculate ρ H one evaluates the nonzero components of the stress-energy tensor \nρ = 1 8 π e -2Φ E 2 + ‖∇ Φ ‖ 2 , (5.14) \nτ = 1 8 π e -2Φ E 2 -‖∇ Φ ‖ 2 , (5.15) \np = 1 8 π e -2Φ E 2 -‖∇ Φ ‖ 2 . (5.16) \nAs one approaches the event horizon it is easy to verify that ‖∇ Φ ‖ → 0, while E → Q/r 2 H , so that ρ → 1 8 π (1 -{ Q 2 / 2 M 2 } )( Q 2 /r 4 H ). Thus \n8 π ρ H r 2 H = ( 1 -Q 2 2 M 2 ) Q 2 r 2 H = M -a M Q 2 2 M (2 M -2 a ) = a 2 M . (5.17) \nCombining this considerable morass yields the simple result \nκ = 1 4 M (5.18) \nAs previously mentioned, this calculation is a particularly obtuse manner in which to compute the surface gravity. This computation is of interest only insofar as it illustrates general principles and serves as a check on the formalism. The inequality κ < 1 / (2 r H ), which previously appeared to be just a random accident of the calculation, is now seen to be intrinsically related to the fact that classical fields satisfy the WEC.', 'VI. DISCUSSION': "For an arbitrary static spherically symmetric blackhole this note has established a general formula for the Hawking temperature in terms of the energy density and radial tension. Adopting Schwarzschild coordinates, and writing \nb ( r ) = r H +8 πG ∫ r r H ρ ˜ r 2 d ˜ r, (6.1) \none finds that \nkT H = ¯ h 4 πr H exp ( -8 πG 2 ∫ ∞ r H ( ρ -τ ) r (1 -b/r ) dr ) ( 1 -8 πG ρ H r 2 H ) . (6.2) \nGeneralizations of this result to axisymmetric spacetimes (for instance, to Kerr-Newman blackholes embedded in an axisymmetric cloud of matter) would clearly be of interest. Generalizations to arbitrary event horizons are probably unmanageable. On the one hand, the Dominant Energy Condition (DEC) guarantees the constancy of the surface gravity (and hence the constancy of the Hawking temperature) over the surface of an arbitrary stationary event horizon. Furthermore, one might conceivably hope to generalize the factor 4 πr H to √ 4 πA H . On the other hand, there is no particular reason to believe that ρ H is constant over the event horizon, nor is it clear how to generalize the notion of φ ( r H ). (Presumably in terms of some line integral from the horizon to spatial infinity?) \nIf the central result of this paper is supplemented by the Weak Energy Condition one may further assert (for static spherically symmetric dirty blackholes) the general inequality \nkT H ≤ ¯ h 4 πr H . (6.3) \nThis inequality may be somewhat strengthened if one explicitly separates out the electromagnetic contribution to the stress-energy. Note that ρ H ≥ ( ρ em ) H ≡ E 2 / 8 π ≡ Q 2 / (8 πr 4 H ). Thus for electrically charged static spherically symmetric dirty blackholes \nkT H ≤ ¯ h 4 πr H ( 1 -GQ 2 r 2 H ) . (6.4) \n(Generalization to magnetic charge and the dyonic case is trivial.) The possibility of further generalizing these inequalities is more promising. I will restrain myself to a single Conjecture: \nFor a stationary dirty blackhole in interaction with matter fields satisfying the Dominant Energy Condition \nkT H ≤ ¯ h √ 4 πA H . (6.5) \nNotes: (1) It should be noted that this inequality is satisfied by the Kerr-Newman geometry. (2) The restrictions 'stationary' and 'Dominant Energy Condition' cannot be dispensed with as they are required merely in order to guarantee the existence of a constant Hawking temperature. (3) With regard to this conjectured inequality, it should be pointed out that a weaker inequality that requires stronger hypotheses can be derived from the 'four laws of blackhole mechanics' [18]. Restricting the results of that paper to the case of zero rotation, one observes the equality ( S H = entropy = (1 / 4) kA H //lscript 2 P ): \nM = ∫ ∞ r H (2 T µ ν -Tδ µ ν ) K ν d Σ µ +2 T H S H . (6.6) \nBy invoking the Strong Energy Condition, the integral can be made positive, in which case one obtains the inequality \nkT H ≤ M 2 S H /k ≡ 2 M/lscript 2 P A H . (6.7) \nWhen restricted to spherical symmetry this reduces to \nkT H ≤ 2 GM r H ¯ h 4 πr H . (6.8) \nWhich is clearly weaker than the inequalities considered above. \nIn summary, this paper has exhibited a general formalism for calculating the surface gravity and Hawking temperature of spherically symmetric static dirty blackholes. The formalism serves to tie together a number of otherwise seemingly accidental results scattered throughout the literature. Clear directions for future research are indicated.", 'ACKNOWLEDGMENTS': 'This research was supported by the U.S. Department of Energy.', 'REFERENCES': "- ∗ Electronic mail: [email protected]\n- [1] G. W. Gibbons and K. Maeda, Nucl. Phys. B298 , 741 (1988).\n- [2] I. Ichinose and H. Yamazaki, Mod. Phys .Lett. A4 , 1509 (1989).\n- [3] H. Yamazaki and I. Ichinose, Class. Quantum Gravit. 9 , 257 (1992).\n- [4] D. Garfinkle, G. T. Horowitz, and A. Strominger, Phys. Rev. D43 , 3140 (1991).\n- [5] A. Shapere, S. Trivedi, and F. Wilczek, Mod. Phys. Lett. A6 , 2677 (1991).\n- [6] T. J. Allen, M. J. Bowick, and A. Lahiri, Phys. Lett. 237B , 47 (1990).\n- [7] B. A. Campbell, N. Kaloper, K. A. Olive Phys. Lett. B263 , 364 (1991).\n- [8] K. Lee and E. J. Weinberg, Phys. Rev. D44 , 3159 (1991).\n- [9] M. Bowick, S. Giddings, J. Harvey, G. Horowitz, and A.Strominger, Phys. Rev. Lett. 61 , 2823 (1988).\n- [10] F. Dowker, R. Gregory, and J. Traschen, Phys. Rev. D45, 2762 (1992).\n- [11] D.V. Galtsov and A.A. Ershov, Phys. Lett. A138 , 160 (1989).\n- [12] N. Straumann and Z.H. Zhou, Phys. Lett. B243 , 33 (1990).\n- [13] P. Bizon Phys. Rev. Lett. 64 , 2844 (1990).\n- [14] P. Bizon and R.M. Wald, Phys. Lett. B267 , 173 (1991).\n- [15] A. Lahiri, \n'An alternative scenario for non-Abelian quantum hair' , \nLos Alamos preprint LA-UR-92-471; hepth/9202045. \n- [16] M. S. Morris and K. S. Thorne, Am. J. Phys. 56 , 395 (1988).\n- [17] G. W. Gibbons and S. W. Hawking, Phys. Rev. D15 , 2752 (1977). \n[18] J. M. Bardeen, B. Carter, and S. W. Hawking, Commun. Math. Phys. 31 , 161 (1973)."}
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2008ApJ...681..925W
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Cosmic Evolution of Black Holes and Spheroids. III. The M<SUB>BH</SUB>-σ<SUB>*</SUB> Relation in the Last Six Billion Years
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2008-01-01
| 17 | 0.46 | 159 |
['accretion', 'accretion disks', 'galaxies active', 'galaxies evolution', 'galaxies quasars', 'astrophysics']
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[]
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We measure the evolution of the correlation between black hole mass and host spheroid velocity dispersion (M<SUB>BH</SUB>-σ<SUB>*</SUB>) over the last 6 billion years, by studying three carefully selected samples of active galaxies at z = 0.57, z = 0.36 and z < 0.1. For all three samples, virial black hole masses are consistently estimated using the line dispersion of Hβ and the continuum luminosity at 5100 Å or Hα line luminosity, based on our cross calibration of the broad-line region size-luminosity relation. For the z = 0.57 sample, new stellar velocity dispersions are measured from high signal-to-noise ratio spectra obtained at the Keck Telescope, while for the two lower redshift samples they are compiled from previous works. Extending our previous result at z = 0.36, we find an offset from the local relation, suggesting that for fixed M<SUB>BH</SUB>, distant spheroids have on average smaller velocity dispersions than local ones. The measured offset at z = 0.57 is Δ log σ<SUB>*</SUB> = 0.12 +/- 0.05 +/- 0.06 (or Δ log M<SUB>BH</SUB> = 0.50 +/- 0.22 +/- 0.25), i.e., Δ log M<SUB>BH</SUB> = (3.1 +/- 1.5) log (1 + z) + 0.05 +/- 0.21. This is inconsistent with a tight and nonevolving universal M<SUB>BH</SUB>-σ<SUB>*</SUB> relation at the 95% CL.
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https://arxiv.org/pdf/0804.0235.pdf
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{'COSMIC EVOLUTION OF BLACK HOLES AND SPHEROIDS. III. THE M BH -σ ∗ RELATION IN THE LAST SIX BILLION YEARS': 'Jong-Hak Woo 1 , Tommaso Treu 1,2 , Matthew A. Malkan 3 , Roger D. Blandford 4 Draft version August 20, 2021', 'ABSTRACT': 'We measure the evolution of the correlation between black hole mass and host spheroid velocity dispersion (M BH -σ ∗ ) over the last 6 billion years, by studying three carefully selected samples of active galaxies at z = 0 . 57, z = 0 . 36 and z < 0 . 1. For all three samples, virial black hole masses are consistently estimated using the line dispersion of H β and the continuum luminosity at 5100 ˚ A or H α line luminosity, based on our cross calibration of the broad line region size-luminosity relation. For the z = 0 . 57 sample, new stellar velocity dispersions are measured from high signal-to-noise ratio spectra obtained at the Keck Telescope, while for the two lower redshift samples they are compiled from previous works. Extending our previous result at z = 0 . 36, we find an offset from the local relation, suggesting that for fixed M BH , distant spheroids have on average smaller velocity dispersions than local ones. The measured offset at z = 0 . 57 is ∆ log σ ∗ = 0 . 12 ± 0 . 05 ± 0 . 06 (or ∆log M BH = 0 . 50 ± 0 . 22 ± 0 . 25), i.e. ∆log M BH = (3 . 1 ± 1 . 5) log(1 + z ) + 0 . 05 ± 0 . 21. This is inconsistent with a tight and non-evolving universal M BH -σ ∗ relation at the 95%CL. \nSubject headings: accretion, accretion disks - black hole physics - galaxies: active - galaxies: evolution - quasars: general', '1. INTRODUCTION': 'Understanding the origin of the black hole mass spheroid velocity dispersion (M BH -σ ∗ ) relation (Ferrarese & Merritt 2000; Gebhardt et al. 2000) is a key goal of unified models of black hole - galaxy coevolution (e.g. Kauffmann & Haenhelt 2000; di Matteo et al. 2005; Ciotti & Ostriker 2007). One of the most powerful observational tests of the proposed explanations is to measure the time evolution of the M BH -σ ∗ relation since various scenarios predict different cosmic evolution. For example, - for a fixed M BH - Robertson et al. (2006) predict an increase of σ ∗ with redshift, Croton (2006) and Bower et al. (2006) predict a decrease, while Granato et al. (2004) expect no evolution. \nIn recent years, a number of groups have investigated the evolution of the M BH -σ ∗ relation, using various techniques to estimate σ ∗ of AGN host galaxies (e.g. Shields et al. 2003; Walter et al. 2004; Salviander et al. 2007). Starting with our pilot study (Treu et al. 2004), we reported the first direct measurement of the M BH -σ ∗ relation beyond the local Universe (Woo et al. 2006, hereafter paper I), and updated it with corrected AGN continuum luminosities using Hubble Space Telescope (HST) images in paper II (Treu et al. 2007). \nBy observing 14 Seyfert 1 galaxies, we determined stellar velocity dispersions in the integrated spectra, and M BH from AGN broad emission line widths, which are thought to measure the gravity of the central mass on sub-parsec scales. We found that the measured M BH -σ ∗ relation at z = 0 . 36 is offset with respect to the local re- \n1 Department of Physics, University of California, Santa Barbara, CA 93106-9530; [email protected], [email protected] \n- 3 Department of Physics and Astronomy, University of California at Los Angeles, CA 90095-1547, [email protected]\n- 4 Kavli Institute for Particle Astrophysics and Cosmology, Stanford, CA, [email protected] \nlationship (∆ log M BH = 0.54 ± 0.12 ± 0.21 at fixed σ ∗ ). In other words black holes of a fixed mass appeared to live in bulges with smaller velocity dispersion 4 Gyrs ago (at 95% CL), consistent with recent growth and evolution of intermediate mass spheroids. Using HST images, we obtained a consistent result, ∆ log M BH > 0 . 51 ± 0 . 14 ± 0 . 17, by measuring the M BH - spheroid luminosity relation of the same sample (paper II). This result may be consistent with a scenario where intermediate-mass blue galaxies undergo merging at relatively recent times and arrive on the local black hole-galaxy relations by becoming more massive red galaxies. However, much work remains to be done due to the small sample size and large uncertainties, before this initial result can become a high precision measurement. \nWe report here our first measurement at the next redshift window ( z = 0 . 57, adding ∼ 50% to the look-back time), so that evolutionary trends can be measured over a longer range in cosmic time. We also improve the local baseline by consistently estimating M BH for a sample of 48 nearby Seyfert 1 galaxies with published stellar velocity dispersion (Greene & Ho 2006). To minimize repetition, readers are referred to our previous works (papers I, II; McGill et al. 2008; hereafter M08) for detailed discussions of the systematics inherent to the measurement. The paper is organized as follows. Section 2 describes sample selection and observations. Section 3 presents our measurements. Section 4 presents the M BH -σ ∗ relation. Discussion and conclusions are presented in § 5. We adopt Ω m = 0 . 3, Ω Λ = 0 . 7, and H 0 = 70 km sec -1 Mpc -1 .', '2. DATA': "A sample of broad-line AGNs was selected from the Sloan Digital Sky Survey Data Release 4 (SDSS DR4). Following our strategy at z = 0 . 36 ± 0 . 01 (paper I), we chose the next redshift window, z = 0 . 57 ± 0 . 01, to avoid strong sky features on the redshifted stellar lines around \nFig. 1.Velocity dispersion measurements. The region including the main stellar features is shown together with the best fit template (red thick line). The regions around narrow AGN emission lines - identified by vertical lines - are masked out before fitting. \n<!-- image --> \nthe Mg-Fe line region, minimizing the uncertainties related to sky subtraction and atmospheric absorption corrections. \nOur selection procedure was slightly modified with respect to that of the lower redshift sample. Initially, 365 broad-line AGNs at z=0.57 ± 0.01 were collected from SDSS DR4, based on the presence of the broad H β line. Out of 365 AGNs, we selected 20 objects with g'-r' > 0.1 and r'-i' > 0.3 (AB), expecting non-negligible stellar light in the observed spectra, based on stellar and AGN spectral models. The effects of this color cut will be modeled in detail in future papers, when Keck and possibly HST data for a larger sample at z = 0 . 57 will be available. However, since the colors of the new sample are similar to those of the z = 0 . 36 sample, we do not expect the color cut to introduce a significant bias. In any case, the color cut will tend to select more massive host galaxies for a given nuclear luminosity. Hence, if any biased is introduced, it should bias against the offset seen in papers I and II. \nHigh signal-to-noise (S/N) ratio spectra for 5 objects were obtained with the LRIS spectrograph (Oke et al. 1995) at the Keck-I telescope during two runs in January 2007 and April 2007. The 831 lines mm -1 grating centered at 7600 ˚ A was used with a 1' wide slit, yielding a pixel scale of 0.92 ˚ A × 0 . '' 215 and a Gaussian resolution ( σ ) ∼ 58 km s -1 . Observing conditions were generally favorable with 0.7-1.2' seeing. Total exposure times for each object ranges between 2.5 and 3.5 hours. The observing strategy, data reduction, calibration, and onedimensional spectra extraction processes were very similar to those described in paper I.", '3. MEASUREMENTS': 'This section describes our measurement of σ ∗ and M BH for the 5 Seyferts at z = 0 . 57 ( § 3.1 and § 3.2), and M BH estimates for the 48 local Seyferts ( § 3.3). The relevant properties of the z = 0 . 57 sample are listed in Table 1.', '3.1. Stellar Velocity Dispersion': 'We used the Mg-Fe region (rest-frame ∼ 5050-5300 ˚ A) to measure velocity dispersion as described in detail in paper I. Here, we briefly summarize the procedure and \nsystematic uncertainties. First, we subtracted broad AGN Fe emission, using a set of I Zw 1 templates. Then, we compared in pixel space the observed spectra with 5 stellar templates (G8, G9, K0, K2, and K5 giant) broadened with a range of Gaussian velocity. AGN narrow emission lines (e.g. [N I] 5201 ˚ A and [Fe XIV] 5304 ˚ A) were masked out before fitting, as shown in Figure 1. Fits were performed for all templates to estimate the effect of template mismatch, yielding comparable measurements within the errors (10-20%). The best-fit template was used for the final dispersion measurements. \nThe Mg-Fe region typically used for dynamical studies is a natural choice for our sample since other strong stellar features such as the CaII triplet are out of the optical spectral range. Feature mismatch due to α -enhancement in massive early-type galaxies is a well-known problem in kinematics studies (e.g. Barth et al. 2003; Woo et al. 2004) and can potentially increase systematic uncertainties. However, in paper I we found that only one out of 14 Seyfert galaxies at z = 0 . 36 shows signs of Mg mismatch, as expected because the inferred stellar velocity dispersions are more typical of a Milky Way type galaxy than of a massive early-type galaxy. As for the lower redshift sample, we do not find significant mismatch in our z = 0 . 57 sample, as shown in Figure 1. \nFollowing the procedure described in paper I, we estimate a total systematic uncertainty of 0.05 dex on σ , combining the effects of template mismatch, potential errors due to the large spectroscopic aperture, and host galaxy morphology and inclination. This translates into 0.20 dex uncertainty of the offset in log M BH from the M BH -σ ∗ relation.', '3.2. Black Hole Mass': "Black hole mass can be estimated using the 'virial' method based on the empirical relation between the size of the broad line region and continuum luminosity of the reverberation sample (Kaspi et al. 2005), and the velocity scale given by the width of the broad emission lines. In practice, we measured the line dispersion of broad H β by fitting the observed line profile with Gauss-Hermite polynomials as described in paper I and in M08. The continuum luminosity around 5100 ˚ A (L 5100 ) was measured by averaging flux in the 5070-5130 ˚ A region. Considering the difficulty of achieving absolute flux calibration for the Keck spectra - due to slit losses, variable seeing and sky transparency- we re-calibrated our spectrophotometry with the extinction corrected i ' band magnitude taken from the SDSS-DR6 archive, by calculating and correcting for the offset between Sloan and our synthetic i ' band magnitude measured from the observed spectra. \nFor low luminosity AGNs (L 5100 < 10 44 erg s -1 ) continuum luminosity can be overestimated due to the significant contribution from host galaxies. Thus, correcting for the host galaxy contamination is crucial to avoid overestimation of M BH . The size-luminosity relation was in fact revised with a lower slope ( ∼ 0.5 as expected in photoionization scenarios) and a higher normalization, after correcting for the galaxy contamination in low luminosity AGNs in the reverberation sample (Bentz et al. 2006a). \nIt requires high resolution HST imaging to correct for the host galaxy contamination for distant AGNs. Since \nTABLE 1 Targets and Measured Properties \n| Name (1) | z (2) | RA (J2000) (3) | DEC (J2000) (4) | i ' mag (5) | Exp. hr (6) | S/N ˚ A - 1 (7) | σ Hβ km s - 1 (8) | λL 5100 10 44 erg s - 1 (9) | log M BH /M /circledot (10) | σ ∗ km s - 1 (11) |\n|------------|---------|------------------|-------------------|---------------|---------------|-------------------|---------------------|-------------------------------|-------------------------------|---------------------|\n| W9 | 0.5651 | 15 52 27.82 | +56 22 36.47 | 19 | 2.5 | 79 | 2598 | 4.31 | 8.64 | 289 ± 19 |\n| W11 | 0.5649 | 1 55 16.18 | -9 45 55.99 | 20.03 | 3 | 32 | 2103 | 1.53 | 8.15 | 126 ± 21 |\n| W14 | 0.5616 | 12 56 31.90 | -2 31 30.62 | 18.71 | 2.5 | 94 | 2192 | 4.94 | 8.54 | 228 ± 20 |\n| W17 | 0.5611 | 10 07 28.38 | +39 26 51.83 | 19.71 | 2.5 | 32 | 2320 | 2 | 8.31 | 165 ± 14 |\n| W22 | 0.5649 | 3 42 29.70 | -5 23 19.49 | 18.6 | 3.5 | 101 | 2442 | 5.77 | 8.68 | 144 ± 21 | \nNote . - Col. (1): Target ID. Col. (2): Redshift from SDSS-DR6. Col. (3): Right Ascension. Col. (4): Declination. Col. (5): Extinction corrected i ' AB magnitude from SDSS photometry. Col. (6): Total exposure time. Col. (7): Signalto-noise ratio of the combined spectrum (average in the 8000-8300 ˚ Aspectral region). Col. (8): Second moment of H β in km s -1 . Typical error is ∼ 10%. Col. (9): Rest frame luminosity at 5100 ˚ A. Typical error is a few %. Col. (10): Logarithm of M BH in solar units. Estimated uncertainty is 0.4 dex. Col. (11): Stellar velocity dispersion. \nthis is not available for our sample at the moment, we cannot but adopt the size-luminosity relation based on the total (observed) luminosity. However, based on our experience at z = 0 . 36, host galaxy contamination is not expected to be a major effect. In paper II, for Seyfert galaxies with similar luminosity, we compared M BH estimates based on the size-luminosity relation of Kaspi et al. (2005) with new estimates based on the revised size-luminosity relations of Bentz et al. (2006a), after correcting for host galaxy contamination using HST images. We found that new M BH estimates are on average 0.09 dex smaller, due to the combined effects of removing host galaxy light while using the new size-luminosity relation with a higher normalization. \nTherefore, we will adopt as our best estimate of M BH , the following equation from Paper I based on Kaspi et al. (2005) and Onken et al. (2004), equivalent to the most recent calibration of empirical M BH estimators from M08: \nM BH = 10 8 . 33 M /circledot × ( σ Hβ 3000kms -1 ) 2 ( λL 5100 10 44 ergs -1 ) 0 . 69 , (1) \nwhere σ H β is the line dispersion (second moment) of H β . We assume 0.4 dex uncertainty on the estimated M BH , based on comparisons of reverberation data and singleepoch data (Vestergaard & Peterson 2006; M08), which dominates the errors on σ H β and L 5100 . \nAs a sanity check, we compared M BH estimates based on Equation 1 with those based on the new sizeluminosity relation (Bentz et al. 2006a) along with the same virial coefficient of Onken et al. (2004): \nM BH = 10 8 . 58 M /circledot × ( σ Hβ 3000kms -1 ) 2 ( λL 5100 ,n 10 44 ergs -1 ) 0 . 518 , (2) \nwhere L 5100 ,n is the nuclear luminosity at 5100 ˚ A after correcting for host galaxy contamination. Since high resolution images needed for an accurate measurement of the nuclear luminosity are not available for our sample, we assume an average AGN fraction in the observed light at 5100 ˚ A. If the host galaxy contamination is negligible (L 5100 , n = L 5100 ), Equation 2 gives 0.16 dex higher M BH compared to Equation 1, while if the AGN fraction is assumed to be 50%, M BH is 0.006 dex higher. Thus, using Equation 1 without correcting for the host galaxy \ncontamination - which we cannot do at the moment does not significantly affect our M BH estimates. As in paper I, we adopt a systematic error of 0.11 dex in M BH estimates, which is dominated by AGN continuum luminosity uncertainty due to host galaxy contamination.", '3.3. Local Seyferts': "To measure the evolution of the M BH -σ ∗ relation, it is important to have a well defined local sample. The sample of 14 Seyfert galaxies with reverberation M BH , and measured stellar velocity dispersion (Onken et al. 2004) is a good local benchmark. However, it is desirable to have a complementary sample for two reasons. First, the reverberation sample is small in size and shows a flattened distribution on the M BH -σ ∗ plane, especially with a new reverberation black hole mass of NGC 4151 (Bentz et al. 2006b; see magenta points in Figure 2). Second, there could be an unknown systematic offset between the reverberation mass and our single-epoch mass due to the uncertainties in measuring velocity and luminosity from single-epoch spectra, potentially caused by, e.g., flux variability, velocity variability, the narrow line subtraction (e.g. Collin et al. 2006; Woo 2008). \nFor these reasons, we estimated M BH for a sample of local Seyferts, using the line dispersion of H β and H α line luminosity (L H α ), and a formula consistently calibrated with that used for M BH estimates at z = 0 . 36 and z = 0 . 57 (M08). We selected 55 Seyfert 1 galaxies at z < 0 . 1 from SDSS-DR6, with published stellar velocity dispersion (Greene & Ho 2006). Seven objects were excluded due to the very faint broad component of H β that prevented us from measuring reliable line widths. For local low luminosity Seyferts, host galaxy light is a significant fraction of the light observed within the Sloan fiber (3' diameter), superimposing strong stellar absorption on the broad H β line profile. Thus, we subtracted the stellar features, using eigenspectra templates based on a principal component analysis of several hundred galaxy spectra (Hao et al. 2005). Since the L 5100 measured from SDSS spectra could be also significantly contaminated by stellar light, we used L H α from Greene & Ho (2006) instead, together with the M BH recipe calibrated by M08 and Green & Ho (2005).", '4. THE M BH -σ ∗ RELATION': "In Figure 2, the M BH -σ ∗ relation for local active galaxies (left panel) and our samples at z = 0 . 36 and z = 0 . 57 \n<!-- image --> \nFig. 2.The M BH -σ ∗ relation of active galaxies. Left panel: local Seyferts with σ ∗ from Greene & Ho (2006) and our own M BH estimates, consistently calibrated with our estimates for distant samples (black circles); local Seyferts with M BH , measured via reverberation mapping (Onken et al. 2004; magenta squares). Right panel: new measurements at z = 0 . 57 (red stars); Seyfert galaxies at z = 0 . 36 from our earlier work (paper I, II; blue circles). The local relationships of quiescent galaxies (Tremaine et al. 2002; black points) are shown for comparison as a solid (Tremaine et al. 2002) and dashed (Ferrarese & Ford 2005) line. \n<!-- image --> \nFig. 3.Offset in M BH with respect to the local quiescent sample (Tremaine et al. 2002) as a function of redshift. Large solid points with error bars represent the average and rms scatter for the four samples of active galaxies. The rms scatter of the z = 0 . 57 sample is 0.5 dex, similar to that of local active galaxies. Note that all 'virial' M BH are based on the second moment of H β and the same calibration of the virial coefficient. The dashed line represent the best fit relation ∆ log M BH = (3 . 1 ± 1 . 5) log(1 + z ) + 0 . 05 ± 0 . 21. \n<!-- image --> \n(right panel) are presented along with local quiescent galaxies. Two local AGN samples (SDSS sample from § 3.3 and the reverberation sample from Onken et al. 2004) are consistent with the M BH -σ ∗ relation of quiescent galaxies, although the scatter is somewhat larger (r.m.s. 0.45 and 0.43 dex, respectively for the SDSS sample and the reverberation sample) compared to that of quiescent galaxies ( ∼ 0.3 dex). The scatter increases as galaxy mass decreases, perhaps consistent with mass- \nent evolution in the sense that less massive galaxies are still evolving to the M BH -σ ∗ relation. This may indicate that the M BH -σ ∗ relation is not as tight for latetype galaxies even at z ∼ 0. Splitting evenly the local sample into two groups below and above σ ∗ = 120km s -1 , and taking into account the measurement errors on σ ∗ , we find that the intrinsic scatter is a factor of 2 larger for the low σ ∗ sample (0.43 vs 0.22). \nThe distant samples are offset from the local M BH -σ ∗ relation. The average offset of the z = 0 . 57 sample is 0 . 50 ± 0 . 22 ± 0 . 25 dex in M BH , corresponding to 0 . 12 ± 0 . 05 ± 0 . 06 in log σ ∗ - in the sense that velocity dispersions were on average smaller for given M BH six Gyrs ago (Figure 3). Using the new size-luminosity relation of Bentz et al. 2006a (Equation 2) and assuming an average AGN fraction ∼ 50%, we find an equivalent offset, ∆logM BH = 0.51. If the AGN fraction is higher, then the offset increases (see Section 3.2), indicating that M BH estimates based on Equation 1 is not significantly overestimated. The result is similar to the average offset of the z = 0 . 36 sample (papers I and II), although the error bars on the measurement are large enough to allow for a variety of redshift trends. We include in our error analysis, in addition to the random errors, a potential systematic error of 0.25 dex, estimated by combining systematic uncertainties in M BH and σ ∗ . \nTo quantify the significance of evolution, we consider the three active samples. We emphasize that M BH was consistently estimated based on the line dispersion of H β and the same virial coefficient (shape factor). Thus, a change in the virial coefficient will move all samples vertically by the same amount, keeping the offset unchanged, unless the kinematics of the broad line region (hence, the virial coefficient) varies as a function of M BH or redshift. Therefore, we consider the systematic error on the relative calibration of M BH to be \nnegligible, leaving systematic errors in the measurement of σ ∗ as the main source of systematic uncertainty in the evolution. Including random and systematic errors in the analysis, we find that the best fit relation is ∆log M BH = (3 . 1 ± 1 . 5) log (1 + z ) + (0 . 05 ± 0 . 21), i.e. the slope is non zero at the two sigma level. However, as discussed in paper I and II, it is important to keep in mind that the observed offset may not represent evolution, if the higher z samples are not direct progenitors of the lower z samples due to, e.g., the somewhat different scales in galaxy mass and M BH .", '5. DISCUSSION AND CONCLUSIONS': 'We investigated the evolution of the M BH -σ ∗ relation using three samples of Seyfert galaxies at z < 0 . 1, z = 0 . 36, and z = 0 . 57, finding evolution in the last 6 Gyr at the 95%CL. This result is consistent with a scenario where black hole growth predates bulge assembly and that bulges grow substantially in the last 6 Gyr - at least at this mass scale - if the local M BH -σ ∗ relation is the universal end-point of black hole-galaxy coevolution. \nAs discussed in paper I, collisional merging of latetype galaxies can drive the evolution of the M BH -σ ∗ relation. The mass and stellar velocity dispersion of the final spheroid will increase, not only by forming new stars but also transforming rotation-supported disk stars into pressure-supported spheroid components. This can potentially overcome the growth in M BH due to merging with the supermassive black hole of the companion, especially if the companion galaxy is not spheroid dominated. \nIn a galaxy merging scenario, the evolution of the M BH -σ ∗ relation could be mass-dependent, similarly to the downsizing trends in galaxy evolution (Cowie et al. 1996) and AGN evolution (Barger et al. 2005). As seen for example in fundamental plane studies (e.g. Treu et al. 2005, Woo et al. 2004, 2005), active and quiescent massive early-type galaxies have relatively old stellar populations in the redshift range considered here ( z ∼ 0 . 4 -0 . 6). Together with the results on the evolution of the mass function (e.g. Bundy et al. 2007), this is consistent with an early epoch of assembly for the most massive spheroids. Thus, the evolution of the M BH -σ ∗ relation could be mass dependent, slower at this redshift for the more massive galaxies (see Peng et al. 2006 for the M BH -spheroid luminosity relation of massive high redshift galaxies, which show evolution in the same sense as our sample since z ∼ 2). Recently, Shen et al. (2008) present the M BH -σ ∗ relation out to z ∼ 0 . 4 based on SDSS spectra, concluding that the offset (in the same direction as the one reported here) with redshift is not significant for their sample. However, since their 28 galaxies with measured σ ∗ at z > 0.3 have an average S/N=18.7 per \n5 The M BH range and measured offset are similar to those of the sample studied in paper II, resulting in the same negligible bias \npixel (and hence the S/N of the stellar spectrum is less than ∼ 10 per pixel if the nuclear light fraction is ∼ 50%) and M BH was based on the FWHM of H β line, direct comparison with our result is not straightforward. As discussed in paper II, the broad observational picture is far from conclusive at the moment, requiring larger samples over a wider mass range than the present sample to test this hypothesis. \nIt is important to consider selection effects. First, since our samples were selected based on the flux and width of the broad lines, they could be biased towards high M BH objects (paper II; see also Lauer et al. 2007b). However, as we showed in paper II with Monte Carlo simulations, this bias is too small to account for the observed offset 5 , unless the intrinsic scatter of the M BH -σ ∗ relation at z = 0 . 57 - which is unknown - is of order 1 dex. Second, although active galaxies are the only target for M BH estimation in the distant universe, they may not represent the general galaxy population, as they are rare objects with a highly accreting and radiatively efficient black hole. However, two pieces of evidence argue against the explanation of the observed evolution purely in terms of systematic differences between active and quiescent galaxies: i) a consistent M BH -σ ∗ relation is found locally for the two active galaxy samples; ii) the M BH -σ ∗ of distant active galaxies is offset from that of the local active sample. \nAn alternative or complementary explanation of the observed offset is that the M BH -σ ∗ relation is not tight for late-type galaxies, as perhaps suggested by the increasing scatter for local active samples, especially at the low mass end. This scenario is consistent with the idea of downsizing, with low mass blue late type-galaxies yet to join the more massive red early-type galaxies on the tight M BH -σ ∗ relation. So far, only a few late-type galaxies are included in the local quiescent galaxy sample that defines the local M BH -σ ∗ relation. A larger sample with more disk-dominant quiescent galaxies is needed to investigate any systematic difference in the local scaling relations. \nThis work is based on data collected at the Keck Observatory, operated by Caltech, UC, and NASA, and is made possible by the public archive of the Sloan Digital Sky Survey. T.T. acknowledges support from the NSF through CAREER award NSF-0642621, from the Sloan Foundation, and from the Packard Foundation. We acknowledge financial support from NASA through HST grant AR-10986. We thank C. Peng and M. Favata for useful discussions, L. Hao for providing the PCA algorithm, and the referee for useful suggestions. \n< 0 . 1 dex.', 'REFERENCES': "Barger, A. J., Cowie, L. L., Mushotzky, R. F., Yang, Y., Wang, W.-H., Steffen, A. T., & Capak, P. 2005, AJ, 129, 578 Bentz, M. C., Peterson, B. M., Pogge, R. W., Vestergaard, M., & Onken, C. A. 2006a, ApJ, 644, 133 Bentz, M. C., et al. 2006b, ApJ, 651, 775 Bower, R. 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1998PhRvD..57.7145I
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Nonlinear metric perturbations and production of primordial black holes
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1998-01-01
| 2 | 0.45 | 159 |
['-', '-', '-', '-', 'particles', 'black hole physics', 'background', 'astrophysics']
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[]
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We consider a simple inflationary model with a peculiarity in the form of a ``plateau'' in the inflaton potential. We use the formalism of a coarse-grained field in order to describe the production of metric perturbations h of an arbitrary amplitude, and obtain a non-Gaussian probability function for such metric perturbations. We associate the spatial regions having large perturbations h~1 with the regions going to primordial black holes after inflation. We show that in our model the nonlinear effects can lead to overproduction of primordial black holes.
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[]
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https://arxiv.org/pdf/astro-ph/9708224.pdf
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{'P. Ivanov': "Theoretical Astrophysics Center, Juliane Maries Vej 30, 2100 Copenhagen, Ø Denmark Astro Space Center of P. N. Lebedev Institute, Profsoyznaya 84/32, 117810 Moscow, Russia \nWe consider the simple inflationary model with peculiarity in the form of 'plateau' in the inflaton potential. We use the formalism of coarse-grained field in order to describe the production of metric perturbations h of an arbitrary amplitude, and obtain non-Gaussian probability function for such metric perturbations. We associate the spatial regions having large perturbations h ∼ 1 with the regions going to primordial black holes after inflation. We show that in our model the non-linear effects can lead to overproduction of the primordial black holes. \nPACS number(s): 98.80.Cq, 97.60.Lf, 98.70. Vc, 98.80.Hw", 'I. INTRODUCTION': "Starting from pioneering works by Zel'dovitch and Novikov [1], and also by Hawking [2], the primordial black holes (hereafter PBH's) were subject of extensive ivestigations. The presence of PBH's may significantly influence on physical processes and effects in the Universe (such as nucleosynthesis, CMBR spectral distortions, or distortions of γ -ray background radiation) due to Hawking effect [3], PBH's may be a component of dark matter (see e.g. [4], [5]). The formation of PBH's is determined by small scale, but large amplitude inhomogeneities in the Early Universe, and the processes of PBH's formation, evolution and decay link the physical conditions of Early Universe with conditions in the radiation-dominated epoch and present-day cosmology. Even the very absence of PBH's may significantly constraint the models of the beginning of cosmological evolution. \nUsually the processes of PBH's formation are associated with production of the scalar mode of perturbations during inflation (see e.g. [5-9]) or phase transitions in the Early Universe [10]. In this paper we are going to discuss the first possibility, which allows to use the powerful and well-elaborated theory of instability of the expanding Universe for analysis of conditions, under which PBH's can form. \nThe theory of generation of adiabatic perturbations during inflation started from pioneering papers [11-13]. It was established that the RMS-amplitude of metric perturbations δ rms is connected with the parameters of inflationary theory by means of relation \nδ rms = 1 2 π H 2 | ˙ φ | , 1 \nwhere H is the Hubble parameter, ˙ φ is the velocity of the field, evolving during inflation. To get PBH's abundance in an observable amount, one should have δ rms ∼ 10 -2 -10 -1 (see, e. g. [14]).On the other hand COBE CMBR data, as well as analysis of Large-Scale Structure formation constraint the amplitude of perturbations δ rms ∼ 10 -5 at super-large scales. Therefore to get PBH's one should increase the amplitude of the perturbations by a factor 10 3 -10 4 at small scales. Unfortunately this cannot be reached in the simplest inflationary models, since in these models δ rms logarithmically grows with increase of scale, and one should use nonstandard models having additional power at small scales to obtain significant PBH's amount. \nRecently, several models of such type were proposed. For instance, Carr and Lidsey [6] proposed toy model having blue type spectrum (the spectrum δ rms ( k ) ∝ k a , where k is the wavenumber, and a is the spectral index), and investigated the constraint on the spectral index a associated with possible PBH's formation in such model. Linde [15] has shown that blue type spectra can be naturally obtained in the two-field model of so-called hybrid inflation. \nAnother type of model having a spike in the power spectrum at some scale k bh was proposed by Ivanov, Naselsky and Novikov ([5], hereafter INN) 1 . They considered one-field inflationary model with inflaton φ and assumed that the potential V ( φ ) has a 'plateau' region at some scale k bh , and has a standard form (say, power-law form) outside the 'plateau' region. The field φ slows down in the 'plateau' region giving increase of the spectrum of perturbations at the scale k bh according to eqn. (1). One can adjust the parameters of 'plateau' region to obtain the desired increase of the spectrum, and consequently the desired PBH's amount. Garcia-Bellido et all [8] and also Randall et all [9] considered more realistic two field models having a saddle point in two-dimensional form of potential V ( φ, ψ ). Like the one-field model, the evolution of the system of fields slows down near the saddle point giving an increase of the spectrum power. Randall et all pointed out that such models solve several fine-tuning problems of the standard inflation, and therefore look very natural from the point of view of high energy physics. Garcia-Bellido et all carefully investigated the process of PBH's formation in such models (see also recent work by Yokoyama, [18]). \nIf the primordial black holes are not super-large, they probably collapse during the radiation dominated epoch of the evolution of the Universe. This means that the amplitude h ∗ of the metric inhomogeneities inside the regions going to PBH's should be of order of unity to overcome the strong pressure forces during collapse of the perturbed region [14]. These large amplitude metric inhomogeneities are assumed to be generated during inflation as rare events in the random field of the metric perturbations. Since the amplitude of the inhomogeneities h ∗ is rather large, the natural question appears: to what extent we can rely on the linear theory of perturbations which usually gives Gaussian probability distribution of PBH's formation? \nTo answer this question we can apply the formalism of coarse-grained fields (introduced by Starobinsky [19]) as an alternative approach to the linear theory that can describe large amplitude deviations of the field and the metric from the background quantities. According to this approach, the spatially inhomogeneous field φ ( /vectorx, t ) is divided into two parts: the large-scale part φ ls , which consists of the modes with physical wavelengths λ ∝ ak -1 greater than some characteristic scale λ c -g ≥ H -1 , and the small scale part which consists of modes with λ < λ c -g . During inflation, the physical wavelengths are stretched, and new perturbations are added to φ ls . This effect may be considered as a new random force f ( t ) in the equation of motion of the field φ ls , and usually the dynamics of φ ls is described in terms of diffusion equation for probability density Ψ( φ ls , t ). This equation was subject of a number of works in connection with problems of Quantum Gravity and Large-Scale Structure formation. Recently, it was pointed out, that this equation can be employed for calculations of the probability to find large amplitude peaks in the random distribution of field φ ls , and it was mentioned that such approach can be applied to the problem of PBH's formation [20]. \nHere we would like to note that when studying the effects originating after the end of inflation, such as PBH's formation, one should use the large scale part of metric instead of large scale part of field. Contrary to the field φ ls , the large scale part of the metric, namely the 'inhomogeneous scale factor a ls ( /vectorx )' (see eqns. (23 -24) for exact definition) is the quantity conserving during the evolution outside the horizon, and this property allows to connect the physical conditions during inflation with the physical conditions during radiation-dominated epoch, when PBH's are formed. Moreover, the criterion for PBH's formation can be directly formulated in terms of a ls ( /vectorx ) ( Refs. [21], [22]). Therefore, the calculation of a ls ( /vectorx ) gives a tool to describe quantitatively the generation of non-linear metric perturbations, and the evolution of these perturbations into PBH's. \nIn this paper we calculate the probability distribution function P ( a ls ( /vectorx )) in the model with almost flat region in the inflaton potential. The main idea of our calculations has already been applied in the models of so-called stochastic inflation (see, e. g. [23] and references therein), and is very simple. When the field φ ls evolves inside the plateau region it slows down, and the random kicks (described by the force f ( t )) significantly influence on its evolution. So, the trajectory of the field inside the plateau region becomes stochastic, and the time ∆ t that the field spends on the plateau, depends on the realization of the stochastic process. The total increase of the scale factor a ls during the field evolution on the plateau, is obviously determined by ∆ t : a ls ∝ e H ∆ t . Since different regions of the Universe separated by distances greater than H -1 evolve independently, the increase of a ls corresponding to different regions is determined by different realizations of the random process. Thus the scale factor a ls varies from one region to another after the field passes the plateau, that is the quantum effects generate the coordinate dependence of the scale factor. The shape of function a ls ( /vectorx ) is conserved during the subsequent evolution of the Universe until the scale of inhomogeneity crosses horizon at the second time. At that time, in the regions with significant contrast of a ls ( /vectorx ) the primordial black holes are formed. \nUsing the approach described above we calculate the probability distribution function P ( a ls ( /vectorx )). With help of a simple criterion of PBH's formation we relate P ( a ls ( /vectorx )) to the probability of PBH's formation. We show that in our \ncase the non-linear effects over-produce PBH's 2 . Although this result is very important qualitatively, it does not significantly change the estimate based on the linear theory. \nWe use the simple one-field model, proposed by INN (see also Refs. [24], [25]). Due to simplicity of this model the bulk of our results are obtained analytically. We hope that our approach provides a reasonable approximation to the case of more complicated two-field models. We are going to discuss these models in our future work. \nThe paper is organized as following. We introduce our model and discuss the classical dynamics of the metric and field in Section 2. In Section 3 we obtain an expression for P ( a ls ( /vectorx )). We consider the role of non-linear effects on the statistics of PBH's production in Section 4. We summarize our conclusions and discuss applicability of our approach in Section 5.", 'II. THE DYNAMICS OF CLASSICAL MODEL': "In this Section we consider the classical dynamics of spatially homogeneous parts of metric and field in the simplest inflationary model with a single scalar field (inflaton) and with the peculiarity in the inflaton potential. In this case the system of dynamical equations contains only two dynamical variables -scale factor a ( t ) and spatially homogeneous part φ 0 ( t ) of the field φ , and reduces to the Hamiltonian constraint equation \nH 2 = 8 π 3 ( V ( φ 0 ) + ˙ φ 2 0 2 ) , 2 \nand to the equation of motion for field φ 0 \n¨ φ 0 +3 H ˙ φ 0 + ∂ ∂φ V ( φ 0 ) = 0 , 3 \nwhere H = ˙ a a , and another symbols have their usual meaning. We hereafter use the natural system of units. We assume that the effective potential V ( φ ) has a small almost flat region ('plateau') between some characteristic values of field φ 1 and φ 2 . The potential is also assumed to be proportional to φ 4 outside the 'plateau' region \nV ( φ ) = λφ 4 4 4 \nat φ < φ 1 \nat φ 1 < φ < φ 2 , and \nV ( φ ) = ˜ λφ 4 4 6 \nat φ > φ 2 . Here V ( φ 1 ) = λφ 4 1 4 , ˜ λ = λ ( φ 1 φ 2 ) 4 + 4 A ( φ 2 -φ 1 ) φ 4 1 . As we will see below the size of the flat region is very small ∆ φ = φ 2 -φ 1 /lessmuch φ , A ( φ 2 -φ 1 ) V ( φ 1 /lessmuch 1 so we can set λ ≈ ˜ λ . At sufficiently large values of φ 0 > 1 the kinetic term in the equation (2) is negligible in comparison with the potential term \n˙ φ 2 0 2 /lessmuch V ( φ 0 ) , 7 \nand the equation (2) reduces to an algebraical relation between H and φ 0 (so-called slow-roll approximation) \nH = √ 8 π 3 V ( φ 0 ) . 8 \nFrom the equation (8) it follows that the Universe expands quasi-exponentially ( H ≈ const and a ∝ e Ht ) at φ 0 > 1. \nV ( φ ) = V ( φ 1 ) + A ( φ -φ 1 ) 5 \nIt can also be easily shown that outside the plateau region the field moves with large friction at φ 0 > 1, so \n| ¨ φ 0 | /lessmuch | 3 H ˙ φ 0 | . 9 \nThe friction dominated condition (9) helps to simplify the integration of the system (2 -3). Integrating the eqns. (2 -3) with help of inequalities (7), (9) at φ 0 > φ 2 , we have \nφ 0 ( t ) = ˜ φ 0 exp -( √ ˜ λ 6 π t ) , 10 \nand \na ( ˜ φ 0 ) = a 0 exp( N ( ˜ φ 0 ) -N ( φ 0 )) , 11 \nwhere ˜ φ 0 and a 0 are some initial values of the field and scale factor. \nN ( φ 0 ) = ∫ φ 0 φ 2 Hdt = π ( φ 2 0 -φ 2 2 ) 12 \nis the number of e-folds of the scale factor during the field rolling down starting from some initial value of φ and down to the field φ 2 . The similar formulae hold at φ end < φ 0 < φ 1 \nφ 0 ( t ) = φ 1 exp( -√ λ 6 π ( t -t 1 )) , 13 \na ( φ 0 ) = a 1 exp( N end ( φ 1 ) -N end ( φ )) , 14 \nwhere φ 0 ( t 1 ) = φ 1 and a 1 = a ( t 1 ), and N end ( φ 0 ) is the number of e-folds up to the end of inflation: N end ( φ 0 ) = π ( φ 2 0 -φ 2 end ), where we assume that inflation ends at standard (for λφ 4 theory) value of φ end = 1 √ 2 π . Note, that N end ( φ 1 ) should be rather large. For example, to get a feature in the spectrum at scales, corresponding to the solar mass, we should have N end ( φ 1 ) ∼ 50 -60. Therefore, the value of φ 1 should be greater than unity ( φ 1 ∼ 4 . 5 for N end ( φ 1 ) 60). \nNow let us consider the dynamics of inflaton in the 'plateau' region φ 1 < φ 0 < φ 2 . In this region the equation (3) is simplified to \n∼ \n¨ φ 0 +3 H 0 ˙ φ 0 + A = 0 , 15 \nwhere H 0 = √ 8 π 3 V 0 . The solution of eqn. (15) can be written as \nφ 0 = φ 2 + 1 3 H 0 ˙ φ in (1 -e -3 H 0 t ) -At 3 H 0 = φ 2 -1 6 πφ 2 (1 -e -3 H 0 t ) -At 3 H 0 , 16 \nand for the field velocity we have \n˙ φ 0 = ˙ φ in e -3 H 0 t -A 3 H 0 , 17 \nwhere ˙ φ in = ˙ φ 0 | φ 0 = φ 2 = -1 3 H 0 ∂ ∂φ V ( φ 2 ) = -√ ˜ λ 6 π φ 2 is the field velocity at the moment t =0 of entrance of the field in the 'plateau' region. The second term in the eqn. (16) and the first term in the eqn. (17) are due to inertial influence of initial velocity ˙ φ in , and the last terms in the both equations are due to nonzero slope of potential in the plateau region. The evolution of the field in the plateau region can be divided into two stages. At first stage the field evolves mainly due to inertial term, and velocity exponentially decreases with time. After some characteristic time t ∗ the nonzero slope of potential A starts to determine the evolution, the velocity tends to the constant value ˙ φ fd = -A 3 H 0 , and the field amplitude starts to decrease linearly with time. The time t ∗ can be estimated by equating the inertial and potential terms in the eqn. (16), and is determined by the condition 3 H 0 t ∗ e 3 H 0 t ∗ = B A , where B = ∂ ∂φ V ( φ 0 = φ 1 ). \nAs we discussed in Introduction, the spectrum amplitude is inversely proportional to the field velocity ( δ rms ≈ 1 2 π H 2 | ˙ φ | ), therefore we need to slow down the velocity approximately by ∼ 10 3 -10 4 times to get the increase of the spectrum \namplitude from the initial value δ rms ( in ) = 1 2 π H 3 B ∼ 10 -5 up to the typical for PBH production δ rms ∼ 10 -2 -10 -1 . For that, we should fix the 'amplification' parameter α = B A 10 3 -10 4 . \n∼ \n-Our model has two possible limiting variants depending on the relation between the time t c of the crossing of plateau region by the field φ 0 ( φ 0 ( t c ) = φ 1 ) and t ∗ . If t c ≈ t ∗ the field crosses the plateau mainly due to inertia. In this case the parameter α determines the number of e-folds during plateau crossing δN ≈ H 0 t c ≈ 1 3 ln α ≈ 2 . 3, and therefore the width of produced bump in the spectrum remains small and fixed. The model of similar type was discussed by INN. Here we consider another possible case t c > t ∗ , where the field spends some time on the plateau, evolving in the friction-dominated approximation. In this case the width of the spectrum is determined by the value of t c , which is the free parameter of our model. Instead of t c we will parameterize our model by the quantity γ -the ratio of wave numbers, corresponding to the fields φ 1 , φ 2 , respectively, t c = H -1 0 ln γ . The parameter γ cannot be too small γ > α 1 / 3 and we take γ ≈ 10 3 in the estimations. If γ is not extremely large ln γ /lessmuch N ( φ 1 ), the size of plateau ∆ φ 0 = φ 2 -φ 1 is of order of typical size ∆ φ ∗ = B 9 H 2 . The typical relative size of plateau is very small \n∆ φ 0 φ 0 = 1 6 πφ 2 0 ≈ 1 6 N ( φ 1 ) ≈ 0 . 003 . 18 \nThus, the correction due to the presence of plateau practically does not influence on the dynamics of the field outside plateau region and we can set λ = ˜ λ . On the other hand, the size of plateau is much greater than H 0 - the typical size of quantum fluctuations, ∆ φ ∗ = H 0 6 πδ rms ( in ) ∼ 10 5 H 0 . \nTypically, the estimate ∆ φ 0 φ 0 /lessmuch 1 holds for arbitrary power-low potentials V ( φ ) ∝ φ p provided power p is not very large. However the opposite limiting case is also possible. For example, Bullock and Primack [20] proposed the potential of the form \nV ( φ ) = λ bp (1 + arctan ( φ )) , φ > 0 \nV ( φ ) = λ bp (1 + 4 ∗ 10 33 φ 21 ) , φ < 0 19 \nwhere the constant λ bp = 6 ∗ 10 -10 is chosen to normalize the large-scale part of spectrum to the RMS-amplitude ≈ 3 ∗ 10 -5 . The flat region in this potential starts from φ = 0 and ends at φ = -1 . 23 ∗ 10 -2 , and inflation ends itself at φ = φ end = -1 . 55 ∗ 10 -2 . It was mentioned by Bullock & Primack that this potential leads to strongly non-Gaussian statistics of field perturbations.", 'III. NON-LINEAR METRIC PERTURBATIONS FROM THE QUANTUM DYNAMICS OF COARSE-GRAINED FIELD': "It is well known that there are two equivalent ways to describe inhomogeneous Universe. The first way is to consider inhomogeneities as a small corrections to the homogeneous space-time and study them in the frameworks of linear theory of perturbations. Another approach splits the metric and the field into large-scale part (coarse-grained over some scale greater than horizon scale), and small-scale part. During inflation, the dynamical equations for coarsegrained field φ ls and coarse-grained scale-factor a ls are equivalent to eqns. (3 , 8) provided the quantum effects are switched off. The quantum effects continuously produce new inhomogeneities of random amplitude with scales greater than the scale of coarse-graining. These inhomogeneities should be added to φ ls and a ls and effectively this leads to the presence of stochastic force term in the equations of motion. Therefore, the dynamics of coarse-grained variables can be described in terms of the distribution functions of φ ls and a ls , and in principal these distribution functions can provide the same information as the power spectrum of perturbations, and furthermore the coarse-grained formalism gives a tool for description of the metric perturbations with amplitude, greater than 1. \nThe effective dynamical equation for the field φ ls has the form [19] 3 \n¨ φ ls +3 H ls ˙ φ ls + ∂ ∂φ V ( φ ls ) = D 1 / 2 f ( t ) , 20 \nwhere D = 9 H 5 ls (2 π ) 2 , and f ( t ) is delta-correlated random force, < f ( t 1 ) f ( t 2 ) > = δ ( t 1 -t 2 ). The equation for coarsegrained scale factor a ls remains unchanged \nH ls = √ 8 π 3 V ( φ ls ) . 21 \nThe solution of the set of eqns. (20 , 21) is extremely difficult problem, and can be done under some additional simplifying assumptions. For example if we choose the featureless potential, and consider the friction-dominated solutions of the eqn. (20), we can obtain the solutions describing self-reproduced inflationary Universe (provided the stochastic term in (20) dominates over potential term, see for example Linde [23]). In our case we cannot use the friction-dominated condition in the beginning of the field evolution inside the plateau region. However we can adopt another simplifying assumptions: first we can set H ls = H 0 = const inside and near the plateau region, and second, we can omit the stochastic term in the eqn. (20) outside the plateau region, assuming the field moves along the classical trajectory there. Under these assumptions the statistics of the scale factor a ls is totally determined by the time ∆ t that field φ ls spends in the plateau region \n∆ N = ln ( a out /a in ) = H 0 ∆ t, 22 \nwhere a in is the value of scale factor at the time t = 0 of entrance of the field in the plateau region, and a out corresponds to the moment ∆ t , when the field leaves the plateau region. To see that let us consider the evolution of the scale factor a ls in the comoving coordinate system. Outside the horizon the hypersurfaces of constant comoving time t com practically coincide with hypersurfaces of constant energy density /epsilon1 = const . On the other hand, the field φ ls evolves slowly during inflation and hypersurfaces of constant energy density are close to hypersurfaces φ ls = const , and therefore we can put a ls ( t com ) = a ls ( φ ls ). After the field passes the plateau region, the evolution of a ls ( φ ls ) can be described by the standard expression (14), so we have \na ls ( φ ls ) = a in exp( π ( φ 2 1 -φ 2 ls ) + ∆ N ) , 23 \nwhere ∆ N is nearly constant inside of coarse-grained regions with comoving scale λ c -g ≈ a out H -1 0 , but changes from one region to another. Thus, the metric outside horizon has the quasi-isotropic form \nds 2 = dt 2 -a 2 ls ( φ 0 ) a ls ( /vectorx ) δ i j dx i dx j , 24 \nwhere we represent the scale factor a ls ( φ ls ) as a multiplication of two factors: a ( φ 0 ) and a ls ( /vectorx ) ≡ e ∆ N . Here a ls ( φ 0 ) and φ 0 ( t ) are determined by the classical equations (13), (14), and the spatial coordinates /vectorx are coarse-grained over the regions with scale λ c -g . To estimate the change of metric from one region to another quantitatively, we introduce the definition of non-linear metric perturbation \nh ≡ a ls ( φ ls ) -a ( φ 0 ) a ( φ 0 ) = exp H 0 (∆ t -t c ) -1 25 \n(remind, that t c = H -1 0 ln γ is the time which the field spends in the plateau region moving along the classical trajectory when the stochastic term in (20) is switched off). Note, that in the limit of small h /lessmuch 1, the metric assumes the form \nds 2 = dt 2 -a 2 ( φ 0 )(1 + 2 h ( /vectorx )) δ i j dx i dx j , 26 \nand the definition (25) is reduced to the standard expression for growing mode of adiabatic perturbation outside the horizon. Namely, in this case h reduces to gauge independent quantities, introduced by a number of authors [11-13], [27] up to a constant factor. The variables (25 , 26) do not depend on time outside the horizon. Therefore, using of these variables is very convenient to match the perturbations, generated during inflation with the perturbations, crossing horizon at the normal stage of the Universe evolution. As one can see from (25) the metric perturbations are determined by stochastic variable ∆ t and the distribution of ∆ t must follow from the solution of eqn. (20). Note, that the definition of non-linear metric perturbations should be taken with a caution. In principal, one can use another definition relating to (25) by some non-linear transformation, and having the same limit (26) in the case of small h . For example, Bond and Salopek [28] used the quantity ˜ h = ln ( a ls ( φ ls ) a ( φ 0 ) ) to define non-linear metric perturbations. However, the criterion for PBH's formation can be directly expressed in terms of the quantity (25) (see next Section), and therefore this quantity is the most natural variable for our purposes. \nAlthough the assumption of constant H 0 greatly simplifies the problem it still remains rather complicated for a simple analytical treatment 4 \nFor further progress we have to make some additional assumptions. We will consider below the plateau region of sufficiently large size. For this case the field approaches to the end of plateau in the friction-dominated approximation, which greatly simplifies the treatment of diffusion processes. To estimate the relevance of friction-dominated approximation we should compare the time t c and the time t ∗ ∼ ln ( α ) of the decay of the inertial term ¨ φ in the eqns. (15 -17 , 20). If t c > t ∗ and therefore γ /greatermuch α 1 / 3 , the inertial term in these equations can be neglected at t ∗ < t < t c . In this regime the solution of the classical equation of motion (15) has the form \nφ 0 ( τ ) ≈ φ 2 -aτ, 27 \nand the equation (20) becomes \ndφ ls dτ + a = d 1 / 2 f ( τ ) , 28 \nwhere β = 3 H 0 , and we introduce the dimensionless time τ = βt , a = A/β 2 and d = D 2 β 3 = H 2 0 24 π 2 . The stochastic equation (28) is associated with simple diffusion type equation, describing the evolution of positions probability distribution Ψ( τ, φ ) \n∂ Ψ ∂τ = d ∂ 2 ∂φ 2 Ψ+ a ∂ ∂φ Ψ , 29 \nNow we assume that the distribution Ψ is not spread out sufficiently before τ ∗ = βt ∗ and take δ -distributed Ψ function at the moment τ = τ ∗ as the initial condition for our problem \nΨ( τ ∗ ) = δ ( φ ls -φ ∗ ) , 30 \nwhere φ ∗ = ∆ φ -aτ ∗ is the value of field corresponding to the beginning of 'friction-dominated' part of plateau region 5 . \nTogether with initial condition (30) we should specify the boundary condition at φ ls = φ 1 . This condition depends on the form of the transition layer between the plateau region and the part of potential with steep slope ∂ ∂φ V ( φ ) = B . We assume this transition to be sharp, and therefore set the condition of absorbing wall at the downstream point φ ls = φ 1 \nΨ( φ 1 , τ ) = 0 , 31 \nNote, that this boundary condition was used by Aryal and Vilenkin [32] for analysis of stochastic inflation in the theory with top-hat potentials. In that paper it was shown that the more reasonable smooth transitions between the flat and steep regions of the potential are unlikely to modify significantly the resulting distribution. \nIn our case the probability density P ( τ ) of time τ relates to the solution of eqn. (27) as \nP ( τ ) = S | φ ls = φ 1 = d ∂ ∂φ Ψ , 32 \nwhere we define by S the probability current S = d ∂ ∂φ Ψ + a Ψ. The conservation of the probability current allows to estimate the correction term to eqn. (32) due to nonzero Ψ( φ 1 ). Assuming that field moves along the classical trajectory after φ ls = φ 1 , we have S ( -φ 1 ) ≈ B/β 2 Ψ ≈ S (+ φ 1 ) ≈ d ∂ ∂φ Ψ. Therefore the correction to the expression (32) is β 2 a B = α -1 ∼ 10 -3 -10 -4 times smaller than the leading term. \nThe conditions (30 , 31) determine the solution of eqn. (29). This solution can be found by standard methods of the theory of diffusion equations (see, e. g. Ref. [33]), and in our case has the form \nΨ( φ, τ ) = 1 √ 4 πd ( τ -τ ∗ ) exp {-1 4 d ( τ -τ ∗ ) ( φ -φ ∗ + a ( τ -τ ∗ ) 2 } \n(1 -exp {-1 d ( τ -τ ∗ ) ( φ -φ 1 )( φ ∗ -φ 1 ) } ) , 33 \nSubstituting (33) to the equation (32) we find the explicit expression for P ( τ ) \nP ( τ ) = 1 (4 πd ( τ -τ ∗ ) ( φ ∗ -φ 1 τ -τ ∗ ) exp {-1 4 d ( τ -τ ∗ ) ( φ 1 -φ ∗ + a ( τ -τ ∗ )) 2 } . 34 \nThe expression for probability distribution of metric can be readily obtained from (34). Using the eqn. (22 -25) to express the time τ in terms of h , taking into account the eqn (27) and the definitions of a , d , and assuming a > 0, we obtain \nP ( h ) = 1 √ 2 πδ 2 pl N cl N st 3 / 2 dN st dh exp {-( N st -N cl ) 2 2 δ 2 pl N st } , 35 \n√ \nwhere δ pl = 3 H 3 0 2 πA = αδ rms ( in ) is the standard metric amplitude calculated for the plateau parameters, and \nN cl = ln γ -τ ∗ / 3 , N st = ln (1 + h ) + N cl 36 \nare the numbers of e-folds for the classical path φ 0 ( t ) and for a random path φ ls ( t ), which start at φ ∗ = φ ( t ∗ ) and end at φ 1 . \nWhen the perturbations are small N st -N cl ≈ h /lessmuch 1, the distribution (35) has the standard Gaussian form \nP ( h ) = P G ( h ) = 1 √ 2 πδ 2 pl N cl exp {-h 2 2 δ 2 pl N cl } , 37 \nand in the opposite case of very large metric perturbations h /greatermuch 1 and N st ∼ ln h > N cl the distribution P ( h ) deviates sharply from the Gaussian law and has the power-law form \nP ( h ) ∝ h 3 / 2+ δ -2 pl / 4 , 38 \nAs seen from eqns. (35 -38), the non-Gaussian effects over-produce the metric perturbations of high amplitude in our model. To understand this fact, let us discuss the origin of non-Gaussian effects in our model. There are two sources for such effects. First, note that the 'effective dispersion' σ 2 eff = δ 2 pl N st in eqn. (35) depends itself on the value of the stochastic variable N st . Qualitatively, it can be explained as follows. In the linear theory the dispersion σ 2 = δ 2 pl N cl is proportional to the time spent by the classical background field φ 0 on the plateau. In non-linear theory the coarse-grained field φ ls ( t ) plays the role of background field, and therefore the distribution of the family of neighboring to φ = φ ls ( t ) paths should be described in terms of the probability distribution with dispersion σ 2 eff , which is proportional to the time spent by field φ ls on the plateau. Second, the amplitude of large metric perturbations h depends on N st exponentially ( h ∼ e N ls ), so order of magnitude increase of N st leads to exponential increase of h . Obviously, these two effects increase the probability of large amplitude metric perturbations.", 'IV. PROBABILITY OF BLACK HOLES FORMATION': "Although the distribution (35) provides very important information about the geometry of spatial part of metric outside horizon, it cannot be directly applied to the estimates of PBH's formation. Indeed, the distribution (35) is formed by the field inhomogeneities with wave-numbers k in the range (∆ k = [ k min ≈ a in H 0 < k < k max ≈ a out H 0 ]). The process of PBH formation is determined mainly by the field modes with wave-numbers ( δk ≈ k bh /lessmuch ∆ k ), where k bh is the typical PBH wavenumber. The modes with k < k bh compose the large-scale background part of metric at the moment of PBH formation, and do not influence on the formation of PBH's significantly. The modes with \nk > k bh lead to high-frequency modulation of the perturbation with k ∼ k bh , which is also unimportant, provided the mode with k ∼ k bh crosses the horizon second time at the radiation-dominated epoch. Therefore, in order to obtain the probability of PBH's formation, we should subtract the contribution of the large-scale and small-scale metric perturbations. \nIn general it is very difficult to separate the perturbations of a given scale in the frameworks of non-linear approach. However, we can estimate the probability density of the perturbations, corresponding to the smallest scale k bh ≈ a out H 0 6 . For that we simply put N cl = 1 in eqns. (35 , 36), assuming that the random process starts when the mode with wavenumber k 1 = e -1 a out H 0 crosses horizon. This procedure automatically subtracts the large-scale contribution of modes with k < k 1 . The small-scale contribution is also absent due to our absorbing boundary condition. We have \nP ( h ) = 1 √ 2 πδ 2 pl 1 ( x +1) 3 / 2 exp {-x 2 2 δ 2 pl ( x +1) } 39 \nfrom the eqn. (35), where x = ln(1 + h ), and in the limit of small h we obtain again the Gaussian distribution \nP ( h ) ≈ P G ( h ) = 1 √ 2 πδ 2 pl exp {-h 2 2 δ 2 pl } . 40 \nThe distribution (39) has nonzero first momentum M 1 = ∫ ∞ -1 dhh P ( h ) = 3 2 δ pl (the lower limit of integration should be -1, since the metric perturbations with h < -1 are cut off). The contribution of M 1 should be added to the background part of metric, and further we will use the renormalized metric perturbation h r = h -3 2 δ pl instead of h . The probability to find the metric perturbations h r with amplitude greater than some threshold value h ∗ P ( h ∗ ) = ∫ ∞ h ∗ dh P ( h ) can be estimated as \nP ( h ∗ ) ≈ 1 √ 2 π ( 2 δ pl ( x ∗ +1) 1 / 2 x ∗ ( x ∗ +2) ) exp {-x 2 ∗ 2 δ 2 pl ( x ∗ +1) } , 41 \nwhere x ∗ = ln(1+ 3 2 δ pl + h ∗ ), and we assume h ∗ /greatermuch δ pl . The same quantity, but calculated for the Gaussian distribution takes the well-known form \nP G ( h ) ≈ 1 √ 2 π δ pl h ∗ exp {-h 2 ∗ 2 δ 2 pl } . 42 \nThe observed quantities (such as, e.g. the matter density of PBH's in different cosmological epochs) can be easily expressed in terms of the probability P ( h ∗ ), provided the mass of PBH's and some criterion for PBH's formation are fixed. In our case the criterion for PBH's formation should give the information about the threshold value h ∗ . Since this criterion plays very important role, let us discuss it in some details. First let us note that PBH's are formed from high amplitude peaks in the density distribution which are approximately spherically-symmetric (see e.g. Ref. [34]). It can also be easily shown that the maxima in the matter density correspond to the maxima in the function a ls ( /vectorx ). The form of a ls ( /vectorx ) totally specifies the number of regions going to PBH's as well as dynamics of the collapsing regions. Therefore we formulate the criterion of PBH's formation in terms of conditions imposed on the function a ls ( /vectorx ). \nThe first criterion was formulated by Carr in his seminal paper [14]. It was shown that an over-dense region forms PBH if the density contrast at the horizon scale δρ ρ lies approximately within the limits 1 3 < δρ ρ < 1. The first part of this inequality tells that the over-dense region should stop expansion before the scale of the region crosses the sound horizon. The second part requires that the over-dense region does not collapse before crossing the causal horizon, and consequently the perturbation does not produce a closed world separated from the rest of the Universe. Then the criterion for PBH's formation was improved by Nadegin, Novikov and Polnarev [21] (hereafter NNP), and also by Biknell and Henriksen [22] with help of numerical computations. The initial condition used by NNP was chosen as a non-linear metric perturbation having the form of a part of the closed Friedman Universe matched with the spatially flat Universe through an intermediate layer of negative density perturbation. The conditions for PBH's formation depend on the size of this part (i.e. the amplitude of the perturbation), as well as on the size of the matching layer. \nThe smaller matching layer is, the larger the pressure gradients needed to prevent collapse will be. Therefore, the amplitude of the perturbation forming PBH must be greater in the case of narrow intermediate layer. In terms of our function a ( /vectorx ) the NNP criterion reads \nh ∗ ≡ a + a --1 > 0 . 75 -0 . 9 43 \nwhere a + is the value of a ( /vectorx ) at the maximum of the perturbation and a -is the same quantity outside the perturbed region 7 . The first number on the right hand side of (43) corresponds to the matching layer of size comparable with size of the over-dense region, and the second number corresponds to the narrow matching layer. Assuming the matching layer to be sufficiently large we take h = 0 . 75 as a criterion of PBH's formation. \nOnce the criterion is specified, we can link the desired PBH's abundance β ( M pbh ) ≈ P ( h ∗ pbh ) with the parameters of our model. For instance, consider the model having the matter density of PBH's equal to the critical one (the density parameter Ω pbh = 1). In this model we have [3], [6] \n∗ \nβ ( M ) = 10 -8 ( M M /circledot ) 1 / 2 . 44 \nEquating the expression (44) to the probability function (39), we have the equation determining the amplitude δ 1 pl required for PBH's abundance (44) as a function of M pbh \nP ( h ∗ pbh , δ 1 pl ) = β ( M pbh ) , 45 \nand equating the expressions (42) and (44) we obtain the analogous equation for determining the reference amplitude δ 2 pl when the non-Gaussian effects are switched off. The solution of these equations is given in Fig. 1. \nFIG. 1. We plot the dependence of plateau parameter δ pl on PBH's mass M pbh assuming that the PBH's abundance is given by the eqn. (44). The solid line represents the solution of eqn. (45) (i.e we calculate δ pl taking into account the non-Gaussian effects in this case). The dashed line represents δ pl calculated in the standard Gaussian theory. The PBH's masses lie in the range: 10 -18 M /circledot < M pbh < 10 6 M pbh . The PBH's of the mass 10 -18 M /circledot ∼ 10 15 g should be evaporated at the present time. Actually, the abundance of these PBH's is limited much stronger than is assumed in our calculations. \n<!-- image --> \nOne can see from this Fig. that the quantities δ 1 pl and δ 2 pl increase with increasing of M pbh and δ 1 pl is always smaller than δ 2 pl . It means that non-Gaussian effects over-produce PBH's in our model (at least when the simple criterion (43) is used), and the slope of potential can be steeper than that required in the Gaussian case. Typically, the ratio δ 2 pl δ 1 pl is about 1 . 5. Say, for the case of M pbh = M /circledot , we have δ 1 pl ( M /circledot ) ≈ 0 . 089 and δ 2 pl ( M /circledot ) ≈ 0 . 134. We plot the probability function P ( h ) for δ 1 pl ( M /circledot ) = 0 . 089 in Fig. 2. \nFIG. 2. The dependence of probability density P ( h ) on the metric amplitude h . The non-Gaussian curve (solid line) is calculated with help of eqn. (39) assuming PBH's abundance β ( M /circledot ) ≈ 10 -8 . That gives δ 1 pl ( M /circledot ) ≈ 0 . 089. The dashed line is the reference Gaussian probability density calculated for the same abundance. For that curve we have δ 2 pl ( M /circledot ) ≈ 0 . 134. The dotted curve represents the Gaussian distribution taken with δ 1 pl ( M /circledot ) ≈ 0 . 089. This distribution strongly under-produces PBH's, and in this case we have β ∼ 10 -17 . \n<!-- image --> \nIn this Fig., we also plot the Gaussian probability function P G ( h ) for δ 2 pl ( M /circledot ) = 0 . 134 (dashed line) and the same quantity for δ 1 pl ( M /circledot ) = 0 . 089 (dotted line). Comparing the curves that correspond to the same PBH's abundance, we see that the non-Gaussian curve is flatter having larger values of P ( h ) at large h . The values of the Gaussian curve with the same plateau parameter δ 1 pl ( M /circledot ) is smaller by many orders of magnitude than the values of the non-Gaussian curve in the case of large h . \nFinally, let us note, that the non-Gaussian effects does not modify significantly the estimates based on the Gaussian theory. As we have seen, the ambiguity in the choice of the plateau slope due to these effects is about 1 . 5. This ambiguity seems to be less than the ambiguity in other parameters and can be obviously absorbed by a small change of the potential slope.", 'V. DISCUSSION': "We have demonstrated that the non-Gaussian effects related to the dynamics of the coarse-grained field (inflaton) and to the evolution of the large-scale part of metric over-produce large-amplitude inhomogeneities of the metric compared to the prediction of the Gaussian (linear) theory of perturbations. We have derived an analytical expression for non-Gaussian probability distribution for non-linear metric perturbations, and estimated the influence of non-linear effects on the probability of primordial black holes formation. We used the simple single field inflationary model with peculiarity in form of the flat region in inflaton potential V ( φ ), and power-law slope of the potential outside the peculiarity region. The key point of our approach is in the using of inhomogeneous coarse-grained metric function a ( /vectorx ) instead of the coarse-grained field φ ls as a basic quantity. This allows to match the physical condition of production of inhomogeneities during inflation with the 'observable' quantities. \nOur results can be considered as semi-qualitative only. The uncertainties come fromthe phenomenological character of our inflationary model as well as from the oversimplified treatment of the process of PBH's formation. The uncertainties related to the choice of parameters of inflationary model are mainly due to unknown form of the potential between the steep and flat regions, and also due to our friction-dominated assumption in the consideration of the stochastic process. These uncertainties can be eliminatedwith help of numerical simulations of stochastic process in more realistic models of inflation. The ambiguities concerning the criterion of PBH's formation are mainly due to the one-point treatment of this process. Actually, PBH formation is nonlocal, and dynamics of collapsing region depends strongly on the form of the spatial profile of the density perturbation (see e.g. Refs. [22], [35] for discussion of this point). The form of the spatial profile can be studied by means of n-point correlation functions of the coarse-grained metric and field. Unfortunately, the formalism of n-point correlation functions is still not elaborated (see, however the Ref. [35] for the first discussion). Note, that probably the influence of the spatial profile of the collapsing region may be taken into account by a redefinition of the threshold value h ∗ , and this value might be effectively less. In this case the role of the non-linear effects would be damped. \nFinally we would like to note that the form of the distribution (35) does not depend explicitly on the specific parameters of our model. This allows to suppose that similar distributions can be obtained in more complicated \nmodels, say, in two-field models proposed in Refs. [8], [9]. We are going to check this very interesting assumption in our future work.", 'Acknowledgments': "The author acknowledges P. Naselsky and I. Novikov for many valuable discussions, and also A. Beloborodov, A. Dolgov and D. Markovic for useful comments. This work was supported in part by the Danish Research Foundation through its establishment of the Theoretical Astrophysic \n- [1] Ya. B. Zeldovich, I.D. Novikov, Astron. Zh. 43 , 758 (1966) [Sov. 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D 54 , 6040 (1996).\n- [9] L. Randall, M. Soljacic, A. Guth, Nucl. Phys. B472 , 377 (1996).\n- [10] M. Crawford, D. N. Schramm, Nature (London) 298 , 538 (1982); L. J. Hall, S. D. Hsu, Phys. Rev. Lett. 64 , 2848 (1990); K. Jedamzik, Phys. Rev. D 55 , 5871 (1997); A. Dolgov, J. Silk, Phys. Rev. D 47 , 4244 (1993).\n- [11] V. N. Lukash, Zh. Eksp. Teor. Fiz. 79 , 1601 (1980) [Sov. Phys. JETP 52 , 807 (1980)].\n- [12] S. W. Hawking, Phys. Lett. 115B , 295 (1982); A. A. Starobinsky, Phys. Lett. 117B , 175 (1982); A. Guth and S. Y. Pi, Phys. Rev. Lett. 49 , 1110 (1982); J. M. Bardeen, P. J. Steinhardt, and M. S. Turner, Phys. Rev. D 28 , 679 (1983).\n- [13] V. F. Mukhanov, Sov. Phys. JETP 67 , 1297 (1988).\n- [14] B. J. Carr, Astrophys. J. 201 , 1 (1975).\n- [15] A. D. Linde, Phys. Lett. 259B , 38 (1991); A. D. Linde, Phys. Rev. D 49 , 748 (1994).\n- [16] H. Hodges, G. R. Blumenthal, Phys. Rev. D 42 , 3329 (1990); H. M. Hodges, G. R. Blumenthal, L. Kofman, and J. R. Primack, Nucl. Phys. B335 , 197 (1990).\n- [17] R. Kates, V. Muller, S. Gottlober, J. P. Muket, J. Retzlaff, Mon. Not. R. Astron. Soc. 277 , 1254 (1995).\n- [18] J. Yokoyama, report No. astro-ph/9604152, 1996 (unpublished).\n- [19] A. A. Starobinsky, in Fundamental Interactions , edited by V. N. Ponamarev (Moscow; MGPI Press, 1984), p. 55; A. A. Starobinsky, in Current Topics in Field Theory, Quantum Gravity, and Strings , proceedings of the V I conference, Meudon and Paris, edited H. J. de Vega and N. Sanchez (Lecture Notes in Physics, Vol. 246) (Springer, New York, 1986), p. 107.\n- [20] J. S. Bullock, J. R. Primack, Phys. Rev D 55 , 7423 (1997).\n- [21] D. K. Nadezhin, I. D. Novikov, and A. G. Polnarev, Astron. Zn. 55 , 216 (1978) [Sov. Astron. 22 , 129 (1978)].\n- [22] G. V. Bicknell, R. N. Henriksen, Astrophys. J. 232 , 670 (1978).\n- [23] A. D. Linde, Particle Physics and Inflationary Cosmology (Harwood, Switzerland, 1990).\n- [24] A. A. Starobinsky, JETP Lett. 55 , 489 (1992).\n- [25] M. Demianski, P. Ivanov, and D. I. Novikov, Phys. Rev. D 50 2488 (1994).\n- [26] O. Buryak, Phys, Rev. D 53 1763 (1996); S. A. Ramsey, B. L. Hu, report No. hep-ph/9706207 (unpublished).\n- [27] V. N. Lukash, I. D. Novikov, in Observational and Physical Cosmology , ed. F. Sanchez, M. Collados, and R. Rebolo, (Cambridge University Press, 1992), p.1.\n- [28] J. R. Bond, D. S. Salopek, Phys. Rev. D 42 , 3936, (1990).\n- [29] M. C. Wang, G. E. Uhlenbeck, Rev. of Mod. Phys. 17 , 323 (1945).\n- [30] T. W. Marshall, E. J. Watson, J. Phys. A18 , 3531 (1985); T. W. Marshall, E. J. Watson, ibid . A20 , 1345 (1987).\n- [31] M. E. Widder, U. M. Titulaer, J. Stat. Phys. 56 , 471 (1989); P. S. Hagen, G. R. Doering, and C. D. Levermore, J. Stat. Phys. 54 , 1321 (1989).\n- [32] M. Aryal, A. Vilenkin, Phys. Lett. 199B , 351 (1987).\n- [33] S. Chandrasekhar, Rev. of Mod. Phys. 15 , 1 (1943).\n- [34] A.G. Doroshkevich, Afz. 6 , 581 (1970); J. M. Bardeen, J. R. Bond, N. Kaizer, and A. S. Szalay, Astrophys. J. 304 , 15 (1986). \n- [35] N. A. Zabotin, P. D. Naselsky, Astron. Zh. 59 , 647 (1982) [Sov. Astron. 26 , 395 (1982)].\n- [36] A. A. Starobinsky, J. Yokoyama, Phys. Rev. D 50 , 6357 (1994)."}
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2001ApJ...551...72K
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Systematic Errors in the Estimation of Black Hole Masses by Reverberation Mapping
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2001-01-01
| 4 | 0.46 | 159 |
['black hole physics', 'galaxies active', 'galaxies nuclei', 'galaxies seyfert', 'methods data analysis', 'astrophysics']
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The mass of the central black hole in many active galactic nuclei has been estimated on the basis of the assumption that the dynamics of the broad emission line gas are dominated by the gravity of the black hole. The most commonly employed method is to estimate a characteristic size-scale r<SUB>*</SUB> from reverberation mapping experiments and combine it with a characteristic velocity v<SUB>*</SUB> taken from the line profiles; the inferred mass is then estimated by r<SUB>*</SUB>v<SUP>2</SUP><SUB>*</SUB>/G. We critically discuss the evidence supporting the assumption of gravitational dynamics and find that the arguments are still inconclusive. We then explore the range of possible systematic error if the assumption of gravitational dynamics is granted. Inclination relative to a flattened system may cause a systematic underestimate of the central mass by a factor of ~(h/r)<SUP>2</SUP>, where h/r is the aspect ratio of the flattening. The coupled effects of a broad radial emissivity distribution, an unknown angular radiation pattern of line emission, and suboptimal sampling in the reverberation experiment can cause additional systematic errors as large as a factor of 3 or more in either direction.
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https://arxiv.org/pdf/astro-ph/0012134.pdf
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{'Systematic Errors in the Estimation of Black Hole Masses by Reverberation Mapping': 'Julian H. Krolik \nDepartment of Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218', 'ABSTRACT': 'The mass of the central black hole in many active galactic nuclei has been estimated on the basis of the assumption that the dynamics of the broad emission line gas are dominated by the gravity of the black hole. The most commonly-employed method is to estimate a characteristic size-scale r ∗ from reverberation mapping experiments and combine it with a characteristic velocity v ∗ taken from the line profiles; the inferred mass is then estimated by r ∗ v 2 ∗ /G . We critically discuss the evidence supporting the assumption of gravitational dynamics and find that the arguments are still inconclusive. We then explore the range of possible systematic error if the assumption of gravitational dynamics is granted. Inclination relative to a flattened system may cause a systematic underestimate of the central mass by a factor ∼ ( h/r ) 2 , where h/r is the aspect ratio of the flattening. The coupled effects of a broad radial emissivity distribution, an unknown angular radiation pattern of line emission, and sub-optimal sampling in the reverberation experiment can cause additional systematic errors as large as a factor of 3 or more in either direction.', '1. Introduction': "The constant variability of AGN lends itself to employment as a diagnostic of their internal structure. Most notably, because the ionizing continuum drives the optical/UV emission lines, the time-lag between fluctuations in the continuum and fluctuations in the lines can be used to constrain the distance between the two emission regions under the assumption that it simply represents light travel-time. This program, called 'reverberation mapping' by analogy with the seismic techniques used in oil exploration, has been extensively implemented, especially in the past decade (see, for example, the reviews in Gondhalekar et al. 1994). \nRecently there have been a number of attempts to combine reverberation-mapping measurements of the broad emission line size scale r ∗ with spectroscopic measurements of the line width v ∗ in order to estimate the mass of the central black hole (Ho 1999, Wandel et al. 1999; Kaspi et al. 2000). These two quantities can be combined to determine the central mass if the spread of projected velocities seen in the line width is due to motions controlled by the gravity of the central black hole. If that is the case, the central mass M ∼ r ∗ v 2 ∗ /G . \nThe logic of the argument depends crucially on our confidence that gravity truly dominates the dynamics of the emission line material. If the estimate M ∼ r ∗ v 2 ∗ /G is to be meaningful, we must be able to rule out significant influence from other forces such as radiation pressure or magnetic fields. The quality of the evidence so far will be discussed in § II. \nIf we grant that gravity is the most important force, the next step is to change the ' ∼ ' in the relation to an '='; i.e., writing M = qr ∗ v 2 ∗ /G , we must evaluate the coefficient q as accurately as possible. In many efforts hitherto (e.g. Wandel et al. 1999, Kaspi et al. 2000), q has been implicitly taken to be the value appropriate to sources following isotropically-oriented circular orbits at radius r ∗ . This value is designated the 'virial' mass. As we shall show, however, there can be a sizable ratio between the true mass and this 'virial' estimate. Several factors combine to determine this ratio. Many of these have been mentioned and discussed qualitatively in the literature (e.g., Wandel et al. 1999, Kaspi et al. 2000), but few have been studied quantitatively. A proper evaluation of the relation between the 'virial' mass and the true mass is important because the 'virial' mass is often compared with other mass estimates (Ho 1999; Wandel 1999; Kaspi et al. 2000; Gebhardt et al. 2000c). \nThe first problem to consider is that there may be a delay between variations in the ionizing continuum (which is genuinely responsible for driving line emission) and the associated variations in the continuum band that is actually observed. Any such delay would artificially enlarge r ∗ by an amount equal to c times the delay. There are some observational indications that delays of this sort exist, but they appear (if real) to be short compared to the continuum-line delay (Collier et al. 1998). \nNext, if all the line emission occurred within a thin spherical shell at radius r , for a fixed definition of v ∗ in terms of the line profile, the parameter q can vary over a wide range depending on the orbital shape distribution and the orbital inclination distribution. This source of systematic error will be discussed in § III. \nThe detailed nature of internal (i.e., non-kinematic) properties of the emitting matter also influence the parameter q , so that, in the absence of information constraining these properties, there is an associated systematic error (see, e.g., the qualitative discussion in Wandel et al. 1999). The breadth of radii across which line-emitting matter is located is one such factor. The angular radiation pattern of the line-emitting material is another. \nOne way to quantify the radial emissivity distribution is through the 'transfer' or 'response' function Ψ defined by \nδL l ( t ) = ∫ ∞ 0 dτ Ψ( τ ) δL c ( t -τ ) . (1) \nThat is, fluctuations in the continuum luminosity δL c predict fluctuations in the line luminosity δL l at later times through a convolution with Ψ( τ ). Although the true relationship between line luminosity and incident continuum flux is often nonlinear, if δL c and δL l are interpreted as fluctuations relative to a long-term mean value, equation 1 is valid provided the fluctuations are small compared \nto the mean. At the order of magnitude level, the amplitude of Ψ( τ ) tells us how much matter there is (and how sensitive its line output is to continuum fluctuations) at radius r ∼ cτ . Although relatively few monitoring experiments produced data of good enough quality to allow inferences of Ψ( τ ) to be made, those that have been analyzed indicate that there is significant response across at least an order of magnitude dynamic range in lag and presumably, therefore, across a comparable dynamic range in radius (Krolik et al. 1991; Done & Krolik 1996; Ulrich & Horne 1996). \nThis fact means that the characteristic r ∗ estimated from comparing the continuum and line light curves is a peculiar weighted average over a wide range of radii. Similarly, the characteristic v ∗ is a different weighted average of the line-of-sight velocities over that same range. Thus, it is unclear how to interpret the combination r ∗ v 2 ∗ , as it mingles contributions from different radii with different weights. \nThe angular radiation pattern can introduce further factors of order unity because gas at radius r but on the near side of the continuum source to us appears to respond more quickly than gas at the same radius but on the far side. As we will show in § 4.2, the angular radiation pattern also interacts with other sources of systematic error. \nFinally, the characteristic scales inferred from monitoring are affected by the shape of the continuum fluctuation power spectrum. The effective power spectrum is determined by a combination of the intrinsic spectrum and the details of the particular finite sampling of the experiment because only certain timescales can be probed by an realizable experiment. Because this limitation amounts to a filtering in frequency space, it effectively alters both the line and continuum fluctuation power spectra. Through this means, the sampling also influences the factor q . \nThe quantitative impact of these factors (assuming regular sampling and noise-free data) will be examined in detail in § IV. We do not examine the consequences of irregular sampling (which can be expected to introduce a further systematic error) or measurement uncertainty (which introduces a random error). We limit ourselves in this paper to systematic errors because they can be computed very efficiently in the frequency domain; estimating the impact of random errors is far more efficiently done in the time domain because the error at each point in the line profile must be evaluated.", '2. How Can We Tell That the Dynamics are Gravity-Dominated?': "One obvious test of whether the dynamics governing the emission line region are dominated by the gravity of a central point-mass is whether the characteristic speed is ∝ r -1 / 2 . The empirical record here is mixed, but indicates at least marginal consistency with this proposition. \nComparing the cross-correlation peaks of different lines to the widths of their mean profiles as measured in the IUE monitoring campaign on NGC 5548 (Clavel et al. 1991), Krolik et al. (1991) found rough agreement with the relation v ( r ) ∝ r -1 / 2 . Repeating this exercise with the widths \nof the rms profiles (for a precise definition, see § 4.1), Peterson & Wandel (1999) likewise found consistency, with somewhat smaller departures. Peterson & Wandel (2000) performed the same test on two other galaxies, NGC 7469 and 3C 390.3, again finding consistency, but with poorer data. There are only three data points for NGC 7469 (one of which has rather large uncertainty in v ∗ ) and four for 3C 390.3 (with one having large error bars in both r ∗ and v ∗ ). In both cases, the dynamic range in radius is only a factor of three. On the other hand, Krolik & Done (1996) found that the best fit to the velocity-resolved monitoring data for the CIV 1549 line from the HST campaign on NGC 5548 (Korista et al. 1995) was a rather slower dependence of velocity on radius, although the increased χ 2 created by forcing v ∝ r -1 / 2 was not large enough to exclude this scaling. \nWhile a significant deviation from r -1 / 2 scaling could rule out point-mass gravitational dynamics, even perfect agreement could not prove it. The reason is that several other kinds of dynamical models make the same prediction. These include such diverse examples as cloud outflows driven by photoionization when the ionization parameter in the clouds is fixed (Blumenthal & Mathews 1975), disk winds driven by line scattering (Murray et al. 1995), and magnetically-driven disk winds (Emmering, Blandford & Shlosman 1992). In fact, at a qualitative level, both of the latter two models are in rough agreement with the NGC 5548 velocity-resolved monitoring data (Chiang & Murray 1996, Bottorff et al. 1997). In this respect, they are in as good agreement with empirical tests as the model of point-mass gravitational dynamics. \nPeterson & Wandel (2000) make the further argument that the mass they infer by assuming gravitational dynamics is so large that the ratio of total luminosity to the Eddington luminosity is too small to permit significant radiative acceleration. This argument is, however, flawed in two important ways. \nFirst, by definition, all rival models have characteristic speeds v ∗ that scale ∝ r -1 / 2 . We may therefore write v ∗ = Av esc , where, as usual, v esc = ( GM/r ) 1 / 2 ; if the model entails outflow, A ≥ 1. If that characteristic speed is then used to infer a mass assuming gravitational dynamics , the inferred mass will be M inf = qv 2 ∗ r/G = qA 2 M . The inferred mass is then an over-estimate of the true mass when A > 1. Because M inf /M ∝ A 2 , the error could be substantial. \nSecond, the correct criterion for driving a radiatively-accelerated wind in emission line gas is not that L > L E . Radiation force is proportional to opacity in optically-thin gas, and the Eddington luminosity criterion assumes that the opacity is purely Thomson. In fact, the opacity to ultraviolet and soft X-ray photons presented by gas capable of radiating the observed emission lines can easily be several orders of magnitude greater (this is, in fact, the basis of the radiatively-driven wind models cited earlier). \nThe values of L/L E inferred by Peterson & Wandel (2000) ranged from ∼ 10 -3 to ∼ 10 -1 . If L E is over-estimated by a factor of 10 (as could easily occur if there are outflows at merely three times the escape speed), the range of L/L E inferred shifts to ∼ 10 -2 -∼ 1. If the effective opacity is ∼ 10 2 times greater than Thomson (e.g., from H photoionization opacity in a gas whose H neutral \nfraction is 10 -4 facing a continuum whose peak value of νL ν occurs near 10 eV), the inferred ratio of radiation force to gravity ranges from ∼ 1 to ∼ 100. Thus, this argument fails to provide any support for dynamics dominated by gravity. \nThe only further support for broad emission line region dynamics being dominated by the gravity of a central point-mass is the rough agreement found by Gebhardt et al. (2000c) and Ferrarese & Merritt (2000b) between the central mass inferred on the basis of the correlation with the host's bulge dispersion (Gebhardt et al. 2000b, Ferrarese & Merritt 2000a) and the mass inferred from reverberation by Ho (1999) and Wandel et al. (1999). A similar relationship between the central mass inferred by reverberation mapping and the narrow line width has been found by Nelson (2000).", '3. The Orbital Shape and Inclination Distribution': 'If we grant for the sake of argument that the emission line dynamics are due to motion in the potential of a point-mass, it remains to determine the proportionality constant q . One factor that determines this quantity is the distribution of orbital shape and inclination. To separate these effects from the ones discussed in subsequent sections, suppose for the moment that the material in question emits line radiation only when located a distance r from the central object. Here, and in the rest of the paper, we will work in terms of the one-dimensional line-of-sight velocity dispersion defined by v 2 ∗ ≡ 〈 u 2 〉 , where u is velocity offset in the line profile and the average is weighted by line flux. \nIn the simplest imaginable case, all the orbits might be circular (at radius r ) and confined to a single plane. Then the mean-square line-of-sight velocity is (1 / 2) GM sin 2 i/r , where i is the inclination angle of the orbital axis relative to the line-of-sight. The factor q is then 2 / sin 2 i . \nOn the other hand, it is almost as simple to posit that the velocities are randomly directed. However, we do not know the orbital shapes. Circular orbits have | /vectorv | 2 = GM/r , so their meansquare speed in any one direction is GM/ (3 r ); in that case, q = 3. By contrast, parabolic orbits have | /vectorv | 2 = 2 GM/r , so that q = 3 / 2 if they are isotropic. \nMore complicated models (e.g., supposing that the velocity is a sum of random and planar components as in McLure & Dunlop 2000) are also possible. These, of course, would yield a different correction factor dependent on the ratio of the magnitudes of the two components as well as on the inclination angle. Efforts to infer nuclear masses from stellar kinematics have found it necessary to employ extensive modeling in order to determine how the uncertainty in orbital shape distribution contributes to uncertainty in the inferred mass (e.g., Gebhardt et al. 2000a). Similar complexities may also apply here. \nWe have now found that values of q predicted by equally simple models range all the way from a minimum of 3 / 2 to a maximum (nominally) of infinity, in the event of a thin orbital plane \nperpendicular to the line-of-sight. Although a very thin orbital plane is a bit implausible, thicknesses of ∼ 10 -1 cannot yet be easily ruled out. If so, the range of uncertainty for q from considerations of orbital shape alone is two orders of magnitude, from 3 / 2 to ∼ 200. \nWe conclude this section with a technical note. Not all authors define v ∗ as the root-meansquare speed. Some measure the FWHM of the profile instead. If the profile is Gaussian with dispersion σ , these quantities are related by the expression v FHWM = 2 √ 2ln 2 σ = 2 . 35 σ (cf. the expression v FWHM = (2 / √ 3) σ 3 d suggested by Netzer 1990 and used by Wandel et al. 1999 and Peterson & Wandel 2000; note that the inferred mass scales as the square of this conversion coefficient).', '4.1. What Kinds of Moments are r ∗ and v ∗ ?': "To describe the way in which the characteristic scales depend on the radial emissivity distribution, we begin with the velocity-dependent response function \nΨ( τ, u ) = ∫ dV 1 4 πr 2 ∂j ∂F ion ( /vectorr ) Φ(ˆ r · ˆ z ) δ [ τ -τ ( /vectorr )] ∫ d 3 v f ( /vectorv , /vectorr ) δ ( u -/vectorv · ˆ z ) , (2) \nwhere j is the local line emissivity, ˆ z is the direction of the line-of-sight, τ ( /vectorr ) = ( r/c )(1 -ˆ r · ˆ z ) is the time lag corresponding to position /vectorr , Φ describes the angular radiation pattern of the emission (we assume that it is oriented with respect to ˆ r ), and f ( /vectorv , /vectorr ) is the velocity distribution function at /vectorr . As we will discuss at greater length later in this paper, it is desirable to define r ∗ and v ∗ so that they correspond, as much as possible, to the same material. For this reason, we will use, as recommended by Peterson & Wandel (1999), the variable part of the line to measure the velocity moment as well as the characteristic lengthscale. \nSeveral different empirical definitions of r ∗ have been used, but all make use of the linecontinuum total-flux cross-correlation function C lc . For example, some have used the peak of C lc , whereas Wandel et al. (1999) and Kaspi et al. (2000) employed the centroid of that portion of C lc whose amplitude is at least 80% of the peak amplitude. Written in terms of Ψ, the cross-correlation function is \nC lc ( τ ) = 1 σ l σ c ∫ du ∫ dt Ψ( u ) ∗ δF c δF c ( t -τ ) , (3) \nwhere σ l,c are the rms fluctuations in the line and continuum flux, F c ( t ) is the continuum flux at time t , and '*' denotes a convolution. The dependence of C lc on the continuum fluctuation power spectrum is more clearly displayed when it is written in terms of Fourier-transformed quantities: \nC lc ( τ ) = ∫ du ∫ df e -2 πifτ ˆ Ψ | ˆ δF c | 2 , (4) \nwhere ˆ over a symbol (except for ˆ z , the unit vector) indicates a Fourier transform. Because Ψ( τ ) is a real function, it is convenient to rewrite this expression as \nC lc ( τ ) = ∫ du ∫ ∞ 0 df 2 Re [ e -2 πifτ ˆ Ψ ] | ˆ δF c | 2 . (5) \nIf τ ∗ is defined as a centroid of the cross-correlation function, we have \nτ ∗ = ∫ dττ ∫ du ∫ ∞ 0 df 2 Re [ e -2 πifτ ˆ Ψ ] | ˆ δF c | 2 ∫ dτ ∫ du ∫ ∞ 0 df 2 Re [ e -2 πifτ ˆ Ψ ] | ˆ δF c | 2 , (6) \nwhere the limits on the τ integrals are chosen according to the particular kind of centroid desired. \nSimilarly, v ∗ may be defined in any of several ways, all based on the rms profile. As discussed in § III, some use its FWHM, others its mean-square speed. Writing the rms profile also in terms of ˆ Ψ and ˆ δF c , we have \nF rms ( u ) = [ T -1 ∫ T 0 dt | Ψ( u ) ∗ δF c | 2 ] 1 / 2 (7) \n= [ T -1 ∫ df | ˆ Ψ ˆ δF c | 2 ] 1 / 2 , (8) \nwhere the second form follows from the convolution and Parseval's theorems. If, for example, v ∗ is defined as the rms speed, \nv ∗ = ∫ duu 2 [ T -1 ∫ df | ˆ Ψ ˆ δF c | 2 ] 1 / 2 ∫ du [ T -1 ∫ df | ˆ Ψ ˆ δF c | 2 ] 1 / 2 . (9) \nContrasting equations 6 and 9 makes it clear that the moments over the radial distribution resulting in r ∗ and v ∗ are different , so that the characteristic scales they refer to need not coincide. These forms also make it obvious that the moments depend on ˆ δF c in addition to the intrinsic nature of the emission line region. \nIn the preceding derivation we have tacitly assumed that an infinite amount of data is available to determine the cross-correlation function and rms line profile. Real experiments do not, of course, yield infinite data trains. However, the pairing between function and Fourier transform can be taken over almost without alteration to discrete Fourier transforms after allowance for a few restrictions: the functions are assumed to be periodic with a period equal to the duration T of the data; the sampling is uniform (we call the interval ∆ t ); the limits on frequency in all integrals are -1 / (2∆ t ) to +1 / (2∆ t ); and the frequency resolution is 1 /T . \nOne consequence of these restrictions is that the data are effectively filtered in such a way as to eliminate all frequencies higher than 1 / (2∆ t ) and lower than 1 /T . In rough terms, there is a mapping between frequency f and distance-scale r ∼ c/f ; therefore, the character of the sampling can bias the weighting given different distance-scales, ultimately leading to a systematic error in the inferred mass.", '4.2. Examples': 'To explore the impact of these moments, we have used equations 6 and 9 to evaluate M inf /M for a variety of choices of continuum variability behavior, underlying physical model, and sampling. Our goal is to separately identify the systematic errors induced by each cause: intrinsic character of the moments, detailed properties of the line emission such as angular radiation pattern or radial distribution, and poor sampling.', '4.2.1. Model definition': 'In all cases, we will use the same basic model, chosen as the simplest non-trivial one permitting an exploration of these effects. In this model the line emissivity depends only on r , the angular radiation pattern Φ( µ ) = (1 -µ ) γ for γ = 0 or 1 (here µ ≡ ˆ r · ˆ z ), and the 1-d velocity distribution is a Gaussian with dispersion v ( r ). So that effects due to orbital shape and inclination may be cleanly separated from the other sources of systematic error, we assume in all cases that v ( r ) = ( GM/ 3 r ) 1 / 2 , i.e., the orbits are circular and isotropically-oriented. This choice implies that the correct value of q is 3 M/M inf . The response function Ψ is then \nΨ( τ, u ) = ∫ ∞ cτ/ 2 dr c 2 r ∂J ∂F ion ( r ) e -u 2 / [2 v 2 ( r )] [ πv 2 ( r )] 1 / 2 Φ(1 -cτ/r ) . (10) \nHere J ≡ ∫ d Ω r 2 j , i.e., the radial emissivity distribution. \nFor the purpose of computing ˆ Ψ, it is convenient to reverse the order of the r and τ integrations, i.e., \nˆ Ψ( f, u ) = ∫ ∞ 0 dr c 2 r ∂J ∂F ion ( r ) ( c/r ) γ √ 2 πv ( r ) e -u 2 / [2 v 2 ( r )] ∫ 2 r/c 0 dτ τ γ e 2 πifτ . (11) \nFor any integral value of γ , the τ integral is easy to evaluate analytically. \nIt is now time to choose a physical model. We have two goals: exploring the consequences of radial distributions that have a range of radial widths, and being at least crudely consistent with what we have learned from detailed studies of the response functions of selected AGN (Krolik et al. 1991; Horne et al. 1991; Wanders et al. 1995; Done & Krolik 1996; Ulrich & Horne 1996). Dependence on the width of the radial distribution is of special interest because there are indications from these detailed studies that the true radial distribution may in fact span an order of magnitude or more in radius. Towards that end, we write \n∂J ∂F ion = J o exp [ -(log r -log r o ) 2 / (∆log r ) 2 ] , (12) \nwhere J o is an (arbitrary and irrelevant) constant, r o is the center of the distribution of line emissivity with respect to radius, and ∆ log r is its characteristic width in terms of log 10 r . We stress, however, that there are substantial uncertainties in the emissivity distributions that have \nbeen measured, and these measurements exist for only a handful of objects; we therefore know very little about the true range of possibilities. For this reason, these simulations must be regarded as purely illustrative; real AGN may have much more complicated emissivity distributions. \nWe also assume for the simulations reported here that the power density spectrum of continuum fluctuations takes a power-law form: | ˆ δF c | 2 ∝ f -n . We have explored the consequences of varying n , and find that in most instances it changes the results in only a minor way. As a result of these preliminary explorations, we decided to fix n = 1 . 5 for all the calculations reported in this paper. \nFinally, we define r ∗ as the centroid of the portion of the cross-correlation curve whose amplitude is greater than 80% of the peak, except in those cases in which the 80% level lies beyond the range of lags where the cross-correlation may be estimated-in those cases (which are few), we use the cross-correlation peak to define the characteristic radius.', '4.2.2. Errors due to the moments and the underlying physics': "We begin by defining the systematic errors due solely to the differing moments. These are defined by assuming an 'ideal' experiment, i.e. one with 900 measurements spaced at an interval ∆ t = (1 / 30)( r o /c ). Two examples are illustrated in Figure 1, one assuming isotropic radiation, the other assuming Φ ∝ 1 -µ . As can be seen, in both cases the inferred mass is biased towards values larger than the true value, but by rather more in the anisotropic radiation case. The reason for this distinction is clear: When we preferentially see regions on the far side of the center, the lag is greater than what it would be if we could see all the emitting regions. Thus, the estimated r ∗ is also greater than it should be, and the estimated mass likewise because M inf ∝ r ∗ . In both cases, the bias grows with increasing width of the radial emissivity distribution. The reason for this, too, is easy to see: differing moments mean little when the underlying distribution is narrow. For the widest distribution we consider (∆ log r = 1 . 5), the error is 60% in the isotropic case, and a factor of 2.7 in the anisotropic case. The method, therefore, has an intrinsic bias toward overestimating the central mass that can be small in favorable cases (sharply peaked radial emissivity distributions) but considerably larger in unfavorable ones.", '4.2.3. Errors due to sampling': "Next we examine the effects of sampling, beginning with experiments that are 'as good as can be hoped'. We define this phrase as denoting an experiment with 64 sampling points and interval ∆ t = (1 / 8)( r o /c ). Judging by the history of the experiments, this seems to be about as many measurements as can be managed (e.g., 60 were obtained in the original IUE campaign: Clavel et al. 1991; 39 were obtained in the HST campaign on NGC 5548: Korista et al. 1995; of the 28 quasars monitored by Kaspi et al. 2000, the median number of observations was 25, although one had as many as 70). \nFig. 1.- Systematic error due solely to the differing moments that define v ∗ and r ∗ . The solid curve refers to isotropic radiation, the dotted to Φ ∝ 1 -µ . \n<!-- image --> \nThe expected level of systematic error from experiments 'as good as can be hoped' is shown in Figure 2. Contrasting that figure to Figure 1, we see that when the emission is anisotropic in the sense chosen, the reduction in sampling makes almost no difference at all. Surprisingly, 'good', but less than 'ideal', sampling actually decreases the level of systematic error in the case of isotropic radiation. We will learn the reason for this when we study poorer sampling. \nTruly sub-optimal sampling increases the opportunity for error. Two kinds of problems are possible. The first kind is a simple shortage of data. Instead of having 64 measurements, as in the 'as good as can be hoped' example, there might be many fewer. The consequences of data sets that are too small are shown in Figure 3. As that figure shows, there is a systematic bias produced by short data sets toward smaller values of M inf /M . When the radial distribution is narrow, the bias is small because only a narrow range of timescales occurs in the data. However, as the radial distribution becomes broader, the effect grows. Because the contrasting moments entering into M inf create a systematic shift toward M inf /M > 1 for large ∆ log r , the statistical bias of short data trains can partially counteract the systematic error induced by the moments. It is then possible for the net systematic error to be fortuitously small. \nThe second kind of deviation from optimal sampling is an offset in timescales sampled. When planning a monitoring experiment, one does not know in advance what the characteristic size of the emission line region is, although one might estimate it by scaling from other examples (e.g., by supposing that r o ∝ L 1 / 2 , as might be expected on the basis of simple photoionization models; but see the doubts raised by Kaspi et al. 2000). Offsets in the sampling interval relative to r o /c are therefore quite likely. We define φ = ( T ∆ t ) 1 / 2 c/r o as the offset parameter; when φ < 1, the scales sampled are too small, when φ > 1, they are too large. Figure 4 illustrates the impact of a factor of 3 error even when a substantial data set is obtained. \nLarger offsets can create still larger errors. After exploring the range 0 . 01 < φ < 100, we find that when the radiation is isotropic, M inf /M /similarequal s ( φ, ∆log r ) φ , where s is a slowly-varying function of φ , such that \ns ( φ, ∆log r ) /similarequal 1 φ < 1, ∆ log r /lessmuch 1 3 φ < 1, ∆ log r > 1 0 . 25 φ > 1, ∆ log r /lessmuch 1 0 . 2 φ > 1, ∆ log r > 1 (13) \nAnisotropic radiation leads to more complicated behavior. Values of φ between 0.1 and 10 bias the result toward M inf /M smaller than unity (for φ < 1) and M inf /M greater than unity (for φ > 1), but by smaller amounts than in the isotropic case. More extreme values of φ push the bias sharply in either direction, smaller values of φ leading to greatly underestimated masses and larger values to numbers that are substantial overestimates. Unfortunately, unlike the case of isotropic radiation, there is no simple approximate expression that encapsulates these dependences. \nThe origin of the sensitivity of M inf /M to φ lies in the mismatch that is created when φ is grossly different from unity between the natural timescales of the system and the timescales probed \nFig. 2.- Systematic error as a function of ∆ log r in the case of 'as good as can be hoped' sampling. The solid curve refers to isotropic radiation, the dotted to Φ ∝ 1 -µ . \n<!-- image --> \nFig. 3.- Systematic error as a function of ∆ log r for several different length experiments, all with timescales centered on r o /c , and assuming isotropic radiation in each case. The solid line is the result of 900 data points, the dashed line is the product of 64 measurements, and the dotted line comes from a simulated experiment with only 25 points. \n<!-- image --> \nFig. 4.- The ratio M inf /M as a function of ∆ log r for four experiments with a large number of points (64), but different sampling intervals. The solid curve pertains to isotropic radiation and ∆ t = (3 / 8)( r o /c ); the dashed curve shows the result for the same model but with ∆ t = (1 / 24)( r o /c ). The dotted and long-dashed curves pertain to a model with Φ ∝ 1 -µ ; ∆ t = (3 / 8)( r o /c ) for the dotted curve, while ∆ t = (1 / 24)( r o /c ) for the long-dashed curve. The jagged breaks, which are particularly noticeable in the solid curve, are due to the relatively small number of points contributing to the cross-correlation centroid integrations. \n<!-- image --> \nby the experiment. As a result, the peak in the profile-integrated cross-correlation function can be severely biased: for example, in the case of isotropic radiation and φ = 0 . 01, the characteristic scale derived from the cross-correlation centroid is /similarequal 0 . 01 times what it is when the sampling is perfectly matched to the true scales. That there is a peak anywhere in the measurable range of lags is due to the existence of some matter close to the line of sight, where it can respond quickly despite its distance from the central source; by contrast, in the case of Φ ∝ 1 -µ , for which matter on the line of sight is essentially invisible, the cross-correlation function is greatest at the largest lag measurable. Because the rms line profile is much less strongly affected by sampling problems, the bias in the mass estimate corresponds very nearly to the bias in the characteristic distance scale estimate. Not surprisingly, tight radial distributions (i.e., small values of ∆ log r ) exacerbate the timescale mismatch effect, but relatively weakly compared to the much larger bias driven by the poor sampling.", "5.1. Deciding Whether the Black Hole's Gravity Dominates Emission Line Dynamics": "Measuring central black hole masses by the reverberation method is possible only if gravity dominates the dynamics of the line-emitting gas and the mass of the black hole is much larger than any other mass within the line-emitting region. As was shown in § 2, the evidence in hand to date neither proves nor disproves this assumption. The question naturally arises as to how this situation might be clarified. \nPoint-mass gravitational dynamics might be discredited if evidence arose showing that v did not scale as r -1 / 2 . One possible way to do so would be to conduct experiments like the HST campaign on NGC 5548, but with better data, so that Ψ( τ, u ) could be more tightly constrained. 'Better' in this context means more epochs of observation and a combination of better signal/noise and velocity resolution. More epochs of observation serves both to reduce statistical scatter and to create a larger dynamic range in timescales (and hence length scales) probed. Better signal/noise and velocity resolution would permit dividing the line profile into more segments in order to achieve a greater dynamic range in u . \nHowever, demonstrating consistency with v ∝ r -1 / 2 is not sufficient to prove that the black hole's gravity controls the dynamics of line-emitting gas because there are other models that make the same prediction. Testing these other models vis-a-vis point-mass gravity requires independent methods. Disk winds might be tested by searching for the correlation between radial and azimuthal velocity that they generically predict (Chiang & Murray 1996). Although this prediction is qualitatively consistent with the relatively rough constraints posed by the HST campaign on NGC 5548, an improved experiment of the sort described in the previous paragraph might sharpen this test. Nonetheless, because the kinematic predictions of so many models are so similar, providing a direct \nproof of gravity-dominated dynamics will be difficult.", '5.2. Systematic Errors Granted Gravitational Dynamics: Magnitude and Mitigation': "If, for the sake of argument, we grant the assumption that the motions of the line-emitting gas are dominated by response to the black hole's gravity, both random and systematic errors of several different varieties may cloud the result. In this paper, we have concentrated on the systematic errors. \nOne sort is due to our ignorance of the orbital shapes. Changing from parabolic to circular orbits alters the inferred mass by a factor of two. Even if the orbital eccentricity distribution is known, flattening of the emission line region can introduce a new systematic error by eliminating our ability to average over the velocity vectors' projections on the line-of-sight. This error can be as large as ∼ ( r/h ) 2 for aspect ratio h/r . In principle, this latter effect could produce a very large error. \nUnfortunately, whether the broad line region is round, flat, or some other shape is still a controversial issue (Wanders et al. 1995, Dumont et al. 1998). It is even possible that the region responsible for some lines (e.g., the high-ionization lines like CIV 1549) is round while the region radiating the Balmer lines is flattened (Rokaki et al. 1992). \nAnother variety of systematic error arises from the interaction between the differing ways in which r ∗ and v ∗ depend on the response function, the details of the radial emissivity distribution, the angular radiation pattern, and the sampling. Although in some sense each of these contributions is logically independent, the magnitude of the combined error cannot be estimated by simply adding them in quadrature. For example, if the line radiation is isotropic, relatively little systematic error is induced by the character of the moments defining r ∗ and v ∗ , wide emissivity distributions, or meager datasets; on the other hand, the results obtained when the line radiation is isotropic are very sensitive to the characteristic sampling timescale. However, it should also be borne in mind that all of these conclusions were reached on the basis of exploring a very simple emissivity model; more complicated geometries might well lead to new dependences for the systematic errors. \nThe central problem, of course, is that with only a single monitoring dataset it is difficult to determine which of these characteristics apply to any particular quasar or line. Although most photoionization models (e.g., Kwan & Krolik 1981, Ferland et al. 1992) predict that H recombination lines are emitted rather anisotropically (more or less in the sense described by Φ ∝ 1 -µ ), we can hardly claim to know this reliably (for an example of complications that might change this prediction, see Kallman & Krolik 1986). It is possible that other lines are also emitted anisotropically, but this suggestion is even more model-dependent (Ferland et al. 1992, O'Brien et al. 1994). Similarly, we do not know anything a priori about the geometric symmetry of the emission line region or the radial dependence of emissivity within it. At the same time, as the curves of Figure 2 illustrated, it is also possible for systematic errors to cancel fortuitously. Consequently, although \nwe may hope that conditions are such that the systematic error is small, it is hard at this stage to be confident. \nA partial step forward could be provided by a two-step process applied to each target object individually: First, the breadth and characteristic scale of the radial emissivity distribution can be estimated by monitoring that spans a truly broad range of timescales; this experiment does not require good quality spectral resolution. Second, with that knowledge in hand, it would be possible to design a new experiment with optimized sampling and spectral resolution that might be better able to control those systematic errors due to inappropriate sampling scale or ignorance of the radial emissivity distribution. \nAnother partial advance could come from making fuller use of the information obtained in emission line monitoring experiments. In the efforts to estimate M so far, only the moments r ∗ and v ∗ have been used. There is potentially much more information contained in the full set of time-dependent line profiles. If good enough data were obtained that Ψ( τ, u ) could be accurately determined, many (although not all) of these uncertainties could be either eliminated or constrained. For example, as we have already remarked, the central assumption that v ∝ r -1 / 2 could be tested directly. At a more detailed level, different orbital shape and inclination distributions can be distinguished by the contrasting shapes they give Ψ in the τ -u plane (Welsh & Horne 1991). Although Ψ( τ, u ) does involve an integration over radius (as in equation 10, for example), it still provides some indication of the radial emissivity distribution.", '5.3. Random Errors': 'We stress that in this paper we have discussed only systematic errors. Random errors arising from flux measurement uncertainties and the fluctuations due to specific realizations of the random processes involved add to the final uncertainty (Peterson et al. 1998; Welsh 1999).', '5.4. Conclusions': 'Given all these considerations, it would seem that, taken in isolation, there are systematic uncertainties in the estimation of black hole masses by the reverberation method that are potentially large, but whose magnitudes are difficult to estimate quantitatively. In order to gain confidence in the results obtained by this method, further efforts to control these systematic errors are essential. \nUnfortunately, the underlying question of whether gravity truly dominates the dynamics is particularly difficult to answer securely and is rendered more difficult by the existence of uncontrolled systematic errors. If other, independent, measurements yield black hole masses that coincide with those inferred from reverberation mapping (as seems possible: Gebhardt et al. 2000c; Ferrarese & Merritt 2000b), this agreement might be taken as evidence in support of gravitational dynamics in \nthe broad line region. Until we understand these systematic errors better, however, it still remains possible that such agreement could be merely the result of a fortuitous cancellation of systematic errors. \nConsider one example of how this might come about: As described in § 2, many models in which non-gravitational dynamics dominate predict a characteristic speed that scales with the escape speed. If the emission line kinematics are interpreted as gravitational, the mass is then consistently overestimated by a fixed ratio. A systematic measurement error that underestimates the mass (as would be induced by a characteristic sampling time that is too short applied to an isotropically-radiated emission line) could then produce an entirely fortuitous agreement. We would then be in the odd position of arriving at the correct mass, but only as a result of mutually cancelling errors. \nI thank Eric Agol, Tim Heckman, and Jerry Kriss for many helpful conversations and suggestions. 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2006JHEP...04..044F
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Excursions beyond the horizon: black hole singularities in Yang-Mills theories (I)
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2006-01-01
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We study black hole singularities in the AdS/CFT correspondence. These singularities show up in CFT in the behavior of finite-temperature correlation functions. We first establish a direct relation between space-like geodesics in the bulk and momentum space Wightman functions of CFT operators of large dimensions. This allows us to probe the regions inside the horizon and near the singularity using the CFT. Information about the black hole singularity is encoded in the exponential falloff of finite-temperature correlators at large imaginary frequency. We construct new gauge invariant observables whose divergences reflect the presence of the singularity. We also find a UV/UV connection that governs physics inside the horizon. Additionally, we comment on the possible resolution of the singularity.
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https://arxiv.org/pdf/hep-th/0506202.pdf
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{'Excursions beyond the horizon: Black hole singularities in Yang-Mills theories (I)': 'Guido Festuccia and Hong Liu \nCenter for Theoretical Physics Massachusetts Institute of Technology Cambridge, Massachusetts, 02139 \nWe study black hole singularities in the AdS/CFT correspondence. These singularities show up in CFT in the behavior of finite-temperature correlation functions. We first establish a direct relation between space-like geodesics in the bulk and momentum space Wightman functions of CFT operators of large dimensions. This allows us to probe the regions inside the horizon and near the singularity using the CFT. Information about the black hole singularity is encoded in the exponential falloff of finite-temperature correlators at large imaginary frequency. We construct new gauge invariant observables whose divergences reflect the presence of the singularity. We also find a UV/UV connection that governs physics inside the horizon. Additionally, we comment on the possible resolution of the singularity.', '1. Introduction': "The AdS/CFT correspondence [1,2,3] provides exciting avenues for exploring various issues in quantum gravity. An important question that the AdS/CFT correspondence may shed light on is that of the nature of spacelike singularities, like the Big Bang or the singularity of a Schwarzschild black hole. A good laboratory for studying spacelike singularities is an eternal black hole in anti-de Sitter (AdS) spacetime. It has been conjectured that quantum gravity in an AdS d +1 black hole background is described by a boundary conformal field theory on S d -1 × IR at a temperature given by the Hawking temperature of the black hole [2,4,5] 1 . \nThe conventional wisdom regarding singularities is that they signal the breakdown of classical gravity and should go away when stringy or quantum gravitational effects are taken into account. Since in AdS/CFT, classical gravity corresponds to the large N and large 't Hooft coupling limit of the boundary theory 2 , one expects that finite N or finite 't Hooft coupling effects may resolve these singularities [6,5]. Such considerations suggest the following strategy: \n- 1. Identify manifestations of the black hole singularity in the large N and large 't Hooft coupling limit of the finite temperature boundary theory;\n- 2. From these manifestations, understand the precise physical mechanism through which the finite N or finite 't Hooft coupling effects may resolve the singularity. \nIn the boundary theory, the physical observables are correlation functions of gauge invariant operators. This means that the physics of singularities should be encoded in the behavior of boundary correlation functions in appropriate limits. \nOne of the obstacles 3 in understanding black hole singularities from finite temperature boundary theory is that the singularities are hidden behind event horizons. The boundary conformal field theory evolves through the bulk Schwarzschild time, i.e. from the point of view of an external observer, and does not appear to directly describe the physics beyond the horizon. In other words, if time evolution inside the horizon of a black hole is to be described by the boundary theory, time has to be holographically generated. This \nmakes the problem particularly challenging, while at the same time exciting. In particular, understanding physics beyond the horizon should shed light on how to holographically describe a Big Bang cosmological spacetime 4 . \nA number of authors [7-17] have explored how to extract physics beyond the horizon from the boundary theory correlation functions 5 . In particular, Fidkowski et al [12] found an interesting but subtle signal of the singularity in the boundary correlators. They found that AdS d +1 black holes with dimension d ≥ 3 contain spacelike geodesics, connecting two asymptotic boundaries, which could get arbitrarily close to the singularity. The authors further argued that such geodesics imply the presence of poles on secondary sheets of the analytically continued coordinate space correlation functions in the large operator dimension limit. \nHere we further explore the manifestations of the black hole singularity in the boundary theory and discuss their implications for the resolution of the singularity. \nWe will establish a direct relation between space-like geodesics in the bulk spacetime and the large operator dimension limit of momentum space Wightman functions in the boundary theory. We show that physics in the region beyond the horizon is encoded in the behaviors of boundary correlation functions along the imaginary frequency axis. In particular, this gives a clear indication that the 'time' inside the horizon is holographically generated from the boundary theory. The presence of the curvature singularity leads to certain exponential falloff of the correlation functions near the imaginary infinity. We also construct new gauge invariant observables which have singularities precisely reflecting the curvature singularity of the black hole. \nIn this paper we will present the main idea using the example of an AdS 5 Schwarzschild black hole, leaving technical details and more extensive discussions of other examples to a longer companion paper [23]. In [24] we develop a new method for computing the large mass quasi-normal frequencies of the black hole, whose knowledge is crucial for the discussion of this paper and [23]. \nThe plan of the paper is as follows. In section 2, we review the relevant black hole geometry and the computation of boundary Wightman function in AdS/CFT. In section \n3, we establish a connection between bulk spacelike geodesics and boundary Wightman functions in the large operator dimension limit. In section 4, we study the manifestation of the singularities in Yang-Mills theory. We conclude in section 5 with a discussion of the possible resolution of singularities at finite N .", '2.1. Black hole geometry': 'We will consider big black holes in AdS 5 , which have a positive specific heat and are the dominant contribution to the thermal canonical ensemble of the boundary Yang-Mills theory when the temperature is sufficiently high [25]. \nThe metric for a Schwarzschild black hole in an AdS 5 spacetime is given by \nds 2 = -f ( r ) dt 2 + f ( r ) -1 dr 2 + r 2 d Ω 2 3 (2 . 1) \nwith \nf ( r ) = r 2 +1 -µ r 2 = 1 r 2 ( r 2 -r 2 0 )( r 2 + r 2 1 ) , (2 . 2) r 2 1 = 1 + r 2 0 , µ = r 2 0 r 2 1 \nwhere µ is proportional to the mass of the black hole and the event horizon is at r = r 0 . r 0 , r 1 can be solved in terms of µ . We have set the curvature radius of AdS to be unity, as we will do throughout the paper. As r →∞ , the metric goes over to that of global AdS with t identified as the boundary time. The fully extended black hole spacetime has two disconnected time-like boundaries, each of topology S 3 × IR. \nFig. 1: Penrose diagram for the AdS black hole. There are two asymptotic AdS regions, which are space-separated from each other. A null geodesic going from the boundary to the singularity is indicated in the figure. \n<!-- image --> \nIt is often convenient to use the tortoise coordinate \nz = ∫ ∞ r dr f ( r ) = -β 4 π log ( r -r 0 r + r 0 ) + ˜ β 2 π tan -1 r 1 r (2 . 3) \nβ = 2 πr 0 r 2 0 + r 2 1 , β = 2 πr 1 r 2 0 + r 2 1 . (2 . 4) \nwith \nThe region outside the horizon corresponds to z ∈ (0 , + ∞ ) with z = 0 at the boundary and z → + ∞ at the horizon. β is the inverse Hawking temperature. To have a feeling of the physical meaning of ˜ β we note that the complex Schwarzschild time that it takes for a radial null geodesic to go from the boundary to the singularity is given by \n˜ \n± ∫ ∞ 0 dr f ( r ) = ± 1 4 ( ˜ β ± iβ ) (2 . 5) \nwhere the imaginary part of the integral arises by going around the pole at r = r 0 in the complex r -plane. A nonzero ˜ β implies that the Penrose diagram for the black hole is not a square, as was first pointed out in [12]. It is convenient to introduce a complex quantity \nB = ˜ β + iβ = 2 π ( r 1 + ir 0 ) r 2 0 + r 2 1 = 2 π r 1 -ir 0 (2 . 6) \n˜ We will consider the complexified Kruskal spacetime in which points related by \nwhich will be important in our discussion. One can invert (2.3) to find r ( z ). In particular, for Re z > ˜ β 4 , r is a one-to-one periodic function of z with period i β 2 . \nt → t + i m + n 2 β, z → z + i m -n 2 β, m,n ∈ Z (2 . 7) \nare identified. The Lorentzian section (fig. 1) of the complexified spacetime can be conveniently described using ( t, z ) with constant imaginary parts. For example, up to identifications (2.7), region III can be specified by \nIm t = -iβ 2 , Im z = 0 . (2 . 8)', '2.2. Boundary Wightman functions': "Now consider an operator O in the boundary theory corresponding to a bulk field φ of mass m . In the supergravity limit, the conformal dimension of O is given by [3,2] \n∆ = d 2 + ν, ν = √ d 2 4 + m 2 . (2 . 9) \nThermal boundary two-point functions of O can be obtained from free bulk Green functions of φ in the Hartle-Hawking vacuum by taking the arguments of φ to the boundary (see e.g. [26,27]). 6 For example, the boundary Wightman function is obtained by \nG + ( x, x ' ) = lim r,r ' →∞ (2 νr ∆ )(2 νr ' ∆ ) G + ( x, r ; x ' , r ' ) (2 . 10) \nwhere G + and G + denote the bulk and boundary correlation functions respectively \nG + ( x, x ' ) = Tr [ e -βH O ( x ) O ( x ' ) ] , (2 . 11) \nG + ( x, r ; x ' , r ' ) = 〈 0 | φ ( r, x ) φ ( r ' , x ' ) | 0 〉 HH . (2 . 12) \nIn the above equations we used the notation x = ( t, e ) with e denoting a point on S 3 and the subscript ' HH ' denotes the Hartle-Hawking vacuum. Going to momentum space (2.10) becomes (see Appendix A for a definition of the Fourier transform) \nG + ( ω, l ) = lim r,r ' →∞ (2 νr ∆ )(2 νr ' ∆ ) G + ( ω, l ; r, r ' ) , (2 . 13) \nwhere l denotes the angular momentum on S 3 . The Feynmann (retarded) propagator in the bulk leads to the Feynmann (retarded) Green function on the boundary by the same procedure. In this paper we will focus on G + for reasons to be commented on later. \nSince the extended black hole background has two asymptotic boundaries, we can also take r and r ' to different boundaries. Such points are always space-like separated in the bulk and lead to boundary correlation functions of complex time separation (see equation (2.8)) \nG 12 ( t ) = Tr e -βH O ( t -iβ/ 2) O (0) ] = G + ( t -iβ/ 2) (2 . 14) \n[ \n] where we have suppressed the boundary spatial coordinates for notational simplicity. G + ( t ) is analytic for -β < Im t < 0 and the two-sided correlator G 12 ( t ) can be obtained from G + ( t ) by simple analytic continuation 7 . \nG + can be found in terms of solutions to the Laplace equation for φ following the standard free field quantization procedure. Let \nφ = e -iωt Y I ( e ) r -d -1 2 ψ ( ω, p ; r ) , \nwith Y I ( e ) denoting scalar spherical harmonics on S 3 (see appendix A for notations on spherical harmonics). The Laplace equation for φ can then be written in terms of the tortoise coordinate as a Schrodinger equation for ψ \n( -∂ 2 z + V l ( z ) -ω 2 ) ψ = 0 . (2 . 15) \nV l ( z ) can be expressed through r ( z ) as \nV l ( z ) = f ( r ) [ ( l +1) 2 -1 4 r 2 + ν 2 -1 4 + 9 µ 4 r 4 ] (2 . 16) \nFor real l > 0, V l ( z ) is a monotonically decreasing function of z ∈ (0 , ∞ ). Near the boundary, \nV l ≈ ν 2 -1 4 z 2 , z → 0 , (2 . 17) \nand near the horizon \nV l ∝ e -4 π β z → 0 , z → + ∞ . (2 . 18) \nFor any given real ω the Schrodinger equation (2.15) has a unique normalizable mode ψ ωl , which we will take to be real. We normalize it at the horizon as ( δ ω is a phase shift) \nψ ωl ( z ) ≈ e -iωz -iδ ω + e iωz + iδ ω , z → + ∞ . (2 . 19) \nAs z → 0, ψ ωl has the form \nψ ωl ≈ C ( ω, l ) z 1 2 + ν + · · · , z → 0 (2 . 20) \nwhere the constant C is fixed by the normalization of (2.19). It is easy to check that ψ ωl is even in ω . \nUsing the mode expansion of φ in the Hartle-Hawking vacuum, the bulk Wightman propagator G + (2.14) in momentum space can be written in terms of ψ ωl as \nG + ( ω, l ; r, r ' ) = 1 2 ω e βω e βω -1 ( rr ' ) -d -1 2 ψ ωl ( r ) ψ ωl ( r ' ) (2 . 21) \nwhich leads to the boundary G + upon using (2.13) and (2.20) \nG + ( ω, l ) = (2 ν ) 2 2 ω e βω e βω -1 C 2 ( ω, l ) (2 . 22) \nG + is to be evaluated for real ω, l and can be analytically continued to general complex ω and l .", '2.3. Analytic properties': "Equations (2.21) and (2.22) indicate that the boundary Yang-Mills theory has a continuous spectrum in the large N and large 't Hooft coupling limit, despite being on a compact space. At finite N , the theory should have a discrete spectrum on S 3 . In the bulk the continuous spectrum arises due to the presence of the horizon. \nThe analytic properties of G + in the complex ω -plane for a given l can be deduced by applying standard techniques of scattering theory to (2.15). The fact that r is a periodic function of the tortoise coordinate with a period i β 2 implies that: [23] \n- 1. The poles in the prefactor 1 ω ( e βω -1) cancel with zeros of C 2 . G + is analytic at ω = 0 and ω = 2 πin β , n ∈ Z . \nThese two features are quite generic, applicable to AdS black holes of all dimensions. The poles of G + in the lower half ω -plane coincide with those of the retarded propagator G R and correspond to quasi-normal frequencies of the black hole background. Since it is not known how to solve the Schrodinger equation (2.15) exactly, the determination of the quasi-normal poles is a difficult mathematical problem. \n- 2. The only singularities of G + in the complex ω -plane are poles. The locations of the poles obey a reflection symmetry: if there is a pole at ω 0 , then there are poles at -ω 0 , ω ∗ 0 , -ω ∗ 0 . \nFig. 2: Poles for G + ( ω, l ) for l = 0 in the complex ω -plane. We use r 0 = 1 , r 1 = √ 2. \n<!-- image --> \nWhen l = 0 (or small compared to ν or ω ), the problem simplifies and various methods [32,33,34,35,36] can be used to determine the locations of poles of G + approximately. One finds that there are four infinite lines of poles as indicated in fig. 2. The poles in the upper right quadrant are given by [24] \nω ≈ 2 π B ν + ω 0 + 4 πn B , n = 0 , 1 , · · · (2 . 23) \nThe other lines are obtained by reflections. ω 0 in (2.23) is a constant of O (1) (independent of ν ) whose exact value is not relevant here 8 . \nThe quasi-normal frequencies for l /negationslash = 0 and other dimensions are more complicated to find and will be discussed in [24].", '3.1. A semi-classical approximation': "We now develop a 'semi-classical' approximation to equation (2.15), in the following large ν limit \nω = νu, l +1 = νk, ν /greatermuch 1 , (3 . 1) \ni.e. we take the mass m of φ to be large and 'measure' the frequency ω and angular momentum l in units of m . With ψ = e νS equation (2.15) becomes \n-( ∂ z S ) 2 -1 ν ∂ 2 z S + V ( z ) + 1 ν 2 Q ( z ) = u 2 (3 . 2) \nwith \nV ( z ) = f ( r ) ( 1 + k 2 r 2 ) , Q ( z ) = f ( r ) [ -1 4 r 2 -1 4 + 9 µ 4 r 4 ] . (3 . 3) \nFig. 3: The potential V ( z ) with µ = 10 and k = 0 is shown. z c is the turning point. Dashed and solid lines indicate the classically forbidden and allowed regions respectively. \n<!-- image --> \nWith k 2 ≥ 0 the leading order potential V ( z ) is a monotonically decreasing function for z ∈ (0 , + ∞ ) as indicated in fig. 3. For scattering sates with u > 0, (3.2) can be solved order by order in 1 /ν expansion using the standard WKB method. In the classically forbidden region (see fig. 3), the exponentially decreasing solution can be written as \nwith 9 \nψ ( wkb ) ωl ( r ) = 1 √ fκ r e ν Z ( 1 + O ( ν -1 )) (3 . 4) \nZ ( r ) = -∫ r r c dr ' κ r ( r ' ) , κ r = 1 f √ V ( r ) -u 2 . (3 . 5) \nV ( r ) = f ( r ) ( 1 + k 2 r 2 ) = u 2 . (3 . 6) \nr c in the lower integration limit of (3.5) is the turning point, given by the real positive root of the equation \nFor u 2 > 0, equation (3.6) has a unique positive root r c > r 0 . Z satisfies the equation \nf Z ' 2 -k 2 r 2 + 1 f u 2 = 1 (3 . 7) \nwith Z ' ( r c ) = 0. Note that we have written the above equations in terms of r for convenience. One can equivalently write them in terms of the tortoise coordinate z . The expressions in terms of r are more convenient to visualize the analytic continuation to be discussed later. \nThe expression for ψ ( wkb ) ωl in the classically allowed region of the potential (3.3) (i.e. for z c < z or r 0 < r < r c ) follows from the standard connection formula, from which one can determine the relative normalization between ψ ( wkb ) ωl of (3.4) and ψ ωl of (2.19) to be \nψ ( wkb ) ωl = 1 √ u ψ ωl , u > 0 , ν →∞ (3 . 8) \nFrom (3.8) we find that in the limit (3.1) the boundary Wightman function G + can be expanded as \nwith \nG + ( ω, l ) ≈ 2 ν e νZ ( u,k ) ( 1 + O ( ν -1 ) ) +subdominant terms , ω > 0 (3 . 9) \nZ ( u, k ) = 2 lim r →∞ ( log r -∫ r r c dr ' κ r ( r ' ) ) . (3 . 10) \nHigher order 1 /ν corrections in (3.9) can also be obtained from (3.2) using the standard WKB procedure. In particular, the term proportional to Q ( z ) will be important at order ν -1 . There could also be subdominant terms in (3.9) coming from reflections at other (complex) turning points of V ( r ). \nWhile equations (3.9)-(3.10) were obtained for u > 0 and k ≥ 0, they can be analytically continued to the full complex u and k -planes. We will show in section 4 that the analytic continuation allows us to probe the region beyond the horizon.", '3.2. Relation with geodesics': "We expect Z ( u, k ) in (3.10) to have a simple interpretation in terms of bulk geodesics. The reason is that in the large mass limit, the propagation of bulk field φ should approximately follow geodesic paths. Thus we expect a direct relation between the WKB approximation of the last subsection with the geodesic approximation. The scaling in (3.1) simply defines u as the 'velocity' in t direction and k as the 'angular velocity' on S 3 . \nDue to translational invariance in t and isometries on S 3 , a bulk spacelike geodesic is characterized by the integrals of motion \nE = f dt ds , q = r 2 dθ ds (3 . 11) \nwhere s is the proper distance and θ denotes the angular coordinate along the geodesic motion on S 3 . We treat geodesics which are related by a translation in t and on S 3 as equivalent. The geodesic satisfies the equation \n1 f ( dr ds ) 2 + q 2 r 2 -1 f E 2 = 1 . (3 . 12) \nEquation (3.12) is precisely (3.7) with the identification 10 \nf Z ' = dr ds , u = iE, k = iq . (3 . 13) \nκ r of (3.5) can be identified as the proper velocity of the geodesic along the r direction. Thus Z ( u, k ) can be associated with a (complex) spacelike geodesic with constants of \nmotion E = -iu and q = -ik , which starts and ends at r = + ∞ . 11 More explicitly, one finds that Z ( u, k ) can be written as \nZ ( u, k ) = -Et ( E,q ) -L ( E,q ) + qd ( E,q ) (3 . 14) \nwhere L ( E,q ) is the (regularized) proper distance of the geodesic, t ( E,q ) is the time separation and d ( E,q ) is the proper distance on S 3 between the final and initial points, \nAlso note the relation \nL ( E,q ) = 2 lim r →∞ ∫ r r c dr √ f + E 2 -f r 2 q 2 -log r t ( E,q ) = 2 E ∫ ∞ r c dr f √ f + E 2 -f r 2 q 2 d ( E,q ) = 2 q ∫ ∞ r c dr r 2 √ f + E 2 -f r 2 q 2 . (3 . 15) \n∂Z ∂E = -t ( E,q ) ∂Z ∂q = d ( E,q ) (3 . 16) \nwhich shows that L ( t, d ) and Z ( E,q ) are related by a Legendre transform. \nNote that E and q do not specify a geodesic uniquely. (3.15) defines a complex geodesic with a choice of root r c ( E,q ) of equation (3.6) as the turning point and a contour from r c ( E,q ) to + ∞ . For the same value of E,q , a different choice of the root or a different contour which cannot be smoothly deformed into the previous one defines a different complex geodesic. The identification (3.14) with the boundary Z ( u, k ) selects a specific one among them. \nIn the above discussions we have concentrated on the Wightman functions 12 . A similar relation with bulk geodesics can be established for retarded and Feynmann functions in momentum space, whose story is more complicated since their dependence on ν is not uniform. Special care is needed when ν is an integer 13 . This makes the large ν limit more subtle. Even at the leading order, one has to take into account of an infinite number of classical paths in the WKB approximation [37]. Fortunately, all these additional complications arise due to the asymptotic behavior of f ∼ r 2 near the boundary of spacetime and do not seem to give additional insight into the question of physics beyond the horizon.", '3.3. Coordinate space correlation functions': "We now look at the Fourier transform of G + ( ω, l ) to the coordinate space correlator (see Appendix A for notations) \nG + ( t ; e, e ' ) = 1 4 π 2 ∞ ∑ l =0 2( l +1) C l ( e · e ' ) ∫ ∞ -∞ dω 2 π e -iωt G + ( ω, l ) . (3 . 17) \nThe two-sided correlator (2.14) can be obtained from (3.17) by taking t → t -i β 2 , while the Euclidean correlator G E ( τ ; e, e ' ) can be obtained by taking t = -iτ with 0 < τ < β . \nIn the large ν limit (3.1), using (3.9) we can approximate the sum over l in (3.17) by an integral over k \nG + ( t, θ ) ≈ ν 3 8 π 3 i sin θ ∫ ∞ -∞ dudk k e -iνut + iνkθ 2 νe νZ ( u,k ) (3 . 18) \nwhere θ = cos -1 ( e · e ' ) and we have extended the integration range for k to ( -∞ , ∞ ) using that Z is an even function of k . (3.18) can be evaluated by the method of steepest descent with the saddle points determined by \n∂Z ∂u = it, ∂Z ∂k = -iθ . (3 . 19) \nUsing equations (3.16) and (3.13) we find that (3.19) become \nt = t ( E,q ) , θ = d ( E,q ) (3 . 20) \ni.e. bulk geodesics with end point separation given by ( t, θ ) appear as saddle points of (3.18). Since from (3.14) the regularized geodesic distance L ( t, θ ) and Z ( u, k ) are related by a Legendre transformation, one finds that \nG + ( t, θ ) ≈ ∑ i 2 νJ 1 2 i ( ν 2 π ) 2 e -νL i ( 1 + O ( ν -1 ) ) (3 . 21) \nwhere i sums over the saddles along the steepest descent contour. The Jacobian J is due to the Gaussian integration around the saddle points.", 'Some remarks:': "- 1. J can be interpreted as the density of the geodesics. It is proportional to the Van Vleck-Morette determinant for the geodesics. One can check that (3.21) agrees precisely (including the prefactor J ) with the expression obtained directly from the geodesic approximation to the coordinate space path integral \nG ( x, r ; x ' , r ' ) = ∑ paths e i ¯ h mS , (3 . 22) \nafter taking the end points to the boundary. \n- 2. In the standard geodesic approximation to (3.22), it is often a subtle question in Lorentzian signature to determine which geodesics contribute to the sum (3.21). In our approach, the steepest descent approximation of the Fourier transform (3.17) gives a precise prescription for determining the sum.\n- 3. Higher order terms in (3.21) can be computed systematically in our approach.", '4. Black hole singularites in Yang-Mills theory': 'In this section we discuss how information about the black hole singularity can be extracted from Z ( u, k ) using its relations with the bulk geodesics developed in the last section. For simplicity, we restrict our discussion to k = 0, in which case the corresponding bulk geodesic has zero angular momentum. The discussion for general k is more involved and will appear in [23]. The k = 0 case already captures many of the essential elements.', '4.1. Probing the physics beyond the horizon and near the singularity': "We first consider the analytic continuation of Z ( u, k = 0) 14 to general complex u . We will see that the analytic continuation allows us to probe the regions beyond the horizon and near the singularity. \nAs discussed in section 2.3, when u > 0, the turning point r c ( u ) in (3.10) lies outside the horizon, i.e. r c > r 0 . The integration contour runs along the positive real r -axis from r c to + ∞ . The analytic continuation of Z ( u ) to the full complex u -plane involves the following two aspects: \n- 1. Analytically continue the turning point r c ( u ) from that for real u > 0;\n- 2. Smooth deformation of the integration contour as r c moves on the complex r -plane. \nBoth steps have some subtleties, which we now discuss in detail. \nFig. 4: The structure of branch cuts of Z ( u ) and r c ( u ) for r 0 = 1 , r 1 = 2. At finite ν , the branch cuts become the pole lines of G + ( ω, l = 0) as in fig. 2. The asymptotic regions are labelled by S ± or B ± , indicating whether the turning point for the corresponding u approaches the singularity ( S ± ) or the boundary ( B ± ). \n<!-- image --> \nThe analytic continuation of r c from the u > 0 region is not unique, since r c ( u ) has branch points in the complex u -plane at which it merges with other solutions of (3.6). These are also branch points of Z ( u ). When k = 0 (3.6) is a quadratic equation for r 2 and r c ( u ) coincides with other roots when \n( u 2 -1) 2 +4 µ = 0 . (4 . 1) \nFrom (4.1) we find that r c ( u ) has four branch points at \nu 0 = ± ( r 1 ± ir 0 ) = ± 2 π , ± 2 π B . (4 . 2) \nB \nB For r c and Z to be single-valued on the u -plane, branch cuts have to be specified. Different choices of the branch cuts correspond to different ways of performing the analytic continuation. The locations of the branch cuts cannot be determined from the integrals (3.10) or (3.15) alone. To determine them we need to use analytic properties of G + ( ω ). As discussed around equation (2.23), the only singularities of G + ( ω ) at finite ν are four lines of poles located at \nu ≈ 2 π + ω 0 ν + 4 πn ν B , n = 0 , 1 , · · · . (4 . 3) \nB \nB and the reflections of (4.3) with respect to the real and imaginary u axes. In the large ν limit, since the spacings between poles go to zero, these lines of poles become branch cuts of Z ( u ). 15 This determines the directions of the branch cuts to be along the radial direction from each branch point to infinity (see fig. 4). \nFig. 5: A radial spacelike geodesic can be described by a particle of energy -u 2 moving in the potential U = -f . The horizon is at r = r 0 . For u 2 < 0, the turning point lies inside the horizon. \n<!-- image --> \nWith the branch cuts precisely specified, r c ( u ) can now be uniquely determined from that for u > 0. In particular, fig. 4 implies that the analytic continuation should be done through the region around u = 0. To to be definite, let us concentrate on real u 2 . In this case, a convenient way to visualize how the turning point changes with u is to treat equation (3.12) as the motion of a one-dimensional particle of energy E 2 = -u 2 , moving in a potential 16 \nU = -V = -f , (4 . 4) \nas in fig. 5. For real u , r c > r 0 , i.e. the turning point lies outside the horizon, while for u pure imaginary, r c < r 0 and the the turning point lies inside the horizon. One can also solve (3.6) explicitly and r c ( u ) is given by the positive branch of the equation \nr 2 c = √ µ + ( 1 -u 2 2 ) 2 -1 -u 2 2 . (4 . 5) \nWith the turning point specified, one can now find the bulk geodesics corresponding to various values of u from simple geometric considerations. For example, real values of u correspond to radial geodesics in the Euclidean section of the spacetime (see fig. 6), since from (3.13) and (3.11) t is pure imaginary along the geodesics. Pure imaginary values of u = iE ( E real) correspond to real geodesics in the Lorentzian section of the spacetime, which connect two asymptotic boundaries (see fig. 7). The turning point of such a geodesic lies in region II for E > 0 and in region IV for E < 0 (see also fig. 1). \nFig. 6: Radial geodesics in the Euclidean section of the spacetime which corresponds to real values of u . The Euclidean section of the r -t plane is a disk with it as the angular coordinate and the origin of the disk at r = r 0 . The solid circle is the boundary. Geodesics with u > 0 (i and ii) and u < 0 (iii) are schematically plotted. Geodesic ii correspond to the large u limit, in which case the tuning point is close to the boundary and the end points of the geodesic are nearly coincidental. \n<!-- image --> \nFig. 7: Radial geodesics in the Lorentzian section of the spacetime corresponding to pure imaginary values of u = iE are schematically plotted. Geodesics i and ii have E > 0 while iii has E < 0. Geodesic ii correspond to the limit E → + ∞ , in which case the tuning point is close to the singularity and the geodesic becomes nearly null. \n<!-- image --> \nThe dependence of r c on u illustrates some interesting features in the relation between bulk and boundary scales. For real u →±∞ , the turning point is given by \nr c ≈ | u | → + ∞ (4 . 6) \ni.e. the turning point approaches the boundary. In this limit, the end points of the geodesic becomes nearly coincidental. When u decreases, r c also decreases. The turning point r c reaches the horizon for u = 0. This behavior reflects a familiar feature of the AdS/CFT correspondence, called IR/UV connection [1,38], which relates long distances in the AdS \nspacetime to high energies in the boundary theory. The turning point r c moves inside the horizon when u moves along the imaginary axis from the origin. Let u = iE . Then as | E | increases, r c decreases (see equation (4.5)). For | E | → + ∞ , we find that \nr c ≈ √ µ | E | → 0 (4 . 7) \ni.e. the turning point approaches the singularity. Thus when dealing with physics inside the horizon, there appears to be a new feature. To probe deeper inside the horizon requires larger E . Since the singularity may heuristically be considered as the UV of the bulk, we find a UV/UV connection. It is important to keep in mind that inside the horizon, r plays the role of the time coordinate. This indicates that the 'time' inside the horizon is indeed holographically generated from the boundary Yang-Mills theory. \nThe above discussions can be easily generalized to all complex values of u using equation (3.6). In particular, from (3.6), | r c | → 0 requires that | u | → ∞ , due to the fact that f blows up at the singularity (large curvature effect). Conversely, | u | → ∞ implies either | r c | → 0 or | r c | → + ∞ . Thus along different directions to infinity in the complex u -plane, the turning point either approaches the boundary or the singularity. The branch cuts in fig. 4 divide the complex infinity of the u -plane into various asymptotic regions. The regions which correspond to the singularity or the boundary are indicated in fig. 4. Near the real u axis, the turning point approaches the boundary as | u | → + ∞ . As | u | → + ∞ near the imaginary u axis, the turning point approaches the singularity. \nFig. 8: The integration contours in the complex r -plane for (a): u > 0, (b): u < 0, (c): u = iE, E > 0, (d): u = iE, E < 0. The solid dot indicates the pole r = r 0 of the integrand (horizon). \n<!-- image --> \nWe now look at the second aspect of the analytic continuation of Z ( u ). This involves the deformation of the integration contour in (3.10) (or (3.15)) as r c ( u ) moves in the complex r -plane. As the contour is deformed, one needs to be careful about the contribution of the pole at r = r 0 of the 1 /f factor in the integrand. To find the contribution of the pole, it is enough to consider when the turning point is close to the pole 17 . This happens when | u | is small, in which case a prescription for the contour deformation can be obtained from the requirement that Z ( u ) be analytic at u = 0. The contours for other values of u can then be obtained by continuous deformation without further subtlety. The resulting contours for real and pure imaginary u 's are plotted in fig. 8, from which the contribution of the pole can be readily obtained. \nFor real u < 0, the contribution from the pole to t ( u ) of (3.15) is -iβ . This is precisely the period of the complex time and matches with the geometric picture of fig. 6. When u = iE for real E , the imaginary part of t ( u ) solely comes from the pole contribution and we find \nIm t ( iE ) = -iβ 2 , E real . (4 . 8) \nThis shows that the end point of the corresponding geodesic lies in the other asymptotic boundary, i.e. region III of fig. 1, again consistent with fig. 7. \nTo summarize, through the function r c ( u ), we establish a correspondence between the complex u - and r -planes. G + ( νu ) evaluated at u in the large ν limit probes the black hole geometry near r c ( u ). The boundary correlation functions encode not only the bulk geometry outside the horizon, but also regions beyond the horizon and near the singularity. \nWe emphasize that the proper identification of the branch cuts for Z ( u ) and r c ( u ) (which in turn depends on the knowledge of poles for G + ( ω )) is crucial for our conclusion above. Different choices of the branch cuts could lead to completely different physical pictures. For example, a different analytic continuation procedure may lead to a r c ( iE ) that for real E is not given by the positive branch of (4.5), but some other root of the turning point equation (3.6). In that case one cannot associate the geodesics in fig. 7 with G + ( ω ) and it is not clear one could probe the physics beyond the horizon. \nGiven Z ( u ) from the large ν limit of the boundary G + ( νu ), the problem of finding the bulk metric is essentially a classical inverse scattering problem. Due to the large number of isometries of the background, the problem is effectively one-dimensional, being that of \na particle moving in (4.4). The equivalent one-dimensional problem can be phrased as follows. Consider sending a particle toward the potential from r = ∞ and waiting for it to come back. L ( E ) is then the (regularized) time interval for this scattering process. With the knowledge of L ( E ) for all values of E , one can in principle reconstruct the potential (4.4). At a given E , L ( E ) probes the behavior of the potential (4.4) near the turning point r c ( u ).", '4.2. Manifestations of singularities in boundary theories': 'We have found that the geometry around the black hole singularity is encoded in the behavior of G + ( ω ) near the imaginary infinity. We now examine the manifestations of the singularity explicitly. \nThe integrals in (3.15) can be evaluated explicitly and one finds 18 \nand \nL ( u ) = -1 2 log( A + ˜ A + A -˜ A -) + 2 log | B | 2 π t ( u ) = β 4 π log ( A + ˜ A -A -˜ A + ) -i ˜ β 4 π log ( A + ˜ A + A -˜ A -) -iβ 2 (4 . 9) \n˜ \nA ± = 1 2 ± u B 4 π , A ± = 1 2 ± u B 4 π (4 . 10) \nIn (4.9) the branch cuts of the logarithms are chosen to be straight lines extending radially from ± 2 π B and ± 2 π B to ∞ , as follows from the discussion in previous subsection (see fig. 4). \nL ( u ) = -2 log ( [ u ] 2 ) + ∞ ∑ n =1 a 2 n 2 n 1 u 2 n t ( u ) = t 0 -i 2 u -i ∞ ∑ n =1 a 2 n 2 n +1 1 u 2 n +1 (4 . 11) \nExpanding (4.9) for large | u | we find that L ( u ) and t ( u ) of (3.15) can be written as \nwhich lead to the expansion for Z ( u ) \nZ ( u ) = iut 0 +2log [ u ] 2 +2 -∞ ∑ n =1 a 2 n 2 n (2 n +1) 1 u 2 n . (4 . 12) \nIn (4.11) and (4.12), \na n = ( 2 π B ) n + ( 2 π B ) n \nand t 0 and [ u ] are given by \nt 0 = 0 u ∈ B + -iβ u ∈ B -B 2 u ∈ S + -B 2 u ∈ S -, [ u ] = u u ∈ B + -u u ∈ B --iu u ∈ S + iu u ∈ S -, (4 . 13) \n where B ± and S ± denote asymptotic regions in fig. 4 whose corresponding turning point approaches the boundary and the singularity respectively. The values of t 0 for various limits can be easily understood from the geometric pictures of the geodesics in fig. 6 and fig. 7. When u → ±∞ , the end points of the Euclidean geodesics become nearly coincidental. The value -iβ for u → -∞ is precisely the full period of the Euclidean circle. When u → ± i ∞ , the bouncing geodesics in fig. 7 become nearly null and the corresponding values of t 0 in (4.13) are twice of those in (2.5). \nEquations (4.12) implies that as ω = νu → ± i ∞ , the boundary correlation function behaves as \nG + ( ω, l = 0) ≈ 1 π (Γ( ν )) 2 ( ∓ i ω 2 ) 2 ν e iω ( ± ˜ β 2 -iβ 2 ) ( 1 + O ( 1 ω 2 )) (4 . 14) \nwhere the upper (lower) sign corresponds to ω → + i ∞ ( -i ∞ ). Note that the correlation function decays exponentially along these directions. For ω →±∞ near the real axis, we find \nG + ( ω, l = 0) ≈ 1 π (Γ( ν )) 2 ( ω 2 ) 2 ν ( 1 + O ( 1 ω 2 )) ω → + ∞ 1 π (Γ( ν )) 2 ( -ω 2 2 ν e βω ( 1 + O 1 ω 2 )) ω →-∞ . (4 . 15) \n( ( )) While equations (4.14) and (4.15) were derived in the large ν limit, they should hold for finite ν , since the | u | → ∞ limit should coincide with the limit | ω | = ν | u | → ∞ regardless of the value of ν . Note that (4.15) is precisely what one would expect of the large frequency behavior of a conformal field theory at finite temperature 19 . The exponential falloff in (4.14) reflects the presence of a curvature singularity in the bulk. The falloff is controlled by the complex parameter B (introduced in (2.6)) which characterizes the black hole geometry. \n) \n(', '4.3. Generalizations to nonzero angular momentum': 'The above discussions can be extended to Wightman functions of nonzero angular momenta. We summarize some main results here, leaving detailed discussions to [23]: \n- 1. For boundary angular velocity k real, which corresponds to geodesics of pure imaginary angular momentum, the structure of the branch cuts for Z ( u, k ) is similar to fig. 4. The locations of the branch points and the directions of the branch cuts depend nontrivially on k . At finite ν , the branch cuts become lines of poles of G + ( ω, l ).\n- 2. For bulk geodesics with real angular momentum q , an important new feature appears: there exist geodesic orbits with constant real r . The existence of such orbits leads to the appearance of virtual states (if there is an orbit lying inside the horizon) or bound states (if there is an orbit lying outside the horizon) in the Schrodinger problem (2.15). These virtual states or bound states lead to two new lines of poles of G + ( ω, l ) along the imaginary ω -axis for pure imaginary l . Thus Z ( u, k ) has two new branch cuts along the imaginary u -axis for k = iq pure imaginary.\n- 3. It remains true that the turning point approaches the boundary (singularity) for | u | → ∞ along the real (imaginary) axis. Furthermore, for any fixed l , in the large ω limit, equations (4.14)-(4.15) remain valid.\n- 4. In the limit that q goes to zero, the branch cuts for Z ( u, iq ) along the imaginary axis move to infinity and fig. 4 is recovered. The fact that there are branch points at u = ± i ∞ at q = 0 leads to interesting behaviors in the expansion of Z ( u, k ) around k = 0. For example, let u = iE with real E , then one finds that to leading order in the limit E → + ∞ , Z ( u, k ) has the following small k expansion 20 \nZ ( iE, k ) ≈ -E B 2 +2log E 2 +2+ 1 E 2 ∞ ∑ l =1 a l ( k 2 E 2 ) l + · · · (4 . 16) \nwith \na 1 = -π 2 µ 1 2 , a 2 = 3 π 16 µ 3 2 , a l ∼ 1 µ 2 l -1 2 (4 . 17) \nwhere µ dependence can be deduced based on dimensional analysis. Note that the expansion parameter for small k is given by k 2 E 2 and the derivatives over k at k = 0 become divergent in the large E limit. We will see in the next section that (4.16) leads to divergences in certain gauge invariant observables in the boundary theory. \nWe also mention by passing a few other generalizations: \n- 5. The discussions of this section can also be generalized to an AdS d +1 black hole of dimension d ≥ 2 [23]. All the essential features for AdS 5 black holes carry over to other dimension d ≥ 3. A special case is a BTZ black hole in AdS 3 , which has an orbifold singularity and the story is somewhat different. In the BTZ case, the corresponding Schrodinger equation (2.15) can be exactly solved and the relation between the large ν limit of Wightman functions and bulk geodesics can be explicitly verified.\n- 6. The subdominant contributions in (3.9) can also be worked out using a more sophisticated WKB method involving more than one turning point. One can show that an infinite number of subdominant contributions become important at the branch cuts 21 where they add up to produce the poles of G + ( ω, l ). In [24] we use this property to derive the positions of poles of G + for general l and dimension in the large ν limit.\n- 7. While we have not examined it in detail, it is interesting to compute the higher order 1 /ν corrections in (3.9). In particular, the function Q ( z ) (3.2) will start contributing at the order O (1 /ν ). Since Q ( z ) becomes singular at r → 0, it would be interesting to see whether it yields new manifestations of the singularity in the boundary theory correlation functions.', '4.4. Coordinate space correlators and alternative indications of curvature singularities': 'In this subsection we consider the Fourier transform of G + ( ω, l ) to coordinate space. To make connection to the result of [12], we consider \nG 12 ( t, θ = 0) = G + ( t -iβ 2 , θ = 0 ) (4 . 18) \nwhich can be obtained from (3.17) by taking t → t -iβ 2 and corresponds to inserting operators on two different boundaries. We restrict to θ = 0 for simplicity. In the large ν limit, G 12 can be evaluated in exact parallel of the discussion of section 3.3. One can approximate the sum over l by an integral \nG 12 ( t ) ≈ ν 4 8 π 3 ∫ ∞ -∞ dudk k 2 e -iνut -1 2 νuβ 2 νe νZ ( u,k ) (4 . 19) \nand perform the integrals using the saddle point method. \nFig. 9: The contour plot for the real part of Z ( u, k = 0) -iu ( t -i β 2 ) for t < ˜ β 2 in the complex u -plane. There is a saddle on the imaginary axis and two complex ones. The steepest descent contour is also shown in figure. The contour does not pass through the saddle on the imaginary axis even if it dominates. \n<!-- image --> \nFrom the discussion of section 3.3, the saddles of (4.19) correspond to geodesics whose end points lie on different boundaries and are separated in time by an amount t and no separation on S 3 . The saddle for the k -integral is simply k = 0. The saddles for the u -integral are given by \n∂Z ( u, k = 0) ∂u = it + β 2 . (4 . 20) \nNote that Z ( u, k = 0) can be obtained from (4.9). The solutions to (4.20) can be visualized conveniently on the contour plot of the real part of Z ( u, k = 0) -iu ( t -i β 2 ) in the complex u -plane. See fig. 9. In the figure we also indicate the steepest descent paths to which the integration contour of (4.19) can be deformed. \nThe dependence of the saddle point structure on t can be summarized as follows. For t < t c = ˜ β 2 , there are three saddles, as indicated in fig. 9. The one on the imaginary axis corresponds to a real geodesic in Lorentzian black hole spacetime with a turning point inside the horizon (see fig. 7). This is the bouncing geodesic discussed by [12]. We will refer to this saddle as the bouncing saddle below. The other two saddles describe complex geodesics, which do not seem to probe the physics beyond the horizon. As t approaches t c , the bouncing saddle moves to infinity along the imaginary axis and the turning point \nof the geodesic approaches the singularity. For t > t c , the bouncing saddle disappears 22 . \nFrom the steepest descent contour, we conclude that the bouncing geodesic does not contribute to coordinate space correlation functions. This result was obtained in [12] by analytical continuation from Euclidean signature 23 . Here we confirm their result.', '4.5. New observables of the boundary theory and manifestations of the curvature singularity': 'While the bouncing geodesic does not contribute to (4.18) directly, from momentum space correlation functions, we can easily construct observables in the boundary theory which are approximated by the bouncing geodesics in the large ν limit. \nFig. 10: The integration contours for G 12 ( t ) ( C 1 ) and H 12 ( τ ) ( C 2 ). \n<!-- image --> \nLet us start with two-sided correlator (4.18) with coincidental spatial coordinates \nG 12 ( t ) = Tr [ e -βH O ( t -iβ/ 2 , e ) O (0 , e ) ] = 1 4 π 2 ∞ ∑ l =0 2( l +1) 2 ∫ C 1 dω 2 π e -iωt G 12 ( ω, l ) (4 . 21) \nwhere the contour C 1 is along the real ω -axis and \nG 12 ( ω, l ) = e -ωβ 2 G + ( ω, l ) . (4 . 22) \n→∞ \nFrom (4.14)-(4.15) and the comments in sec. 4.3 regarding their generalizations to nonzero l , we find that the large ω behaviors of G 12 ( ω, l ) are given by \nG 12 ( ω, l ) ≈ 1 π (Γ( ν )) 2 ( ∓ i ω 2 ) 2 ν e ± iω ˜ β 2 ω →± i ∞ 1 π (Γ( ν )) 2 ( ± ω 2 ) 2 ν e ∓ βω 2 ω →±∞ (4 . 23) \nWe now construct new observables in the boundary theory \nH 12 ( τ ) = 1 4 π 2 ∞ ∑ l =0 2( l +1) 2 ∫ C 2 dω 2 π e -iωτ G 12 ( ω, l ) (4 . 24) \nwhere the contour C 2 are along the imaginary axis. H 12 ( τ ) can be defined for -˜ β 2 < τ < ˜ β 2 due to the exponential falloff (4.23) of G 12 ( ω ) along the imaginary axis. Note that H 12 ( τ ) is gauge invariant by definition. While its existence depends on the asymptotic behavior of G 12 ( ω, l ) along the imaginary ω -axis, it is an object which can in principle be intrinsically defined in the boundary theory 24 . \nIn the large ν limit (4.24) can be evaluated in exact parallel as (4.18)-(4.19) by approximating the sum over l by an integral and performing the saddle point approximation in the resulting integrals. The only difference is that the integration contour for ω is now along the imaginary axis rather than the real axis. Thus instead of picking up the two complex saddles in fig. 9, (4.24) is given by expanding around the bouncing saddle on the imaginary axis. \nAs δt = t c -τ → 0, the bouncing saddle moves to the imaginary infinity and the turning point r c of the corresponding geodesic approaches the singularity. In this limit, we can use (4.16) to approximate Z ( u, k ). Then the integrals for H 12 can be written as ( u = iE ) \nH 12 ( τ ) ∼ ∫ dE ∫ dkk 2 exp [ -νEδt +2 ν log E 2 + ν E 2 ∞ ∑ l =1 a l ( k 2 E 2 ) l ] ∼ ( E c ) 2 ν +1 ( 1 + ∞ ∑ n =1 c n ( E 2 c ν √ µ ) n ) ∼ 1 ( δt ) 2 ν +1 ( 1 + ∞ ∑ n =1 b n ( ν √ µδt 2 ) n ) (4 . 25) \nwhere the saddle for the k -integral is k = 0 and for the E integral is \nE c ≈ 2 δt →∞ . (4 . 26) \nNote that in (4.25) we have only worked out the qualitative δt dependence in the small δt limit rather than attempting a precise evaluation. The order of limits should be first ν →∞ and then δt → 0. In evaluating the high order 1 /ν terms in (4.25), there are two other sources of 1 /ν corrections that we did not take into account: 1 /ν corrections in (3.9) and those in turning the sum over l in (4.24) to an integral. Since we are only interested in power counting, apart from magic cancellations, this will not affect the qualitative behaviors in (4.25). The divergences in 1 /ν n terms arise from the k -integral and are due to the structure of the small k expansion in (4.16). \nNote that (4.25) is precisely what one expects of a bouncing geodesic as its turning point approaches the singularity. From the bulk point of view, one expects the contribution from the geodesic should take the form 25 \nJ 1 2 e -νL ( 1 + ∞ ∑ n =1 R n ν n ) (4 . 27) \nwhere L is the regularized the geodesic distance, J is the Van Vleck-Morette determinant. Higher order terms in (4.27) arise from cubic and higher order terms in the sum over paths around the geodesic and R n can be expressed in terms of bulk geometric quantities in the form of components of curvature tensors (and their derivatives) integrated along the geodesic. In general the explicit expressions for R n are very complicated (see e.g. [40]). On dimensional ground, one expects that in the limit that the turning point approaches the singularity R n ∼ 1 /epsilon1 n , with /epsilon1 the proper time between the turning point and the singularity. This leads to \nH 12 ∼ J 1 2 e -νL ( 1 + ∞ ∑ n =1 b n ( ν/epsilon1 ) n ) . (4 . 28) \nEquation (4.28) precisely agrees with (4.25) since from the metric (2.2) \n/epsilon1 ∼ ∫ r c 0 dr √ f ∼ r 2 c √ µ ∼ √ µ E 2 c ∼ √ µδt 2 \nwhere we have used (4.7) and (4.26). \nTo summarize we have constructed new gauge invariant observables (4.24) in the boundary theory that are sensitive to the physics beyond the horizon and in particular their behaviors near τ → ˜ β 2 precisely reflect the curvature divergence of the singularity. These observables are related nonlocally in time with (4.18). It would be interesting to better understand their meaning in Yang-Mills theories 26 .', '5. Discussions: Resolution of black hole singularities at finite N ?': "In this paper we established a direct relation between space-like geodesics in the bulk and the large operator dimension limit of the boundary Wightman functions G + ( ω, l ) in momentum space. The results present an intriguing picture on how physics beyond the horizon is encoded in the boundary theory. In particular, it gives a clear indication that the 'time' inside the horizon is holographically generated from the thermal Yang-Mills theory. \nThe poles of G + ( ω, l ) separate the asymptotic region of the complex ω -plane into several sectors (see fig. 2 and fig. 4). The sectors near the real axis describe the physics near the boundary while the sectors near the imaginary axis describe the physics near the singularity 27 . We found the following signals of the singularity in the boundary theory: \n- 1. G + ( ω, l ) falls off exponentially for ω → ± i ∞ (see equation (4.14) or (4.23)). The falloff is controlled by the complex parameter B (2.6), which characterizes the black hole geometry.\n- 2. We constructed new observables H 12 ( τ ) (equation (4.24)) in the boundary theory which are related nonlocally in time with coordinate space Wightman functions. The curvature divergence of the singularity is reflected in the divergences of H 12 ( τ ) as τ →± ˜ β 2 . While the leading order divergence of H 12 ( τ ) (i.e. the prefactor in (4.25)) can be attributed to 1. above, the divergences in higher order 1 /ν terms are due to more delicate behavior of Z ( u, k ) for small k and u →± i ∞ . \nWhile the above results were derived in the large ν expansion, the essential features should persist for finite ν . For example, we expect equations (4.14) and (4.16) should be valid for finite ν as well 28 . \nWe now comment on the implications of our results in resolving the black hole singularity. \nThe rich analytic behavior observed for G + in the complex ω -plane is tied to the fact that in the large N limit, the boundary theory has a continuous spectrum, even though it lives on a compact space. In the bulk, the continuous spectrum arises because of the presence of the horizon. In Yang-Mills theory, the continuous spectrum should be related to the fact that in the high temperature phase, typical states in the thermal ensemble have energy of order N 2 . 29 \nAt finite N , no matter how large, the boundary theory on S 3 has a discrete spectrum. In particular, the finite temperature Wightman function should have the form \nG + ( ω ) = 2 π ∑ m,n e -βE m ρ mn δ ( ω -E n + E m ) (5 . 1) \nwhich is a sum of delta functions along the real ω -axis, where m,n sum over the physical states of the theory. G + ( ω ) in equation (5.1) does not have an unambiguous continuation off the real axis. In particular, the procedures of analytically continuing G + to complex ω and taking the large N limit do not commute. Equation (4.14) arises by taking the large N limit first and then doing the analytic continuation. This appears to imply that at finite N , geometric notions associated with a black hole, such as the event horizon and the singularity, no longer exist. This is not surprising since the black hole geometry arises as a saddle point in the path integral of the boundary theory in a 1 /N expansion. If one does not use such an expansion, the geometric notions lose their meaning. Thus the singularity appears to be resolved at finite N . \nThe above arguments, however, do not tell us how the singularity is resolved. There are several possibilities according to which the singularity can be resolved in AdS/CFT: \n- I. The singularity is already resolved by α ' -effects in perturbative string theory, i.e. at finite 't Hooft coupling and infinite N . \nIIa. The singularity is resolved only at finite N . But at infinite N there is a large N phase transition at certain value of the 't Hooft coupling. \nIIb. The singularity is resolved only at finite N and there is no large N phase transition for any 't Hooft coupling. \nTo see which possibility is realized, it is important to investigate whether the signals of the singularity found this paper and in [12] persist to weak coupling in the large N limit. If the answer is yes, it would strongly suggest possibility IIb above. This would be a very desirable situation since one would then be able to study the black hole singularity in string theory using weakly coupled Yang-Mills theory and focusing on the large N limit. If the answer is no, then both I and IIa are possible. In the event that IIa is realized, one should still be able to detect signals of the singularity at weak coupling, even though the precise signals may not be directly obtainable from the results at strong coupling. \nIn any case, we believe the results in the paper should provide a valuable guide for understanding the black hole singularity in AdS/CFT. \nFinally, it would be interesting to apply the techniques we developed in this paper to other backgrounds, like charged or rotating black holes.", 'Acknowledgments': 'We would like to thank O. Aharony, M. Douglas, Q. Ejaz, D. Freedman, G. Horowitz, R. Jaffe, M. Kruczenski, S. Mathur, J. Maldacena, S. Minwalla, J. Negele, K. Rajagopal, M. Rozali, A. Scardicchio, R. Schiappa, N. Seiberg, A. Sen, S. Shenker, D. Son, L. Susskind, W. Taylor, B. Wecht and B. Zwiebach for very useful discussions. We also want to thank Center of Mathematical Sciences at Zhejiang university for hospitality during part of the work. This work is supported in part by Alfred P. Sloan Foundation and funds provided by the U.S. Department of Energy (D.O.E) under cooperative research agreement #DFFC02-94ER40818.', 'Appendix A. Fourier transform on S 3': "A complete set of scalar harmonics on S 3 can be written as Y lmm ' ( e ) transforming under ( l/ 2 , l/ 2) representations of SO (4) = SU (2) × SU (2) with -l/ 2 ≤ m,m ' ≤ l/ 2. We \nuse e to denote a point on S 3 and I = ( l, m, m ' ) to denote the full set of indices. Y I is normalized so that \n∫ S 3 Y I ( e ) Y J ( e ) = δ IJ ∑ m,m ' Y I ( e 1 ) Y I ( e 2 ) = 1 4 π 2 2( l +1) C l ( e 1 · e 2 ) \nwith \nAlso note that \nC l (cos θ ) = sin( l +1) θ sin θ \n∇ 2 S 3 Y I = -l ( l +2) Y I \nwhere ∇ 2 S 3 is the Laplace operator on S 3 . \nConsider a correlation function G ( e, e ' ) on S 3 which only depends on the geodesic distance between two points e and e ' . It can be expanded as \nG ( e, e ' ) = 1 4 π 2 ∞ ∑ l =0 2( l +1) C l ( e · e ' ) G ( l ) = ∑ I G ( l ) Y I ( e ) Y I ( e ' ) .", 'References': "- [1] J. M. Maldacena, 'The large N limit of superconformal field theories and supergravity,' Adv. Theor. Math. Phys. 2 , 231 (1998) [Int. J. Theor. Phys. 38 , 1113 (1999)] [arXiv:hep-th/9711200].\n- [2] E. Witten, 'Anti-de Sitter space and holography,' Adv. Theor. Math. Phys. 2 , 253 (1998) [arXiv:hep-th/9802150].\n- [3] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, 'Gauge theory correlators from non-critical string theory,' Phys. Lett. B 428 , 105 (1998) [arXiv:hep-th/9802109].\n- [4] E. Witten, 'Anti-de Sitter space, thermal phase transition, and confinement in gauge theories,' Adv. Theor. Math. Phys. 2 , 505 (1998) [arXiv:hep-th/9803131].\n- [5] J. M. Maldacena, 'Eternal black holes in Anti-de-Sitter,' arXiv:hep-th/0106112.\n- [6] G. T. Horowitz and S. F. 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2008PhRvL.101p1101S
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High-Energy Collision of Two Black Holes
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2008-01-01
| 29 | 0.45 | 159 |
['-', '-', '-', 'methods numerical', 'methods numerical', '-', '-', '-', '-']
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[]
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We study the head-on collision of two highly boosted equal mass, nonrotating black holes. We determine the waveforms, radiated energies, and mode excitation in the center of mass frame for a variety of boosts. For the first time we are able to compare analytic calculations, black-hole perturbation theory, and strong field, nonlinear numerical calculations for this problem. Extrapolation of our results, which include velocities of up to 0.94c, indicate that in the ultrarelativistic regime about 14±3% of the energy is converted into gravitational waves. This gives rise to a luminosity of order 10<SUP>-2</SUP>c<SUP>5</SUP>/G, the largest known so far in a black-hole merger.
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https://arxiv.org/pdf/0806.1738.pdf
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{'The high-energy collision of two black holes': "Ulrich Sperhake 1 , Vitor Cardoso 2 , 3 , Frans Pretorius 4 , Emanuele Berti 5 , Jos'e A. Gonz'alez 6 1 Theoretisch Physikalisches Institut, Friedrich Schiller Universitat, 07743 Jena, Germany 2 CENTRA, Departamento de F'ısica, Instituto Superior T'ecnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal 3 Department of Physics and Astronomy, The University of Mississippi, University, MS 38677-1848, USA 4 Department of Physics, Princeton University, Princeton, NJ 08544, USA 5 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109, USA and 6 Instituto de F'ısica y Matem'aticas, Universidad Michoacana de San Nicol'as de Hidalgo, Edificio C-3, Ciudad Universitaria. C. P. 58040 Morelia, Michoac'an, M'exico \nWe study the head-on collision of two highly boosted equal mass, nonrotating black holes. We determine the waveforms, radiated energies, and mode excitation in the center of mass frame for a variety of boosts. For the first time we are able to compare analytic calculations, black hole perturbation theory, and strong field, nonlinear numerical calculations for this problem. Extrapolation of our results, which include velocities of up to 0 . 94 c , indicate that in the ultra-relativistic regime about 14 ± 3% of the energy is converted into gravitational waves. This gives rise to a luminosity of order 10 -2 c 5 /G , the largest known so far in a black hole merger. \nPACS numbers: 04.25.D-, 04.25.dc, 04.25.dg, 04.50.-h, 04.50.Gh, 04.60.Cf, 04.70.-s \nI. Introduction. An important and long-standing problem in general relativity concerns the ultra-relativistic scattering of black holes (BHs). This is one of the most violent events one can conceive of in the theory. The lack of solutions has spurred much speculation about what may happen in this regime. For example, these events are a natural testing ground for the cosmic censorship conjecture: is there a class of initial conditions where they generically lead to the formation of a naked singularity, or do event horizons always form to cloth singular behavior in the geometry? \nRelated questions concern the ultra-relativistic scattering of particles. If the center of mass (CM) energy is beyond the Planck scale, gravity is expected to dominate the interaction. Furthermore, since the kinetic energy dominates over the rest mass energy, the gravitational interaction should be rather insensitive to the structure of the particles, implying that the trans-Planckian scattering of point particles should be well described by BH scattering [1]. This is of particular relevance for recent proposals to solve the hierarchy problem by adding 'large' extra dimensions [2], or an extra dimension with a warp factor [3], thus producing an effective electroweak Planck scale. This offers the exciting possibility that BHs could be produced in particle colliders and ultra high-energy cosmic ray interactions with the atmosphere [1, 4]. A naive estimate of the cross section for M Pl ∼ 1 TeV predicts that super-TeV particle colliders will produce BHs at a rate of a few per second, making the Large Hadron Collider (LHC) at CERN a potential black hole factory. An important element to search for BH production signatures is to understand the BH scattering process, and in particular the energy lost to gravitational radiation. Given that the beam commissioning to 7 TeV is scheduled for late 2008, this is a timely research topic. Further interesting applications of high-speed BH collisions to high-energy physics have recently been suggested by the AdS/CFT correspondence conjecture [5]. Particu- \nrly intriguing is the possibility of using this duality to understand properties of the quark-gluon plasma formed in gold ion collisions at Brookhaven's Relativistic Heavy Ion Collider (RHIC) through a study of ultra-relativistic BH collisions in AdS [6]. \nEarly attempts to understand the ultra-relativistic BH scattering problem were based on work by Penrose [7] in the 1970s. He modeled the spacetime metric as the union of two Aichelburg-Sexl waves [8], describing the collision of two infinitely boosted Schwarzschild BHs, and found a closed trapped surface at the moment of collision, giving an upper limit of roughly 29% of the initial energy of the spacetime radiated in gravitational waves. Beyond the collision event the solution is unknown. Given the extreme conditions of high-speed scattering it is unlikely that analytic solutions describing the full dynamics of the spacetime will be found, and therefore numerical methods must be employed. Only recently have long-term stable numerical evolutions of black-hole binaries been achieved [9]. The flurry of subsequent activity exploring the merger process has so far exclusively focused on rest-mass dominated scenarios (see [10] for a review). \nIn this Letter we report the first numerical solutions describing the collision of two equal mass BHs in the regime where the initial energy of the system is dominated by the kinetic energy of the BHs. In Sec. II we describe the problem setup, including the numerical code and initial conditions. We also review some existing analytical approximations to aspects of the problem, which will be important both to interpret the numerical results and to give some confidence in extrapolations of the results to infinite boost. In Sec. III we present the primary results, focusing on the gravitational waves emitted during the collision. Concluding remarks are given in Sec. IV. Unless stated otherwise, we use geometrical units G = c = 1. \n- II. Numerical Setup and Analysis Tools. The numerical simulations presented here have been performed with the Lean code, described in detail in [11], where \nhead-on collisions of different classes of initial data were compared. Here we exclusively study evolutions of puncture initial data [12] describing two equal mass, nonspinning, boosted BHs colliding with zero impact parameter in the CM frame. The initial coordinate separation between the punctures is set to r 0 , and the boosts are prescribed in the form of non-vanishing Bowen-York [13] parameters ± P for the initial linear momentum of either BH. The Hamiltonian constraint is solved using Ansorg's spectral solver TwoPunctures [14]. The irreducible masses M irr1 , 2 of the BHs are estimated from their apparent horizon areas, calculated using Thornburg's apparent horizon finder AHFinderDirect [17]. This enables us to calculate the BH masses M 1 , 2 from Christodoulou's [15] relation M 2 1 , 2 = M 2 irr1 , 2 + P 2 , from which we define the Lorentz boost parameter γ ≡ M 1 , 2 /M irr1 , 2 (cf. [16]). From a numerical point of view, simulations with large values of γ are challenging, partly because the Lorentz contraction decreases the smallest length scale that needs to be resolved. Thus mesh-refinement is essential, and here it is provided via the Carpet package [18]. \nWe use the Newman-Penrose scalar Ψ 4 to measure gravitational radiation. At an extraction radius r from the center of the collision we decompose Ψ 4 into multipole modes ψ lm of the spherical harmonics of spin-weight -2, -2 Y lm , according to Ψ 4 ( t, r, θ, φ ) = ∑ ∞ l =2 ∑ l m = -l -2 Y lm ( θ , φ ) ψ lm ( t, r ). Due to the symmetries of this problem, the only non-vanishing multipoles all have even l , m = 0, and are purely real, corresponding to a single polarization state h + . The energy spectrum and luminosity of the radiation are given by \ndE dω = ∑ l 1 16 π 2 | ˆ ψ l 0 ( ω ) | 2 ω 2 ≡ ∑ l dE l dω , (1) \n∣ \ndE dt = ∑ l lim r →∞ r 2 16 π ∣ ∣ ∣ ∫ t -∞ ψ l 0 ( ˜ t ) d ˜ t ∣ ∣ ∣ ∣ 2 ≡ ∑ l dE l dt , (2) \n∣ respectively, where a hat denotes Fourier transform. \nOur results are affected by three main sources of uncertainties: finite extraction radius, discretization and spurious initial radiation. We reduce the error arising from finite extraction radius by measuring the waveform components at several radii, and fitting them to an expression of the form ψ lm ( r, t ) = ψ (0) lm ( t )+ ψ (1) lm ( t ) /r . The waveform 'at infinity' ψ (0) lm ( t ) is the quantity reported throughout this work and used to calculate related quantities, such as the radiated energy. The uncertainty in this extrapolated value is estimated by performing a second fit including also a quadratic term ψ (2) lm /r 2 , and taking the difference between the first- and second-order fits. The resulting uncertainty in the radiated energy is typically ∼ 3 -5 %. \nTo estimate discretization errors we evolved the most challenging simulation with γ ≈ 3 with resolutions h = M/ 174, M/ 209 and M/ 244, where M = M 1 + M 2 . We observe convergence slightly below second order in the total radiated energy, and use a conservative estimate of \n10% for the resulting error near γ ≈ 3, which drops to a few percent in the non-boosted case (cf. [11]). \nFinally, the conformally flat puncture initial data is known to contain spurious gravitational radiation, which increases strongly with boost γ (from a few times 10 -5 for BHs at rest, to about 8% of the total ADM mass of the system for γ ≈ 3). In order to extract physically meaningful information, one has to separate the spurious radiation from the radiation generated by the collision itself. This is done by 'waiting' for the spurious radiation to pass the last extraction radius, and then discarding the earlier part of the wave signal. For large boosts, the amount of time between the trailing edge of the spurious radiation and the leading edge of the waves emitted during the collision is roughly r 0 / (4 γ 2 ). Thus, the initial separation required to cleanly extract the emitted signal increases rapidly with γ . Because large separations require larger computational domains and longer run-times, the spurious radiation effectively limits our ability to study very large γ . With current resources, we were able to use initial separations of up to 66 M for γ > 2, leading to an uncertainty in the total radiated energy which grows rapidly with boost, reaching a value of 5% for γ ≈ 3. By combining all errors, we estimate the total uncertainty in the radiated energy to be about 15% for γ ≈ 3, about 10% near γ = 2 and a few percent for simulations with small velocities. \nHigh-energy collisions are uncharted territory for numerical relativity. It is helpful, therefore, to have alternative methods for guidance and consistency checks. Besides Penrose's bound, we will make extensive use of extrapolations of Smarr's 'zero-frequency limit' (ZFL) [19] and of point particle (PP) calculations [20], where one considers a small object of mass m colliding with a massive BH of mass M BH to linear order in m/M BH . \nIII. Results. We ran a series of simulations from γ = 1 to γ ≈ 3, with initial separations as discussed in the previous section. In all cases the collision results in a single BH plus gravitational radiation, i.e. there is no sign of any violation of cosmic censorship. The final BH is born highly distorted. We measure the distortion by taking the ratio C of the proper equatorial to polar circumferences of the common apparent horizon (CAH). For the range of boosts studied here, the peak value is well fitted by the relation C peak ∼ 1 . 5 -0 . 5 /γ . Thus in the largeγ limit C peak ∼ 1 . 5, in agreement with Penrose's result of C = π/ 2 for a CAH consisting of two flat disks. After birth, the BH settles down to a Schwarzschild solution, and the gravitational radiation can be described as a superposition of quasinormal modes (QNMs) of the resulting BH. \nIn Fig. 1 we show the dominant component ψ 20 of the waveform from collisions with γ = 1 . 07 , 1 . 3 , 1 . 7 , 3 . 0 (corresponding to β = v/c /similarequal 0 . 36 , 0 . 64 , 0 . 82 , 0 . 94, respectively). The origin of the ( t -r ) axis roughly corresponds to the instant of formation of a CAH. One can identify three main parts in the waveforms: a precursor, a main burst at the onset of the CAH formation \nFIG. 1: Dominant multipolar component ψ 20 ( t -r ) for different values of β , as indicated in the inset. \n<!-- image --> \nFIG. 2: Energy spectrum for l = 2 and different values of β . Horizontal lines are the corresponding ZFL-PP predictions, vertical lines are the QNM frequencies of the final BH. \n<!-- image --> \nand the final ringdown tail. These seem to be universal properties of collisions involving BHs and were observed in the past in different settings [20, 21]. The start of ringdown, roughly associated with the absolute maxima | ψ peak 20 | in | ψ 20 | , occurs ∼ 15 M after the CAH formation, independently of γ . Except for a small neighborhood around γ ∼ 1, the maximal wave amplitude | ψ peak 20 | increases monotonically with the boost factor. The small dip in the wave amplitude for small, but non-zero velocities has been seen before both in numerical simulations and analytic predictions [22]. For moderate boosts, we observe the absolute maxima in ψ 20 to be well approximated by | Mrψ peak 20 | ≈ 0 . 26 + 0 . 48 γ -2 [1 / 4 + log(1 / 2 γ )] [cf. Eq. (3) below]. The peak amplitude in the waveform h 20 is roughly h peak 20 ∼ ψ peak 20 /ω 2 QNM , where ω QNM is the lowest ringdown frequency for the mode [23]. \nFigure 2 shows the energy spectrum (1) for collisions with different CM energy. For large CM energies, the spectrum is nearly flat up to some cutoff frequency. A flat spectrum is predicted by the ZFL and PP approaches, as indicated by the dotted lines in the figure. The cutoff frequency is well approximated by the least-damped QNM of the final hole, marked by a vertical line. The spectrum increases at small frequencies because of initial data con- \nFIG. 3: Total radiated energy (including error bars) as a function of β , and best fit using the ZFL prediction. \n<!-- image --> \ntamination and finite-distance effects. \nOur numerical results indicate that the peak luminosity (2) is attained approximately 10 M after the CAH formation. The peak luminosity is about 5 × 10 -3 for β = 0 . 9, and may be as large as 10 -2 as γ → ∞ . Restoring units, we get 10 -2 c 5 /G ∼ 3 . 6 × 10 57 erg s -1 , the largest luminosity from a BH merger known to date. This is two orders of magnitude larger than for the infall from rest of two equal mass BHs, and one order of magnitude larger than for the inspiral of equal mass binaries. Nevertheless, it is still two orders of magnitude below the universal limit suggested by Dyson, dE/dt /lessorsimilar 1 [24]. \nThe total energy E radiated as a function of boost parameter is shown in Fig. 3. Error bars on the radiated energies are determined as described in Sec. II. We have verified that E calculated from the radiation (2) is consistent with alternative estimates obtained by directly measuring the mass of the final hole from the CAH properties, and by using the ringdown frequency to estimate the mass of the final hole [23]. The ZFL predicts the following functional form for the total radiated energy as a function of CM boost γ : \nE M = E ∞ ( 1 + 2 γ 2 2 γ 2 + (1 -4 γ 2 ) log ( γ + √ γ 2 -1) 2 γ 3 γ 2 -1 ) . (3) \nThe quantity E ∞ is some unknown cutoff parameter, which is also the total fraction of energy radiated as γ → ∞ . By fitting Eq. (3) to the numerical data we obtain E ∞ = 0 . 14 ± 0 . 03. The ZFL is a perturbative calculation about ω = 0, and its validity for our scattering problem is not obvious. However, given the good agreement with our numerical results in the kinetic-energy dominated regime γ > 2, the extrapolation procedure should provide a reasonably accurate estimate for E ∞ . \n√ \nWith regard to the multipolar contributions of the radiated energy, we find that E 4 is at least one order of magnitude smaller than E 2 for slow-motion collisions. This observation is consistent with the PP results for an infall from rest [20], which predict an exponential decrease of E l with l . For larger boosts the ZFL and PP approach predict a strong increase in the relative contri- \nTABLE I: Relative multipolar contribution (in %) and, in parentheses, the ZFL prediction. \n| β | 0.64 | 0.75 0.82 0.86 0.90 0.94 |\n|-------------|--------|----------------------------------------------------|\n| E 4 /E 2 | | 1.0(1.4) 2.4(3.4) 3.9(5.4) 5.0(7.3) 7.3(10) 11(14) |\n| 10 E 6 /E 2 | | 0.2(0.3) 1.1(1.5) 2.1(4.0) 4.2(7.5) 11(16) 33(30) | \nbution of higher multipoles, with E l ∼ M/l 2 as γ →∞ . Our numerical results are in reasonable agreement with these calculations, as demonstrated in Table I. The discrepancies still present are due to the relatively large uncertainties in the energy carried by higher multipoles and to the breakdown of the ZFL prediction for small boosts. IV. Conclusions. In 1971, Hawking [25] placed an upper limit of 29% on the total energy radiated when two BHs, initially at rest, coalesce. 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Lett. 96 , 111102 (2006).\n- [10] F. Pretorius, arXiv:0710.1338 [gr-qc].\n- [11] U. Sperhake, Phys. Rev. D 76 , 104015 (2007). \nclose to the estimate of D'Eath and Payne computed using perturbative techniques [26]. Even though our calculations are in 4D, a consequence of this to searches for BH formation at the LHC is a warning that estimates of the 'missing energy' based upon trapped surface calculations could significantly overestimate this effect. \nThis long overdue study represents an important step towards a full understanding of high-energy BH collisions. More accurate evolutions using significantly larger boosts are mainly inhibited by the junk radiation in the initial data. More work is also needed to study scattering with non-zero impact parameter, unequal masses and non-zero spins. For applications to LHC and RHIC physics, including the effects of extra dimensions, charge and AdS asymptotics (for RHIC) will be necessary. \nAcknowledgements. This work was supported in part by DFG grant SFB/TR 7, by FCT - Portugal through projects PTDC/FIS/64175/2006 and POCI/FP/81915/2007, and by the Fulbright Foundation (V.C.). E.B. was supported by the NASA Postdoctoral Program at JPL/Caltech, administered by Oak Ridge Associated Universities through a contract with NASA. F.P. was supported by the Alfred P. Sloan Foundation and NSF PHY-0745779. Computations were performed at LRZ Munich, Milipeia at CFC in Coimbra, and the Woodhen cluster at Princeton University. \n- [12] S. Brandt and B. Brugmann, Phys. Rev. Lett. 78 , 3606 (1997).\n- [13] J. M. Bowen and J. W. York, Phys. Rev. D 21 , 2047 (1980).\n- [14] M. Ansorg, B. Brugmann and W. Tichy, Phys. Rev. D 70 , 064011 (2004).\n- [15] D. Christodoulou, Phys. Rev. Lett. 25 , 1596 (1970).\n- [16] G. B. Cook, J. W. York, Jr., Phys. Rev. D 41 , 1077 (1990).\n- [17] J. Thornburg, Phys. Rev. D 54 , 4899 (1996); Class. Quant. Grav. 21 , 743 (2004).\n- [18] Carpet Code homepage, http://www.carpetcode.org/\n- [19] L. Smarr, Phys. Rev. D 15 , 2069 (1977).\n- [20] M. Davis, R. Ruffini, W. H. Press and R. H. Price, Phys. Rev. Lett. 27 , 1466 (1971); C. O. Lousto and R. H. Price, Phys. Rev. D 55 , 2124 (1997); V. Cardoso and J. P. S. Lemos, Phys. Lett. B 538 , 1 (2002).\n- [21] P. Anninos, D. Hobill, E. Seidel, L. Smarr and W. M. Suen, Phys. Rev. Lett. 71 , 2851 (1993).\n- [22] J. G. Baker et al. , Phys. Rev. D 55 , 829 (1997).\n- [23] E. Berti, V. Cardoso and C. M. Will, Phys. Rev. D 73 , 064030 (2006).\n- [24] F. Dyson, Interstellar Communication , ed. A. G. W. Cameron (Benjamin, NY, 1963), as quoted by K. S. Thorne in Gravitational Radiation , eds. N. Deruelle and T. Piran (Amsterdam, Netherlands: North-holland, 1983).\n- [25] S. W. Hawking, Phys. Rev. Lett. 26 , 1344 (1971).\n- [26] P. D. D'Eath and P. N. Payne, Phys. Rev. D 46 , 694 (1992)."}
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2008PhRvD..78b4009D
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Effective one body approach to the dynamics of two spinning black holes with next-to-leading order spin-orbit coupling
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2008-01-01
| 10 | 0.45 | 159 |
['-', '-', '-', 'theory', '-', 'perturbation theory', '-', '-']
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[]
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Using a recent, novel Hamiltonian formulation of the gravitational interaction of spinning binaries, we extend the effective one body (EOB) description of the dynamics of two spinning black holes to next-to-leading order (NLO) in the spin-orbit interaction. The spin-dependent EOB Hamiltonian is constructed from four main ingredients: (i) a transformation between the “effective” Hamiltonian and the “real” one; (ii) a generalized effective Hamilton-Jacobi equation involving higher powers of the momenta; (iii) a Kerr-type effective metric (with Padé-resummed coefficients) which depends on the choice of some basic “effective spin vector” S<SUB>eff</SUB>, and which is deformed by comparable-mass effects; and (iv) an additional effective spin-orbit interaction term involving another spin vector σ. As a first application of the new, NLO spin-dependent EOB Hamiltonian, we compute the binding energy of circular orbits (for parallel spins) as a function of the orbital frequency, and of the spin parameters. We also study the characteristics of the last stable circular orbit: binding energy, orbital frequency, and the corresponding dimensionless spin parameter a^<SUB>LSO</SUB>≡cJ<SUB>LSO</SUB>/(G(H<SUB>LSO</SUB>/c<SUP>2</SUP>)<SUP>2</SUP>). We find that the inclusion of NLO spin-orbit terms has a significant “moderating” effect on the dynamical characteristics of the circular orbits for large and parallel spins.
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https://arxiv.org/pdf/0803.0915.pdf
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{'Effective one body approach to the dynamics of two spinning black holes with next-to-leading order spin-orbit coupling': "Thibault Damour ∗ \nInstitut des Hautes ' Etudes Scientifiques, 91440 Bures-sur-Yvette, France \nPiotr Jaranowski † \nFaculty of Physics, University of Bia/suppresslystok, Lipowa 41, 15-424 Bia/suppresslystok, Poland", 'Gerhard Schafer ‡': "Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universitat, Max-Wien-Pl. 1, 07743 Jena, Germany (Dated: October 31, 2018) \nUsing a recent, novel Hamiltonian formulation of the gravitational interaction of spinning binaries, we extend the Effective One Body (EOB) description of the dynamics of two spinning black holes to next-to-leading order (NLO) in the spin-orbit interaction. The spin-dependent EOB Hamiltonian is constructed from four main ingredients: (i) a transformation between the 'effective' Hamiltonian and the 'real' one, (ii) a generalized effective Hamilton-Jacobi equation involving higher powers of the momenta, (iii) a Kerr-type effective metric (with Pad'e-resummed coefficients) which depends on the choice of some basic 'effective spin vector' S eff , and which is deformed by comparable-mass effects, and (iv) an additional effective spin-orbit interaction term involving another spin vector σ . As a first application of the new, NLO spin-dependent EOB Hamiltonian, we compute the binding energy of circular orbits (for parallel spins) as a function of the orbital frequency, and of the spin parameters. We also study the characteristics of the last stable circular orbit: binding energy, orbital frequency, and the corresponding dimensionless spin parameter ˆ a LSO ≡ cJ LSO / ( G ( H LSO /c 2 ) 2 ) . We find that the inclusion of NLO spin-orbit terms has a significant 'moderating' effect on the dynamical characteristics of the circular orbits for large and parallel spins. \nPACS numbers: 04.25.-g, 04.25.Nx", 'I. INTRODUCTION': "Coalescing black hole binaries are among the most promising sources for the currently operating ground-based network of interferometric detectors of gravitational waves. It is plausible that the first detections concern binary systems made of spinning black holes, because (as emphasized in [1]) the spin-orbit interaction can increase the binding energy of the last stable orbit, and thereby lead to larger gravitational wave emission. This makes it urgent to have template waveforms accurately describing the gravitational wave emission of spinning binary black holes. These waveforms will be functions of at least eight intrinsic real parameters: the two masses m 1 , m 2 and the two spin vectors S 1 , S 2 . Due to the multi-dimensionality of the parameter space, it seems impossible for state-of-the-art numerical simulations to densely sample this parameter space. This gives a clear motivation for developing analytical methods for computing the needed, densely spaced, bank of accurate template waveforms. \nAmong existing analytical methods for computing the motion and radiation of binary black hole systems, the most complete, and the most promising one, is the Effective One Body (EOB) approach [1, 2, 3, 4]. This method was the first to provide estimates of the complete waveform (covering inspiral, plunge, merger, and ring-down) of a coalescing black hole binary, both for non-spinning systems [3], and for spinning ones [5]. Several recent works [6, 7, 8, 9, 10] have shown that there was an excellent agreement 1 between the EOB waveforms (for non-spinning systems) and the results of recent numerical simulations (see [11] for references and a review of the recent breakthroughs in numerical relativity). In addition, the EOB method predicted, before the availability of reliable numerical relativity (NR) results, a value for the final spin parameter ˆ a fin of a coalescing black hole binary [3, 5] which agrees within ∼ 10% with the \nresults of recent numerical simulations (see [11] for a review and references). Recently, it has been shown that the introduction of some refinements in the EOB approach, led to an EOB/NR agreement for ˆ a fin at the 2% level [12]. \nIn a previous paper [1] the EOB method (originally developed for non-spinning systems) has been generalized to the case of spinning black holes. It was shown there that one could map the third post-Newtonian (3PN) orbital dynamics, together with the leading order (LO) spin-orbit and spin-spin dynamical effects of a binary system onto an 'effective test particle' moving in a Kerr-type metric . In the present paper, we extend and refine the EOB description of spinning binaries by using a recently derived [13] Hamiltonian description of the spin-orbit interaction valid at the next to leading order (NLO) in the PN expansion. (The NLO spin-orbit effects in the harmonic-gauge equations of motion were first obtained in [14, 15].) Let us recall that LO spin-orbit effects are proportional to G/c 2 , while NLO ones contain two sorts of contributions: ∝ G/c 4 and ∝ G 2 /c 4 . Regarding the spin-spin coupling terms, we shall use here only the LO results which are made of two different contributions: the LO S 1 S 2 terms [16] (which have been recently extended to NLO in [17]), and the LO S 2 1 and S 2 2 terms. The latter are specific to Kerr black holes, being related to the quadrupole gravitational moment of a rotating black hole. 2 It was shown in [1] that the complete LO spin-spin terms (the sum of S 1 S 2 , S 2 1 , and S 2 2 terms) admitted a remarkable rewriting involving a particular linear combination S 0 , defined below, of the two spin vectors. This fact, together with the more complicated structure of spin-orbit terms at the NLO, will lead us below to define a particular, improved EOB description of spinning binaries. \nThe present paper consists of two parts: In the first part (Sections 2 and 3) we shall develop the formalism needed to finally define (in Section 4) our improved EOB description of spinning binaries. In the second part (Section 5), we shall consider one of the simplest 'applications' of our EOB Hamiltonian: a discussion of the energetics of circular, equatorial orbits for systems with parallel spins. In this section, we shall make contact with previous related analytical investigations, notably [15], and prepare the ground for making contact with numerical data. \nA few words about our notation: We use the letters a, b = 1 , 2 as particle labels. Then, m a , x a = ( x i a ), p a = ( p ai ), and S a = ( S ai ) denote, respectively, the mass, the position vector, the linear momentum vector, and the spin vector of the a th body; for a /negationslash = b we also define r ab ≡ x a -x b , r ab ≡ | r ab | , n ab ≡ r ab /r ab , | · | stands here for the Euclidean length of a 3-vector.", 'II. PN-EXPANDED HAMILTONIAN': "Our starting point is the PN-expanded (or 'Taylor-expanded') two-body Hamiltonian H which can be decomposed as the sum of: (i) an orbital part H o , (ii) a spin-orbit part H so (linear in the spins), and (iii) a spin-spin term H ss (quadratic in the spins), \nH ( x a , p a , S a ) = H o ( x a , p a ) + H so ( x a , p a , S a ) + H ss ( x a , p a , S a ) . (2.1) \nThe orbital Hamiltonian H o includes the rest-mass contribution and is explicitly known (in ADM-like coordinates) up to the 3PN order [18, 19]. Its structure is \nH o ( x a , p a ) = ∑ a m a c 2 + H oN ( x a , p a ) + 1 c 2 H o1PN ( x a , p a ) + 1 c 4 H o2PN ( x a , p a ) + 1 c 6 H o3PN ( x a , p a ) + O ( 1 c 8 ) . (2.2) \nThe spin-orbit Hamiltonian H so can be written as \nH so ( x a , p a , S a ) = ∑ a Ω a ( x b , p b ) · S a , (2.3) \nHere, the quantity Ω a is the sum of a LO contribution ( ∝ 1 /c 2 ) and a NLO one ( ∝ 1 /c 4 ), \nΩ a ( x b , p b ) = Ω LO a ( x b , p b ) + Ω NLO a ( x b , p b ) . (2.4) \nThe 3-vectors Ω LO a and Ω NLO a were explicitly computed in Ref. [13]. They are given, for the particle label a = 1, by \nΩ LO 1 = G c 2 r 2 12 ( 3 m 2 2 m 1 n 12 × p 1 -2 n 12 × p 2 ) , (2.5a) \nΩ NLO 1 = G 2 c 4 r 3 12 ( ( -11 2 m 2 -5 m 2 2 m 1 ) n 12 × p 1 + ( 6 m 1 + 15 2 m 2 ) n 12 × p 2 ) + G c 4 r 2 12 ( ( -5 m 2 p 2 1 8 m 3 1 -3( p 1 · p 2 ) 4 m 2 1 + 3 p 2 2 4 m 1 m 2 -3( n 12 · p 1 )( n 12 · p 2 ) 4 m 2 1 -3( n 12 · p 2 ) 2 2 m 1 m 2 ) n 12 × p 1 + ( ( p 1 · p 2 ) m 1 m 2 + 3( n 12 · p 1 )( n 12 · p 2 ) m 1 m 2 ) n 12 × p 2 + ( 3( n 12 · p 1 ) 4 m 2 1 -2( n 12 · p 2 ) m 1 m 2 ) p 1 × p 2 ) . (2.5b) \nThe expressions for Ω LO 2 and Ω NLO 2 can be obtained from the above formulas by exchanging the particle labels 1 and 2. Let us now focus our attention on the dynamics of the relative motion of the two-body system in the center-of-mass frame , which is defined by the requirement p 1 + p 2 = 0 . It will be convenient in the following to work with suitably rescaled variables. We rescale the phase-space variables R ≡ x 1 -x 2 and P ≡ p 1 = -p 2 of the relative motion as follows \nr ≡ R GM ≡ x 1 -x 2 GM , p ≡ P µ ≡ p 1 µ = -p 2 µ , (2.6) \nwhere M ≡ m 1 + m 2 and µ ≡ m 1 m 2 /M . Note that this change of variables corresponds to rescaling the action by a factor 1 / ( GMµ ). It is also convenient to rescale the original time variable T and any part of the Hamiltonian according to \nt ≡ T GM , ˆ H NR ≡ H NR µ , (2.7) \nwhere H NR ≡ H -Mc 2 denotes the 'non relativistic' version of the Hamiltonian, i.e. the Hamiltonian without the rest-mass contribution. It has the structure ˆ H NR = 1 2 p 2 -1 r + O ( 1 c 2 ) . \n( ) It will be convenient in the following to work with the following two basic combinations of the spin vectors: \nS ≡ S 1 + S 2 = m 1 c a 1 + m 2 c a 2 , (2.8a) \nS ∗ ≡ m 2 m 1 S 1 + m 1 m 2 S 2 = m 2 c a 1 + m 1 c a 2 , (2.8b) \nwhere we have introduced (as is usually done in the general relativistic literature) the Kerr parameters 3 of the individual black holes, a 1 ≡ S 1 / ( m 1 c ) and a 2 ≡ S 2 / ( m 2 c ). Note that, in the 'spinning test mass limit' where, say, m 2 → 0 and S 2 → 0, while keeping a 2 = S 2 / ( m 2 c ) fixed, we have a 'background mass' M /similarequal m 1 , a 'background spin' S bckgd ≡ Mc a bckgd /similarequal S 1 = m 1 c a 1 , a 'test mass' µ /similarequal m 2 , and a 'test spin' S test = S 2 = m 2 c a 2 /similarequal µc a test [with a test ≡ S test / ( µc )]. Then, in this limit the combination S /similarequal S 1 = m 1 c a 1 /similarequal Mc a bckgd = S bckgd measures the background spin, while the other combination, S ∗ /similarequal m 1 c a 2 /similarequal Mc a test = M S test /µ measures the (specific) test spin a test = S test / ( µc ). The quantities S and S ∗ are the two simplest symmetric (under the permutation 1 ↔ 2) combinations of the two spin vectors which have these properties. \nIn view of the rescaling of the action by a factor 1 / ( GMµ ), corresponding to the rescaled phase-space variables above, it will be natural to work with correspondingly rescaled spin variables 4 \n¯ S X ≡ S X GMµ , (2.9) \nfor any label X (X = 1 , 2 , ∗ , · · · ). \nMaking use of the definitions (2.6)-(2.9) one easily gets from Eqs. (2.3)-(2.5) the center-of-mass spin-orbit Hamiltonian (divided by µ ) expressed in terms of the rescaled variables: \nˆ H so ( r , p , ¯ S , ¯ S ∗ ) = H so ( r , p , ¯ S , ¯ S ∗ ) µ = 1 c 2 ˆ H so LO ( r , p , ¯ S , ¯ S ∗ ) + 1 c 4 ˆ H so NLO ( r , p , ¯ S , ¯ S ∗ ) + O ( 1 c 6 ) , (2.10) \nwhere (here n ≡ r / | r | ) 5 \n))) \nˆ H so LO ( r , p , ¯ S , ¯ S ∗ ) = ν r 2 { 2 ( ¯ S, n, p ) + 3 2 ( ¯ S ∗ , n, p ) } , (2.11a) ˆ H so NLO ( r , p , ¯ S , ¯ S ∗ ) = ν r 3 { -(6 + 2 ν ) ( ¯ S, n, p ) -(5 + 2 ν ) ( ¯ S ∗ , n, p ) } + ν r 2 { ((( 19 8 ν p 2 + 3 2 ν ( n · p ) 2 ))) ( ¯ S, n, p ) + (((( -5 8 +2 ν ) p 2 + 3 4 ν ( n · p ) 2 ( ¯ S ∗ , n, p } , (2.11b) \n) \nwith ν ≡ µ/M ranging from 0 (test-body limit) to 1/4 (equal-mass case). \nNote that the structure of the rescaled spin-orbit Hamiltonian is \nˆ H so ( r , p , ¯ S , ¯ S ∗ ) = ν c 2 r 2 ( g ADM S ( ¯ S, n, p ) + g ADM S ∗ ( ¯ S ∗ , n, p ) ) . (2.12) \nThis corresponds to an unrescaled spin-orbit Hamiltonian of the form \nH so = G c 2 L R 3 · ( g ADM S S + g ADM S ∗ S ∗ ) , (2.13) \nwhere R = GMr is the unrescaled relative distance (in ADM coordinates), L ≡ R × P = GMµ r × p the relative orbital angular momentum, and where we have introduced two dimensionless coefficients which might be called the 'gyro-gravitomagnetic ratios', because they parametrize the coupling between the spin vectors and the 'apparent' gravitomagnetic field \nv ×∇ GM c 2 R ∝ R × P R 3 \nseen in the rest-frame of a moving particle (see, e.g., Refs. [20, 21] for a discussion of the expression of the 'gravitomagnetic field' in the rest-frame of a moving body). The explicit expressions of these two gyro-gravitomagnetic ratios are \ng ADM S = 2 + 1 c 2 ((( 19 8 ν p 2 + 3 2 ν ( n · p ) 2 -( 6 + 2 ν ) 1 r ))) , (2.14a) \ng ADM S ∗ = 3 2 + 1 c 2 ((( ( -5 8 +2 ν ) p 2 + 3 4 ν ( n · p ) 2 -( 5 + 2 ν ) 1 r ))) . (2.14b) \nIn the following we shall introduce two related 'effective' 'gyro-gravitomagnetic ratios', that enter the effective EOB Hamiltonian (in effective coordinates). The label 'ADM' on the gyro-gravitomagnetic ratios (2.14) is a reminder of the fact that the NLO value of these ratios depend on the precise definition of the radial distance R (which is coordinate dependent). Let us, however, briefly discuss the origin of the (coordinate-independent) LO values of these ratios, namely \ng LO S = 2 , g LO S ∗ = 3 2 = 2 -1 2 . (2.15) \nHere the basic ratio 2 which enters both g LO S and g LO S ∗ comes from the leading interaction, predicted by the Kerr metric, between the orbital angular momentum of a test particle and the background spin. See Eq. (4.17) below. As for the -1 2 'correction' in the coupling of the 'test mass' spin combination S ∗ it can be seen (e.g. from Eq. (3.6b) of [22]) to come from the famous 1 2 factor in the Thomas precession (which is a universal, special relativistic effect, separate from the effects which are specific to the gravitational interaction, see Eqs. (3.2) and (3.3) in [22]). \nTo complete this Section, let us recall the remarkable form [found in Ref. [1], see Eq. (2.54) there] of the leadingorder spin-spin Hamiltonian H ss (including S 2 1 , S 2 2 as well as S 1 S 2 terms). The unrescaled form of the spin-spin Hamiltonian reads \nH ss ( R , S 0 ) = ν 2 G c 2 S i 0 S j 0 ∂ ij 1 R , (2.16) \nwhile its rescaled version reads \nˆ H ss ( r , ¯ S 0 ) ≡ H ss ( R , S 0 ) µ = 1 2 ν 2 c 2 ¯ S i 0 ¯ S j 0 ∂ ij 1 r = 1 2 ν 2 c 2 3( n · ¯ S 0 ) 2 -¯ S 2 0 r 3 . (2.17) \nThe remarkable fact about this result is that it is entirely expressible in terms of the specific combination of spins S 0 ≡ GMµ ¯ S 0 defined as: \nS 0 ≡ S + S ∗ = ( 1 + m 2 m 1 ) S 1 + ( 1 + m 1 m 2 ) S 2 . (2.18) \nWe shall come back below to the remarkable properties of the combination S 0 , which will play a central role in our EOB construction.", "III. EFFECTIVE HAMILTONIAN AND 'EFFECTIVE GYRO-GRAVITOMAGNETIC' RATIOS": "We have obtained in the previous Section the expression of the full center-of-mass-frame Hamiltonian (2.1), in PNexpanded form. In order to transform this Hamiltonian into a format which can be resummed in a manner compatible with previous work on the EOB formalism, we need to perform two operations on the Hamiltonian (2.1). First, we need to transform the phase-space coordinates ( x a , p a , S a ) by a canonical transformation compatible with the one used in previous EOB work. Second, we need to compute the effective Hamiltonian corresponding to the (canonically transformed) real Hamiltonian (2.1). \nWe start by performing the purely orbital canonical transformation which was found to be needed in Refs. [2, 4] to go from the ADM coordinates (used in the PN-expanded dynamics) to the coordinates used in the EOB dynamics. This orbital canonical transformation is (implicitly) given by \nx ' i = x i + ∂G o ( x, p ' ) ∂p ' i , p ' i = p i -∂G o ( x, p ' ) ∂x i . (3.1) \nHere the orbital generating function G o ( q, p ' ) has been derived to 2PN accuracy in [2], and to 3PN accuracy in [4]. In the present paper, as we are only concerned with the additional spin-orbit terms, treated to 1PN fractional accuracy, it is enough to work with the 1PN-accurate generating function G o ( x, p ' ). In terms of the rescaled variables, the rescaled 1PN-accurate orbital generating function reads \n¯ G o ( r , p ) ≡ G o ( r , p ) GMµ \n= 1 c 2 ( r · p ) ( -1 2 ν p 2 + ( 1 + 1 2 ν ) 1 r ) . (3.2) \nThis transformation changes the phase-space variables from ( r , p , ¯ S , ¯ S ∗ ) to ( r ' , p ' , ¯ S , ¯ S ∗ ). At the linear order in the transformation (which will be enough for our purpose), the effect of the transformation on any of the phase-space variable, say y , is y ' = y + { y, G o } , where {· , ·} denotes the Poisson bracket. As G o is independent of time, it leaves the Hamiltonian numerically invariant: H ' ( y ' ) = H ( y ). This means that it changes the functional form of the Hamiltonian according to H ' ( y ' ) = H ( y ' -{ y, G o } ) = H ( y ' ) -{ H,G o } . Note the appearance of the opposite sign in front of the Poisson bracket, with respect to the effect of the generating function on the phase-space variables. \nAs G o is of order 1 /c 2 , its explicit effect on the two separate terms, H so LO and H so NLO , in the PN expansion of the spin-orbit Hamiltonian is given by: \nH ' so LO ( r ' , p ' , ¯ S , ¯ S ∗ ) = H so LO ( r ' , p ' , ¯ S , ¯ S ∗ ) , (3.3a) \nH ' so NLO ( r ' , p ' , ¯ S , ¯ S ∗ ) = H so NLO ( r ' , p ' , ¯ S , ¯ S ∗ ) -{ H so LO , ¯ G o } ( r ' , p ' , ¯ S , ¯ S ∗ ) . (3.3b) \nIt will be convenient in the following to further transform the phase-space variables by performing a secondary, purely spin-dependent canonical transformation, affecting only the NLO spin-orbit terms. The associated new generating function, G s ( r , p , ¯ S , ¯ S ∗ ) (assumed to be proportional to the spins and of order 1 /c 4 ) will change the variables ( y ' ) ≡ ( r ' , p ' , ¯ S , ¯ S ∗ ) into ( y '' ) ≡ ( r '' , p '' , ¯ S '' , ¯ S ''∗ ) according to the general rule 6 y '' = y ' + { y ' , G s } . For the same reason as above, the (first-order) effect of G s on the functional form of the Hamiltonian will involve a Poisson bracket with the opposite sign: H '' ( y '' ) = H ( y '' ) -{ H,G s } . \nWe shall consider a generating function whose unrescaled form reads \nG s ( R , P , S , S ∗ ) = G µc 4 1 R 3 ( R · P )( R × P ) · ( a ( ν ) S + b ( ν ) S ∗ ) , (3.4) \nwhile its rescaled form reads \n¯ G s ( r , p , ¯ S , ¯ S ∗ ) ≡ G s ( R , P , S , S ∗ ) GMµ = 1 c 4 ν ( n · p ) r ( a ( ν ) ( ¯ S, n, p ) + b ( ν ) ( ¯ S ∗ , n, p ) ) . (3.5) \nHere a ( ν ) and b ( ν ) are two arbitrary, ν -dependent dimensionless coefficients. 7 Similarly to the result above, the explicit effect of this new canonical transformation on the two separate terms, H ' so LO and H ' so NLO , in the PN expansion of the spin-orbit Hamiltonian reads: \nH '' so LO ( r '' , p '' , ¯ S '' , ¯ S ''∗ ) = H ' so LO ( r '' , p '' , ¯ S '' , ¯ S ''∗ ) , (3.6a) \nH '' so NLO ( r '' , p '' , ¯ S '' , ¯ S ''∗ ) = H ' so NLO ( r '' , p '' , ¯ S '' , ¯ S ''∗ ) -{ H oN , ¯ G s } ( r '' , p '' , ¯ S '' , ¯ S ''∗ ) , (3.6b) \nwhere H oN is the Newtonian orbital Hamiltonian. In the following, we shall, for simplicity of notation, omit the double primes on the new phase-space variables (and on the corresponding Hamiltonian). \nThe second operation we need to do is to connect the 'real' Hamiltonian H to the 'effective' one H eff , which is more closely linked to the description of the EOB quasi-geodesic dynamics. The relation between the two Hamiltonians is quite simple [2, 4]: \nH eff µc 2 ≡ H 2 -m 2 1 c 4 -m 2 2 c 4 2 m 1 m 2 c 4 , (3.7) \nwhere we recall that the real Hamiltonian H contains the rest-mass contribution Mc 2 = ( m 1 + m 2 ) c 2 . Let us also note that Eq. (3.7) is equivalent to \nH eff µc 2 = 1 + H NR µc 2 + 1 2 ν ( H NR ) 2 µ 2 c 4 , (3.8) \nwhere H NR denotes the 'non relativistic' part of the total Hamiltonian H , i.e., H NR ≡ H -Mc 2 , or more explicitly \nH NR = ( H oN + H o1PN c 2 + H o2PN c 4 + H o3PN c 6 ) + ( H so LO c 2 + H so NLO c 4 ) . (3.9) \nBy expanding (in powers of 1 /c 2 and in powers of the spins) the exact effective Hamiltonian (3.7), one easily finds that the 'spin-orbit part' of the effective Hamiltonian H eff (i.e. the part which is linear-in-spin) differs from the corresponding part H so in the 'real' Hamiltonian by a factor /similarequal 1 + ν ˆ H NR /c 2 /similarequal 1 + ν ˆ H oN /c 2 , so that we get, for the explicit PN expansion of H so eff , \nH so eff µ = 1 c 2 ˆ H so LO + 1 c 4 ( ˆ H so NLO + ν ˆ H oN ˆ H so LO ) . (3.10) \nCombining this result with the effect of the two generating functions discussed above (and omitting, as we already said, the double primes on the new phase-space variables ( r '' , p '' , ¯ S '' , ¯ S ''∗ )), we get the transformed spin-orbit part of the effective Hamiltonian in the form \nH so eff µ = ν c 2 r 2 ( n × p ) · ( g eff S ¯ S + g eff S ∗ ¯ S ∗ ) , (3.11) \nwhich corresponds to the following unrescaled form (with L ≡ R × P ): \nH so eff = G c 2 L R 3 · ( g eff S S + g eff S ∗ S ∗ ) . (3.12) \nHere the two 'effective gyro-gravitomagnetic' ratios g eff S and g eff S ∗ differ from the 'ADM' ones introduced above by three effects: (i) a factor /similarequal 1 + ν ˆ H NR /c 2 /similarequal 1 + ν ˆ H oN /c 2 due to the transformation from H to H eff , (ii) the effect of the orbital generating function G o going from ADM to EOB coordinates, and (iii) the effect of the spin-dependent generating function G s , which involves the gauge parameters a ( ν ) and b ( ν ). Their explicit expressions are then found to read \ng eff S ≡ 2 + 1 c 2 ((( ( 3 8 ν + a ( ν ) ) p 2 -( 9 2 ν +3 a ( ν ) ) ( n · p ) 2 -1 r ( ν + a ( ν ) ) ))) , (3.13a) \ng eff S ∗ ≡ 3 2 + 1 c 2 ((( ( -5 8 + 1 2 ν + b ( ν ) ) p 2 -( 15 4 ν +3 b ( ν ) ) ( n · p ) 2 -1 r ( 1 2 + 5 4 ν + b ( ν ) ) ))) . (3.13b) \nThe choice of the two 'gauge' parameters a ( ν ) and b ( ν ) is arbitrary, and physical results should not depend on them. 8 This would be the case if we were dealing with the exact Hamiltonian. However, as we work only with an approximation to the exact Hamiltonian, there will remain some (weak) dependence of our results on the choice of a ( ν ) and b ( ν ). We can use this dependence to try to simplify, and/or to render more accurate, the spin-orbit effects implied by the above expressions. In particular, we shall focus in this paper on a special simplifying choice of these gauge parameters: namely, the values \na ( ν ) = -3 8 ν, b ( ν ) = 5 8 -1 2 ν, (3.14) \nwhich suppress the dependence of the effective gyro-gravitomagnetic ratios on p 2 . With this particular choice, the explicit expressions of these ratios become \ng eff S ≡ 2 + 1 c 2 ((( -27 8 ν ( n · p ) 2 -5 8 ν 1 r ))) , (3.15a) \ng eff S ∗ ≡ 3 2 + 1 c 2 ((( -( 15 8 + 9 4 ν ) ( n · p ) 2 -( 9 8 + 3 4 ν ) 1 r ))) . (3.15b)", 'IV. SPIN-DEPENDENT EFFECTIVE-ONE-BODY HAMILTONIAN': "Up to now we only considered PN-expanded results. In this Section, we shall generalize the approach of [1] in incorporating, in a resummed way, the spin-dependent effects within the EOB approach. Let us first recall that the approach of [1] consists in combining three different ingredients: \n- · a generalized Hamilton-Jacobi equation involving higher powers of the momenta (as is necessary at the 3PN accuracy [4]);\n- · a ν -deformed Kerr-type metric g αβ eff , which depends on the choice of some basic 'effective spin vector' S i eff ;\n- · the possible consideration of an additional spin-orbit interaction term ∆ H so ( r , p , S 0 , σ ) in the effective Hamiltonian, whose aim is to complete the spin-dependent interaction incorporated in the definition of the HamiltonJacobi equation based on a certain choice of 'effective spin vector' S i eff . \nAt the LO in spin-orbit and spin-spin interactions, Ref. [1] showed that one had the choice between two possibilities: (i) use as effective spin vector the combination S + 3 4 S ∗ which correctly describes the LO spin-orbit effects, but only approximately describes the LO spin-spin effects; 9 or \n- (ii) use as effective spin vector the combination \nS 0 ≡ S + S ∗ = ( 1 + m 2 m 1 ) S 1 + ( 1 + m 1 m 2 ) S 2 , (4.1) \nwhich correctly describes the full LO spin-spin interaction (see (2.17) above), and complete the description of the LO spin-orbit effects by adding a term ∆ H so ( r , p , S 0 , σ ) involving a suitably defined spin combination σ . (At LO, Ref. [1] defined σ LO = -1 4 S ∗ .) \nIntuitively speaking, the second possibility consists in considering that the 'effective particle' is endowed not only with a mass µ , but also with a 'spin' proportional to σ , so that it interacts with the 'effective background spacetime' both via a geodesic-type interaction (described by the generalized Hamilton-Jacobi equation), and via an additional spin-dependent interaction proportional to its spin ∝ σ . \nAt the present, NLO approximation, where it is crucial to accurately describe the spin-orbit interaction, as well as, by consistency, the LO spin-spin ones, we have chosen to follow the second possibility, which offers more flexibility, and which looks natural in view of the remarkably simple LO result (2.17) for the spin-spin interaction (see, however, the suggestion at the end of the concluding Section 6). \nTherefore we shall successively introduce the ingredients needed to define \n- · the Hamilton-Jacobi equation (describing the basic 'geodesic-type' part of the effective Hamiltonian);\n- · the effective, ν -deformed Kerr-type metric g αβ eff ;\n- · the 'effective spin vector' S i eff entering the previous Kerr-type metric;\n- · the additional spin-orbit interaction ∆ H so ( r , p , S 0 , σ ) involving a new, specific NLO spin combination σ . \nThe modified Hamilton-Jacobi equation [4] is of the form \ng αβ eff P α P β + Q 4 ( P i ) = -µ 2 c 2 , (4.2) \nwhere Q 4 ( P i ) is a quartic-in-momenta term (which only depends on the space momentum components P i ). For circular orbits Q 4 ( P i ) will be zero (see [1, 4]), so that we will not need its explicit expression in the present paper. \nThe role of the Hamilton-Jacobi equation above is to allow one to compute the main part (modulo the additional spin-orbit interaction added later) of the effective Hamiltonian H main eff = E eff ≡ -P 0 c by solving (4.2) with respect to P 0 . The result can be written as \nH main eff = E eff = β i P i c + αc √ µ 2 c 2 + γ ij P i P j + Q 4 ( P i ) , (4.3) \nwhere we have introduced the auxiliary notation \nα ≡ ( -g 00 eff ) -1 / 2 , β i ≡ g 0 i eff g 00 eff , γ ij ≡ g ij eff -g 0 i eff g 0 j eff g 00 eff . (4.4) \nThe next crucial ingredient consists in defining the (spin-dependent) effective metric entering the Hamilton-Jacobi equation, and thereby the effective Hamiltonian (4.3). We shall follow here Ref. [1] in employing an effective co-metric of the form (here P t ≡ cP 0 ) \ng αβ eff P α P β = 1 R 2 + a 2 cos 2 θ ((( ∆ R ( R ) P 2 R + P 2 θ + 1 sin 2 θ ( P φ + a sin 2 θ P t c ) 2 -1 ∆ t ( R ) ( ( R 2 + a 2 ) P t c + aP φ ) 2 ))) , (4.5) \nwhere the functions ∆ t and ∆ R are defined as \n∆ t ( R ) ≡ R 2 P n m [ A ( R ) + a 2 R 2 ] , (4.6a) \n∆ R ( R ) ≡ ∆ t ( R ) D -1 ( R ) , (4.6b) \nand where the Kerr-like parameter a is defined as a ≡ S eff / ( Mc ), where S eff denotes the modulus of the 'effective spin vector' S i eff entering the definition of the Kerr-like metric above. We shall come back below to the choice of this vector S i eff (which is one of the ingredients in the definition of a spin-dependent EOB formalism). In Eq. (4.6a) P n m denotes the operation of taking the ( n, m )-Pad'e approximant, 10 and the PN expansions of the metric coefficients A and D -1 equal (here ˆ u ≡ GM/ ( Rc 2 )) \nA (ˆ u ) = 1 -2ˆ u +2 ν ˆ u 3 + ( 94 3 -41 32 π 2 ) ν ˆ u 4 , (4.7a) \nD -1 (ˆ u ) = 1 + 6 ν ˆ u 2 +2(26 -3 ν ) ν ˆ u 3 . (4.7b) \nFor pedagogical clarity, we have given above the expression of the effective EOB metric in a Boyer-Lindquist-type coordinate system aligned with the instantaneous direction of the (time-dependent) effective spin vector S i eff . This expression will suffice in the present paper where we will only consider situations where the spin vectors are aligned with the orbital angular momentum, so that they are fixed in space. As emphasized in [1], when applying the EOB formalism to more general situations (non aligned spins) one must rewrite the effective co-metric components in a 'fixed' Cartesian-like coordinate system. This is done by introducing \nn i ≡ x i /R, s i ≡ S i eff S eff , cos θ ≡ n i s i , ρ ≡ √ R 2 + a 2 cos 2 θ, (4.8) \nand rewriting the co-metric components as \ng 00 eff = -( R 2 + a 2 ) 2 -a 2 ∆ t ( R ) sin 2 θ ρ 2 ∆ t ( R ) , (4.9a) \ng 0 i eff = -a ( R 2 + a 2 -∆ t ( R )) ρ 2 ∆ t ( R ) ( s × R ) i , (4.9b) \ng ij eff = \n1 ρ 2 ( ∆ R ( R ) n i n j + R 2 ( δ ij -n i n j ) ) \n-a 2 ρ 2 ∆ t ( R ) ( s × R ) i ( s × R ) j . (4.9c) \nMaking use of Eqs. (4.9) one computes \nα = ρ √ ∆ t ( R ) ( R 2 + a 2 ) 2 -a 2 ∆ t ( R ) sin 2 θ , (4.10a) \nβ i = a ( R 2 + a 2 -∆ t ( R )) ( R 2 + a 2 ) 2 -a 2 ∆ t ( R ) sin 2 θ ( s × R ) i , (4.10b) \nγ ij = g ij eff + β i β j α 2 . (4.10c) \nReplacing the latter expressions in the general form of the effective energy (4.3) yields the most general form of the main part of the effective Hamiltonian H main eff ( x , P , S a ). \nThe definition of H main eff ( x , P , S a ) crucially depends on the choice of effective Kerr-type spin vector. In order to automatically incorporate, in a correct manner, the LO spin-spin terms, we shall use here \nMc a ≡ S eff ≡ S 0 = S + S ∗ = ( 1 + m 2 m 1 ) S 1 + ( 1 + m 1 m 2 ) S 2 . (4.11) \nNote that, besides its usefulness in treating spin-spin effects, this definition has several nice features. For example, if we introduce the Kerr parameters of the individual black holes, a 1 ≡ S 1 / ( Mc ), a 2 ≡ S 2 / ( Mc ), the Kerr parameter a 0 ≡ S 0 / ( Mc ) (where we naturally take m 0 = M = m 1 + m 2 ) associated to the spin combination (4.1) is simply \na 0 = a 1 + a 2 . (4.12) \nLet us also note that the corresponding dimensionless spin parameters (with, again, m 0 = M = m 1 + m 2 ) \nˆ a i ≡ c S i Gm 2 i , i = 0 , 1 , 2 , (4.13) \nsatisfy \nˆ a 0 = X 1 ˆ a 1 + X 2 ˆ a 2 , (4.14) \nwhere X 1 ≡ m 1 /M and X 2 ≡ m 2 /M are the two dimensionless mass ratios (with X 1 + X 2 = 1 and X 1 X 2 = ν ). This last result shows that, in ˆ a -space, the 'point' ˆ a 0 is on the straight-line segment joining the two 'points' ˆ a 1 and ˆ a 2 . The individual Kerr bounds tell us that each point ˆ a 1 and ˆ a 2 is contained within the unit Euclidean sphere. By convexity of the unit ball, we conclude that the 'effective' dimensionles spin parameter ˆ a 0 will also automatically satisfy the Kerr bound | ˆ a 0 | ≤ 1. This is a nice consistency feature of the definition of the associated Kerr-type metric. \nIt remains to define the additional 'test-spin' vector σ , and the associated additional effective spin-orbit interaction term. Following the logic of [1] (and generalizing the LO results given in Eqs. (2.56)-(2.58) there), these quantities are defined by \nand \nσ ≡ 1 2 g eff S S + 1 2 g eff S ∗ S ∗ -S eff = 1 2 ( g eff S -2 ) S + 1 2 ( g eff S ∗ -2 ) S ∗ , (4.15) \n∆ H so ( x , P , S 0 , σ ) ≡ R 2 + a 2 0 -∆ t ( R ) ( R 2 + a 2 0 ) 2 -a 2 0 ∆ t ( R ) sin 2 θ 0 ( P, σ, R ) M , (4.16) \nwhere a 0 ≡ S 0 / ( Mc ) and cos θ 0 ≡ n i S i 0 / | S 0 | . The justification for these definitions is that the 'main' Hamilton-Jacobi part of the effective Hamiltonian contains, as spin-orbit (i.e. linear-in-spin) part, the following term \nH maineff so = cP i ( β i ) linear-in-spin = cP i ( R 2 + a 2 0 -∆ t ( R ) ( R 2 + a 2 0 ) 2 -a 2 0 ∆ t ( R ) sin 2 θ 0 ( a 0 × R ) i ) linear-in-spin = 2 GM cR 3 P i ( a 0 × R ) i +(NNLO corrections) = 2 G c 2 L R 3 · S 0 +(NNLO corrections) , (4.17) \nwhere the factor 2 GM comes from the second term in the PN expansion of ∆ t ( R ) = R 2 -2 GMR/c 2 + 2 ν ( GM ) 3 / ( Rc 6 ) + (quadratic-in-spin terms). Note that the absence of c -4 correction in the effective metric function A ( R ) means that the leading term ∝ 2 GM in the spin-orbit part of H main is valid both to LO and to NLO, i.e., up to 'next to next to leading order' (NNLO). \nWhen comparing this result to the NLO result (3.12), we see that the 'main' part of the effective Hamiltonian contains a spin-orbit piece which is equivalent to having effective gyro-gravitomagnetic ratios equal to g maineff S = 2 and g maineff S ∗ = 2, instead of the correct values derived above. One then easily checks that the definition above of σ and of the associated supplementary spin-orbit interaction ∆ H so ( x , P , S 0 , σ ) has the effect of including the full result for the NLO spin-orbit interaction. It is also to be noted that the additional spin-orbit interaction ∆ H so goes to zero proportionally to ν in the test mass limit m 2 → 0 because, on the one hand, g eff S -2 is proportional to ν (if a ( ν ) is), and, on the other hand, though g eff S ∗ -2 does not tend to zero with ν , the second spin combination S ∗ does tend to zero proportionally to ν [see Eqs. (5.2) below]. \nSummarizing: we propose to define a total effective spin-dependent Hamiltonian of the form \nH eff ( x , P , S 1 , S 2 ) ≡ H main eff ( x , P , S 0 ) + ∆ H so ( x , P , S 0 , σ ) , (4.18) \nwhere H main eff ( x , P , S 0 ) is given by the right-hand side of Eq. (4.3) computed for the effective spin variable equal to S 0 [defined in Eq. (4.1)] and where ∆ H so ( x , P , S 0 , σ ) is the additional spin-orbit interaction term defined above [with a 0 ≡ S 0 / ( Mc )]. \nFinally, the real EOB-improved Hamiltonian (by contrast to the 'effective' one) is defined by solving Eq. (3.7) with respect to H real = H NR + Mc 2 : \nH real = Mc 2 √ 1 + 2 ν ( H eff µc 2 -1 ) , (4.19) \nwhere H eff is given in Eq. (4.18).", 'V. DYNAMICS OF CIRCULAR ORBITS': "In this Section we shall apply the construction of the NLO spin-dependent EOB Hamiltonian to the study of the dynamics of circular orbits of binary black hole systems. \nBesides the dimensionless spin parameters ˆ a 1 and ˆ a 2 already introduced above, it is convenient to introduce the dimensionless spin variables corresponding to the basic spin combinations S and S ∗ , namely \nˆ a ≡ c S GM 2 , ˆ a ∗ ≡ c S ∗ GM 2 . (5.1) \nLet us note in passing the various links between the dimensionless spin parameters that one can define [including ˆ a 0 ≡ c S 0 / ( GM 2 ) already introduced above], \nˆ a = X 2 1 ˆ a 1 + X 2 2 ˆ a 2 , ˆ a ∗ = ν ˆ a 1 + ν ˆ a 2 , (5.2a) \nˆ a 0 = ˆ a +ˆ a ∗ = X 1 ˆ a 1 + X 2 ˆ a 2 . (5.2b) \nHere as above we use the mass ratios X 1 ≡ m 1 /M , X 2 ≡ m 2 /M such that X 1 + X 2 = 1 and X 1 X 2 = ν . Let us note that for equal-mass binaries ( m 1 = m 2 , X 1 = X 2 = 1 2 ), with arbitrary (possibly unequal) spins, one has ˆ a = ˆ a ∗ = 1 4 (ˆ a 1 + ˆ a 2 ) = 1 2 ˆ a 0 . Note also that, in the test-mass limit, say m 1 /greatermuch m 2 so that X 1 → 1 and X 2 → 0, one has \nˆ a = ˆ a 0 = ˆ a 1 , ˆ a ∗ = 0 . (5.3) \nIn the general case where the spin vectors are not aligned with the (rescaled) orbital angular momentum vector 11 /lscript , \n/lscript = r n × p , (5.4) \nthere exist no circular orbits. However, there exist (at least to a good approximation) some 'spherical orbits', i.e. orbits that keep a constant value of the modulus of the radius vector r , though they do not stay within one fixed plane. As discussed in [1] one can analytically study these spherical orbits within the EOB approach, and discuss, in particular, the characteristics of the last stable spherical orbit . \nFor simplicity, we shall restrict ourselves here to the situation where both individual spins are parallel (or antiparallel) to the orbital angular momentum vector /lscript . In that case, we can consistently set everywhere the radial momentum to zero, p r = n · p = 0, and express the (real) EOB Hamiltonian as a function of r , /lscript = p ϕ (using p 2 = /lscript 2 /r 2 , where /lscript ≡ | /lscript | ), and of the two scalars ˆ a, ˆ a ∗ measuring the projections of our basic spin combinations on the direction of the orbital angular momentum /lscript . They are such that \nˆ a · /lscript = ˆ a/lscript, ˆ a ∗ · /lscript = ˆ a ∗ /lscript. (5.5) \nThe scalars ˆ a and ˆ a ∗ can be either positive or negative, depending on whether, say, ˆ a is parallel or antiparallel to /lscript . The sequence of circular (equatorial) orbits is then determined by the constraint \n∂H real ( r, /lscript, ˆ a, ˆ a ∗ ) ∂r = 0 . (5.6) \nThen, the angular velocity along each circular orbit is given by \nΩ ≡ 1 GMµ ∂H real ( r, /lscript, ˆ a, ˆ a ∗ ) ∂/lscript . (5.7) \nAs mentioned above, we have chosen the special values a ( ν ) = -3 8 ν , b ( ν ) = 5 8 -1 2 ν of the two gauge parameters, to simplify the expression of the Hamiltonian. \nIn Figs. 1-4 we explore several aspects of the dynamics of circular orbits, using as basic diagnostic the relation between the energy and the angular velocity along the sequence of circular orbits ('binding energy curve'). More precisely, we plot the dimensionless 'non relativistic' energy \ne ≡ H real Mc 2 -1 , (5.8) \nas a function of the dimensionless angular velocity: \nˆ Ω ≡ GM c 3 Ω . (5.9) \nFor simplicity, we shall restrict most of our studies to symmetric binary systems, i.e. systems with m 1 = m 2 and a 1 = a 2 . For such systems the dimensionless effective spin parameter is ˆ a 0 = ˆ a 1 = ˆ a 2 . The information contained in these figures deals with the following aspects of the description of the dynamics: \n- · As a warm up, and a reminder, Fig. 1 considers the case of non-spinning binaries (i.e. ˆ a 0 = 0). This figure contrasts the behaviour of the successive PN versions of the EOB dynamics, with that of the successive PN versions of the non-resummed, 'Taylor-expanded' Hamiltonian. The numbers 1,2,3 refer to 1PN, 2PN, and \nFIG. 1: Binding energy curves for circular orbits of symmetric non-spinning binaries ( m 1 = m 2 and ˆ a 1 = ˆ a 2 = 0 ): dimensionless 'non relativistic' energy e versus dimensionless angular frequency ˆ Ω. The notation E( n, ∗ ) means computation of the energy using the EOB-improved real Hamiltonian (4.19) with the n PN-accurate metric function ∆ t ( R ); the function ∆ t ( R ) was computed by means of Eq. (4.6a) using the (1 , n ) Pad'e approximant at the n PN order. Here n = 1 , 2 , 3 , 4, where n = 4 refers to the '4PN' case where a term + a 5 ν ˆ u 5 is added to the function A (ˆ u ). For the curves labelled by T( n, ∗ ) the computation was done with the direct PN-expanded (ADM-coordinates) orbital Hamiltonian (2.2) with the terms up to the n PN order included. \n<!-- image --> \n3PN, while the letter 'E' refers to 'EOB' and the letter 'T' refers to 'Taylor'. For instance, E(3 , ∗ ) refers to the e ( ˆ Ω) binding energy curve computed with the 3PN-accurate EOB Hamiltonian. [The star in E(3 , ∗ ) replaces the label we shall use below to distinguish LO versus NLO treatment of spin-orbit effects. In the present non-spinning case we are insensitive to this distinction.] To be precise, the notation E( n, ∗ ) refers to a computation of the circular orbits using the ˆ a 0 → 0 limit 12 of the EOB-improved real Hamiltonian (4.19) with the n PN-accurate metric function ∆ t ( R ); where ∆ t ( R ) was computed by means of Eq. (4.6b) using the following Pad'e approximants: (1,1) at the 1PN order, (1,2) at the 2PN order, and (1,3) at the 3PN order. As for the Taylor-based approximants to the binding energy curve, T( n, ∗ ), they were computed by using as basic Hamiltonian (to define the dynamics) the n PN-accurate Taylor-expanded Hamiltonian, in ADM coordinates, (2.2), without doing any later PN re-expansion. 13 \nIt is interesting to note that the successive PN-approximated EOB binding energy curves are stacked in a monotonically decreasing fashion, when increasing the PN accuracy, and all admit a minimum at some value of the orbital frequency. This minimum corresponds to the last stable circular orbit (see below). The monotonic stacking of the EOB energy curves therefore implies that a higher PN accuracy predicts circular orbits which are more bound, and can reach higher orbital frequencies. Let us note in this respect that recent comparisons between EOB and numerical relativity data have found the need to add a positive 4PN additional term + a 5 ν ˆ u 5 in the basic EOB radial potential A (ˆ u ) of Eq. (4.7a) above, with a 5 somewhere between +10 and +80 [7, 8, 9, 10]. Though we do not know yet what is the 'real' value of the 4PN coefficient a 5 we have included in Fig. 1 two illustrative 14 values of this '4PN' orbital parameter, namely a 5 = +25 and a 5 = +60. Note that the effect of such positive values of a 5 is to push the last few circular orbits towards more bound, higher orbital frequency orbits. This effect will compound itself with the effects of spin explored below, and should be kept in mind when looking at our other plots. \nBy contrast with the 'tame' and monotonic behaviour of successive EOB approximants, we see on Fig. 1 that \nFIG. 2: Binding energy curves for circular orbits of symmetric parallely spinning binaries ( m 1 = m 2 and ˆ a 1 = ˆ a 2 ∝ r × p ): dimensionless energy e versus dimensionless angular frequency ˆ Ω along circular orbits for various values of the dimensionless effective spin parameter ˆ a 0 ≡ c S 0 / ( GM 2 ) = ˆ a 1 = ˆ a 2 within the effective-one-body approach. The label E(3 , 1) means that we use the EOB Hamiltonian with 3PN-accurate orbital effects and NLO spin-orbit coupling, i.e. Eq. (4.15) was used with the NLO gyro-gravitomagnetic ratios g eff S and g eff S ∗ , Eqs. (3.13). \n<!-- image --> \nthe successive Taylor-Hamiltonian approximants T( n, ∗ ) have a more erratic behaviour. Note in particular, that the 3PN-accurate energy curve does not admit any minimum as the orbital frequency increases (in other words, there is no 'last' stable circular orbit). In view of this bad behaviour of the 3PN-accurate orbital TaylorHamiltonian, we shall not consider anymore in the following figures the predictions coming from such Taylor Hamiltonians. 15 \n- · In Fig. 2 we study the effect of changing the amount of spin on the black holes of our binary system. We use here our new, NLO spin-orbit EOB Hamiltonian, as indicated by the notation E(3 , 1), where the first label, 3, refers to the 3PN accuracy, and the second label, 1, to the 1PN fractional accuracy of the spin-orbit terms (i.e., the NLO accuracy). Note that the EOB binding energy curves are stacked in a monotonically decreasing way as the dimensionless effective spin ˆ a 0 increases from ˆ a 0 = -1 (maximal spins antiparallel to the orbital angular momentum) to ˆ a 0 = +1 (maximal spins parallel to the orbital angular momentum). Note also that this curve confirms the finding of [1] that parallel spins lead to the possibility of closer and more bound circular orbits.\n- · Fig. 3 contrasts the effect of using the NLO spin-orbit interaction instead of the LO one in the EOB Hamiltonian. We use the full 3PN accuracy, and include the LO spin-spin interaction. E(3 , 0) denotes a result obtained with the 3PN-accurate EOB Hamiltonian using the LO (or 0PN-accurate) spin-orbit terms, while E(3 , 1) uses the 3PNaccurate EOB Hamiltonian with NLO (1PN-accurate) spin-orbit terms. Each panel in the Figure corresponds to a specific value of the dimensionless effective spin ˆ a 0 . To guide the eye we use in all our figures a solid line to denote our 'best' description, i.e. the 3PN-NLO EOB E(3 , 1). Note that the addition of the NLO effects in the spin-orbit interaction has the clear effect of moderating the influence of the spins (especially for positive spins). While the binding energy curves using the LO spin-orbit effects tend to abruptly dive down towards very negative energies when the spins are large and positive, 16 the corresponding NLO curves have a much more moderate behaviour. \nAmong the binding energy curves shown above, all the EOB ones (at least when the effective spin is not too large and positive), and some of the Taylor ones, admit a minimum for a certain value of the orbital frequency ˆ Ω. This minimum corresponds to an inflection point in the corresponding (EOB or Taylor) Hamiltonian considered as a \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \n<!-- image --> \nFIG. 3: Energy e versus angular frequency ˆ Ω along circular orbits for various values of the parameter ˆ a 0 , as predicted by the EOB Hamiltonian. We have assumed m 1 = m 2 , a 1 = a 2 , and θ 0 = π/ 2. As before E( n, s ) refers to an EOB Hamiltonian, with n PN accuracy in the orbital terms, and an accuracy in the spin-orbit coupling equal to the LO one if s = 0, and the NLO one if s = 1. In all cases, we include the full LO spin-spin coupling. \n<!-- image --> \nfunction of r . In other words, the minimum is the solution of the two equations 17 \n∂H real ∂r = 0 , ∂ 2 H real ∂r 2 = 0 . (5.10) \nTABLE I: LSO parameters for symmetric binary systems (with m 1 = m 2 and ˆ a 1 = ˆ a 2 = ˆ a 0 ) for the 3PN-NLO EOB Hamiltonian E(3 , 1). \n| ˆ a 0 | e | ˆ Ω |\n|----------|-------------|-----------|\n| - 1 . 00 | - 0 . 01039 | 0 . 04473 |\n| - 0 . 75 | - 0 . 01143 | 0 . 05139 |\n| - 0 . 50 | - 0 . 01270 | 0 . 05989 |\n| - 0 . 25 | - 0 . 01437 | 0 . 07143 |\n| 0 . 00 | - 0 . 01670 | 0 . 08822 |\n| 0 . 25 | - 0 . 02026 | 0 . 11521 |\n| 0 . 50 | - 0 . 02660 | 0 . 16444 |\n| 0 . 75 | - 0 . 03701 | 0 . 23249 |\n| 1 . 00 | - 0 . 03826 | 0 . 22210 | \nThe solutions of these two simultaneous equations correspond to what we shall call here the Last Stable (circular) Orbit (LSO). 18 Several methods have been considered in the literature [4, 23] for using PN-expanded results to estimate the characteristics of the LSO. One of these methods consists in considering the minima in the Taylor expansion of the function e ( ˆ Ω). These minima (called 'Innermost Circular Orbit' (ICO) in Refs. [15, 23], where they were used to estimate the LSO of spinning binaries) differ from the minima in the Taylor energy curves considered in Fig. 1 above, which were based on using a Taylor-expanded Hamiltonian. The advantage of consistently working (as we do here) within a Hamiltonian formalism is that we are guaranteed that the minima in the corresponding energy curves, when they exist, do correspond to a Last Stable orbit (and an associated inflection point) for some well-defined underlying dynamics. By contrast the dynamical meaning (if any) of a minimum of the Taylor-expanded function e Taylor ( ˆ Ω) is unclear. Anyway, as we saw above that the 3PN-accurate Taylor-expanded orbital Hamiltonian does not admit any Last Stable Orbit, we have not plotted in Fig. 4 the Taylor-based predictions for spinning binaries because they do not seem to lead to reasonable results. \nConcerning the dynamical meaning of the LSO, let us recall that it had been analytically predicted in [3] (and confirmed in recent numerical simulations [11]) that the transition between inspiral and plunge is smooth and progressive, so that the passage through the LSO is blurred . In spite of the inherent 'fuzziness' in the definition of the LSO, it is still interesting to delineate its dynamical characteristics because they strongly influence some of the gross features of the GW signal emitted by coalescing binaries (such as the total emitted energy, and the frequency of maximal emission). \nLet us comment on the results of our study of the characteristics of LSO's: \n- · In Fig. 4 we plot the LSO binding energy, predicted by the EOB approach, as a function of the dimensionless effective spin parameter ˆ a 0 . We contrast LO spin-orbit versus NLO spin-orbit (1 versus 0). We use 3PN accuracy (for the orbital effects) in all cases, and always include the LO spin-spin interaction. The upper panel shows that the use of LO spin-orbit interactions leads to dramatically negative LSO binding energies when the spins become moderately large. [The middle panel is a close-up of the upper one, and focuses on spins ˆ a 0 ≤ +0 . 2.] We find that the 3PN-LO EOB Hamiltonian E(3 , 0) admits an LSO only up to spins as large as: ˆ a 0 ≤ +0 . 9. However, as first found in [1], spin effects become dramatically (and suspiciously) large already when ˆ a 0 ≥ +0 . 5. By contrast, as we found above, the inclusion of NLO spin-orbit interactions has the effect of moderating the dynamical influence of high (positive) spins. The bottom panel focusses on our 'best bet' 3PN-NLO Hamiltonian E(3 , 1). \nAs mentioned above, Ref. [15] has considered, instead of the Taylor-Hamiltonian LSO, the minimum of the Taylor-expanded function e Taylor ( ˆ Ω) (or 'ICO'). For the two cases ˆ a 0 = -1 , 0 (corresponding to their κ i = -1 , 0), they found, in the 3PN-NLO case, energy minima equal to e ≡ E ICO /m = -0 . 0116 , -0 . 0193 for corresponding orbital frequencies ˆ Ω ≡ mω ICO = 0 . 059 , 0 . 129. These numerical values should be compared with the numerical values we quote in Table I below. On the other hand, for the large and parallel spin case ˆ a 0 = +1 Ref. [15] found that the Taylor-expanded function e Taylor ( ˆ Ω) has no minimum. Finally, note that the qualitative shape of the curve giving the (EOB) LSO energy as a function of the spin parameters ˆ a 0 is similar both to the corresponding \nFIG. 4: Binding energy of the Last Stable (circular) Orbit (LSO) predicted by the EOB approach. We study the effect of including NLO spin-orbit terms by contrasting the LO and NLO predictions. We plot the dimensionless energy e LSO of the LSO versus ˆ a 0 . We have assumed m 1 = m 2 , a 1 = a 2 , and θ 0 = π/ 2. For E(3 , 0) a LSO exists up to ˆ a 0 ≤ +0 . 9. \n<!-- image --> \ncurve for a spinless test-particle in a Kerr background (see, e.g., Fig. 7 below), and to the curve giving the LSO energy of a spinning test particle in a Kerr background, as a function of the test spin (see Fig. 4 in Ref. [24]). \nTo complement the information displayed in Figs. 1-4, we give in Table I the numerical values of the main LSO characteristics (binding energy and orbital frequency) for our 'best bet' Hamiltonian, namely the 3PN-NLO EOB one E(3 , 1). \nIn Figs. 5 and 6 we study the effective-spin-dependence of another LSO-related physical quantity of relevance for the dynamics of coalescing binaries: the total (orbital plus spin) angular momentum of the binary when it reaches \nFIG. 5: The dimensionless total angular momentum Kerr parameter ˆ a LSO J , Eq. (5.13), at the LSO, versus ˆ a 0 . We have assumed m 1 = m 2 , a 1 = a 2 , and θ 0 = π/ 2. The parameter ˆ a LSO J is computed from Eq. (5.13) with ˆ j LSO = ˆ /lscript LSO +ˆ a 1 +ˆ a 2 = ˆ /lscript LSO +2ˆ a 0 . We compare the various EOB predictions obtained either by improving the accuracy of spin-orbit terms [E(3 , 1) versus E(3 , 0)], or by improving the accuracy of orbital terms [E(4 , 1) versus E(3 , 1)]. We use two representative values of the 4PN parameter a 5 = +25 and a 5 = +60. For comparison, we also include a fit to recent numerical estimates of the final Kerr parameter of the black hole resulting from the coalescence of the two constituent black holes. \n<!-- image --> \nthe LSO [i.e., at the end of the (approximately) adiabatic inspiral, just before the plunge], \nJ ≡ L + S 1 + S 2 . (5.11) \nIn terms of rescaled dimensionless variables, this becomes \nˆ j ≡ c GMµ J = ˆ /lscript + m 1 m 2 ˆ a 1 + m 2 m 1 ˆ a 2 , (5.12) \nwhere ˆ /lscript ≡ c /lscript . Actually, the most relevant quantity is the dimensionless Kerr parameter associated to the total LSO mass-energy and the total LSO angular momentum, i.e., the value at the LSO of the ratio \nwhere ˆ j is the modulus of ˆ j . \nˆ a J ≡ cJ G ( H real /c 2 ) 2 = ν ˆ j ( H real / ( Mc 2 ) ) 2 , (5.13) \n- · In Fig. 5 we contrast the dependence of ˆ a LSO J on the dimensionless effective spin parameter ˆ a 0 for several EOB models: the two 3PN-accurate ones [E(3 , 0) using LO-accurate spin-orbit, and E(3 , 1) using NLO-accurate spinorbit], and two illustrative [7, 9] '4PN-accurate' NLO-spin-orbit models E(4 , 1) (using either a 5 = +25 or a 5 = +60, as in Fig. 1). [Here, we are still considering fully symmetric systems with m 1 = m 2 and a 1 = a 2 , so that ˆ a 0 = ˆ a 1 = ˆ a 2 .] Again we see the moderating influence of NLO corrections. The EOB-LO curve E(3 , 0) exhibits a sudden drop down (pointed out in [1]) before rising up again (and disappearing at ˆ a 0 = +0 . 9 when the LSO ceases to exist). By contrast, the NLO curve E(3 , 1) exhibits a much more regular dependence on ˆ a 0 , which is roughly linear over the entire range of values -1 ≤ ˆ a 0 ≤ 1. The two illustrative E(4 , 1) curves exhibit a 'mixed' behaviour where a 'drop' similar to the one featuring in the LO curve is still present, though it is moderated by NLO spin-orbit effects. This sensitivity to the inclusion of a 4PN contribution in A (ˆ u ) is due to a delicate interplay between the modified shape of the basic spin-independent 'radial potential' A (ˆ u, a 5 ) and the use of a (1,4) Pad'e resummation of the 'effective spin-dependent radial potential' ∆ t ( R ), Eq. (4.6a). Indeed, the additional contributions proportional to a 5 and a 2 are both repulsive , and tend to compound their effect, which is to push the LSO toward closer, more bound orbits [1]. \nWe have also indicated in Fig. 5 the final (i.e., after coalescence) dimensionless Kerr parameter of (symmetric) spinning binaries, as obtained in recent numerical simulations [26, 27, 28, 29]. For simplicity, we have shown the simple analytic fit proposed in [28]. The fact that the 3PN-NLO-accurate EOB LSO Kerr parameter [E(3 , 1)] is systematically above the final Kerr parameter is in good agreement with the fact that, after reaching the \nFIG. 6: The dimensionless total angular momentum Kerr parameter ˆ a LSO J at the E(3,1) LSO versus ˆ a 2 for various values of the parameter ˆ a 1 . Here we consider spin-dissymmetric systems with a 1 /negationslash = a 2 (but still m 1 = m 2 and θ 0 = π/ 2). The parameter ˆ a LSO J is computed from Eq. (5.13) with ˆ j LSO = ˆ /lscript LSO +ˆ a 1 +ˆ a 2 . \n<!-- image --> \nFIG. 7: Comparison of the test-mass limit m 2 /similarequal µ → 0 (with fixed ˆ a 2 ; so that S 2 /m 2 → 0) for two Hamiltonians. We consider the specific 'non relativistic' binding energy ˆ e LSO ≡ e LSO /ν = ( E LSO -Mc 2 ) / ( µc 2 ) at the LSO versus ˆ a 0 . The solid curve is the result of taking the test-mass limit of the EOB Hamiltonian, while the short-dashed curve is the result for a test particle moving in the Kerr spacetime. \n<!-- image --> \nLSO, the system will still loose a significant amount of angular momentum 19 during the plunge and the mergerplus-ringdown. In the case of non-spinning binaries, it has been shown that, by using the EOB formalism up to the end of the process [i.e., by taking into account the losses of J and E during plunge, as well as during merger-plus-ringdown], there was a good agreement (better than ∼ 2%) between EOB and numerical relativity for the final spin parameter [12]. We hope that the same type of agreement will hold also in the case of spinning binaries considered here. \n- · In Fig. 6 we plot the LSO dimensionless Kerr parameter of Eq. (5.13) for spin-dissymmetric systems, namely a 1 /negationslash = a 2 (but with m 1 = m 2 ), computed with the 3PN-NLO EOB Hamiltonian model E(3 , 1). This plot illustrates that the LSO spin parameter is a smooth (and essentially linear) function of the two individual spins.\n- · Finally, we compare in Fig. 7 the spinless test particle limit [i.e., m 2 → 0, together with a 2 = S 2 / ( m 2 c ) → 0, as appropriate to black holes for which ˆ a ≤ 1] for two Hamiltonians: the 3PN-NLO EOB one E(3 , 1), and the exact \none, as known from the geodesic action of a spinless test particle in the Kerr metric. For non-spinning systems the EOB Hamiltonian is constructed so as to reduce to the exact Schwarzschild-derived one in the test-particle limit. However, for spinning systems, we have chosen in Eq. (4.6a) to define the crucial metric coefficient ∆ t ( R ) by Pad'e-resumming the sum of A ( R ; ν )+ a 2 /R 2 . This Pad'e-resummation is indeed useful for generally ensuring, for comparable mass systems, that ∆ t ( R ) have a simple zero at some 'effective horizon' r H . However, in the test-mass limit ν → 0, while the Taylor-approximant to A ( R ; ν ) + a 2 /R 2 would coincide with the exact Kerr answer, the Pad'e-resummed version of A ( R ; ν ) + a 2 /R 2 differs from it. We see, however, on Fig. 7 that the resulting difference has a very small effect on the LSO energy per unit ( µ ) mass, except when the dimensionless effective spin ˆ a 0 is very close to +1. On the other hand, as we saw above when discussing Fig. 5, the issue of the Pad'e resummation of ∆ t ( R ) becomes more subtle when one considers the comparable-mass case, together with the inclusion of a repulsive 4PN parameter a 5 .", 'VI. CONCLUSIONS': "The main conclusions of this work are: \n- · We have prepared the ground for an accurate Effective One Body (EOB) description of the dynamics of binary systems made of spinning black holes by incorporating the recent computation of the next-to-leading order (NLO) spin-orbit interaction Hamiltonian [13] (see also Refs. [14, 15]) into a previously developed extension of the EOB approach to spinning bodies [1].\n- · We found that the inclusion of NLO spin-coupling terms has the quite significant result of moderating the effect of the LO spin-coupling, which would, by itself (as found in Ref. [1]), predict that the Last Stable (circular) Orbit (LSO) of parallely-fast-spinning black holes can reach very large binding energies of the order of 30% of the total rest-mass energy Mc 2 . By contrast, the inclusion of NLO spin-orbit terms predicts that the LSO of parallely-fast-spinning systems, though significantly more bound than that of non-spinning holes, can only reach binding energies of the order of 4% of the total rest-mass energy Mc 2 (see Fig. 4 above). This reduction in the influence of the spin-orbit coupling is due to the fact that the (effective) 'gyro-gravitomagnetic ratios' are reduced by NLO effects from their LO values g LO S = 2, g LO S ∗ = 3 2 to the values (here considered along circular orbits) \ng circ eff S = 2 -5 8 νx, g circ eff S ∗ = 3 2 -( 9 8 + 3 4 ν ) x, (6.1) \nwhere x /similarequal GM/ ( Rc 2 ) /similarequal ( GM Ω /c 3 ) 2 / 3 . This reduction then reduces the repulsive effect of the spin-orbit coupling which is responsible for allowing the binary system to orbit on very close, and very bound, orbits (see discussion in Section 3C of Ref. [1]). \n- · We studied the dependence of the dimensionless Kerr parameter of the binary system, ˆ a J ≡ cJ/ ( G ( H real /c 2 ) 2 ) , computed at the LSO, on the spins of the constituent black holes. Again the moderating effect of including NLO spin-orbit terms is very significant (compare the solid and the dashed 20 lines in Fig. 5). Thanks to this moderating effect the LSO Kerr parameter ˆ a LSO J is found to have a monotonic, and roughly linear, dependence on the spin parameters of the individual black holes (see solid line in Fig. 5 and the various curves in Fig. 6). We also studied the effect of including the type of 4PN parameter a 5 found useful in recent work [7, 8, 9, 10] for improving the agreement between EOB waveforms and numerical ones.\n- · We leave to future work the analog of what was initiated for spinning systems in Ref. [5], and recently completed for the case of non-spinning black holes in Ref. [12], i.e., a full dynamical study, within the EOB approach, of the Kerr parameter of the final black hole resulting from the merger of spinning black holes which takes into account the angular momentum losses that occur after the LSO, during the plunge, the merger, and the ringdown. Let us also note that Ref. [30] has recently proposed an approximate analytical approach (which is similar in spirit \nto the approximation used in Refs. [1, 3, 5] and above, namely that of considering the Kerr parameter of an effective test particle at, or after, the LSO) towards estimating the final spin of a binary black hole coalescence. The resulting prediction is, however, only in coarse agreement ∼ 10% with numerical results. Note in this respect that, as displayed in Fig. 5, the 'zeroth order' EOB result [corresponding to using the Kerr parameter for E(3 , 1) at the LSO, without taking into account the later losses of angular momentum] is already in ∼ 20% agreement with the fit to the numerical data [28]. The fact (displayed on Fig. 5) that the E(3,1) EOB LSO Kerr parameter is systematically above the final (after coalescence) Kerr parameter determined by recent numerical simulations [26, 27, 28, 29] is in qualitative agreement with the fact that the system will loose a significant amount of angular momentum during the plunge and the merger-plus-ringdown. Note, however, the sensitivity of ˆ a LSO J to a '4PN deformation' of the EOB Hamiltonian by the parameter a 5 . As said above, this sensitivity is due to the fact that the radial function ∆ t ( R ) /R 2 combines the additional repulsive effects of both a positive 4PN contribution + a 5 ν ( GM/ ( c 2 R ) ) 5 and a positive spin-dependent contribution + a 2 /R 2 . We leave to future work an exploration of this issue, which might need the use of a different Pad'e resummation than the (1,4) one used in (4.6a). \nIt remains to be seen whether the EOB/Numerical Relativity comparison for the final Kerr parameter of spinning systems will be as good as it was found to be for the non-spinning case [12], i.e., at the 2% level. If this is the case, it will establish the physical relevance of the improved EOB Hamiltonian constructed in the present paper. \n- · Let us finally note that there is some flexibility in the improved spin-dependent EOB Hamiltonian proposed above (besides the flexibility in the choice of the Pad'e resummation mentioned above). On the one hand, the choice (3.14) for the gauge parameters a ( ν ) and b ( ν ) might be replaced by other choices. On the other hand, the choice (4.11) for the effective spin vector might also be replaced by other ones. In particular, it might be interesting to consider the alternative definition \nMc a new ≡ S eff new ≡ 1 2 g eff new S S 0 = 1 2 g eff new S ( S + S ∗ ) . (6.2) \nThis definition coincides with the one used above at LO in spin-orbit effects (because g eff new S = 2 + O ( ν/c 2 )), and allows one to use a simplified supplementary spin-orbit contribution, built with \nσ new ≡ 1 2 ( g eff S ∗ -g eff S ) S ∗ , (6.3) \ninstead of (4.15). It might be interesting to explore which of these possible definitions exhibits the best agreement with current numerical results.", 'Acknowledgments': "This work was supported in part by the KBN Grant no 1 P03B 029 27 (to P.J.) and by the Deutsche Forschungsgemeinschaft (DFG) through SFB/TR7 'Gravitational Wave Astronomy'. \n- [1] T. Damour, 'Coalescence of two spinning black holes: An effective one-body approach,' Phys. Rev. D 64 , 124013 (2001) [arXiv:gr-qc/0103018]. \n- [7] A. Buonanno, Y. Pan, J. G. Baker, J. Centrella, B. J. Kelly, S. T. McWilliams, and J. R. van Meter, 'Toward faithful templates for non-spinning binary black holes using the effective-one-body approach,' Phys. Rev. D 76 , 104049 (2007) [arXiv:0706.3732 [gr-qc]].\n- [8] T. Damour and A. Nagar, 'Comparing Effective-One-Body gravitational waveforms to accurate numerical data,' Phys. Rev. D 77 , 024043 (2008) [arXiv:0711.2628 [gr-qc]].\n- [9] T. Damour, A. Nagar, E. N. Dorband, D. Pollney, and L. Rezzolla, 'Faithful Effective-One-Body waveforms of equal-mass coalescing black-hole binaries,' arXiv:0712.3003 [gr-qc].\n- [10] T. Damour, A. Nagar et al., in preparation.\n- [11] F. Pretorius, 'Binary black hole coalescence,' arXiv:0710.1338 [gr-qc].\n- [12] T. Damour and A. Nagar, 'Final spin of a coalescing black-hole binary: An effective-one-body approach,' Phys. Rev. D 76 , 044003 (2007) [arXiv:0704.3550 [gr-qc]].\n- [13] T. Damour, P. Jaranowski, and G. Schafer, 'Hamiltonian of two spinning compact bodies with next-to-leading order gravitational spin-orbit coupling,' arXiv:0711.1048 [gr-qc].\n- [14] G. Faye, L. Blanchet, and A. Buonanno, 'Higher-order spin effects in the dynamics of compact binaries. I: Equations of motion,' Phys. Rev. D 74 , 104033 (2006) [arXiv:gr-qc/0605139].\n- [15] L. Blanchet, A. Buonanno, and G. Faye, 'Higher-order spin effects in the dynamics of compact binaries. II: Radiation field,' Phys. Rev. D 74 , 104034 (2006) [Erratum-ibid. D 75 , 049903 (2007)] [arXiv:gr-qc/0605140].\n- [16] B. M. Barker and R. F. O'Connell, 'Derivation of the equations of motion of a gyroscope from the quantum theory of gravitation,' Phys. Rev. D 2 , 1428 (1970).\n- [17] J. Steinhoff, S. Hergt, and G. Schafer, 'On the next-to-leading order gravitational spin(1)-spin(2) dynamics,' arXiv:0712.1716 [gr-qc].\n- [18] T. Damour, P. Jaranowski, and G. Schafer, 'Poincar'e invariance in the ADM Hamiltonian approach to the general relativistic two-body problem,' Phys. Rev. D 62 , 021501 (2000) [Erratum-ibid. D 63 , 029903 (2001)] [arXiv:gr-qc/0003051].\n- [19] T. Damour, P. Jaranowski, and G. Schafer, 'Dimensional regularization of the gravitational interaction of point masses,' Phys. Lett. B 513 , 147 (2001) [arXiv:gr-qc/0105038].\n- [20] T. Damour, M. Soffel, and C. Xu, 'General relativistic celestial mechanics. 1. Method and definition of reference systems,' Phys. Rev. D 43 , 3273 (1991).\n- [21] T. Damour, M. Soffel, and C. Xu, 'General relativistic celestial mechanics. 3. Rotational equations of motion,' Phys. Rev. D 47 , 3124 (1993).\n- [22] T. Damour and J. H. Taylor, 'Strong field tests of relativistic gravity and binary pulsars,' Phys. Rev. D 45 , 1840 (1992).\n- [23] L. Blanchet, 'Innermost circular orbit of binary black holes at the third post-Newtonian approximation,' Phys. Rev. D 65 , 124009 (2002) [arXiv:gr-qc/0112056].\n- [24] S. Suzuki and K. Maeda, 'Innermost stable circular orbit of a spinning particle in Kerr spacetime,' Phys. Rev. D 58 , 023005 (1998) [arXiv:gr-qc/9712095].\n- [25] M. Campanelli, C. O. Lousto, Y. Zlochower, B. Krishnan, and D. Merritt, 'Spin flips and precession in black-hole-binary mergers,' Phys. Rev. D 75 , 064030 (2007) [arXiv:gr-qc/0612076].\n- [26] F. Herrmann, I. Hinder, D. M. Shoemaker, P. Laguna, and R. A. Matzner, 'Binary black holes: Spin dynamics and gravitational recoil,' Phys. Rev. D 76 , 084032 (2007) [arXiv:0706.2541 [gr-qc]].\n- [27] P. Marronetti, W. Tichy, B. Brugmann, J. Gonzalez, and U. Sperhake, 'High-spin binary black hole mergers,' arXiv:0709.2160 [gr-qc].\n- [28] L. Rezzolla, E. N. Dorband, C. Reisswig, P. Diener, D. Pollney, E. Schnetter, and B. Szilagyi, 'Spin diagrams for equal-mass black-hole binaries with aligned spins,' arXiv:0708.3999 [gr-qc].\n- [29] L. Rezzolla, P. Diener, E. N. Dorband, D. Pollney, C. Reisswig, E. Schnetter, and J. Seiler, 'The final spin from the coalescence of aligned-spin black-hole binaries,' arXiv:0710.3345 [gr-qc].\n- [30] A. Buonanno, L. E. Kidder, and L. Lehner, 'Estimating the final spin of a binary black hole coalescence,' Phys. Rev. D 77 , 026004 (2008) [arXiv:0709.3839 [astro-ph]]."}
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2008PhRvD..78h4017L
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Binary-black-hole initial data with nearly extremal spins
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2008-01-01
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['-', '-', '-', '-', 'methods numerical', 'techniques spectroscopic', '-', 'methods numerical', '-']
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There is a significant possibility that astrophysical black holes with nearly extremal spins exist. Numerical simulations of such systems require suitable initial data. In this paper, we examine three methods of constructing binary-black-hole initial data, focusing on their ability to generate black holes with nearly extremal spins: (i) Bowen-York initial data, including standard puncture data (based on conformal flatness and Bowen-York extrinsic curvature), (ii) standard quasiequilibrium initial data (based on the extended-conformal-thin-sandwich equations, conformal flatness, and maximal slicing), and (iii) quasiequilibrium data based on the superposition of Kerr-Schild metrics. We find that the two conformally flat methods (i) and (ii) perform similarly, with spins up to about 0.99 obtainable at the initial time. However, in an evolution, we expect the spin to quickly relax to a significantly smaller value around 0.93 as the initial geometry relaxes. For quasiequilibrium superposed Kerr-Schild data [method (iii)], we construct initial data with initial spins as large as 0.9997. We evolve superposed Kerr-Schild data sets with spins of 0.93 and 0.97 and find that the spin drops by only a few parts in 10<SUP>4</SUP> during the initial relaxation; therefore, we expect that superposed Kerr-Schild initial data will allow evolutions of binary black holes with relaxed spins above 0.99. Along the way to these conclusions, we also present several secondary results: the power-law coefficients with which the spin of puncture initial data approaches its maximal possible value; approximate analytic solutions for large spin puncture data; embedding diagrams for single spinning black holes in methods (i) and (ii); nonunique solutions for method (ii). All of the initial-data sets that we construct contain subextremal black holes, and when we are able to push the spin of the excision boundary surface into the superextremal regime, the excision surface is always enclosed by a second, subextremal apparent horizon. The quasilocal spin is measured by using approximate rotational Killing vectors, and the spin is also inferred from the extrema of the intrinsic scalar curvature of the apparent horizon. Both approaches are found to give consistent results, with the approximate-Killing-vector spin showing the least variation during the initial relaxation.
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https://arxiv.org/pdf/0805.4192.pdf
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{'Binary-black-hole initial data with nearly-extremal spins': 'Geoffrey Lovelace, 1,2 Robert Owen, 1, 2 Harald P. Pfeiffer, 2 and Tony Chu 2 \n1 Center for Radiophysics and Space Research, Cornell University, Ithaca, NY 14853 2 Theoretical Astrophysics 130-33, California Institute of Technology, Pasadena, CA 91125 (Dated: November 26, 2024) \nThere is a significant possibility that astrophysical black holes with nearly-extremal spins exist. Numerical simulations of such systems require suitable initial data. In this paper, we examine three methods of constructing binary-black-hole initial data, focusing on their ability to generate black holes with nearly-extremal spins: (i) Bowen-York initial data, including standard puncture data (based on conformal flatness and Bowen-York extrinsic curvature), (ii) standard quasi-equilibrium initial data (based on the extended-conformal-thin-sandwich equations, conformal flatness, and maximal slicing), and (iii) quasi-equilibrium data based on the superposition of Kerr-Schild metrics. We find that the two conformally-flat methods (i) and (ii) perform similarly, with spins up to about 0.99 obtainable at the initial time . However, in an evolution , we expect the spin to quickly relax to a significantly smaller value around 0.93 as the initial geometry relaxes. For quasi-equilibrium superposed Kerr-Schild (SKS) data [method (iii)], we construct initial data with initial spins as large as 0.9997. We evolve SKS data sets with spins of 0.93 and 0.97 and find that the spin drops by only a few parts in 10 4 during the initial relaxation; therefore, we expect that SKS initial data will allow evolutions of binary black holes with relaxed spins above 0.99. Along the way to these conclusions, we also present several secondary results: the power-law coefficients with which the spin of puncture initial data approaches its maximal possible value; approximate analytic solutions for large spin puncture data; embedding diagrams for single spinning black holes in methods (i) and (ii); non-unique solutions for method (ii). All of the initial data sets that we construct contain sub-extremal black holes, and when we are able to push the spin of the excision boundary surface into the super-extremal regime, the excision surface is always enclosed by a second, sub-extremal apparent horizon. The quasilocal spin is measured by using approximate rotational Killing vectors, and the spin is also inferred from the extrema of the intrinsic scalar curvature of the apparent horizon. Both approaches are found to give consistent results, with the approximate-Killing-vector spin showing least variation during the initial relaxation. \nPACS numbers: 04.25.D-,04.25.dg,04.20.Ex,02.70.Hm', 'I. INTRODUCTION': "There is a significant possibility that black holes with nearly-extremal spins exist; by 'nearly-extremal', we mean that the spin S and mass M of the hole satisfy 0 . 95 /lessorsimilar S/M 2 /lessorsimilar 1. Some models of black-hole accretion [1, 2, 3] predict that most black holes will have nearly-extremal spins, and observational evidence for black holes with nearly-extremal spins includes, e.g., estimates of black-hole spins in quasars [4] and estimates of the spin of a black hole in a certain binary Xray source [5]. There is considerable uncertainty about whether black holes do in fact typically have nearlyextremal spins; e.g., some models [6, 7, 8] of black-hole accretion do not lead to large spins. This uncertainty could be reduced by measuring the holes' spins directly using gravitational waves. \nThis prospect of detecting the gravitational waves emitted by colliding black holes, possibly with nearlyextremal spins, motivates the goal of simulating these spacetimes numerically. Indeed, one focus of intense research has been spinning black hole binaries, including the discovery of dramatic kicks when two spinning black holes merge [9, 10, 11, 12, 13, 14, 15, 16, 17] as well as some initial exploration of the orbital dynamics of spinning binaries [18, 19, 20, 21, 22, 23]. All of these simu- \nlations start from puncture initial data as introduced by Brandt and Brugmann [24]. \nThe simplifying assumptions employed in puncture initial data make it impossible to construct black holes with spins arbitrarily close to unity. The numerical value of the fastest obtainable spin depends on which dimensionless ratio is chosen to characterize 'black hole spin.' Often, dimensionless spin is defined based on quasilocal properties of the black hole, \nχ := S M 2 , (1) \nwhere S is taken to be nonnegative and is a suitable quasilocal spin (e.g., obtained using approximate rotational Killing vectors on the apparent horizon as described, for example, in Appendix A) and M is a suitable quasilocal mass. The latter may be obtained from Christodoulou's formula relating spin, area and mass of a Kerr black hole, \nM 2 := M 2 irr + S 2 4 M 2 irr , (2) \nwhere we define the irreducible mass in terms of the area A of the apparent horizon by M irr := A/ 16 π . \nThe quantity χ is not preserved during an evolution. Specifically, most black hole initial data are not exactly \n√ \nin equilibrium, which leads to transients and emission of an artificial pulse of gravitational radiation early in numerical simulations. The geometry in the vicinity of the black holes relaxes on a time-scale t relax (typically a few M ), and during this relaxation, the spin changes by \n∆ χ := χ ( t = 0) -χ ( t relax ) . (3) \nWhen constructing a single spinning black hole with standard puncture data [24], for instance, χ ( t = 0) /lessorsimilar 0 . 98, which seems encouragingly large. However Dain et al. [25, 26] evolved standard puncture data with initial spin close to this limit, and they find that the spin rapidly drops to χ ( t relax ) 0 . 93, i.e. ∆ χ ≈ 0 . 05. \n≈ \n≈ For single-black-hole spacetimes, another widely used dimensionless spin-measure is the ratio of total angular momentum 1 J ADM and Arnowitt-Deser-Misner (ADM) energy E ADM , \nε J := J ADM E 2 ADM . (4) \nDain et al. noted that χ ( t relax ) is close to ε J and explained this result as follows: the spacetime is axisymmetric, which implies that the angular momentum J ADM is conserved and that the black hole's spin equals J ADM . Moreover, so long as a negligible fraction of the spacetime's energy is carried off by the spurious radiation, the hole's quasi-local mass will relax to a value of E ADM , giving χ ( t relax ) ≈ ε J . Thus conformally-flat Bowen-York data cannot be used to simulate black holes with nearlyextremal equilibrium spins, even though the initial spins can be made fairly close to χ = 1. \nThis paper examines three different approaches of constructing black hole initial data with nearly-extremal spin. First, we revisit puncture initial data and inversionsymmetric Bowen-York initial data. We show that for a single, spinning black hole at rest, both approaches are identical, and we determine spin-limits based purely on initial data more accurately than before: \nε J ≤ 0 . 928200 , χ ( t = 0) ≤ 0 . 9837 . (5) \nWe show that the limiting values of ε J and χ ( t = 0) are approached as power-laws of the spin-parameter (curiously, with different powers). We furthermore give insight into the geometric structure of these high-spin Bowen-York initial data sets through numerical study and approximate analytical solutions and find that a cylindrical throat forms which lengthens logarithmically with the spin-parameter. \nSecond, we investigate the high-spin limit of another popular approach of constructing initial data, the quasiequilibrium formalism [27, 28, 29, 30, 31] based on the \nconformal thin sandwich equations [32, 33]. For the standard choices of conformal flatness and maximal slicing, we are able to construct initial data with spins somewhat larger than the standard Bowen-York limits given in Eq. (5): \nε J /lessorsimilar 0 . 94 , χ ( t = 0) /lessorsimilar 0 . 99 . (6) \nOnce again ε J is much lower than χ ( t = 0), which suggests that these data sets lead to equilibrium spins of approximate magnitude χ ≈ 0 . 94. Interestingly, these families of initial data are found to exhibit non-unique solutions [34, 35, 36], and the largest spins are obtained along the upper branch. \nThe third approach also utilizes the quasi-equilibrium formalism [27, 28, 29, 30, 31], but this time we make use of the freedom to chose an arbitrary background data. Specifically, we choose background data as a superposition of two Kerr-Schild metrics. This approach is based on the original proposal of Matzner and collaborators [37, 38] and was first carried over into the conformal thin sandwich equations in Ref. [39]; also, background data consisting of a single, non-spinning KerrSchild black hole was used to construct initial data for a black-hole-neutron-star binary in Ref. [40]. For single black holes, this data simply reduces to the analytical Kerr solution. For binary black holes, we construct initial data with spins as large as \nχ ( t = 0) = 0 . 9997 . (7) \nWe also present evolutions, demonstrating that our rapidly-spinning initial data sets remain rapidly-spinning after the numerical evolution relaxes. In particular, we evolve an orbiting binary with χ ( t = 0) = 0 . 9275 and a head-on merger with χ ( t = 0) = 0 . 9701. In both cases, | ∆ χ/χ ( t = 0) | is significantly smaller than 10 -3 . We conclude that the conformally-curved SKS initial data we present in this paper, in contrast with conformally-flat Bowen-York data, is suitable for simulating binary black holes with nearly-extremal spins. \nThroughout the paper, we use two different techniques to measure the dimensionless spin of black holes, which are described in the appendices. The first (Appendix A) technique uses the standard surface-integral based on an approximate rotational Killing vector of the apparent horizon. We compute the approximate Killing vector with a variation of the technique introduced by Cook and Whiting [41], extended with new normalization conditions of the approximate Killing vector, and we denote the resulting spin 'AKV spin', χ AKV . The second approach (Appendix B) is based on the shape of the horizon in the form of its scalar curvature; specifically, the spin magnitudes are inferred from the minimum and maximum of the intrinsic Ricci scalar curvature of the horizon. We call the spin inferred in this way the 'scalar curvature spin,' and we label the spin magnitudes inferred from the scalar curvature minimum and maximum as χ min SC and χ max SC , respectively. Typically, binary-blackhole initial data produces holes that are initially not in \nequilibrium. Therefore, we use only the AKV spin to measure the initial black hole spin (Secs. III-IV.) We use both the AKV and the scalar-curvature spin when we measure the spin after the holes have relaxed to equilibrium (Sec. V). \nWe also monitor whether any of the constructed initial data sets have super-extremal spins, as this may shed light, for example, on the cosmic censorship conjecture. When using the Christodoulou formula [Eq. (2)] to define M , the quasilocal dimensionless spin χ is by definition bounded [42], χ ≤ 1. This can be seen most easily by introducing the parameter ζ , defined as \nζ := S 2 M 2 irr , (8) \nand then rewriting χ as \nχ = 1 -(1 -ζ ) 2 1 + ζ 2 . (9) \nThe ratio χ is therefore not useful to diagnose superextremal black holes. A more suitable diagnostic is found in the parameter ζ . For Kerr black holes, the first term on the right-hand-side of Eq. (2) is always smaller or equal to the second, with equality only for extremal spin; i.e., ζ ≤ 1, with equality for extremal spin. This motivates an alternative definition of extremality [42]: a black hole is said to be superextremal if the second term in Eq. (2) is larger than the first one, i.e. if ζ > 1. In this paper, we monitor ζ , which we call the spin-extremality parameter, along with the dimensionless spin χ . We find instances where ζ exceeds unity. Before this happens, however, a larger, subextremal ( ζ < 1) apparent horizon appears, enclosing the smaller, superextremal horizon (Sec. IV B, Fig. 12). \nThis paper is organized as follows. Section II summarizes the various formalisms that we use to construct initial data. Section III investigates single black hole initial data, followed by the construction of binary-black-hole initial data in Sec. IV. Section V presents binary black hole evolutions that show the good properties of superposed Kerr-Schild data, and the various spin-diagnostics. We summarize and discuss our results in Sec. VI. Finally, Appendix A and Appendix B present our techniques to define black hole spin.", 'II. INITIAL DATA FORMALISM': 'Before constructing initial data for rapidly-spinning single (Sec. III) and binary (Sec. IV) black holes, we first summarize the initial data formalisms we will use. After laying some general groundwork in Sec. II A, we describe Bowen-York initial data (including puncture initial data) in Sec. II B and quasi-equilibrium extended-conformalthin-sandwich data in Sec. II C. \nM', 'A. Extrinsic curvature decomposition': "Initial data sets for Einstein's equations are given on a spatial hypersurface Σ and must satisfy the constraint equations \nR + K 2 -K ij K ij = 0 , (10) \n∇ j K ij -g ij K ) = 0 . (11) \n( \n) Here, g ij is the induced metric of the slice Σ, with covariant derivative ∇ i , R := g ij R ij denotes the trace of the Ricci-tensor R ij , and K ij denotes the extrinsic curvature of the slice Σ as embedded into the space-time manifold . \nThe constraint equations (10) and (11) can be transformed into elliptic partial differential equations using a conformal transformation, e.g. [33]. One introduces a conformal metric, ˜ g ij via \ng ij = ψ 4 ˜ g ij , (12) \nwith the strictly positive conformal factor ψ > 0. Substituting Eq. (12) into Eq. (10) yields an elliptic equation for ψ . One furthermore decomposes the extrinsic curvature into trace and tracefree part, \nK ij = A ij + 1 3 g ij K, (13) \nand splits off a longitudinal part from the tracefree extrinsic curvature, \nA ij = 1 σ ( L V ) ij + M ij . (14) \nIn Eq. (14), σ is a strictly positive weight-function, the longitudinal operator is defined as ( L V ) ij = 2 ∇ ( i V j ) -2 3 g ij ∇ k V k , and M ij is symmetric and trace-free 2 . Finally, one introduces the conformally scaled quantities σ = ψ 6 ˜ σ , M ij = ψ -10 ˜ M ij , which allows the momentum constraint [Eq. (11)] to be rewritten completely in terms of conformal quantities: \nA ij = ψ -10 ˜ A ij , (15) \n˜ A ij = 1 ˜ σ ( ˜ L V ) ij + ˜ M ij . (16) \nThe Hamiltonian and momentum constraints then become \n˜ ∇ 2 ψ -1 8 ˜ R -1 12 K 2 ψ 5 + 1 8 ˜ A ij ˜ A ij ψ -7 = 0 , (17) \n˜ ∇ j ( 1 ˜ σ ( ˜ L V ) ij ) -2 3 ψ 6 ˜ ∇ i K + ˜ ∇ j ˜ M ij = 0 . (18) \nGiven choices for ˜ M ij , K , ˜ g ij and ˜ σ , and also boundary conditions, one can solve Eqs. (17) and (18) for ψ and V i , and then assemble the (constraint-satisfying) initial data g ij and K ij . \nMany important approaches to construct binary black hole initial data can be cast in this form. The various approaches differ in the choices for the freely specifiable parts and the boundary conditions. Some choices of free data aim for simplicity, such as Bowen-York initial data. Other approaches aim to preserve freedom, resulting in more complicated sets of equations but also more flexibility to control properties of the resulting initial data. The quasi-equilibrium extended-conformal-thin-sandwich approach falls into this second category, and we will exploit precisely its inherent freedom in choosing the free data to construct black holes with nearly-extremal spins.", 'B. Bowen-York initial data': "In this section, we describe two approaches of constructing initial data based on the well-known BowenYork extrinsic curvature. These two approaches, puncture data and inversion-symmetric data, differ in how they treat the coordinate singularity at r = 0; both can be obtained from the general procedure outlined in Sec. II A by setting ˜ σ ≡ 1, K ≡ 0, ˜ M ij ≡ 0 and by using a conformally flat metric \n˜ g ij = f ij . (19) \nThe momentum constraint [Eq. (18)] then reduces to ˜ ∇ j ( ˜ L V ) ij = 0, which is solved by choosing the analytical Bowen-York solutions [43, 44]. \nThe Bowen-York solutions can be written down most conveniently in Cartesian coordinates, f ij = δ ij : \nV i P = -1 4 r 7 P i + n i P k n k ] , (20) \n[ \n] V i S = -1 r 2 /epsilon1 i lm S l n m , (21) \nwhere r = ( x i x j δ ij ) 1 / 2 is the coordinate distance to the origin and n i = x i /r is the coordinate unit vector pointing from the origin to the point under consideration. The spatially-constant vectors P i and S i parametrize the solutions 3 \n[ \n˜ A ij P = 3 2 r 2 2 P ( i n j ) -( δ ij -n i n j P k n k ] , (22) \n( ] ˜ A ij S = 6 r 3 n ( i /epsilon1 j ) kl S k n l . (23) \n) \nThe conformal factor ψ is then determined by the Hamiltonian constraint [Eq. (17)], which simplifies to \n˜ ∇ 2 ψ + 1 8 ψ -7 ˜ A ij ˜ A ij = 0 . (24) \nWe would like to recover an asymptotically flat space; this implies the boundary condition ψ 1 as r →∞ . \n→ \n→∞ This boundary condition makes it possible to evaluate the linear ADM-momentum and ADM-like angular momentum of Bowen-York initial data without solving Eq. (24). These quantities are defined by surface integrals at infinity, \nJ ( ξ ) = 1 8 π ∮ ∞ ( K ij -g ij K ) ξ i s j dA, (25) \nwhere s i is the outward-pointing unit-normal to the integration sphere 4 . By letting ψ → 1 in Eq. (15), one can replace K ij by ˜ A ij and then evaluate the resulting integrals. The choice of vector ξ i determines which quantity is computed: For instance, ξ = ˆ e x corresponds to the x-component of the linear ADM-momentum, ξ = ∂ φ = -x ˆ e y + y ˆ e x yields the z-component of the ADM-like angular momentum 5 . For Eqs. (22) and (23), the results are P i ADM = P i and J i ADM = S i , respectively. \nThe ADM energy is given by the expression \nE ADM = 1 16 π ∮ ∞ ∇ j ( G i j -δ i j G ) s i dA, (26) \nwhere G ij := g ij -f ij , G := G ij g ij . For conformal flatness, Eq. (26) reduces to \nE ADM = -1 2 π ∮ ∞ ∂ r ψ dA. (27) \nThe derivative of the conformal factor is known only after Eq. (24) is solved; therefore, in contrast with the linear and angular momenta, E ADM can be computed only after solving the Hamiltonian constraint. \nWe now turn our attention to inner boundary conditions. ˜ A ij P and ˜ A ij S are singular at r = 0. This singularity is interpreted as a second asymptotically flat universe; when solving Eq. (24), this can be incorporated in two ways: \n- · Inversion Symmetry : The demand that the solution be symmetric under inversion at a sphere with radius R inv centered on the origin [44] results in a boundary condition for ψ at r = R inv , namely ∂ψ/∂r = -ψ/ (2 R inv ). The Hamiltonian constraint Eq. (24) is solved only in the exterior of the sphere, r ≥ R inv , and the solution in the interior can be recovered from inversion symmetry [44], e.g. \nψ ( x i ) = R inv r ψ ( R 2 inv r 2 x i ) . (28) \n- · Puncture data : One demands [24] the appropriate singular behavior of ψ for r → 0 to ensure that the second asymptotically flat end is indeed flat. That is, ψ must behave as \nψ ( x i ) = m p 2 r +1+ u ( x i ) (29) \nfor some positive parameter m p (the 'puncture mass') and function u ( x i ) that is finite and continuous in R 3 and approaches 0 as r →∞ . Equation (24) then implies an equation for u that is finite everywhere and can be solved without any inner boundaries: \n˜ ∇ 2 u = -1 8 ˜ A ij ˜ A ij r 7 r + m p 2 + ur ) 7 . (30) \n( \n) The majority of binary black hole simulations use puncture data, see, e.g., Refs. [9-23]. \nBoth approaches allow specification of multiple blackholes at different locations, each with different spin and momentum parameters S i and P i . For puncture data this is almost trivial; this accounts for the popularity of puncture data as initial data for black hole simulations. In contrast, for inversion-symmetric data, one needs to employ a rather cumbersome imaging procedure 6 (see e.g. [47] for details). \nFor a single spinning black hole at the origin, the extrinsic curvature ˜ A ij S given by Eq. (23) is identical for inversion-symmetric and puncture data. For inversionsymmetric data, the conformal factor has the usual falloff at large radii, \nψ ( x i ) = 1 + E ADM 2 r + O ( r -2 ) , as r →∞ . (31) \nUsing Eq. (28) we find the behavior of ψ as r → 0: \nψ ( x i ) = R inv r + E ADM 2 R inv + O ( r ) , as r → 0 . (32) \nComparison with Eq. (29) shows that this is precisely the desired behavior for puncture data, if one identifies R inv = m p / 2 and E/ (2 R inv ) = 1 + u (0). Because puncture data has a unique solution, it follows that for single spinning black holes, puncture data and inversionsymmetric data are identical , provided m p = 2 R inv . \nFor inversion-symmetric initial data for a single, spinning black hole, it is well-known [48] that the apparent horizon coincides with the inversion sphere, r AH = R inv . Therefore, we conclude that for puncture data for a single, spinning black hole, the apparent horizon is an exact coordinate sphere with radius r AH = m p / 2, despite ˜ A ij S and u ( x i ) not being spherically symmetric.", 'C. Quasi-equilibrium extended-conformal-thin-sandwich initial data': "Another popular approach of constructing binaryblack-hole initial data is the quasi-equilibrium extendedconformal-thin-sandwich (QE-XCTS) formalism [27, 28, 29, 30, 31]. Instead of emphasizing the extrinsic curvature, the conformal thin sandwich formalism [32] emphasizes the spatial metric g ij and its time-derivative . Nevertheless, it is equivalent [33] to the extrinsic curvature decomposition outlined in Sec. II A. The vector V i is identified with the shift β i , \nV i ≡ β i , (33) \nand the weight-functions σ and ˜ σ are identified (up to a factor 2) with the lapse and the conformal lapse, respectively, \nσ ≡ 2 α, ˜ σ ≡ 2˜ α. (34) \nThe tensor ˜ M ij is related to the time-derivative of the spatial metric, ˜ u ij := ∂ t ˜ g ij by \n˜ M ij ≡ 1 2˜ α ˜ u ij . (35) \nBecause M ij is trace free [Eqs. (13) and (15)-(16)], we require ˜ u ij to be trace free. \nThe conformal thin sandwich equations allow control of certain time-derivatives in the subsequent evolution of the constructed initial data. If the lapse α and shift β i from the initial data are used in the evolution, for instance, then the trace-free part of ∂ t g ij will be proportional to ˜ u ij . Therefore (see Refs. [27, 30]) \n˜ u ij ≡ 0 (36a) \nis a preferred choice for initial data sets that begin nearly in equilibrium, such as binary black holes quasi-circular orbits. \nThe evolution equation for K can be used to derive an elliptic equation for the conformal lapse ˜ α (or, equivalently, for αψ ). Upon specification of \n∂ t K ≡ 0 , (36b) \nthis fifth elliptic equation is to be solved for ˜ α simultaneously with Eqs. (17) and (18), cf. [27, 30]. \nOur numerical code uses the conformal factor ψ , the shift β i , and the product of lapse and conformal factor αψ = ˜ αψ 7 as independent variables, in order to simplify the equation for ∂ t K . Thus, the actual equations being solved take the form \n0 = ˜ ∇ 2 ψ -1 8 ˜ Rψ -1 12 K 2 ψ 5 + 1 8 ψ -7 ˜ A ij ˜ A ij , (37a) 0 = ˜ ∇ j ( ψ 7 2( αψ ) ( ˜ L β ) ij ) -2 3 ψ 6 ˜ ∇ i K -˜ ∇ j ( ψ 7 2( αψ ) ˜ u ij ) , (37b) \n0 = ˜ ∇ 2 ( αψ ) -( αψ ) [ ˜ R 8 + 5 12 K 4 ψ 4 + 7 8 ψ -8 ˜ A ij ˜ A ij ] + ψ 5 ( ∂ t K -β k ∂ k K ) , (37c) \nwith \n˜ A ij = ψ 7 2 αψ ( ( ˜ L β ) ij -˜ u ij ) . (37d) \nThese equations can be solved only after \n- 1. specifying the remaining free data: i.e., the conformal metric ˜ g ij and the trace of the extrinsic curvature K (we chose already ˜ u ij ≡ 0 and ∂ t K ≡ 0),\n- 2. choosing an inner boundary S which excises the black holes' singularities, and also an outer boundary B , and\n- 3. choosing boundary conditions for ψ , αψ , and β i on B and S . \nThe initial data is required to be asymptotically flat, and the outer boundary B is placed at infinity 7 . If ˜ g ij is asymptotically flat, the outer boundary conditions are then \nψ = 1 on B , (38a) \nαψ = 1 on , (38b) \nβ i = ( Ω 0 × r ) i + ˙ a 0 r i on B . (38c) \nB \nHere r i is the coordinate position vector. The shift boundary condition consists of a rotation (parametrized by the orbital angular velocity Ω 0 ) and an expansion (parametrized by ˙ a 0 ); the initial radial velocity is necessary for reducing orbital eccentricity in binary-black-hole initial data [49]. \nThe inner boundary condition on the conformal factor ψ ensures that the excision surfaces S are apparent \nhorizons [27]: \n˜ s k ∂ k ψ = -ψ -3 8˜ α ˜ s i ˜ s j [ ( ˜ L β ) ij -˜ u ij ] -ψ 4 ˜ h ij ˜ ∇ i ˜ s j + 1 6 Kψ 3 on S . (39) \nHere ˜ s i := ψ 2 s i , s i is unit vector normal to S , and ˜ h ij := ˜ g ij -˜ s i ˜ s j is the induced conformal 2-metric on S . \n-The inner boundary condition on the shift is \nβ i = αs i -Ω r ξ i on S , (40) \nwhere ξ i s i = 0. The first term on the right-hand-side ensures that the apparent horizons are initially at rest; the tangential term determines the black hole's spin [27, 28, 29]. \nReferences [27, 28, 29] chose the sign of the last term in Eq. (40) such that positive values of Ω r counteract the spin of the corotating holes that are obtained with Ω r = 0. Here, we are interested in large spins, and we reverse the sign of the last term in Eq. (40) so that positive, increasing Ω r results in increasing spins. \nTwo sets of choices for ˜ g ij , K , S , and the boundary condition for αψ on S are discussed in the next subsections. Each set of choices will be used to construct binary-black-hole initial data in Sec. IV.", '1. Conformal flatness & maximal slicing (CFMS)': 'The simplest choice for ˜ g ij is a flat metric, \n˜ g ij ≡ f ij . (41) \nThis choice has been used almost exclusively in the previous formulations of binary-black-hole initial data. \nThe simplest choice for K , also commonly used in prior formulations of binary-black-hole initial data, is maximal slicing, i.e. \nK ≡ 0 . (42) \nAlso for simplicity, we choose to make the excision surface S consist of coordinate spheres: \nS = n ⋃ a =1 S a , (43) \nwhere S a are surfaces of constant Euclidean distance r exc about the center of each excised hole, and n = 1 or 2 is the number of black holes present in the initial data. \nThe boundary condition for the lapse on S determines the temporal gauge; we adopt the condition given in Eq. (59a) of Ref. [28]: \n∂ ∂r a ( αψ ) = 0 on S a , (44) \nwhere r a is the Euclidean distance from the center of hole a . This type of initial data is used in Refs. [49, 50, 51].', '2. Superposed Kerr Schild (SKS)': "Single black holes with angular [52, 53] or linear [54] momentum do not admit conformally-flat spatial slicings; therefore, conformal flatness [Eq. (41)] is necessarily deficient. This has motivated investigations of binary-blackhole initial data whose free data have stronger physical motivation, e.g. Refs. [37, 38, 55, 56, 57, 58, 59, 60, 61]. \nIn this subsection, we consider conformally-curved data that are in the same spirit as the SKS data of Refs. [37, 38] although here i) we apply the idea to the QE-XCTS formalism, and ii) as discussed below, our free data is very nearly conformally-flat and maximally-sliced everywhere except in the vicinity of the black holes . \nThe choices we make here generalize the conformallycurved data in chapter 6 of Ref. [39] to nonzero spins. Specifically, the free data and lapse boundary condition will be chosen so that the conformal geometry near each hole's horizon is that of a boosted, spinning, Kerr-Schild black hole. The conformal metric ˜ g ij and the mean curvature K take the form \n˜ g ij := f ij + n ∑ a =1 e -r 2 a /w 2 a ( g a ij -f ij ) , (45) \nK := n ∑ a =1 e -r 2 a /w 2 a K a . (46) \nHere g a ij and K a are the spatial metric and mean curvature, respectively, of a boosted, spinning Kerr-Schild black hole with mass ˜ M a , spin ˜ S a , and speed ˜ v a . \nFar from each hole's horizon, the conformal metric is very nearly flat; this prevents the conformal factor from diverging on the outer boundary [39]. The parameter w a is a weighting factor that determines how quickly the curved parts of the conformal data decay with Euclidean distance r a ( a = 1 , 2 , ... ) from hole a ; in this paper, the weight factor w a is chosen to be larger than the size scale of hole a but smaller than the distance d to the companion hole (if any): M a /lessorsimilar w a /lessorsimilar d a . This is similar to the 'attenuated' superposed-Kerr-Schild data of Refs. [38, 62], except that here the weighting functions are Gaussians which vanish far from the holes, while in Refs. [38, 62] the weighting functions go to unity far from the holes. \nThe excision surfaces S a are not coordinate spheres unless ˜ S a = 0 and ˜ v a = 0. Instead they are deformed in two ways. i) They are distorted so that they are surfaces of constant Kerr radius r Kerr , i.e. \nx 2 + y 2 r 2 Kerr + ˜ S a 2 / ˜ M a 2 + z 2 r 2 Kerr = 1 (47) \nwhere x , y , and z are Cartesian coordinates on the S . Then, ii) the excision surfaces are Lorentz-contracted along the direction of the boost. \nThe boundary condition for the lapse α on S a is a Dirichlet condition that causes α (and, consequently, the \ntemporal gauge) in the vicinity of each hole to be nearly that of the corresponding Kerr-Schild spacetime, i.e. \nαψ = 1 + n ∑ a =1 e -r 2 a /w 2 a ( α a -1) on S a , (48) \nwhere α a is the lapse corresponding to the Kerr-Schild spacetime a .", 'III. SINGLE-BLACK-HOLE INITIAL DATA WITH NEARLY-EXTREMAL SPINS': "In this section, we examine to which extent the formalisms presented in Sec. II can generate single black hole initial data with nearly-extremal spin. We consider first Bowen-York initial data and then conformally-flat quasi-equilibrium data. Since superposed-Kerr-Schild data can represent single Kerr black holes exactly, there is no need to investigate single-hole superposed-Kerr-Schild data. In Sec. IV, we will both consider conformally-flat and superposed-Kerr-Schild data for binary black holes. \nTo orient the reader, the initial data sets constructed in this section, as well as the binary-black-hole data sets constructed in Sec. IV, are summarized in Table I. \nUnless noted otherwise, all spins presented in this section are measured using the approximate-Killing-vector spin χ AKV described in Appendix A. Therefore, the subscript 'AKV' in χ AKV will be suppressed for simplicity.", 'A. Bowen-York (puncture) initial data': "As discussed in Sec. II B, for a single spinning black hole at rest, puncture initial data is identical to inversionsymmetric initial data. Such solutions have been examined in the past (e.g. [48, 63]), and additional results were obtained (partly in parallel to this work) in the study by Dain, Lousto, and Zlochower al [25]. \nWe revisit this topic here to determine the maximum possible spin of Bowen-York (BY) initial data more accurately than before, to establish the power-law coefficients for the approach to these limits with increasing spin parameter S , and to present new results about the geometric structure of Bowen-York initial data with very large spin parameter. \nWe solve Eq. (30) with the pseudo-spectral elliptic solver described in Ref. [64]. The singular point of u at the origin is covered by a small rectangular block extending from ± 10 -4 m p along each coordinate axis. This block overlaps four concentric spherical shells with radii of the boundaries at 8 · 10 -5 m p , 0 . 005 m p , 0 . 3 m p , 50 m p , and 10 9 m p . The equations are solved at several different resolutions, with the highest resolution using 20 3 basisfunctions in the cube, L = 18 in the spheres and 26 and 19 radial basis-functions in the inner and outer two spherical shells, respectively. \nTABLE I: Summary of the initial data sets constructed in this paper. The first row (BY-Single) represents Bowen-York initial data for single black holes of various spins. The next two rows (CFMS-Single and CFMS) are quasi-equilibrium, conformallyflat, maximally-sliced initial data for single and binary spinning black holes, respectively. All other data sets employ superposed Kerr-Schild quasi-equilibrium data with the second block of rows representing families of initial data sets for various spins and the last block of rows representing individual data sets to be evolved. The data sets SKS-0.93-E0 to SKS-0.93-E3 demonstrate eccentricity removal, and SKS-HeadOn is used in a head-on evolution. The first block of columns gives the label used for each data set, and the relevant section of this paper devoted to it. The next block of columns lists the most important parameters entering the initial data. The last block of columns lists some properties of those data sets that we evolve in Sec. V. \n| Label | Section | Figures n | d Ω 0 | ˙ a 0 × 10 4 | Ω r or S/m 2 p | ˜ S | | χ AKV | | M irr | M E ADM |\n|-------------|-----------|-----------------------------------|---------------|----------------|-------------------------|-------|-------------|--------------------------------|-----------|\n| BY-Single | III A | 1-5, 8, 19 1 | - - | - | 0 . 01 ≤ S/m 2 p ≤ 10 4 | - | | | |\n| CFMS-Single | III B | 6-8, 19 1 | - - | - | 0 ≤ Ω r ≤ 0 . 191 | - | | | |\n| CFMS | IVA | 9, 13 | 2 32 0.007985 | 0 | 0 ≤ Ω r ≤ 0 . 1615 | - | | | |\n| SKS-0.0 | IVB | 11, 13 | 2 32 0.006787 | 0 | 0 ≤ Ω r ≤ 0 . 24 | 0 | | | |\n| SKS-0.5 | IVB | 11, 13 | 2 32 0.006787 | 0 | 0 ≤ Ω r ≤ 0 . 27 | 0.5 | | | |\n| SKS-0.93 | IVB | 11-13 | 2 32 0.006787 | 0 | 0 ≤ Ω r ≤ 0 . 35 | 0.93 | | | |\n| SKS-0.99 | IVB | 10-13 | 2 32 0.007002 | 3.332 | 0 . 28 ≤ Ω r ≤ 0 . 39 | 0.99 | | | |\n| SKS-0.93-E0 | VB | 14 | 2 32 0.006787 | 0 | 0.28 | | | 0.93 0.9278 0.9371 1.131 2.243 | |\n| SKS-0.93-E1 | VB | 14 | 2 32 0.007 | 0 | 0.28 | | | 0.93 0.9284 0.9375 1.132 2.247 | |\n| SKS-0.93-E2 | VB | 14 | 2 32 0.006977 | 3.084 | 0.28 | | | 0.93 0.9275 0.9395 1.134 2.249 | |\n| SKS-0.93-E3 | | VC 10-11, 13-16, 19 2 32 0.007002 | | 3.332 | 0.28 | | | 0.93 0.9275 0.9397 1.134 2.250 | |\n| SKS-HeadOn | | VD 10-11, 13, 17-19 2 100 | 0 | 0 | 0.3418 | | | 0.97 0.9701 0.8943 1.135 2.257 | | \nFIG. 1: Convergence test for a single puncture black hole with a very large spin parameter S/m 2 p = 10000. Plotted are results vs. resolution N , which is the total number of basis-functions. The solid lines show the relative differences of three angular momentum measures to the analytically expected value 10000. The dashed lines show differences from the next-higher resolution of two dimensionless quantities for which no analytic answer is available. \n<!-- image --> \nBecause of the axisymmetry of the data-set, the rotational Killing vector of the apparent horizon is simply ∂ φ . The integral for the quasilocal spin, Eq. (A1) \nturns out to be independent of ψ and can be evaluated analytically with a result equal to the spin-parameter, S . Thus we can use this initial data set to check how well our spin-diagnostics and our ADM angular momentum diagnostic works (recall that J ADM is also equal to the spin-parameter S ). This comparison is performed in Fig. 1, which shows relative differences between the numerically extracted values for the approximate-Killingvector (AKV) spin, the coordinate spin (defined with the AKV spin in Appendix A), and the ADM angular momentum J ADM relative to the expected answer, S . The figure also shows differences between neighboring resolutions for the two quantities of interest below, S/M 2 = χ and S/E 2 ADM = J ADM /E 2 ADM = ε J . \nFigure 1 seems to show exponential convergence with increased resolution N . Since puncture data is only C 2 at the puncture, one would rather expect polynomial convergence. The effect of the non-smoothness at the puncture is mitigated by choosing a very high resolution close to the puncture (a small cube with sides ± 10 -4 m p with 20 3 basis-functions). Therefore, for the resolutions considered in Fig. 1, the truncation error is dominated by the solution away from the puncture, and exponential convergence is visible. If we used infinite-precision arithmetic and were pushing toward higher resolution than shown in Fig. 1, then we would expect to eventually see polynomial convergence dominated by the cube covering the puncture. \nNext, we construct a series of initial data sets with increasing spin-parameter S , and compute χ , ε J , and ζ for each initial data set. The results are plotted in Fig. 2 and confirm earlier results [26, 63]. In addition, the inset shows that the asymptotic values χ max = 0 . 9837 and \nFIG. 2: Properties of single, spinning puncture black holes with spin-parameter S and puncture mass m p . The dimensionless spin χ := S/M 2 , ADM angular momentum ε J := J ADM /E 2 ADM , and spin-extremality parameter ζ := S/ ' 2 M 2 irr ' are plotted against the spin parameter S/m 2 p . The horizon mass M is related to the spin S and irreducible mass M irr in Eq. (2). \n<!-- image --> \nε u, max = 0 . 928200 are approached as power-laws in the spin-parameter, \nχ max -χ ∝ ( S m 2 p ) -0 . 75 , (49) \nε J, max -ε J ∝ ( S m 2 p ) -1 . 4 . (50) \nThe exponents of these power-laws are computed here for the first time. \nTo confirm that the apparent horizon is indeed at r = R inv , we ran our apparent horizon finder on the high-spin puncture initial data sets. The horizon finder had great difficulty converging, and the reason for this becomes clear from Fig. 3. The main panel of this figure shows the area of spheres with coordinate radius r . The area is minimal at r = m p / 2, as it must be, since m p / 2 = R inv is the radius of the inversion sphere. However, the area is almost constant over a wide range in r -for S/m 2 p = 10000 over about two decades in either direction: 0 . 01 /lessorsimilar r/R inv /lessorsimilar 100. Thus, the EinsteinRosen bridge (the throat) connecting the two asymptotically flat universes lengthens as the spin increases, giving rise to an ever-lengthening cylinder. If this were a perfect cylinder, then the expansion would be zero for any r = const cross-section. Because the geometry is not perfectly cylindrical, the expansion vanishes only for r = m p / 2 = R inv , but remains very small even a sig- \nFIG. 3: Properties of coordinate spheres with radius r for high-spin puncture initial data. Main panel: Area of these spheres. Inset: residual of the apparent horizon equation on these spheres. The area is almost constant over several orders of magnitude in r . The apparent-horizon-residual vanishes at r = R inv , but is very small over a wide range of r . \n<!-- image --> \nnificant distance away from r = m p / 2 = R inv . This is shown in the inset, which plots the residual of the apparent horizon finder at different radii. \nWith the lengthening of the throat, the interval in r with small expansion lengthens, and the value of the expansion within this interval reduces. Both effects make it harder for the apparent horizon finder to converge. In Fig. 2, we have used our knowledge of the location of the apparent horizon to set r AH = m p / 2, rather than to find this surface numerically. Without this knowledge, which arises due to the identification of puncture data and inversion symmetric data, computation of Fig. 2 would have been significantly harder, perhaps impossible. \nLet us assume for the moment that the solution ψ ( r ) = m p 2 r +1+ u ( r ) is spherically symmetric (we give numerical evidence below that this is indeed a good approximation). Because g ij = ψ 4 f ij , the area of coordinate spheres is then given by \nA ( r ) = 4 πψ 2 ( r ) r. (51) \nIn the throat region, where A ( r ) ≈ const, the conformal factor must therefore behave like 1 / √ r , as also argued independently by Dain, Lousto, and Zlochower [25]. \nTo extend on Dain et al. 's analysis, let us substitute Eq. (23) into Eq. (24) to obtain the well-known equation \n˜ ∇ 2 ψ = -9 S 2 sin 2 θ 4 r 6 ψ -7 , (52) \nwhere θ is the angle between the spin-direction and the point x i . Continuing to assume that ψ is approximately \nspherically symmetric, we can replace the factor sin 2 θ by its angular average (4 π ) -1 sin 2 θ d Ω = 2 / 3, and obtain \nd 2 ¯ ψ dr 2 + 2 r d ¯ ψ dr = -3 S 2 2 r 6 ¯ ψ -7 . (53) \n∫ \nHere, we introduced an overbar ¯ ψ to distinguish the spherically symmetric solution ¯ ψ ( r ) of Eq. (53) from the full solution ψ ( x i ) of puncture/inversion-symmetric initial data. Following Dain et al. [25] we assume that the conformal factor behaves as a power-law ( ¯ ψ ( r ) = Ar α ) and substitute this into Eq. (53). We find that Eq. (53) determines the power-law exponent α = -1 / 2 and the overall amplitude A = (6 S 2 ) 1 / 8 , so that \n¯ ψ ( r ) = ( 6 S 2 ) 1 / 8 √ r = 96 1 / 8 ( S m 2 p ) 1 / 4 ( r R inv ) -1 / 2 . (54) \nIn Eq. (54), we chose the scaling S/m 2 p which is commonly used in the puncture-data literature, but kept r/R inv to emphasize the inversion symmetry of the data in our figures (in a log-plot using r/R inv , the solution will appear symmetric, see e.g. Fig. 3). While ¯ ψ ( r ) solves the spherically symmetric Eq. (53) exactly, it must deviate from ψ ( x i ) for sufficiently large r because ¯ ψ → 0 as r → ∞ , whereas ψ → 1. The deviation will become significant when ¯ ψ ∼ 1, i.e. at radius r x ∼ √ S/m 2 p . Because of inversion symmetry, this implies a lower bound of validity at 1 /r x , so that Eq. (54) holds for \n( S m 2 p ) -1 / 2 /lessorsimilar r R inv /lessorsimilar ( S m 2 p ) 1 / 2 . (55) \nThe circumference of the cylindrical throat is \nC = 2 π ¯ ψ ( r ) 2 r = 2 π 96 1 / 4 √ S m 2 p R inv , (56) \nand its length is \nL = ∫ ( S/m 2 p ) 1 / 2 ( S/m 2 p ) -1 / 2 ¯ ψ 2 ( r ) dr = 96 1 / 4 √ S m 2 p ln ( S m 2 p ) R inv . (57) \nTherefore, the ratio of length to circumference, \nL C = 1 2 π ln ( S m 2 p ) , (58) \ngrows without bound as S/m 2 p becomes large, albeit very slowly. The scaling with ( S/m 2 p ) 1 / 2 in Eqs. (55)-(57) might seem somewhat surprising. However, in the large spin limit, S/M 2 is just a constant close to unity (namely χ max = 0 . 9837). Therefore, S 1 / 2 ≈ M , i.e. the scaling S 1 / 2 is effectively merely a scaling with mass. \nFigure 4 shows the conformal factor ψ , the 'puncture function' u , and the estimate ¯ ψ of Eq. (54) for three different values of S/m 2 p . There are several noteworthy features in this figure. First, both ψ and u show clearly three different regimes: \nFIG. 4: Solutions of high-spin puncture initial data. Plotted are the conformal factor ψ and puncture function u in the equatorial plane as a function of radius r . Furthermore, the approximate solution ¯ ψ is included, with solid circles denoting the range of validity of this approximation, cf. Eq. (55). Three curves each are plotted, corresponding from top to bottom to S/m 2 p = 10000 , 1000 , 100. \n<!-- image --> \n- · For large r , ψ ≈ 1 and u ∝ 1 /r . This is the upper asymptotically-flat end.\n- · For intermediate r , ψ ∝ 1 / √ r and u ∝ 1 / √ r . This is the cylindrical geometry extending symmetrically around the throat. This region becomes more pronounced as S increases.\n- · For small r , ψ ∝ 1 /r and u ≈ const. This is the lower asymptotically-flat end. \nFigure 4 also plots the approximate solution ¯ ψ [cf. Eq. (54)] for its range of validity [given by Eq. (55)]. Note that slope and amplitude of ¯ ψ fit very well the numerical solution ψ . In fact, the agreement is much better than with u . \nOne could also have started the calculation that led to Eq. (54) with Eq. (30). Assuming spherical symmetry, and assuming that u /greatermuch m p / (2 r ) + 1, we would have derived Eq. (53), but with ¯ ψ replaced by u . We would then have found the approximate behavior Eq. (54) for u . The disadvantage of this approach is the need for additional approximations, which reduce the accuracy of the result. From Fig. 4 we see that, in the throat region, the dotted lines representing ¯ ψ are close to the dashed lines of u . But the agreement between ψ and ¯ ψ is certainly better. \nFinally, we note that the limits of validity of ¯ ψ [Eq. (55)] match very nicely the points where the numerical ψ diverges from ¯ ψ . \nFIG. 5: Angular decomposition of the conformal factor ψ ( r, θ, φ ) for single black hole puncture data. \n<!-- image --> \nTo close this section, we present numerical evidence that indeed ψ is approximately spherically symmetric, the assumption that entered into our derivation of Eq. (54). We decompose the conformal factor of the numerical puncture data solutions into spherical harmonics, \nψ ( r, θ, φ ) = ∞ ∑ l =0 l ∑ m = -l ψ lm ( r ) Y lm ( θ, φ ) , (59) \nand plot in Fig. 5 the sizes of the l /negationslash = 0 modes relative to the spherically symmetric mode ψ 00 . Because of the symmetries of the problem, the only non-zero modes have m = 0 and even l . In the throat region, the largest non-spherically symmetric mode ψ 20 is about a factor of 65 smaller than the spherically symmetric mode. With increasing l , ψ lm decays very rapidly. Also, in both asymptotically flat ends, the non-spherically symmetric modes decay more rapidly than the l = 0 mode, as expected for asymptotically flat data. This figure again shows nicely the inversion symmetry of the data, under r/R inv → ( r/R inv ) -1 . Given the simple structure of the higher modes, it should be possible to extend the analytical analysis of the throat to include the non-spherical contributions. To do so, one would expand ψ as a series in Legendre polynomials in θ ; the ψ -7 -term on the right hand side of Eq. (52) would result in a set of ordinary differential equations for those coefficients. In the throat region, the radial behavior of each mode should be ∝ 1 / √ r , and the ordinary differential equations should simplify to algebraic relations. \nFIG. 6: Conformally-flat, maximally-sliced, quasiequilibrium initial data sets with a single, spinning black hole. We plot the horizon mass M , irreducible mass M irr , and the (approximateKilling-vector) spin S against the rotation parameter Ω r [cf. Eq. (40)]. Only Ω r is varied in this figure; all other parameters are held fixed. The upper and lower points with the same Ω r are obtained numerically by choosing different initial guesses. The inset shows a close-up view of the turning point, which occurs at Ω r ≈ 0 . 191 . \n<!-- image -->", 'B. Quasi-equilibrium extended-conformal-thin-sandwich data': "We have seen in Sec. III A that puncture initial data for single, spinning black holes can be constructed for holes with initial spins of χ ≤ 0 . 9837. In this section, we address the analogous question for excision blackhole initial data: how rapid can the initial spin be for a single, spinning black hole constructed using quasiequilibrium, extended-conformal-thin-sandwich (QE-XCTS) initial data? \nAs noted previously, if the free data ˜ g ij and K are chosen to agree with the analytic values for a Kerr black hole, g Kerr ij and K Kerr , then the QE-XCTS initial data can exactly represent a single Kerr black hole. In this case, χ = 1 is obtained trivially by choosing ˜ S = ˜ M 2 = 1, where ˜ M and ˜ S are the mass and spin, respectively, of the Kerr black hole described by the conformal metric. \nSetting aside this trivial solution, we construct conformally-flat, maximally-sliced (CFMS) data for a single, spinning hole. We construct a family of QEXCTS initial data sets for single spinning black holes by numerically solving the XCTS equations [in the form stated in Eqs. (37a)-(37c)] using the same spectral elliptic solver [64] as in Sec. III A. The free data are given by Eqs. (41)-(42) and by Eqs. (36a)-(36b). \nOn the outer boundary B , we impose Eqs. (38a)-(38c). \nSo that the coordinates are asymptotically inertial, we choose Ω 0 = ˙ a 0 = 0 in Eq. (38c). \nWe excise a coordinate sphere of radius r exc about the origin, where \nr exc = 0 . 85949977 (60) \nis chosen such that for zero spin M = 1. On this inner boundary S , we impose Eqs. (39)-(40) and Eq. (44). The spin is determined by Eq. (40): first, the vector ξ i is chosen to be the coordinate rotation vector ∂ φ , making the spin point along the positive z axis; then, the rotation parameter Ω r is varied while the other parameters are held fixed. The spin is measured on the apparent horizon using the approximate-Killing-vector spin (Appendix A); because in this case the space is axisymmetric, the 'approximate' Killing vector reduces to the corresponding exact rotational Killing vector. \nFigure 6 show how the mass M and AKV spin S depend on Ω r . At Ω r = 0, we find the sphericallysymmetric solution with S = 0 and M irr = M = 1 (the mass is proportional to the excision radius, and Eq. (60) sets it to unity). Using this spherically-symmetric solution as an initial guess for the elliptic solver, we find solutions for increasing Ω r with spin increasing initially linearly with Ω r and with approximately constant mass. Beyond some critical Ω r , crit , the elliptic solver fails to converge, and close to this point, all quantities vary in proportion to √ Ω r, crit -Ω r . These symptoms indicate a critical point where the solutions 'turn over' and continue towards smaller Ω r . Analogous non-unique solutions of the XCTS equations have been discovered before in Ref. [34]. To construct solutions along the upper branch, one must choose a sufficiently close initial guess for the elliptic solver; we follow the steps outlined in Ref. [34] and are able to find solutions along the upper branch for a wide range of Ω r < Ω r , crit . As Fig. 6 shows, mass and spin of the horizon in solutions along the upper branch increase with decreasing Ω r , analogous to the findings in [34, 35]. \nFigure 7 shows the dependence of χ = S/M 2 , ε J = J ADM /E 2 ADM , and ζ = S/ ( 2 M 2 irr ) on Ω r . The curves reflect again the non-unique solutions. The dimensionless spin χ increases continuously along the lower branch, and reaches χ ≈ 0 . 85 at the critical point. As Ω r is decreased along the upper branch, χ continues to increase, eventually reaching values larger than 0 . 99. It appears χ continues to increase as Ω r → 0. To find the limiting value, consider that the behavior of the extremality parameter ζ in the inset of Fig. 7. Assuming that ζ can be extrapolated to Ω r → 0, we find a limiting value of ζ ≈ 0 . 88. By Eq. (9), this implies a maximal value of χ 0 . 992. \nIn Figs. 6-7, the data sets on the lower branch appear to be physically reasonable. For spins χ /lessorsimilar 0 . 85, the mass M is nearly constant, and the dimensionless spin χ increases linearly with Ω r . Furthermore, as Ω r → 0 the lower branch continuously approaches the exact Schwarzschild spacetime (see [28]). The upper branch \n≈ \nFIG. 7: Conformally-flat, maximally-sliced quasiequilibrium initial data sets with a single spinning black hole: The dimensionless spin χ , dimensionless ADM angular momentum ε J , and spin-extremality parameter ζ plotted against Ω r [cf. Eq. (40)]. Only Ω r is varied in this figure; all other parameters are held fixed. The inset enlarges the area in the upper left corner; we are able to generate data sets with χ > 0 . 99, whereas the largest spin obtainable on the lower branch is χ ≈ 0 . 85. \n<!-- image --> \nappears to be physically less reasonable; for instance, the spin χ increases for decreasing horizon frequency Ω r . Comparing Figs. 2 and 7, we see that the QE-XCTS data leads to somewhat larger values of χ and ε J relative to puncture data. However, the values are not too different, and similar trends remain. For instance, χ is much closer to unity than ε J . \nTo investigate differences or similarities between puncture data and QE-XCTS data further, we compute embedding diagrams of the equatorial planes of these data sets. The initial data for single black holes have rotational symmetry about the z-axis, so the metric (12) on the initial data hypersurface, when restricted to the equatorial plane, can be written as \nds 2 = ψ 4 ( dr 2 + r 2 dφ 2 , (61) \n( where r and φ are the usual polar coordinates. This metric is now required to equal the induced metric on the 2-D surface given by Z = Z(R) embedded in a 3-D Euclidean space with line-element \n) \nds 2 Euclidean = d R 2 +R 2 dφ 2 + d Z 2 . (62) \nSetting d Z = d Z d R d R, we obtain the induced metric on the Z = Z(R) surface \nds 2 = [ 1 + ( d Z d R ) 2 ] d R 2 +R 2 dφ 2 . (63) \nFIG. 8: Embedding diagrams for puncture and quasiequilibrium initial data. Plotted is the embedding height Z as a function of the embedding radius R, both scaled by the mass M . For quasiequilibrium data (dashed lines), Z=0 at r = r exc ; for puncture data (solid lines), Z=0 at r = R inv . The thin solid purple curve represents the embedding of a plane through a Schwarzschild black hole in Schwarzschild slicing. \n<!-- image --> \nEquating Eqs. (61) and (63), we find \nR = ψ 2 r (64) \nand \n[ 1 + ( d Z d R ) 2 ] d R 2 = ψ 4 dr 2 . (65) \nCombining (65) and (64) results in \n( d Z dr ) 2 = -4 rψ 2 dψ dr ( ψ + r dψ dr ) . (66) \nSince the pseudo-spectral elliptic solver gives ψ as a function of r , Eqs. (64) and (66) allow us to solve for the embedding radius R and the embedding height Z in terms of r . \nFigure 8 shows embedding diagrams for three sets of QE-XCTS and puncture data. We have set Z=0 at r = r exc for QE-XCTS data and at r = R inv for puncture data. This figure also contains the embedding of a plane through Schwarzschild in Schwarzschild coordinates (i.e. the S = 0 limit of BY puncture data), given by R/M = Z 2 / (8 M 2 ) + 2. Both puncture data and CFMS data exhibit a lengthening throat with increasing spin S/M 2 . For puncture data, this lengthening can be deduced from the analytical results in Sec. III A: as the spin parameter S of the puncture data increases by a factor of \n10 while m p ≡ 1 is held constant, we find from Eq. (57) that L /S 1 / 2 should increase by \n∆ L /S 1 / 2 = 96 1 / 4 2 ln 10 ≈ 3 . 60 , (67) \nwhere the factor 1 / 2 arises because R inv = m p / 2 = 0 . 5. The embedding diagram shows only the top half of the throat, and S 1 / 2 ≈ M [cf. the discussion after Eq. (58)]. Therefore in Fig. 8 the S = 100 , 1000 , 10000 lines for BY (puncture) data should be spaced by ∆ Z/M ≈ 1 . 80 for large R/M . This indeed is the case. \nThe CFMS datasets appear to scale proportionally to √ S , which is similar to the puncture data's behavior. Furthermore, the CFMS initial data sets also develop a lengthening throat as S becomes large (the effect is not as pronounced as for puncture data, owing to the smaller maximal S we achieved.) Thus it appears that large spin CFMS data might be similar to large spin puncture data. However, the throats of the QE-XCTS data show a bulge near the bottom, because for these data sets R actually decreases with r in the immediate vicinity of r exc . This is unlike the puncture data, which very clearly exhibit cylindrical throats, consistent with the discussion leading to (58).", 'IV. BINARY-BLACK-HOLE INITIAL DATA WITH NEARLY-EXTREMAL SPINS': "In this section, we construct binary-black-hole initial data with rapid spins, confining our attention to the special case of spins aligned with the orbital angular momentum. In the limit of large separation, binary-blackhole puncture initial data will behave like two individual puncture initial data sets. Specifically, we expect that it should be possible to construct puncture binary-blackhole initial data with initial spins χ ( t = 0) /lessorsimilar 0 . 98, but the spins will rapidly drop to χ /lessorsimilar 0 . 93 as the black holes settle down. For this reason, and also because puncture data is not well-suited to our pseudospectral evolution code, we will restrict our attention to binary black holes constructed with the QE-XCTS approach. \nAs laid out in Table I, we first construct a family (labelled CFMS) of standard conformally-flat initial data on maximal slices; then, we turn our attention to families (labelled SKS) of superposed Kerr-Schild initial data. Finally, we construct a few individual SKS initial-data sets which we evolve in Sec. V. All of the data sets represent equal-mass, equal-spin black holes with spins parallel to the orbital angular momentum. \nIn this section, unless otherwise indicated, all dimensionless spins are the approximate-Killing-vector spin χ AKV (Appendix A), and the subscript 'AKV' will be suppressed for simplicity. \nFIG. 9: Main panel: Dimensionless spin χ [Eq. (1)] and spin-extremality parameter ζ [Eq. (8)] for the family CFMS of spinning binary-black-hole initial data. Inset: Enlargement of χ toward the end of the upper branch, with circles denoting the individual initial data sets that were constructed. Compare with Fig. 7. \n<!-- image -->", 'A. Conformally flat, maximal slicing data (CFMS)': 'To construct conformally-flat binary-black-hole data, we solve the same equations and boundary conditions as for the single-black-hole case, as described in Sec. III B, with the main difference being that we excise two spheres with radius r exc [cf. Eq. (60)] with centers on the xaxis at x = ± d/ 2. The initial spins of the holes are set by adjusting Ω r , just as in the single-hole case. The parameters Ω 0 and ˙ a 0 in the outer boundary condition on the shift [Eq. 38c] determine the initial angular and radial motion of the holes, which in turn determine the initial eccentricity e of the orbit. We set Ω 0 = Ω 0 e z , where e z is a unit vector that points along the positive z axis. For the CFMS family of data sets considered here, we use values for Ω 0 and ˙ a 0 that should result in closed, fairly circular orbits, since our choices of Ω 0 and ˙ a 0 lead to data sets that approximately satisfy the Komar-mass condition E ADM = M K (cf. [29]). Specifically, on the lower branch of the resulting non-unique family of initial data, \n| E ADM -M K | E ADM /lessorsimilar 1% , (68) \nwhere the Komar mass is defined by (e.g., Eq. (35) of Ref. [29]) \nM K := 1 4 π ∮ ∞ ( ∇ i α -β j K ij ) dA. (69) \n(On the upper branch, E ADM and M K differ by up to 3%.) \nAs the rotation parameter Ω r is varied (with the coordinate separation d held fixed), we find that the CFMS- \nmily of binary-black-hole initial data behaves qualitatively similarly to the analogous single-black-hole initial data discussed in Sec. III B. There is a maximal Ω r, crit such that no solutions can be found for Ω r > Ω r, crit ; for values of Ω r below Ω r, crit , two solutions exist. Figure 9 plots the dimensionless spin χ and the spin-extremality parameter ζ against Ω r for this family of initial data. We only show values for one of the holes, since the masses and spins are equal. Spins larger than χ ≈ 0 . 85 appear on the upper branch. The highest spin we have been able to construct is larger than χ = 0 . 97.', 'B. Superposed-Kerr-Schild data': "In this section, we solve the same equations and boundary conditions as in the conformally flat case, except that we use SKS free data (Sec. II C 2) instead of conformallyflat free data. To construct the individual Kerr-Schild data, we need to choose for each black hole the coordinate location of its center, its conformal mass ˜ M , conformal spin ˜ S , and its boost-velocity. We center the black holes on the x-axis at x = ± d/ 2, use the same mass ˜ M = 1 for both black holes, and set the boost velocity to (0 , ± d Ω 0 / 2 , 0). The conformal spins are always equal and are aligned with the orbital angular momentum of the holes. \nIn contrast to the CFMS data, there are now two parameters that influence the black holes' spins: i) the rotation parameter Ω r in Eq. (40), and ii) the conformal spin ˜ S . For concreteness, we choose to construct data for four different values of the conformal spin: ˜ S/ ˜ M 2 = 0 , 0 . 5 , 0 . 93 , and 0 . 99. For each choice, we construct a family of initial data sets for different values of Ω r , which we label as SKS-0.0, SKS-0.5, SKS-0.93, and SKS-0.99 respectively. \nOther choices that went into the construction of the SKS initial data sets are as follows: \n- · The excision boundaries are chosen to be the coordinate locations of the horizons of the individual Kerr-Schild metrics, i.e. they are surfaces of constant Kerr-radius \nr Kerr exc = ˜ r + := ˜ M + √ ˜ M 2 -˜ S 2 , (70) \nlength-contracted by the Lorentz-factor appropriate for the boost velocity of each black hole. This length-contraction accounts for the tangential motion of the hole but neglects the much smaller radial motion. \n- · When superposing the individual Kerr-Schild metrics, we use a damping length scale w = 10 r Kerr exc [cf. Eqs. (45) and (46)], except for the SKS-0.99 family, which uses w = d/ 3.\n- · The orbital frequency Ω 0 and radial expansion ˙ a 0 are held fixed along each family. We expect that \nFIG. 10: Convergence of the spectral elliptic solver. Left panel : The residual constraint violation as a function of the total number of grid-points N when running the elliptic solver at several different resolutions. Right panel: Convergence of the black hole dimensionless spin χ [Eq. (1)] with increasing resolution L AH of the apparent horizon finder, applied to the highest-resolution initial data set of the left panel. The three curves in each panel represent three different initial data sets: One from the family SKS-0.99, as well as the two initial data sets that are evolved in Sec. V. \n<!-- image --> \nour choices for Ω 0 and ˙ a 0 will lead to bounded, fairly circular orbits, since \n| E ADM -M K | E ADM /lessorsimilar 3% . (71) \nIn Sec. V B we reduce the orbital eccentricity for one data set in the family SKS-0.93. \nWe again solve the XCTS equations using the spectral elliptic solver of Ref. [64]; the families of SKS initial data sets that we construct are summarized in Table I. The elliptic solver needs some initial guess for the variables to be solved for; we superpose the respective single-black hole Kerr-Schild quantities, i.e. \nψ = 1 , (72a) \nαψ = 1 + n ∑ a =1 e -r 2 a /w 2 a ( α a -1) , (72b) \nβ i = n ∑ a =1 e -r 2 a /w 2 a β i a , (72c) \nwhere n = 2 and α a and β i a are the lapse and shift corresponding to the boosted, spinning Kerr-Schild metrics g a ij used in the conformal metric ˜ g ij . Convergence of the elliptic solver and spin are demonstrated in Fig. 10 by showing the decreasing constraint violation 8 and differ- \nFIG. 11: The mass M ( upper panel ) and dimensionless spin χ ( lower panel ) of one of the holes for Superposed-Kerr-Schild, binary-black-hole initial data sets with spins aligned with the orbital angular momentum. The mass and spin are plotted against Ω r [Eq. (40)] for four different choices of the conformal spin: ˜ S = 0, 0 . 5, 0 . 93, and 0 . 99. Also shown are the data sets SKS-0.93-E3-identical to the Ω r = 0 . 28 ˜ M, ˜ S = 0 . 93 ˜ M 2 data set on the solid curve but with lower eccentricity-and SKS-Headon; both sets are evolved in Sec. V. The inset in the lower panel shows a close-up of the spins as they approach unity, with symbols denoting the individual data sets. \n<!-- image --> \nes in spin with increasing resolution. \nWe now turn our attention to the physical properties of the SKS initial data sets. Figure 11 shows the horizon mass M and the dimensionless spin χ of either black hole for the four families of SKS initial data. As expected, we find that generally the spin χ increases with increasing Ω r . For each of the SKS-families, we find that the elliptic solver fails to converge for sufficiently large Ω r . We suspect that the SKS-families exhibit a turning point, similar to the CFMS-single and binary black hole initial data shown in Figs. 7 and 9. If this is the case, Fig. 11 only shows the lower branch of each family, and an additional branch of solutions will be present. Because we are satisfied with the spin magnitudes that are possible along the lower branch, we do not attempt to find the upper branch here. \nIn contrast to the CFMS data sets (where the lower branch only allowed spins as large as χ /lessorsimilar 0 . 85), the SKS initial data allows spins that are quite close to unity. For the different SKS families, we are able to construct initial data with spins as large as \n- · χ ≈ 0 . 95 for SKS-0,\n- · χ ≈ 0 . 985 for SKS-0.5, \nFIG. 12: The irreducible mass M irr and Euclidean coordinate radius r ( upper panels ) and dimensionless spin χ := S/M 2 and spin-extremality parameter ζ := S/ (2 M 2 irr ) ( lower panels ) for one of the black holes in the SKS-0.93 ( left ) and SKS-0.99 ( right ) initial-data-set families. These quantities are computed on two surfaces: i) the apparent horizon (solid lines), and ii) the excision boundary of the initial data (dashed lines). Because we enforce that the excision surface is a marginally trapped surface, typically the apparent horizon and excision boundary coincide. However, if Ω r is increased beyond the values where χ approaches unity, the apparent horizon lies outside of the excision surface. The excision surface can obtain superextremal spins ( ζ > 1), but only when it is enclosed by a subextremal horizon. \n<!-- image -->", '· χ ≈ 0 . 9997 for SKS-0.99.': "These spins are far closer to extremal than possible with Bowen-York initial data [ χ /lessorsimilar 0 . 984 (Fig. 2)] or conformally flat, maximally sliced XCTS initial data [ χ /lessorsimilar 0 . 85 or /lessorsimilar 0 . 99 along the lower and upper branch, respectively (Fig. 7)]. \nWe note that the spins in the SKS binary-black-hole initial data families are only weakly dependent on the orbital parameters Ω 0 and ˙ a 0 . This can be seen from the individual data-point labeled SKS-0.93-E3 shown in Fig. 11. This data-set uses different values for Ω 0 and ˙ a 0 but is nevertheless close to the family SKS-0.93. The initial data sets SKS-0.93-E3 and SKS-HeadOn will be discussed in detail in Sec. V. \nThe inset of Fig. 11 highlights a remarkable feature of the SKS-0.93 and SKS-0.99 families: with increasing Ω r , the spin initially increases but eventually decreases . Figure 12 investigates this behavior in more detail, where this effect is more clearly visible in the lower two panels: both the spin χ and the extremality parameter ζ of the apparent horizon change direction and begin to decrease. For Ω r smaller than this critical value, the apparent horizon finder always converges onto the excision surfaces, \nwhich by virtue of the boundary condition Eq. (39), are guaranteed to be marginally trapped surfaces. As Ω r is increased through the critical value (at which χ and ζ change direction), a second marginally trapped surface (solid line) splits off from the excision surface (dashed line) and moves continuously outward. This can be seen in the upper panels of Fig. 12, which plot the minimal and maximal coordinate radius and the irreducible mass of both the excision surface and the outermost marginally trapped surface, which is by definition the apparent horizon. \nBut what about the excision surface? The boundary condition Eq. (39) forces the excision surface to be a marginally trapped surface, independent of the value of Ω r . For sufficiently large Ω r , however, the excision surface is surrounded by a larger marginally trapped surface and thus is not the apparent horizon. The dashed lines in Fig. 12 present data for the excision surface. These lines continue smoothly across the point where the second marginally trapped surface forms. The extremality parameter ζ for the excision surface continues to increase and eventually becomes larger than unity; the excision surface can then be thought of as having a superextremal spin. However, for the outer marginally trapped surface-the true apparent horizon-the extremality parameter always satisfies ζ < 1. The irreducible mass M irr of this surface increases faster than the spin, and therefore ζ = S/ (2 M 2 irr ) decreases with increasing Ω r . \nOne might interpret these results as support of the cosmic censorship conjecture. The XCTS boundary conditions (39) and (40) control the location and the spin of the excision surface. By appropriate choices for the shift boundary condition (40), we can force the excision surface to become superextremal. However, before this can happen, a new horizon appears, surrounding the excision surface and hiding it from 'our' asymptotically flat end of the spacetime. The newly formed outer horizon always remains subextremal.", 'C. Suitability for evolutions': "In the previous sections, we have constructed a wide variety of binary-black-hole initial data sets. To get some indication about how suitable these are for evolutions, we consider the initial time-derivatives of these data sets, ∂ t g ij and ∂ t K ij . Recall that solutions of the XCTS equations give a preferred initial lapse and shift for the evolution of the initial data; hence, the time derivatives ∂ t g ij and ∂ t K ij can be computed by simply substituting the initial data into the ADM evolution equations. We expect initial data with smaller time-derivatives to be closer to quasi-equilibrium and to have less initial spurious radiation. \nFigure 13 presents the L2 norms of the time derivatives, ‖ ∂ t g ij ‖ L 2 and ‖ ∂ t K ij ‖ L 2 where the L2 norm of a tensor T ijk ··· ( x ) evaluated at N gridpoints x i is defined \nFIG. 13: The time derivatives of the metric ( left panel ) and extrinsic curvature ( right panel ). In the superposed-KerrSchild (SKS) data sets, ‖ ∂ t K ij ‖ L 2 has minima near values of Ω r for which the dimensionless spin χ is approximately equal to the spin ˜ S of the conformal metric (cf. Fig. 11). On the upper branch of the conformally-flat, maximally-sliced (CFMS) excision data, where the spin is χ > 0 . 83 (Fig. 9), the time derivatives become much larger than the SKS time derivatives. The data sets SKS-0.93-E3 (with χ ≈ ˜ S = 0 . 93) and SKS-Headon (with χ ≈ ˜ S = 0 . 97) are evolved in Secs. V CVD; the time derivatives are significantly lower for the set SKS-Headon because of the larger coordinate separation of the holes ( d = 100 vs. d = 32). \n<!-- image --> \nas \nwhere \n‖ T ijk ··· ‖ L 2 := √ √ √ √ 1 N N ∑ i =0 ¯ T 2 ( x i ) , (73) \n¯ T := √ T ijk ··· T i ' j ' k ' ··· δ ii ' δ jj ' δ kk ' · · · . (74) \nFigure 13 shows that generally ∂ t K ij is larger than ∂ t g ij . This has also been found in previous work, e.g. [65], and is not surprising, because the XCTS formalism allows some control over the time derivative of the metric through the free data ˜ u ij = ∂ t ˜ g ij , whereas there is less control of ∂ t K ij . We note that for CFMS data, the time derivatives are larger and grow more rapidly with χ than for SKS data; in particular, the time derivatives on the upper branch are ∼ 10 times larger than for SKS-initial data, suggesting that these data are much farther from equilibrium. \nIn the SKS case, the time derivatives of K ij have local minima at particular values of Ω r ; comparison with Fig. 11 gives spins χ at these minima of ‖ ∂ t K ij ‖ L 2 as follows: \n- · SKS-0.5: Ω r ≈ 0 . 1, χ ≈ 0 . 45,\n- · SKS-0.93: Ω r ≈ 0 . 28, χ ≈ 0 . 93,\n- · SKS-0.99: Ω r ≈ 0 . 34, χ ≈ 0 . 98. \nNote that these minima occur at values of Ω r such that χ ≈ ˜ S/ ˜ M 2 ; that is, transients in the initial data and presumably the spurious radiation are minimized when the conformal spin and AKV spin are consistent. For this reason, we conclude that SKS initial data with χ ≈ ˜ S/ ˜ M 2 is preferable; this is the type of initial data we will evolve in the next section. \nAlso note that minimizing the spurious radiation has purely numerical advantages: the spurious radiation typically has finer structure (and thus requires higher resolution) that the physical radiation. If such radiation is minimized, the numerical evolutions may require less resolution and will be more efficient. Conformally-curved initial data has been found to reduce the amount of spurious radiation in Refs. [39, 66].", 'V. EXPLORATORY EVOLUTIONS OF SUPERPOSED KERR-SCHILD (SKS) INITIAL DATA': 'So far, we have confined our discussion to black hole spins in the initial data . In this section, we compare the initial spin to the value to which the spin relaxes after the initial burst of spurious radiation, when the holes have settled down. Recall, for instance, that for Bowen-York puncture initial data with spins close to the maximal possible value [ χ ( t = 0) ≈ 0 . 98], the spins quickly relax by about ∆ χ ≈ 0 . 05 to a maximal possible relaxed value of χ ( t relax ) ≈ 0 . 93 (cf. [25]). While the SKS data presented in Sec. IV B can achieve larger initial spins [ χ ( t = 0) = 0 . 9997] than conformally-flat puncture data, only evolutions can determine ∆ χ and χ ( t relax ). \nTherefore, in this section we perform brief, exploratory evolutions of some SKS initial data sets to determine ∆ χ for those data sets. 9 Besides determination of χ ( t relax ), these evolutions will also allow us to demonstrate that the technique of eccentricity reduction developed in Ref. [49] is applicable to SKS initial data as well as to compare the spin measures defined in Appendices A and B. The focus here lies on initial data, and we evolve only long enough for our purposes. Longer simulations that continue through merger and ringdown are the subject of ongoing research. \nThis section is organized as follows. In Sec. VA, we summarize the evolution code that we will use. In Sec. V B, we perform eccentricity reduction on one of the data sets in the SKS-0.93 family, which corresponds to an orbiting binary black hole with equal masses and equal spins (of magnitude χ ≈ 0 . 93) aligned with the orbital angular momentum. Then, in Sec. V C, we evolve the resulting low-eccentricity data set (labeled SKS-0.93- \nE3). Finally, In Sec. V D, we evolve a head-on plunge of SKS initial data (labeled SKS-Headon) representing two widely-separated black holes with initial spins of magnitude χ = 0 . 970 and direction normal to the equatorial plane.', 'A. Description of evolution code': "The initial data are evolved using the Caltech-Cornell pseudospectral evolution code SpEC [50]. The details of the evolution methods, equations, and boundary conditions that we use are the same as those described in Ref. [67]. The singularities are excised, with the excision surfaces chosen to lie slightly inside the black hole horizons. Note that whereas Ref. [67] excises coordinate spheres inside the black holes' apparent horizons, here we use Lorentz-contracted ellipsoidal excision boundaries which are adapted to the shape of the initial apparent horizons. \nThe highest-resolution initial data set (with N ≈ 85 3 gridpoints) is interpolated onto evolution grids labelled N 1, N 2, and N 3 with approximately 61 3 , 67 3 , and 74 3 gridpoints, respectively. The outer boundary is at a coordinate radius of r = 32 d for the orbiting simulation discussed in Secs. V B and V C and at r = 14 d for the head-on simulations discussed in Sec. V D. This translates to about r = 450 E ADM and r = 620 E ADM for the orbiting and head-on simulations, respectively. As in earlier simulations [49, 50, 67], a small region of the evolution grid lies inside the horizon and is not covered by the initial data grid; we extrapolate ψ , αψ , and β i into this region and then compute g ij and K ij .", 'B. Eccentricity removal for orbiting SKS-binaries': "We obtain initial data with small orbital eccentricity using the iterative method of Ref. [49], as refined in Ref. [67], applied here for the first time to binary-blackhole data with rapid spin. In this method, the choice of Ω 0 and ˙ a 0 for the next iteration are made so that if the orbit were Newtonian, the eccentricity would vanish. For the non-Newtonian orbit here, successive iterations succeed in reducing the orbital eccentricity. \nThis procedure is based on the proper separation s between the apparent horizons, measured along a coordinate line connecting the geometric centers of the apparent horizons. The time derivative ds/dt is fitted to a fiveparameter curve that, together with the initial proper separation s ( t = 0) is used to define the eccentricity e and to define improved values for Ω 0 and ˙ a 0 . Specifi- \nFIG. 14: Color online. Eccentricity reduction for evolutions of superposed-Kerr-Schild binary-black-hole initial data. The proper separation s ( upper panel ) and its time derivative ds/dt ( lower panel ) are plotted for initial data sets SKS-0.93E0, -E1, -E2, and -E3, which have successively smaller eccentricities e . All evolutions are performed at resolution N 1. \n<!-- image --> \nly, \nds dt := A 0 + A 1 t + B cos ( ωt + ϕ ) , (75a) \ne := B ωs ( t = 0) , (75b) \nΩ 0 , new := Ω 0 + B sin φ 2 s ( t = 0) , (75c) \n˙ a 0 , new := ˙ a 0 -B cos φ s ( t = 0) (75d) \nHeuristically, the eccentricity is embodied by the oscillating part of ds/dt . \nFigure 14 illustrates the eccentricity reduction for one of the data sets in family SKS-0.93. Plotted are the proper separation s and its derivative ds/dt for evolutions of several initial data sets (summarized in Table I): \n- · set SKS-0.93-E0, which is identical to the set in family SKS-0.93 with Ω r = 0 . 28 (Fig. 11);\n- · set SKS-0.93-E1, which is the same as SKS-0.93E0 except that the orbital frequency Ω 0 is manually adjusted to lower the orbital eccentricity somewhat; and\n- · sets SKS-0.93-E2 and SKS-0.93-E3, which are successive iterations (starting from set SKS-0.93-E1) of the eccentricity-reduction scheme Eqs. (75). \nThe ad hoc adjustment of Ω 0 was somewhat effective, reducing e by about 50%. The subsequent iterations using Eqs. (75) reduced e by factors of about 5 and 8, respectively. Surprisingly, the lowest eccentricity, corresponding to a smooth inspiral trajectory is obtained with a \nFIG. 15: Convergence test of the evolution of the initial data set SKS-0.93-E3. Shown are evolutions on three different resolutions, N 1, N 2, and N 3, with N 3 being the highest resolution. The top panel shows the approximate-Killing-vector (AKV) spin of one of the holes as a function of time, with the top inset showing the spin's initial relaxation; the bottom panel shows the constraint violation as a function of time. \n<!-- image --> \npositive ˙ a 0 = 3 . 332 × 10 -4 . This is not due to insufficient resolution; for SKS-0.93-E3, we have verified that we obtain the same eccentricity e ∼ 0 . 001 for all three numerical resolutions N1, N2, N3. \nNote that we choose to stop the evolutions at about t = 670 E ADM , which corresponds to about 1.9 orbits; this is sufficient for reducing the eccentricity and for measuring ∆ χ . In the next subsection, we discuss the evolution of the low-eccentricity set SKS-0.93-E3 in detail, focusing on the relaxation of the spin χ .", 'C. Low-eccentricity inspiral with χ ≈ 0 . 93': "We evolved the data-set SKS-0.93-E3 at three different numerical resolutions for a duration of about 670 E ADM , corresponding to about 1.9 orbits. From post-Newtonian theory [68], we estimate that this simulation would proceed through about 20 orbits to merger. \nFigure 15 presents a convergence test for this run. The lower panel of Fig. 15 shows the normalized constraint violation (see Eq. (71) of Ref. [69] for the precise definition.) While the constraints are small, the convergence seems poor until t ≈ 500 E ADM . For this time-period the constraint violations at high resolution N 3 are dominated by the outgoing pulse of spurious radiation-i.e. far away from the black holes-which we have not attempted to adequately resolve. At t ≈ 500 E ADM , the pulse of spurious radiation leaves the computational domain through the outer boundary; afterwards, the constraints decrease exponentially with increasing resolution, as expected. \nThe upper panel of Figure 15 shows the AKV spin χ AKV = S/M 2 for the three runs with different resolutions N 1, N 2, and N 3. Based on the difference be- \nFIG. 16: A comparison of different definitions of the spin. The top panel shows the spin as a function of time for several different measures of the spin; the bottom panel shows the fractional difference between χ AKV and alternative spin definitions. Note that for t < 30 E ADM , the time-axis has a different scaling to make the initial transients visible. \n<!-- image --> \nween N 2 and N 3, the spin of the evolution N 3 should be accurate to a few parts in 10 4 . For the time-interval 5 < t/E ADM < 670, the measured spin on resolution N 3 is consistent with begin constant within its estimated accuracy. Very early in the simulation, t < 5 E ADM , the spin χ changes convergently resolved from its initial value χ ( t = 0) = 0 . 927 48 to a relaxed value χ ( t relax ) = 0 . 927 14 (see inset of Fig. 15). Therefore, for SKS-0.93-E3, we find ∆ χ = 0 . 000 34. \nContrast this result with the evolution of a binary black hole puncture initial data set with large spins, which is reported in Ref. [25]: for that particular evolution, χ ( t = 0) = 0 . 967, χ ( t relax ) = 0 . 924, i.e. ∆ χ = 0 . 043, more than a factor 100 larger than for the evolution of SKS-0.93-E3 reported here. This comparison is somewhat biased against the puncture evolution in [25], which starts at a smaller separation possibly resulting in larger initial transients. However, even in the limit that the black holes are infinitely separated (i.e., in the singleblack-hole limit), the spins in Bowen-York puncture data relax to values near ε J = J ADM /E 2 ADM ; to achieve a final spin of χ ( t relax ) ≈ 0 . 93, the initial spin of Bowen-York data must be χ ( t = 0) ≈ 0 . 98 (cf. Fig. 2 of Ref. [25]). We conclude that the spin relaxes by a much smaller amount in the SKS case than in Bowen-York puncture or inversion symmetric data. \nFigure 15 and the discussion in the previous paragraph only addresses the behavior of the AKV spin, where the approximate Killing vectors are computed from the minimization problem [cf. Eq. (A10)]. We now compare the different spin-definitions we present in Appendices A and B. Figure 16 compares these different definitions of the black hole spin for the N 3 evolution of initial data set SKS-0.93-E3. Shown are the AKV spin of one hole in the \nbinary, the scalar curvature (SC) spins χ min SC and χ max SC of Appendix B [Eqs. (B2a) and (B2b)], and also the spin obtained by using Eq. A1 with a coordinate rotation vector instead of an approximate Killing vector (which we call the 'coordinate spin' here). After the holes have relaxed, the SC spins track the AKV spin more closely than does the coordinate spin. However, during very early times, as the holes are relaxing and the horizon shape is very distorted, the SC spins show much larger variations. Consequently, the SC spin is a poorer measure of the spin at early times than even the coordinate spin.", 'D. Head-on plunge with χ ≈ 0 . 97': 'In the previous subsection, we have seen that for SKS binary-black hole-initial data with χ = 0 . 93, the initial spins change by only a few parts in 10 4 . A spin χ ≈ 0 . 93 is roughly the largest possible equilibrium spin that is obtainable using standard conformally-flat, Bowen-York puncture data (cf. the discussion at the beginning of Sec. V). We now begin to explore binary-black-hole simulations with spin-magnitudes that are not obtainable with Bowen-York initial data methods. \nWe construct and evolve SKS binary-black-hole data for a head-on plunge of two equal mass black holes with spins of equal magnitude χ = 0 . 97 and with the spins orthogonal to the line connecting the black holes. This data set, labelled SKS-Headon, is summarized in Table I and was briefly discussed in Sec. IV B, cf. Figs. 10, 11 and 13. As for the orbiting evolution SKS-0.93-E3, we adjust the rotation parameter Ω r so that conformal spin ˜ S/ ˜ M 2 and AKV spin χ are approximately equal. Starting such a simulation at close separation results in rapid coordinate motion of the apparent horizons during the first few E ADM of the evolution. These motions are currently difficult to track with our excision code; therefore, we begin at a larger separation d than we used in the nearly-circular data sets described previously. \nFigure 17 presents a convergence test of the constraints (lower panel) and the AKV spin χ AKV (upper panel) during the subsequent evolution. Again, we are interested in the initial relaxation of the spins; therefore, we choose to stop evolution at t ≈ 120 E ADM . During this time, the black hole proper separation decreased from s ( t = 0) = 47 . 6 E ADM to s ( t = 120) = 44 . 1 E ADM . \nDuring the first ∼ 10 E ADM , χ AKV shows (a numerically resolved) decrease of about 3 × 10 -5 ; this change arises due to initial transients as the black holes and the full geometry of the spacetime relax into an equilibrium configuration. Subsequently, the spin remains constant to within about 10 -4 , where these variations are dominated by numerical truncation error. \nFigure 18 compares our various spin-measures for the head-on simulation. Interestingly, the spin χ coord computed from coordinate rotation vectors agrees much better with χ AKV than for the SKS-0.93-E3 evolution, perhaps because the black holes here are initially at rest. \nFIG. 17: Convergence test of the head-on evolution SKSHeadOn. Shown are evolutions at three different resolutions, N 1, N 2, and N 3, with N 3 being the highest-resolution. The top panel shows the approximate-Killing-vector (AKV) spin of one of the holes as a function of time; the bottom panel shows the constraint violations as a function of time. \n<!-- image --> \nFIG. 18: A comparison of various measures of the spin for the head-on evolution of data set SKS-Headon, which is a plunge of two equal-mass black holes with with parallel spins of magnitude χ AKV = S/M 2 = 0 . 970 pointed normal to the equatorial plane. The top panel shows various measures of the spin as a function of time, and the bottom panel shows the fractional difference between the approximate-Killing-vector (AKV) spin χ AKV and alternative spin definitions. \n<!-- image --> \nThe scalar-curvature (SC) spins χ min SC and χ max SC , derived from the scalar curvature of the apparent horizon [Eqs. (B2a) and (B2b)], show some oscillations at early times; after the initial relaxation, the SC spin agrees with the AKV spin to about 1 part in 10 4 .', 'A. Maximal possible spin': 'In this paper, we have examined a variety of methods for constructing black hole initial data with a particular emphasis on the ability to construct black holes with nearly-extremal spins. These are spins for which the dimensionless spin χ = S/M 2 and spin-extremality parameter ζ = S/ (2 M 2 irr ) are close to unity. \nWhen discussing black hole spin, one needs to distinguish between the initial black hole spin and the relaxed spin of the holes after they have settled down. Using conformally-flat Bowen-York (BY) data (both puncture data or inversion symmetric data) for single black holes, the largest obtainable spins are χ ≈ 0 . 984 , ζ ≈ 0 . 833 (cf. Ref. [63] and Fig. 2). With conformally-flat, maximallysliced (CFMS), quasi-equilibrium extended-conformalthin-sandwich (QE-XCTS) data, we are able to obtain initial spins as large as χ ≈ 0 . 99 , ζ ≈ 0 . 87 for single black holes (Fig. 7). The limitations of BY puncture data and CFMS QE-XCTS data are already present when constructing highly spinning single black holes; therefore, we expect the methods to be able to construct binary-blackhole data with similar spins as for single holes-i.e., up to about 0 . 98. Construction of CFMS QE-XCTS binaryblack-hole initial data confirms this conjecture (compare Fig. 9 with Fig. 7). \nFor superposed-Kerr-Schild (SKS) initial data, the situation is different. For single black holes, SKS data reduce to the analytical Kerr solution, without any limitations on the spin magnitude. Thus limitations of SKS data will only be visible for binary-black hole configurations. As Sections IV and V show, however, those limitations are quite minor. SKS data can indeed achieve initial spins that are much closer to extremality than what is possible with BY data or CFMS QE-XCTS data; we have explicitly demonstrated this by constructing SKS data for binary black holes with χ ≈ 0 . 9997 , ζ ≈ 0 . 98, as can be seen from Figs. 11 and 12. \nAs the black hole spacetimes settle into equilibrium and emit spurious gravitational radiation, the initial spin χ decreases to a smaller relaxed spin χ ( t relax ). Thus an interesting quality factor for high-spin black hole initial data is ∆ χ = χ ( t = 0) -χ ( t relax ) [Eq. (3)] considered as a function of the relaxed spin. The magnitude of ∆ χ is indicative of the amplitude of any initial transients, whereas the maximally achievable χ ( t relax ) gives the largest possible spin which can be evolved with such initial data. Figure 19 presents this plot, with the circle and cross representing the two evolutions of SKS data which were described in Sec. V. \nWe have not evolved high-spin puncture data, nor high-spin CFMS-XCTS data; therefore, we do not know precisely ∆ χ for these initial data. We estimate ∆ χ for puncture data by noting that evolutions of single-hole, BY puncture data with large spins show [25] that the black hole spin χ := S/M 2 relaxes approximately to the \nFIG. 19: The change ∆ χ in black hole spin χ during the initial relaxation of black hole initial data plotted as a function of the black-hole spin after relaxation. The SKS initial data constructed in this paper have smaller transients and allow for larger relaxed spins. \n<!-- image --> \ninitial value of ε J := J ADM /E 2 ADM . Therefore, for BY puncture data, we approximate \n∆ χ ≈ ε J -χ ( t = 0) , (76) \nχ ( t relax ) ≈ ε J . (77) \nThis curve is plotted in Fig. 19. Because high-spin singleblack-hole, CFMS QE-XCTS initial data and BY puncture data have quite similar values of χ ( t = 0) and ε J , as well as similar embedding diagrams (cf. Fig. 8), we conjecture that Eqs. (76)-(77) are also applicable to CFMS QE-XCTS data. This estimate is also included in Fig. 19. We see that both types of initial data result in a ∆ χ of similar magnitude which grows rapidly with χ relaxed . \nPerhaps the most remarkable result of Fig. 19 is the extremely small change in black hole spin during the relaxation of SKS initial data, even at spins as large as χ = 0 . 97. The small values of ∆ χ combined with the ability to construct initial data with initial spins χ ( t = 0) as large as 0 . 9997 (cf. Fig. 11) makes it highly likely that SKS initial data are capable of constructing binary black holes with relaxed spins significantly closer to unity than 0.97. Evolutions of initial data with spins χ much closer to unity, i.e., farther into the regime that is inaccessible to conformally-flat data, are a subject of our ongoing research. \nIn summary, the two main results of this paper are as follows: \n- · SKS initial data can make binary black holes that initially have nearly-extremal spins, and\n- · for SKS initial data, the relaxed spin is quite close to the initial spin, even when the spin is large.', 'B. Additional results': "While working toward the main results discussed in the previous subsection, we have also established several additional interesting results. We have considered spinning, single-black-hole, puncture data which is identical to single-black-hole, spinning, inversion-symmetric data. Using this correspondence and our accurate spectral elliptic solver, we revisited the relation between black-hole spin χ , specific total angular momentum of the spacetime ε J , and the spin-parameter S for BY puncture data, and established in Fig. 2 that both χ and ε J approach their limits for S → ∞ as power-laws , cf. Eqs. (49) and (50). We have also extended the analytical analysis of Dain, Lousto, and Zlochower [25] of the throat region of high-spin puncture data toward more quantitative results, including the precise amplitudes of the conformal factor, throat circumference and throat length, as well as their scaling with spin-parameter S and puncture mass m p [Eqs. (54)-(58)]. Furthermore, Ref. [25] implicitly assumed that the throat-region is approximately spherically symmetric; our Fig. 5 presents explicit evidence in support of this assumption, but also shows that the throat is not precisely spherically symmetric. \nWe have also examined high-spin QE-XCTS initial data employing the common approximations of conformal flatness and maximal slicing (CFMS). With increasing angular frequency Ω r of the horizon, we discover nonunique solutions. Thus, the non-uniqueness of the XCTS equations can not only be triggered by volume terms (as in [34]) but also through boundary conditions [in this case, by Eq. (40)]. Interestingly, CFMS QE-XCTS data appears to be very similar to BY puncture data, in regard to nearly-extremal spins. Both data formalisms result in similar maximal values of χ ( t = 0) and ε J (Figs. 2, 7 and 19) and have embedding diagrams which develop a lengthening throat as the spin is increased (Fig. 8). \nWe also have found an interesting property of the horizon geometries for SKS data, which one might interpret as support of the cosmic censorship conjecture. Specifically, we find that by increasing Ω r sufficiently, we can in fact force the excision boundaries of the initial data to be 'horizons' (i.e. marginally trapped surfaces) with superextremal spin ( ζ > 1). However, these superextremal surfaces are always enclosed by a larger, subextremal ( ζ < 1) apparent horizon. \nTo measure black hole spins, we have employed and compared several different techniques to measure black hole spin. Primarily, we use a quasilocal spin definition based on (approximate) Killing vectors [Eq. (A1)]. This formula requires the choice of an 'approximate' Killing vector, and we have used both straightforward coordinate rotations to obtain χ coord and solved Killing's equation in a least-squares sense to obtain χ AKV (see Appendix A or details). Furthermore, we introduced a new technique to define black-hole spin which does not require choice of an approximate Killing vector and is invariant under spatial coordinate transformations and transformations \nassociated with the boost gauge ambiguity of the dynamical horizon formalism. This new technique is based on the extrema of the scalar curvature of the apparent horizon. Figures 16 and 18 show that all four spin measures agree to good precision, but differences are noticeable. The spin-measures based on the horizon curvature exhibit more pronounced variations during the initial transients, and the quasilocal spin based on coordinate rotations is off by several tenths of a percent. The quasilocal spin based on approximate Killing vectors χ AKV has the smallest initial variations. \nFinally, we would like to point out that a modified version of the SKS-initial data has been very successfully used to construct black hole-neutron star initial data [70].", 'Acknowledgments': 'It is a pleasure to acknowledge useful discussions with Ivan Booth, Gregory Cook, Stephen Fairhurst, Lawrence Kidder, Lee Lindblom, Mark Scheel, Saul Teukolsky, and Kip Thorne. The numerical calculations in this paper were performed using the Spectral Einstein Code ( SpEC ), which was primarily developed by Lawrence Kidder, Harald Pfeiffer, and Mark Scheel. We would also like to acknowledge the anonymous referee for reminding us of an important technical caveat. Some equations in this paper were obtained using Mathematica . This work was supported in part by grants from the Sherman Fairchild Foundation to Caltech and Cornell and from the Brinson Foundation to Caltech; by NSF grants PHY0652952, DMS-0553677, PHY-0652929, and NASA grant NNG05GG51G at Cornell; and by NSF grants PHY0601459, PHY-0652995, DMS-0553302 and NASA grant NNG05GG52G at Caltech.', 'APPENDIX A: QUASILOCAL SPIN USING APPROXIMATE KILLING VECTORS (AKV SPIN)': "In this appendix and the one that follows, we address the task of defining the spin of a dynamical black hole, given g ij and K ij . We use two different measures. The first, defined here, is a standard quasilocal angular momentum defined with approximate Killing vectors which correspond to approximate symmetries of a black hole's horizon. The second measure, defined in Appendix B, infers the spin from geometrical properties (specifically, from the intrinsic scalar curvature ˚ R ) of the apparent horizon, assuming that the horizon is that of a single black hole in equilibrium, (i.e., that the horizon is that of a Kerr black hole). Note that quantities relating to the geometry of the two-dimensional apparent horizon surface H are denoted with a ring above them, to avoid confusion with the analogous quantities on the spatial slice, Σ. \nIt has become standard in the numerical relativity community to compute the spin angular momentum of a black hole with the formula [71, 72, 73] \nS = 1 8 π ∮ H φ i s j K ij dA, (A1) \nwhere s i is the outgoing normal of H embedded in Σ and /vector φ is an 'azimuthal' vector field, tangent to H . The azimuthal vector field /vector φ carries information about the 'axis' about which the spin is being computed. There are, however, far more vector fields on a two sphere than there are axes in conventional Euclidean space. We must find suitable criteria for fixing these azimuthal vector fields in numerical simulations, so that they reduce to the standard rotation generators when considered on a metric sphere. \nBecause angular momentum is generally thought of as a conserved charge associated with rotation symmetryand indeed the quantity given in (A1) can be shown to be conserved under time evolution [71, 73] when /vector φ is a Killing vector of the dynamical horizon worldtube-it makes sense to consider Killing's equation to be the essential feature of the azimuthal vector field. If a Killing vector on a dynamical horizon is tangent to each (twodimensional) apparent horizon, then the vector field must be a Killing vector of each apparent horizon. However in a general spacetime, on an arbitrary apparent horizon, there is no reason to expect any Killing vectors to exist. So in the cases of most interest to numerical relativity, when there are no true rotation symmetries, we must relax the symmetry condition and find those vector fields that come 'closest' to generating a symmetry of the apparent horizon. In other words, we seek optimal 'approximate Killing vectors' of the apparent horizon. \nIn [74], a practical method for computing approximate Killing vectors was introduced, which has since been applied on numerous occasions, e.g. [18, 19, 29, 75] This method involves integrating the Killing transport equations along a predetermined network of coordinate paths. The resulting vector field is guaranteed to be a Killing vector field if such a field exists and coincides with the computed field at any point on the network. However if no true Killing field exists, the integral of the Killing transport equations becomes path dependent. This means that the computed vector field will depend in an essential way on the network of paths chosen for the integral. Perhaps even more serious, if there is no true Killing field, then the transport of a vector around a closed path will not necessarily be an identity map. As a result, the computed vector field cannot be expected to reduce to any smooth vector field in the limit that the network becomes more refined. This kind of approximate Killing vector field is simply not mathematically well-defined in the continuum limit. \nHere we will describe a kind of approximate Killing vector field that, as well as having a well-defined continuum limit, is actually easier to construct than those of \nthe Killing transport method, at least in our particular code. Our method is extremely similar to that described by Cook and Whiting [41], but was actually developed independently by one of the current authors [76].", '1. Zero expansion, minimal shear': "Killing's equation, \nD ( A φ B ) = 0 , (A2) \nhas two independent parts: the condition that /vector φ be expansion-free, \nΘ := ˚ g AB D A φ B = 0 , (A3) \nand the condition that it be shear-free, \nσ AB := D ( A φ B ) -1 2 ˚ g AB Θ = 0 , (A4) \nwhere uppercase latin letters index the tangent bundle to the two-dimensional surface, ˚ g AB is the metric on that surface, and D A is the torsion-free covariant derivative compatible with that metric. \nWhen constructing approximate Killing vectors, a question arises: which condition is more important, zero expansion or zero shear? Shear-free vector fields (conformal Killing vectors) are simply coordinate rotation generators in the common case of coordinate spheres in a conformally flat space. They are therefore readily available in that context. A very interesting and systematic approach to their use has been given by Korzynski [77], and they have been used in the construction of conformally flat binary black hole initial data sets [28, 29]. However, in the case of a general surface in a general spatial slice, the conformal Killing vectors are not known a priori , and they are more difficult to construct than expansion-free vector fields. Expansion-free vector fields have the additional benefit of providing a gauge-invariant spin measure on a dynamical horizon [73] 10 , so we restrict attention to the expansion-free case. \nAny smooth, expansion-free vector field tangent to a topological two-sphere can be written as \nφ A = /epsilon1 AB D B z, (A5) \nwhere /epsilon1 AB is the Levi-Civita tensor and z is some smooth potential function. \nWe assume that the function z has one local maximum, one local minimum, and no other critical points. This is equivalent to the assumption that the orbits of /vector φ are simple closed loops. In order for φ A φ A to have the proper dimensions, z must have dimensions of area. For the case of the standard rotation generators of the metric twosphere, the three z functions are the three /lscript = 1 spherical harmonics, multiplied by the square of the areal radius of the sphere. \nWithin this space of expansion-free vector fields, we would now like to minimize the following positive-definite norm of the shear: \n‖ σ ‖ 2 := ∮ H σ BC σ BC dA. (A6) \nSubstituting Eq. (A5) for /vector φ in this expression and integrating twice by parts, ‖ σ ‖ 2 takes the form of an expectation value: \n‖ σ ‖ 2 = ∮ H zHz dA, (A7) \nwhere H is the self-adjoint fourth-order differential operator defined by \nHz = D 4 z + ˚ RD 2 z + D A ˚ RD A z, (A8) \nand D 2 is the Laplacian on the (not necessarily round) sphere, D 4 is its square, and ˚ R is the Ricci scalar curvature of the sphere. In our sign convention, ˚ R = 2 on the unit sphere, so we can immediately see that Hz = 0 when z is an /lscript = 1 spherical harmonic, and therefore that their associated vector fields are shear-free. \nIt is now tempting to minimize the functional ‖ σ ‖ 2 in (A7) with respect to z . However, doing so will simply return the condition that z lie in the kernel of H . If there are no true Killing vectors, this will mean that z is a constant, and therefore that /vector φ vanishes. We need to restrict the minimization procedure to cases that satisfy some normalization condition. In this case, we require that the norm of the vector field, \n∮ H φ A φ A dA, (A9) \ntake some given positive value. This restriction can be made with the use of a Lagrange multiplier. Specifically, the functional we wish to minimize is \nI [ z ] := ∮ H zHz dA + λ (∮ H D A z D A z dA -N ) (A10) \nfor some yet undetermined positive parameter N . Note that λ is the Lagrange multiplier and we have made use of the fact that Eq. (A5) implies that /vector φ · /vector φ = /vector Dz · /vector Dz . Minimizing the functional I with respect to z returns a generalized eigenvalue problem: \nHz = λD 2 z. (A11) \nIt is at this point that we can most easily clarify the difference between our construction of approximate Killing vectors and that of Cook and Whiting in [41]. The difference lies in the choice of norm in which the minimization problem is restricted. Rather than fixing the norm (A9) to take some fixed value in the minimization, Cook and Whiting instead fix the dimensionless norm: \n∮ H ˚ Rφ A φ A dA. (A12) \nIn general, we see no particular reason to prefer either norm over the other, but for the current purposes we have at least an aesthetic preference for (A9), which is positive-definite even at high spin, whereas (A12) is not, because the scalar curvature ˚ R of the horizon becomes negative near the poles at high spin. If the norm (A9) in Eq. (A10) is replaced by (A12), the result is the problem described in [41]: \nHz = λ ( ˚ RD 2 z + D A ˚ RD A z ) . (A13) \nIn our numerical code, we discretize (A11) (or, optionally, (A13), but not for any results published here) and solve the resulting linear algebra problem with a LAPACK routine [78]. Note, however, one technical peculiarity: the operators H and D 2 in (A11) share a kernel, the space of constant functions. This means that this generalized eigenvalue problem is singular , a fact that can cause considerable difficulties for the numerical solution [79]. The same can be said of (A13). For our purposes, this complication is easily evaded. Since we are working with a spectral code, it is easiest to discretize the problem using expansion into the spectral basis functions (coordinate spherical harmonics). When this is done, the space of constant functions-the shared kernel of the two operators-is simply the span of a single basis function: the constant, Y 00 . This basis function can easily be left out of the spectral expansion, and thereby removed from the numerical problem. \nExpansion into coordinate spherical harmonics has another practical advantage. As noted earlier, for metric spheres in standard coordinates, the Killing vectors arise when z is given by an /lscript = 1 spherical harmonic. Thus, assuming our horizon is nearly round, and noticeably so in the given coordinates, the lowest basis functions (the /lscript = 1 spherical harmonics) should nicely approximate the intended eigenfunctions. The higher basis functions should simply provide small corrections. \nIn summary, the approach that we take to finding approximate Killing vectors begins with a spectral decomposition of Eq. (A11). This problem, of course, provides as many eigenvectors as there are elements of the spectral decomposition. We restrict attention to the three eigenvectors with smallest eigenvalues (ignoring the vector corresponding to the constant eigenfunction, which is physically irrelevant and removed from discretization), as these are the ones corresponding to vector fields with the smallest shear, and at least for spheres that are only \nslightly deformed, the orbits of these vector fields are smooth closed loops. \nIt must be noted that only the eigenvector with the smallest eigenvalue corresponds to a vector field with strictly minimum shear: even locally, all other eigenvectors are saddle points of the minimization problem. The three of them taken together, however, provide a geometrically-defined subspace of the vector space of expansion-free vector fields, a natural generalization of the rotation generators on metric spheres. Using these three vector fields (normalized as described in the next subsection), one can define 'components' of the spin angular momentum of a black hole 11 , and from these components infer the spin around an arbitrary axis or even a spin 'magnitude' using a metric on this threedimensional space of generalized rotation generators. In practice, we have found no need to go quite so far. As mentioned in [41], the approximate Killing vectors generally adapt themselves so well to the horizon that one of the components is much larger than the other two, so this is considered the spin magnitude, and the associated approximate Killing vector is considered to define the spin axis.", '2. Normalization': 'Solutions to the eigenproblem (A11) can only determine the approximate Killing vectors up to a constant scaling. Fixing this scaling is equivalent to fixing the value of N in (A10). The standard rotation generators of metric spheres are normalized such that, when considered as differential operators along their various orbits, they differentiate with respect to a parameter that changes by a value of 2 π around each orbit. Naively one would like to fix the normalization of approximate Killing vectors in the same way, but a subtlety arises: we can only rescale the vector field by a fixed, constant value. Rescaling differently along different orbits would introduce extraneous shear and would remove the vector field from the pure eigenspace of (A11) in which it initially resided. If an approximate Killing vector field has different parameter circumferences around different orbits, then it is impossible to rescale it such that the parameter distance is 2 π around every orbit. The best one can ask is that 2 π is the average of the distances around the various orbits. \nTo consider this in detail, introduce a coordinate system, topologically the same as the standard spherical coordinates on the metric sphere, but adapted to the potential function z so that the latitude lines are the level sur- \nfaces of z (and, in particular, the poles are at the two critical points we have assumed z to have). More precisely, choose z for the zenith coordinate on the sphere, and an arbitrary rotational coordinate-say, the azimuthal angle in the encompassing spatial slice, describing rotations about the axis connecting the critical points of z -for the azimuthal coordinate ϕ on the sphere. If the parameter τ is defined such that /vector φ = ( d/dτ ) z =const . , then in the basis related to these coordinates, the components of /vector φ are: \nφ z ( z, ϕ ) = ( dz dτ ) z =const . = 0 , (A14) \nφ ϕ ( z, ϕ ) = ( dϕ dτ ) z =const . . (A15) \nAround a closed orbit C ( z ), at fixed z , the parameter τ changes by a value of: \nτ ( z ) = ∫ C ( z ) dϕ φ ϕ ( z, ϕ ) (A16) \n= ∫ C ( z ) √ ˚ gdϕ, (A18) \n= ∫ C ( z ) dϕ /epsilon1 ϕz ∂ z z (A17) \nwhere ˚ g is the determinant of the surface metric, evaluated in the ( z, ϕ ) coordinates. Note that Eq. (A18) follows from Eq. (A17) by the fact that the condition ˚ g AB ˚ g CD /epsilon1 AC /epsilon1 BD = 2 implies /epsilon1 ϕz = 1 / √ ˚ g . The average value of τ , over the various orbits, is: \n〈 τ 〉 = 1 z max -z min ∫ z max z min ∫ C ( z ) √ ˚ gdϕdz (A19) = A z max -z min , (A20) \nwhere A is the surface area of the apparent horizon. Requiring this average to equal 2 π , we arrive at the normalization condition: \n2 π ( z max -z min ) = A. (A21) \nThis normalization condition requires finding the minimum and maximum values of the function z , which is only computed on a discrete grid. In our spectral code, in particular, this numerical grid is quite coarse, so numerical interpolation is needed, in combination with an optimization routine. We have implemented such routines to search for z min and z max , but a numerically-cheaper normalization condition would be of interest. Such a condition arises when one assumes that the black hole under consideration is approximately Kerr. In the Kerr metric, for the function z generating the true rotation generator of the Kerr horizon, the following identity holds: \n∮ H ( z -〈〈 z 〉〉 ) 2 dA = A 3 48 π 2 , (A22) \nFIG. 20: Error, relative to the analytic solution, of the spin on the horizon of a Kerr black hole in slightly deformed coordinates. The vertical axis represents | χ computed -χ analytic | , and data are shown for the spin computed with the standard coordinate rotation vector (in deformed coordinates, so not a true Killing vector), and with our approximate Killing vectors (AKV) using both the extremum norm, Eq. (A21), and the integral norm, Eq. (A22). The spin computed from the coordinate rotation vector quickly converges to a physically inaccurate result. The spin from approximate Killing vectors converges in resolution L AH to the correct value χ = 1 / 2. Curves are also shown for the two spin measures defined in the Appendix B. These spin measures also converge exponentially to the physically correct result. \n<!-- image --> \nwhere 〈〈 z 〉〉 is the average of z over the sphere. The existence of an identity of this form is somewhat nontrivial: the fact that the right side is given purely by the horizon area, and that it does not involve the spin of the Kerr hole, is what makes this identity useful as a normalization condition. This normalization is much easier to impose, and requires significantly less numerical effort. \nTo close the discussion of spin computed from approximate Killing vectors, we demonstrate the effectiveness of the method in a simple test case: an analytic Kerr black hole in slightly deformed coordinates. We begin with a Kerr black hole of dimensionless spin parameter χ = 1 / 2, in Kerr-Schild coordinates, but we rescale the x -axis by a factor of 1 . 1. This rescaling of the x coordinate causes the coordinate rotation vector x∂ y -y∂ x to no longer be the true, geometrical rotation generator. And indeed, when we compute the quasilocal angular momentum (A1) on the horizon using this coordinate vector, the result converges to a physically inaccurate value, as demonstrated by the black dotted curve in Fig. 20. If, however, the approximate Killing vectors described above are used, the result is not only convergent, but physically accurate. Because the accuracy is slightly better with the normalization condition of Eq. (A22), that is the condition we use for all results presented in this paper.', 'APPENDIX B: SCALAR-CURVATURE SPIN (SC SPIN)': "In this appendix, we define a spin measure in terms of the intrinsic geometry of the horizon, which we compare with the AKV spin in Sec. V. The AKV spin described in Appendix A is a well-defined measure of black hole spin, even when the holes' horizons have only approximate symmetries. At times sufficiently before or after the holes merge, however, the horizons will not be too tidally distorted and thus will not be too different from the exactly-axisymmetric horizons of Kerr black holes. \nBy assuming that the geometric properties of the horizon behave precisely as they do for a Kerr black hole, one can infer the hole's spin from those properties. For instance, it is common to measure polar and equatorial circumferences of the apparent horizon; the spin is then obtained by finding the Kerr spacetime with the same circumferences [80, 81, 82]. \nTo avoid introducing coordinate dependence by defining 'polar' and 'equatorial' planes, we infer the spin from the horizon's intrinsic scalar curvature ˚ R . The horizon scalar curvature ˚ R has previously been studied analytically for Kerr-Newman black holes [83] and for Kerr black holes perturbed by a distant moon [84]. Numerical studies of ˚ R have focused attention on the quasinormal ringing of single, perturbed, black holes [80] as well as on the shape of the individual and common event horizons in Misner data [85]. To our knowledge, the scalar curvature ˚ R has not been previously used to infer the horizon spin in numerical simulations. \nAt a given point on a Kerr black hole's horizon, the horizon scalar curvature ˚ R depends only on the hole's mass M and spin S . The extrema of ˚ R can be expressed in terms of the irreducible mass and dimensionless spin of the Kerr black hole via Eqs. (1)-(2) as \nmin( ˚ R ) = -1 + 2 √ 1 -χ 2 2 M 2 irr , (B1a) \nmax( ˚ R ) = -2 M 2 irr χ 4 ( -2 + χ 2 +2 √ 1 -χ 2 ) . (B1b) \nSolving for χ and requiring it to be real yields χ as a function of M irr and either min( ˚ R ) or max( ˚ R ). We take these functions as definitions of the spin, even when the space-time is not precisely Kerr: \n( \nχ min SC ) 2 := 1 -[ 1 2 + M 2 irr min( ˚ R ) ] 2 , (B2a) \n) ( χ max SC ) 2 := -2 + 2 √ 2 M 2 irr max( ˚ R ) M 2 irr max( ˚ R ) (B2b) \nThe definitions of the spin given by Eqs. (B2a)-(B2b) are manifestly independent of spatial coordinates and are well-defined for black holes that are tidally deformed. 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2008CQGra..25q5015N
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Hawking radiation as quantum tunneling from a noncommutative Schwarzschild black hole
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2008-01-01
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We study the tunneling process through the quantum horizon of a Schwarzschild black hole in noncommutative spacetime. This is done by considering the effect of smearing of the particle mass as a Gaussian profile in flat spacetime. We show that even in this noncommutative setup there will be no correlation between the different modes of radiation, which reflects the fact that information does not come out continuously during the evaporation process at least at late time. However, due to spacetime noncommutativity, information might be preserved by a stable black hole remnant.
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https://arxiv.org/pdf/0801.4074.pdf
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{'Hawking Radiation as Quantum Tunneling from Noncommutative Schwarzschild Black Hole': 'Kourosh Nozari a , ∗ and S. Hamid Mehdipour a , b , † \na Department of Physics, Faculty of Basic Sciences, University of Mazandaran, P. O. Box 47416-1467, Babolsar, IRAN \nand \nb Islamic Azad University, Lahijan Branch, P. O. Box 1616, Lahijan, IRAN', 'Abstract': "We study tunneling process through quantum horizon of a Schwarzschild black hole in noncommutative spacetime. This is done by considering the effect of smearing of the particle mass as a Gaussian profile in flat spacetime. We show that even in this noncommutative setup there will be no correlation between the different modes of radiation which reflects the fact that information doesn't come out continuously during the evaporation process at least at late-time. However, due to spacetime noncommutativity, information might be preserved by a stable black hole remnant. PACS : 04.70.-s, 04.70.Dy, 11.10.Nx \nKey Words : Quantum Tunneling, Hawking Radiation, Noncommutative Spacetime, Black Hole Entropy, Information Loss Paradox", '1 Introduction': 'In 1975, Hawking proposed a scenario in which black hole can radiate from its event horizon as a black body with a purely thermal spectrum at the temperature T H = ¯ hc 3 κ 2 πk B G , utilizing the procedure of quantum field theory in curved spacetime ( κ is the surface gravity that demonstrates the strength of the gravitational field near the black hole surface). This leads us to a non-unitary quantum evolution where maps a pure state to a mixed state. In 2000, Parikh and Wilczek [2] proposed the method of null-geodesic to derive Hawking temperature as a quantum tunneling process. In this quantum tunneling framework, the form of the corrected radiation is not exactly thermal which yields a unitary quantum evolution. However, their form of the correction for emission is not adequate by itself to retrieve information since it fails to find the correlations between the emission rates of different modes in the black hole radiation spectrum. Possibly, spacetime noncommutativity [3-5], that is, an inherent trait of the manifold by itself and the fact that spacetime points might be noncommutative, opens the way to find a solution to the black hole information paradox that can be solved by ceasing the black hole to decay beyond a minimal mass M 0 . In 2003, Smailagic and Spallucci [6-8] postulated a new attractive model of noncommutativity in terms of coherent states which satisfies Lorentz invariance, Unitarity and UV-finiteness of quantum field theory. In 2005, Nicolini, Smailagic and Spallucci (NSS) [9] by using this method have found the generalized line element of Schwarzschild spacetime based on coordinate coherent state noncommutativity. It has been shown that the generalized line element does not permit the black hole to decay lower than M 0 . Thus, the evaporation process finishes when black hole approaches a Planck size remnant with zero temperature, which does not diverge at all, rather it reaches a maximum value before shrinking to absolute zero. Since spacetime noncommutativity can eliminate some kind of divergences (which appear in General Relativity), and also is an intrinsic property of the manifold itself (even in the absence of gravity), we hope to cure a step further and modify the tunneling paradigm utilizing the noncommutative field theory. In this manner, we would like to proceed the Parikh-Wilczek tunneling process using a fascinating formulation of noncommutativity of coordinates that is carried out by the Gaussian distribution of coherent states.', '2 Noncommutative Schwarzschild Black Hole': "A valuable test of spacetime noncommutativity is its possibly observable effects on the properties of black holes. To inquire into this issue, one would require to prosperously build the noncommutativity corresponding to the General Relativity. Although this issue has been considered in the literature [10], but no perfect and wholly convincing theory of this model yet exists. There are plenty formulations of noncommutative field theory established upon the Weyl-Wigner-Moyal /star -product [11] that conduct to downfall in finding a solution to the some prominent difficulties, such as Lorentz invariance breaking, defeat of unitarity and UV divergences of quantum field theory. The incident of noncommutativity at a observable scale has the ability which leads to important effects in the expected properties of the black holes. Although a perfect noncommutative theory of gravity does not yet exist, it becomes essential to model the noncommutativity effects in the frame of the commutative General Relativity. Lately, the authors in Ref. [6-8] have regarded a physically inspired and obedient type of the noncommutativity amendments to Schwarzschild black hole solutions (coordinate coherent states formalism), that can be released from the difficulties mentioned above. In this formalism, General Relativity in its common commutative case as characterized by the Einstein-Hilbert action stays appropriate. If noncommutativity effects can be behaved in a perturbative manner, then this comes into view defensible, at least to a good approximation. The authors in Ref. [10] have really demonstrated that the leading noncommutativity amendments to the form of the Einstein-Hilbert action are at least second order in the noncommutativity parameter, θ . The generalization of quantum field theory by noncommutativity based on coordinate coherent state formalism is also interestingly curing the short distance behavior of pointlike structures [6-9] (see also [12]). In this approach, the particle mass M , instead of being completely localized at a point, is dispensed throughout a region of linear size √ θ , that the implementation of these arguments leads to substitution of position Dirac-delta function, describing pointlike structures, with Gaussian function, describing smeared structures. On the other hand, the mass density of a static, spherically symmetric, particle-like gravitational source cannot be a delta function distribution but will be given by a Gaussian distribution of minimal width √ θ as follows \nρ θ ( r ) = M (4 πθ ) 3 2 exp ( -r 2 4 θ ) . (1) \nThe Schwarzschild solution of the Einstein equations associated with these smeared mass Gaussian function sources leads to line element as \nds 2 = -( 1 -2 M θ r ) dt 2 + ( 1 -2 M θ r ) -1 dr 2 + r 2 d Ω 2 , (2) \nwhere the smeared mass distribution is implicity given in terms of the lower incomplete Gamma function as \nM θ = ∫ r 0 ρ θ ( r )4 πr 2 dr = 2 M √ π γ ( 3 2 , r 2 4 θ ) ≡ 2 M √ π ∫ r 2 4 θ 0 t 1 2 e -t dt. (3) \n(Hereafter we set the fundamental constants equal to unity; ¯ h = c = k B = 1.) In the limit of θ → 0, one recovers the complete Gamma function Γ( 3 2 ), \nlim θ → 0 M θ = M, (4) \nand the modified Schwarzschild solution reduces to the ordinary Schwarzschild solution. The line element (2) characterizes the geometry of a noncommutative inspired Schwarzschild black hole. The radiating behavior of such a modified Schwarzschild black hole can now be investigated and can easily be shown by plotting g 00 as a function of r , for different values of M ( hereafter, for plotting the figures we set the value of the noncommutativity parameter equal to unity; θ = 1). Fig. 1 shows that coordinate noncommutativity leads to the existence of a minimal non-zero mass which black hole (due to Hawking radiation and evaporation) can shrink to it. The event horizon of this line element can be found where g 00 ( r H ) = 0, that is implicity written in terms of the upper incomplete Gamma function as \nr H = 2 M θ ( r H ) = 2 M ( 1 -2 √ π Γ ( 3 2 , r 2 H 4 θ ) ) . (5) \nThe noncommutative Schwarzschild radius versus the mass can approximately be calculated by setting r H = 2 M into the upper incomplete Gamma function as \nr H = 2 M ( E ( M √ θ ) -2 M √ πθ exp ( -M 2 θ ) ) , (6) \nwhere E ( x ) shows the Gauss Error Function defined as \nE ( x ) ≡ 2 √ π ∫ x 0 e -t 2 dt. \nFigure 1: g 00 versus the radius r for different values of mass M . The figure shows the possibility of having extremal configuration with one degenerate event horizon when M = M 0 ≈ 1 . 9 ( i.e. , the existence of a minimal non-zero mass), and no event horizon when the mass of the black hole is smaller than M 0 . Also as figure shows, the distance between the horizons will increase by increasing the black hole mass (two event horizons). \n<!-- image --> \nFor very large masses, the E ( M √ θ ) tends to unity and second term on the right will exponentially be reduced to zero and one retrieves the classical Schwarzschild radius, r H ≈ 2 M . \nT H = 1 4 π dg 00 dr | r = r H = M [ E ( r H 2 √ θ ) 2 πr 2 H -exp ( -r 2 H 4 θ ) 4( πθ ) 3 2 ( r H + 2 θ r H ) ] . (7) \nWhen such a noncommutative black hole radiates, its temperature can be calculated to find \nFor the commutative case, M √ θ → ∞ , one recovers the classical Hawking temperature, T H = 1 8 πM . The numerical calculation of the modified Hawking temperature as a function of the mass is presented in Fig. 2. In this modified (noncommutative) version, not only T H does not diverge at all but also it reaches a maximum value before dropping to absolute zero at a minimal non-zero mass, M = M 0 ≈ 1 . 9, which black hole shrink to it. \nTo find the analytical form of the modified (noncommutative) entropy, S NC , we should note that our calculation for obtaining the modified Hawking temperature, equation (7), is exact and no approximation has been made. But there is no analytical solution for entropy from the first law of classical black hole thermodynamics dM = T H dS with T H given as (7), even if we set r H = 2 M in this relation. Nevertheless, to obtain an approximate \nFigure 2: Black hole temperature, T H , as a function of M . The existence of a minimal non-zero mass and disappearance of divergence are clear. \n<!-- image --> \nanalytical form of entropy we can use the following expression as an approximation for noncommutative Hawking temperature \nT H = 1 4 πr H , (8) \nwhere r H is given by equation (6). Eventually, the entropy of the black hole can be obtained as analytical form using the first law, \nS NC = ∫ dM T H = 2 π ∫ dM κ ( M ) = 4 πM 2 E ( M √ θ ) -6 πθ E ( M √ θ ) +12 √ πθM exp ( -M 2 θ ) . (9) \nWhere κ ( M ) is the horizon noncommutative surface gravity and is given by \nκ ( M ) = [ 4 M ( E ( M √ θ ) -2 M √ πθ exp ( -M 2 θ ) )] -1 (10) \nBehavior of the entropy S NC , as a function of the mass is depicted in Fig. 3. As this figure shows, at the final stage of the black hole evaporation, the black hole ceases to radiate and its entropy reaches zero and the existence of a minimal non-zero mass is clear again. In the large mass limit i.e. , M √ θ /greatermuch 1, one recovers the standard Bekenstein-Hawking Entropy plus θ -corrections, which leads to \nS NC = 4 πM 2 +12 √ πθM exp ( -M 2 θ ) . (11) \nFigure 3: Black hole entropy, S NC , as a function of M . Note that the figure is plotted approximately by the equation (24). \n<!-- image --> \nIt should be noted that if we had picked out a different form for the probability of matter distribution, instead of distribution (1), solely the smeared mass distribution M θ would be altered however their general properties would be directed to entirely comparable consequences to those above. For instance, we consider a Lorentzian distribution of smeared particle \nρ θ ' ( r ) = M θ ' π 2 ( r 2 + θ ' 2 ) 2 . (12) \n√ \nHere the noncommutativity parameter, θ ' , is actually not identical to θ . The smeared mass distribution is now given by \nM θ ' = ∫ r 0 ρ θ ' ( r )4 πr 2 dr = 2 M π ( tan -1 ( r √ θ ' ) -r √ θ ' ( r 2 + θ ' ) ) . (13) \nIn the limit of θ ' going to zero, we get M θ ' → M . As expected, the smeared mass Lorentzian distribution, M θ ' , has the same limiting properties and is completely comparable to the smeared mass Gaussian distribution, M θ , qualitatively. Then, many of the outcomes that we achieved stay applicable if we take the other kind of probability distribution (see [13]). The lack of responsiveness of these consequences to our Gaussian formalism of the smearing can easily be exhibited by plotting g 00 as a function of r for \nFigure 4: g 00 versus the radius r for different values of mass M utilizing Lorentzian smearing. We set θ ' = 1 (which is not exactly the same as θ = 1). This figure is the same as the Fig. 1 with feasibility of having extremal configuration with one degenerate event horizon when M = M 0 ≈ 2 . 2 ( i.e. , the existence of a minimal non-zero mass), and no event horizon when the mass of the black hole is smaller than M 0 . Also as figure again displays, the distance between the horizons will grow by increasing the black hole mass (two event horizons)in the same way as Gaussian profile. These features show the lack of responsiveness of these consequences to Gaussian formalism of the smearing. \n<!-- image --> \ndifferent values of M (see Fig. 4). We set θ ' = 1 (which is not equivalent to θ = 1). Comparing these results with the results of Fig. 1 demonstrates the close similarity of consequences in these two setup at least in asymptotic values r . In both situations a minimum g 00 happens at comparatively small r with slightly comparable values of M . The M 0 value is seen to be entirely similar in the two situations. The preeminent distinction in the two approaches is comprehended to take place around less than M 0 where there is the mainly responsiveness to noncommutativity influences and the detailed form of the matter distribution. However, we actually should not have confidence to the details of our modeling when r is excessively small. In fact, in the area where noncommutativity effects precisely begin to be sensed, the detailed nature of the sharpened mass distribution is not actually being explored. In a recent paper [14], we have reported some outcomes about extraordinary thermodynamical treatment for Planck-sized black hole evaporation, i.e. , when M is less than M 0 . ‡ In this manner, one encounters some uncommon ther- \nmodynamical features goes to negative entropy, negative temperature and abnormal heat capacity where the mass of the black hole becomes of the order of Planck mass or less. It is also in this extreme situation that majority of the distinctions between, e.g. , the Gaussian, Lorentzian or some other forms of the smeared mass distribution would be anticipated to commence to come into view. Therefore, we will henceforth use the Gaussian-smeared mass distribution in our calculations just in the circumstance that M ≥ M 0 .", '3 Quantum Tunneling Near the Horizon': "We are now ready to discuss the quantum tunneling process in the noncommutative framework. In accordance with Ref. [17], one can express the general spherically symmetric line element in the form \nds 2 = -[ N t ( t, r ) dt ] 2 + L ( t, r ) 2 [ dr + N r ( t, r ) dt ] 2 + R ( t, r ) 2 d Ω 2 . (14) \nWhen we insert this expression into the Einstein-Hilbert action, due to some restrictions and advantages e.g. , no time derivative in the action, invariance of the action under reparametrization and no singular behavior at the horizon, one finds \n N t ( t, r ) = N t ( r ) N r ( t, r ) = N r ( r ) L ( t, r ) = r R ( t, r ) = 1 \n \n N t ( r ) = 1 N r ( r ) = √ 2 M θ r \n To describe noncommutative quantum tunneling process where a particle moves in dynamical geometry and pass through the horizon without singularity on the path, we should use a coordinates system that, unlike Schwarzschild coordinates, are not singular at the horizon. These simply choices for L and R mentioned above (first indicated by Painlev'e § ) can prepare this purpose. Thus for a noncommutative Schwarzschild solution one can easily acquire \nThe noncommutative line element now immediately reads \nds 2 = -( 1 -2 M θ r ) dt 2 +2 √ 2 M θ r dtdr + dr 2 + r 2 d Ω 2 , (15) \nThe metric in these new coordinates is now stationary, non-static, and there are neither coordinate nor intrinsic singularities (due to noncommutativity). The equation of motion for a massless particle (the radial null geodesic) is ˙ r ≡ dr dt = ± N t -N r , where the upper sign (lower sign) corresponds to an outgoing (ingoing) geodesic respectively. Since the horizon, r = r H , is concluded from the condition N t ( r H ) -N r ( r H ) = 0, in the vicinity of horizon N t -N r treats as \nN t -N r /similarequal ( r -r H ) κ ( M ) + O ( ( r -r H ) 2 ) , (16) \nIf we suppose that t increases towards the future, then the above equations should be modified by the particle's self-gravitation effect. Kraus and Wilczek [19] studied the motion of particles in the s -wave as spherical massless shells in dynamical geometry and developed self-gravitating shells in Hamiltonian gravity. Further elaborations was performed by Parikh and Wilczek [2]. On the other hand, Shankaranarayanan et al have applied the tunneling approach to obtain the Hawking temperature in different coordinates within a Complex paths scenario [20]. This technique has been successfully applied to obtain a global temperature for multi-horizon spacetimes [21]. In this paper, we are going to develop Parikh-Wilczek method to noncommutative coordinate coherent states. We keep the total ADM mass ( M ) of the spacetime fixed, and allow the hole mass fluctuated, due to the fact that we take into account the response of the background geometry to an emitted quantum of energy E which moves in the geodesics of a spacetime with M replaced by M -E . Thus we should replace M by M -E both in the equations (15) and (16). \nSince the characteristic wavelength of the radiation is always haphazardly small near the horizon due to the infinite blue-shift there, the wave-number reaches infinity and therefore WKB approximation is reliable close to the horizon. In the WKB approximation, the probability of tunneling or emission rate for the classically forbidden region as a function of the imaginary part of the particle's action at stationary phase would take the following form \nΓ ∼ exp( -2Im I ) . (17) \nTo calculate the imaginary part of the action we consider a spherical shell consist of components massless particles each of which travels on a radial null geodesic. We use these \nradial null geodesics like an s -wave outgoing positive energy particle which pass through the horizon outwards from r in to r out to compute the Im I , as follows (on the condition that r in > r out , where we should have: r in = 4 M √ π γ ( 3 2 , r 2 in 4 θ ) and r out = 4( M -E ) √ π γ ( 3 2 , r 2 out 4 θ )), \nIm I = Im ∫ r out r in p r dr = Im ∫ r out r in ∫ p r 0 dp ' r dr, (18) \none can alter the integral variable from momentum in favor of energy by using Hamilton's equation ˙ r = dH dp r | r , where the Hamiltonian is H = M -E ' . We now evaluate the integral without writing out the explicit form for the radial null geodesic. The r integral can be done first by deforming the contour, \nIm I = Im ∫ M -E M ∫ r out r in dr ˙ r dH = -Im ∫ E 0 ∫ r out r in drdE ' ( r -r H ) κ ( M -E ' ) (19) \nThe r integral has a pole at the horizon which lies along the line of integration and this yields ( -πi ) times the residue. Therefore, \nIm I = π ∫ E 0 dE ' κ ( M -E ' ) , (20) \nHere, reutilizing the first low of black hole thermodynamics, dM = κ 2 π dS , one can find the imaginary part of the action as [22] \nIm I = -1 2 ∫ S NC ( M -E ) S NC ( M ) dS = -1 2 ∆ S NC (21) \nHawking radiation as tunneling from the black hole event horizon was also investigated in the context of string theory [22], and it was exhibited that the emission rates in the high energy corresponds to a difference between counting of states in the microcanonical and canonical ensembles. In fact, the emission rates in the tunneling approach just to first order in E , replace the Boltzmann factor in the canonical ensemble Γ ∼ exp( -βE ), which is described by the inverse temperature as the coefficient β . So, the emission rates in the high energy are proportional to exp(∆ S ) (see also [23]), \nΓ ∼ exp( -2Im I ) ∼ e S final e S initial = exp(∆ S ) = exp[ S ( M -E ) -S ( M )] , (22) \nwhere ∆ S is the difference in black hole entropies before and after emission. In other words, at higher energies the emission probability depends on the final and initial number of microstates available for the system. Thus, at higher energies the emission spectrum \ncannot be precisely thermal due to the fact that the high energy corrections arise from the physics of energy conservation with noncommutativity corrections. In this model, one takes into account the back-reaction results in a finite separation between the initial and final radius as a result of self-gravitation effects of outgoing shells that is the classically forbidden trajectory i.e. , the barrier. On the other hand, according to energy conservation the tunneling barrier is produced by a change in the radius (the decreasing of the black hole horizon) just by the emitted particle itself. \nLet us now insert our result for noncommutative black hole entropy, equation (9), into above equation and write the new noncommutative-corrected tunneling probability as follows \nΓ ∼ exp(∆ S NC ) = exp[ S NC ( M -E ) -S NC ( M )] = exp ( 4 π [ ( M -E ) 2 -3 2 θ ] E ( M -E √ θ ) + \n12 √ πθ ( M -E ) exp ( -( M -E ) 2 θ ) -4 π [ M 2 -3 2 θ ] E ( M √ θ ) -12 √ πθM exp ( -M 2 θ ) ) . (23) \nIt is simply observed that to linear order in E , two expressions for Γ in the microcanonical and canonical ensembles coincide. So, manifestly the emission rate (23) deviates from the pure thermal emission but is consistent with an underlying unitary quantum theory [24]. We must note that the tunneling probability can also be derived by writing out the explicit metric in the tunneling computation, which would be leaded to the same result in spite of more complicated calculations. \nAt this stage, we want to demonstrate that there are no correlations between emitted particles even with the inclusion of the noncommutativity corrections at least at late-times. (However, there might be short-time correlations between the quanta emitted earlier and the quanta emitted later on that decay to zero after the black hole is equilibrated at latetimes). This means it can be exhibited that the probability of tunneling of two particles of energy E 1 and E 2 is precisely similar to the probability of tunneling of one particle with their compound energies, E = E 1 + E 2 , i.e. \n∆ S E 1 +∆ S E 2 = ∆ S ( E 1 + E 2 ) ⇒ χ ( E 1 + E 2 ; E 1 , E 2 ) = 0 , (24) \nwhere the emission rate for a first quanta emitted, E 1 , yields \n∆ S E 1 = ln Γ E 1 = 4 π [ ( M -E 1 ) 2 -3 2 θ ] E ( M -E 1 √ θ ) +12 √ πθ ( M -E 1 ) exp ( -( M -E 1 ) 2 θ ) - \n4 π [ M 2 -3 2 θ ] E ( M √ θ ) -12 √ πθM exp ( -M 2 θ ) , (25) \nand similarly the emission rate for a second quanta emitted, E 2 , is given by \n∆ S E 2 = ln Γ E 2 = 4 π [ ( ( M -E 1 ) -E 2 ) 2 -3 2 θ ] E ( ( M -E 1 ) -E 2 √ θ ) + 12 √ πθ ( ( M -E 1 ) -E 2 ) exp ( -( ( M -E 1 ) -E 2 ) 2 θ ) -4 π [ ( M -E 1 ) 2 -3 2 θ ] E ( ( M -E 1 ) √ θ ) -12 √ πθ ( M -E 1 ) exp -( M -E 1 ) θ . (26) \nFinally, the emission rate for a single quanta emitted with the same total energy, E , is given by \n( \n2 ) \n∆ S ( E 1 + E 2 ) = ln Γ ( E 1 + E 2 ) = 4 π [ ( M -( E 1 + E 2 ) ) 2 -3 2 θ ] E ( M -( E 1 + E 2 ) √ θ ) + 12 √ πθ ( M -( E 1 + E 2 ) ) exp ( -( M -( E 1 + E 2 ) ) 2 θ ) -4 π [ M 2 -3 2 θ ] E ( M √ θ ) -12 √ πθM exp ( -M 2 θ ) . (27) \nIt can be easily confirmed that these probabilities of emission are uncorrelated. On the other hand, the statistical correlation function, χ ( E ; E 1 , E 2 ) is zero which leads to the independence between different modes of radiation during the evaporation. Hence, in this method the form of the corrections as back-reaction effects even with inclusion of noncommutativity effects are not adequate by themselves to retrieve information because there are no correlations between different modes at least at late-times and information doesn't come out with the Hawking radiation (for reviews on resolving the so-called information loss paradox , see [25-27]). Nevertheless, noncommutativity effect is adequate by itself to preserve information due to the fact that in the noncommutative framework black hole doesn't evaporate completely and this leads to the existence of a minimal non-zero mass ( e.g. , a Planck-sized remnant containing the information ) which black hole can reduce to it. So information might be preserved in this remnant. \nIn string theory and loop quantum gravity, the entropy of black hole has been achieved by direct microstate counting as follows (in units of the Planck scale), \nS QG = 4 πM 2 + α ln(16 πM 2 ) + O ( 1 M 2 ) . (28) \nIt was recently suggested by the authors of Ref. [28] that the Planck scale corrections to the black hole radiation spectrum via tunneling can be written as \nΓ ∼ exp(∆ S QG ) = exp ( S QG ( M -E ) -S QG ( M ) ) = ( 1 -E M ) 2 α exp ( -8 πME ( 1 -E 2 M ) ) . (29) \nSince, loop quantum gravity anticipates a negative value for α (see e.g. [29]) which yields diverging emission rate if E → M , this leads to no suppressing the black hole emission (although the suppression can only occur when α > 0, which is not recommended at least by loop quantum gravity). But our outcome is actually sensible, comparing the noncommutative result for the emission rate (equation (23)) with the quantum gravity result (equation (29)) shows that the noncommutative result is reasonably successful in ceasing the black hole emission when ( M -E ) → M 0 . In fact, the cases ( M -E ) < M 0 are the noncommutativity-forbidden regions that the tunneling particle can not be traversed through it. Therefore, the limit ( M -E ) → 0 can not be applied by our process because of existence of non-vanishing mass at final phase of black hole evaporation.", '4 Summary and Remarks': "We summarize this paper with some remarks. In this paper, generalization of the standard Hawking radiation via tunneling through the event horizon based on the solution of the equation (17) within the context of coordinate coherent state noncommutativity has been studied and then the new corrections of the emission rate based on spacetime noncommutativity has been achieved. In this study, we see that there aren't any correlations between the tunneling rates of different modes in the black hole radiation spectrum even in noncommutative framework at least at late-times. In our opinion, if we really believe the idea of stable black hole remnants due to the fact that there are some exact continuous global symmetries in nature [30], then we should accept that the information stays inside the black hole and can be retained by a stable Planck-sized remnant. In principle, there are four main outcomes of the black hole evaporation: \n- · The black hole can evaporate completely, and information would disappear from our world.\n- · The black hole can completely disappear, but information emerges in the final burst \nof radiation when the black hole shrinks to the Planck size. \n- · There are correlations between different modes of radiation during the evaporation that information appears continuously through them.\n- · The black hole never disappears completely, and information would preserve in a stable black hole remnant. \nIndeed, it is not conceivable to date to give a clear answer to the question of the black hole information paradox and this is reasonable because there is no complete self-consistent quantum theory of evaporating black holes (and gravity). In this paper we have studied the reliability of the forth conjecture within a noncommutative framework. We have shown that although there is no correlation between the tunneling rates of different modes in the black hole radiation spectrum in noncommutative spacetime at least at late times, but noncommutativity has the capability to overcome the information loss paradox via existence of stable black hole remnant. At this stage we should stress that there is another point of view on using relation (17). There is a problem here known as 'factor 2 problem. Some authors such as Chowdhury [31] and Pilling [32] have argued that relation (17) is not invariant under canonical transformations but the same formula with a factor of 1 / 2 in the exponent is canonically invariant. As final remark we emphasize that some authors have treated black hole thermodynamics in noncommutative framework adapting a coordinate noncommutativity against coherent state approach( see [33] and references therein). A question then arises: what is the difference between these two procedure? The standard way to handle noncommutative problems is through the use of WignerWeyl-Moyal /star -product. That means to use complex number commuting coordinates and shift non-commutativity in the product between functions. This is mathematically correct, but it is physically useless since any model written in terms of star product, even the simplest field theory, becomes non-local and it is not obvious how to handle non-local quantum field theory. One proposed approach is perturbation in the θ parameter[34]. This is physically sensible since once expanded up to a given order in θ , the resulting field theory becomes local. The smeared picture of particles based on coordinate coherent states defines complex number coordinates as quantum mean values of the original non-commuting ones between coordinate coherent states. In other words, in this setup one can see commuting coordinates as classical limit(in the quantum mechanical sense) of the non-commuting ones. In this framework, the emergent semi-classical geometry keeps memory of its origin. For example, free propagation of a point-like object is described \nby a minimal width Gaussian wave-packet as has been considered in our setup. So, the difference between two approaches lies in the definition of quantum field theoretical propagators. \nNote added: After we have completed this work, Banerjee et al have reported a similar treatment of the problem [35].", 'References': "- [1] S. W. Hawking, Particle creation by black holes, Commun. Math. Phys. 43 , 199 (1975).\n- [2] M. K. Parikh and F. Wilczek, Hawking radiation as tunneling , Phys. Rev. Lett. 85 , 5042 (2000), [arXiv:hep-th/9907001].\n- [3] H. S. Snyder, Quantized Spacetime, Phys. Rev. 71 , 38 (1947).\n- [4] N. Seiberg and E. Witten, String theory and noncommutative geometry, JHEP 9909 , 032 (1999).\n- [5] M. R. Douglas and N. A. Nekrasov, Noncommutative field theory, Rev. Mod. Phys. 73 , 977 (2001).\n- [6] A. Smailagic and E. 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Aschieri et al , A Gravity Theory on Noncommutative Spaces, Class. Quant. Grav. 22 , 3511 (2005), [arXiv:hep-th/0504183]. \n- [11] J. E. Moyal, Quantum mechanics as a statistical theory, Proc. Cambridge Phil. Soc. 45 (1949) 99; M. S. Barlett and J. E. Moyal, The exact transition probabilities of quantum-mechanical oscillators calculated by the phase-space method, Proc. Cambridge Philos. 45 (1949) 545.\n- [12] P. Nicolini, Vacuum energy momentum tensor in (2+1) NC scalar field theory , [arXiv:hep-th/0401204] ; A. Gruppuso, Newtons law in an effective non-commutative space-time, J. Phys. A 38 , 2039 (2005), [arXiv:hep-th/0502144].\n- [13] T. G. Rizzo, Noncommutative inspired black holes in extra dimensions, JHEP 0609 , 021 (2006), [arXiv:hep-ph/0606051].\n- [14] K. Nozari and S. H. Mehdipour, Failure of Standard Thermodynamics in Planck Size Black Hole System, Chaos, Solitons and Fractals 32 , 1 (2007), [arXiv:hepth/0610076].\n- [15] M. S. El Naschie, The concepts of E-infinity: An elementary introduction to the Cantorian-fractal theory of quantum physics, Chaos, Solitons and Fractals 22 , 495 (2004).\n- [16] L. Nottale, Fractal space-time and microphysics: towards a theory of scale relativity , World Scientific, Singapore, 1993.\n- [17] P. Kraus and F. Wilczek, Some Applications of a Simple Stationary Line Element for the Schwarzschild Geometry, Mod. Phys. Lett. A 40 , 3713 (1994), [arXiv:grqc/9406042].\n- [18] P. Painlev'e, La m'ecanique classique et la th'eorie de la relativit'e, Compt. Rend. Acad. Sci. (Paris) 173 , 677 (1921).\n- [19] P. Kraus and F. Wilczek, Self-Interaction Correction to Black Hole Radiance, Nucl. Phys. B 433 , 403 (1995), [arXiv:gr-qc/9408003]; P. Kraus and F. Wilczek, Some Applications of a Simple Stationary Line Element for the Schwarzschild Geometry, Mod. Phys. Lett. A 9 , 3713 (1994), [arXiv:gr-qc/9406042]. \n- [20] S. Shankaranarayanan, T. Padmanabhan and K. Srinivasan, Hawking radiation in different coordinate settings: Complex paths approach, Class. Quant. Grav. 19 , 2671 (2002), [arXiv:gr-qc/0010042]; S. Shankaranarayanan, K. Srinivasan and T. Padmanabhan, Method of complex paths and general covariance of Hawking radiation , Mod. Phys. Letts. A 16 , 571 (2001), [arXiv:gr-qc/0007022].\n- [21] S. Shankaranarayanan, Temperature and Entropy of Schwarzschild-de Sitter spacetime, Phys. Rev. D 67 084026 (2003), [arXiv:gr-qc/0301090].\n- [22] E. Keski-Vakkuri and P. Kraus, Microcanonical D-branes and Back Reaction, Nucl. Phys. B 491 , 249 (1997), [arXiv:hep-th/9610045].\n- [23] S. Massar and R. Parentani, How the Change in Horizon Area Drives Black Hole Evaporation, Nucl. Phys. B 575 , 333 (2000), [arXiv:gr-qc/9903027].\n- [24] M. K. Parikh, A Secret Tunnel Through The Horizon, Int. J. Mod. Phys. D 13 , 2351 (2004), [arXiv:hep-th/0405160]; M. K. Parikh, Energy Conservation and Hawking Radiation , [arXiv:hep-th/0402166].\n- [25] J. Preskill, Do Black Holes Destroy Information? , Publication: An international symposium on Black Holes, Membranes, Wormholes and Superstrings, Houston Advanced Research Center, 16-18 January 1992. Edited by Sunny Kalara and D. V. Nanopoulos. Singapore: World Scientific, 1993, p.22 , [arXiv:hep-th/9209058].\n- [26] D. N. Page, Information in black hole radiation, Phys. Rev. Lett. 71 , 3743 (1993), [arXiv:hep-th/9306083].\n- [27] J. G. Russo, The Information Problem in Black Hole Evaporation: Old and Recent Results , [arXiv:hep-th/0501132].\n- [28] A. J. M. Medved and E. Vagenas, On Hawking Radiation as Tunneling with Logarithmic Corrections, Mod. Phys. Lett. A 20 , 1723 (2005), [arXiv:gr-qc/0505015]; M. Arzano, A. Medved and E. Vagenas, Hawking Radiation as Tunneling through the Quantum Horizon, JHEP 0509 , 037 (2005), [arXiv:hep-th/0505266].\n- [29] K. A. Meissner, Black hole entropy in Loop Quantum Gravity, Class. Quant. Grav. 21 , 5245 (2004), [arXiv:gr-qc/0407052]. \n- [30] J. D. Bekenstein, Nonexistence of baryon number for static black holes, Phys. Rev. D 5 , 1239 (1972).\n- [31] Borun D. Chowdhury, Problems with Tunneling of Thin Shells from Black Holes, Pramana 70 , 593 (2008), [arXiv:hep-th/0605197].\n- [32] T. Pilling, Black hole thermodynamics and the factor of 2 problem, Phys. Lett. B 660 , 402 (2008), [arXiv:gr-qc/0709.1624].\n- [33] K. Nozari and B. Fazlpour, Thermodynamics of Noncommutative Schwarzschild Black Holes, Mod. Phys. Lett. A 22 , 2917 (2007), [arXiv:hep-th/0605109].\n- [34] M. Chaichian, M. M. Sheikh-Jabbari and A. Tureanu, Hydrogen Atom Spectrum and the Lamb Shift in Noncommutative QED, Phys.Rev.Lett. 86 , 2716 (2001), [arXiv:hepth/0010175].\n- [35] R. Banerjee, B. Ranjan Majhi and S. Samanta, Noncommutative Black Hole Thermodynamics , [arXiv:0801.3583]."}
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2008PhRvD..77f4010M
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High-spin binary black hole mergers
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2008-01-01
| 19 | 0.45 | 159 |
['-', '-', '-', '-', 'methods numerical', '-', '-', 'black hole physics', '-']
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[]
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We study identical mass black hole binaries with spins perpendicular to the binary’s orbital plane. These binaries have individual spins ranging from s/m<SUP>2</SUP>=-0.90 to 0.90, (s<SUB>1</SUB>=s<SUB>2</SUB> in all cases) which is near the limit possible with standard Bowen-York puncture initial data. The extreme cases correspond to the largest initial spin simulations to date. Our results expand the parameter space covered by Rezzolla et al., and when combining both data sets, we obtain estimations for the minimum and maximum values for the intrinsic angular momenta of the remnant of binary black hole mergers of J/M<SUP>2</SUP>=0.341±0.004 and 0.951±0.004, respectively. Note, however, that these values are reached through extrapolation to the singular cases |s<SUB>1</SUB>|=|s<SUB>2</SUB>|=1 and thus remain as estimates until full-fledged numerical simulations provide confirmation.
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https://arxiv.org/pdf/0709.2160.pdf
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{'High-spin binary black hole mergers': "Pedro Marronetti, 1 Wolfgang Tichy, 1 Bernd Brugmann, 2 Jose Gonz'alez, 3, 2 and Ulrich Sperhake 2 \n1 Department of Physics, Florida Atlantic University, Boca Raton, FL 33431, USA \n2 Theoretical Physics Institute, University of Jena, 07743 Jena, Germany \n3 Instituto de F'ısica y Matem'aticas, Universidad Michoacana de San Nicol'as de Hidalgo, Morelia, Mexico \nWe study identical mass black hole binaries with spins perpendicular to the binary's orbital plane. These binaries have individual spins ranging from s/m 2 = -0 . 90 to 0 . 90, ( s 1 = s 2 in all cases) which is near the limit possible with standard Bowen-York puncture initial data. The extreme cases correspond to the largest initial spin simulations to date. Our results expand the parameter space covered by Rezzolla et al. and, when combining both data sets, we obtain estimations for the minimum and maximum values for the intrinsic angular momenta of the remnant of binary black hole mergers of J/M 2 = 0 . 341 ± 0 . 004 and 0 . 951 ± 0 . 004 respectively. Note, however, that these values are reached through extrapolation to the singular cases | s 1 | = | s 2 | = 1 and thus remain as estimates until full-fledged numerical simulations provide confirmation. \nPACS numbers: 04.25.Dm, 04.70.Bw, 95.30.Sf, 97.60.Lf", 'I. INTRODUCTION': "The existence of black holes, originally introduced as a family of solutions to the vacuum Einstein field equations, was a matter of speculation for the best part of the 20th century. In the past decade, however, astronomical observations placed them as the most promising models for objects detected in X-ray binaries (with sizes of a few to tens of solar masses) and for the supermassive entities at the center of some galaxies (with millions to billions of solar masses) [1]. Any black hole can be fully specified by its mass, angular momentum and charge. Since electrically charged black holes are quickly neutralized by free charges found in their vicinity (i.e, from accretion disks, interstellar plasma, etc.), only their mass and angular momentum are of astrophysical relevance. In a dynamical environment, several factors can determine the black hole's angular momentum: the characteristics of its progenitor (in case of stellar collapse formation), the merger with other black holes and neutron stars of comparable mass, the merger with smaller objects, and accretion of matter from a surrounding disk. (For a review on how these situations can alter the black hole rate of rotation and the bounds for the maximum spin attainable see, for instance, [2] and references therein.) Highly spinning black hole binaries are of particular interest given their astrophysical relevance. X-ray spectroscopic studies of accretion disks around the supermassive galactic black holes may provide evidence of spin parameters larger than 0 . 9 [3] (however, see for instance [4] on how this may only be a tentative estimate). \nRecent progress in numerical relativistic simulations of binary black holes (BBH) [5, 6, 7] makes now possible long and stable evolutions that were impractical a few years ago. In this paper, we studied identical black holes with spins perpendicular to the orbital plane. In general, black hole spins would have arbitrary directions. However, it has been recently suggested [8] that supermassive galactic BBH may favor configurations with spin alignments like the ones studied here, due to the dynamical \ninteraction of the holes with their galactic environments. Several groups have recently studied BBH with spins perpendicular to the orbital plane [9, 10, 11, 12, 13]. Here, we extend those studies, with binaries with initial individual spins ranging from s/m 2 = -0 . 90 to 0 . 90 (these extreme values are the largest modeled to date) with m 1 = m 2 and s 1 = s 2 . The binaries in this sequence have not been considered before in such detail. Note also that, due to the symmetry of these systems, no gravitational recoil is present in the merger remnant. \nWe perform a least-square data fit of our data following the quadratic formula used by Campanelli et al. [9] and Rezzolla et al. [13] and also of both Rezzolla et al. and our data sets combined. The last fit predicts minimum and maximum values for the spin parameter of the black hole remnant of 0 . 341 ± 0 . 004 and 0 . 951 ± 0 . 004 respectively. \nWe find that current numerical techniques for the evolution of black holes with spins s/m 2 > 0 . 75 are limited in that they produce an artificial loss of angular momentum that increases with the magnitude of the spin. While this effect can be diminished by increasing the grid resolution, even relatively large resolutions such as M/ 85 present loss rates larger than 1% per 100 M for the case s/m 2 = 0 . 90. This effect is clearly seen in the long term evolution of BBH that results in a highly spinning black hole and also on single black hole simulations. \nIn Sec. II, we present the numerical details and tests of our simulations. Sections III and IV present our results and conclusions.", 'II. NUMERICAL TECHNIQUES AND TESTS': "All the binary systems studied here consist of identical black holes with spins aligned with the orbital angular momentum. In our coordinates, the orbit develops in the xy plane and the only non-vanishing component of the spins is in the ˆ z direction. We simulated binaries with spin parameters ranging from s/m 2 = -0 . 90 to 0 . 90. \nAs starting points of our simulations, we use stan- \nTABLE I: Initial data parameters. Here m b is the bare mass parameter of each puncture and M = 2 m is the sum of the ADM masses m measured at each puncture. The holes have coordinate separation D , with puncture locations (0 , ± D/ 2 , 0), linear momenta ( ∓ P, 0 , 0), and spins (0 , 0 , s ). We also list the initial values of the ADM mass M i and the angular momentum J i . The 2PN angular velocity is set to Ω M = 0 . 05550 in each case. \n| s/m 2 | m/M m b /M D/M | P/M | M i /M J | i /M 2 |\n|---------|------------------|------------------------------------|------------|----------|\n| | | -0.90 0.5000 0.1767 6.6965 0.14162 | 0.989 | 0.498 |\n| | | -0.75 0.5000 0.3307 6.6372 0.14041 | 0.988 | 0.557 |\n| | | -0.50 0.5000 0.4246 6.5366 0.13838 | 0.987 | 0.655 |\n| | | -0.25 0.5000 0.4656 6.4337 0.13631 | 0.986 | 0.752 |\n| 0.00 | | 0.5000 0.4777 6.3286 0.13419 | 0.986 | 0.849 |\n| 0.25 | | 0.5000 0.4654 6.2212 0.13204 | 0.985 | 0.94 |\n| 0.50 | | 0.5000 0.4240 6.1117 0.12983 | 0.985 | 1.043 |\n| 0.62 | | 0.5000 0.3888 6.0583 0.12875 | 0.985 | 1.09 |\n| 0.75 | | 0.5000 0.3299 6.0000 0.12756 | 0.985 | 1.14 |\n| 0.82 | | 0.5000 0.2810 5.9684 0.12691 | 0.985 | 1.167 |\n| 0.90 | | 0.5000 0.1764 5.9320 0.12616 | 0.985 | 1.198 | \ndard puncture initial data [14] with the momentum and spin parameters in the extrinsic curvature given by second-order post-Newtonian (2PN) estimates [15]. It is sufficient to use 2PN estimates because standard puncture data are inconsistent with PN theory beyond ( v/c ) 3 [16, 17, 18, 19]. These parameters along with the initial ADM mass M i and angular momentum J i are shown in Table I. The coordinate distance D , the linear momenta P and spin parameters s are directly used in the Bowen-York extrinsic curvature, while the bare mass parameter is obtained from the condition that the ADM masses measured at each puncture should be m = M/ 2. As in [20, 21, 22] we assume that m is a good approximation for the initial individual black hole masses. \nTo complete the definition of the initial data, we also need to specify initial values for the lapse α and shift vector β i . At time t = 0 we use \nα = ( 1 + m b 4 r 1 + m b 4 r 2 ) -4 , β i = 0 , \nwhere r A is the coordinate distance from puncture A . Lapse and shift evolve according to \n( ∂ t -β i ∂ i ) α = -2 αK, ( ∂ t -β k ∂ k ) β i = 3 4 B i , ( ∂ t -β k ∂ k ) B i = ( ∂ t -β k ∂ k ) ˜ Γ i -ηB i . \nThe gravitational fields are evolved using the 'moving punctures' method [6, 7] with the implementation discussed in [23, 24], with the exception that as in [25] the dynamical variable φ has been replaced by the variable \nW = e -2 φ , \nTABLE II: Characteristics of our numerical grids. The values between brackets show the number of inner (moving) levels times the number of grid points per level per dimension, plus the number of outer (fixed) levels times the number of grid points. h min is the finest grid spacing and OB gives the coordinate distance to the grid's outer boundary. The last entry corresponds to a grid used with the code LEAN [26]. \n| Grid | Structure |\n|--------|-----------------------------------------------------------|\n| 1 | [5 × 48 : 5 × 54] [ h min = M/ 56 . 9 : OB = 238 . 5 M ] |\n| 2 | [5 × 56 : 5 × 63] [ h min = M/ 66 . 4 : OB = 235 . 3 M ] |\n| 3 | [5 × 64 : 5 × 72] [ h min = M/ 75 . 9 : OB = 246 . 4 M ] |\n| 4 | [5 × 72 : 5 × 81] [ h min = M/ 85 . 3 : OB = 243 . 0 M ] |\n| 5 | [5 × 80 : 5 × 90] [ h min = M/ 94 . 8 : OB = 240 . 3 M ] |\n| 6 | [5 × 80 : 5 × 90] [ h min = M/ 85 . 3 : OB = 267 . 0 M ] |\n| L | [3 × 69 : 6 × 149] [ h min = M/ 80 . 0 : OB = 256 . 0 M ] | \nwhich obeys the evolution equation \n∂ t W = 1 3 W ( αK -∂ i β i ) + β i ∂ i W. \nThe new variable W has two advantages. First, our simulations seem to converge better when we use W instead of φ or χ = e -4 φ [6]. This may be related to the fact that W grows linearly near the black hole centers after some time of evolving the system. Such linear behavior leads to more accurate finite differencing derivatives near the punctures. In addition, the determinant of the 3-metric det( γ ij ) = W -6 remains always positive, even if W becomes slightly negative due to numerical error. This property is not ensured if one evolves, for example, the variable χ . In the latter case one has to explicitly guard against this to prevent code crashes. \nThe simulations presented here were performed using the BAM code, details of which are described in [23, 27]. BAM is based on three-dimensional Cartesian coordinates and can achieve high spatial resolution near the punctures using a structure of nested refinement levels. The outermost of these levels are fixed, while the innermost track the movement of the punctures. For the runs presented here, we used 10 levels of refinement with the outer boundaries located about 240 M away from the grid center. Since the black holes are identical, we use quadrant symmetry. Table II lists the characteristics of the different grid layouts used in this paper. BAM characteristics make it ideal for quick and relatively inexpensive runs [24]. Most of the simulations presented here were done on dual processor workstations with characteristics speeds of 0.9, 1.7 and 2.5 days per orbit when using grids 1, 2 and 3 respectively. \nTable III enumerates the binary simulations using the nomenclature SXX Y , where XX indicates the value of s/m 2 and Y indicates which one of the numerical grids of Table II was used. We measure the mass and angular momentum of the remnant of the binary black hole merger using techniques based on volume integrals (see [24] for details). These values of mass and angular momentum are labeled M f and J f respectively. \nTABLE III: Binary black hole simulations performed for this paper. The runs are named SXX Y , where XX indicates the value of s/m 2 and Y is the numerical grid used for that run. Table II describes the different grids. \n| Run | M f /M J f /M 2 J f /M 2 f | M f /M J f /M 2 J f /M 2 f | M f /M J f /M 2 J f /M 2 f |\n|-----------------------|------------------------------|------------------------------|------------------------------|\n| S - 0 . 90 1 | 0.976 | 0.348 | 0.365 |\n| S - 0 . 90 2 | 0.970 | 0.359 | 0.382 |\n| S - 0 . 90 3 | 0.970 | 0.358 | 0.38 |\n| S - 0 . 90 4 | 0.970 | 0.358 | 0.38 |\n| S - 0 . 75 1 | 0.970 | 0.405 | 0.431 |\n| S - 0 . 75 2 | 0.968 | 0.408 | 0.436 |\n| S - 0 . 75 3 | 0.968 | 0.409 | 0.437 |\n| S - 0 . 50 1 | 0.963 | 0.483 | 0.52 |\n| S - 0 . 50 2 | | 0.486 | 0.524 |\n| S - | 0.963 | 0.551 | 0.6 |\n| 0 . 25 1 S - 0 . 25 2 | 0.958 0.958 | 0.555 | 0.606 |\n| S +0 . 00 1 | 0.951 | 0.605 | 0.669 |\n| S +0 . 00 2 | 0.951 | 0.614 | 0.679 |\n| S +0 . 00 3 | 0.951 | 0.619 | 0.684 |\n| S +0 . 00 4 | 0.951 | 0.619 | 0.684 |\n| S +0 . 25 1 | 0.944 | 0.671 | 0.753 |\n| S +0 . 25 2 | 0.944 | 0.676 | 0.759 |\n| S +0 . 50 1 | 0.934 | 0.721 | 0.827 |\n| S +0 . 50 2 | 0.933 | 0.72 | 0.826 |\n| S +0 . 62 1 | 0.927 | 0.731 | 0.851 |\n| S +0 . 62 2 | 0.926 | 0.737 | 0.859 |\n| S +0 . 62 3 | 0.926 | 0.737 | 0.859 |\n| S +0 . 75 1 | 0.917 | 0.729 | 0.866 |\n| S +0 . 75 2 | 0.916 | 0.733 | 0.874 |\n| S +0 . 75 3 | 0.916 | 0.736 | 0.877 |\n| S +0 . 75 4 | 0.916 | 0.738 | 0.88 |\n| S +0 . 82 1 | 0.911 | 0.688 | 0.828 |\n| S +0 . 82 2 | 0.910 | 0.703 | 0.85 |\n| S +0 . 82 3 | 0.909 | 0.701 | 0.848 |\n| S +0 . 90 1 | 0.905 | 0.645 | 0.788 |\n| S +0 . 90 2 | 0.905 | 0.64 | 0.781 |\n| S +0 . 90 3 | 0.903 | 0.643 | 0.788 |\n| S +0 . 90 4 | 0.902 | 0.643 | 0.79 |\n| S +0 . 90 5 | 0.902 | 0.646 | 0.794 |\n| S +0 . 90 6 | 0.906 | 0.644 | 0.785 | \nWe study the effect of different approximations and limitations inherent to our simulations: grid resolution, grid structure, measurement of the final mass and angular momentum, and the intrinsic characteristics of our initial data sets. We start by varying the spatial resolution of our grids, while keeping the grid size and structure (i.e., the layout of the refinement levels) unchanged. Grids 1 to 5 cover maximum spatial resolutions going from M/ 56 . 9 to M/ 94 . 8. The results, presented in Table III, do not seem to vary significantly with the spatial resolution: the values of J f /M 2 f obtained for grids 2 to 5 agree with each other at a level of about 1 . 5% or better. We test the convergence of the calculations of mass and angular momentum, using four runs for the binary with spin s/m 2 = 0 . 90 (labeled S +0 . 90 Y , Y =1 to 4). These runs shared the same grid layout but with vary- \nFIG. 1: Mass and angular momentum plots for three different resolutions. The dashed curves have been scaled according to a factor corresponding to 4th order convergence in spatial resolution. \n<!-- image --> \ning spatial resolution, ranging from h min = M/ 56 . 9 to h min = M/ 85 . 3 [36]. We find that the lowest resolution run ( S +0 . 90 1) fell outside the convergence regime. The results of the other three runs, presented in Fig. 1, seem to show that these runs are in the convergent regime. \nTo evaluate the influence of the grid layout, we compare the results from two grids with identical resolution but different number of grid points per box (Grids 4 and 6 from Table II). The box size has been found crucial in previous work [23]: boxes smaller than critical sizes tend to change drastically the binary dynamics, altering orbits and merger times. Runs S + 0 . 90 4 and S + 0 . 90 6 returned values of mass and angular momentum with differences of ∆ M f = 4 10 -3 (0 . 4%) and ∆ J f = 5 10 -4 (0 . 1%) respectively. \nAdditionally, and as an independent check, we compare our results with those of Campanelli et al. [28] (shown in Fig. 2 as empty circles) and Pollney et al. [12] (empty square). \nTo evaluate the accuracy of the algorithms used to measure mass and angular momentum, we compare our volume-integral based results with calculations done using surface integrals [24] and find differences of up to 0 . 5% in magnitude. We also test the satisfaction of the Christodoulou formula J c = 2 M irr ( M 2 f -M 2 irr ) 1 / 2 ( M irr being the irreducible mass of the final black hole). The relative difference in the values of J f and J c for the run S +0 . 90 5 was less than 0 . 5%.", 'III. RESULTS': "Figure 2 summarizes the results of Table III. The spin parameter of the binary remnant J f /M 2 f is shown as a function of the initial black hole spins s/m 2 . These values were measured at 500 M after the merger (which cor- \nFIG. 2: Spin of the merger remnant as a function of the initial black hole spins measured 500 M after the black hole merger. The binaries are composed of identical black holes with initial spins of magnitude s/m 2 aligned with the orbital angular. The values calculated in [28] ([12]) are shown as empty circles (square). The solid square represents the remnant spin calculated right after the merger for the case s/m 2 = 0 . 90, using Grid L. The solid line is a quadratic interpolation of the values corresponding to Grid 2 for s/m 2 < 0 . 75 plus the Grid L value. \n<!-- image --> \nresponds approximately to the side-to-side light-crossing time of our grids). The most striking feature of the plot is the existence of an apparent maximum at s/m 2 ≈ 0 . 75. As we will see, this maximum is merely a numerical artifact caused by an artificial loss of angular momentum in the case of highly spinning black holes if we wait for 500 M after the merger. Shortly after the merger, the spin of the remaining black hole is still larger. This is confirmed by a simulation of the s/m 2 = 0 . 90 case using the LEAN code (Grid L in Table II) that tracks the common apparent horizon and the emission of energy and angular momentum in the form of gravitational waves. We calculate the spin about 50 M after the merger in four different ways: from the balance of gravitational wave energy and angular momentum loss, from the ringdown frequencies (using the tabulated results given in [29]), from the isolated horizon geometry (using the techniques from [9]), and using Christodoulou's formula. The corresponding results are J f /M 2 f = 0 . 95 , 0 . 925 , 0 . 913 , 0 . 918 respectively. The first result is the least accurate, given that the wave extraction radius (50 M ) was not considerably far from the center. The rest cluster around 0.92 (plotted in Fig. 2 as a solid square) that agrees better with an extrapolation from lower s/m 2 results. Figure 3 shows the evolution of the angular momentum obtained using isolated horizon techniques which indicates a loss rate of about 0 . 5% per 100 M . This last result together with the measurements at 500 M after the merger seem to indi- \nFIG. 3: Evolution of the angular momentum for the binary with s/m 2 = 0 . 90 obtained using the isolated horizon techniques of [9]. The top plot corresponds to the angular momentum of the remnant of the merger, while the bottom plot tracks the spin of the individual black holes before the merger (occurring at 265 M ). \n<!-- image --> \nte an artificial loss in angular momentum for high spin black holes. Also note that this effect is not seen at the lower spin end of Fig. 2. This is simply due to the fact that the merger remnant of binaries with individual spins s/m 2 < 0 . 75 is a low spinning black hole, and thus not affected by this effect. \nTo study this effect in more detail, we perform single black hole evolutions. Figure 4 compares evolutions for the cases with s/m 2 = 0 . 53 and 0 . 90. In order to facilitate the comparison, we plot the relative differences from the initial values of the mass, angular momentum and intrinsic angular momentum parameter, these quantities measured using volume integrals methods. The angular momentum loss, while negligible in the 0 . 53 case, is more pronounced in the larger spin case. Note however that the mass is largely unaffected in both cases. Figure 5 shows the results for four different grid resolutions, indicating firstly that the mass is better conserved than the angular momentum and secondly that this loss gets smaller with increasing resolution, albeit quite slowly. The angular momentum curves are still consistent with 4th order convergence, however, Richardson extrapolating these curves to arbitraily large resolution still shows loss of angular momentum. For BBH evolutions with many orbits before the merger, this loss could in principle produce an 'effective' value of s/m 2 at the time of the merger smaller than the initially intended. In order to evaluate this effect, we track the evolution of the individual spins before the merger as shown in the bottom panel of Fig. 3. An angular momentum loss of about 3% is detected before the merger, which occurs at t = 265 M . \nFIG. 4: Single black hole evolutions with s/m 2 = 0 . 53 (dashed) and 0 . 90 (solid), on Grid 4. The plots present the relative differences with respect to the initial values, denoted with the 0 sub-index. The bottom plot presents the relative variation of intrinsic angular momentum j ≡ J f /M 2 f . \n<!-- image --> \nFIG. 5: Evolution of single black holes with s/m 2 = 0 . 90 for different spatial resolutions. \n<!-- image --> \nAt first sight, part of this loss could also be attributed to our choice of initial data. Traditionally, puncture initial data sets are constructed by solving the Hamiltonian constraint under the assumption of conformal flatness for the spatial metric. The momentum constraint is analytically satisfied by the use of the Bowen-York (BY) formula for the extrinsic curvature which, in principle, allows for the arbitrary specification of the linear and angular momentum of the black holes. In practice, the amount of \nFIG. 6: Puncture tracks of one of the holes for two simulations with s/m 2 = 0 . 90 that start out at different coordinate separations; D = 8 M (dashed) and D = 6 M (solid). The D = 6 M orbits have been rotated to coincide with the ones from the D = 8 M case in the final orbits. Both runs were performed using Grid 3. \n<!-- image --> \n'junk radiation' associated with these initial data sets increases with the magnitude of the spin of the hole. The amount of this extra radiation has been studied perturbatively by Gleiser et al. [30] and numerically by Hannam et al. [31] and Dain et al. [32]. The last shows that, for the case s/m 2 = 0 . 90, a single BY black hole would relax into a Kerr black hole after emitting about 0 . 1% of the initial gravitational mass. \nTo further study this effect in the evolution prior to the merger, we perform a simulation of a binary with s/m 2 = 0 . 90 that starts out at a coordinate separation D = 8 M and compare it with one starting at D = 6 M . The former simulation lasted more than two orbits and a half longer than the latter, corresponding to an additional simulation time of 395 M . Figure 6 shows the orbits of one of the punctures for these simulations. Both runs were performed using Grid 3. Figure 7 shows the corresponding evolution of the mass and angular momentum, where the curves corresponding to the D = 8 M run were shifted in time by 395 M . Naively, one would expect the D = 8 M to dissipate more angular momentum before the merger, given that it is more than two orbits longer than its D = 6 M counterpart. This would result in the longer run remnant having a smaller intrinsic spin than the shorter evolution. However, the resulting spin for the D = 8 M run is about 3% larger for the D = 6 M case, but within the error margin for these measurements, making it difficult to draw any conclusions. \nNext, we fit the values of the highest resolution runs from Table III, ignoring the results for s/m 2 > 0 . 75, but adding the result at s/m 2 = 0 . 90 from Grid L (solid \nFIG. 7: Evolution of mass and angular momentum for the runs of Fig. 6. The solid (dotted) lines corresponds to the separation D = 6 M ( D = 8 M ). The dashed line corresponds to the D = 8 M run shifted in time by 395 M . \n<!-- image --> \nTABLE IV: Least-squares fit of Eq. (1) from Campanelli et al. [9] ( C ), Rezzolla et al. [13] ( R ), this paper ( M ), and Rezzolla et al. and our data sets combined ( R+M ). The last two rows show the extrapolation to initial critical black holes aligned (counter-aligned) with the orbital angular momentum that corresponds to the maximum (minimum) possible intrinsic spin for the remnant black hole. \n| | C | R | M | R+M |\n|-----------|--------|-----------|------------------------------------------|-----------|\n| p 0 | 0.6879 | 0.6883(4) | 0.6855(16) | 0.6888(4) |\n| p 1 | 0.1476 | 0.1530(4) | 0.1499(8) | 0.1525(5) |\n| p 2 | | | -0.0093 -0.0088(5) -0.0110(8) -0.0106(5) | |\n| Max J/M 2 | 0.946 | 0.959(2) | 0.941(6) | 0.951(4) |\n| Min J/M 2 | 0.355 | 0.347(2) | 0.342(6) | 0.341(4) | \nsquare). We follow the fitting formula for the final black hole intrinsic spin used in Campanelli et al. [9] and Rezzolla et al. [13] (Eq. (8)) \nJ f /M 2 f = p 0 + p 1 ( s 1 /m 2 1 + s 2 /m 2 2 ) + p 2 ( s 1 /m 2 1 + s 2 /m 2 2 ) 2 , (1) \nwhere p 0 , p 1 and p 2 are the fitting parameters. We present in Table IV a comparison of our best fit parameters with those from [9] and [13] and with the ones obtained by fitting [13] and our data sets together. For these fits, we assume a nominal error of 0 . 02 for all our values. We see that the fitting parameters are in an agreement consistent with small-number statistics in all cases. Table IV also shows the extrapolation to maximally rotating black holes, aligned and counter-aligned with the orbital angular momentum. The maximum and minimum values for the intrinsic angular momentum of the \nremnant predicted by the fit of the combined data sets is J f /M 2 f = 0 . 341 ± 0 . 004 and 0 . 951 ± 0 . 004 respectively [37]. Note that, while our data set is smaller than the one in [13], it contains measurements that are closer to the extrapolated values for critical BBH. \nRecently, Boyle et al. [33] introduced a formalism that predicts any final quantity resulting from the merger using a Taylor expansion on the initial binary mass ratio q ≡ m 1 /m 2 and the components of the initial spins. For the highly symmetric binaries considered in this paper, their expansion for J f /M 2 f to second order reduces to the polynomial of Eq. (1) with the equivalences p 0 = s 000 | 000 3 , p 1 = s 001 | 000 3 and p 2 = (1 / 4) (2 s 002 | 000 3 + s 001 | 001 3 ), where the parameters on the right-hand sides are those from the corresponding expansion in [33]. \nFinally, we would like to highlight the ability of the code BAM to probe accurately and efficiently BBH parameter space. Our lowest resolution runs (Grid 1), performed on dual-processor workstations, are good enough to capture the main characteristics of the results of Fig. 2.", 'IV. CONCLUSIONS': 'We studied the effect of the initial spins of the black holes in a binary system on the mass and angular momentum of the black hole that results from the merger. We concentrated on equal-mass binaries with spins aligned with the orbital angular momentum ( s 1 = s 2 ), covering a range of initial spin parameters going from s/m 2 = -0 . 90 to 0 . 90. The runs at the extrema of the range are the highest spin simulations to date. The main results of the paper are presented in Fig. 2, where the spin parameter of the remnant ( J f /M 2 f ) is given as a function of the initial spin parameters. \nWe combined our results with those of Rezzolla et al. [13] in a quadratic least-square fit and obtain, by extrapolation to initial critical black holes, predicted maximum and minimum values of J/M 2 for the black hole remnant of 0 . 951 ± 0 . 004 and 0 . 341 ± 0 . 004 respectively. These error bounds are simply derived from the uncertainty of the fitting parameters provided by their standard error. The small size of the samples studied here plus the fact that the limits to critical black hole results are obtained through extrapolation to points in parameter space with singular properties may lead to revisions in these estimates once binaries with spins larger than 0.90 can be accurately simulated. For this, new recipes for initial data sets that allow for initial black holes with spin parameters larger than 0.928 (the limit of BY data) are needed. The methods introduced by Dain [34, 35] and studied in head-on BBH simulations by Hannam et al. [31] may hold the key to this problem. \nWe also find a problem for the simulations starting with spins s/m 2 > 0 . 75. Current evolutions based on the moving punctures methods present losses of angular \nmomentum at non-negligible rates for highly spinning black holes, even when relatively high grid resolutions are employed. Measurements of the merger remnant intrinsic spin 500 M after the merger of two s/m 2 = 0 . 90 black holes show the spurious loss of more than 10% of the angular momentum. This effect increases with the magnitude of the black hole spin. This loss does not affect strongly the calculation of gravitational wave templates, since they cover only up to a short period after the merger. However, it becomes more important, for instance, for simulations of black hole-neutron star or neutron star-neutron star binaries, where the potential formation of an accretion disk around a black hole has to be followed for much longer periods.', 'Acknowledgments': "It is a pleasure to thank to Mark Hannam and Sascha Husa for their help and insight all along this project. \n- [1] R. Narayan, New J. Phys. 7 , 199 (2005), gr-qc/0506078.\n- [2] C. F. Gammie, S. L. Shapiro, and J. C. McKinney, Astrophys. J. 602 , 312 (2004), astro-ph/0310886.\n- [3] C. S. Reynolds, L. W. Brenneman, and D. Garofalo, Astrophys. Space Sci. 300 , 71 (2005), astro-ph/0410116.\n- [4] K. Nandra, P. M. O'Neill, I. M. George, J. N. Reeves, and T. J. Turner, Astron. Nachr. 327 , 1039 (2006), astroph/0610585.\n- [5] F. Pretorius, Phys. Rev. Lett. 95 , 121101 (2005), grqc/0507014.\n- [6] M. Campanelli, C. O. Lousto, P. Marronetti, and Y. Zlochower, Phys. Rev. Lett. 96 , 111101 (2006), grqc/0511048.\n- [7] J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and J. van Meter, Phys. Rev. Lett. 96 , 111102 (2006), grqc/0511103.\n- [8] T. Bogdanovic, C. S. Reynolds, and M. C. Miller, Ap. J. Lett. 661 , L147 (2007), astro-ph/0703054.\n- [9] M. Campanelli, C. O. Lousto, Y. Zlochower, B. Krishnan, and D. Merritt, Phys. Rev. D75 , 064030 (2007), gr-qc/0612076.\n- [10] F. Herrmann, I. Hinder, D. Shoemaker, P. Laguna, and R. A. Matzner, Phys. Rev. D 76 , 084032 (2007), arXiv:0706.2541.\n- [11] M. Koppitz, D. Pollney, C. Reisswig, L. Rezzolla, J. Thornburg, P. Diener, and E. Schnetter, Phys. Rev. Lett. 99 , 041102 (2007), gr-qc/0701163.\n- [12] D. Pollney, C. Reisswig, L. Rezolla, B. Szilagyi, M. Ansorg, B. Deris, P. Diener, E. Dorband, M. Koppitz, A. Nagar, et al., Phys. Rev. D 76 , 124002 (2007), arXiv:0707.2559.\n- [13] E. L. Rezzolla, N. Dorband, C. Reisswig, P. Diener, D. Pollney, E. Schnetter, and B. Szilagyi (2007), arXiv:0708.3999.\n- [14] S. Brandt and B. Brugmann, Phys. Rev. Lett. 78 , 3606 (1997), gr-qc/9703066.\n- [15] L. E. Kidder, Phys. Rev. D52 , 821 (1995), gr- \nWe would also like to thank C. Lousto, L. Boyle and M. Kesden for useful discussions. This work was supported by NSF grants PHY-0555644 and PHY-0652874. We also acknowledge partial support by the National Computational Science Alliance under Grants PHY050016N and PHY060021P and by DFG grant SFB/Transregio 7 'Gravitational Wave Astronomy'. We thank the DEISA Consortium (co-funded by the EU, FP6 project 508830), for support within the DEISA Extreme Computing Initiative (www.deisa.org). J.G. and U.S. acknowledge support from the ILIAS Sixth Framework Programme. The high resolution simulations were performed at the Charles E. Schmidt College of Science computer cluster Boca 5 , at the Cray XT3 MPP system (BigBen) at the Pittsburgh Supercomputer Center, and at LRZ Munich. \n- qc/9506022.\n- [16] W. Tichy, B. Brugmann, M. Campanelli, and P. Diener, Phys. Rev. D 67 , 064008 (2003), gr-qc/0207011.\n- [17] N. Yunes and W. Tichy, Phys. Rev. D 74 , 064013 (2006), gr-qc/0601046.\n- [18] N. Yunes, W. Tichy, B. J. Owen, and B. Bruegmann, Phys. Rev. D 74 , 104011 (2006), gr-qc/0503011.\n- [19] B. J. Kelly, W. Tichy, M. Campanelli, and B. F. Whiting, Phys. Rev. D76 , 024008 (2007), arXiv:0704.0628.\n- [20] W. Tichy, B. Brugmann, and P. Laguna, Phys. Rev. D 68 , 064008 (2003), gr-qc/0306020.\n- [21] W. Tichy and B. Brugmann, Phys. Rev. D 69 , 024006 (2004), gr-qc/0307027.\n- [22] M. Ansorg, B. Brugmann, and W. Tichy, Phys. Rev. D 70 , 064011 (2004), gr-qc/0404056.\n- [23] B. Brugmann, J. Gonzalez, M. Hannam, S. Husa, U. Sperhake, and W. Tichy, Phys. Rev. D 77 , 024027 (2008), gr-qc/0610128.\n- [24] P. Marronetti, W. Tichy, B. Bruegmann, J. Gonzalez, M. Hannam, S. Husa, and U. Sperhake, Class. Quantum Grav. 24 , S34 (2007), gr-qc/0701123.\n- [25] W. Tichy and P. Marronetti, Phys. Rev. D 76 , 061502 (2007), arXiv:gr-qc/0703075.\n- [26] U. Sperhake, Phys. Rev. D 76 , 104015 (2007), grqc/0606079.\n- [27] B. Brugmann, W. Tichy, and N. Jansen, Phys. Rev. Lett. 92 , 211101 (2004), gr-qc/0312112.\n- [28] M. Campanelli, C. O. Lousto, and Y. Zlochower, Phys. Rev. D 74 , 041501 (2006), gr-qc/0604012.\n- [29] E. Berti, V. Cardoso, and C. M. Will, Phys. Rev. D 73 , 064030 (2006), gr-qc/0512160.\n- [30] R. J. Gleiser, C. O. Nicasio, R. H. Price, and J. Pullin, Phys. Rev.D 57 , 3401 (1998).\n- [31] M. Hannam, S. Husa, B. Brugmann, J. A. Gonzalez, and U. Sperhake, Class. Quant. Grav. 24 , S15 (2007), grqc/0612001.\n- [32] S. Dain, C. O. Lousto, and R. Takahashi, Phys. Rev. D\n- 65 , 104038 (2002), gr-qc/0201062.\n- [33] L. Boyle, M. Kesden, and S. Nissanke (2007), arXiv:0709.0299.\n- [34] S. Dain, Phys. Rev. Lett. 87 , 121102 (2001), grqc/0012023.\n- [35] S. Dain, Phys. Rev. D 64 (2001), gr-qc/0103030.\n- [36] Run S+0.90 5 was not used because the high order of our algorithms (fourth) made the difference in results fall \n- too close to our calculation round-off error for meaningful convergence tests.\n- [37] To asses the influence of the LEAN data point in our results, we re-calculated our predictions for the maximum and minimum values of J f /M 2 f without it. The new results are within the error bands of the values quoted in Table IV"}
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2000PhRvD..62l4022O
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Transition from inspiral to plunge for a compact body in a circular equatorial orbit around a massive, spinning black hole
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2000-01-01
| 9 | 0.45 | 159 |
['-', '-', '-', 'waves', '-', 'black hole physics', '-']
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[]
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There are three regimes of gravitational-radiation-reaction-induced inspiral for a compact body with mass μ, in a circular, equatorial orbit around a Kerr black hole with mass M>>μ: (i) the adiabatic inspiral regime, in which the body gradually descends through a sequence of circular, geodesic orbits; (ii) a transition regime, near the innermost stable circular orbit (isco); (iii) the plunge regime, in which the body travels on a geodesic from slightly below the isco into the hole's horizon. This paper gives an analytic treatment of the transition regime and shows that, with some luck, gravitational waves from the transition might be measurable by the space-based LISA mission.
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https://arxiv.org/pdf/gr-qc/0003032.pdf
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{'The Transition from Inspiral to Plunge for a Compact Body in a Circular Equatorial Orbit Around a Massive, Spinning Black Hole': "Amos Ori, (1) and Kip S. Thorne (2) \n(1) Department of Physics, Technion-Israel Institute of Technology, Haifa, 32000, Israel (2) Theoretical Astrophysics, California Institute of Technology, Pasadena, CA 91125 (Received 21 February 2000) \nThere are three regimes of gravitational-radiation-reaction-induced inspiral for a compact body with mass µ , in a circular, equatorial orbit around a Kerr black hole with mass M /greatermuch µ : (i) The adiabatic inspiral regime , in which the body gradually descends through a sequence of circular, geodesic orbits. (ii) A transition regime , near the innermost stable circular orbit (isco). (iii) The plunge regime , in which the body travels on a geodesic from slightly below the isco into the hole's horizon. This paper gives an analytic treatment of the transition regime and shows that, with some luck, gravitational waves from the transition might be measurable by the space-based LISA mission. \nPACS numbers: 04.30.Db, 04.80.Nn, 97.60.Lf", 'I. INTRODUCTION AND SUMMARY': "The space-based Laser Interferometer Space Antenna (LISA) [1], if it flies, is likely to detect and study the gravitational waves from white dwarfs, neutron stars and small black holes with masses µ > ∼ 1 M /circledot , spiraling into Massive ( M ∼ 10 5 -10 8 M /circledot /greatermuch µ ) black holes in the nuclei of distant galaxies [2-4]. In preparation for these studies, it is necessary to understand, theoretically, the radiationreaction-induced evolution of the inspiral orbits, and the gravitational waveforms that they emit. \nRegardless of an orbit's shape and orientation, when µ /lessmuch M the orbital evolution can be divided into three regimes: (i) The adiabatic inspiral regime , in which the body gradually descends through a sequence of geodesic orbits with gradually changing 'constants' of the motion E = (energy), L = (polar component of angular momentum), and Q = (Carter constant). (ii) A transition regime , in which the character of the orbit gradually changes from inspiral to plunge. (iii) A plunge regime , in which the body plunges into the horizon along a geodesic with (nearly) unchanging E , L and Q . \nThe plunge regime, being (essentially) ordinary geodesic motion, is well understood; and the adiabatic inspiral regime is the focus of extensive current research (see, e.g. [4-6]). By contrast, so far as we are aware, there have been no publications dealing with the transition regime. \nIn 1990-91, we carried out an initial exploration of the transition regime for the special case of circular, equatorial orbits; but due to the press of other projects we did not publish it. Now, with prospects for LISA looking good, we have resurrected our work and present it here, in the context of LISA. \nWe begin, in Sec. II, by summarizing some key, wellknown details of the inspiral and plunge regimes. Then in Sec. III A we present a qualitative picture of the transition from inspiral to plunge, based on the motion of a \nparticle in a slowly changing effective potential (Fig. 1). With the aid of this qualitative picture, in Sec. III B we derive a non-geodesic equation of motion for the transition regime, and in Sec. III C we construct the solution to that equation of motion (Figs. 2 and 3). Then in Sec. IV, with the aid of our solution, we estimate the gravitational-wave signal strength from the transition regime and the signal-to-noise ratio that it would produce in LISA. We conclude that, with some luck, LISA may be able to detect and study the transition waves. In Sec. V we make concluding remarks about the need for further research.", 'II. ADIABATIC INSPIRAL AND PLUNGE': "Throughout this paper we use Boyer-Lindquist coordinates ( t, r, θ, φ ) [7] for the massive hole's Kerr metric, and we use geometrized units, with G = c = 1. The hole's mass is M and the inspiraling body's mass is µ ≡ ηM . We use M and µ to construct dimensionless versions (denoted by tildes) of many dimensionfull quantities; for example, ˜ r = r/M , and ˜ t = t/M . The hole's dimensionless spin parameter is a ≡ (spin angular momentum) /M 2 (with -1 < a < +1). The body moves around its circular, equatorial orbit in the + φ direction, so a > 0 corresponds to an orbit that is prograde relative to the hole's spin, and a < 0 to a retrograde orbit. \nWhen the inspiraling body is not too close to the innermost stable circular orbit (isco), it moves on a circular geodesic orbit with dimensionless angular velocity [8] \n˜ Ω ≡ M Ω = dφ d ˜ t = 1 ˜ r 3 / 2 + a (2.1) \n(where φ is angle around the orbit) and with orbital energy [8] \nE = -ηM 1 -2 / ˜ r + a/ ˜ r 3 / 2 √ 1 -3 / ˜ r +2 a/ ˜ r 3 / 2 . (2.2) \nAs it moves, the body radiates energy into gravitational waves at a rate given by [4] \n˙ E GW = -˙ E = 32 5 η 2 ˜ Ω 10 / 3 ˙ E , (2.3) \nwhere ˙ E is a general relativistic correction to the Newtonian, quadrupole-moment formula (Table II of Ref. [4]). This energy loss causes the orbit to shrink adiabatically at a rate given by \ndr dt = -˙ E GW dE/dr . (2.4) \nThe inspiral continues adiabatically until the body nears the isco, which is at the dimensionless radius ˜ r isco = r isco /M given by [8] \n˜ r isco = 3 + Z 2 -sign( a )[(3 -Z 1 )(3 + Z 1 +2 Z 2 )] 1 / 2 , Z 1 ≡ 1 + (1 -a 2 ) 1 / 3 [(1 + a ) 1 / 3 +(1 -a ) 1 / 3 ] , Z 2 ≡ (3 a 2 + Z 1 2 ) 1 / 2 ; (2.5) \ncf. Table I. The circular geodesic orbit at the isco has dimensionless angular velocity (Table I), energy, and angular momentum given by [8,7] \n˜ Ω isco ≡ M Ω = 1 ˜ r 3 / 2 isco + a , (2.6) \n˜ E isco ≡ E isco µ = E isco ηM = 1 -2 / ˜ r isco + a/ ˜ r 3 / 2 isco √ 1 -3 / ˜ r isco +2 a/ ˜ r 3 / 2 isco , (2.7) \n˜ L isco ≡ L isco µM = L isco ηM 2 = 2 √ 3˜ r isco ( 3 √ ˜ r isco -2 a ) . (2.8) \nAs the body nears the isco, its inspiral gradually ceases to be adiabatic and it enters the transition regime (Sec. III). Radiation reaction (as controlled by ˙ E GW ) continues to drive the orbital evolution throughout the transition regime, but gradually becomes unimportant as the transition ends and pure plunge takes over. \nThe plunge is described to high accuracy by reactionfree geodesic motion; Eqs. (33.32) of Ref. [7]. Up to fractional corrections of order η 4 / 5 , the orbital energy and angular momentum of the plunging body are equal to E isco and L isco throughout the plunge. [cf. Eq. (3.26) below].", 'A. Qualitative Explanation of Transition': "As the body nears its innermost stable circular orbit, r = r isco , the adiabatic approximation begins to break \ndown. This breakdown can be understood in terms of the effective potential, which governs geodesic radial motion via the equation \n( d ˜ r d ˜ τ ) 2 = ( dr dτ ) 2 = ˜ E 2 -V (˜ r, ˜ E, ˜ L ) , (3.1) \nwhere ˜ E ≡ E/µ = E/ ( ηM ), ˜ L ≡ L/ ( µM ) = L/ ( ηM 2 ), and ˜ τ ≡ τ/M are the body's dimensionless energy, angular momentum, and proper time. The explicit form of the effective potential can be inferred from Eqs. (33.32) and (33.33) of MTW [7]: \nV (˜ r, ˜ E, ˜ L ) = ˜ E 2 -1 ˜ r 4 ( [ ˜ E (˜ r 2 + a 2 ) -˜ La ] 2 -(˜ r 2 -2˜ r + a 2 )[˜ r 2 +( ˜ L -˜ Ea ) 2 ] ) . (3.2) \nFor a Schwarzschild black hole, this reduces to \nV (˜ r, ˜ E, ˜ L ) = ( 1 -2 ˜ r ) ( 1 + ˜ L 2 ˜ r 2 ) for a = 0 (3.3) \n(cf. Eq. (25.16) of MTW [7]). \nThroughout the inspiral and transition regimes, the body moves along a nearly circular orbit; its change of radius during each circuit around the hole is ∆ r /lessmuch r . (Only after the body is well into its final plunge toward the hole does ∆ r become comparable to r .) This nearcircular motion guarantees that the ratio of the energy radiated to angular momentum radiated is equal to the body's orbital angular velocity [9]: \nd ˜ E d ˜ τ = ˜ Ω d ˜ L d ˜ τ . (3.4) \nCorrespondingly, in and near the transition regime, which occupies a narrow range of radii around ˜ r isco , the body's energy and angular momentum are related by ∗ \n˜ E = ˜ E isco + ˜ Ω isco ξ , ˜ L = ˜ L isco + ξ . (3.5) \nBy combining Eqs. (3.5) and (3.2), we can regard the body's effective potential as a function of ˜ r and the difference ξ ≡ ˜ L -˜ L isco of its orbital angular momentum from that of the isco. \nFIG. 1. The gradually changing effective potential V (˜ r, ξ ) for radial geodesic motion. Each curve is for a particular value of ξ ≡ ˜ L -˜ L isco . As ξ decreases due to radiation reaction, the body, depicted by the large dot, at first remains at the minimum of the effective potential ( ξ 1 ; 'adiabatic regime'). However, as ξ nears zero (at ξ /similarequal ξ 2 ), the body cannot keep up with the rapid inward motion of the minimum; it lags behind in a manner described by the transition-regime analysis of Sec. III. At ξ /similarequal ξ 5 the effective potential has become so steep that radiation reaction is no longer important, the transition regime ends, and the body plunges toward the black hole with nearly constant energy and angular momentum. \n<!-- image --> \nFigure 1 shows V (˜ r, ξ ) for a sequence of angular momenta ξ 1 , . . . , ξ 5 around ξ = 0. As ξ decreases to ξ = 0, the minimum of the potential flattens out and disappears; and just when it is disappearing, the minimum's radius r min is moving inward at an infinite rate: dr min /dξ →∞ as ξ 0. \nIn the adiabatic regime of large ξ , the body sits always at the minimum of the effective potential. Its orbit is a slowly shrinking circle, guided inward by the motion of the minimum. As ξ nears zero and the minimum's inward speed grows large, the body's inertia prevents it from continuing to follow the minimum. The body begins to lag behind, as depicted at ξ = ξ 2 in Fig. 1. This lag invalidates the adiabatic inspiral analysis of Sec. II and initiates the transition regime. \n→ \nAs ξ continues to decrease, there comes a point (near ξ 5 in Fig. 1) at which the effective potential has become so steep that its inward force on the body dominates strongly over radiation reaction. There the transition regime ends, and the body begins to plunge inward rapidly on a nearly geodesic orbit with nearly constant ˜ E and ˜ L . The objectives of the following subsections are to derive a set of equations describing the transition regime (Sec. III B), and show how the transition matches smoothly onto the adiabatic regime at large positive ξ and to the plunge regime at large negative ξ (Sec. III C).", 'B. Equation of Motion for Transition Regime': "Throughout the transition regime, because the body moves on a nearly circular orbit with radius close to r isco , and because the body's small mass µ ≡ ηM /lessmuch M keeps its radiation reaction weak, its angular velocity remains very close to Ω isco \ndφ d ˜ t ≡ ˜ Ω /similarequal ˜ Ω isco , (3.6) \nand its proper time ticks at very nearly the standard isco rate \nd ˜ τ d ˜ t /similarequal ( d ˜ τ d ˜ t ) isco = √ 1 -3 / ˜ r isco +2 a/ ˜ r 3 / 2 isco 1 + a/ ˜ r 3 / 2 isco ; (3.7) \ncf. Eq. (5.4.5a) of [10]. \nThis nearly circular motion at ˜ r /similarequal ˜ r isco produces gravitational waves which carry off angular momentum and energy at very nearly the same rate as they would for circular geodesic motion at ˜ r isco . This means that ˜ E and ˜ L evolve in accord with Eqs. (3.5), where \ndξ d ˜ τ = -κη , (3.8) \nand \nκ = 32 5 ˜ Ω 7 / 3 isco 1 + a/ ˜ r 3 / 2 isco √ 1 -3 / ˜ r isco +2 a/ ˜ r 3 / 2 isco ˙ E isco ; (3.9) \n∝ In the transition regime, the body's radial motion is described by the geodesic equation of motion with a radial self force per unit mass † η ˜ F self inserted on the right-hand side: \ncf. Eqs. (2.3), (3.5), (3.7), and Table I. It is the smallness of η ≡ µ/M (e.g., η = 10 -5 for the realistic case of a 10 M /circledot black hole spiraling into the 10 6 M /circledot black hole) that makes the angular momentum ξ evolve very slowly and keeps the body in a nearly circular orbit throughout the transition regime [cf. the factors of η that appear in Eqs. (3.8) and (3.20)-which with Eqs. (3.11) and (3.22) imply d ˜ r/d ˜ τ ∝ η 3 / 5 .] \nd 2 ˜ r d ˜ τ 2 = -1 2 ∂V (˜ r, ξ ) ∂ ˜ r + η ˜ F self . (3.10) \n† This radial self force, like the radiation reaction force that drives the inspiral, is produced by interaction of the body with its own gravitational field-that field having been influenced by the black hole's spacetime geometry; see, e.g., Ref. [6]. The contravariant radial component of the self force, with dimensionality restored using r = M ˜ r and τ = M ˜ τ , is ( dp r /dτ ) self = ( µd 2 r/dτ 2 ) self = ( µ/M )( d 2 ˜ r/d ˜ τ 2 ) self = η 2 ˜ F self . \n(We write it as η ˜ F self because its magnitude is proportional to η = µ/M .) \nThe radial self force η ˜ F self is nondissipative (since it has hardly any radial velocity with which to couple). This contrasts with the φ -directed radiation-reaction force, which couples to the orbital angular velocity to produce a shrinkage of the body's angular momentum [Eq. (3.8)] and a corresponding decrease of its energy, d ˜ E/d ˜ τ = ˜ Ω dξ/d ˜ τ . Because the radial force is nondissipative, it is of little importance. It can be absorbed into the nondissipative effective potential term -1 2 ∂V/∂ ˜ r in the equation of motion. Doing so will not change the general character of the effective potential, as depicted in Fig. 1; it will merely change, by fractional amounts proportional to η , the various parameters that characterize the effective potential: the location ˜ r isco of the innermost stable circular orbit (at which the ξ = 0 effective potential curve has its inflection point), the values at the isco of the orbital energy and angular momentum ˜ E isco and ˜ L isco , and the constant α defined below. There will be other O( η ) changes in ˜ r isco , ˜ E isco , ˜ L isco and α caused by the body's own perturbation of the hole's spacetime geometry [11-13]. In this paper, we shall ignore all such changes, and correspondingly we shall neglect the radial self force η ˜ F self . \nWe shall describe the body's location in the transition regime by \nR ≡ ˜ r -˜ r isco . (3.11) \nThroughout the transition regime both R and ξ are small, and correspondingly the effective potential can be expanded in powers of R and ξ . Up through cubic terms in R and linear terms in ξ (the order needed for our analysis), the effective potential takes the form \nV ( R,ξ ) = 2 α 3 R 3 -2 βRξ +constant , (3.12) \nwhere α and β are positive constants that we shall evaluate below. Note that for ξ = 0, this is a simple cubic potential with inflection point at R = 0, i.e. at ˜ r = ˜ r isco ; and note that for ξ > 0, it acquires a maximum and a minimum, while for ξ < 0 it is monotonic; cf. Fig. 1. By inserting Eq. (3.12) into Eq. (3.10), setting ˜ r = ˜ r isco + R , and neglecting the radial self force or absorbing it into ˜ r isco , α and β as described above, we obtain the following radial equation of motion: \nd 2 R d ˜ τ 2 = -αR 2 + βξ . (3.13) \nBy then setting ˜ τ ≡ 0 at the moment when ξ = 0 and using Eq. (3.8) for the rate of change of ξ , so \nξ = -ηκ ˜ τ , (3.14) \nwe bring our equation of motion into the form \nd 2 R d ˜ τ 2 = -αR 2 -ηβκ ˜ τ . (3.15) \nWe shall explore the consequences of this equation of motion in the next subsection, but first we shall deduce the values of α and β . \nThe constants α and β can be evaluated from the following relations, which follow directly from (3.12), (3.11) and (3.5): \nα = 1 4 ( ∂ 3 V (˜ r, ˜ E, ˜ L ) ∂ ˜ r 3 ) isco , (3.16) \nβ = -1 2 ( ∂ 2 V (˜ r, ˜ E, ˜ L ) ∂ ˜ L∂ ˜ r + ˜ Ω ∂ 2 V (˜ r, ˜ E, ˜ L ) ∂ ˜ E∂ ˜ r ) isco . (3.17) \nBy inserting expression (3.2) into these relations, we obtain α and β in the limit η ≡ µ/M → 0: \nα = 3 ˜ r 6 isco ( ˜ r 2 +2[ a 2 ( ˜ E 2 -1) -˜ L 2 ]˜ r +10( ˜ L -a ˜ E ) 2 ) isco = 1 1296 for a = 0 , (3.18) β = 2 ˜ r 4 isco ( ( ˜ L -a 2 ˜ E ˜ Ω)˜ r -3( ˜ L -a ˜ E )(1 -a ˜ Ω) ) isco = 1 36 √ 3 for a = 0 . (3.19) \nHere ˜ r isco and ˜ Ω isco are given by Eqs. (2.5) and (2.6); and ˜ L isco and ˜ E isco are expressed in terms of ˜ r isco by Eqs. (2.7) and (2.8). Numerical values of α and β , computed from these equations, are tabulated in Table I.", 'C. Solution for Motion in the Transition Regime': 'The equation of motion in the transition regime, Eq. (3.15), can be converted into dimensionless form by setting \nR = η 2 / 5 R o X , ˜ τ = η -1 / 5 τ o T , (3.20) \nwhere \nR o = ( βκ ) 2 / 5 α -3 / 5 , τ o = ( αβκ ) -1 / 5 ; (3.21) \ncf. Table I. The resulting dimensionless equation of motion is \nd 2 X dT 2 = -X 2 -T . (3.22) \nWe seek the unique solution of this differential equation which, at early times T /lessmuch -1, joins smoothly onto the adiabatic inspiral solution of Sec. II. In that adiabatic inspiral, the orbit is the circle at the minimum of the effective potential of Fig. 1 and Eq. (3.12), i.e., the circle at R = βξ/α = √ -βκητ/α , which translates into \n√ \n√ X = √ -T for adiabatic inspiral near the isco. (3.23) \nFIG. 2. Dimensionless orbital radius X as a function of dimensionless proper time T for an orbit near the isco. Adiabatic Inspiral: The analytic solution (3.23) for adiabatic inspiral outside the isco. Transition: The numerical solution to the dimensionless equation of motion (3.22) for the transition regime in the vicinity of the isco. Plunge: The analytic solution (3.25) for the orbital plunge inside the isco. \n<!-- image --> \nFIG. 3. Same as Fig. 2, but drawn on a different scale. \n<!-- image --> \nWe have not been able to find an analytic formula for the solution to the equation of motion (3.22) that joins smoothly onto this adiabatic solution, but it is easy to construct the unique solution numerically. It is plotted in Figures 2 and 3, along with the adiabatic inspiral solution (3.23) and the plunge solution [Eq. (3.25) below]. \nThe transition solution is well approximated by adiabatic inspiral at times T < -1, but at T > -1 it deviates from adiabatic inspiral and evolves smoothly into a plunge. The solution diverges ( X → -∞ ) at a finite time T = T plunge 3 . 412. ‡ \nIn the plunge regime, radiation reaction is unimpor- \n/similarequal \ntant; i.e., the orbit evolves inward with (very nearly) constant orbital angular momentum ˜ L /similarequal ˜ L final and energy ˜ E /similarequal ˜ E final (which we evaluate below); i.e., the orbit is well approximated by geodesic free fall. In the dimensionless equation of motion (3.22), the free-fall approximation translates into neglecting the last term, T , so d 2 X/dT 2 = -X 2 , which has the analytic first integral \ndX/dT = -√ constant -2 3 X 3 . (3.24) \nFor large | X | , the constant can be neglected and we obtain the analytic solution \nX = -6 ( T plunge -T ) 2 for plunge near the isco, (3.25) \nwhich is plotted in Figs. 2 and 3. \nCombining Eqs. (3.5), (3.14), and (3.20), one finds that throughout the transition regime, the energy and angular-momentum deficits (i.e., the deviations of ˜ E and ˜ L from their isco values) scale as η 4 / 5 . In particular, the final deficits in the plunge stage are given by \n˜ L final -˜ L isco = -( κτ 0 T plunge ) η 4 / 5 , ˜ E final -˜ E isco = -˜ Ω isco ( κτ 0 T plunge ) η 4 / 5 , (3.26) \nwhere, as was noted above, \nT plunge = 3 . 412 . (3.27)', 'IV. GRAVITATIONAL WAVES FROM TRANSITION REGIME, AND THEIR OBSERVABILITY': "The gravitational waves emitted in the transition regime are all near the orbital frequency 2 π Ω isco and its harmonics. The strongest waves are at the second harmonic (twice the orbital frequency): \nf /similarequal 2 Ω isco 2 π = ˜ Ω isco πM . (4.1) \nin the Taylor expansion (3.12) become important and stop the divergence. Well before this (in fact, throughout the range 1 /lessmuch -X /lessmuch X break ) both the transition approximation and the free-fall approximation ( ˜ E = ˜ E final = constant, ˜ L = ˜ L final = constant) are valid, so these two approximations can be matched in this regime to obtain a solution valid all the way down to the horizon. The same type of breakdown also occurs at the other asymptotic limit + X > ∼ X break : The transition regime's adiabatic-inspiral equation (3.23) breaks down and must be replaced, via matching at 1 /lessmuch X /lessmuch X break , by the exact Kerr metric's adiabatic inspiral formulae [4]. \nWe shall compute their properties. \nThe transition waves last for a proper time ∆ τ = M ∆˜ τ = Mη -1 / 5 ˜ τ o ∆ T , during which the body spirals inward through a radial distance ∆ r = M ∆ R = Mη 2 / 5 R o ∆ X , where ∆ T covers the range T /similarequal -1 to /similarequal 2 . 3 and ∆ X covers the range X /similarequal 1 to X /similarequal -5 (Fig. 3); i.e., \n∆ T = 3 . 3 , ∆ X = 6 . (4.2) \nCorrespondingly, neglecting any cosmological redshift, the duration of the transition waves as seen at Earth is \n∆ t = M ( d ˜ τ/d ˜ t ) isco η -1 / 5 ˜ τ o ∆ T , (4.3) \nand their frequency band is ∆ f = (1 /πM )( d ˜ Ω /d ˜ r ) isco ∆˜ r , which, using the above expression for ∆ r and Eq. (2.1) for ˜ Ω(˜ r ), gives \n∆ f = 3 2 πM ˜ Ω 2 isco √ ˜ r isco η 2 / 5 R o ∆ X . (4.4) \nThe total number of cycles of these transition waves is \nN cyc = f ∆ t = ˜ Ω isco ˜ τ o π ( d ˜ τ/d ˜ t ) isco η -1 / 5 ∆ T . (4.5) \nThese second-harmonic waves arriving at Earth have the form h + = h +amp cos(2 π ∫ fdt + ϕ + ), h × = h × amp cos(2 π ∫ fdt + ϕ × ), where ϕ + and ϕ × are constant phases. The amplitudes h +amp and h × amp depend on the source's orientation. When one squares and adds these amplitudes and then averages over the sky (' 〈 ... 〉 '), one obtains an rms amplitude \nh rms amp = 〈 h 2 +amp + h 2 × amp 〉 1 / 2 , (4.6) \nwhich is related to the power being radiated into the second harmonic by ˙ E 2 = 4 πD 2 ( h rms amp ) 2 (2 πf ) 2 / (32 π ); cf. Eq. (35.27) of MTW [7]. Here D is the distance to the source. Equating this to the radiated power ˙ E 2 = (32 / 5) η 2 ˜ Ω 10 / 3 isco ˙ E ∞ , 2 [4], where ˙ E ∞ , 2 is a relativistic correction factor listed on the first line of Table IV of [4], we obtain the following expression for the waves' rms amplitude \nh rms amp = 8 √ 5 Mη D ˜ Ω 2 / 3 isco √ ˙ E ∞ , 2 . (4.7) \nThe signal to noise ratio S/N that these waves produce in LISA depends on the orientations of LISA and the source relative to the line of sight between them. When one squares S/N and averages over both orientations, then takes the square root, one obtains [14] \n( S N ) rms = h rms amp √ 5 S h ( f ) / ∆ t . (4.8) \nHere 5 S h ( f ) is the spectral density of LISA's strain noise inverse-averaged over the sky § and 1 / ∆ t is the band width associated with the waves' duration ∆ t . \nThe noise spectral density S h ( f ) for the current strawman design of LISA has been computed by the LISA Mission Definition Team [15]. An analytic fit to this S h ( f ), after averaging over some small-amplitude oscillations that occur at f > 0 . 01 Hz, is the following: \nS h ( f ) = [ (4 . 6 × 10 -21 ) 2 +(3 . 5 × 10 -26 ) 2 ( 1Hz f ) 4 +(3 . 5 × 10 -19 ) 2 ( f 1Hz ) 2 ] Hz -1 . (4.9) \nThe rate for µ ∼ 10 M /circledot black holes to spiral into M ∼ 10 6 M /circledot black holes in galactic nuclei has been estimated by Sigurdsson and Rees [2]; their 'very conservative' result is ∼ one event per year out to 1 Gpc. The inspiraling holes are likely to be in rather eccentric, nonequatorial orbits [16], for which our analysis needs to be generalized. If, however, the orbit is circular and equatorial and the holes are at 1 Gpc distance, then the above formulas give the numbers shown in Table II. \nAs shown in the table, the signal to noise for this source is of order unity. With some luck in the orientation of LISA, the orientation of the source, the distance to the source, and/or the holes' masses, a S/N of a few might occur. Since the signal would already have been detected from the much stronger adiabatic inspiral waves, this signal strength could be enough to begin to explore the details of the transition from inspiral to plunge.", 'V. CONCLUSIONS': "Our analysis of the transition regime has been confined to circular, equatorial orbits. This is a serious constraint, since there is strong reason to expect that most inspiraling bodies will be in orbits that are strongly noncircular and nonequatorial [16]. Our estimated signal-to-noise ratio, S/N ∼ 1, for LISA's observations of the transition \n§ i.e., 1 / (5 S h ) ≡ average over the sky of 1/(spectral density). The factor 5 in this definition is to produce accord with the conventional notation for ground-based interferometers, where S h ( f ) denotes the spectral density for waves with optimal direction and polarization. In the case of LISA, at frequencies above about 0.01 Hz, the beam pattern shows sharp frequency-dependent variations with direction due to the fact that the interferometer arms are acting as one-pass delay lines rather than optical cavities, and this produces a more complicated dependence of sensitivity on angle than for ground-based interferometers. As a result, S h (as we have defined it) is the spectral density for optimal direction and polarization only below about 0.01 Hz, not above. \nregime from a circular, equatorial orbit at the plausible distance ∼ 1 Gpc suggests that for more realistic orbits the transition regime might be observable. This prospect makes it important to generalize our analysis to more realistic orbits. \nFull analyses for equatorial, noncircular orbits and for nonequatorial, circular orbits can be carried out using techniques now in hand: the Teukolsky formalism, and computations of the orbital evolution based on the energy and angular momentum radiated down the hole and off to infinity (see, e.g., Ref. [5] and references therein). For nonequatorial, noncircular orbits, the analysis should also be possible with existing techniques - up to an unknown radiation-reaction-induced rate of evolution of the Carter constant. That unknown quantity could be left as a parameter in the analysis, to be determined when current research on gravitational radiation reaction [17,18,6] has reached fruition. \nWhen this paper was in near final form, we became aware of a similar analysis, by Buonanno and Damour [19], of the transition from inspiral to plunge. Whereas we treat the case of infinitesimal mass ratio η /lessmuch 1 and finite black-hole spin -1 < a < +1, Buananno and Damour treat finite η (0 < η ≤ 1 / 4) and vanishing spins a = 0. Both analyses give the same dimensionless equation of motion (3.22) for the transition regime.", 'ACKNOWLEDGMENTS': "We thank James Anderson for helpful discussions, Theocharis Apostolatos and Richard O'Shaughnessy for checking some details of our analysis, and Sam Finn for helpful advice and for permission to use unpublished results from his numerical solutions of the Teukolsky equation; see Tables I and II. This work was supported in part by NASA grant NAG5-6840, and in view of its potential applications to LIGO, also by NSF grant AST-9731698. \n- [1] For details of the LISA mission see http://lisa.jpl.nasa.gov/.\n- [2] S. Sigurdsson and M. J. Rees, Mon. Not. R. Astron. Soc. 284 , 318 (1997).\n- [3] S. Sigurdsson, Class. Quant. Grav. 14 , 1425 (1997)\n- [4] L. S. Finn and K. S. Thorne, paper in preparation.\n- [5] S. A. Hughes, Phys. Rev. D, in press; astro-ph/9909269.\n- [6] L. Barack and A. Ori, Phys. Rev. D, submitted; grqc/9912010.\n- [7] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (W. H. Freeman, San Francisco, 1973).\n- [8] J. M. Bardeen, W. H. Press, and S. A. Teukolsky, Astrophys. J. 178 , 347 (1972); J. M. Bardeen, in Black Holes ,\n- eds. C. DeWitt and B. S. DeWitt (Gordon and Breach, New York, 1973), p. 241.\n- [9] Sec. 10.7 of Ya. B. Zel'dovich and I. D. Novikov, Relativistic Astrophysics, vol. I, Stars and Relativity (U. Chicago Press, Chicago, 1971); also available as a Dover Reprint Volume.\n- [10] I. D. Novikov and K. S. Thorne, in Black Holes , eds. C. DeWitt and B. S. DeWitt (Gordon and Breach, New York, 1973), p. 343.\n- [11] L. E. Kidder, C. M. Will and A. G. Wiseman, Phys. Rev. D 47 , 3281 (1993).\n- [12] T. Damour, B. R. Iyer and B. S. Sathyaprakash, Phys. Rev. D 57 , 885 (1998).\n- [13] A. Buonanno and T. Damour, Phys. Rev. D 60 , 023517 (1999).\n- [14] K. S. Thorne, in Three Hundred Years of Gravitation , eds. S. W. Hawking and W. Israel (Cambridge University Press, Cambridge U. K. 1987), p. 330.\n- [15] LISA Mission Definition Team and LISA Science Study Team, LISA: Laser Interferometer Space Antenna (March Press, Boulder, CO, 1999). The spectral density S h ( f ), as we have defined it, is related to the noise curve h noise ( f ) shown in the figure on page 6 by S h = 3 . 156 × 10 7 ( h noise ) 2 / 125 (P. Bender, private communication). Here 3 . 156 × 10 7 is the number of seconds in a year, and 125 = (5 √ 5) 2 where the √ 5 comes from averaging the noise over the sky and the 5 from the fact that the figure in this document is for a signal to noise ratio of 5.\n- [16] D. Hils and P. L. Bender, Astrophys. J. 445 , L7 (1995); Y. Mino, M. Sasaki, M. Shibata, H. Tagoshi, and T. Tanaka, Prog. Theor. Phys. Suppl. 128 , chapter 1 (1998).\n- [17] Y. Mino, M. Sasaki and T. Tanaka, Phys. Rev. D 55 , 3457 (1997).\n- [18] T. C. Quinn and R. M. Wald, Phys. Rev. D, 56 , 3381 (1997).\n- [19] A. Buonanno and T. Damour, Phys. Rev. D, submitted; gr-qc/0001013. \nTABLE I. Dimensionless parameters characterizing the isco and the transition regime of inspiral. The values of ˙ E are from numerical solutions of the Teukolsky equation by L. S. Finn (first line of Table II of Ref. [4]).TABLE II. Properties of the second-harmonic, transition-regime gravitational waves from a µ = 10 M /circledot black hole spiraling into a M = 10 6 M /circledot black hole (so η = µ/M = 10 -5 ) at r = 1 Gpc distance. The values of ˙ E ∞ , 2 are from numerical solutions of the Teukolsky equation by L. S. Finn (first line of Table IV of Ref. [4]). \n| a | ˜ r isco | ˜ Ω isco | ˙ E | α | β | κ | R o | τ o |\n|--------|------------|------------|--------|-----------|----------|----------|--------|--------|\n| -0.99 | 8.972 | 0.03863 | 1.24 | 0.0001543 | 0.006626 | 0.005013 | 3.129 | 45.5 |\n| -0.9 | 8.717 | 0.04026 | 1.233 | 0.0001732 | 0.00707 | 0.005527 | 3.117 | 43.04 |\n| -0.5 | 7.555 | 0.04935 | 1.197 | 0.000307 | 0.00973 | 0.008966 | 3.048 | 32.69 |\n| 0 | 6 | 0.06804 | 1.143 | 0.0007716 | 0.01604 | 0.01955 | 2.925 | 21.05 |\n| 0.2 | 5.329 | 0.07998 | 1.114 | 0.00124 | 0.02057 | 0.02914 | 2.852 | 16.82 |\n| 0.5 | 4.233 | 0.1086 | 1.053 | 0.003115 | 0.0327 | 0.06291 | 2.687 | 10.93 |\n| 0.8 | 2.907 | 0.1737 | 0.9144 | 0.01401 | 0.06446 | 0.2123 | 2.326 | 5.539 |\n| 0.9 | 2.321 | 0.2254 | 0.7895 | 0.03447 | 0.09039 | 0.4214 | 2.041 | 3.77 |\n| 0.99 | 1.454 | 0.3644 | 0.4148 | 0.2234 | 0.1289 | 1.531 | 1.284 | 1.867 |\n| 0.999 | 1.182 | 0.4379 | 0.2022 | 0.5127 | 0.09568 | 2.594 | 0.8551 | 1.51 | \n| a | f , Hz | ∆ f f | ∆ t , sec | N cyc | ˙ E ∞ , 2 | h rms amp | ( S N ) rms , 10 - |\n|--------|----------|---------|-------------|---------|-------------|-------------|----------------------|\n| -0.99 | 0.002496 | 0.033 | 9300 | 23 | 1.029 | 2 | 1.2 |\n| -0.9 | 0.002601 | 0.033 | 8800 | 23 | 1.02 | 2 | 1.2 |\n| -0.5 | 0.003188 | 0.037 | 7000 | 22 | 0.9734 | 2.3 | 1.4 |\n| 0 | 0.004396 | 0.044 | 4800 | 21 | 0.8957 | 2.7 | 1.6 |\n| 0.2 | 0.005167 | 0.047 | 4100 | 21 | 0.8535 | 2.9 | 1.6 |\n| 0.5 | 0.007016 | 0.054 | 2900 | 21 | 0.7653 | 3.4 | 1.6 |\n| 0.8 | 0.01123 | 0.062 | 1900 | 22 | 0.5914 | 4.1 | 1.3 |\n| 0.9 | 0.01457 | 0.063 | 1700 | 24 | 0.4617 | 4.3 | 1.1 |\n| 0.99 | 0.02354 | 0.051 | 1800 | 43 | 0.1656 | 3.6 | 0.72 |\n| 0.999 | 0.02829 | 0.037 | 3400 | 96 | 0.06128 | 2.4 | 0.58 |"}
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2013JHEP...10..107V
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Black hole entanglement and quantum error correction
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2013-01-01
| 36 | 0.45 | 159 |
['methods statistical', 'black hole physics', '-', '-']
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It was recently argued in [1] that black hole complementarity strains the basic rules of quantum information theory, such as monogamy of entanglement. Motivated by this argument, we develop a practical framework for describing black hole evaporation via unitary time evolution, based on a holographic perspective in which all black hole degrees of freedom live on the stretched horizon. We model the horizon as a unitary quantum system with finite entropy, and do not postulate that the horizon geometry is smooth. We then show that, with mild assumptions, one can reconstruct local effective field theory observables that probe the black hole interior, and relative to which the state near the horizon looks like a local Minkowski vacuum. The reconstruction makes use of the formalism of quantum error correcting codes, and works for black hole states whose entanglement entropy does not yet saturate the Bekenstein-Hawking bound. Our general framework clarifies the black hole final state proposal, and allows a quantitative study of the transition into the "firewall" regime of maximally mixed black hole states.
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https://arxiv.org/pdf/1211.6913.pdf
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{'Quantum Error Correction': 'Erik Verlinde 1 ∗ and Herman Verlinde 2 , 3 † \n1 Institute for Theoretical Physics, University of Amsterdam, Amsterdam, The Netherlands \n2 Department of Physics, Princeton University, Princeton, NJ 08544, USA \n2 Princeton Center for Theoretical Science, Princeton, NJ 08544, USA', 'Abstract': "It was recently argued in [1] that black hole complementarity strains the basic rules of quantum information theory, such as monogamy of entanglement. Motivated by this argument, we develop a practical framework for describing black hole evaporation via unitary time evolution, based on a holographic perspective in which all black hole degrees of freedom live on the stretched horizon. We model the horizon as a unitary quantum system with finite entropy, and do not postulate that the horizon geometry is smooth . We then show that, with mild assumptions, one can reconstruct local effective field theory observables that probe the black hole interior, and relative to which the state near the horizon looks like a local Minkowski vacuum. The reconstruction makes use of the formalism of quantum error correcting codes, and works for black hole states whose entanglement entropy does not yet saturate the Bekenstein-Hawking bound. Our general framework clarifies the black hole final state proposal, and allows a quantitative study of the transition into the 'firewall' regime of maximally mixed black hole states. \nOctober 2012", 'Contents': '| 1 | Introduction | 2 |\n|-------------------------------|------------------------------------------------------------------------------------|-----|\n| 2 | Setting the Stage | 4 |\n| 2.1 | Black Holes States: Old and Young . . . . . . . . . . . . . . . . . . . . . . . | 4 |\n| 2.2 | The Firewall Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 6 |\n| 2.3 | Evaporation in the Interaction Picture . . . . . . . . . . . . . . . . . . . . . | 10 |\n| 2.4 | Black Hole Entanglement Revisited . . . . . . . . . . . . . . . . . . . . . . . | 15 |\n| 3 Passing through the Horizon | | 18 |\n| 3.1 | Quantum Error Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 18 |\n| 3.2 | Construction of the Interior Operators . . . . . . . . . . . . . . . . . . . . . | 22 |\n| 3.3 | Ancilla versus Black Hole Final State . . . . . . . . . . . . . . . . . . . . . . | 26 |\n| 3.4 . | Fidelity and Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 28 |\n| 3.5 | Approaching the Firewall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . | 30 |\n| 4 Summary | 4 Summary | 34 |', '1 Introduction': "The question of how black holes absorb, store, share and release quantum information remains among the most vexing mysteries in theoretical physics [2, 3]. Logical consistency of quantum black hole physics with the Bekenstein-Hawking entropy [4, 2] imposes tight constraints on the microscopic theory. The discovery of the holographic principle [5][6] and its realization in string theory [7] suggests that black holes satisfy all rules of quantum mechanics. The principle of black hole complementarity [8] is designed to reconcile the seemingly conflicting observations of an outside observer and an infalling observer. It postulates that \n- i) black hole formation and evaporation are described via unitary quantum evolution\n- ii) the region outside the stretched horizon is well described by QFT in curved space\n- iii) to an outside observer, the stretched horizon behaves like a quantum mechanical membrane with microscopic entropy bounded by the Bekenstein-Hawking formula\n- iv) an infalling observer can cross the event horizon without encountering any trouble. \nIn a recent paper [1], Almheiri et al formulated an interesting counter-argument against the simultaneous validity of these four assumptions. The reasoning of [1], referred to as the 'firewall argument', goes as follows. At late times the quantum state of the black hole is maximally entangled with the early radiation. Because entanglement can not be shared, and the information eventually has to come out, the late radiation can not be entangled with the black hole. Since the latter property is believed to be a necessary requirement for the quantum state to look like the local Minkowski vacuum near the horizon, one seems forced to conclude that the infalling observer can not pass through the horizon without experiencing dramatic events. This provocative conclusion has led to a lively debate [9][10]. \nThe firewall argument is the old information paradox in reverse. The original paradox was that locality and the smoothness of the horizon seems to imply that quantum mechanics falls short in describing black hole formation and evaporation. So if instead one adopts the rules of quantum mechanics and quantum information theory, smoothness of the black hole horizon becomes the real mystery [11]. The value added by the reasoning of [1], and of other recent [12, 17, 13, 14] and less recent [15, 16] works, is that it aims to bring the paradox more sharply into focus, by adopting a more precise quantum information theoretic language. \nThe ramifications of the firewall argument are somewhat disturbing, since it could endanger the validity of Hawkings derivation of the evaporation process, which stands at the very basis of this scientific discussion. Therefore, before accepting its conclusion, one should carefully examine the logic and assumptions that go into it. This requires a precise enough framework, that preserves all the essential physics and elements of the paradox. \nMotivated by this challenge, we develop a systematic description of the evaporation process that keeps the rules of quantum mechanics manifest. The idea is to treat the black hole as an ordinary quantum system that interacts with a radiation field through absorption and emission, much like an atom would. We will not adopt postulate iv), that the horizon is smooth, but instead we will investigate if we can derive postulate iv) from the other three postulates. The argument of [1] indicates that this derivation may be possible for young black holes, that do not yet saturate the Bekenstein-Hawking bound, but should break down for old black holes that do saturate the bound. \nBesides the postulates i), ii) and iii), our approach relies on one additional physical assumption: we assume that the quantum mechanical amplitudes, that describe the transition between different black hole states as a result of the emission of Hawking quanta, are ergodic matrices [17]. Their detailed form depends on unknown Planck scale physics, but their coarse grained properties are determined via standard thermodynamic reasoning. \nAs another helpful tool, we will make use of the parallel, already exploited in [12], between the physics of black hole radiation and the loss of coherence of a quantum computer, due to its interaction with a noisy environment. This analogy allows one to apply inventive theoretical techniques of quantum information theory [22, 23], such as quantum error correcting codes (QECC's), to organize the internal state of the black hole [12]. QECC's are designed to counteract the quantum information loss. We will show that in the black hole context, the same technique can be used to safeguard the smoothness of the horizon. It enables the construction of microscopic observables, that play the role of Hawking partners of the outside radiation, relative to which the black hole state looks like a local horizon vacuum state. Smoothness of the horizon and the Hawking evaporation process are thus consistent with each other, and with the laws of quantum mechanics - at least for black hole states that do not saturate the BH entropy. \nQEC codes are not full proof [23]. As long as the black hole state has a less-thanmaximal entropy, the code works with high fidelity: the local horizon geometry is smooth, and effective QFT is valid on both sides. When the black hole starts saturating the BH entropy bound, however, the QEC procedure becomes less reliable and sometimes fails this are the instances when quantum information is leaking out [12]. Since the code is also responsible for the reconstruction of semi-classical interior observables, the effective QFT description of the black hole interior breaks down in this limit. \nIn section 2, we state our man assumptions and develop an interaction picture for black hole evaporation. In section 3, we present the reconstruction of the black hole interior via QEC technology, and analyze the firewall limit. For the well-informed reader that prefers the short version of our story, we have summarized our main arguments in section 4.", '2 Setting the Stage': 'In this section and the next, we will describe the basic setup and main assumptions. Our guiding principle is that a black hole is an ordinary quantum mechanical system, - much like an atom, albeit a very large one and with an enormous level density of states - that evolves and interacts with its environment via standard hamiltonian evolution. We start by setting up some notation that will help us model the evaporation process in a way that keeps the rules of quantum mechanics manifestly intact, in particular it obeys unitarity. Along the way, we present a version of the firewal argument, that can be directly addressed and tested within our framework.', '2.1 Black Holes States: Old and Young': "Consider a macrosopic black hole. Suppose that we have measured its mass M with some high accuracy. For now, we will ignore the uncertainty in M , and any other macroscopic properties such as angular momentum and charge. We associate a Hilbert space H M to the black hole with mass M , whose states \n∣ i 〉 ∈ H M (1) \n∣ \nare indistinguishable from a macroscopic perspective. Each state ∣ ∣ i 〉 describes the interior of a black hole, with microscopic properties that are hidden from the outside observer by the stretched horizon. The number of black holes micro-states is determined by the Bekenstein Hawking entropy 1 \nN = dim H M = e S BH , (2) \nwith S BH = 4 πM 2 . We will assume that the time evolution of every micro state describes a black hole that slowly evaporates via the Hawking process. So as time progresses, every state ∣ ∣ i 〉 ∈ H M will evolve into a new state given by the product of a micro-state of a black hole with smaller mass M ' = M -E and a radiation state with energy E . \nTo describe this evaporation process we need to enlarge our Hilbert space, so that it incorporates all possible intermediate situations. So we introduce the total Hilbert space \nH , given by the tensor product \nH = H BH ⊗H R , (3) \nwhere H BH is the space spanned by all black hole states with mass less than M \nH BH = ⊕ E ≤ M H M -E , (4) \nand H R is the Hilbert space of the radiation field outside of the stretched horizon. In the next subsection, we will set up a general description of the evaporation process as a unitary Hamiltonian evolution on the total Hilbert space (3). Note that the black hole states ∣ ∣ i 〉 are not part of the asymptotic past and future Hilbert space, but are to be regarded as very long lived resonances. It is standard in quantum mechanics to include resonances as an additional Hilbert space sector, even if they do not appear as true stable asymptotic states. Unitarity is preserved under time evolution, because the resonant black hole states eventually all evolve into the unique zero mass ground state ∣ ∣ 0 〉 BH times an asymptotic radiation state. We first summarize what one would expect the time evolution to look like from a somewhat coarse grained perspective. \nFirst consider a young black hole of mass M . Let us imagine that it was formed in a pure state ∣ ∣ i 〉 . The total state of the black hole with its environment then takes the form \n∣ Ψ young 〉 = ∣ ∣ i 〉 ∣ Φ 〉 . (5) \n∣ Here ∣ ∣ Φ 〉 denotes the state of the environment. Since the black hole is assumed to be formed in a pure state, there is no entanglement between the black hole and its environment. Tracing over the environment yields the pure state density matrix \n∣ \nρ young BH = ∣ ∣ i 〉〈 i ∣ ∣ . (6) \n∣ \nNext consider an old black hole of mass M that has formed in the far past and has evaporated for a very long time. At this late stage, the black hole and radiation are described by a very-close-to-maximally entangled quantum state of the form \n∣ ∣ Ψ old 〉 = 1 √ N ∑ i ∣ ∣ i 〉 ∣ ∣ Φ i 〉 . (7) \nHere the sum runs over all basis states ∣ ∣ i 〉 of the Hilbert space H M . The precise form of \nthe states ∣ ∣ Φ i 〉 ∈ H R depends on the early history of the black hole. Since the dimension of H R is much larger than that of H M , the states ∣ ∣ Φ i 〉 can be safely assumed to form an orthonormal set. Hence when we trace the density matrix ρ = | Ψ 〉〈 Ψ | over the radiation Hilbert space H R , the resulting black hole density matrix ρ BH becomes, to a very good approximation, proportional to the identity matrix on the Hilbert space H M \nρ old BH = 1 N ∑ i ∣ ∣ i 〉〈 i ∣ ∣ ≡ 1 N 1 M . (8) \nIn the following, we will consider the quantum states of both old as well as young black holes and take them as our initial condition at some time t = 0. Since hamiltonian evolution is a linear process one can express the density matrix of an entangled black hole at time t in terms of the evolved quantum states of initially unentangled black holes. We will make use of this fact to study the evolution of the density matrix ρ BH as the black hole continues to evaporate, and to see if it is possible to uncover an effective description of the semi-classical vacuum state near the horizon, as seen by an infalling observer. Note that even for young black holes this is not automatic, since we have not assumed the existence of an interior part. One of the main results of our paper is that this can indeed be done, with only a minimal set of assumptions. \nThe maximally entangled old black holes and completely unentangled young black holes are the extreme limits of a broad class of intermediate cases going from slightly to almost maximally entangled situations. We will also consider these intermediate cases, since they are more realistic. Indeed one could argue that even young black holes can never be close to a pure state. As emphasized e.g. in [18], in order for two different space time regions to be connected parts of the same space time, these regions have in terms of the microscopic physics to be maximally entangled. So in this sense, even young black holes may already come close to saturating the BH entropy bound.", '2.2 The Firewall Argument': "We now briefly summarize one version of the argument in [1]. Consider the old black hole. Following [1], we factorize the total Hilbert space into an early and late part \nH = H late ⊗H early R . (9) \nHere H early R contains all radiation emitted during the early life of the black hole before t = 0. The late Hilbert space H late decomposes into the product of the Hilbert space of black hole \nstates with mass M ' < M , times the Hilbert space of all radiation that will be emitted at late times t > 0. For notational convenience we will denote the two tensor factors of H late by \nH late = H A ⊗H B , H A = H BH , H B = H late R . (10) \nIndeed, the firewall argument involves an Alice and Bob gedanken experiment. Alice passes through the horizon at some time t = τ > 0 and observes the interior black hole state in H A . Bob, on the other hand, initially stays outside and then falls through the horizon a small time later. He then catches up with Alice to confirm that she safely fell through the horizon. Suppose he succeeds and concludes that both are unharmed, for now. Why is this a problem? To explain the issue, we need a bit more notation. \nAt t = τ , the moment that Alice falls in, the black hole state (8) has evolved into a mixed state on the tensor product Hilbert space H late and also contains some radiation. Let us denote the orthonormal basis of late radiation states by \n∣ n 〉 ∈ H B . \n∣ \n∣ ∣ n 〉 b = ∏ ω 1 √ n ω ! ( b † ω ) n ω ∣ ∣ 0 〉 b . (11) \nThe states ∣ ∣ n 〉 span the Hilbert space of a quantum field theory in the curved space-time background in the neighborhood of the black hole. In what follows, we will not need any detailed information about this QFT: all our arguments and calculations will go through regardless of whether it is strongly interacting or free. But for concreteness, consider the case of a free field theory. We can then expand the fields as usual in terms of creation and annihilation modes via φ ( t ) = ∑ ω 1 √ ω ( b † ω e iωt + b ω e -iωt ) . Our short hand notation ∣ ∣ n 〉 is then short-hand for the usual particle number eigenstates \nIn particular, the states ∣ n 〉 b are energy eigenstates of the Schwarzschild Hamiltonian H . \n∣ 〉 The central player in the firewall argument, and our main object of study, is the density matrix of the black hole and late radiation at time t = τ > 0. It is defined on the tensor product Hilbert space H A ⊗H B and takes the general form \nρ AB = ∑ n , m ρ nm ∣ ∣ n 〉 b 〈 m ∣ ∣ . (12) \n∣ \nThe partial density matrix ρ nm acts only on H A , and specifies the correlation between the black hole state and external radiation. We do not have much a priori information about its detailed form, although in the following we will find out quite a bit about its coarse grained properties. Indeed, we have one important piece of knowledge: thanks to Hawking, or more directly (given that we already postulated the size of the BH Hilbert space) by application of standard rules of statistical physics, we know that if we take the partial trace of ρ AB over the internal Hilbert space, ρ B = tr A ( ρ AB ) , we obtain a thermal density matrix \nwith Z = ∑ n e -βE n and β = 8 πM the inverse Hawking temperature. Here we have normalized the Boltzman weights W n such that \nρ B = ∑ n W n ∣ ∣ n 〉 b 〈 n ∣ ∣ ; W n = e -βE n Z , (13) \ntr B ( ρ B ) = ∑ n W n = 1 . (14) \nNote that equation (13), although it looks static, in fact represents a time dependent state, by virtue of the fact that the level density of allowed frequencies ω increases inversely proportional with the time interval between 0 and τ . So (13) describes a radiation state with total average energy ¯ E = ∑ n E n W n that increases linearly with time τ . In the following we will assume that the time τ is short compared to the black hole life time, so that ¯ E glyph[lessmuch] M . \nThe statistical mechanical derivation of the density matrix (13) makes use of the fact that the number N n of microscopic black hole states after the emitting radiation equals \ndim H M -E n = N n = Ne -βE n , \nwhile the total number of states in H A ⊗H B with total mass M is given by \ndim( H A ⊗H B ) M = ∑ n dim H M -E n = NZ. (15) \nEquation (13) thus follows from the fact that each black hole state is occupied with the same uniform probability. \nLet us now return to the firewall argument [1]. The initial condition that the black hole starts at t = 0 in the maximally entangled state (8) heavily restricts the possible form of the density matrix at the later time t = τ of the black hole and the radiation that was emitted during that time. It was argued in [1] that, after letting the state evolve and emit \nan extra segment of radiation, the maximally mixed form is essentially preserved. Suppose that this is true. One then arrives at the following form of ρ nm : \nρ AMPS nm = 1 NZ δ nm 1 M -E n . (16) \nNow comes the firewall argument: a density matrix of the factorized form (16) is clearly inconsistent with the assumption that Alice can fall through the horizon without disturbance. A direct way to see the problem is to compute the mutual information \nI AB = S A + S B -S AB (17) \nbetween the interior Hilbert space H A and the outside Hilbert space H B . Using eqns (16) and (13), one finds \nS AMPS AB = log( NZ ) , S A = log N -β ¯ E, S B = log Z + β ¯ E, (18) \nwhere ¯ E = ∑ n W n E n is the average amount of energy contained in the late radiation. Notice the appearance of the extra term log Z = -βF , with F the free energy of the radiation. 2 We will comment on its significance in a moment. \nEquation (18) tells us that the black hole interior and the late radiation share zero mutual information I AMPS AB = S A + S B -S AMPS AB = 0 . A local vacuum state near the black hole horizon, on the other hand, encodes measurable entanglement between modes on each side. The density matrix (16) evidently does not. The authors of [1] argue that this discrepancy is a logical consequence of quantum information theory: since all black hole degrees of freedom are entangled with the early radiation, and entanglement can only be shared once, there is no room left for entanglement between the black hole interior and the late radiation. This contradiction led the authors of [1] to conclude that the horizon of an old black hole can not be a smooth region of space time. \nThe above version of the firewall argument is not completely precise. In particular, the maximally mixed state (16) is not consistent with unitarity. To see this, note that the entropy S AMPS AB = log NZ of the state (16) is larger, by an extra term of log Z , than the BH entropy S BH = log N of the initial state (8). The origin of the extra term is easily understood. The time evolution takes place in Hilbert space H late = H A ⊗ H B which is much bigger than the interior Hilbert space H A = H BH by itself. The initial state (8) \nthus spreads out into an enlarged Hilbert space, and its coarse grained entropy gains an extra contribution proportional to the associated Helmholtz free energy F . Unitary time evolution, on the other hand, preserves microscopic entropy. So (16) can not be exactly equal to the density matrix (8) evolved to a later time t = τ . There must be off diagonal corrections that restore the unitarity. \nWe thus have found a small gap in the reasoning of [1]. In [1] the black hole and emitted Hawking radiation combined were treated as if it forms a closed system, with a Hilbert space of some fixed size N . However, as the above simple analysis indicates, such a framework is too restrictive to describe the Hawking emission process. Eqns (13) and (12) only make sense as relations on a tensor product space H late = H A ⊗H B on which the b -modes can act independently. The local environment of the black hole, into which it slowly releases radiation and information, has a separate Hilbert space that is spanned by states that do not evolve from the black hole states themselves. A radiating black hole is not a closed system, but an open system, just like a radiating atom. Monogamy of entanglement holds in closed systems, but becomes less restrictive in open systems.", '2.3 Evaporation in the Interaction Picture': 'Clearly, we need a more precise framework to model the evaporation of the black hole states (6) and (8) in a way that manifestly preserves unitarity. We have already characterized a Hilbert space. We also need to characterize our Hamiltonian that describes the time evolution. We will choose to describe the time evolution in terms of the Schwarzschild time t , the time coordinate used by an outside observer. \nWe first consider time evolution on the full Hilbert space, including the early radiation. In accordance with the decomposition (3), we write the Hamiltonian H as \nH = H BH + H R + H int , (19) \nwhere H BH acts only on interior black hole states, H R governs the evolution of the radiation outside the stretched horizon. H int represents the interaction Hamiltonian, that is responsible for the Hawking radiation process. The precise form of all terms in the Hamiltonian depend on unknown micro-physical details. We will therefore make only minimal assumptions about the specific form of each term. All that we need to know is that H BH and H R each act within the respective Hilbert spaces, and that H int acts on the tensor product in a way that allows the two sectors to reach a quasi-static thermal equilibrium. \nWe can now use the familiar interaction picture, in which the time dependence of op- \nFigure 1: A stationary element of the Hilbert space on which the interaction Hamiltonian acts is a tensor product of an eternal black hole state ∣ ∣ i 〉 A with a state representing the radiation outside the stretched horizon. The radiation state at t = 0 factorizes into a product of a state ∣ ∣ Φ i 〉 early containing the early radiation sand the vacuum state ∣ ∣ 0 〉 B of the late radiation. \n<!-- image --> \nerators is governed by the free Hamiltonian H BH + H R , and states evolve through the interaction Hamiltonian H int . Following standard time-dependent perturbation theory, we may then expand the time-dependent states ∣ ∣ Ψ( t ) 〉 as linear combinations, with time dependent coefficients, of static eigenstates of the non-interacting Hamiltonian. The static states are of the form ∣ ∣ j 〉 | n 〉 where ∣ ∣ j 〉 ∈ H BH is a microstate of a non-radiating eternal black hole of mass M -E n , that exist in the fiducial reference world in which the interaction Hamiltonian has been turned off 3 , and ∣ ∣ n 〉 ∈ H R represents a static radiation state with energy E n that lives outside the stretched horizon of the static eternal black hole. This sitation is depicted in the left diagram in fig. 1. \nFollowing [1], we assume that the time evolution of states in H R are well described by ordinary quantum field theory on curved space-time, and that all interactions with the black hole microstates take place at the stretched horizon. In the context of the interaction picture this means that the time evolution of the radiation field can be represented in the static black hole background. Thus, all complications of the evaporation process, including the backreaction, are contained in the time evolution of the quantum states generated by the interaction Hamiltonian H int ( t ). \nTo proceed, we need to take slightly better look at the radiation states ∣ ∣ Φ i 〉 . Superficially, ∣ ∣ Φ i 〉 specifies initial conditions only for radiation on a spatial slice outside the stretched horizon. However, not all modes in the outside region need to pass through this \n∣ \nslice. As indicated in fig. 1, the stretched horizon is a semi-permeable membrane from which right-moving modes can emanate. Hence, the Cauchy data need to include initial conditions at or inside the future stretched horizon. The natural and only consistent choice is to impose vacuum boundary conditions, as measured in the Schwarzschild time, for the outgoing modes at t = 0. In this way we guarantee that the mass M -E n of the eternal black hole is all contained in the interior part ∣ j of the wave function. \n∣ From a semi-classical space-time perspective, it looks perhaps a bit suspect to assume that the radiation modes start out in their ground state as measured in the static frame, since this represents a singular state to an infalling observer. It is important to keep in mind, however, that smoothness of the horizon is not an input assumption of our set-up. Our basic principle is to preserve the rules of quantum mechanics, and to describe the evaporation process in the same way as one would describe the emission of a photon by an atom. With this starting point, Hawking radiation is not a consequence of a geometric regularity condition, but the result of turning on the interaction Hamiltonian H int . From this perspective, the fact that the late radiation starts in the vacuum state means that one is dealing with spontaneous emission, rather than stimulated emission. \n〉 \nWith this motivation, we postulate that the quantum states of the radiation at t = 0 factorize into the tensor product of state ∣ ∣ Φ i 〉 early , describing the early radiation, times a vacuum state ∣ ∣ 0 〉 b that specifies the initial condition for the late radiation. \n∣ ∣ Φ i 〉 = ∣ 0 〉 b ∣ ∣ Φ i 〉 early . (20) \n∣ \n〉 \n∣ \n∣ 〈 We are interested in the time evolution of the quantum state on H late = H A ⊗ H B , describing the interior black hole together with the late radiation, starting from t = 0. It is sufficient to consider this time evolution for pure initial states ∣ ∣ i 〉 | 0 〉 . This state evolves through the interaction hamiltonian H int , and thus after time τ it takes the form \n∣ ∣ Here ∣ ∣ 0 〉 b denotes the Boulware vacuum of the late radiation. The state ∣ ∣ Φ i 〉 early is stationary, since the early radiation is assumed to have already left the interaction region at t = 0. In other words, the early state is entirely decoupled from the late evolution. In the following we mostly forget about ∣ ∣ Φ i 〉 early . All we need to know is that after tracing ρ = | Ψ 〉〈 Ψ | over H early R we end up with a maximally mixed state of the form (8), multiplied with the vacuum density matrix ∣ 0 b 〈 0 | for the late radiation. \nU ( τ ) ∣ ∣ i 〉∣ ∣ 0 〉 b \nwhere U ( τ ) denotes the evolution operator \nU ( τ ) = T exp ( -i glyph[planckover2pi1] ∫ τ 0 dt H int ( t ) ) . (21) \nSince the interaction Hamiltonian couples the black hole interior to the radiation space H b , the time evolved state decomposes as \nU ( τ ) ∣ ∣ i 〉∣ ∣ 0 〉 b = ∑ n , j C i n , j ∣ ∣ j 〉 ∣ ∣ n 〉 b . (22) \nThe coefficients C i n , j represent the time-dependent probability amplitude for finding the black hole in the lower energy state ∣ ∣ j 〉 , combined with Hawking radiation in the state ∣ ∣ n 〉 b , at the time t = τ . The states ∣ ∣ j 〉 in (22) are black holes with energy M -E n , which span a Hilbert space of dimension dim H M -E n = e -βE n N . So for given n , the transition amplitudes C i n ,j combine into a non-invertible matrix. \nIn the following, it will be convenient to write the C n coefficients as explicit matrix elements C i n ,j = 〈 j ∣ ∣ Cn ∣ ∣ i 〉 of operators acting on the black hole Hilbert space. 4 \nCn ≡ b 〈 n ∣ U ( τ ) ∣ ∣ 0 〉 b (23) \n∣ \n∣ Equation (22) can then be succinctly written as (here we simplify U ( τ ) to U ) \nU ∣ ∣ i 〉∣ ∣ 0 〉 = ∑ n Cn ∣ ∣ i 〉 ∣ ∣ n 〉 . (24) \nThe unitarity constraint U † U = 1 implies that \n∑ n , j C i n , j C ∗ k n , j = δ ik , ∑ n C † n Cn = 1 M . (25) \nHere and in what follows, C n and C † n will always denote maps from H M to H M -E n and back. For the rest, quite little is known about the C i n , j coefficients: their precise form depends on secret details of the micro-physics. Our main assumption will be that they look like large ergodic matrices [17], with statistical properties consistent with black hole thermodynamics. We will see that this will enable us to determine their coarse grained structure with sufficient accuracy to decode, with the help of some familiar quantum information technology [23], \nthe local semi-classical physics on both sides of the black hole horizon. Equation (24) will form the starting point for much of the rest of our discussion. \nOur main story is quite independent of the detailed form of the interaction Hamiltonian H int ( t ). But for concreteness and to gain some intuition, it may be useful to have a specific model in mind. Suppose that the radiation is described by a free field φ . One of our postulates states that the black hole and radiation interact only at the stretched horizon. Locality then implies that H int ( t ) is expressed in the radiation field φ ( t ), and its first derivative, at the stretched horizon. This leads to a simplified model of the form \nH int ( t ) = ∫ dt ( φ in ( t ) V in ( t ) + φ out ( t ) V out ( t ) ) . (26) \nThe interaction contains both incoming and outgoing fields, and hence describes black hole formation and evaporation. Here we focus on the evaporation process, which just involves the outgoing field φ out = φ . The corresponding interaction operator V ( t ) can be expanded in modes V ω and V † ω , that map black hole states with mass M to states with mass M ± ω . The matrix elements V ij ( ω ) = 〈 j ∣ ∣ V ω ∣ ∣ i 〉 determine the emission and absorption rates of particles of frequency ω . Semi-classically this process is described by the creation of a Hawking pair. Hence it is natural to conjecture that the operators V ( ω ) and V † ( ω ) have something to do with the creating or annihilating particles behind the horizon. \nIn principle one can express the Cn operators in terms of the V operators as follows: one inserts the Fock states of the radiation and commutes the annihilation operators through the evolution operator. One obtains an expression involving products of a number of V operators, one for every emitted particle. It is suggestive that the number of V insertions matches the number of infalling partner modes of the Hawking pairs. This suggests that it is natural to view the transition amplitudes Cn as creating a state with a collection of particles behind the horizon. Our aim is to make this intuition more precise. \nThere exists of course a close parallel between the above discussion and the GKPW dictionary that underlies the AdS/CFT correspondence [7]. Indeed, we may compare the process of falling into a black hole with entering the near-horizon geometry of a stack of D-branes. In this comparison, the AdS space replaces the black hole interior geometry, the CFT represents the interior Hilbert space, the value of the scalar field φ at the stretched horizon plays the role of the asymptotic boundary value of the bulk field in AdS, and the V operators represent the CFT operator dual to φ .', '2.4 Black Hole Entanglement Revisited': "As a first application of our set up, let us revisit the time evolution of the young and old black hole states (6) and (8). \nFirst consider the young black hole. At time τ it has evolved into a state of the form \nρ young AB = U ∣ i 〉 | 0 〉 b 〈 0 ∣ ∣ 〈 i ∣ U † , (27) \n∣ \n∣ which we may expand in terms of the radiation basis as in equation (12), with \n∣ \nρ young nm = Cn ∣ ∣ i 〉〈 i ∣ C † m . (28) \n∣ We now make the assumption that the amplitudes C n are large ergodic matrices. In practice this means that we can view their matrix elements as being randomly chosen from an appropriate ensemble. The properties of this ensemble are completely fixed by quantum statistics, and should be consistent with the usual laws of thermodynamics. \n∣ \nIn particular, if we start with a generic pure black hole state, the density matrix on the radiation states is expected to take the thermal form (13). Thus statistically one should have the following identity \n〈 i ∣ ∣ C † m Cn ∣ i 〉 = W n δ mn , (29) \n∣ \n∣ with W n the normalized Boltzmann weight (13). This should be viewed as a coarse grained equality that holds for generic states ∣ i 〉 . \n∣ 〉 Let us check this relation by writing the left hand side explicitly in components \n∑ j C i n ,j C ∗ i m ,j = W n δ mn , (30) \nand use our knowledge about the size of the black hole Hilbert spaces. First take n = m : we are then summing Ne -βE n positive terms - so the answer is clearly non-zero - and since all amplitudes are statistically of equal size, the result must be proportional to e -βE n . The unitarity condition (25) tells us that the sum over n of the entire expression is equal to 1, so this fixes the normalization. Next suppose that n glyph[negationslash] = m . Then we are summing of the order of Ne -βE n terms with random phases. This sum is strongly suppressed relative to the diagonal term. The relative suppression of the off-diagonal terms is of order 1 / √ N . We thus find that statistically the above identity (30) indeed holds. \n∣ \nBefore proceeding to the old black hole, let us determine the mutual information between the radiation and the black hole state. Since the initial state is pure, the entropy the black hole interior A , the radiation region B , and their union AB , is easily calculated, and gives \nyoung BH : S AB = 0 , S B = log Z + β ¯ E, S A = S B , I AB = 2 S B . (31) \nWe see that the radiation region is maximally entangled with the internal black hole and carries the thermal entropy, as expected for a near horizon state. Indeed, on information theoretic grounds, there seems to be no obstruction to identify the young black hole state (27) with a smooth local vacuum state. We will make this identification explicit in the next section. \nNow let us look at the old black hole (8). After time τ it has evolved into the state \nρ old AB = U ρ (0) U † , with ρ (0) = 1 N 1 M ⊗| 0 〉 b 〈 0 ∣ ∣ (32) \nThe density matrix is obtained from that of the young black hole (27)by averaging over all initial black hole states. It can be expanded in the radiation basis as in (12) with \nρ old nm = 1 N CnC † m . (33) \nThe fact that the density matrix ρ B of the radiation region B is again thermal follows from 1 N tr A ( C † n Cm ) = W n δ mn which is automatically obtained from (29) by averaging over all black hole microstates. \nLet us again compute the entropies associated with the three regions A , B and A ∪ B . First we note that the density matrix (32) manifestly satisfies \n( ρ old BH ) 2 = 1 N ρ old BH , (34) \nfrom which we immediately see that it still carries entropy S AB = log N , as it should. This result should be contrasted with the entropy S AMPS AB = log NZ of the diagonal mixed state (16). The relation between the two density matrices can be understood if we introduce a phase averaging procedure, that averages over all possible phases of the transition amplitudes C i j, n . Performing this phase average eliminates all off diagonal terms of ρ old AB , resulting \nin the maximally mixed density matrix ρ AMPS \nρ old mn = 1 N C n C † m = 1 NZ δ nm 1 M -En . (35) \nThe extra term log Z in the entropy is thus due to coarse graining as a result of taking the phase average. One should keep in mind that there are many off-diagonal corrections to (35). Individually, these are all small but there are many of them. So collectively, they can have a substantive effect. \nThe diagonal form (35) is still appropriate for computing the entropy of the black hole interior region A . We thus find the following result for the entropies and mutual information \nold BH : S A = log N -β ¯ E, S AB = log N S B = log Z + β ¯ E, I AB = log Z. (36) \nWe see that the radiation region B and the interior region A now do share mutual information! This means that there is a non-zero amount of entanglement between the two. \nHow is it possible that the maximally mixed state (8) evolves into a state with nonzero mutual information between region A and B ? Clearly, this would not be possible if we would think of Hawking particles as quantum bits that were previously contained inside the black hole. The key point is that the state (8) is embedded inside a larger Hilbert space that, besides the black hole states, also contains the radiation modes. Since monogamy of entanglement does not hold in open systems, the time evolution can generate new entanglement. Note however that, comparing with eqn (31), the amount of shared information is a bit less than for the young black hole. For a free photon gas, log Z = βE/ 3, so the amount of entanglement for the old black hole is roughly a factor of 8 less than for the young black hole. The question we will need to investigate is if this amount of shared entropy is sufficient to describe an approximate vacuum state near the horizon. \nAbove we have used quantum statistical arguments to determine sums of products of transition amplitudes C i n,j and their complex conjugates. We will use this same reasoning repeatedly in the following sections. One may try to formalize this using a random matrix approach. One practical choice would be to pick a gaussian matrix ensemble, where one can use Wick's theorem. A more accurate characterization of the matrix ensemble is as the space of all matrices C i n,j that satisfy the unitarity constraint (25). The language of Wick contractions still serves a useful terminology for labeling and estimatingthe different resonant contributions. We will not try develop this formalism further here, since for the cases of interest the relevant result can be determined by simple reasoning as above.", '3 Passing through the Horizon': 'In this section we will take a more detailed look at this question. Our goal is to first make explicit how a relatively young quantum black hole, given either by a pure initial state (6) or a partially entangled quantum state can in fact describe a smooth horizon state. From a quantum information perspective, this result is not a real surprise.. Nonetheless, it is still an important first step. It has up to now been far from obvious how any quantum mechanical model of a black hole can be consistent with having a smooth horizon. This logical tension is at the heart of the black hole information paradox. \nSpecifically, we will show how the transition amplitudes Cn can be used to reconstruct a mirror of the QFT Hilbert space, spanned by the Hawking pairs of the outside radiation modes. Our construction makes use of the close analogy between our description of the evaporation process and the loss of coherence of a quantum computer [12, 23]. In the case of a quantum computer, one can recover the lost quantum information via the use of quantum error correction (QEC). This new tool will then enable us to investigate the issues raised by the information paradox, and the firewall argument in more quantitative terms.', '3.1 Quantum Error Correction': "Many of the quantities and relations introduced above have direct cousins in the quantum information context. Let us briefly mention a few entries of the dictionary [22, 23]. \nOne can think of the black hole as a quantum storage device exposed to environmental noise. Eqn (24) then describes the time evolution, the leakage of quantum information from the device into the environment, written in the so-called Kraus representation. The transition matrices Cn are known as Kraus operators. The emission of a Hawking quantum corresponds to the occurrence of an error, and the Kraus operators are typically denoted by En . We will continue to use the notation Cn . The evolution operator U = ∑ n Cn ⊗ | n 〉 b 〈 0 | is known as the error super operator, and the image of the Cn operators is called an error subspace. \nNot all states in the Hilbert space can be simultaneously protected from error. A QEC code includes the specification of a code subspace, the space of encoded states that one wishes to protect. We will denote states in the code subspace by \n∣ ∣ i 〉 ∈ H code . (37) \nThe code subspace is chosen independently of the actual state of the system. In the black hole context, one may be tempted to try to identify the whole interior Hilbert space with \nthe code subspace, but this is not realistic. The presence of extra states and degrees of freedom outside the code space is essential, so that one can build in sufficient redundancy to safeguard the encoded information. \nWe choose the encoded states to span a subspace within the Hilbert space H M of mass M , of dimension N code glyph[lessmuch] N . We will comment later on the corrections that come into play when we try to take the limit N code → N . \nNot all types of quantum errors can be corrected. A necessary and sufficient condition is that for any pair of Kraus operators Cn and Cm , and for any pair of states ∣ ∣ i 〉 and ∣ ∣ ¯ k 〉 in the code subspace, one has [22, 23] \n∣ \n〈 i ∣ C † n Cm ∣ ∣ ¯ k 〉 = W n δ nm δ i ¯ k . (38) \n∣ Here we fixed the normalization via eqn (29). If the property (38) does not hold, then errors can ruin the distinguishability of different encoded states and quantum information could be irrevocably damaged. For us, (38) is more than just a pre-condition. As we will see, it is the key algebraic identity that will enable us to decode the interior Hilbert state and exhibit the semi-classical horizon geometry. \nIn the black hole context, the condition (38) is naturally satisfied: it generalizes eqn (29) to the case where ∣ ∣ i 〉 and ∣ ∣ k 〉 are different states, and follows from the same considerations outlined below eqn (29). It expresses the fact that Cn and Cm are statistically independent complex random matrices, that only interfere constructively when all terms in the component expansion of the product are of the form ∣ ∣ C i n,j | 2 . Equation (38) should therefore not be read as an exact identity, but as an equation that for generic states holds with very high accuracy. Its corrections are suppressed by a relative factor of 1 / √ N . However, even while essentially true for any pair of states in H A , eqn (38) can not be read as a true global statement of the form C † n Cm = W n δ nm , since this would appear to imply that C † n Cn is an invertible matrix, which it is not: its image has dimension N n = Ne -βE n < N . So for given E n , we can use eqn (38) with confidence as long as we choose \nN code glyph[lessmuch] N n = Ne -βE n . \nWe will encounter the necessity of this inequality at many other places. \nA quantum error correcting code is specified by the pair ( H code , { Cn } ) , the code subspace and the collection of errors it tries to correct. Given eqn (38), one can naturally associate a notion of entropy to a given QECC, given by S code = ∑ n W n log W n \nLet us now describe the error correcting operation. It makes use of the recovery operators \nR n = 1 √ W n ∑ i ∣ ∣ i 〉〈 i ∣ ∣ C † n . (39) \nThese operators R n can be combined into a single super-operator R with the help of an 'ancillary' Hilbert space H a , or simply 'the ancilla', spanned by basis states ∣ ∣ n 〉 a in one-toone correspondence with the basis states ∣ ∣ n 〉 b of the late radiation. Hence H a is isomorphic to H B . As an example, in case the radiation field is described by a free scalar field φ , the ancilla H a is the Fock space spanned by creation and annihilation operators a † and a . \nThe recovery super-operator R acts on the tensor product H A ⊗H a via \nR ∣ ∣ j 〉∣ ∣ 0 〉 a = ∑ n R n ∣ ∣ j 〉∣ ∣ n 〉 a (40) \nThe ancillary Hilbert space is an important ingredient of quantum error correction, since as we will see shortly, it provides the depository into which it dumps the entanglement between the code space and the environment [23]. We emphasize, however, that in our context unlike for the usual application of QEC codes - the ancillary Hilbert space H a is just an intermediate algebraic device, that helps us put the recovery operator in a convenient form. Since our main purpose is to keep track of information flow, it is therefore important keep in mind that the ancillary Hilbert space never really exists and needs to be projected out in the end, in a way that manifestly preserves unitarity. We will return to this point, once the physical meaning of the ancilla has become clear. \nThe recovery operator is designed to reverse the error: applying R to a time evolved state U ∣ ∣ i 〉∣ ∣ 0 〉 a , one recovers the original state ∣ ∣ i 〉 times some state on the Hilbert space H B ⊗H a , provided that the initial state ∣ ∣ i 〉 lies inside the code subspace. The verification of this claim is straightforward, and makes use of the property (38) of the Cn 's: 5 \nRU ∣ ∣ i 〉∣ ∣ 0 〉 a ∣ ∣ 0 〉 b = ∑ m , n R mCn ∣ ∣ i 〉∣ ∣ m 〉 a | n 〉 b = ∑ m , n , ¯ k 1 √ W m ∣ ∣ ¯ k 〉〈 ¯ k ∣ ∣ C † m Cn ∣ ∣ i 〉∣ ∣ m 〉 a ∣ ∣ n 〉 b (41) = ∑ ¯ k | ¯ k 〉〈 ¯ k ∣ ∣ i 〉 ∑ n √ W n ∣ ∣ n 〉 a ∣ ∣ n 〉 b . \nWe read off that \nRU ∣ ∣ i 〉∣ ∣ 0 〉 a ∣ ∣ 0 〉 b = ∣ ∣ i 〉 ∣ ∣ 0 U 〉 if ∣ ∣ i 〉 ∈ H code 0 if ∣ i 〉 / ∈ H code (42) \n∣ \n∣ ∣ 0 U 〉 = ∑ n √ W n ∣ ∣ n 〉 a ∣ ∣ n 〉 b . (43) \nwhere ∣ ∣ 0 U 〉 is the following familiar looking state, defined on the tensor product of the Hilbert space of the outside radiation region with the ancillary Hilbert space \nGiven that W n are the Boltzmann weights, we recognize this state as the Unruh vacuum state, with the ancilla states playing the role of the interior radiation region. We see explicitly how the R operation works: it sweeps all entanglement between the time evolved black hole and the emitted radiation, encoded in U ∣ ∣ i 〉∣ ∣ 0 〉 b , into entanglement of the radiation with the ancilla. Thus the recovery operation restores the purity of the black hole state, provided it started out within the code space. \nWe see from (42) that the recovery operator R is not a unitary map, since it projects out all states that do not map back into the code space. This means that the operation R as defined in (40)-(39) in fact is not really a valid superoperator. Preferrably, one would want the recovery operation to be a unitary operation. To achieve this, consider the operator \nΠ = a 〈 0 ∣ ∣ R † R ∣ 0 〉 a . (44) \n∣ We may also write Π = ∑ n R † n R n = ∑ i , n 1 W n Cn ∣ ∣ i 〉〈 i ∣ ∣ C † n . If R had been a valid superoperator, then Π would have been equal to the identity operator. Instead it defines a projection operator on the space of all states that can be reached by acting with the Cn operators on the code subspace. In other words, it is the projection of all states that the code states can evolve into as a result of time evolution over time τ . \n∣ \nA straightforward calculation, similar to the one done in equation (41), shows that Π indeed has the property of a projection operator: \nΠ 2 = Π . (45) \nTo complete the recovery super operator, one thus adds one extra term to the right-hand \nside of (40) proportional to the projection operator 1 -Π on the orthogonal complement. \n( 1 -Π ) U ∣ ∣ i 〉∣ ∣ 0 〉 b = 0 if ∣ ∣ i 〉 ∈ H code U ∣ i 〉 | 0 〉 b if ∣ ∣ i 〉 / ∈ H code (46) \n∣ \n∣ We see that the QECC has indeed succeeded in its task of protecting all states within the code subspace from loss of coherence.", '3.2 Construction of the Interior Operators': "We have seen that the combined operation RU , time evolution followed by quantum error correction, associates to every black hole initial state ∣ ∣ i 〉 ∈ H code an Unruh state of the form (43). Since for any state ∣ ∣ i 〉 , there is some recovery operator R for which ∣ ∣ i 〉 lies within its code subspace, we have shown that ∣ ∣ i 〉 (when evolved forward in time) always contains a factor that can be decoded into a local vacuum state near the horizon. \nWe now define interior operators, that form the Hawking partner of the outgoing radiation modes. Specifically, we associate to every operator on H B a partner operator that acts on the interior black hole states. Our construction works for every state in the code subspace. Suppose that the black hole together with the radiation at time t = 0 are described by some mixed state of the form \nρ (0) = ∑ i , ¯ k ρ i ¯ k ∣ ∣ i 〉〈 ¯ k ∣ ∣ ⊗ ∣ ∣ 0 〉 b 〈 0 ∣ ∣ , (47) \nwhere the sum runs over some small subspace H small of the total black hole Hilbert space. We are then free to choose a code subspace such that H small ⊂ H code . At time t = τ , the state (47) has evolved into the density matrix \nρ ( τ ) = U ρ (0) U † . (48) \nWe wish to show that this density matrix describes a black hole with a smooth horizon. \nWe will now define the interior operators. In the previous section, we introduced the ancillary Hilbert space H a , as a carbon copy of the outside radiation Hilbert space H B . For us, this ancilla does not really exist, but is just introduced as a helpful algebraic tool. Even so, we are free to interpret H a as representing all states on the hidden Rindler wedge, the spatial region on the other side of the horizon from H B . Now consider some operator A that acts on H a . We wish to identify A with the Hawking partner of an identical operator \nacting on the outside radiation space H B . But we can not use the ancilla as our Hilbert space, so we need to transform A into an operator acting on the actual internal Hilbert space of the black hole. We now associate to A the following operator acting in the interior black hole Hilbert space \nA = a 〈 0 ∣ ∣ R † A R ∣ ∣ 0 〉 a . (49) \n∣ ∣ 〉 As an example, in case the radiation field is described by a free scalar field φ , we can choose to consider A = φ ( y ), where y is a point on the left Rindler wedge behind the horizon, that is, φ ( y ) is defined as a mode expansion in terms of ancillary oscillators a † and a . The general definition (49) then associates an effective local field operator φ ( y ) , acting on the interior black hole Hilbert space, via \nwhere R is the recovery superoperator defined in (39)-(40). Note that this is a vacuum expectation value in H a . So A is a proper linear operator acting on H A , the state space of the black hole, and the ancilla is indeed just a virtual device. Eqn (49) is modeled after the definition (44) of the projection operator onto the code subspace. Indeed, Π = a 0 ∣ R † 1 a R ∣ 0 〉 a assumes the role of the identity operator on the interior Hilbert space H a . \nφ ( y ) = a 〈 0 ∣ R † φ ( y ) R ∣ ∣ 0 〉 a . (50) \n∣ \n〈 \n∣ \n∣ \n∣ Eqns (49)-(50) are our proposal for a microscopic realization of the interior QFT operators. \nTo support this identification, we first need to verify that the map (49) between A and A is an isomorphism between operator product algebras. In other words, for any two operators A 1 and A 2 , we need to show that \nA 1 A 2 = a 〈 0 ∣ ∣ R † A 1 A 2 R ∣ ∣ 0 〉 a . (51) \nThis equation tells us that applying the map (49) after taking the product gives the same result as taking the product after applying the map. If this holds, then the A operators indeed have the same operator product algebra as the effective QFT operators. \nLet us check that (51) indeed holds. 6 It is useful to expand out the definition (40) of the recovery superoperator. The definition (49) can be expanded as \nA = ∑ n , m R † n A nm R m , (52) \nwhere A nm = a 〈 n ∣ ∣ A ∣ ∣ m 〉 a . We see that the combination R † n R m takes over the role of the operator ∣ ∣ n 〉 a 〈 m | in the ancilliary Hilbert space. As a free field example: with this notation, we may define the effective interior creation and annihilation operators a † and a via \nThe a and a † operators act purely within the internal black hole Hilbert space H A , and thus commute with the creation and annihilation modes b † and b of the radiation. \na † = ∑ n , m R † n 〈 n ∣ ∣ a † ∣ ∣ m 〉 R m and a = ∑ n , m R † n 〈 n ∣ ∣ a ∣ ∣ m 〉 R m . \nWith the definition (52), the requirement (51) that A satisfies the QFT operator product algebra amounts to the identity \nR † n R s R † t R m = δ st R † n R m . (53) \nAs shown in the Appendix, this property indeed holds with great accuracy, as long as N code glyph[lessmuch] N . The precision with which it is valid is a measure for the successfulness of the quantum error correcting procedure R . For us, it measures the faithfulness with which we can recover the operator algebra of the interior low energy effective field theory. It is a strength of our formalism that we can in principle compute the corrections and limits of validity of the effective theory; we will consider the correction to the operator algebra when discussing the limit N code /N → 1 in the next section. \nNext we compute the expectation value of the operators in the state ρ ( τ ) given in (48). Here we assume that the initial state (47) at t = 0 is contained inside the code subspace, and that N code glyph[lessmuch] N . Let A and B denote some general operators in terms of the a and b -modes, respectively, and let A be the interior operator defined via (49). The computation of the expectation value tr( ρ ( τ ) A B ) is straightforward, and summarized in the Appendix. We find that the expectation value coincides with that of the effective field theory in the local Unruh vacuum state \ntr ( ρ ( τ ) A B ) = 〈 0 U ∣ AB ∣ ∣ 0 U 〉 . (54) \n∣ \n∣ This is the main result of our paper. It shows that the black hole state can look like a smooth horizon state, without straining the rules of quantum information theory. \nAnother way to obtain this result is to use the expansion (52) in terms of matrix elements. In the Appendix it is shown that \ntr A ( ρ R † n R m ) = √ W n W m ∣ ∣ n 〉 b 〈 m ∣ ∣ . (55) \nuiding principle is that a black hole is an ordinary quantum mechanical system, - much ike an atom, albeit a very large one and with an enormous level density of states - that volves and interacts with its environment via standard hamiltonian evolution. We start ubsequent study. .1 provided it started out within the code space. We see from (42) that the recovery operator R is not a unitary map, since it projects out all states that do not map back into the code space. This means that the operation R as (44) had been a valid suAppendix A: Some calculations † A B U | 0 b 〉 = 〈 0 b ∣ ∣ U † 〈 0 a | R † A = 〈 0 b ∣ ∣ a 〈 0 | U † R † AB 〈 0 U ∣ ∣ AB | 0 U 〉 cod wledgements n Given that W n are the Boltzmann weights, we recognize this state as the Unruh vacuum state, with the ancilla states playing the role of the interior radiation region. We see explicitly how the R R as hole states (6) and (8). ider the young black hole. At time τ it has evolved into a state of the form ρ young AB = U ∣ ∣ i 〉 | 0 〉 b (26) rsion of the firewal argument put forward in [1], that is most directly addressed by our ubsequent study. .1 M with some The situation is of course similar to other holographic dualities, like AdS/CFT. From the dual perspective, the interior space-time emerges in a mysterious way. But for Alice, the smooth local space-time is a classical reality, relative to which the operators (52) behave like ordinary observables. So she has no trouble using them. Note, however, that Alice ω , but reassembles them (61) \n.1 We may also write Π = ρ young nm = Cn ∣ ∣ i lack hole with mass M , whose states \nC † n . If R = (27) H M to the x ) | 0 U 〉 \n(1) \nn | n 〉 a | n 〉 b . [1] M with some , and any other macroscopic H M to the lack hole with mass M ∣ (1) Cn Π 2 = Π (45) B ) = ∑ i, ¯ k ρ i ¯ k 〈 i ∣ ∣ 〈 0 b ∣ ∣ U † A B U | = ∑ i, ¯ k ρ i ¯ k 〈 i ∣ ∣ 〈 0 U ∣ ∣ AB | 0 U 〉| ¯ k 〈 0 b ∣ ∣ U † A B U | 0 b 〉 = b (69) = (70) (44) We may also write Π = ∑ n R had been a valid suprojection operator on the space of all states that can be reached by acting with the Cn 15 awking entropy N (2) ith = 4 a -modes, and φ b from b -modes. To really get the complete set of modes behind the horizon, we would need to extend our framework a bit, by adding \n<!-- image --> \nS BH \ntate ∣ ∣ i 〉 ∈ H M \n| i 〉〈 i | C \nN = dim H M = e S BH , (2) ith S BH = 4 πM 2 . 21 π ns π tm = 1 √ W n W s W t W m ∑ i, ¯ k Cn | i 〉〈 i | tr ( ρ ( τ ) A B ) = ∑ i, ¯ k ρ i ¯ k 〈 i ∣ ∣ 〈 0 b ∣ ∣ U A B U | 0 b 〉| k 〉 (72) Π 2 = Π (45) ncorporates all possible intermediate situations. So we introduce the total Hilbert space , given by the tensor product incoming modes live behind the past horizon, and are obtain by analytic continuation of modes on the left region in fig 1. To define them, we would need to introduce transition From this one directly computes (here B nm = b 〈 n ∣ ∣ B ∣ ∣ m 〉 b ) \n[2] re indistinguishable from a macroscopic perspective. Each state | i 〉 describes the interior f a black hole, with microscopic properties that are hidden from the outside observer by the tretched horizon. The number of black holes micro-states is determined by the Bekenstein awking entropy To complete the recovery super operator, one thus adds one extra term to the right-hand π nm = 1 √ W n W m ∑ i Cn ∣ = 〈 0 U ∣ ∣ AB | 0 U 〉 code (71) ∑ ∣ 〈 † ¯ operators on the code subspace. In other words, it is the projection of all states that the code states can evolve into as a result of the evaporation process. A straightforward calculation, similar to the one done in eqn (41), shows that Π indeed has the property of a projection operator: πM 2 . We will assume that the time evolution of every micro state describes black hole that slowly evaporates via the Hawking process. So as time progresses, every will evolve into a new state given by the product of a micro state of a black ole with smaller mass M ' = M -E and a radiation state with energy E . To describe this evaporation process we need to enlarge our Hilbert space, so that it the infalling modes. Since the horizon state looks smooth, the infalling modes can be analytically continued to the region behind the horizon. So we have indeed explained how a QFT mode, and by extension any object, can fall into a black hole. Note that we can also try to go beyond the past horizon, by switching time direction. The outgoing modes behind the past horizon are obtained by analytic continuation of outgoing modes on the right region in fig 1. The time-flipped Hawking partners of the Figure 2: Steps in the quantum computation of the expectation value tr( ρ A B ) for a pure state density matrix ρ (0) = ∣ ∣ i 〉〈 i ∣ ∣ . The initial state ∣ ∣ i 〉∣ ∣ 0 〉 b first evolves via the evolution operator U . The interior operator then acts via a recovery operation R followed by the operator A acting on the ancilla. The outside operator B acts directly on the time evolved radiation state. At the instant that A and B act, the interior state is identical to the initial state ∣ ∣ i 〉 , and the a and b states are in the local vacuum state ∣ ∣ 0 U 〉 . Then follows the conjugate of the recovery operation a 〈 0 ∣ ∣ R † , which eliminates the ancilla. Finally, one projects back onto the final state b 〈 0 ∣ ∣ 〈 i ∣ ∣ U † . \n= \nH ∑ ¯ ∣ 〈 ∣ Π , given by the tensor product From this one directly computes (here nm b n ∣ ∣ m b \ntate ∣ ∣ i 〉 ∈ H M will evolve into a new state given by the product of a micro state of a black ole with smaller mass M ' = M E and a radiation state with energy W n W m i 〉〈 i | C † m i, k 3.3 Ancilla versus Black Hole Final State This is the expectation value in the local Unruh vacuum (43). \nH a , \n' M -and a radiation state with energy E . To describe this evaporation process we need to enlarge our Hilbert space, so that it ncorporates all possible intermediate situations. So we introduce the total Hilbert space , given by the tensor product H = H BH ⊗H R (3) 32 π nm = 1 √ W n W m ∑ i C n | i 〉〈 i | C † m (74) π = 1 √ ∑ C | i 〉〈 i | C † C | ¯ k 〉〈 ¯ k | C † (75) 21 5 Our use of the ancillary Hilbert space H a is somewhat reminiscent of the black hole final state proposal of [2]. However, there is a key difference: unlike the final state proposal, our procedure is completely in accord with the conventional rules of quantum mechanics. We have stated explicitly in what Hilbert space we are working, and time evolution is unitary. Our internal operators involve an expectation value inside an auxilary Hilbert space 26 What made this magic work? The definition (49) of the interior modes involves a QECC operation, which requires detailed knowledge of the emission amplitudes C n , whose precise form depends on unknown details of Planck scale physics. It looks like cheating to assume that Alice can access these observables, without actually having such detailed knowledge. The situation is of course similar to other holographic dualities, like AdS/CFT. From the dual perspective, the interior space-time emerges in a mysterious way. But for Alice, the smooth local space-time is a classical reality, relative to which the operators (49) behave like ordinary observables. So she has no trouble using them. Note, however, that Alice \n= \nWe will assume that the time evolution of every micro state describes black hole that slowly evaporates via the Hawking process. So as time progresses, every = δ st √ ∑ Cn | i ∑ ¯ ρ i ¯ k 〈 i ∣ ∣ 〈 0 U ∣ ∣ AB | 0 U 〉| ¯ k 〉 = 〈 0 U ∣ ∣ AB | 0 U 〉 (73) To complete the recovery super operator, one thus adds one extra term to the right-hand H = H BH ⊗H R (3) amplitudes for the ingoing modes involved in the black hole formation process. ∣ tr ( ρ A B ) = ∑ n , m A nm tr ( ρ R † n R m B ) = √ W n W m A nm B nm . (56) \nπ ns \ntm \nW n W s W t W m \nδ st \n√ W n W m \nn \n5 \ns \n= \nδ st π nm \n(76) \nt \nm \ni, ¯ k \n∑ i C n | i 〉〈 i | C † m \ndoes not organize the modes in terms of their Schwarschild energy ω , but reassembles them into local Minkowski modes, that make no reference to the precise location of the horizon. \nTo make this a bit more concrete, consider the two point function of local field operators, say one behind and one in front of the horizon. The general result (54) gives that \ntr ( ρ φa ( y ) φ b ( x ) ) = 〈 0 U ∣ φ a ( y ) φ b ( x ) ∣ ∣ 0 U 〉 , (57) \n∣ \n∣ where the subscript indicates that φ a is made up from a -modes, and φ b from b -modes. What is the meaning of this expectation value? Strictly speaking, it does not yet describe what we want. In our discussion so far, we have only included the outgoing modes that are emitted by the black hole. The ancillary a modes play their guest role as the Hawking partner of the outgoing b modes: they initially live on the left region in fig 1, but can be analytically continued to live just behind the future horizon. To really get the complete set of modes behind the horizon, we would need to extend our framework a bit, by adding the infalling modes. Since the horizon state looks smooth, the infalling modes can be analytically continued to the region behind the horizon. So we have indeed explained how a QFT mode, and by extension any object, can fall into a black hole. \nNote that we can also try to go beyond the past horizon, by switching time direction. The outgoing modes behind the past horizon are obtained by analytic continuation of outgoing modes on the right region in fig 1. The time-flipped Hawking partners of the incoming modes live behind the past horizon, and are obtain by analytic continuation of modes on the left region in fig 1. To define them, we would need to introduce transition amplitudes for the ingoing modes involved in the black hole formation process.", '3.3 Ancilla versus Black Hole Final State': "Our use of the ancillary Hilbert space H a is somewhat reminiscent of the black hole final state proposal of [19]. However, there is a key difference: unlike the final state proposal, our procedure is completely in accord with the conventional rules of quantum mechanics. We have stated explicitly in what Hilbert space we are working, and time evolution is unitary. Our internal operators (49)] involve an expectation value inside an auxilary Hilbert space H a , but we could have avoided that altogether by directy using the definition (52). So we work at all times in the same Hilbert space, without tensoring in or projecting out any extra degrees of freedom. \nThe black hole final state proposal of [19] works differently. There the ancilla H a and the radiation space H b are new Hilbert space sectors that arise out of nowhere whenever a \nback hole gets formed, and are postulated to start out in the state ∣ 0 U 〉 = ∑ n √ W n ∣ ∣ n 〉 a ∣ n 〉 b . \n∣ \n∣ \nThis is a crude prescription, however, that immediately obscures the rules of quantum mechanics. Where was the Hilbert space H a ⊗H b before the black hole was formed? How did it end up in a maximally entangled state ∣ ∣ 0 U 〉 with the radiation modes? How can one keep track of quantum information if such things can happen? \n∣ ∣ ∣ i 〉 → ∣ ∣ i 〉∣ ∣ 0 U 〉 . (58) \nSuperficially, our equation (42) looks very similar to (58). For us, however, equation (42) does not represent time evolution, and the state ∣ ∣ i 〉 | 0 U 〉 does not exist in our physical Hilbert space H A ⊗ H b . So let us interpret the proposal of [19] in the same spirit. In equation (58), we should not think of the ancilla states ∣ ∣ n 〉 a as real, but virtual. Indeed we can undo the damage done by the rash introduction of the extra Hilbert space and restore unitarity by writing 7 \nU ∣ ∣ i 〉∣ 0 〉 b = a 〈 0 ∣ ∣ R † ∣ i 〉∣ ∣ 0 U 〉 . (59) \n∣ \n∣ \nThis interpretation immediately circumvents the main substantive objection against the final state proposal made in [20], where it was pointed out that any non-trivial dynamics between the two sectors would immediately destroy the coherence of the physically observable state. But as the above discussion makes clear, the ancillary Hilbert space should be viewed as dynamically completely decoupled from the physical Hilbert space: the evolution operator U and Hamiltonian H acts only on H A ⊗ H b and leaves H a inert. That the dynamics exactly factorizes in this way would look very unreasonable if H a had been physically real, but is obvious in our set-up. The final state proposal is only salvageable but then loses much of its explanatory potential - by realizing that the ancillary Hilbert space H a is an artificial construct. \n∣ ∣ ∣ The left-hand side is a manifestly unitary evolution of the initial state ∣ ∣ i 〉∣ ∣ 0 〉 b , while the right-hand side looks like a non-unitary final state projection a 〈 0 ∣ ∣ R † applied to the state ∣ ∣ i 〉∣ ∣ 0 U 〉 . The apparent non-unitarity of the final state prescription cancels out against the initial unitarity violation of introduction of the additional Hilbert space sector. \nIf the ancilla states are not real, how then are we are able to still employ quantum error correcting technology? QEC involves the reversal of decoherence. When the code block interacts with its environment, the two become entangled and the Von Neumann entropy of both systems increases, as seen from (31) The recovery operation R implements \nan entanglement swap: it purifies the code space by sweeping all its entanglement with the radiation modes into the ancilla. Hence reality of the ancilla is absolutely essential for quantum error correction, since where else can the entanglement entropy go? \nThe resolution of this apparent conflict is as follows. As seen from the definition (49), A = a 〈 0 | R † A R | 0 〉 a , the interior operators make only temporary use of the recovery procedure. Reading from right to left, they tensor in the auxiliary ground state ∣ ∣ 0 〉 a , and perform the recovery operation R . This step removes entropy from the code block and temporarily stores it inside H a . Then it applies A , and instantly undoes the recovery by acting with the 'final state projection' a 〈 0 ∣ ∣ R † . This final state projection - which is necessary for restoring unitarity - erases the ancilla and dumps the entropy back into the interior Hilbert space H a . 8 So the QEC procedure is able to perform an entanglement swap, but at a price: the interior black hole Hilbert space space clogs up with entropy as time passes by. \nThis looks like a recipe for trouble. As the entropy of the black hole density matrix increases, one needs a bigger and bigger code space for performing the recovery procedure on the whole density matrix. Eventually N code starts approaching the size N of the full Hilbert space. But as we have seen, the quantum error correcting code becomes less and less reliable with increasing N code /N , and as a result, our identification of the interior effective QFT operators starts breaking down. This is a manifestation of the firewall problem.", '3.4 Fidelity and Errors': "Why do quantum error correcting codes have any useful application in the study of black hole information and entanglement? In the context of their application to quantum computers, QEC codes are designed to protect the quantum coherence of a particular part of the Hilbert space, the code subspace, by correcting the errors that occur by emission of particular quanta. It does so by undoing the transformation that is responsible for the emitted quanta of radiation, and moving the entanglement with these quanta to another part of the Hilbert space that is outside the code subspace. \nIn the black hole context, the quantum recovery operation acts in an exactly identical way on the Hilbert states, but the QEC code plays a rather different role [12]. The code subspace is typically already in some highly entangled with the early radiation outside the black hole. The error correcting procedure still works by undoing the transformation due to the emission of radiation. If it succeeds, it entangles the radiation again with another part of the Hilbert space. As we have just shown, this entanglement helps preserve the \nentanglement across the smooth the horizon. \nIn any QEC procedure there is always a small but finite chance that it fails. In the situation of quantum computer in an approximate pure state, the error will result into some partial decoherence: a small part of the code subspace will become entangled with the outside. As a result, the corresponding internal quantum state is no longer suitable for doing quantum computations. In the black hole application, failure of the QEC code means the opposite. Occurrence of an error means that the code did not preserve the entanglement between the code subspace with the early radiation. Therefore the QEC error will cause a swap in entanglement: it releases a qubit of information from the code subspace and lets it escape to the outside. This is how the information leaks out of the black hole [12]. \nWhat is the fidelity and how big is the error of the QECC? To determine this, we need to look at how well the recovered state approximates the original state. A good way to compute this error is to determine how well the projection operator Π, when evolved backwards in time, approximates the identity operator on the code subspace \nΠ U ∣ ∣ i 〉∣ ∣ 0 〉 b = U ∣ ∣ i 〉∣ ∣ 0 〉 b + error . (60) \nThis equality is without error for a perfect QECC with 100% fidelity. Plugging in the definitions for the left-hand side gives \n∑ ¯ k, n , m 1 W n Cn ∣ ∣ ¯ k 〉〈 ¯ k ∣ ∣ C † n Cm ∣ ∣ i 〉∣ ∣ m 〉 b . (61) \nWe evaluate this expression via the same procedure as before: we keep only those terms in which every matrix element C i n ,j constructively interferes with its complex conjugate. In the limit N code glyph[lessmuch] N n , the dominant interference occurs between C † n Cm for the term with n = m . This leads to the first term on the right-hand side of (60). This is the good term. \nBut there is also a bad term: when N code /N n becomes non-negligible, the other interference term becomes important: C † n on the right can constructively interfere with C n on the left. In this way we find that the error is given by \nerror = EU ∣ ∣ i 〉∣ ∣ 0 〉 b , with E = ∑ n N code N n 1 M -E n . (62) \nThe operator E will be called the 'error operator'. Hence to have an accurate recovery procedure, we must choose the code subspace small enough so that this error is negligibly small. Thus we again encounter the condition that N code glyph[lessmuch] N n .", '3.5 Approaching the Firewall': "The firewall argument concerns the limit in which the black hole density matrix approximates the maximally mixed state (8). There are several motivations to study this limit. As emphasized in [18], two space time regions can be connected if they are maximally entangled at a microscopic level. These arguments suggest that to be able to fall through the horizon without drama, one actually needs the maximal entanglement between the inside and the outside. Other related perspectives are the interpretation of black holes as information mirrors [12]and fast scramblers [17], and Mathur's argument that, as far as quantum information goes, order one corrections are needed to the usual semi-classical picture. \nMotivated by these ideas, we will now study what happens as one approaches the maximally mixed limit. We will still assume that the initial state ρ (0) in (47) fits inside a code subspace of size N code < N , which we parametrize via \nN code N = e -βµ . (63) \nThe maximally mixed case corresponds to setting N code /N = 1, so that S code = log N code saturates the Bekenstein-Hawking bound S BH = log N . We will call this the firewall limit. \nBased on our set up, we can try to give a somewhat more precise characterization of the density matrix of an old black hole, as follows. As pointed out above, the quantum error correction procedure involved in reproducing the internal effective field theory is not perfect, but has a finite probability of producing an error. In the previous section, we saw that, when considering operators that act on an internal state with energy M -E n , this error is of the order \nN code N n = e β ( E n -µ ) . (64) \nThe occurrence of an error coincides with an event in which the black hole releases information about its interior state into the environment. So an important first question is: what is the expected maximal rate at which these errors can occur? At the late stages of evaporation, the number of information carrying quanta should exceed the number of quanta that increase the black hole entropy. Therefore, the error rate eventually exceeds the frequency of good recoveries. Our proposed definition of the density matrix of an old black hole is that its entropy has grown to the level such that the probability that any given Hawking particle carries out a qubit of quantum information into the environment is of order 1. This amounts to an error rate of about one qubit per black hole crossing time M . So in the following we will set our time step τ equal to one crossing time. \nThe errors of the QECC give rise to corrections to the internal low energy effective field theory. Consider the operator combination R † n R m featuring in the definition (52) of the internal operators. The erroneous Wick contraction produces an extra contribution \nR † n R m = 1 √ W n W m ∑ i Cn ∣ ∣ i 〉〈 i ∣ ∣ C † m = δ nm e β ( E n -µ ) 1 M -E n . (65) \nHere the bar indicates the phase average. The normalization of the right-hand side is determined by unitarity, c.f. eqn (35). Note that it indeed vanishes when e -βµ → 0, i.e. when the code space becomes negligibly small. This extra interference term deforms the algebra (53), that underlies the isomorphism between the internal operator algebras and that of the radiation field. Thus we need to reinvestigate our definition of the interior operators, and pick the definition that best preserves the effective QFT description. \nThe most reasonable prescription is to introduce a normal ordered product : R † n R m :, defined as the operator product with the phase averaged contribution R † n R m subtracted. In other words, the normal ordered operators are traceless. This recipe preserves the identity tr A ( ρ : R † m R n : ) = √ W n W m | n 〉 b 〈 m | , which ensures that, if we keep our definition of the internal operators of the form (52) but now as normal ordered expression : A :, the expectation values of the effective QFT still look like those in the local vacuum state ∣ ∣ 0 U 〉 . So from the semi-classical perspective, it looks like the horizon is still smooth - in the sense that all expectation values are well behaved. \nThe error has not gone away, however: the normal ordering procedure still leads to a deformation of the operator product algebra (53) \n: R † n R s : : R † t R m : = δ st ( : R † n R m : + δ nm e β ( E n -µ ) 1 M -E n ) . (66) \nTo verify this relation, observe that the product of two normal ordered operators is not normal ordered; this leads to an extra contribution equal to R † n R m . This extra term can be ignored as long as µ glyph[greatermuch] E n . When the energy E starts approaching the value of µ , however, it starts to show up as a manifestation of the error of the QEC operation. Before we investigate the form of the deformation, let us determine how big or small it is. \nWhat is the probability that the QEC procedure produces an error? Consider the operator Π introduced in (44). It can be expanded as Π = ∑ n : R † n R n :, and in the low energy effective theory, it represents the unit operator 1 . It satisfies Π 2 = Π provided that the recovery precedure works perfectly. But once we include the possibility of errors, Π is \nno longer a perfect projection operator. Using (66), we find a correction term given by \nΠ 2 = Π + E , E = ∑ n e β ( E n -µ ) 1 M -E n . (67) \nWe assume that e -βµ is adjusted so that E is still small. We may then interpret E as the operator that produces the error in the recovery operation. \nTo determine the relative magnitude of the 'good' first term and the 'bad' error term, let us take their expectation value. For concreteness, we assume that the black hole density matrix at time t = 0 takes the form ρ (0) = 1 N code ∑ i ∣ i 〉〈 i ∣ ∣ . We then have \n∣ \ntr A ( ρ ( τ ) E ) = ∑ n e -βµ ∣ ∣ n 〉 b 〈 n ∣ ∣ = e -βµ 1 B . (69) \n∣ tr A ( ρ ( τ )Π) = ∑ n W n ∣ ∣ n 〉 b 〈 n ∣ ∣ = 1 Z e -βH b (68) \nThe good term is the thermal density matrix ρ B of the radiation, and thus its trace is equal to 1. So it is normalized as a probability measure. When there is a non-zero error rate, however, this normalization needs to be adjusted, to account for the fact that the QEC operation is no longer has 100% fidelity. It seems reasonable to interpret the second equation as the relative probability that an error takes place in the quantum channel. It tells us that the errors have an infinite temperature: every state ∣ ∣ n 〉 in the radiation Hilbert space has , regardless of its total energy E n , equal probability p glyph[similarequal] e -βµ to be the carrier of some irretrievable error. This probability increases linearly with the N code , as expected. \nAs we increase the size of the code space, we see that there is a natural cross over point, where the error probability becomes of order 1. This crossover takes place when the dimension of the code space reaches the critical value \nN code N = e -βµ glyph[similarequal] 1 dim H B . (70) \nOne may think of this as a detailed balance equation: it picks the saturation value for the size of the code subspace, such that the QEC procedure starts producing approximately one error per crossing time. Notice that the right-hand side is a UV sensitive quantity: it depends on the size of the short-distance cut-off of the effective field theory that describes the Hawking radiation 9 and on the location of the stretched horizon. \nWhen the error correction works, we get a semi-classical description of the evaporation process. But when it fails it produces modifications to the low energy effective field theory. What do they look like? Consider the identity (51) that verifies that the mapping (49)-(52) preserved operator product relations. We now find that the right-hand side receives an extra diagonal term, which we may think of as a result of normal ordering. We find that the operator product algebra is deformed according to \n: A 1 : : A 2 : = : A 1 A 2 : + ∑ n ( A 1 A 2 ) nn e β ( E n -µ ) 1 M -E n . (71) \nHow would a low energy observer interpret this deformation? It is tempting to look for a way to translate this algebra in pure effective field theory language, perhaps with some non-locality. However, the extra term is not of the form (49), and acts on a bigger Hilbert space than H code , the space we used for our re-construction of the effective QFT. Because of this, we have not found a sensible way to answer this question. \nPerhaps this is indeed the wrong question to ask. The presence of the extra term is dictated by unitarity of the underlying microscopic theory, and the microscopic mechanisms that are responsible for releasing the information from the black hole are likely to be not describable by semi-classical physics. In our QEC terminology, they produce error terms that in magnitude are of order one. Moreover, once the black hole entropy starts to decrease, the number of information carrying radiation modes must exceed the number of modes that increase the entanglement, the error rate must exceed the frequency of successful recovery operations. This indeed indicates that in this regime the semi-classical description can no longer be trusted. We are then led to conclude that when corrections to the semi-classical theory becomes important, the effective QFT disintegrates. This is a manifestation of the firewall problem, and perhaps the manifestation of the formation of an actual firewall or fuzzball. [25]. \nThis last conclusion is still a bit premature, however. It is quite conceivable that the breakdown of the effective QFT is simply a consequence of our attempt to capture a closeto-maximally entangled state of a large system in terms of a single semi-classical reality. Indeed, what we have shown is that a black hole with small enough non Neumann entropy relative to its BH entropy bound, i.e. with e -βµ glyph[lessmuch] 1, supports a semi-classical geometry with a smooth horizon. Since a maximally mixed black hole is an incoherent sum of black hole states with less than maximal entropy, we have shown that it represents an incoherent sum of different (in the sense that they are incompatible with each other) semi-classical states. Usually this means that the semi-classical states are the states that we measure.", '4 Summary': "In this final section we give an executive summary of our main results. Consider the total quantum state of the black hole (with internal states ∣ ∣ j 〉 ) after emission of quanta of late radiation (described by states ∣ n 〉 ) \n∣ \nHere ∣ ∣ Φ i 〉 is the state of the early radiation. Here we assume that the late radiation started coming out at some time t = 0, and (72) is the state at some later time t = τ . Since the early radiation is assumed to be dynamically decoupled, we are allowed to treat the states ∣ Φ i 〉 as static. \n∣ ∣ Ψ 〉 = 1 √ N ∑ n ,i,j C i n ,j ∣ ∣ j 〉 ∣ ∣ n 〉 ∣ ∣ Φ i 〉 . (72) \n∣ 〉 We would like to investigate with which part of the wave function the radiation state ∣ ∣ n 〉 is entangled, the black hole state or the early radiation state. In [1] it is argued that, since for an old black hole the Hilbert space of the early radiation is so much larger than that of the black hole itself, the radiation state must be maximally entangled with the environment state ∣ ∣ Φ i 〉 and has no entanglement with the black hole state ∣ ∣ j 〉 . 10 This appears to exclude the existence of a smooth horizon. The question that we will first try to answer is: \n∣ \n〉 \n∣ \nUnder which conditions can an observer who has only access to the internal black hole states ∣ j accurately determine in which state ∣ n 〉 b the late radiation has been emitted? \n∣ \n∣ ∣ ∣ ∣ n 〉〉 = 1 √ N n ∑ i,j C i n ,j ∣ ∣ j 〉 ∣ ∣ n 〉 ∣ ∣ Φ i 〉 , (73) \n∣ ∣ 〉 To focus the discussion, let us assume that an outside observer has detected the radiation in a particular state ∣ ∣ n 〉 . Thus instead of the full quantum state ∣ Ψ 〉 we now look at \n∣ \nwhere we renormalized the state so that its norm is equal to one. Is there a measurement that acts on the states ∣ ∣ j 〉 whose outcome is going to tell us about the quantum numbers n of the late radiation? \nThe key point on which our paper is based is that the coefficients C i n ,j are not just some arbitrary set of numbers: for given n they are transition amplitudes between black hole states, as a result of emitting the radiation in state ∣ ∣ n 〉 . Hence the C i n ,j are determined by the microphysics, and represent the matrix elements of particular operators Cn , that act on \nthe interior part of the black hole Hilbert space . We assume that the C i n ,j are all statistically independent random matrices. Now pick some arbitrary state ∣ ∣ m 〉 , with the same energy as ∣ ∣ n 〉 , and consider the operator Π m = 1 W m : Cm 1 code C † m : with matrix elements \n(Π m ) jk = 1 W m ∑ i C ∗ i m ,j C i m ,k -trace (74) \n∣ \nHere W m = 1 Z e -βE m is the Boltzmann weight. The sum over i runs over some code subspace of the black hole Hilbert space of dimension N code < N . We will now show that this operator, which only acts on the black hole interior, is capable of recovering the information about the radiation state ∣ n . \n∣ Applying this operator Π m to (73) gives \n〉 \nΠ m ∣ ∣ ∣ ∣ n 〉〉 = 1 √ N n ∑ i,j,k C i n ,j (Π m ) jk ∣ ∣ k 〉 ∣ ∣ n 〉 ∣ ∣ Φ i 〉 . (75) \nSince the C i n,j 's are ergodic matrices, whose coefficients all arbitrary complex phases, we can evaluate this sum with the help of standard statistical reasoning. We need to look for the resonant contributions. Inserting (74) into (75), we obtain an expression with a sum over two internal indices: j and i . The index j labels black hole states of mass M -E n and thus runs over N n indices and i labels the code subspace and thus runs over N code indices. We evaluate both sums by keeping only resonant terms, by using eqns like (38) and (65). We call the sum over j the 'good contraction' and the sum over i the 'bad contraction'. The second term in (74) is chosen such that it cancels the 'bad contraction'. The good contraction gives back the state ∣ ∣ ∣ ∣ n 〉〉 when m = n , but with a projection on to the code subspace states \n∣ \n∣ \nΠ m ∣ ∣ ∣ n 〉〉 = δ mn Π code ∣ ∣ ∣ n 〉〉 , (76) \n∣ ∣ where Π code = ∑ n Π n . So it looks like we have been able to establish that the black hole state is maximally entangled with the radiation mode ∣ ∣ n 〉 , provided that the black hole part of the initial state lies in the code subspace. So why can't we just take the code space equal to the complete interior Hilbert space and set N code = N ? \nThere's the catch. Eqn (76) is true as a coarse grained equation. In computing the sums in (75) we ignored many off-diagonal terms, which individually are all very small. We should worry, however, about their collective effect. One place where this enters is in the verification that the operator Π m really acts like a projection operator with Π 2 m = Π m . By \n<!-- image --> \nFigure 3: The QECC protects the coherence of a subspace H code within the black hole Hilbert space H BH . It safeguards the smoothness of the horizon for every state in H code . As the density matrix ρ spreads out, the code subspace needs to grow along with it. This degrades the fidelity of the code and spoils the semi-classical correspondence. The state decoheres into a sum of semiclassical components, each of which fits inside a smaller code space and has a smooth horizon. \n<!-- image --> \nthe same calculation method, we find \nΠ 2 m = Π m + Em , Em = N code N m 1 M -E m , (77) \nwhere 1 M -E m is the unit operator on the black hole Hilbert space with mass M -E m . The extra term Em the error of the recovery operation. So to have an accurate recovery without appreciable errors, we need that the code subspace is small. \nWe conclude that the black hole is still maximally entangled with the radiation mode when the state of the black hole fits inside of some suitably chosen code subspace with \nΠ code ∣ ∣ ∣ ∣ n 〉〉 = ∣ ∣ ∣ ∣ n 〉〉 and N code glyph[lessmuch] N n . (78) \n∣ \n〉 \n∣ However, we have also found that when the von Neumann entropy of the internal black hole state becomes too large, then the quantum recovery operation receives an error contribution of the form (77). The low energy effective field theory breaks down. This is consistent with the firewall argument. But we have learned more: with our methods we \nIn section 3.2 we show that in this case one can reconstruct the interior QFT observables in terms of the recovery operators. This is compatible with information theoretic constraints: the reconstruction works only as long as the entanglement entropy with the early radiation is sufficiently smaller than the entropy log N n of the space of the states ∣ j . \nhave been able to qualify and quantify how the effective field theory description begins to break down as the entanglement with the external environment increases. \nWhat we have just described is a standard issue in quantum measurement theory. Suppose we think of the black hole as a measuring device. We can then ask under which circumstance can it measure the state of the radiation. Measurement in general happens when the measured object decoheres by entangling itself with the internal quantum states of the measuring device. The firewall argument invokes the situation where the complete state of a big measuring apparatus A (black hole) itself has been measured in the past by an even bigger system Z (early radiation). Monogamy of entanglement then seems to imply that the smaller apparatus A can no longer do any measurements on an even smaller system B (late radiation). This is not true, in general. \nIt is useful to compare two situations: (a) the bigger system Z remains in sufficient contact with the smaller apparatus A , so that it keeps actively measuring the complete state of A , or (b) A was once maximally entangled with Z in the far past, but A and Z are dynamically decoupled at present. In case (a), A can indeed not do any measurements itself. This follows from the dynamics of dephasing: in other to measure some other small system, B , it needs to build phase coherence with B . But while A is being measured, all components of its state are randomized in ways that prevent its ability to build phase coherence with anything else. So in this case, entanglement is indeed monogamous. In case (b), however, the measuring apparatus A can in general still do measurements. Once it is dynamically decoupled from Z , it can start building new phase coherence with some other system B . So even if A started out in a maximally mixed state, over time it can build new entanglement via interactions with its environmen, and measure B . The black hole situation is more like case (b).", 'Acknowledgements': 'We thank Bartek Czech, Borun Chowdhury, Jan de Boer, Daniel Harlow, Steve Jackson, Juan Maldacena and Andrea Puhm for helpful discussions. The work of E.V. is supported by a Spinoza Grant of the Dutch Science Foundation (NWO), an Advanced Grant by the European Research Council (ERC) and the Foundation for Fundamental Research on Matter (FOM). The work of H.V. is supported by NSF grant PHY-0756966.', 'Appendix A: Some calculations': 'In this Appendix we outline some of the calculations, the were left out of the main text in Chapter 3. We first do the calculation of the expectation value (54) of the product A B of an internal operator A given in (49) and an external operator B made out of b -modes. We first do an intermediate calculation \n∣ \n∣ \n〉 \nb 〈 0 ∣ ∣ U † A B U ∣ ∣ 0 〉 b = b 0 ∣ U † a 〈 0 ∣ R † A R ∣ 0 a B U ∣ 0 〉 b (79) \n〈 〈 \n∣ 〈 ∣ ∣ ∣ 〉 = b 0 ∣ a 〈 0 ∣ U † R † AB RU ∣ 0 a ∣ 0 〉 b (80) \n∣ ∣ \n∣ \n∣ \n〉 \n∣ ∣ \nHere we first use the definition (49), then move the B operators inwards (since they commute with the recovery operators), and finally use the result (42) for the action of RU on states of the form ∣ ∣ i 〉∣ ∣ 0 〉 a ∣ ∣ 0 〉 b . Here 1 code denotes the unit operator on H code . Now we take the trace with the density matrix ρ (0) given in (47). Since ρ (0) is assumed to act within H code , we immediately find \n∣ 〈 ∣ ∣ ∣ 〉 = 〈 0 U ∣ ∣ AB ∣ ∣ 0 U 〉 1 code . (81) \ntr ( ρ ( τ ) A B ) = ∑ i , ¯ k ρ i ¯ k 〈 i ∣ ∣ 〈 0 b ∣ ∣ U † A B U ∣ ∣ 0 b 〉∣ ∣ ¯ k 〉 (82) \n= ∑ i , ¯ k ρ i ¯ k 〈 i ∣ ∣ 〈 0 U ∣ ∣ AB 0 U 〉 | ¯ k 〉 = 〈 0 U ∣ ∣ AB 0 U 〉 . (83) \nThis reproduces the expectation value in the local Minkowski vacuum. \nNext we will verify eqn (53). We will work in the limit that N code glyph[lessmuch] N . Using the definition (39), we have \nR † n R m = 1 √ W n W m ∑ i Cn ∣ ∣ i 〉〈 i ∣ ∣ C † m . (84) \nWe now compute \nR † n R s R † t R m = 1 √ W n W s W t W m ∑ i , ¯ k Cn ∣ ∣ i 〉〈 i ∣ ∣ C † s Ct ∣ ∣ ¯ k 〉〈 ¯ k ∣ ∣ C † m (85) \n= δ st √ W n W m ∑ i Cn ∣ ∣ i 〉〈 i ∣ ∣ C † m = δ st R † n R m , (86) \nwhere we used eqn (38). In the same way, we derive (55) \ntr A ( ρ R † n R m ) = ∑ i , ¯ k, ¯ s ρ i ¯ k 1 √ W t W s 〈 i ∣ ∣ C † m Cs ∣ ∣ ¯ s 〉〈 ¯ s ∣ ∣ C † t Cn ∣ ∣ ¯ k 〉 (87) \n= √ W n W m δ nt δ ms , (88) \nwhere we use (38) twice.', 'References': "- [1] A. Almheiri, D. Marolf, J. Polchinski and J. Sully, Black Holes: Complementarity or Firewalls?, 1207.3123\n- [2] S. Hawking, Particle Creation by Black Holes, Commun.Math.Phys. 43 (1975) 199220\n- [3] W. Unruh, Notes on black hole evaporation, Phys.Rev. D14 (1976) 870\n- [4] J. D. Bekenstein, 'Black holes and entropy,' Phys. Rev. D 7 , 2333 (1973).\n- [5] G. 't Hooft, 'Dimensional reduction in quantum gravity,' gr-qc/9310026.\n- [6] L. Susskind, 'The World as a hologram,' J. Math. Phys. 36 , 6377 (1995) [hepth/9409089].\n- [7] J. Maldacena, The large N limit of superconformal field theories and supergravity , Adv. Theor. Math. Phys. 2 (1998) 231, hep-th/9711200; S. Gubser, I. Klebanov and A. Polyakov, Gauge Theory Correlators from Non-Critical String Theory , Phys.Lett.B428 (1998) 105, hep-th/9802109; E. Witten, Anti-de Sitter Space and Holography , Adv.Theor.Math.Phys.2 (1998) 253, hep-th/9802150;\n- [8] L. Susskind, L. Thorlacius and J. Uglum, The Stretched horizon and black hole complementarity, Phys.Rev. D48 (1993) 37433761, hep-th/9306069; Y. Kiem, H. L. Verlinde and E. P. Verlinde, Black hole horizons and complementarity, Phys.Rev. D52 (1995) 70537065, hep-th/9502074; D. A. Lowe, J. Polchinski, L. Susskind, L. Thorlacius and J. Uglum, Black hole complementarity versus locality, Phys.Rev. D52 (1995) 69977010, hep-th/9506138\n- [9] R. Bousso, Complementarity Is Not Enough, 1207.5192; S. D. Mathur and D. Turton, Comments on black holes I: The possibility of complementarity, 1208.2005; B. D. Chowdhury and A. Puhm, Is Alice burning or fuzzing?, 1208.2026; L. Susskind, \nSingularities, Firewalls, and Complementarity, 1208.3445; I. Bena, A. Puhm and B. Vercnocke, Non-extremal Black Hole Microstates: Fuzzballs of Fire or Fuzzballs of Fuzz ?, 1208.3468; T. Banks and W. Fischler, Holographic Space-Time Does Not Predict Firewalls, 1208.4757; A. Ori, Firewall or smooth horizon?, 1208.6480; S. Hossenfelder, Comment on the black hole firewall, 1210.5317 \n- [10] L. Susskind, The Transfer of Entanglement: The Case for Firewalls, 1210.2098\n- [11] S. D. Mathur, The Information paradox: A Pedagogical introduction, Class.Quant.Grav. 26 (2009) 224001, 0909.1038. The information paradox: conflicts and resolutions, 1201.2079.\n- [12] P. Hayden and J. Preskill, Black holes as mirrors: Quantum information in random subsystems, JHEP 0709 (2007) 120, 0708.4025\n- [13] S. G. Avery, Qubit Models of Black Hole Evaporation, 1109.2911\n- [14] S. B. Giddings and Y. Shi, Quantum information transfer and models for black hole mechanics, 1205.4732\n- [15] D. N. Page, Average entropy of a subsystem, Phys.Rev.Lett. 71 (1993) 12911294, grqc/9305007\n- [16] D. N. Page, Black hole information, hep-th/9305040\n- [17] Y. Sekino and L. Susskind, Fast Scramblers, JHEP 0810 (2008) 065, 0808.2096\n- [18] M. Van Raamsdonk, Comments on quantum gravity and entanglement, 0907.2939\n- [19] G. T. Horowitz and J. M. Maldacena, The Black hole final state, JHEP 0402 (2004) 008, hep-th/0310281\n- [20] D. Gottesman and J. Preskill, 'Comment on 'The Black hole final state',' JHEP 0403 , 026 (2004) [hep-th/0311269].\n- [21] W. H. Zurek, Entropy Evaporated by a Black Hole, Phys. Rev. Lett. 49 (Dec, 1982) 16831686\n- [22] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information. Cambridge University Press, 2000\n- [23] J. Preskill, Lectures on Quantum Information and Quantum Computation, http://www.theory.caltech.edu/ ∼ preskill/ph229/#lecture \n- [24] T. Jacobson, 'Thermodynamics of space-time: The Einstein equation of state,' Phys. Rev. Lett. 75 , 1260 (1995) [gr-qc/9504004]\n- [25] S. D. Mathur, The Fuzzball proposal for black holes: An Elementary review, Fortsch.Phys. 53 (2005) 793827, hep-th/0502050"}
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2020PhRvD.101d4031G
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Lensing by Kerr black holes
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2020-01-01
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Interpreting horizon-scale observations of astrophysical black holes demands a general understanding of null geodesics in the Kerr spacetime. These may be divided into two classes: "direct" rays that primarily determine the observational appearance of a given source, and highly bent rays that produce a nested sequence of exponentially demagnified images of the main emission: the so-called "photon ring." We develop heuristics that characterize the direct rays and study the highly bent geodesics analytically. We define three critical parameters γ , δ , and τ that respectively control the demagnification, rotation, and time delay of successive images of the source, thereby providing an analytic theory of the photon ring. These observable parameters encode universal effects of general relativity, independent of the details of the emitting matter.
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https://arxiv.org/pdf/1910.12873.pdf
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{'Lensing by Kerr Black Holes': "Samuel E. Gralla 1, ∗ and Alexandru Lupsasca 2, 3, † \n1 Department of Physics, University of Arizona, Tucson, Arizona 85721, USA 2 Center for the Fundamental Laws of Nature, Harvard University, Cambridge, Massachusetts 02138, USA 3 Society of Fellows, Harvard University, Cambridge, Massachusetts 02138, USA \nInterpreting horizon-scale observations of astrophysical black holes demands a general understanding of null geodesics in the Kerr spacetime. These may be divided into two classes: 'direct' rays that primarily determine the observational appearance of a given source, and highly bent rays that produce a nested sequence of exponentially demagnified images of the main emission-the so-called 'photon ring'. We develop heuristics that characterize the direct rays and study the highly bent geodesics analytically. We define three critical parameters γ , δ , and τ that respectively control the demagnification, rotation, and time delay of successive images of the source, thereby providing an analytic theory of the photon ring. These observable parameters encode universal effects of general relativity, independent of the details of the emitting matter.", 'I. INTRODUCTION': "With the advent of horizon-scale observations of astrophysical black holes [1-6], the intricate properties of null geodesics in the Kerr spacetime [7-11] are fast becoming a matter of practical relevance to astronomy. Thanks to ray-tracing codes now operating with exquisite accuracy and speed [12-14], determining the observational appearance of a specified emission model is a quick and routine task. However, given the enormous uncertainty in the nature of the emission arising from the present targets M87* and Sgr A*, the 'inverse problem' may be more relevant: Given an observation, what can one learn about the emission profile of the source? \nAnswering this kind of question demands a general understanding of the effects of gravitational lensing in the Kerr spacetime. The authors of the present manuscript have been involved in separate, recent efforts in this direction [15, 16]. Reference [15] argued that bright rings of emission from optically thin matter [17-19] (hereafter, 'photon rings' 1 ) should be understood as superposed, exponentially demagnified images of the main emission, predicting a distinctive multipeak structure and giving the first quantitative estimate of the typical brightness enhancement (a factor of about 2-3). Soon after, Ref. [16] obtained a formula for the asymptotic demagnification factor as a function of black hole spin and observer inclination, confirmed the typical brightness enhancement and multipeak structure in state-of-the-art models [5] ray-traced at higher resolution than previously considered, and proposed an experimental method for detecting the discrete peaks using space-based interferometry. \nIn this paper, we unite our perspectives on the problem and significantly generalize these results, with the aim of presenting a complete guide to understanding lensing by Kerr black holes. \nWe have developed two new analytic tools in service of this goal: 1) a complete, fully explicit solution of the Kerr null geodesic equation expressed in terms of Legendre elliptic integrals and Jacobi elliptic functions (presented in a companion paper [11]), and 2) a logarithmic approximation valid for highly bent photons (derived in App. B). We use the first tool to explore general properties of null geodesics, and exploit the second to provide a detailed analytic theory of the photon ring. \nIt is helpful to organize the analysis by the number of orbits that an emitted photon executes before reaching the detector (Fig. 7). For 'direct' photons that complete of order half-an-orbit or less, we find that the spin of the black hole has little influence on the trajectory. For example, we show that for an equatorial (i.e., spinaligned or antialigned) disk of emission viewed face-on, the arrival impact parameter b of a photon emitted from Boyer-Lindquist radius r s is given by 'just adding one', \nb M ≈ r s M +1 , (1) \nwith this formula holding empirically to 10% accuracy at all spin (Fig. 5 left). For observers inclined relative to the disk, the spin still has little effect on the arrival position from a fixed equatorial radius, although it does shrink the apparent size of the equator of the black hole (Fig. 6). For models with emission extending to the horizon, the observed central dark area will correspondingly shrink. \nPhotons that make of order half-an-orbit to one orbit contribute a demagnified image of the source over a band surrounding a critical curve on the image plane. For a diffuse, optically thin source near the horizon, this image superposes onto the direct emission to produce a thin ring with diameter ∼ 10 M , width ∼ M , and about twice the \nbackground intensity [15], 2 a striking feature in simulated images [5]. Here, we show that the precise width of this band varies significantly with spin (Fig. 5 left), especially in the region corresponding to photons emitted from the vicinity of the horizon. For models with emission near the horizon, the demagnified image will therefore be broader, and contribute significantly more flux, when the black hole spins rapidly. \nPhotons executing of order one orbit or more contribute a sequence of highly demagnified images near the critical curve [15, 16, 20, 21]. We derive an asymptotic expansion for the number of orbits as a function of the (perpendicular) distance from the critical curve, and show that the resulting logarithmic approximation is excellent even for photons executing of order only a single orbit (Fig. 3). We develop a precise analytic theory of the demagnified images based on three key quantities defined for the bound photon orbits: \n- · The Lyapunov exponent γ characterizing the instability of the bound orbit, defined relative to a half-libration in polar angle θ [16].\n- · The change δ in azimuthal angle φ over a polar half-libration [22].\n- · The period τ of a polar half-libration. \nWe show that for an equatorial disk of emission viewed face-on, each successive image is demagnified by a factor of e -γ , rotated by an angle δ , and delayed by a time τ . These images alternate between showing the front side and the backside of the disk (Fig. 7). For nonequatorial sources, we instead distinguish two families of images, each with demagnification e -2 γ , rotation 2 δ , and time delay 2 τ . These simple associations break down when the observer is significantly inclined, but we are still able to make precise statements about the origin of emission as a function of observed position near the critical curve. \nThese results unite and generalize our previous treatments of the demagnification factor [15, 16], while also introducing δ and τ as additional key quantities characterizing the demagnified images. The spin-dependent critical parameters γ , δ and τ control universal (matterindependent) features of general relativity that could in principle be observed with future detectors. \nThis paper is organized as follows. In Sec. II, we review and present a useful formalism for Kerr null geodesics. Next, in Sec. III, we analyze the bound photon orbits, and define their critical parameters γ , δ , and τ . Then, in Sec. IV, we discuss the screen of a distant observer, presenting new details about the map from conserved quantities to position in the image plane. We describe properties of complete rays in the Kerr exterior in Sec. V, and study segments of rays that represent propagation \nfrom source to observer in Sec. VI. Finally, in Sec. VII, we develop the analytic theory of the photon ring in terms of the critical parameters γ , δ , and τ .", 'II. GENERAL FRAMEWORK': "We work with Boyer-Lindquist coordinates ( t, r, θ, φ ) on the spacetime of a Kerr black hole with mass M and angular momentum J = Ma , and define \nΣ( r, θ ) = r 2 + a 2 cos 2 θ, ∆( r ) = r 2 -2 Mr + a 2 . (2) \nThe roots of ∆( r ) correspond to the outer/inner horizons \nr ± = M ± √ M 2 -a 2 . (3) \nWe assume that 0 < a < M , such that the coordinate φ increases in the sense of rotation of the black hole. The nonrotating ( a → 0) and extremal ( a → M ) limits may be taken after final observables are computed. \nIn discussing null geodesics, we will make a distinction between 'rays' and 'photons'. By a ray , we will mean a complete null geodesic in the Kerr exterior, which enters from the white hole or the celestial sphere, before eventually leaving via the black hole or the celestial sphere. By a photon , we will mean a portion of a ray, which represents the emission and absorption (or observation) of light. In radiative transport, one considers rays that propagate through a medium, gaining and losing photons (according to the local emissivity and absorptivity) on their way to the detector. \nWe will adopt the 'integral' approach to the study of null geodesics in the Kerr spacetime. In this approach, pioneered by Carter [7] and Bardeen [8], one reduces the equations to quadratures using conserved quantities. Building on important earlier developments [23-26], in a companion paper [11] we have classified all motions, reduced all integrals to real elliptic form, and inverted the equations to provide explicit, parameterized trajectories. Herein, we only summarize the results needed for this paper; complete derivations may be found in Ref. [11]. \nEach Kerr photon trajectory possesses two conserved quantities λ and η , corresponding to the energy-rescaled angular momentum and Carter integral, respectively. These allow the four-momentum p µ along the trajectory to be reconstructed as \nΣ E p r = ± r √ R ( r ) , (4a) \nΣ E p θ = ± θ √ Θ( θ ) , (4b) \nΣ E p φ = a ∆ ( r 2 + a 2 -aλ ) + λ sin 2 θ -a, (4c) \nΣ E p t = r 2 + a 2 ∆ ( r 2 + a 2 -aλ ) + a ( λ -a sin 2 θ ) , (4d) \nwhere E = -p t is the constant 'energy at infinity' 3 and \nR ( r ) = ( r 2 + a 2 -aλ ) 2 -∆( r ) [ η +( λ -a ) 2 ] , (5) \nΘ( θ ) = η + a 2 cos 2 θ -λ 2 cot 2 θ. (6) \nThe symbols ± r and ± θ indicate the sign of p r and p θ , respectively. Turning points in r and θ occur at zeros of the radial and angular 'potentials' R ( r ) and Θ( θ ), respectively. \nConsider a null geodesic connecting spacetime events ( t s , r s , θ s , φ s ) and ( t o , r o , θ o , φ o ), where s and o stand for source and observer. By integrating along the trajectory, the geodesic equation (4) may be recast in integral form, 4 \nI r = G θ , (7a) \n∆ φ := φ o -φ s = I φ + λG φ , (7b) \n∆ t := t o -t s = I t + a 2 G t , (7c) \nwhere we define \nI r = r o r s d r ± r √ R ( r ) , (8a) \nG θ = θ o θ s d θ ± θ √ Θ( θ ) , (8b) \nI φ = r o r s a (2 Mr -aλ ) ± r ∆( r ) √ R ( r ) d r, (8c) \nG φ = θ o θ s csc 2 θ ± θ √ Θ( θ ) d θ, (8d) \nI t = r o r s r 2 ∆( r ) + 2 Mr ( r 2 + a 2 -aλ ) ± r ∆( r ) √ R ( r ) d r, (8e) \nG t = θ o θ s cos 2 θ ± θ √ Θ( θ ) d θ. (8f) \nHere, the notation GLYPH<31> indicates that these integrals are to be understood as path integrals along the photon trajectory, with the signs ± r = sign( p r ) and ± θ = sign( p θ ) switching at radial and angular turning points, respectively. In particular, all path integrals increase monotonically along the trajectory.", 'A. Angular integrals': "The analysis of the angular integrals differs depending on the region of conserved quantity space. In this paper, unless otherwise specified, we will restrict to positive η , \nη > 0 , (9) \nthereby excluding the so-called 'vortical' geodesics with η < 0. (This excludes only a small portion near the middle of an observer's screen, where the image is normally dark-see Fig. 1 below. Furthermore, equatorial sources cannot emit vortical photons, as these never intersect the equatorial plane.) The η > 0 geodesics librate between turning points θ ± above and below the equatorial plane, \nθ ± = arccos ( ∓ √ u + ) , (10) \nwhere \nu ± = glyph[triangle] θ ± √ glyph[triangle] 2 θ + η a 2 , glyph[triangle] θ = 1 2 ( 1 -η + λ 2 a 2 ) . (11) \nTo aid in the expression of the angular path integrals G θ , G φ , and G t , we introduce the notation \nF i = F ( arcsin ( cos θ i √ u + )∣ ∣ ∣ ∣ u + u -) , (12) \nΠ i = Π ( u + ; arcsin ( cos θ i √ u + )∣ ∣ ∣ ∣ u + u -) , (13) \nE ' i = E ' ( arcsin ( cos θ i √ u + )∣ ∣ ∣ ∣ u + u -) , (14) \nwhere i ∈ { s, o } can be either source or observer. Here, F ( ϕ | k ), E ( ϕ | k ), and Π( n ; ϕ | k ) respectively denote the incomplete elliptic integrals of the first, second, and third kind, 5 while the prime denotes a derivative with respect to k , E ' ( ϕ | k ) := ∂ k E ( ϕ | k ) = [ E ( ϕ | k ) -F ( ϕ | k )] / (2 k ). These integrals vanish at the equator, \nF i = Π i = E ' i = 0 , ( θ i = 0) (15) \nand become complete at turning points, \nF i = ∓ K, Π i = ∓ Π , E ' i = ∓ E, ( θ i = θ ± ) (16) \nwhere our notation for the complete elliptic integrals is \nK = K ( u + u -) = F ( π 2 ∣ ∣ ∣ ∣ u + u -) , (17) \nΠ = Π ( u + ∣ ∣ ∣ ∣ u + u -) = Π ( u + ; π 2 ∣ ∣ ∣ ∣ u + u -) , (18) \nE ' = E ' ( u + u -) = E ' ( π 2 ∣ ∣ ∣ ∣ u + u -) . (19) \nThe η > 0 angular path integrals may be written in terms of these quantities and the number m of angular turning points encountered along the trajectory as [11, 26] \nG θ = 1 a √ -u -[2 mK ± s F s ∓ o F o ] , (20) \nG φ = 1 a √ -u -[2 m Π ± s Π s ∓ o Π o ] , (21) \nG t = -2 u + a √ -u -[2 mE ' ± s E ' s ∓ o E ' o ] , (22) \nwith ± i denoting the sign of p θ at the source ( i = s ) or observer ( i = o ) point, \n± i = sign ( p θ i ) . (23) \nSince p θ changes sign after each turning point, these signs obey the constraint \n± s = ± o ( -1) m . (24) \nFinally, note that the integral for G θ can be inverted to solve for θ o or θ s as a function of G θ [11, 25, 26]. Since in this paper, we mainly fix the observer point (a telescope at infinity), we present θ s in terms of θ o and G θ . This may be inferred from expressions for θ o ( G θ , θ s ) by interchanging s and o , before sending G θ → -G θ to compensate. 6 From Eq. (71) of Ref. [11] (noting that τ therein denotes G θ , while ν θ therein denotes ± s ), we find \ncos θ s √ u + = sn ( F o ± o sign( η ) a √ -u -G θ ∣ ∣ ∣ ∣ u + u -) , (25) \nwhere sn( ϕ | k ) denotes the Jacobi elliptic sine function. This formula holds regardless of the sign of η [11, 26].", 'B. Radial integrals': 'In this paper, we will consider a distant observer at \nr o →∞ . (26) \nGeodesics that reach this far observer have at most one radial turning point outside the horizon. Given a choice of conserved quantities ( λ, η ), a simple way to test whether the ray has a turning point is to compute r 4 ( λ, η ) via Eq. (A8d) below. If r 4 is real and outside the horizon, then the ray has a turning point at radius r 4 ; otherwise, the ray never encounters a turning point. \nFor the rays with no turning point, the radial integrals I r , I φ , and I t are single-valued functions of r s , while for the rays with a turning point, these radial integrals must be double-valued in order to track whether or not the turning point has been reached. We will denote the number of turning points of a photon (portion of a ray) by w ∈ { 0 , 1 } . The radial integral I r may then be written \nI r = GLYPH<2> ∞ r s d r √ R ( r ) +2 w GLYPH<2> r s r 4 d r √ R ( r ) , (27) \nand likewise for I φ and I t with the appropriate integrands. 7 We may relate w to the emission direction by \nw = { 0 p r s > 0 , 1 p r s < 0 (and r + < r 4 < r s ) . (28) \nThe conditions r s > r 4 > r + ensure that r s lies along a ray that reaches infinity after passing through a turning point r 4 (such that negative initial radial momentum is allowed). If these conditions are not both satisfied for a given choice of conserved quantities, then only w = 0 is allowed for those quantities, i.e., only photons emitted outward will reach infinity. \nA ray reaching infinity originates either from the event horizon (of the white hole) or from infinity. We denote the associated radial integral I r by I total r , \nI total r = 2 GLYPH<2> ∞ r 4 d r √ R ( r ) r + < r 4 ∈ R , GLYPH<2> ∞ r + d r √ R ( r ) otherwise , (29) \nwhere we remind the reader that a ray reaching infinity began at infinity if r 4 ( λ, η ) is real and greater than the horizon, and otherwise began at the (white hole) horizon. \nThe full set of radial integrals were evaluated and reduced to elliptic form in Ref. [11], building on previous work in Refs. [24, 25]. The necessary antiderivatives for computing Eqs. (27) and (29) are given in App. A below. \nAs in Eq. (25) for θ s ( G θ ), one may derive an inversion formula for r s ( I r ) [11, 25]. Eq. (B119) of Ref. [11] gives a formula for r o , and we may infer the formula for r s as described above Eq. (25), i.e., by interchanging o and s and then sending I r →-I r . Noting that τ = I r therein, and letting r o →∞ , the emission radius is given by \nr s = r 4 r 31 -r 3 r 41 sn 2 ( 1 2 √ r 31 r 42 I r -F o ∣ ∣ k ) r 31 -r 41 sn 2 ( 1 2 √ r 31 r 42 I r -F o ∣ ∣ k ) , (30) \nwith \nF o = F ( arcsin √ r 31 r 41 ∣ ∣ ∣ ∣ k ) , k = r 32 r 41 r 31 r 42 . (31) \nHere, we introduced the notation \nr ij = r i -r j , (32) \nwith the roots { r 1 , r 2 , r 3 , r 4 } given in Eqs. (A8) below. This formula is contingent on the radial integral I r being in the allowed range, \n0 < I r < I total r . (33) \nProvided that Eq. (33) is satisfied, Eq. (30) gives the emission radius of a photon reaching infinity with conserved quantities ( λ, η ). This formula holds even when (some of) the radial roots are complex [11].', 'C. Fractional number of orbits': 'It is useful to have some measure of the total number of orbits executed by a given photon. However, since the spatial trajectory is three-dimensional, there is some arbitrariness in the definition of an orbit. As in Ref. [16], \nwe define the journey from the equator to a polar turning point θ ± to be one quarter of an orbit, so that beginning and ending at the same turning point constitutes one full orbit. For a measure of the fractional number of orbits, we seek a quantity that grows monotonically from zero, increasing by 1 after completing an orbit as defined above. Since the path integral G θ satisfies the requisite monotonicity property, we simply normalize by its value G 1 θ over one orbit, 8 defining the fractional number of orbits n to be \nn = G θ G 1 θ , (34) \nwith \nG 1 θ = 2 GLYPH<2> θ + θ -d θ √ Θ( θ ) = 4 K a √ -u -. (35) \nUsing I r = G θ [Eq. (7a) above], we equivalently have \nn = a √ -u -4 K I r . (36) \nNote that I r = G θ is also the Mino time parameter [27] that decouples the differential equations (4). Our parameter n is proportional to the Mino time and provides a new physical interpretation of this quantity.', 'III. CRITICAL RAYS': "For generic values of λ and η , the radial potential (5) possesses four distinct roots (A8), of which the real subset corresponds to radial turning points. At special 'critical' values ˜ λ and ˜ η , the radial potential may develop a double root at some special radius ˜ r , \nR (˜ r ) = R ' (˜ r ) = 0 . (37) \nThis occurs for ˜ r > r + if [8] and only if [11] \n˜ λ = a + ˜ r a [ ˜ r -2 ˜ ∆ ˜ r -M ] , (38) \n˜ η = ˜ r 3 a 2 [ 4 M ˜ ∆ (˜ r -M ) 2 -˜ r ] , (39) \nwhere ˜ r must lie in the range ˜ r ∈ [˜ r -, ˜ r + ], with \n˜ r ± = 2 M [ 1 + cos ( 2 3 arccos ( ± a M ) )] . (40) \nHere and below, we use the notation ˜ Q for a quantity Q evaluated at criticality, i.e., at r = ˜ r , λ = ˜ λ , and η = ˜ η . \nThe double root (37) indicates the existence of orbits with fixed Boyer-Lindquist radius ˜ r , i.e., bound photon orbits . At the boundaries (40) of the allowed range, the orbits are circular, equatorial, and prograde (˜ r -) or retrograde (˜ r + ), whereas for intermediate radii the orbits also librate between turning points θ -and θ + given in Eq. (10) above [note from Eqs. (39) and (40) that ˜ η ≥ 0]. The pole-crossing orbits ˜ λ = 0 (where the turning points approach the poles) lie at the radius ˜ r = ˜ r 0 given by \n˜ r 0 = M +2 √ M 2 -a 2 3 cos 1 3 arccos ( 1 -a 2 M 2 ) ( 1 -a 2 3 M 2 ) 3 / 2 . (41) \nThus, the region of the Kerr spacetime spanned by bound photon orbits takes the shape of a spherical shell of variable thickness (the 'photon shell'), which is thickest at the equator and vanishingly thin at the pole (e.g., Fig. 2 of Ref. [16]). This shell is largest in the extremal limit a → M , in which its range extends from ˜ r -= M to ˜ r + = 4 M at the equator. In the nonrotating limit a → 0, the shell is vanishingly thin everywhere, degenerating to the 'photon sphere' r = 3 M . \nSince there are no orbits that oscillate between two radial turning points outside the horizon, the bound photon orbits are unstable. The rate of deviation of nearby orbits may be characterized by a Lyapunov exponent, which is usually defined with respect to a coordinate or affine time (e.g., as in Ref. [28]). We will instead follow Ref. [16] and define the exponent using the fractional number of orbits as a parameter. Consider a precisely critical ray with conserved quantities ˜ λ (˜ r ) and ˜ η (˜ r ), but that is not precisely at the radius ˜ r . (Such rays approach the critical radius in the asymptotic future or past.) In the regime | r -˜ r | glyph[lessmuch] ˜ r , a simple calculation (App. A1 of [16]) gives 9 \nr 2 -˜ r r 1 -˜ r ≈ e 2 γ ( n 2 -n 1 ) , (42) \nwhere r 1 and r 2 denote the photon radius after executing n 1 and n 2 fractional orbits, respectively, while the Lyapunov exponent is \nγ = 4˜ r √ ˜ χ a √ -˜ u -˜ K. (43) \nHere, ˜ K = K (˜ u + / ˜ u -) is evaluated using the critical conserved quantities according to the convention established above, while ˜ χ is defined as \n˜ χ = 1 -M ∆(˜ r ) ˜ r (˜ r -M ) 2 . (44) \nWe will see below that γ controls the demagnification of successive images of an isotropically emitting source, as first realized in Ref. [16]. \nIt is useful to know the change in φ accrued over each orbit (period in the θ -motion) of a bound photon. This quantity was computed by Teo [22], and may also be be inferred from an r → ˜ r limit of the integral formulation above, as follows. First, note from Eqs. (8) that for r ≈ ˜ r , we have \nI φ ≈ a ( ˜ r + M ˜ r -M ) I r = a ( ˜ r + M ˜ r -M ) G θ , (45) \nwhere the last step follows from Eq. (7a). Letting r → ˜ r in Eq. (7b) after using Eq. (45), the change in φ for a bound photon is given in terms of angular integrals as \n∆ φ = a ( ˜ r + M ˜ r -M ) G θ + λG φ . (46) \nTo determine the change in φ over a complete orbit, we use the formulas (20) and (21) with θ s = θ o and m = 2. Denoting this change in φ by 2 ˆ δ , we find \nˆ δ = 2 a √ -˜ u -[ a ( ˜ r + M ˜ r -M ) ˜ K + ˜ λ ˜ Π ] , (47) \nin agreement with Eq. (18) of Ref. [22]. This quantity ˆ δ encodes the change in φ completed by a bound photon over each half-orbit. \nAs discussed in Ref. [22], this expression for ˆ δ is not a smooth function of ˜ r , but rather has a jump discontinuity of 2 π at the pole-crossing orbit ˜ r = ˜ r 0 . This can be understood by imagining two photons passing nearly over the pole, but on opposite sides. The photon moving in a locally counterclockwise direction is regarded as having accumulated approximately π radians during the passage, whereas the clockwise photon passing on the other side is regarded as having accumulated -π radians. This discontinuity is essential to the mathematics of the integral formulation of the equations, but for presenting final results it will be convenient to define a continuous function by adding 2 π to the ˜ r > ˜ r 0 branch of ˆ δ . We will denote this smooth version by δ , \nδ = ˆ δ +2 πH (˜ r -˜ r 0 ) , (48) \nwhere H ( x ) denotes the Heaviside function. Combining Eqs. (47) and (48) gives \nδ = 2 √ -˜ u -[ ( ˜ r + M ˜ r -M ) ˜ K + ˜ λ ˜ Π a ] +2 πH (˜ r -˜ r 0 ) . (49) \nWe will see below that δ controls the apparent rotation of successive images of an isotropically emitting source. \nFinally, consider the elapsed time t over a full libration. By a similar argument as used for Eq. (45), we find \nI t ≈ ˜ r 2 ( ˜ r +3 M ˜ r -M ) G θ . (50) \nPlugging this into Eq. (7c) and letting r → ˜ r leads to \n∆ t = ˜ r 2 ( ˜ r +3 M ˜ r -M ) G θ + a 2 G t (51) \nfor a bound photon orbit. Using Eqs. (20) and (22) with θ s = θ o and m = 2 gives the lapse in t for a full orbit. Denoting this time lapse over a full orbit by 2 τ , we find \nτ = 2 a √ -˜ u -[ ˜ r 2 ( ˜ r +3 M ˜ r -M ) ˜ K -2 a 2 ˜ u + ˜ E ' ] . (52) \nThis quantity τ gives the change in t over each half-orbit of a bound photon. We will see below that τ controls the time-delay between the arrival of successive images of an isotropically emitting source.", 'IV. THE SCREEN OF A DISTANT OBSERVER': "Now consider a distant observer with inclination θ o relative to the spin axis of the black hole. We will exclude the equatorial case and use the reflection symmetry of the spacetime to place the observer in the upper hemisphere, \nθ o ∈ [0 , π/ 2) . (53) \nFirst, consider the off-axis case θ o glyph[negationslash] = 0. We use the axisymmetry of the spacetime to set the observer azimuthal angle to zero, \nObserver θ o glyph[negationslash] = 0 : r o →∞ , φ o = 0 . (54) \nOrthogonal impact parameters ( α, β ) of photons reaching the observer (54) are proportional to direction cosines on the observer's sky, and may therefore be regarded as image plane Cartesian coordinates. Expressed in terms of photon conserved quantities, a convenient choice is [8, 29] \nα = -λ sin θ o , β = ± o √ Θ( θ o ) (55) = ± o √ η + a 2 cos 2 θ o -λ 2 cot 2 θ o . \nThis defines a 'line of sight' α = β = 0 to the black hole, with the β -axis regarded as the projection of the spin axis onto the plane perpendicular to this line of sight. 10 The projected black hole rotation is in the counterclockwise direction as seen by the observer. In comparing to an observed image, one may rescale α and β to adjust for angular size, translate or rotate to adjust for the position and orientation of the source, and reverse the handedness \n<!-- image --> \nFIG. 1. The 2-1 mapping from conserved quantities ( λ, η ) to image coordinates ( α, β ). The curve C + of critical rays separates the regions of ( λ, η )-space where rays have no radial turning points (blue and yellow) from the region where they have a single radial turning point (green). (Yellow rays are vortical, while blue rays are ordinary.) Rays can reach an observer at inclination θ o only in the darker portion inside the gray parabola. The 2-1 image of this portion of C + defines the image-plane critical curve C . As θ o → 0 , the parabola closes to the vertical half-line λ = 0, η > -a 2 , while as θ o → π/ 2, it opens up to a horizontal line η = 0, such that the entire blue and green regions (and none of the yellow region) map to the image. As a → 0, the vortical region disappears from both plots (no vortical geodesics exist). In these plots, we chose a/M = 94%, θ o = 17 · , and set M = 1. \n<!-- image --> \nα → -α to account for the projected black hole spin direction. Finally, notice that we have \n± o = sign( β ) . (56) \nRays that reach our distant observer may have two qualitatively different origins: they either came from the white hole, or else from the celestial sphere. Equivalently, we may imagine tracing a photon back in time from the observer and asking whether it 'ends up' (started) at the horizon r = r + , or at infinity r → ∞ . 11 The boundary between these two behaviors corresponds to a ray that, when traced backwards in time, orbits indefinitely as it approaches a bound orbit at some radius ˜ r . Such rays must have the same conserved quantities ˜ λ (˜ r ) and ˜ η (˜ r ) [given in Eqs. (38) and (39) above] as the bound photon orbits. This condition defines the critical curve C . \nThe radius of the associated photon orbit provides a convenient parameterization of C , \n˜ α = α ( ˜ λ (˜ r )) , ˜ β = β ( ˜ λ (˜ r ) , ˜ η (˜ r )) , (57) \ndefined using Eqs. (38), (39), and (55). In light of the sign ± o = sign( β ) in Eqs. (55), Eq. (57) really refers to two separate parameterized curves (one in the upper halfplane and one in the lower half), whose union gives rise to the closed curve C on the image plane. Put differently, the critical curve is a 2-1 mapping from the critical locus in conserved quantity space (Fig. 1). In particular, C is reflection-symmetric about the α axis. The range of the parameter ˜ r is determined by the requirement that ˜ β be real, which restricts to bound photon orbits for which nearby photons can escape to infinity at the observer inclination θ o (see Fig. 2 of Ref. [16]). In the edge-on case θ o = π/ 2, this corresponds to the full range ˜ r ∈ [˜ r -, ˜ r + ] of bound orbits in the photon shell [Eq. (40)], whereas at smaller inclinations, there is a smaller range that can be determined numerically by finding the roots of ˜ β (˜ r ). \nThe shape of the critical curve depends on the black hole spin a and the observer inclination θ o . However, it is very nearly circular everywhere across this parameter space, except in the extremal, edge-on limit, where it becomes flattened on one side [8, 16, 18, 19, 29]. \nIt is useful to have a simple test of whether a given screen position ( α, β ) lies inside the critical curve. One method is to compute ( λ, η ) via the inversion of Eq. (55), \nλ = -α sin θ o , (58) \nη = ( α 2 -a 2 ) cos 2 θ 0 + β 2 , (59) \nand then plug these parameters into the formula (A8d) for the radial root r 4 , which is always the outermost turning point outside the horizon (when it exists). That is, \nThe screen point ( α, β ) is outside C if r 4 ( α, β ) is real and outside the horizon; otherwise it lies inside C [ r 4 is constructed from Eqs. (58), (59), and (A8d)].", 'A. On-axis observer': "In the special case θ o = 0 of an on-axis observer, it is more convenient to use polar coordinates ( b, ϕ ) on the image plane. Here, b is the impact radius b = √ α 2 + β 2 and ϕ is the angle of arrival, \nϕ = φ o ( θ 0 = 0 , r o →∞ ) . (60) \nSince photons that reach the pole must have vanishing azimuthal angular momentum ( λ = 0), it follows from Eqs. (55) that \nb = √ η + a 2 . (61) \nMoreover, since all photons reach a polar observer with negative p θ o , we also have from Eq. (56) that \n± o = -1 . (62) \nTo simplify expressions in the case of a polar observer, we send \nλ → 0 , η → b 2 -a 2 , (63) \nwhich in particular sends \nu + → 1 , u -→ 1 -b 2 a 2 , (64) \nas well as \nθ -→ 0 , θ + → π. (65) \nIn most expressions, one can simply set these values, but more care is needed near turning points (pole crossings). In particular, the angle φ jumps by φ → φ + π discontinuously at each turning point. This coordinate artefact is reflected in the mathematics as a divergence of the angular integral G φ at each turning point. The relevant finite limit (recalling that η > 0) is \nlim λ → 0 ± 2 λ Π a √ -u -= ± π. (66) \nThe critical curve of a polar observer is a perfect circle centered at the origin. The range of ˜ r degenerates to a single value ˜ r = ˜ r 0 , which is the unique radius (41) in the photon shell [˜ r -, ˜ r + ] that admits pole-crossing bound orbits ( ˜ λ = 0). That is, from the perspective of a polar observer, the only visible portion of the photon shell is a \nphoton sphere. The critical curve radius ˜ b = √ ˜ η + a 2 is given by \n˜ b = √ √ √ √ ˜ r 3 0 a 2 [ 4 M ∆(˜ r 0 ) (˜ r 0 -M ) 2 -˜ r 0 ] + a 2 . (67) \nIn this case, the angle ϕ = φ o may be viewed as the parameter along C . \nUsing Eqs. (63), (64), and (66), the critical parameters γ , δ , and τ reduce to \nγ 0 = 4˜ r 0 √ ˜ b 2 -a 2 √ 1 -M ∆(˜ r 0 ) ˜ r 0 (˜ r 0 -M ) 2 K ( a 2 a 2 -˜ b 2 ) , (68) δ 0 = π + 2 a √ ˜ b 2 -a 2 ( ˜ r 0 + M ˜ r 0 -M ) K ( a 2 a 2 -˜ b 2 ) , (69) τ 0 = 2 √ ˜ b 2 -a 2 [ ˜ r 2 0 ( ˜ r 0 +3 M ˜ r 0 -M ) K ( a 2 a 2 -˜ b 2 ) -2 a 2 E ' ( a 2 a 2 -˜ b 2 )] . (70) \nIn the limit a → 0 of a nonspinning black hole (where any observer can be made polar by rotational symmetry), these quantities simplify tremendously: \n˜ r 0 = 3 M, ˜ b = 3 √ 3 M, (71) \nγ 0 = δ 0 = π, τ 0 = 3 3 πM. (72) \n√ \nThese critical parameters characterize the critical orbits in the photon spheret of the Schwarzschild spacetime. \nIt is helpful to contrast the cases of on-axis and offaxis observers. In the off-axis case θ o glyph[negationslash] = 0, we set the azimuthal coordinate to a fiducial value φ o = 0, and the two conserved quantities λ and η (together with the sign ± o ) encode the arrival position of photons via Eqs. (55). On the other hand, in the on-axis case θ o = 0, one conserved quantity λ always vanishes, and the arrival position is encoded by the second conserved quantity η together with the azimuthal coordinate φ o via Eqs. (60) and (61). The critical curve has a similar shape in each case but a rather different mathematical description: for off-axis observers, we parameterize it by ˜ r , while for on-axis observers, we have ˜ r = ˜ r 0 and the curve is instead parameterized by ϕ (and given by b = ˜ b ).", 'V. BEHAVIOR OF RAYS': "We now make some general comments about the properties of rays, i.e., complete null geodesics in the Kerr exterior. Their radial integral I r is the total integral discussed in Eq. (29) above. Plugging Eqs. (7a) and (56) into Eq. (25), we find that (regardless of the sign of η ) \ncos θ s √ u + = sn ( F o +sign( ηβ ) a √ -u -I total r ∣ ∣ ∣ ∣ u + u -) . (73) \nFIG. 2. Latitude bands of the event horizon and celestial sphere, as seen by a distant observer. Rays from the horizon (emitted just outside the black hole, or emerging from the white hole) arrive within the critical curve (black), while rays from the celestial sphere arrive outside of it. We show the screen position of these rays, colored by the latitude of emission on the event horizon or celestial sphere, as shown in central inset (colors change every 30 · , with orange/green dots depicting the north/south poles). The observer sees infinitely many 'unfoldings' of both the horizon and the celestial sphere. Here, we show an extreme black hole ( a = M ) as viewed by a distant observer at inclinations (clockwise from top left) θ o = 0 · , 17 · , 60 · , and 90 · . \n<!-- image --> \nFIG. 3. Fractional number of orbits n as a function of signed perpendicular distance d from the critical curve C on a distant observer's image plane. Top: black hole spin a/M = 94% and observer inclination θ o = 17 · ; bottom: spin a/M = 99 . 9% and inclination θ o = 90 · . The curve C is parameterized in two separate segments above and below the α -axis by the radius ˜ r that rays asymptotically approach. (The directions of increasing ˜ r are indicated on each segment by red arrows. The range of ˜ r is determined by the condition β 2 ≥ 0; only the equatorial observer θ o = π/ 2 sees the entire range ˜ r ∈ [˜ r -, ˜ r + ] of bound photon orbits.) Physically, the coordinate system (˜ r, d ) labels (nearly) bound photons by the Boyer-Lindquist radius ˜ r of their (nearby) spherical photon orbit. The fractional number of orbits diverges logarithmically as | d | → 0. The logarithmic approximation [Eq. (74)] is excellent within a distance ∼ M of the critical curve (we set M = 1 in all the plots), except near the vertical straight line ('NHEKline') that appears in the extremal limit for θ o glyph[greaterorsimilar] 47 · and requires a separate analytic treatment [29]. \n<!-- image --> \nThe formula (73) gives the latitude at which the ray arriving at screen coordinate ( α, β ) entered the spacetime (either from the white hole if arriving inside C , or from the celestial sphere if arriving outside C ). The level sets of this function show how the horizon and celestial sphere are 'unfolded' infinitely many times on the image plane, converging to the critical curve (Fig. 2). \nEach successive unfolding corresponds to a photon that has undergone an additional half-orbit before reaching the observer. To study this effect quantitatively, we consider the total (fractional) number of oribts n , which is proportional to I r by Eq. (36). The results of App. B provide an asymptotic expansion valid for near-critical rays. From Eqs. (36), (B45), (B49) and (B56), we have \nn ≈ -1 2 γ (˜ r ) log [ ˆ C ± (˜ r ) d ] , d → 0 ± , (74) \nwhere d is the signed perpendicular distance from the \nclosest point ˜ r on the critical curve, γ (˜ r ) is the Lyapunov exponent (43), and we also introduced coefficients \nˆ C + (˜ r ) = ( 1 + √ ˜ χ 8˜ χ ) 2 ∆(˜ r ) 2˜ r 4 ˜ χ √ ˜ β 2 + ˜ ψ 2 , (75) \nˆ C -(˜ r ) = -√ 1 -˜ χ 1 + √ ˜ χ √ 1 + Q 2 ( δr + , 0) 1 -Q 2 ( δr + , 0) ˆ C + (˜ r ) . (76) \nSee Eqs. (44), (57), (B23), (B24) and (B54) for definitions of the various quantities that appear. In the nonrotating limit a → 0, Eqs. (74), (75) and (76) agree with Eqs. (2), (3), and (4) of Ref. [15]. The exact and approximate fractional number of orbits are shown in Fig. 3. \nAs depicted in Fig. 3, we may think of (˜ r, d ) as a set of coordinates for the image plane that are defined in the neighborhood of C for which there is a unique line segment connecting any point p to C , with the line intersecting C perpendicularly. The coordinate ˜ r of the point \nFIG. 4. Behavior of photons emitted from a source sphere r = r s and received at the pole θ o = 0. We show the cosine of the emission latitude θ s as a function of screen radius b . Each oscillation from +1 to -1 represents an image of the source sphere. When the source sphere is inside the photon sphere ( r s < ˜ r 0 ), the images do not overlap-the sphere is 'unwrapped' infinitely many times on the image plane. When the source sphere is outside the photon sphere, its first image is folded on itself, and subsequent images are superposed on this first image. If the sphere is optically thick, emission corresponding to dashed lines will not be visible. In the flat spacetime this corresponds to the statement that one sees only the top half of the sphere. \n<!-- image --> \np is the Boyer-Lindquist radius of the associated photon orbit where C is intersected, and the coordinate d is the signed length of the segment (i.e., | d | is the length, with d positive/negative when the point p is outside/inside C ). This actually defines two coordinate charts-one in the upper half-plane and one in the lower half-plane-since each radius ˜ r corresponds to two points on C related by β → -β . That is, points near C are uniquely described by (˜ r, d, sign( β )). We will generally leave the sign( β )-dependence implicit, regarding (˜ r, d ) as a single chart. In the case of an on-axis observer θ o = 0, for whom the ˜ r -parameterization breaks down, we would instead use ( ϕ, d ), where ϕ = φ o and d = b -˜ b , with ˜ b given by Eq. (67). \nThe formula (74) may be compared with Eq. (11) of Ref. [16]. Accounting for a factor of two difference in the definition of n , the prefactors agree exactly, but the argument in the log differs in two ways. First, we include the coefficients ˆ C ± associated with a definite physical quantity, the total (fractional) number of orbits outside the horizon. Strictly speaking, these are subleading to the dominant log d term, but nonetheless they are necessary to attain any reasonable degree of accuracy. The second difference is that the dependence on the deviation from the critical curve appears as the normal distance d in place of the unspecified displacement δρ/ρ c in Ref. [16], making precise the scaling argument given therein.", 'VI. BEHAVIOR OF PHOTONS': "We now make some general comments about the behavior of photons reaching the observer, i.e., portions of null geodesics corresponding to emission and observation of light. We will consider the apparent positions (location on the observer screen) of various simple geometric \nsources. A given source has infinitely many apparent positions (arising from photons making arbitrarily many orbits around the black hole), but throughout this section, we confine our attention to the first one or two, deferring discussion of higher-order images to Sec. VII below. We use the term 'position' even when discussing extended sources; for example, the apparent positions of a source ring ( r s , θ s ) are closed curves on the image plane.", 'A. Spheres observed from the pole': "We begin by discussing the apparent positions of latitude lines on a sphere of some radius r s , as viewed from above ( θ o = 0). Recall from Sec. IV A that we use polar coordinates ( b, ϕ ) on the image plane for such an observer. Using Eqs. (7a), (63) and (64), Eq. (25) becomes \ncos θ s = cd ( √ b 2 -a 2 I r ∣ ∣ ∣ ∣ a 2 a 2 -b 2 ) , (77) \nwhere cd is the Jacobi elliptic function cd( ϕ | k ). The integral I r may be computed either numerically, or using elliptic integrals; we use expressions given in Ref. [11]. For fixed r s , the radial integral I r is a function of b that is single-valued for b < ˜ b and double-valued for b > ˜ b [see Eq. (27)]. Thus, for b < ˜ b there is a unique emission latitude θ s for each radius b , whereas for b > ˜ b there are two, corresponding to outward and inward emission [see Eq. (28)]. (The emission from these different points on the sphere would be superposed if the sphere is optically thin. In flat spacetime, this would be tantamount to looking straight down through a sphere.) The emission latitude(s) as a function of b are shown in Fig. 4 for a selection of sphere radii r s and black hole spins a . \n<!-- image --> \nFIG. 5. Behavior of photons emitted from the equatorial plane θ s = π/ 2 and received at the pole θ o = 0. Solid lines correspond to 'direct' photons with no angular turning points ( m = 0), while dashed lines correspond to 'backward-emitted' photons that bend around the black hole before reaching the observer ( m = 1). The color bands on the horizontal axis show the range over which the backward-emitted photons can reach the observer [the apparent m = 1 range of r s ∈ (+ , ∞ )], and the colored ticks represent the critical curve radius ˜ b . Higher-order photons ( m ≥ 2) produce essentially vertical lines at the critical radius (e.g., Fig. 4 of Ref. [15]) and are not shown here. On the left, we show the emission radius r s as a function of screen radius b . On the right, we show the frame dragging integral I φ , with the curves cut off at the apparent position of the ergosphere, where time-delay effects become essential (see further discussion in the main text). \n<!-- image -->", 'B. Equatorial plane observed from the pole': "We now consider the apparent positions of rings lying on the equatorial plane ( θ s = π/ 2) and observed from directly above ( θ o = 0). Using Eqs. (62), (63), and (64), Eq. (20) becomes 12 \nG θ = 2 m +1 √ b 2 -a 2 K ( a 2 a 2 -b 2 ) = I r , (78) \nwhere the second equality follows from the geodesic equation I r = G θ [Eq. (7a)]. The condition 0 < I r < I total r [Eq. (33)] is thus \n0 < 2 m +1 √ b 2 -a 2 K ( a 2 a 2 -b 2 ) < I total r . (79) \nThis condition provides the range of integers m for which there exist trajectories linking the equator and the polar observer with m turning points, as a function of the image radius b . For most values of b , only m = 0 is allowed, with higher-order values of m becoming allowed near the critical curve ˜ b , where I r diverges logarithmically. For any value of m ∈ { 0 , 1 , 2 , . . . } satisfying the condition (79), Eq. (30) for r s ( I r ) with Eq. (78) for I r ( b, m ) provides the emission radius r s ( b, m ). These maps r s ( b, m ) for m ∈ N were called 'transfer functions' in Ref. [15]. \nIn Fig. 5, we show the first ( m = 0) and second ( m = 1) transfer functions, which correspond to the main images of the front and the back of an equatorial disk, respectively. As discussed in Ref. [15], the 'backside image' ( m = 1) is highly demagnified, appearing only in a thin band near the critical curve. 13 Subsequent (further demagnified) images will be discussed in Sec. VII below. \nThe angle of arrival ϕ of a photon is given by Eqs. (7b), (21), (60) and (66) as 14 \nϕ = φ s + I φ + mπ, (80) \nwhere we absorb the ± from Eq. (66) using ϕ ∼ ϕ + 2 π . The last term reflects the m passages of the photon through the pole before it reaches the observer. In the zero-spin limit, the middle term vanishes, showing that successive images of a single source appear on alternating, opposite sides of the image plane. The middle term I φ introduces an additional, spin-dependent shift in image plane angle ϕ , which we regard as the effect of frame dragging. \nIn Fig. 5, we plot I φ for the front side ( m = 0) and backside ( m = 1) images for a selection of spins. For a static disk of emission with a nonaxisymmetric profile, the observed images will be rotated by this b -dependent factor; for example, a 'color wheel' will appear 'swirled'. However, a static disk cannot exist inside \nthe ergoradius r = 2 M (where rotation is inevitable), and we have therefore chosen to cut off the curves at the associated apparent radius b . If the curves were continued inside, they would display a divergence at the apparent position of the event horizon due to the irregularity of the coordinate φ . In a physical model, time-delay effects would compensate this divergence (∆ t diverges as well) to give a regular appearance to the source.", 'C. Equatorial plane: Inclined observer': 'We now consider equatorial sources ( θ s = π/ 2) seen by inclined observers ( θ o glyph[negationslash] = 0). Noting that F s = 0 and ± o = sign( β ), Eqs. (7a) and (20) become \n√ -u -a 2 I r +sign( β ) F o = 2 mK. (81) \nFor each r s and m , this equation defines a relationship between α and β , i.e., a curve on the image plane. However, if this curve intersects the α -axis, then it will be discontinuous there on account of the sign( β ) appearing in Eq. (81). 15 This jump can be simply compensated by sending m → m +1 whenever the α -axis is crossed from below, since the incomplete elliptic integral F o becomes the complete elliptic integral K at β = 0. 16 That is, smooth curves on the image plane are labeled by integers ¯ m defined using the Heaviside function H ( x ) by \n¯ m = m -H ( β ) . (82) \nThis reflects the geometric fact that, since the observer is assumed to lie above the equatorial plane, emission arriving from above the line of sight must have an additional angular turning point relative to the corresponding emission arriving from below (Fig. 7). \nEach source ring r s maps to an infinite number of observed rings labeled by ¯ m ∈ { 0 , 1 , 2 , . . . } . Even ¯ m corresponds to emission towards the observer (i.e., from the front of an equatorial disk), while odd ¯ m corresponds to emission away from the observer (i.e., from the back of a disk). In Fig. 6, we show the first ( ¯ m = 0) and second ( ¯ m = 1) rings in the form of equatorial contour plots for various values of black hole spin and inclination. Subsequent rings ( ¯ m ≥ 2) appear very near the critical curve and are discussed in Sec. VII below.', 'VII. THE PHOTON RING': "We now discuss universal properties of photons arriving near the critical curve C . Our discussion will be \nframed in terms of the three key quantities γ , δ , and τ that characterize the critical orbits (Sec. III above). We will first derive expressions of the form \nd ∝ e -2 nγ , (83) \n∆ φ = 2 nδ +(corrections) , (84) \n∆ t = 2 nτ +(corrections) , (85) \nwhere n is the fractional number of orbits (Sec. II C) and d is the signed perpendicular distance from the critical curve (Fig. 3). These formulas help make conceptual points about how the critical parameters { γ, δ, τ } of bound photon orbits influence image plane observables, but for quantitative claims, it it necessary to relate to the turning point number m and discuss the corrections in detail. For these purposes, it will be helpful to introduce the notation \nf i = ˜ F i ˜ K , π i = ˜ Π i ˜ Π , e ' i = ˜ E i ˜ E ' , (86) \nwhere as usual, i ∈ { s, o } stands for source or observer. These quantities range between -1 and +1 at θ + and θ -, respectively, while vanishing at the equator θ i = π/ 2.", 'A. Distance from critical curve ( γ )': "The analysis in App. B shows that near criticality, the radial integral I r evaluated from r s to r o →∞ grows as \nI r ≈ -1 2˜ r √ ˜ χ log[ C ± ( r s , ˜ r ) d ] , (87) \nwhere ˜ χ is given in Eq. (44), while C ± ( r s , ˜ r ) (with ± the sign of d ) can be inferred from the equations in Secs. B 5 and B6 together with the expression (B56) for d . 17 \nUsing the geodesic equation I r = G θ [Eq. (7a)] in Eq. (87) and solving for d , we obtain \nd ≈ 1 C ± ( r s , ˜ r ) exp [ -2˜ r √ ˜ χG θ ( m,θ s , θ o ) ] . (88) \nA more illuminating form of this equation is \nd ≈ 1 C ± e -2 nγ , (89) \nwhere n is the fractional number of orbits [Eq. (36)] and γ (˜ r ) is the Lyapunov exponent of the photon orbit at radius ˜ r [Eq. (43)]. Thus, for each factor e -2 γ closer to the critical curve, the observed photon has executed one additional orbit. We may relate n = G θ /G 1 to the number of polar turning points m by using Eqs. (20) and \nFIG. 6. Apparent positions of source rings of constant Boyer-Lindquist radius r s in the equatorial plane θ s = π/ 2, as a function of black hole spin and observer inclination. (We set M = 1.) Solid lines are the front side image ¯ m = 0, while dashed lines are the backside image ¯ m = 1 (Fig. 7). The apparent position of the horizon is a filled gray line, while the apparent positions of r s = 3, 5, and 7 are blue, green, and purple, respectively. From top to bottom, the rows are spin a/M = 1%, 50%, 94%, 99.9%; from left to right, the columns are observer inclination θ o = 1 · , 17 · , 60 · , 80 · . \n<!-- image --> \n(35), and setting the conserved quantities equal to their critical values, \nn ≈ m 2 ± o 1 4 [( -1) m f s -f o ] , (90) \nwhere the geometric factor f i was introduced in Eq. (86). \nEqs. (88) and (89) are valid for d glyph[lessmuch] M , or equivalently for n glyph[greatermuch] 1 or m glyph[greatermuch] 1. In practice, we find that the agreement is reasonable even for d ∼ M (Fig. 3), and hence for n ∼ 1. In particular, the logarithmic approximation is already useful at m = 1, and it becomes excellent for all higher m ∈ { 2 , 3 , 4 , . . . } . \n<!-- image --> \nFIG. 7. Illustration of the meaning of m and ¯ m in the case of equatorial sources ('the disk'). For a polar observer (left), even/odd values of m correspond to emission from the front/back of the disk, and arrive on opposite sides of the image. For the inclined observer (right), we instead use ¯ m [Eq. (82)], and again even/odd values come from the front/back of the disk. Solid lines are front side images, while dashed lines are backside images. For the left source on the right figure, we omit the ¯ m = 2 front side image (green) for clarity. These curves are schematic and do not represent actual trajectories. \n<!-- image --> \nFor each value of ˜ r , around the the curve C , and for each choice of integer m (typically accurate for m glyph[greaterorsimilar] 1), Eq. (89) provides the signed perpendicular distance d of an arriving photon that originated on the poloidal ring ( r s , θ s ) and encountered m angular turning points on its way to the observer. The emission angle along the ring, as well as the emission time, may be found from ∆ φ and ∆ t , which we now discuss.", 'B. Lapse in azimuthal angle ( δ )': 'Now, consider the lapse in φ [Eq. (7b)], \n∆ φ = I φ + λG φ ( m,θ s , θ o ) . (91) \nThe analysis of App. B shows that near criticality, the integral I φ takes the asymptotic form \nI φ ≈ a ( ˜ r + M ˜ r -M ) I r + D ± (˜ r, r s ) , (92) \nwhere the precise form of D ± (˜ r ) may be inferred from the expressions in App. B. For our present purposes, the only important property of D ± (˜ r ) is that it is independent of d , except via the sign ± = sign( d ). Using the geodesic equation I r = G θ [Eq. (7a)], Eqs. (91) and (92) give \n∆ φ ≈ a ( ˜ r + M ˜ r -M ) G θ + ˜ λG φ + D ± (˜ r ) , (93) \nsuch that the d -dependence drops out entirely, other than via ± = sign( d ). A more illuminating form of this expression is [combining Eqs. (20), (21), (47), and (90)] \n∆ φ ≈ 2 n ˆ δ -J φ ± ( m, ˜ r ) , (94) \nwhere 2 ˆ δ (˜ r ) is the lapse in φ per orbit of a bound photon at radius ˜ r [Eq. (49)], and \nJ φ ± = ± o ˜ λ ˜ Π a √ -˜ u -[( -1) m ( f s -π s ) -( f o -π o )] -D ± (˜ r ) . (95) \nOnce again, we remind the reader that here, the subscript ± is the sign of d , encoding whether one is inside ( -) or outside (+) the critical curve.', 'C. Lapse in time ( τ )': "Finally, consider the lapse in t [Eq. (7c)], \n∆ t = I t + a 2 G t ( m,θ s , θ o ) . (96) \nThe analysis of App. B shows that near criticality, the integral I t takes the asymptotic form \nI t ≈ ˜ r 2 ( ˜ r +3 M ˜ r -M ) I r + H ± (˜ r, r s ) , (97) \nwhere the precise form of H ± (˜ r ) may be inferred from the expressions in App. B. For our present purposes, the only important property of H ± (˜ r ) is once again that it is independent of d , except via its sign ± = sign( d ). Using the geodesic equation I r = G θ [Eq. (7a)], Eqs. (96) and (97) become \n∆ t ≈ ˜ r 2 ( ˜ r +3 M ˜ r -M ) G θ + a 2 G t + H ± (˜ r ) , (98) \nsuch that the d -dependence drops out entirely, other than via ± = sign( d ). A more illuminating form of this expression is [combining Eqs. (20), (22), (52), and (90)] \n∆ t ≈ 2 nτ -J t ± ( m, ˜ r ) , (99) \nwhere 2 τ (˜ r ) is the lapse in t per orbit of a bound photon at radius ˜ r [Eq. (52)], and \nJ t ± = ∓ o 2 a ˜ u + ˜ E ' √ -˜ u -[( -1) m ( f s -e ' s ) -( f o -e ' o )] -H ± (˜ r ) . (100) \nYet again, we remind the reader that here, the subscript ± is the sign of d , encoding whether one is inside ( -) or outside (+) the critical curve.", 'D. Equatorial sources viewed from the pole': "To unpack the physics of the photon ring, we begin with the simplest case of an equatorial source ( θ s = π/ 2) and a polar observer ( θ o = 0). In this case, the source integrals vanish ( f s = π s = e ' s = 0), and the observer integrals become complete ( f o = π o = e ' o = 1). Together with ± o = -1 [Eq. (62)], this reduces Eq. (90) to \nn ≈ m 2 + 1 4 . (101) \nLikewise, Eqs. (89), (94), and (99) simplify to \nd ≈ 1 C ± exp [ -( m + 1 2 ) γ ] , (102) \n∆ φ ≈ ( m + 1 2 ) ˆ δ + D ± , (103) \n∆ t ≈ ( m + 1 2 ) τ + H ± . (104) \nBecause of the discontinuity in ˆ δ , at this stage, we consider θ o to be small but finite. \nThe formulas (102), (103), and (104) encode the arrival position and time of the infinitely many apparent positions of a given source. The details are determined by the dependence of the coefficients C ± , D ± , and H ± on the source radius r s . However, these terms are independent of the image number m , and hence cancel out of appropriate ratios and differences, \nd m +1 d m ≈ e -γ , (105) \n(∆ φ ) m +1 -(∆ φ ) m ≈ ˆ δ, (106) \n(∆ t ) m +1 -(∆ t ) m ≈ τ. (107) \nWe may now replace ˆ δ with δ since the two agree modulo 2 π [i.e., the difference can be absorbed into the left-hand side of Eq. (106)]. Then all quantities are continuous and we may take the full limit θ o → 0. Recalling that d = b -˜ b \nFIG. 8. The critical parameters δ 0 , τ 0 , and γ 0 for an onaxis observer. Above, we show their dependence on black hole spin, and below, we schematically illustrate their effects. Successive images are demagnified by e -γ 0 , rotated by δ 0 , and delayed by τ 0 . The image labeled m (top left) is shown artificially large, but the demagnified images are then to scale. \n<!-- image --> \nand φ o = ϕ , and additionally denoting the observation time t o by t , we thus obtain \nb m +1 -˜ b b m -˜ b ≈ e -γ 0 , (108) \nϕ m +1 -ϕ m ≈ δ 0 , (109) \nt m +1 -t m ≈ τ 0 , (110) \nwhere γ 0 , δ 0 , and τ 0 were given in Eqs. (68), (69), and (70), respectively. \nEqs. (108), (109) and (110) show that the successive apparent positions of a source ( r s , θ s = π/ 2 , φ s , t s ) move a factor of e γ 0 closer to the critical curve for every additional half-orbit, while rotating an angle δ 0 around the curve and appearing a time τ 0 later. Recalling that \nFIG. 9. Variation of the critical parameters γ , δ , and τ around the critical curve. We show the value of these parameters as a function of polar angle tan ˜ ϕ = ˜ β/ ˜ α around the curve. (For δ , we plot modulo 2 π .) The rotation and delay parameters δ and τ become large near the NHEKline (Fig. 3) present for inclined observers of rapidly rotating black holes. The demagnification parameter γ becomes small near the edges of the NHEKline (see also Fig. 6 of Ref. [16]). The time delay τ ∼ 16 M has been seen previously in numerical simulations of emitting sources near black holes [30, 31]. \n<!-- image --> \nδ 0 = π for a nonspinning black hole, we see that successive images appear on opposite sides of the critical curve. This is easily understood from the geometry of the source (Fig. 7 left). \nNote that the arrival positions can be neatly represented in terms of a complex coordinate z = ( b -˜ b ) e iϕ , 18 such that \nz m +1 = e -γ 0 + iδ 0 z m . (111) \nThus we may view -γ 0 + iδ 0 as a single complex exponent. Now consider an equatorial source of some finite extent from r -s to r + s , as in the emitting portion of an accretion disk. Let b ± m (˜ r ) represent the m th observed position of the inner and outer edges. At some sufficiently high m (typically m ≥ 1 is sufficient), we may compute b ± m using \nthe approximation (108). Denoting the apparent width of each image by ∆ b m = b + m -b -m , from Eq. (102) we have \n∆ b m +1 ∆ b m ≈ e -γ 0 . (112) \nThat is, successive images of the equatorial disk are demagnified (narrower) by a factor of e -γ 0 . The total flux associated with each image also decreases by the same typical factor, i.e., the flux is exponentially suppressed in the orbit number. Each successive image also rotates on the screen by an angle δ 0 , an effect which would be visible for nonaxisymmetric source profiles. Finally, each successive image arrives a time τ 0 later, an effect that would be observable for time-variable source profiles. Some of these properties are illustrated in Fig. 8.", 'E. General sources viewed from the pole': "Suppose now that the source is not equatorial, but the observer is still on the pole. From Eq. (90) using ± o = -1 \nand f o = π o = e ' o = 1 yet again, we have \nn = m 2 + 1 4 -( -1) m 4 f s . (113) \nRepeating the same procedure that led to Eqs. (108), (109) and (110), we now find 19 \nb m +1 -˜ b b m -˜ b ≈ e -x m γ 0 , (114) \nϕ m +1 -ϕ m ≈ x m δ 0 -( -1) m πf s . (115) \nt m +1 -t m ≈ x m τ 0 +( -1) m 4 a ˜ u + ˜ E ' √ -˜ u -( f s -e ' s ) , (116) \nwith \nx m = 1 + ( -1) m f s . (117) \nThus, although γ 0 , δ 0 , and τ 0 no longer give precisely the demagnification, rotation, and time delay (respectively), they still encode these effects in a relatively straightforward way, depending on whether m is even or odd. We again obtain simple expressions if we advance m by two instead of one, \nb m +2 -˜ b b m -˜ b ≈ e -2 γ 0 , (118) \nϕ m +2 -ϕ m ≈ 2 δ 0 , (119) \nt m +2 -t m ≈ 2 τ 0 . (120) \nThus, a given source gives rise to two families of images (one for even m and one for odd m ), each of which has demagnification 2 γ 0 , rotation 2 δ 0 , and time delay 2 τ 0 . These are just the Lyapunov exponent, lapse in φ , and lapse in t for a complete bound photon orbit, respectively. That is, each successive image of each family differs by one orbit around the black hole. Roughly speaking, the two families correspond to emission towards and away from the observer; for an equatorial disk, they are images of the front and back of the disk, respectively. \nRecall that δ 0 = π in Schwarzschild. As such, each family of images approaches the critical curve radially, since the rotation of each successive image is 2 π ∼ 0.", 'F. Inclined observer: Equatorial sources': "Next, suppose that the source is equatorial ( θ s = π/ 2), so that f s = 0, while the observer is inclined ( θ o glyph[negationslash] = 0), so that ± o = sign( β ). Then Eq. (90) becomes \nn = m 2 -1 4 sign( β ) f o . (121) \nRecalling that we set φ o = 0 for the inclined observer, it follows from Eqs. (89), (94), and (99) that \nd ≈ 1 C ± exp [ -γ ( m -1 2 sign( β ) f o )] , (122) \nφ s ≈ -( m -1 2 sign( β ) f o ) ˆ δ \n-sign( β ) ˜ λ ˜ Π a √ -˜ u -( f o -π o ) -D ± , (123) \nt -t s ≈ ( m -1 2 sign( β ) f o ) τ -sign( β ) (2 a ˜ u + ˜ E ' ) √ -˜ u -( f o -e ' o ) + H ± . (124) \nAs our observer is now inclined, the quantities γ , ˆ δ , and τ depend nontrivially on ˜ r , which together with the sign of β specifies a point on the critical curve. Selecting a position (˜ r, sign( β )) on the critical curve, Eq. (122) gives the perpendicular distance of a photon that originated at ( r s , θ s = π/ 2) and encountered m polar turning points on its way. The emission angle φ s of this photon is given by Eq. (123), and the emission time t s by Eq. (124) (in terms of the observation time t o = t ). \nWe may again take a ratio to find \nd m +1 d m ≈ e -γ , (125) \nwhich may be compared with (108) above. Fixing a position (˜ r, sign β ) along the critical curve, Eq. (125) shows that photons from a given equatorial source ring ( r s , θ s = π/ 2) arrive at perpendicular distances d that successively decrease by a factor of e -γ . Fixing the observation time t , these photons originated from angles φ m s and times t m s related by \nφ m +1 s -φ m s ≈ -δ, (126) \nt m +1 s -t m s ≈ -τ, (127) \nwhere now we have switched to the continuous quantity δ , absorbing the jump of 2 π into the φ coordinate. \nRecall that γ , δ , and τ depend on the critical curve position ˜ r under consideration. For a stationary, axisymmetric source, we may regard e -γ as a demagnification factor that varies over the critical curve. For a general equatorial source, we see no simple way to describe the properties of the images in terms of those of the source, but it is clear from the exceptionally simple formulas (125), (126), and (127) that γ , δ , and τ still encode universal features of high-order images. The variation of these critical parameters is shown in Fig. 9.", 'G. General source and observer': 'For nonequatorial sources sources observed at nonzero inclination, Eqs. (122), (123) and (124) are supplemented \nby terms involving dependence on m through ( -1) m , as in Eqs. (114), (115), and (116) above. These terms give rise to separate behavior for even and odd values of m , as described in Sec. VII E above in the case of a polar observer. Rather than present these details, we instead merely note that in the general case we still have simple expressions when m is shifted by two, \nd m +2 d m ≈ e -2 γ , (128) \nφ m +2 s -φ m s ≈ -2 δ, (129) \nt m +2 s -t m s ≈ -2 τ. (130) \nThat is, given any source ring ( r s , θ s ) observed at any inclination θ o at some time t , and choosing any perpendicular (˜ r, sign( β )) to the image-plane critical curve, photons arrive in two separate families (even and odd m ) at distances decreasing by factors of e -2 γ , which were emitted at successively earlier times (with delay -2 τ ) as well as different positions around the ring (with increment -2 δ ). Although these properties do not translate in any simple way into a description of the distortion and demagnification of a general source observed at a general inclination, it is clear from the exceptionally simple formulas (128), (129), and (130) that γ , δ , and τ still encode universal features of high-order images.', 'ACKNOWLEDGMENTS': 'SEG was supported in part by NSF grant PHY1752809 to the University of Arizona. Portions of this work were completed at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611. AL was supported in part by the Jacob Goldfield Foundation.', 'Appendix A: Radial roots and integrals': 'In Ref. [11], we derived analytic formulas for the roots of the radial potential (5) which are ordered when the roots are real. We reproduce these formulas here for convenience. We introduce \nA = a 2 -η -λ 2 , (A1) \nB = 2 M [ η +( λ -a ) 2 ] > 0 , (A2) \nC = -a 2 η, (A3) \nand further define \nP = -A 2 12 -C , (A4) \nQ = -A 3 [ ( A 6 ) 2 -C ] -B 2 8 , (A5) \nas well as \nz = √ ω + + ω -2 -A 6 > 0 , (A6) \nω ± = 3 √ √ √ √ -Q 2 ± √ ( P 3 ) 3 + ( Q 2 ) 2 . (A7) \nThe four roots are then given by \nr 1 = -z -√ -A 2 -z 2 + B 4 z , (A8a) \nr 2 = -z + √ -A 2 -z 2 + B 4 z , (A8b) \nr 3 = z -√ -A 2 -z 2 -B 4 z , (A8c) \nr 4 = z + √ -A 2 -z 2 -B 4 z . (A8d) \nThese roots always satisfy r i ≤ r j when i < j and both r i and r j are real. On the critical curve C , we have r 3 = r 4 , but otherwise r 4 is always the largest real root outside the horizon. Thus, rays reaching infinity either have a turning point at r 4 , are asymptotic to a photon orbit at r 3 = r 4 , or have no turning point at all (when r 4 is complex, or real but inside the horizon). \nWe now present the results from Ref. [11] needed to compute the radial integrals of interest to this paper. Rays that arrive outside the critical curve are case (2) of Ref. [11]. The antiderivative is given by Eqs. (B35)-(B40) therein, \nI (2) r ( r ) = 2 √ r 31 r 42 F ( arcsin √ r -r 4 r -r 3 r 31 r 41 ∣ ∣ ∣ ∣ r 32 r 41 r 31 r 42 ) . (A9) \nIn particular, the complete radial integral (29) is \nI total r = 4 √ r 31 r 42 F ( arcsin √ r 31 r 41 ∣ ∣ ∣ ∣ r 32 r 41 r 31 r 42 ) . (A10) \nRays that arrive inside the critical curve are also case (2) when all roots are real; otherwise, if r 3 = ¯ r 4 are complex conjugate roots, then the rays are case (3). For case (2), the antiderivative is again Eq. (A9), whereas for case (3), the antiderivative is given by Eqs. (B55) and (B67)-(B71) of Ref. [11], \nI (3) r ( r ) = 1 √ AB F ( arccos A ( r -r 1 ) -B ( r -r 2 ) A ( r -r 1 ) + B ( r -r 2 ) ∣ ∣ ∣ ∣ k 3 ) , A = √ r 32 r 42 > 0 , B = √ r 31 r 41 > 0 , (A11) \nk 3 = ( A + B ) 2 -r 2 21 4 AB ∈ (0 , 1) . (A12) \nIn particular, the complete radial integral (29) is \nI total r = 2 √ r 31 r 42 F ( arcsin √ r 31 r 41 ∣ ∣ ∣ ∣ r 32 r 41 r 31 r 42 ) -I (2) r ( r + ) , (A13) \nif all roots are real; otherwise, when r 3 = ¯ r 4 , it is \nI total r = 1 √ AB F ( arccos A -B A + B ∣ ∣ ∣ ∣ k 3 ) -I (3) r ( r + ) . (A14)', 'Appendix B: Asymptotic approximation for the radial integrals': "The integrands of the fundamental radial integrals I r , I φ , and I t involve (the square root of) the radial potential (5) in their denominators. Single roots of R ( r ) correspond to turning points where the integral remains finite. However, for critical conserved quantities λ = ˜ λ (˜ r ) and η = ˜ η (˜ r ), the roots r 3 and r 4 [Eqs. (A8)] coalesce, rendering the integral logarithmically divergent at the double root ˜ r = r 3 = r 4 . Physically, this represents a critical photon asymptotically approaching its associated photon orbit radius ˜ r . If the conserved quantities are not precisely critical but only nearly so, then the total integral I r is finite for each such ray, but the value diverges logarithmically in the deviation of the conserved quantities from their critical values. Physically, this represents a near-critical photon spending an asymptotically large amount of time orbiting near its associated bound photon orbit at ˜ r . In this situation, one expects the integral to break into two contributions, one from near the photon orbit and one from far away, such that the near-critical integral can be estimated by the method of matched asymptotic expansions. In this appendix, we compute the relevant approximations to the radial integrals using this method. \nAll bound photon orbits cross the equatorial plane and hence have η > 0. Therefore, in this appendix, we will use \nq = √ η > 0 , (B1) \nin lieu of η . Consider a null geodesic whose conserved quantities λ and q are nearly equal to those of a bound photon orbit. We may then write \nλ = ˜ λ (1 + δλ ) , q = ˜ q (1 + δq ) , (B2) \nwhere ˜ λ and ˜ η = ˜ q 2 are associated to the radius ˜ r of the photon orbit by Eqs. (38) and (39), and | δλ | ∼ | δq | glyph[lessmuch] 1. We also introduce a new radial coordinate δr by \nr = ˜ r (1 + δr ) , (B3) \nand use it to define 'near' and 'far' zones as follows: \nNear: \n| δr | glyph[lessmuch] 1 , \n(B4) \nFar: \n| δλ | ∼ | δq | glyph[lessmuch] | δr | . (B5) \nThese regimes overlap when | δλ | ∼ | δq | glyph[lessmuch] | δr | glyph[lessmuch] 1. We emphasize that throughout this discussion, 'near' and 'far' refer to distance from the photon orbit radius ˜ r , rather than distance from the black hole. The far-zone region is disjoint, consisting of a 'right' region containing asymptotic infinity, and a 'left' region containing the event horizon. The radial potential R ( r ) has different approximations in the near and far zones. In the near-zone, it is well approximated by the scaling regime δr 2 ∼ δλ ∼ δq , in which \nR ( r ) ≈ R n ( δr ) := 4˜ r 4 ˜ χ ( δr 2 -δr 2 0 ) , (B6) \nwhere ˜ χ is as defined in Eq. (44), and in the last step we also introduced a dimensionless quantity 20 \nδr 2 0 = ∆(˜ r ) 2˜ r 2 ˜ χ [ -( ˜ r -3 M ˜ r -M ) ˜ λ a δλ + ˜ q 2 ˜ r 2 δq ] , ˜ χ = 1 -M ∆(˜ r ) ˜ r (˜ r -M ) 2 . (B7) \nNotice that the quadratic near-zone potential R n ( δr ) has zeros at δr = ± δr 0 ; these correspond to radial turning points provided that δr 2 0 > 0. For photons that reach infinity, only the outer root is relevant. Note also that \n˜ χ = 3 4 -( a ˜ q 2˜ r 2 ) 2 ∈ ( 0 , 3 4 ] . (B8) \nIn the far-zone, the radial potential R ( r ) is instead well-approximated by its value at λ = ˜ λ and q = ˜ q , \nR ( r ) ≈ R f ( δr ) := 4˜ r 4 δr 2 ( δr 2 4 + δr + ˜ χ ) = 4˜ r 4 ˜ χδr 2 Q ( δr ) , (B9) \nwhere in the last step, we introduced for future convenience a function \nQ ( δr ) = 1 + δr ˜ χ + δr 2 4˜ χ . (B10) \nThe double root δr = 0 of R f ( δr ) is outside the regime of validity of the far-zone approximation and does not correspond to a physical turning point. (It is the far-zone remnant of the two roots δr = ± δr 0 that are separately resolved by the near-zone approximation.) The quartic potential R f ( δr ) has two other negative roots δr -0 < δr + 0 < 0, where δr ± 0 = 2 ( -1 ± √ 1 -˜ χ ) , which a photon that comes in from infinity cannot encounter. \nIf a light ray with conserved quantities (B2) reaches infinity, then by definition it arrives near the closed curve C . Rays arriving inside C have no radial turning points, while rays arriving outside have a single turning point. The preceding analysis shows that these cases correspond to δr 2 0 < 0 and δr 2 0 > 0, respectively: \nInside C : \nδr 2 0 < 0 , (B11) \nOutside C : δr 2 0 > 0 . (B12) \nIn Sec. B 7 below, we show that δr 2 0 is actually proportional to the (signed) perpendicular distance from C . We have now laid the groundwork to compute the geodesic path integrals involving the radial potential. To do so, it will suffice to evaluate the definite integrals \nI ab r = GLYPH<2> r b r a d r √ R ( r ) , I ab φ = GLYPH<2> r b r a a (2 Mr -aλ ) ∆( r ) √ R ( r ) d r, I ab t = GLYPH<2> r b r a r 2 ∆( r ) + 2 Mr ( r 2 + a 2 -aλ ) ∆( r ) √ R ( r ) d r, (B13) \nfor all combinations of in/out for the conserved quantities, and near/far for each of r a and r b . We will organize the calculation in sections based on the near/far split, considering only the cases that arise when photons reach infinity. We will present integrals in terms of the inverse hyperbolic tangent, defined as \narctanh x = 1 2 log ( 1 + x 1 -x ) , (B14) \nwhich is manifestly real whenever x ∈ [ -1 , 1].", '1. Both points in the near-zone': "When both endpoints of the geodesic are in the near-zone, the radial integrals (B13) simplify to \nI nn r = 1 2˜ r √ ˜ χ GLYPH<2> δr b δr a d( δr ) √ δr 2 -δr 2 0 , I nn φ = a ( ˜ r + M ˜ r -M ) I nn r , I nn t = ˜ r 2 ( ˜ r +3 M ˜ r -M ) I nn r , (B15) \nwhere the label 'nn' stands for 'near-near' (i.e., both points in the near-zone). Here and hereafter, δr a and δr b denote the δr -coordinate values of the Boyer-Lindquist radii r a and r b , respectively, with δr b < δr b . \nWe can now evaluate I nn r . Photons arriving outside C have 0 < δr 0 ≤ δr and the manifestly real integral \nI nn , out r ( δr a , δr b ) = 1 2˜ r √ ˜ χ arctanh ( √ δr 2 -δr 2 0 δr )∣ ∣ ∣ ∣ ∣ δr b δr a . (B16) \nOn the other hand, photons arriving inside C have δr 2 0 < 0 and the manifestly real integral \nI nn , in r ( δr a , δr b ) = 1 2˜ r √ ˜ χ arctanh ( δr √ δr 2 -δr 2 0 )∣ ∣ ∣ ∣ ∣ δr b δr a . (B17) \nThese results can be combined into a single formula \nI nn r ( δr a , δr b ) = sign( δr ) 4˜ r √ ˜ χ log sign ( δr 2 0 ) 1 + √ 1 -δr 2 0 δr 2 1 -√ 1 -δr 2 0 δr 2 ∣ ∣ ∣ ∣ ∣ ∣ δr b δr a . (B18)", '2. Both points in one region of the far-zone': 'The far-zone consists of two disjoint regions on either side of the near-zone, with one region containing the horizon, and the other region containing asymptotic infinity. When both points endpoints of the geodesic are in the same region of the far-zone, the radial integrals (B13) reduce to \nI ff r = 1 2˜ r √ ˜ χ GLYPH<2> δr b δr a d( δr ) √ δr 2 Q ( δr ) , (B19) \nI ff φ = aM ˜ r 2 √ ˜ χ GLYPH<2> δr b δr a c 0 +(1 + δr ) ( δr -δr + )( δr -δr -) d( δr ) √ δr 2 Q ( δr ) , (B20) \nI ff t = ˜ r 2 √ ˜ χ GLYPH<2> δr b δr a c 1 (1 + δr ) + c 2 (1 + δr ) 2 +(1 + δr ) 4 ( δr -δr + )( δr -δr -) d( δr ) √ δr 2 Q ( δr ) , (B21) \nwhere we introduced dimensionless coefficients \nc 0 = -a ˜ λ 2 M ˜ r , c 1 = 2 aM ˜ r 3 ( a -˜ λ ) , c 2 = a 2 ˜ r 2 , (B22) \nand δr ± denotes the δr -coordinate of the outer/inner event horizon, \nδr ± = M ± √ M 2 -a 2 ˜ r -1 ∈ ( -1 , 0) . (B23) \nNow define a symmetric function of two variables \nQ 2 ( δr a , δr b ) = 2 √ Q ( δr a ) √ Q ( δr b ) Q ( δr a ) + Q ( δr b ) -( δr a -δr b ) 2 4˜ χ ∈ (0 , 1] , (B24) \nwhose range (0 , 1], which assumes that both δr a and δr b are outside the event horizon δr + (but not that they are positive), is derived in Sec. B 8 below. This range guarantees that the following functions are manifestly real outside the horizon: \nQ φ ( δr ) = c 0 +(1 + δr + ) δr + ( δr + -δr -) √ Q ( δr + ) arctanh Q 2 ( δr, r + ) -c 0 +(1 + δr -) δr -( δr + -δr -) √ Q ( δr -) arctanh Q 2 ( δr, r -) , (B25) Q t ( δr ) = -4˜ χ √ Q ( δr ) -4 M √ ˜ χ ˜ r arctanh Q 2 ( δr, ∞ ) + c 1 (1 + δr + ) + c 2 (1 + δr + ) 2 +(1 + δr + ) 4 δr + ( δr + -δr -) √ Q ( δr + ) arctanh Q 2 ( δr, r + ) -c 1 (1 + δr -) + c 2 (1 + δr -) 2 +(1 + δr -) 4 δr -( δr + -δr -) √ Q ( δr -) arctanh Q 2 ( δr, r -) . (B26) \nManifestly real forms of the far integrals are then \nI ff r ( δr a , δr b ) = -sign( δr ) 2˜ r √ ˜ χ arctanh Q 2 ( δr, 0) ∣ ∣ ∣ ∣ δr b δr a (B27) = -sign( δr ) 2˜ r √ ˜ χ arctanh ( √ Q ( δr ) 1 + δr 2˜ χ )∣ ∣ ∣ ∣ ∣ δr b δr a , I ff φ ( δr a , δr b ) = -sign( δr ) aM ˜ r 2 √ ˜ χ [ ˜ r 2 M ( ˜ r + M ˜ r -M ) arctanh Q 2 ( δr, 0) + Q φ ( δr ) ]∣ ∣ ∣ ∣ δr b δr a (B28) = a ( ˜ r + M ˜ r -M ) I ff r ( δr a , δr b ) -sign( δr ) aM ˜ r 2 √ ˜ χ Q φ ( δr ) ∣ ∣ ∣ ∣ δr b δr a , I ff t ( δr a , δr b ) = -sign( δr )˜ r 2 √ ˜ χ [ ˜ r +3 M ˜ r -M arctanh Q 2 ( δr, 0) + Q t ( δr ) ]∣ ∣ ∣ ∣ δr b δr a (B29) = ˜ r 2 ( ˜ r +3 M ˜ r -M ) I ff r ( δr a , δr b ) -sign( δr )˜ r 2 √ ˜ χ Q t ( δr ) ∣ ∣ ∣ ∣ δr b δr a .', '3. One point in the near-zone and one point in the far-zone': "We now wish to consider the case where one point is in the near-zone and the other point is in the far-zone. This requires the method of matched asymptotic expansions, which we implement as follows. First, we choose an arbitrary matching radius δR . We then split the integral into a portion from δr a to δR , and a remaining portion from δR to δr b . The arbitrary point δR is assumed to be in the overlap region | δλ | ∼ | δq | glyph[lessmuch] | δR | glyph[lessmuch] 1, so that the first integral may be computed with the near-zone approximation (presented in Sec. B 1), while the second integral may be computed with the far-zone approximation (presented in Sec. B 2). Using the relevant definite integrals computed in these sections, and taking into account their various approximations, the arbitrary radius δR disappears from the final expressions. \nWe begin with I r . Photons arriving outside C necessarily have 0 < δr 0 < δr a glyph[lessmuch] 1 and δr a glyph[lessmuch] δr b , and the answer is \nI nf , out r ( δr a , δr b ) = -1 2˜ r √ ˜ χ [ arctanh ( √ Q ( δr b ) 1 + δr b 2˜ χ ) +arctanh ( √ δr 2 a -δr 2 0 δr a ) + 1 2 log ( 1 -˜ χ (8˜ χ ) 2 δr 2 0 )] . (B30) \nThis expression simplifies when the bounds of integration cover the entire range [ δr 0 , + ∞ ) of allowed radial motion. Note that the second term vanishes as the lower bound of integration δr a → δr 0 . Moreover, the argument of the first term goes to √ ˜ χ as δr b →∞ , leaving \nI nf , out r ( δr 0 , ∞ ) = -1 2˜ r √ ˜ χ [ arctanh √ ˜ χ + 1 2 log ( 1 -˜ χ (8˜ χ ) 2 δr 2 0 )] = -1 4˜ r √ ˜ χ log [ ( 1 + √ ˜ χ 8˜ χ ) 2 δr 2 0 ] . (B31) \nFor photons arriving inside C , we must separately consider the two regions of the far-zone. In the right region containing asymptotic infinity, we integrate from a near-zone point 0 < δr a glyph[lessmuch] 1 to a far-zone point δr b glyph[greatermuch] δr a > 0, so we label this definite integral 'nf' for 'near-far'. The answer is \nI nf , in r ( δr a , δr b ) = -1 2˜ r √ ˜ χ [ arctanh ( √ Q ( δr b ) 1 + δr b 2˜ χ ) +arctanh ( δr a √ δr 2 a -δr 2 0 ) + 1 2 log ( 1 -˜ χ (8˜ χ ) 2 ∣ ∣ δr 2 0 ∣ ∣ )] . (B32) \nIn the left region containing the event horizon, we instead integrate from a far-zone point δr a < 0 to a near-zone point δr b < 0, with | δr b | glyph[lessmuch] 1 and | δr b | glyph[lessmuch] | δr a | , so we label this integration 'fn' for far-near. The answer involves a single change of sign, \nI fn , in r ( δr a , δr b ) = -1 2˜ r √ ˜ χ [ arctanh ( √ Q ( δr a ) 1 + δr a 2˜ χ ) -arctanh ( δr b √ δr 2 b -δr 2 0 ) + 1 2 log ( 1 -˜ χ (8˜ χ ) 2 ∣ ∣ δr 2 0 ∣ ∣ )] . (B33) \nThe calculation proceeds identically for I φ and I t , which are conveniently expressed in terms of the I r integrals: \nI nf , in / out φ ( δr a , δr b ) = a ( ˜ r + M ˜ r -M ) I nf , in / out r ( δr a , δr b ) -aM ˜ r √ ˜ χ [ Q φ ( δr b ) -Q φ (0)] , (B34) \nI fn , in φ ( δr a , δr b ) = a ( ˜ r + M ˜ r -M ) I fn , in r ( δr a , δr b ) -aM ˜ r √ ˜ χ [ Q φ ( δr a ) -Q φ (0)] , (B35) \nI nf , in / out t ( δr a , δr b ) = ˜ r 2 ( ˜ r +3 M ˜ r -M ) I nf , in / out r ( δr a , δr b ) -˜ r 2 √ ˜ χ [ Q t ( δr b ) -Q t (0)] , (B36) \nI fn , in t ( δr a , δr b ) = ˜ r 2 ( ˜ r +3 M ˜ r -M ) I fn , in r ( δr a , δr b ) -˜ r 2 √ ˜ χ [ Q t ( δr a ) -Q t (0)] . (B37)", '4. One point in the left far-zone and the other point in the right far-zone': "The last remaining case of relevance is when the geodesic has a lower endpoint δr a in the left far-zone ( δr a < 0) and an upper endpoint δr b in the right far-zone ( δr b > 0). In this case, the photon passes through the near-zone, \nand we may obtain the radial integral by adding together the expressions for the near-far and far-near cases derived above. For I r , summing Eqs. (B32)-(B33) results in \nI lr r ( δr a , δr b ) = I fn , in r ( δr a , δR ) + I nf , in r ( δR,δr b ) = -1 2˜ r √ ˜ χ [ arctanh ( √ Q ( δr a ) 1 + δr a 2˜ χ ) +arctanh ( √ Q ( δr b ) 1 + δr b 2˜ χ ) +log ( 1 -˜ χ (8˜ χ ) 2 ∣ ∣ δr 2 0 ∣ ∣ )] , (B38) \nfrom which the arbitrary radius δR has cancelled out. Here, the label 'lr' stands for 'left-right'. Likewise, \nI lr φ ( δr a , δr b ) = a ( ˜ r + M ˜ r -M ) I lr r ( δr a , δr b ) -aM ˜ r √ ˜ χ [ Q φ ( δr a ) + Q φ ( δr b ) -2 Q φ (0)] , (B39) \nI lr t ( δr a , δr b ) = ˜ r 2 ( ˜ r +3 M ˜ r -M ) I lr r ( δr a , δr b ) -˜ r 2 √ ˜ χ [ Q t ( δr a ) + Q t ( δr b ) -2 Q t (0)] . (B40)", '5. Full answer for I r outside C': 'We have now computed all the basic definite integrals that are needed to obtain the full radial integrals I r , I φ , and I t for a photon reaching a distant observer at large radius r o →∞ . As an example of how to glue them together, we now explicitly consider the radial integral I r . It is straightforward to similarly assemble I φ and I t . \nFirst, consider a photon arriving outside C (i.e., with δr 2 0 > 0). Tracing back in time from the detector, the photon reaches a radial turning point δr 0 in the near-zone and then returns to infinity. Its radial motion in the allowed range [ δr 0 , + ∞ ) can thus be divided into four stages, as follows. \nBefore the photon reaches the near-zone, the integral is given by \nI r ≈ I ff r ( δr s , ∞ ) = -1 2˜ r √ ˜ χ [ arctanh √ ˜ χ -arctanh ( √ Q ( δr s ) 1 + δr s 2˜ χ )] . (B41) \nOnce the photon reaches the near-zone, but before it reaches the turning point, the integral is given by the limit δr o →∞ of Eq. (B30): \nI r ≈ I nf , out r ( δr s , ∞ ) = -1 2˜ r √ ˜ χ [ arctanh √ ˜ χ +arctanh ( √ δr 2 s -δr 2 0 δr s ) + 1 2 log ( 1 -˜ χ (8˜ χ ) 2 δr 2 0 )] . (B42) \nOnce the photon reaches the turning point, but before it exits the near-zone, \nI r ≈ I nn , out r ( δr 0 , δr s ) + I nf , out r ( δr 0 , ∞ ) = -1 2˜ r √ ˜ χ { -arctanh ( √ δr 2 s -δr 2 0 δr s ) + 1 2 log [ ( 1 + √ ˜ χ 8˜ χ ) 2 δr 2 0 ]} , (B43) \nwhere the first term is obtained from Eq. (B16) and the second from Eq. (B31). Once the photon exits the near-zone, the integral is given by \nI r ≈ I nf , out r ( δr 0 , δr s ) + I nf , out r ( δr 0 , ∞ ) = -1 2˜ r √ ˜ χ { arctanh ( √ Q ( δr s ) 1 + δr s 2˜ χ ) + 1 2 log ( 1 -˜ χ (8˜ χ ) 2 δr 2 0 ) + 1 2 log [ ( 1 + √ ˜ χ 8˜ χ ) 2 δr 2 0 ]} . (B44) \nWhen the photon finally reaches infinity again, the complete integral is \nI r ≈ 2 I nf , out r ( δr 0 , ∞ ) = -1 2˜ r √ ˜ χ log [ ( 1 + √ ˜ χ 8˜ χ ) 2 δr 2 0 ] . (B45)', '6. Full answer for I r inside C': 'Now consider a photon arriving inside C (i.e., with δr 2 0 < 0). Tracing back in time from the detector, the photon passes through the near-zone on its way to the event horizon, never encountering a radial turning point. Its radial motion in the allowed range [ δr + , + ∞ ) can thus be divided into three stages, as follows. \nBefore the photon reaches the near-zone, the integral is given by \nI r ≈ I ff r ( δr s , ∞ ) = -1 2˜ r √ ˜ χ [ arctanh √ ˜ χ -arctanh ( √ Q ( δr s ) 1 + δr s 2˜ χ )] , (B46) \nwhich is the same expression as outside C , Eq. (B41). Once the photon enters the near-zone, but before it exits the near-zone, the integral is given by Eq. (B32): \nI r ≈ I nf , in r ( δr s , ∞ ) = -1 2˜ r √ ˜ χ [ arctanh √ ˜ χ +arctanh ( δr s √ δr 2 s -δr 2 0 ) + 1 2 log ( 1 -˜ χ (8˜ χ ) 2 ∣ ∣ δr 2 0 ∣ ∣ )] . (B47) \nOnce the photon exits the near-zone, but before it crosses the horizon, the integral is given by the δr b →∞ limit of Eq. (B38), \nI r ≈ I lr r ( r s , ∞ ) = -1 2˜ r √ ˜ χ [ arctanh ( √ Q ( δr s ) 1 + δr s 2˜ χ ) +arctanh √ ˜ χ +log ( 1 -˜ χ (8˜ χ ) 2 ∣ ∣ δr 2 0 ∣ ∣ )] . (B48) \nFinally, when the photon crosses the horizon, the complete integral is given by \nI r ≈ I lr r ( r + , ∞ ) = -1 2˜ r √ ˜ χ [ arctanh ( √ Q ( δr + ) 1 + δr + 2˜ χ ) +arctanh √ ˜ χ +log ( 1 -˜ χ (8˜ χ ) 2 ∣ ∣ δr 2 0 ∣ ∣ )] = -1 2˜ r √ ˜ χ log [ √ 1 -˜ χ ( 1 + √ ˜ χ ) (8˜ χ ) 2 √ 1 + Q 2 ( δr + , 0) 1 -Q 2 ( δr + , 0) ∣ ∣ δr 2 0 ∣ ∣ ] . (B49) \nInterestingly, note that the square root containing Q 2 can be pulled out of the logarithm, since \n√ 1 + Q 2 ( δr + , 0) 1 -Q 2 ( δr + , 0) = e arctanh Q 2 ( δr + , 0) . (B50)', '7. Perpendicular distance from C': "The logarithmic approximations for the radial integrals presented thus far are written in terms of the variable δr 2 0 defined in Eq. (B7) above. For each choice of ˜ r , this quantity encodes the arrival positions of photons near the associated point (˜ α, ˜ β ) on the curve C [Eq. (57)], expressed in terms of their fractional deviations in conserved quantities δλ and δq . Since the point ˜ r is arbitrary, we are in effect using three coordinates (˜ r, δλ, δq ) to describe positions on a two-dimensional image plane. A convenient way to remove this large redundancy is to consider only perpendicular displacements from C , denoting the signed distance by d (i.e., d < 0 inside and d > 0 outside the closed curve C ). We expect this choice to provide the best approximation for a given point near the curve C , since the line segment intersecting the curve perpendicularly is the shortest. In this appendix, we relate δr 2 0 to d [Eq. (B56) below], restricting to perpendicular displacements. Plugging into the above logarithmic approximations gives the desired expressions in terms of the coordinates (˜ r, d ) depicted in Fig. 3. \nSince δr 2 0 = 0 corresponds to the curve C , the gradient of δr 2 0 in the image plane ( α, β ) is perpendicular to C . The norm of the gradient therefore gives the rate of change in the perpendicular direction, \nδr 2 0 d ≈ ∣ ∣ ∣ glyph[vector] ∇ ( δr 2 0 ) ∣ ∣ ∣ C , (B51) \nwhere we restrict δr 2 0 to perpendicular displacements. To compute the gradient, we first express δr 2 0 in terms of α and β . Using the inverse of Eq. (55), \nλ = -α sin θ o , q = √ ( α 2 -a 2 ) cos 2 θ o + β 2 , (B52) \none finds that, to leading order in a small deviation from the curve C with | α/ ˜ α -1 | ∼ | β/ ˜ β -1 | glyph[lessmuch] 1, \nδλ = λ ˜ λ -1 ≈ α ˜ α -1 , δq = q ˜ q -1 ≈ ˜ α cos 2 θ o ( α -˜ α ) + ˜ β ( β -˜ β ) ˜ q 2 . (B53) \nPlugging these relations into Eq. (B7) results in \nδr 2 0 ≈ ∆(˜ r ) 2˜ r 4 ˜ χ [ ˜ ψ ( α -˜ α ) + ˜ β ( β -˜ β )] , ˜ ψ = ˜ α -( ˜ r + M ˜ r -M ) a sin θ o , (B54) \nfrom which we may read off the gradient as \nglyph[vector] ∇ ( δr 2 0 ) = ∆(˜ r ) 2˜ r 4 ˜ χ ( ˜ ψ∂ α + ˜ β ∂ β ) . (B55) \n(In light of the flat metric ds 2 = d α 2 +d β 2 on the image plane, the vector fields { ∂ α , ∂ β } coincide with the unit vectors { ˆ α, ˆ β } .) From Eqs. (B51) and (B55), it therefore follows that, when δr 2 0 is evaluated on a perpendicular displacement, \nd = 2˜ r 4 ˜ χ ∆(˜ r ) δr 2 0 √ ˜ β 2 + ˜ ψ 2 . (B56) \nFinally, we also present expressions for the unit tangent and normal to C . The parameter derivatives are given by \nα ' (˜ r ) = 2˜ r ˜ χ a sin θ o > 0 , β ' (˜ r ) = -˜ ψ ˜ β α ' (˜ r ) . (B57) \nThe unit tangent vector to C is therefore \nˆ T = ± o α ' (˜ r ) ∂ α + β ' (˜ r ) ∂ β √ [ α ' (˜ r )] 2 +[ β ' (˜ r )] 2 = ˜ β ∂ α -˜ ψ∂ β √ ˜ β 2 + ˜ ψ 2 , (B58) \nwhere the inclusion of the sign ± o = sign( β ) guarantees that ˆ T points clockwise around C , which corresponds to the direction of increasing/decreasing ˜ r in the upper/lower half of the image plane (see Fig. 3). As such, the outward normal is obtained by rotating ˆ T by 90 · counterclockwise in the image plane: \nˆ n = ˜ ψ∂ α + ˜ β ∂ β √ ˜ β 2 + ˜ ψ 2 . (B59) \nWe thus confirm directly that the gradient of δr 2 0 is proportional to ˆ n , \nglyph[vector] ∇ ( δr 2 0 ) = ∆(˜ r ) 2˜ r 4 ˜ χ √ ˜ β 2 + ˜ ψ 2 ˆ n. (B60)", '8. Range of Q 2': "In this section, we prove that the range of the bivariate function Q 2 ( δr a , δr b ) defined in Eq. (B24) is (0 , 1]. This guarantees that the expressions derived in Sec. B 2 for the far-zone integrals are indeed (manifestly) real, as claimed. We assume that both δr a and δr b are outside the event horizon. That is, we assume that δr a ≥ δr + and δr b ≥ δr + , though neither δr a nor δr b need be positive. In that case, we have both Q ( δr a ) > 0 and Q ( δr b ) > 0, since the roots δr ± 0 = 2 ( -1 ± √ 1 -˜ χ ) of Q ( δr ) always obey \nδr -0 < δr + 0 ≤ δr -< δr + < 0 . (B61) \nFirst, we wish to prove that 0 ≤ Q 2 ( δr a , δr b ), or equivalently, that \n0 < Q ( δr a ) + Q ( δr b ) -( δr a -δr b 2 √ ˜ χ ) 2 . (B62) \nExpanding and canceling terms leaves \n0 < 2 + 1 ˜ χ ( δr a + δr b + δr a δr b 2 ) . (B63) \nIn terms of the positive quantities p a = δr a -δr + 0 > 0 and p b = δr b -δr + 0 > 0, this reduces to the inequality \n0 < √ 1 -˜ χ ˜ χ ( p a + p b ) + p a p b 2˜ χ , (B64) \nwhich is manifestly satisfied since p a , p b , ˜ χ , and √ 1 -˜ χ are all positive. \nNext, we need to show that Q 2 ( δr a , δr b ) ≤ 1, or equivalently, that \n2 √ Q ( δr a ) √ Q ( δr b ) ≤ Q ( δr a ) + Q ( δr b ) -( δr a -δr b 2 √ ˜ χ ) 2 . (B65) \nCompleting the square and rearranging yields \n( δr b -δr a 2 √ ˜ χ ) 2 ≤ [ √ Q ( δr b ) -√ Q ( δr a ) ] 2 . (B66) \nFrom now on, we assume without loss of generality that δr b ≥ δr a . Then taking the square root of both sides leaves \n0 ≤ δr b -δr a 2 √ ˜ χ ≤ √ Q ( δr b ) -√ Q ( δr a ) , (B67) \nor equivalently, \n0 ≤ ( √ Q ( δr b ) -δr b 2 √ ˜ χ ) -( √ Q ( δr a ) -δr a 2 √ ˜ χ ) . (B68) \nA simple way to establish that this inequality holds is by noting that the function \nP ( δr ) = √ Q ( δr ) -δr 2 √ ˜ χ (B69) \nis monotonic on the radial range δr > δr + of interest. Indeed, letting p = δr -δr + 0 > 0, one finds that \nP ' ( δr ) = 2 √ 1 -˜ χ -√ ˜ χ + p 2 √ ˜ χ > 2 √ 1 -˜ χ -√ ˜ χ 2 √ ˜ χ > 0 , (B70) \nwhere the last inequality follows from the range of ˜ χ ∈ (0 , 3 / 4]. \nFinally, note that Q 2 ( δr, δr ) = 1 for all δr , so the upper bound may be saturated. 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2016PhRvL.117t1601B
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Smooth Horizonless Geometries Deep Inside the Black-Hole Regime
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2016-01-01
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We construct the first family of horizonless supergravity solutions that have the same mass, charges, and angular momenta as general supersymmetric rotating D 1 -D 5 -P black holes in five dimensions. This family includes solutions with arbitrarily small angular momenta, deep within the regime of quantum numbers and couplings for which a large classical black hole exists. These geometries are well approximated by the black-hole solution, and in particular exhibit the same near-horizon throat. Deep in this throat, the black-hole singularity is resolved into a smooth cap. We also identify the holographically dual states in the N =(4 ,4 ) D 1 -D 5 orbifold conformal field theory (CFT). Our solutions are among the states counted by the CFT elliptic genus, and provide examples of smooth microstate geometries within the ensemble of supersymmetric black-hole microstates.
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https://arxiv.org/pdf/1607.03908.pdf
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{'Smooth horizonless geometries deep inside the black-hole regime': "Iosif Bena, 1 Stefano Giusto, 2 Emil J. Martinec, 3 Rodolfo Russo, 4 Masaki Shigemori, 5 David Turton, 1 and Nicholas P. Warner 6 \n1 Institut de Physique Th'eorique, Universit'e Paris Saclay, CEA, CNRS, F-91191 Gif sur Yvette, France \n2 Dipartimento di Fisica ed Astronomia, Universit'a di Padova & INFN Sezione di Padova, Via Marzolo 8, 35131 Padova, Italy 3 Enrico Fermi Inst. and Dept. of Physics, University of Chicago, 5640 S. Ellis Ave., Chicago, IL 60637-1433, USA \n4 Centre for Research in String Theory, School of Physics and Astronomy, \nQueen Mary University of London, Mile End Road, London, E1 4NS, United Kingdom \n5 Yukawa Institute for Theoretical Physics, Kyoto University, \nKitashirakawa-Oiwakecho, Sakyo-ku, Kyoto 606-8502 Japan \n6 Department of Physics and Astronomy and Department of Mathematics, \nUniversity of Southern California, Los Angeles, CA 90089, USA \[email protected], [email protected], [email protected], [email protected], [email protected], [email protected], [email protected] \nWeconstruct the first family of horizonless supergravity solutions that have the same mass, charges and angular momenta as general supersymmetric rotating D1-D5-P black holes in five dimensions. This family includes solutions with arbitrarily small angular momenta, deep within the regime of quantum numbers and couplings for which a large classical black hole exists. These geometries are well-approximated by the black-hole solution, and in particular exhibit the same near-horizon throat. Deep in this throat, the black-hole singularity is resolved into a smooth cap. We also identify the holographically-dual states in the N = (4 , 4) D1-D5 orbifold CFT. Our solutions are among the states counted by the CFT elliptic genus, and provide examples of smooth microstate geometries within the ensemble of supersymmetric black-hole microstates.", '1. INTRODUCTION': "The black-hole information paradox reveals a profound conflict between Quantum Mechanics and General Relativity [1]. Quantum mechanically, a black hole has an entropy given by the horizon area in Planck units, while in General Relativity the black hole is unique for a given mass, charge and angular momentum. Unitarity is violated because the enormous black-hole entropy is not visible at the black-hole horizon and so the information about the black-hole state cannot be encoded in the Hawking radiation. Thus, unitarity can only be preserved if there is new physics at the scale of the horizon [2]. However, constructing structure at the scale of the horizon is no easy task: The horizon is a null surface, and any classical matter or wave that can carry information will either fall in or dilute very fast. \nOne of the great successes of string theory has been a precise accounting of the entropy of certain black holes [3], and the identification of the microstates that give rise to this entropy, albeit in a regime of coupling where the classical black-hole solution is not valid. However, this is not enough to solve the information paradox. In order to create the required structure at the horizon, all the typical microstates of the black hole must become horizon-sized bound states that have the same mass and conserved charges as the black hole, and that exist in the same regime of parameters in which the classical blackhole solution is valid. Furthermore, microstates that are describable in supergravity should be horizonless. \nFor supersymmetric black holes it has been possible to construct large classes of supergravity solutions cor- \nresponding to such horizonless bound states, and these are known as 'fuzzball' or microstate geometries [4, 5]. These solutions correspond to some of the microstates of the black hole, but have limitations, as we now discuss. \nThe microstate geometries constructed in [6-9], although carrying the same charges and angular momenta as a large black hole, have the following issues: (i) In all examples, these solutions carry an angular momentum that is a large fraction of the maximally allowed value for the black hole; (ii) Their CFT dual is not known and so their role in the ensemble of black-hole microstates remains unclear; (iii) It is not clear whether these configurations are generic and represent typical microstates of a black hole [10], nor whether the states of the black hole will continue to be described by such geometries when the black hole becomes non-extremal. \nAnother class of microstate geometries relevant for large supersymmetric black holes in five dimensions is discussed in [11-14]. While these solutions have known CFT duals, they also carry macroscopic five-dimensional angular momenta j, ˜ j . \nThe purpose of this Letter is to simultaneously resolve the first two issues described above by (i) constructing the first microstate geometries of rotating, supersymmetric D1-D5-P (BMPV) black holes in string theory [15] in which the angular momenta take arbitrary finite values, in particular including arbitrarily small values; and by (ii) identifying the dual CFT states. In doing so we also demonstrate, via an explicit example, that adding momentum to a two-charge solution describing a microstate of a string-size black hole can result in a large-scale, lowcurvature supergravity solution.", '2. BLACK-HOLE MICROSTATE GEOMETRIES': "We work in type IIB string theory on R 4 , 1 × S 1 × M , where M is T 4 or K 3. We take the size of M to be microscopic, and that of S 1 to be macroscopic. The S 1 is parameterized by the coordinate y . We wrap n 1 D1-branes on the S 1 and n 5 D5-branes on S 1 ×M , and consider momentum charge, P, along the y direction. We work in the low-energy, six-dimensional supergravity theory obtained by reduction on M . \nThe near-horizon geometry of a six-dimensional rotating, supersymmetric black string with the foregoing charges is S 3 fibered over the extremal BTZ black hole [16], whose metric is: \nds 2 BTZ = glyph[lscript] 2 AdS [ ρ 2 ( -dt 2 + dy 2 ) + dρ 2 ρ 2 + ρ 2 ∗ ( dt + dy ) 2 ] . (1) \nThis metric is locally AdS 3 and it asymptotes to the standard AdS 3 form for ρ glyph[greatermuch] ρ ∗ . It can be written as a circle of radius ρ ∗ fibered over AdS 2 in the near-horizon region ρ glyph[lessmuch] ρ ∗ (see, for example, [17]). Dimensional reduction on this circle yields the AdS 2 of the near-horizon BMPV solution. Following the usual abuse of terminology, we will refer to this region as the AdS 2 throat. \nThe BTZ parameters are related to the supergravity D1, D5, and P charges Q 1 , 5 ,P and the radial coordinate r (to be used later) via ρ = r/ √ Q 1 Q 5 and glyph[lscript] 2 AdS = √ Q 1 Q 5 . The horizon radius, ρ ∗ , of the BTZ solution (1) determines the onset of the AdS 2 throat (and thus the radius of the fibered S 1 ) and is given by ρ 2 ∗ = Q P / ( Q 1 Q 5 ). This value is determined by a competition between the momentum charge that exerts pressure on the geometry, and the D1 and D5 charges that exert tension. \nTypical black-hole microstates should be very wellapproximated by the black-hole solution until very close to the horizon. This requires a long, large, BTZ-like AdS 2 throat. To obtain such a throat, prior work has used bubbling solutions with multiple Gibbons-Hawking (GH) centers [6, 7]; the moduli space of these solutions includes 'scaling' regions [8, 9, 18] in which the GH centers approach each other arbitrarily closely, whereupon the solution develops an arbitrarily long AdS 2 throat. It has been argued that quantum effects set an upper bound on the depth of such throats [9, 19], and a corresponding lower bound on the energy gap, which matches the lowest energy excitations of the (typical sector of the) dual CFT. This suggests that microstate geometries are capable of sampling typical sectors of the dual CFT. \nUnfortunately, all the previously-known scaling microstate geometries involve at least three GH centers, whose dual CFT states are currently unknown. The holographic dictionary between supergravity solutions and CFT states has been constructed only for two-centered solutions [20]; we therefore construct new black-hole microstate solutions by adding momentum excitations to a certain two-charge seed solution. We do this using 'superstratum' technology [13, 14, 21] to introduce deformations, with specific angular dependence, so as to modify \nthe momentum and the angular momenta of the solution. \nA particular sub-class of our deformations has the effect of reducing the angular momenta of the two-charge seed solution, while introducing no additional angular momentum. These deformations therefore allow us to obtain solutions that have arbitrarily small angular momenta and describe microstates of the non-rotating D1D5-P (Strominger-Vafa) black hole. The solutions have an AdS 2 throat, which becomes longer and longer as the angular momenta j, ˜ j → 0, thus classically approximating the non-rotating black hole to arbitrary precision.", '3. THE NEW CLASS OF SOLUTIONS': "The metric, axion and dilaton of our 1 4 -BPS solutions are determined by four functions, Z 1 , Z 2 , Z 4 , F and two vector fields β, ω [22]: \nds 2 6 = -2 √ P ( dv + β ) ( du + ω + 1 2 F ( dv + β ) ) + √ P ds 2 4 , (2) \nwhere ds 2 4 is the flat metric on R 4 written in spherical bipolar coordinates, \nds 2 4 = Σ dr 2 r 2 + a 2 +Σ dθ 2 +( r 2 + a 2 ) sin 2 θ dφ 2 + r 2 cos 2 θ dψ 2 , \n(3) \nwith 0 ≤ θ ≤ π/ 2 and 0 ≤ φ, ψ < 2 π . The coordinates u and v are light-cone variables related to the asymptotic time t and the S 1 coordinate y via: \nu ≡ ( t -y ) / √ 2 , v ≡ ( t + y ) / √ 2 , y ∼ = y +2 πR y . (4) \nThe functions Σ and P are defined by: \nΣ ≡ r 2 + a 2 cos 2 θ , P ≡ Z 1 Z 2 -Z 2 4 , (5) \nand the dilaton and axion are given by: \ne 2Φ = Z 2 1 P -1 , C 0 = Z 4 Z -1 1 . (6) \nThe tensor gauge fields are also related to these functions but we will not discuss their explicit form here. \nWe consider solutions that have a simple v -fibration: \nβ = 2 -1 / 2 a 2 R y Σ -1 (sin 2 θ dφ -cos 2 θ dψ ) . (7) \nWe begin with the background of a maximally-rotating D1-D5 supertube [23, 24] and add deformations that depend upon the angles ( v, φ, ψ ) via the phase dependence: \nˆ v k,m,n ≡ √ 2 R -1 y ( m + n ) v +( k -m ) φ -mψ, (8) \nwhere k ∈ Z > 0 and m,n ∈ Z ≥ 0 . These fluctuations modify the angular momenta j, ˜ j and the momentum number n P = p y R y with p y the momentum along the y circle. In order to obtain smooth solutions whose holographic duals \nwe can identify, we add a fluctuating mode with strength b k,m,n using the 'coiffuring' technique of [12-14, 25]: \nZ 1 = Q 1 Σ + R 2 y 2 Q 5 b 2 k,m,n ∆ 2 k, 2 m, 2 n Σ cos ˆ v 2 k, 2 m, 2 n , (9) \nZ 2 = Q 5 Σ , Z 4 = b k,m,n R y ∆ k,m,n Σ cos ˆ v k,m,n , (10) \nwhere \n∆ k,m,n ≡ a k r n ( r 2 + a 2 ) -( k + n ) / 2 cos m θ sin k -m θ . (11) \nThis coiffuring ensures that, while the tensor fields depend on ˆ v k,m,n , the metric does not. The remaining parts of the solution are given by \nF = b 2 k,m,n F k,m,n , ω = ω 0 + b 2 k,m,n ω k,m,n , (12) \nwhere ω 0 is the value that ω takes in the undeformed supertube solution: \nω 0 ≡ 2 -1 / 2 a 2 R y Σ -1 (sin 2 θ dφ +cos 2 θ dψ ) . (13) \nThe general expressions for F k,m,n and ω k,m,n are given in Appendix A and we leave the expressions of the tensor gauge fields to a subsequent publication. \nRegularity and absence of closed timelike curves (CTCs) requires \nQ 1 Q 5 /R 2 y = a 2 + b 2 / 2 , b 2 = x k,m,n b 2 k,m,n , (14) \nwith x -1 k,m,n ≡ ( k m )( k + n -1 n ) . The conserved charges of the solution are \nj = N 2 ( a 2 + m k b 2 ) , ˜ j = N 2 a 2 , n P = N 2 m + n k b 2 (15) \nwhere N ≡ n 1 n 5 R 2 y / ( Q 1 Q 5 ), with n 1 , n 5 the numbers of D1 and D5 branes. \nRotating D1-D5-P black holes with regular horizons exist when n 1 n 5 n P -j 2 > 0 and this cosmic censorship bound defines the 'black-hole regime' for these parameters. Our solutions lie within this bound for \nb 2 a 2 > k n + √ ( k -m + n )( m + n ) . (16) \nHence, in this regime of parameters, these solutions correspond to horizonless microstates of large-horizon-area BMPVblack holes. They span the whole range of angular momenta that these black holes can have. This is a dramatic improvement over the earlier solutions [8, 9], which only have j glyph[greaterorsimilar] 0 . 88 √ n 1 n 5 n P . The solutions with m = 0 are also remarkable because, as a → 0, they give the first family of microstate geometries of the non-rotating D1D5-P black hole. An explicit example (with k = 1 , m = 0 and general n ) is given by: \nF 1 , 0 ,n = -a -2 ( 1 -r 2 n ( r 2 + a 2 ) -n ) ω 1 , 0 ,n = 2 -1 / 2 R y Σ -1 ( 1 -r 2 n ( r 2 + a 2 ) -n ) sin 2 θ dφ . (17) \nOne can easily show that the corresponding metrics are regular and have no CTCs. For our more general class of solutions, this proof becomes increasingly complicated, however our construction explicitly removes CTCs in the most dangerous regions (near r = 0 and θ = 0 or π/ 2), and there is little reason to expect problems elsewhere.", '4. THE DUAL CFT STATES': "Our geometries are asymptotically AdS 3 × S 3 and correspond holographically to 1 / 2-BPS states in a (4 , 4) twodimensional CFT with central charge c = 6 n 1 n 5 ≡ 6 N . Since these states are supersymmetric, they should have a simple description at the locus in moduli space at which the CFT is realized as the symmetric orbifold M N /S N . \nThe untwisted sector of this theory consists of N copies of the CFT with target space M . The theory also contains twisted sectors, in which the elementary fields have non-trivial periodicities connecting different CFT copies: When k copies are cyclically permuted by the boundary conditions, we call the corresponding state a 'strand of winding k '. Following the conventions of [26], we denote by | ++ 〉 1 a strand of length 1 in the RR ground state that has j = ˜ j = 1 / 2. To describe our states we will also need the twisted-sector RR ground state, | 00 〉 k , which has winding k and is a scalar under all symmetries. \nIn our solutions, the momenta are carried by excitations of the | 00 〉 k strands. These excitations can be described by (4 , 4) superconformal algebras with central charge 6 k living on each strand. Denoting the Virasoro generators as L n and the R-symmetry SU (2) generators as J i n , we excite the | 00 〉 k strands with two mutuallycommuting, momentum-carrying perturbations: J + -1 = ( J 1 -1 + iJ 2 -1 ) and ( L -1 -J 3 -1 ). \nThe charges of our solutions (15) support the identification of the dual CFT states with coherent superpositions of states of the form: \n( | ++ 〉 1 ) N 1 ( ( J + -1 ) m m ! ( L -1 -J 3 -1 ) n n ! | 00 〉 k ) N k,m,n , (18) \nfor all values of N 1 such that N 1 + kN k,m,n = N . \nTo find the exact coefficients of this superposition of states, one can straightforwardly generalize the derivation of [26] to states with n > 0. These coefficients are thereby determined in terms of the supergravity parameters a and b k,m,n . \nFrom this calculation one finds that the average numbers of | ++ 〉 1 and | 00 〉 k strands are given by N a 2 and N b 2 / (2 k ) respectively, from which the strand quantum numbers immediately yield the supergravity momentum and angular momenta in (15). It is also possible to compute 3-point correlators between our heavy states and BPS states of low conformal dimension. These correlators depend not only on the average numbers of strands but also on the spread of the coherent superposition, providing an even more stringent check of the identification between the CFT states and the supergravity solutions. \nFIG. 1: Sketch of the superstratum spatial geometry in the r -y plane, compared to the extremal BTZ geometry. \n<!-- image -->", '5. THE STRUCTURE OF THE METRIC': 'In the AdS/CFT limit, one takes R y to be the largest scale in the problem, implying that Q P glyph[lessmuch] √ Q 1 Q 5 . We further focus on the regime a 2 glyph[lessmuch] Q P = ( m + n ) b 2 / (2 k ), in which the structure of the cap lies deep inside the AdS 2 region discussed above. \nIn the a → 0 limit, the AdS 2 throat tends to infinite depth and our solutions tend to the BMPV solution. Furthermore, when m = 0, the angular momentum vanishes and the solutions tend to that of the non-rotating D1-D5-P (Strominger-Vafa) black hole. \nIf, instead, we keep a 2 glyph[lessmuch] Q P small but finite, then the leading terms in the metric for r glyph[greatermuch] a are those of the corresponding black hole. The AdS 2 throat extends in the radial direction for a proper length of order glyph[lscript] AdS log ( Q P /a 2 ), and the geometry caps off smoothly in the region r glyph[lessmuch] a , as shown in Fig. 1. In string units, the proper length of the y circle in the AdS 2 throat is of order ( g s n P ) 1 / 2 /N 1 / 4 when the volume of the compact space M is of order one. Thus one can easily arrange that the proper length of the y circle in the AdS 2 region is large in string units, whereupon the supergravity approximation is valid. \nThe momentum charge is carried by a superstratum deformation (supergravity wave) concentrated deep inside the AdS 2 region. The wave profile is determined by the functions ∆ k,m,n [28]. Inside the support of the wave, the momentum density that stabilized the size of the y -circle quickly dilutes, and the circle starts to shrink until one gets to r = 0, where the coiffuring relations guarantee that the geometry caps off smoothly. \nOur solutions therefore provide examples, with arbitrary finite angular momenta, of how the horizon of a D1-D5-P black hole can be replaced by a smooth cap. The solutions only differ significantly from the corresponding black hole metric near the cap; the difference is suppressed in the AdS 2 throat and further out into the asymptotic AdS 3 region. For example, in the k = 1, m = 0, general n solution, the leading corrections to the \ncorresponding black hole metric have magnitude (in a local orthonormal frame) of order √ na 2 /r 2 in the AdS 2 throat, and of order a 2 /r 2 in the asymptotic AdS 3 region. \nWhen a is exactly zero, from the dictionary (15) one can see that for any value of j and n P there exists a one-parameter family of CFT states that should correspond to a bulk solution with a = 0. Since this solution is exactly the classical black-hole solution with an event horizon, one might naively conclude that certain pure CFT states have a bulk dual with an event horizon, which would contradict the intuition expressed in the Introduction. However, several hints indicate that the strong-coupling description of these particular states (and also of two-charge states of the form ( | 00 〉 k ) N k ) requires ingredients beyond supergravity. For example, the supergravity approximation to the sequence of dualities used to derive the geometry [27] is not valid in these instances. Moreover, in the D1-D5 CFT, this class of states can be distinguished from the thermal ensemble only by the VEVs of non-chiral primary operators.', '6. DISCUSSION': "In this Letter, we have constructed a new family of black hole microstate geometries that solve the ten-yearold problem of lowering the angular momentum j arbitrarily below the cosmic censorship bound, and we have identified the dual CFT states. Our results demonstrate how adding momentum can transform a two-charge solution describing a microstate of a string-size black hole into a smooth low-curvature solution with a long AdS 2 throat. We are confident that all the solutions one can build by generalizing the present work to include more general fluctuations will continue to share these properties. The generic black-hole microstate differs from the states we have constructed in the distribution and type of momentum carriers - our solutions correspond in the CFT to using a very limited set of generators of the chiral algebra (see (18)) to carry the momentum. It is a very interesting question to ask how closely one can approach the generic state using our techniques. \nAcknowledgments . We thank Samir Mathur for discussions. The work of IB and DT was supported by the John Templeton Foundation Grant 48222. The work of EJM was supported in part by DOE grant DESC0009924. The work of SG was supported in part by the Padua University Project CPDA144437. The work of RR was partially supported by the STFC Consolidated Grant ST/L000415/1 'String theory, gauge theory & duality' . The work of MS was supported in part by JSPS KAKENHI Grant Number 16H03979. The work of DT was supported in part by a CEA Enhanced Eurotalents Fellowship. The work of NPW was supported in part by the DOE grant DE-SC0011687. SG, EM, RR, MS and NPW are very grateful to the IPhT, CEA-Saclay for hospitality while a substantial part of this work was done.", 'Appendix A: Details of the general solution': 'The form of F k,m,n and ω k,m,n for general k, m, n is \nF k,m,n = 4 [ m 2 ( k + n ) 2 k 2 F 2 k, 2 m, 2 n + n 2 ( k -m ) 2 k 2 F 2 k, 2 m +2 , 2 n -2 ] , ω k,m,n = µ k,m,n ( dψ + dφ )+ ζ k,m,n ( dψ -dφ ) , (A1) \nµ k,m,n = R y √ 2 [ ( k -m ) 2 ( k + n ) 2 k 2 F 2 k, 2 m +2 , 2 n + m 2 n 2 k 2 F 2 k, 2 m, 2 n -2 -r 2 + a 2 sin 2 θ 4 Σ F k,m,n -∆ 2 k, 2 m, 2 n 4 Σ + x k,m,n 4 Σ ] , (A2) \nwhere \nF 2 k, 2 m, 2 n = -j 1 + j 2 + j 3 ≤ k + n -1 ∑ j 1 ,j 2 ,j 3 =0 ( j 1 + j 2 + j 3 j 1 , j 2 , j 3 ) ( k + n -j 1 -j 2 -j 3 -1 k -m -j 1 ,m -j 2 -1 ,n -j 3 ) 2 ( k + n -1 k -m,m -1 ,n ) 2 ∆ 2( k -j 1 -j 2 -1) , 2( m -j 2 -1) , 2( n -j 3 ) 4( k + n ) 2 ( r 2 + a 2 ) , (A3) \nand where \n( j 1 + j 2 + j 3 j 1 , j 2 , j 3 ) ≡ ( j 1 + j 2 + j 3 )! j 1 ! j 2 ! j 3 ! . (A4) \nIt should be understood that in F k,m,n and µ k,m,n , when the coefficient of an F function is zero, the term is zero. \nThe expression for ζ k,m,n can be obtained from µ k,m,n by quadrature using the BPS equations for ω , which now reduce to an integrable system of differential equations, as was the case for the n = 0 solutions studied in [13]. \nFor regularity, µ k,m,n must vanish at r = 0 , θ = 0; this fixes x k,m,n to the value given below Eq. (14). \n- [1] S. W. Hawking, Phys. Rev. D 14 , 2460 (1976).\n- [2] S. D. Mathur, Class. Quant. Grav. 26 , 224001 (2009).\n- [3] A. Strominger and C. Vafa, Phys. Lett. B 379 , 99 (1996).\n- [4] S. D. Mathur, Fortsch. Phys. 53 , 793 (2005).\n- [5] I. Bena and N. P. Warner, Lect. Notes Phys. 755 , 1 (2008).\n- [6] I. Bena and N. P. Warner, Phys. Rev. D 74 , 066001 (2006).\n- [7] P. Berglund, E. G. Gimon and T. S. Levi, JHEP 0606 , 007 (2006).\n- [8] I. Bena, C.-W. Wang and N. P. Warner, JHEP 0611 , 042 (2006).\n- [9] I. Bena, C.-W. Wang and N. P. Warner, JHEP 0807 , 019 (2008).\n- [10] J. de Boer, S. El-Showk, I. Messamah and D. Van den Bleeken, JHEP 1002 , 062 (2010).\n- [11] O. Lunin, S. D. Mathur and D. Turton, Nucl. Phys. B 868 , 383 (2013).\n- [12] S. Giusto and R. Russo, JHEP 1403 , 007 (2014).\n- [13] I. Bena, S. Giusto, R. Russo, M. Shigemori and N. P. Warner, JHEP 1505 , 110 (2015).\n- [14] I. Bena, E. Martinec, D. Turton and N. P. Warner, JHEP 1605 , 064 (2016).\n- [15] J. C. Breckenridge, R. C. Myers, A. W. Peet and C. Vafa, Phys. Lett. B 391 , 93 (1997).\n- [16] M. Banados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 69 , 1849 (1992); M. Banados, M. Henneaux, \nOne might worry that the warp factor Z 1 could become negative and render the solution singular if the amplitude of the fluctuations becomes too large. However, the minimal value of Z 1 occurs when cos ˆ v 2 k, 2 m, 2 n = -1. Then the regularity conditions in Eq. (14) and the identity \n∆ 2 k, 2 m, 2 n x k,m,n ≤ ∑ m,n =0 k =1 δ k + n -1 ,p ∆ 2 k, 2 m, 2 n x k,m,n = a 2 ( r 2 + a 2 ) ≤ 1 \nensure that b 2 k,m,n ∆ 2 k, 2 m, 2 n < b 2 and hence Z 1 > 0. \n- C. Teitelboim and J. Zanelli, Phys. Rev. D 48 , 1506 (1993), Erratum: [Phys. Rev. D 88 , 069902 (2013)].\n- [17] A. Strominger, JHEP 9901 , 007 (1999).\n- [18] F. Denef, JHEP 0210 , 023 (2002).\n- [19] J. de Boer, S. El-Showk, I. Messamah and D. Van den Bleeken, JHEP 0905 , 002 (2009).\n- [20] S. Giusto, O. Lunin, S. D. Mathur and D. Turton, JHEP 1302 , 050 (2013).\n- [21] I. Bena, M. Shigemori and N. P. Warner, JHEP 1410 , 140 (2014).\n- [22] S. Giusto, L. Martucci, M. Petrini and R. Russo, Nucl. Phys. B 876 , 509 (2013).\n- [23] V. Balasubramanian, J. de Boer, E. Keski-Vakkuri and S. F. Ross, Phys. Rev. D 64 , 064011 (2001).\n- [24] J. M. Maldacena and L. Maoz, JHEP 0212 , 055 (2002).\n- [25] I. Bena, S. F. Ross and N. P. Warner, Class. Quant. Grav. 31 , 165015 (2014).\n- [26] S. Giusto, E. Moscato and R. Russo, JHEP 1511 (2015) 004.\n- [27] O. Lunin and S. D. Mathur, Nucl. Phys. B 610 , 49 (2001); O. Lunin, J. M. Maldacena and L. Maoz, hepth/0212210; I. Kanitscheider, K. Skenderis and M. Taylor, JHEP 0706 , 056 (2007).\n- [28] To see this, note that the ∆ k,m,n are peaked around r max ∼ a √ n/k (and around a band of latitude on S 3 ), so for instance for n ∼ k and n , k large, the wave profile is sharply peaked near r ∼ a (the blue band in Fig. 1).'}
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2016ApJ...825..126D
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Supermassive Black Holes with High Accretion Rates in Active Galactic Nuclei. V. A New Size-Luminosity Scaling Relation for the Broad-line Region
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2016-01-01
| 33 | 0.54 | 159 |
['accretion', 'accretion disks', 'galaxies active', 'galaxies nuclei', '-', '-']
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This paper reports results of the third-year campaign of monitoring super-Eddington accreting massive black holes (SEAMBHs) in active galactic nuclei (AGNs) between 2014 and 2015. Ten new targets were selected from the quasar sample of the Sloan Digital Sky Survey (SDSS), which have generally been more luminous than the SEAMBH candidates in the last two years. Hβ lags ({τ }<SUB>{{H</SUB>}β }) in five of the 10 quasars have been successfully measured in this monitoring season. We find that the lags are generally shorter, by large factors, than those of objects with same optical luminosity, in light of the well-known R <SUB>H</SUB> <SUB>β</SUB>-L <SUB>5100</SUB> relation. The five quasars have dimensionless accretion rates of \dot{{M}\quad }=10-10<SUP>3</SUP>. Combining these with measurements of the previous SEAMBHs, we find that the reduction of Hβ lags depends tightly on accretion rates, {τ }<SUB>{{H</SUB>}β }/{τ }<SUB>R-L</SUB>\propto {\dot{{M}}}<SUP>-0.42</SUP>, where {τ }<SUB>R-L</SUB> is the Hβ lag from the normal R <SUB>H</SUB> <SUB>β</SUB>-L <SUB>5100</SUB> relation. Fitting 63 mapped AGNs, we present a new scaling relation for the broad-line region: {R}<SUB>{{H</SUB>}β }={α }<SUB>1</SUB>{{\ell }}<SUB>44</SUB><SUP>{β </SUP><SUB>1</SUB>} {min} [1,{(\dot{{M}}/{\dot{{M}}}<SUB>c</SUB>)}<SUP>-{γ </SUP><SUB>1</SUB>}], where {{\ell }}<SUB>44</SUB>={L}<SUB>5100</SUB>/{10}<SUP>44</SUP> {erg} {{{s}}}<SUP>-1</SUP> is the 5100 Å continuum luminosity, and the coefficients are {α }<SUB>1</SUB>={29.6}<SUB>-2.8</SUB><SUP>+2.7</SUP> lt-day, {β }<SUB>1</SUB>={0.56}<SUB>-0.03</SUB><SUP>+0.03</SUP>, {γ }<SUB>1</SUB>={0.52}<SUB>-0.16</SUB><SUP>+0.33</SUP>, and {\dot{{M}}}<SUB>c</SUB>={11.19}<SUB>-6.22</SUB><SUP>+2.29</SUP>. This relation is applicable to AGNs over a wide range of accretion rates, from 10<SUP>-3</SUP> to 10<SUP>3</SUP>. Implications of this new relation are briefly discussed.
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https://arxiv.org/pdf/1604.06218.pdf
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{'SUPERMASSIVE BLACK HOLES WITH HIGH ACCRETION RATES IN ACTIVE GALACTIC NUCLEI. V. A NEW SIZE-LUMINOSITY SCALING RELATION FOR THE BROAD-LINE REGION': 'PU DU 1 , KAI-XING LU 2,1 , ZHI-XIANG ZHANG 1 , YING-KE HUANG 1 , KAI WANG 1 , CHEN HU 1 , JIE QIU 1 , YAN-RONG LI 1 , XU-LIANG FAN 6 , XIANG-ER FANG 9 , JIN-MING BAI 6 , WEI-HAO BIAN 8 , YE-FEI YUAN 9 , 4,5 1,3,* \nLUIS C. HO \nAND JIAN-MIN WANG \n(SEAMBH COLLABORATION) \nReceived 2015 December 13; accepted 2016 April 20', 'ABSTRACT': 'This paper reports results of the third-year campaign of monitoring super-Eddington accreting massive black holes (SEAMBHs) in active galactic nuclei (AGNs) between 2014 -2015. Ten new targets were selected from quasar sample of Sloan Digital Sky Survey (SDSS), which are generally more luminous than the SEAMBH candidates in last two years. H β lags ( τ H β ) in five of the 10 quasars have been successfully measured in this monitoring season. We find that the lags are generally shorter, by large factors, than those of objects with same optical luminosity, in light of the well-known R H β -L 5100 relation. The five quasars have dimensionless accretion rates of ˙ M = 10 -10 3 . Combining measurements of the previous SEAMBHs, we find that the reduction of H β lags tightly depends on accretion rates, τ H β /τ R -L ∝ ˙ M -0 . 42 , where τ R -L is the H β lag from the normal R H β -L 5100 relation. Fitting 63 mapped AGNs, we present a new scaling relation for the broad-line region: R H β = α 1 /lscript β 1 44 min [ 1 , ( ˙ M / ˙ M c ) -γ 1 ] , where /lscript 44 = L 5100 / 10 44 erg s -1 is 5100 Å continuum luminosity, and coefficients of α 1 = (29 . 6 + 2 . 7 -2 . 8 ) lt-d, β 1 = 0 . 56 + 0 . 03 -0 . 03 , γ 1 = 0 . 52 + 0 . 33 -0 . 16 and ˙ M c = 11 . 19 + 2 . 29 -6 . 22 . This relation is applicable to AGNs over a wide range of accretion rates, from 10 -3 to 10 3 . Implications of this new relation are briefly discussed. Subject headings: black holes: accretion - galaxies: active - galaxies: nuclei', '1. INTRODUCTION': 'This is the fifth paper of the series reporting the ongoing large campaign of monitoring Super-Eddington Accreting Massive Black Holes (SEAMBHs) in active galaxies and quasars starting from October 2012. One of the major goals of the campaign is to search for massive black holes with extreme accretion rates through reverberation mapping (RM) of broad emission lines and continuum. Results from the campaigns in 2012-2013 and 2013-2014 have been reported by Du et al. (2014, 2015, hereafter Papers I and IV), Wang et al. (2014, Paper II) and Hu et al. (2015, Paper III). This paper carries out the results of SEAMBH2014 sample, which was monitored from September 2014 to June 2015. With the three monitoring years of observations, we build up a new scaling relation of the broad-line region (BLR) in this paper. \n- 1 Key Laboratory for Particle Astrophysics, Institute of High Energy Physics, Chinese Academy of Sciences, 19B Yuquan Road, Beijing 100049, China\n- 2 Astronomy Department, Beijing Normal University, Beijing 100875, China\n- 3 National Astronomical Observatories of China, Chinese Academy of Sciences, 20A Datun Road, Beijing 100020, China\n- 4 Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China\n- 5 Department of Astronomy, School of Physics, Peking University, Beijing 100871, China\n- 6 Yunnan Observatories, Chinese Academy of Sciences, Kunming 650011, China\n- 8 Physics Department, Nanjing Normal University, Nanjing 210097, China\n- 9 Department of Astronomy, University of Science and Technology of China, Hefei 230026, China \nReverberation mapping (RM) technique, measuring the delayed echoes of broad lines to the varying ionizing continuum (Bahcall et al. 1972; Blandford & McKee 1982; Peterson 1993), is a powerful tool to probe the kinematics and geometry of the BLRs in the time domain. Countless clouds, which contribute to the smooth profiles of the broad emission lines (e.g., Arav et al. 1997), are distributed in the vicinity of supermassive black hole (SMBH), composing the BLR. As an observational consequence of photonionization powered by the accretion disk under the deep gravitational potential of the SMBH, the profiles of the lines are broadened, and line emission from the clouds reverberate in response to the varying ionizing continuum. The reverberation is delayed because of light travel difference between H β and ionizing photons and is thus expected to deliver information on the kinematics and structure of the BLR. The unambiguous reverberation of the lines, detected by monitoring campaigns from ultraviolet to optical bands since the late 1980s, supports this picture of the central engine of AGNs (e.g., Clavel et al. 1991; Peterson et al. 1991, 1993; Maoz et al. 1991; Wanders et al. 1993; Dietrich et al. 1993, 1998, 2012; Kaspi et al. 2000; Denney et al. 2006, 2010; Bentz et al. 2009, 2014; Grier et al. 2012; Papers I-IV; Barth et al. 2013, 2015; Shen et al. 2015a,b). The R H β -L 5100 relation was first discussed by Koratkar & Gaskell (1991) and Peterson (1993). Robust RM results for 41 AGNs in the last four decades lead to a simple, highly significant correlation of the form \nR H β ≈ α 0 /lscript β 0 44 , (1) \nwhere /lscript 44 = L 5100 / 10 44 erg s -1 is the 5100 Å luminosity in units of 10 44 erg s -1 (corrected for host galaxy contamination) and R H β = c τ H β is the emissivity-weighted radius of the BLR (Kaspi et al. 2000; Bentz et al. 2013). We refer to this type \nTable 1 The SEAMBH project: observational results \n| Object | α 2000 | δ 2000 | redshift | monitoring period | N spec | Comparison stars | Comparison stars |\n|---------------------------------|---------------------------------|---------------------------------|---------------------------------|---------------------------------|---------------------------------|---------------------------------|---------------------------------|\n| | α 2000 | δ 2000 | redshift | monitoring period | N spec | R ∗ | P.A. |\n| First phase: SEAMBH2012 sample | First phase: SEAMBH2012 sample | First phase: SEAMBH2012 sample | First phase: SEAMBH2012 sample | First phase: SEAMBH2012 sample | First phase: SEAMBH2012 sample | First phase: SEAMBH2012 sample | First phase: SEAMBH2012 sample |\n| Mrk 335 | 00 06 19.5 | + 20 12 10 | 0.0258 | Oct., 2012 - Feb., 2013 | 91 | 80 \'\' . 7 | 174 . 5 · |\n| Mrk 1044 | 02 30 05.5 | - 08 59 53 | 0.0165 | Oct., 2012 - Feb., 2013 | 77 | 207 \'\' . 0 | - 143 . 0 · |\n| IRAS 04416+1215 | 04 44 28.8 | + 12 21 12 | 0.0889 | Oct., 2012 - Mar., 2013 | 92 | 137 \'\' . 9 | - 55 . 0 · |\n| Mrk 382 | 07 55 25.3 | + 39 11 10 | 0.0337 | Oct., 2012 - May., 2013 | 123 | 198 \'\' . 4 | - 24 . 6 · |\n| Mrk 142 | 10 25 31.3 | + 51 40 35 | 0.0449 | Nov., 2012 - Apr., 2013 | 119 | 113 \'\' . 1 | 155 . 2 · |\n| MCG + 06 - 26 - 012 | 11 39 13.9 | + 33 55 51 | 0.0328 | Jan., 2013 - Jun., 2013 | 34 | 204 \'\' . 3 | 46 . 1 · |\n| IRAS F12397+3333 | 12 42 10.6 | + 33 17 03 | 0.0435 | Jan., 2013 - May., 2013 | 51 | 189 \'\' . 0 | 130 . 0 · |\n| Mrk 486 | 15 36 38.3 | + 54 33 33 | 0.0389 | Mar., 2013 - Jul., 2013 | 45 | 193 \'\' . 8 | - 167 . 0 · |\n| Mrk 493 | 15 59 09.6 | + 35 01 47 | 0.0313 | Apr., 2013 - Jun., 2013 | 27 | 155 \'\' . 3 | 98 . 5 · |\n| Second phase: SEAMBH2013 sample | Second phase: SEAMBH2013 sample | Second phase: SEAMBH2013 sample | Second phase: SEAMBH2013 sample | Second phase: SEAMBH2013 sample | Second phase: SEAMBH2013 sample | Second phase: SEAMBH2013 sample | Second phase: SEAMBH2013 sample |\n| SDSS J075101.42+291419.1 | 07 51 01.4 | + 29 14 19 | 0.1208 | Nov., 2013 - May., 2014 | 38 | 133 \'\' . 3 | - 41 . 3 · |\n| SDSS J080101.41+184840.7 | 08 01 01.4 | + 18 48 40 | 0.1396 | Nov., 2013 - Apr., 2014 | 34 | 118 \'\' . 8 | - 98 . 2 · |\n| SDSS J080131.58+354436.4 | 08 01 31.6 | + 35 44 36 | 0.1786 | Nov., 2013 - Apr., 2014 | 31 | 100 \'\' . 0 | 145 . 2 · |\n| SDSS J081441.91+212918.5 | 08 14 41.9 | + 21 29 19 | 0.1628 | Nov., 2013 - May., 2014 | 34 | 79 \'\' . 0 | 73 . 9 · |\n| SDSS J081456.10+532533.5 | 08 14 56.1 | + 53 25 34 | 0.1197 | Nov., 2013 - Apr., 2014 | 27 | 164 \'\' . 5 | - 172 . 9 · |\n| SDSS J093922.89+370943.9 | 09 39 22.9 | + 37 09 44 | 0.1859 | Nov., 2013 - Jun., 2014 | 26 | 175 \'\' . 1 | - 139 . 0 · |\n| Third phase: SEAMBH2014 sample | Third phase: SEAMBH2014 sample | Third phase: SEAMBH2014 sample | Third phase: SEAMBH2014 sample | Third phase: SEAMBH2014 sample | Third phase: SEAMBH2014 sample | Third phase: SEAMBH2014 sample | Third phase: SEAMBH2014 sample |\n| SDSS J075949.54+320023.8 | 07 59 49.5 | + 32 00 24 | 0.1880 | Sep., 2014 - May., 2015 | 27 | 109 \'\' . 2 | - 48 . 3 · |\n| SDSS J080131.58+354436.4 | 08 01 31.6 | + 35 44 36 | 0.1786 | Oct., 2014 - May., 2015 | 19 | 139 \'\' . 2 | - 85 . 3 · |\n| SDSS J084533.28+474934.5 | 08 45 33.3 | + 47 49 35 | 0.3018 | Sep., 2014 - Apr., 2015 | 18 | 205 \'\' . 5 | - 126 . 4 · |\n| SDSS J085946.35+274534.8 | 08 59 46.4 | + 27 45 35 | 0.2438 | Sep., 2014 - Jun., 2015 | 26 | 169 \'\' . 8 | - 89 . 1 · |\n| SDSS J102339.64+523349.6 | 10 23 39.6 | + 52 33 50 | 0.1364 | Oct., 2014 - Jun., 2015 | 26 | 123 \'\' . 2 | 108 . 1 · | \nNote . -This table follows the contents Table 1 in Paper IV. We denote the samples monitored during the 2012-2013, 2013-2014 and 2014-2015 observing seasons as SEAMBH2012, SEAMBH2013 and SEAMBH2014, respectively. N spec is the numbers of spectroscopic epochs, R ∗ is the angular distance between the object and the comparison star and PA the position angle from the AGN to the comparison star. We marked the time lag of J080131 as \'uncertain" in PaperIV, however we pick it up here because its lag reported in Paper IV is highly consistent with the number measured in the present paper. \nof correlation as the normal R H β -L 5100 relationship. The constants α 0 and β 0 differ slightly from one study to the next, depending on the number of sources and their exact luminosity range (e.g., Kilerci Eser et al. 2015). For sub-Eddington accreting AGNs, α 0 = 35 . 5 ltd and β 0 = 0 . 53, but for SEAMBHs they are different (see Paper IV). \nAs reported in Paper IV, some objects from the SEAMBH2012 and SEAMBH2013 samples have much shorter H β lags compared with objects with similar luminosity, and the R H β -L 5100 relation has a large scatter if they are included. In particular, the reduction of the lags increases with the dimensionless accretion rate, defined as ˙ M = ˙ M · / L Edd c -2 , where ˙ M · is the accretion rate, L Edd is the Eddington luminosity and c is the speed of light. Furthermore, it has been found, so far in the present campaigns, that SEAMBHs have a range of accretion rates from a few to ∼ 10 3 . This kind of shortened H β lags was discovered in the current SEAMBH project (a comparison with previous campaigns is given in Section 6.5). Such high accretion rates are characteristic of the regime of slim accretion disks (Abramowicz et al. 1988; Szuszkiewicz et al. 1996; Wang & Zhou 1999; Wang et al. 1999; Mineshige et al. 2000; Wang & Netzer 2003; Sadowski 2009). These interesting properties needed to be confirmed with observations. We aim to explore whether we can define a new scaling relation, R H β = R H β ( L 5100 , ˙ M ), which links the size of the BLR to both the AGN luminosity and accretion rate. \nWe report new results from SEAMBH2014. We describe target selection, observation details and data reduction in §2. H β lags, BH mass and accretion rates are provided in §3. Properties of H β lags are discussed in §4, and a new scaling relation of H β lags is established in §5. Section 6 introduces the fundamental plane, which is used to estimate accretion rates from single-epoch spectra, for application of the new size-luminosity scaling relation of the BLR. Brief discus- \nf the shortened lags are presented in §7. We draw conclusions in §8. Throughout this work we assume a standard Λ CDM cosmology with H 0 = 67 km s -1 Mpc -1 , Ω Λ = 0 . 68 and Ω M = 0 . 32 (Ade et al. 2014).', '2.1. Target Selection': "We followed the procedures for selecting SEAMBH candidates described in Paper IV. We used the fitting procedures to measure H β profile and 5100 Å luminosity of SDSS quasar spectra described by Hu et al. (2008a,b). Following the standard assumption that the BLR gas is virialized, we estimate the BH mass as \nM · = f BLR R H β V 2 FWHM G = 1 . 95 × 10 6 f BLR V 2 3 τ 10 M /circledot , (2) \nwhere R H β = c τ H β , τ H β is the H β lag measured in the rest frame, τ 10 = τ H β / 10days, G is the gravitational constant, and V 3 = V FWHM / 10 3 km s -1 is the full-width-half-maximum (FWHM) of the H β line profile in units of 10 3 km s -1 . We take the virial factor f BLR = 1 in our series of papers (see some discussions in Paper IV). \nIn order to select AGNs with high accretion rates, we employed the formulation of accretion rates derived from the standard disk model of Shakura & Sunyaev (1973). In the standard model it is assumed that the disk gas is rotating with Keplerian angular momentum, and thermal equilibrium is localized between viscous dissipation and blackbody cooling. Observationally, this model is supported from fits of the socalled big blue bump in quasars (Czerny & Elvis 1987; Wandel & Petrosian 1988; Sun & Malkan 1989; Laor & Netzer 1989; Collin et al. 2002; Brocksopp et al. 2006; Kishimoto et al. 2008; Davis & Laor 2011; Capellupo et al. 2015). The \nTable 2 Light curves of J075949 and J080131 \n| J075949 | J075949 | J075949 | J075949 | J075949 | J080131 | J080131 | J080131 | J080131 | J080131 |\n|------------|--------------------|-----------|-------------------|-------------------|------------|--------------------|-----------|-------------------|-------------------|\n| Photometry | Photometry | Spectra | Spectra | Spectra | Photometry | Photometry | Spectra | Spectra | Spectra |\n| JD | mag | JD | F 5100 | F H β | JD | mag | JD | F 5100 | F H β |\n| 29.374 | 17 . 373 ± 0 . 007 | 76.365 | 2 . 773 ± 0 . 029 | 2 . 469 ± 0 . 038 | 60.324 | 17 . 757 ± 0 . 010 | 112.294 | 2 . 096 ± 0 . 016 | 0 . 853 ± 0 . 027 |\n| 30.360 | 17 . 375 ± 0 . 008 | 80.319 | 2 . 864 ± 0 . 017 | 2 . 461 ± 0 . 031 | 62.314 | 17 . 720 ± 0 . 010 | 116.394 | 2 . 076 ± 0 . 016 | 0 . 893 ± 0 . 029 |\n| 32.352 | 17 . 406 ± 0 . 009 | 83.314 | 2 . 724 ± 0 . 038 | 2 . 366 ± 0 . 041 | 63.299 | 17 . 734 ± 0 . 010 | 119.336 | 2 . 106 ± 0 . 021 | 0 . 831 ± 0 . 031 |\n| 33.339 | 17 . 416 ± 0 . 009 | 86.422 | 2 . 795 ± 0 . 011 | 2 . 535 ± 0 . 023 | 68.397 | 17 . 721 ± 0 . 013 | 135.324 | 2 . 086 ± 0 . 026 | 0 . 807 ± 0 . 033 |\n| 34.331 | 17 . 408 ± 0 . 009 | 89.378 | 2 . 888 ± 0 . 012 | 2 . 402 ± 0 . 032 | 77.318 | 17 . 746 ± 0 . 009 | 139.319 | 2 . 074 ± 0 . 028 | 0 . 862 ± 0 . 035 | \nNote . - JD: Julian dates from 2,456,900; F 5100 and F H β are fluxes at (1 + z )5100 Å and H β emission lines in units of 10 -16 erg s -1 cm -2 -1 and 10 -14 erg s -1 cm -2 . (This table is available in its entirety in a machine-readable form in the online journal. A portion is shown here for guidance regarding its form and content.) \ndimensionless accretion rate is given by \n˙ M = 20 . 1 ( /lscript 44 cos i ) 3 / 2 m -2 7 , (3) \nwhere m 7 = M · / 10 7 M /circledot (see Papers II and IV) and i is the inclination angle to the line of sight of the disk. We take cos i = 0 . 75, which represents a mean disk inclination for a type 1 AGNs with a torus covering factor of about 0.5 (it is assumed that the torus axis is co-aligned with the disk axis). Previous studies estimate i ≈ 0 -45 · [e.g., Fischer et al. (2014) find a inclination range of i ≈ 10 · -45 · , whereas Pancoast et al. (2014) quote i ≈ 5 · -45 · ; see also supplementary materials in Shen & Ho (2014)], which results in ∆ log ˙ M = 1 . 5 ∆ logcos i /lessorsimilar 0 . 15 from Equation (3). This uncertainty is significantly smaller than the average uncertainty on ˙ M ( ∼ 0 . 3 -0 . 5 dex) in the present paper, and is thus ignored. Equation (3) applies to AGNs that have accretion rates 10 -2 /lessorsimilar ˙ M /lessorsimilar 3 × 10 3 , namely excluding the regimes of advection-dominated accretion flows (ADAF; Narayan & Yi 1994) and of flows with hyperaccretion rates ( ˙ M ≥ 3 × 10 3 ; see Appendix A for the validity of Equation 3 for SEAMBHs). Using the normal R H β -L 5100 relation (Bentz et al. 2013), we fitted all the quasar spectra in SDSS Data Release 7 by the procedures in Hu et al. (2008a, b) and applied Equations (2) and (3) to select high -˙ M targets. We ranked quasars in terms of ˙ M and chose ones as candidates with the highest ˙ M . We found that the high -˙ M quasars are characterized by 1) strong optical Fe II lines; 2) relatively narrow H β lines ( /lessorsimilar 2000km s -1 ); 3) weak [O III] lines; and 4) steep 2-10 keV spectra (Wang et al. 2004). These properties are similar to those of NLS1s (Osterbrock & Pogge 1987; Boroson & Green 1992), but most of the candidates have more extreme accretion rates (a detailed comparison of SEAMBH properties with normal quasars will be carried out in a separate paper). Considering that the lags of all targets should be measured within one observing season, and taking into consideration the limitations of the weather of the Lijiang Station of Yunnan Observatory (periods between June and September are raining seasons there), we only chose objects with maximum estimated lags of about 100 days or so (the monitoring periods should be at least a few times the presumed lags). Also, to ensure adequate signal-to-noise ratio (S/N) for measurements of light curves, we restricted the targets to a redshift range of z = 0 . 1 -0 . 3 and magnitudes r ' ≤ 18 . 0. The fraction of radio-loud objects with ˙ M > 3 is not high. We \ndiscarded radio-loud objects 10 based on available FIRST observations, in order to avoid H β reverberations potentially affected by nonthermal emission from relativistic jets, or optical continuum emission strongly contaminated by jets. We chose about 20 targets for photometry monitoring, which served as a preselection to trigger follow-up spectroscopic monitoring. The photometric monitoring yielded 10 targets with significant variations ( /greaterorsimilar 0 . 1 magnitudes), and time lags were successfully measured for 5 objects (Table 1). For an overview of our entire ongoing campaign, Table 1 also lists samples from SEAMBH2012 and SEAMBH2013. \nTo summarize: we have selected about 30 targets for spectroscopic monitoring during the last three years (2012-2014). The successful rate of the monitoring project is about 2/3. Our failure to detect a lag for the remaining 1/3 of the sample are either due to low-amplitude variability or bad weather that leads to poor monitoring cadence. In particular, the SEAMBH2014 observations were seriously affected by the El Niño phenomenon.", '2.2. Photometry and Spectroscopy': "The SEAMBH project uses the Lijiang 2.4m telescope, which has an alt-azimuth Ritchey-Chrétien mount with a field de-rotator that enables two objects to be positioned along the same long slit. It is located in Lijiang and is operated by Yunnan Observatories. We adopted the same observational procedures described in detail in Paper I, which also introduces the telescope and spectrograph. We employed the Yunnan Faint Object Spectrograph and Camera (YFOSC), which has a back-illuminated 2048 × 4608 pixel CCD covering a field of 10 ' × 10 ' . During the spectroscopic observation, we put the target and a nearby comparison star into the slit simultaneously, which can provide high-precision flux calibration. As in SEAMBH2013 (Paper IV), we adopted a 5 '' -wide slit to minimize the influence of atmospheric differential refraction, and used Grism 3 with a spectral resolution of 2.9 Å/pixel and wavelength coverage of 3800-9000 Å. To check the accuracy of spectroscopic calibration, we performed differential photometry of the targets using some other stars in the same field. We used an SDSS r ' -band filter for photometry to avoid the potential contamination by emission lines such as H β and H α . Photometric and spectroscopic exposure times are typically 10 and 60 min, respectively. \n10 It has been realised that high-accretion rate AGNs are usually radioquiet (Greene & Ho 2006), although there are a few NLS1s reported to be radio-loud. The fraction of radio-loud AGNs decreases with increasing accretion rate (Ho 2002, 2008). \nTable 3 Light curves of J084533 and J085946 \n| J084533 | J084533 | J084533 | J084533 | J084533 | J085946 | J085946 | J085946 | J085946 | J085946 |\n|------------|--------------------|-----------|-------------------|-------------------|------------|--------------------|-----------|-------------------|-------------------|\n| Photometry | Photometry | Spectra | Spectra | Spectra | Photometry | Photometry | Spectra | Spectra | Spectra |\n| JD | mag | JD | F 5100 | F H β | JD | mag | JD | F 5100 | F H β |\n| 29.417 | 17 . 803 ± 0 . 008 | 75.402 | 1 . 728 ± 0 . 026 | 1 . 145 ± 0 . 047 | 30.428 | 17 . 401 ± 0 . 008 | 90.382 | 2 . 651 ± 0 . 015 | 1 . 693 ± 0 . 024 |\n| 30.398 | 17 . 797 ± 0 . 008 | 80.395 | 1 . 721 ± 0 . 019 | 1 . 085 ± 0 . 033 | 33.403 | 17 . 399 ± 0 . 007 | 97.433 | 2 . 593 ± 0 . 033 | 1 . 728 ± 0 . 038 |\n| 33.376 | 17 . 777 ± 0 . 009 | 84.365 | 1 . 680 ± 0 . 020 | 1 . 046 ± 0 . 035 | 36.410 | 17 . 406 ± 0 . 005 | 104.293 | 2 . 392 ± 0 . 024 | 1 . 835 ± 0 . 034 |\n| 36.368 | 17 . 780 ± 0 . 009 | 91.326 | 1 . 673 ± 0 . 016 | 1 . 086 ± 0 . 028 | 48.357 | 17 . 422 ± 0 . 008 | 111.261 | 2 . 583 ± 0 . 022 | 1 . 595 ± 0 . 037 |\n| 38.411 | 17 . 816 ± 0 . 017 | 104.409 | 1 . 681 ± 0 . 015 | 1 . 069 ± 0 . 029 | 51.353 | 17 . 440 ± 0 . 006 | 116.448 | 2 . 512 ± 0 . 025 | 1 . 697 ± 0 . 027 | \nNote . - This table is available in its entirety in a machine-readable form in the online journal. A portion is shown here for guidance regarding its form and content. \nThe reduction of the photometry data was done in a standard way using IRAF routines. Photometric light curves were produced by comparing the instrumental magnitudes to those of standard stars in the field (see, e.g., Netzer et al. 1996, for details). The radius for the aperture photometry is typically ∼ 4 '' (seeing ∼ 1 . 5 '' -2 '' ), and background is determined from an annulus with radius 8 '' . 5 to 17 '' . The uncertainties on the photometric measurements include the fluctuations due to photon statistics and the scatter in the measurement of the stars used. \nThe spectroscopic data were also reduced with IRAF. The extraction width is fixed to 8 '' . 5, and the sky regions are set to 7 '' . 4 -14 '' . 1 on both sides of the extracted region. The average S/N of the 5100 Å continuum of individual spectra are from ∼ 16 to ∼ 22, except for J085946, which only has S/N ≈ 12. The flux of spectroscopic data was calibrated by simultaneously observing a nearby comparsion star along the slit (see Paper I). The fiducial spectra of the comparison stars are generated using observations from several nights with the best weather conditions. The absolute fluxes of the fiducial spectra are calibrated using additional spectrophotometric standard stars observed in those nights. Then, the in-slit comparison stars are used as standards to calibrate the spectra of targets observed in each night. The sensitivity as a function of wavelength is produced by comparing the observed spectrum of the comparison star to its fiducial spectrum. Finally, the sensitivity function is applied to calibrate the observed AGN spectrum 11 . The procedures adopted here resemble the method used by Maoz et al. (1990) and Kaspi et al. (2000). In order to illustrate the invariance of the comparison stars, we show the light curves from the differential photometry of the comparison stars in Appendix B. It is clear that their fluxes are very stable and the variations are less than ∼ 1%. \nThe calibration method of van Groningen & Wanders (1992), based on the [O III] emission line and popularily used in many RM campaigns (e.g., Peterson et al. 1998; Bentz et al. 2009; Denney et al. 2010; Grier et al. 2012; Barth et al. 2015), is not suitable for SEAMBHs. [O III] λ 5007 tends to be weak in SEAMBHs (especially for the objects in SEAMBH2013-2014), and, even worse, is blended with strong Fe II around 5016 Å. Applying the calibration method to SEAMBHs results in large statistical (caused by the weakness of [O III]) and systematic (caused by the variability of Fe II; see Paper III) uncertainties. The method based on inslit comparison stars, used in our campaign, does not rely on \n[O III] and provides accurate flux calibration for the spectra of SEAMBHs. For comparison, in Paper I we measured the [O III] fluxes in the calibrated spectra of three objects in SEAMBH2012 with relatively strong [O III]; the variation of their [O III] flux is on the order of ∼ 3%. This clearly demonstrates the robustness of our flux calibration method based on in-slit comparison stars. \nThe procedures to measure the 5100 Å and H β flux are nearly the same as those given in Paper I. The continuum beneath H β line is determined by interpolation of two nearby bands (4740-4790 Å and 5075 -5125 Å) in the rest frame. These two bands have minimal contamination from emission lines. The flux of H β is measured by integrating the band between 4810 and 4910 Å after subtraction of the continuum; the H β band is chosen to avoid the influence from Fe II lines. The 5100 Å flux is taken to be the median over the region 5075-5125 Å. Detailed information of the observations is pr ovided in Table 1. All the photometry and continuum and H β light curves for the five objects with successfully detected lags are listed in Tables 2 -4 and shown in Figure 1. We also calculated the mean and RMS (root mean square) spectra and present them in Appendix C.", '2.3. Host Galaxies': "Like the SEAMBH2013 sample, we have no observations that can clearly separate the host galaxies of the AGNs in SEAMBH2014. Shen et al. (2011) propose the following empirical relation to estimate the fractional contribution of the host galaxy to the optical continuum emission: L host 5100 / L AGN 5100 = 0 . 8052 -1 . 5502 x + 0 . 912 x 2 -0 . 1577 x 3 , for x < 1 . 053, where x = log ( L tot 5100 / 10 44 erg s -1 ) and L tot 5100 is the total emission from the AGN and its host at 5100 Å. For x > 1 . 053, L host 5100 /lessmuch L AGN 5100 , and the host contamination can be neglected. The host fractions at 5100 Å for the objects (J075949, J080131, J084533, J085946 and J102339) are (27.5%, 37.1%, 14.2%, 19.0% and 31.8%). The values of L 5100 listed in Table 5 are the host-subtracted luminosities. We note that this empirical relation is based on SDSS spectroscopic observations with a 3 '' fiber, whereas we used a 5 '' -wide slit. It should apply to our observations reasonably well (see Paper IV for additional discussions on this issue). We will revisit this issue in the future using high-resolution images that can more reliably separate the host.", '3. MEASUREMENTS OF H β LAGS, BLACK HOLE MASSES AND ACCRETION RATES': '<!-- image --> \n<!-- image --> \n<!-- image --> \nFigure 1. Light curves and cross-correlation results. Each object has six panels: ( a, b, c ) are light curves of SDSS r \' -band magnitude, 5100 Å continuum and H β emission, respectively; ( d, e, f ) are auto-correlation function (ACF) of the r \' -band magnitude (5100 Å continuum for J080131; see the main t ext), cross-correlation function (CCF) of the r \' -band magnitude and H β line emission (5100 Å and H β for J080131) and the Monte Carlo simulations of peak (red) and centroid (blue) of the lags, respectively. In panels d and e , the solid lines show the results of the ICCF method, and the points with error bars are from ZDCF ( Z -transformed discrete correlation function). F 5100 and F H β are in units of 10 -16 erg s -1 cm -2 -1 and 10 -14 erg s -1 cm -2 , respectively. Bars with terminals denote systematic errors and are plotted in the corners of the panels (see Paper I for details). For J084533, the systematic error bars are so small that the caps of error bars merge in panels b and c ; the same holds for J102339 in panel b . \n<!-- image --> \nAs in Papers I-IV, we used cross-correlation analysis to determine H β lags relative to photometric or 5100 Å continuum light curves. We use the centroid lag for H β . The uncertainties on the lags are determined through the \'flux randomization/random subset sampling" method (RS/RSS; Peterson et al. 1998, 2004). The cross-correlation centroid distribution (CCCD, described in Appendix E) and cross-correlation peak distribution (CCPD) generated by the FR/RSS method (Maoz & Netzer 1989; Peterson et al. 1998, 2004; Denney et al. 2006, 2010, and references therein) are shown in Figure 1. We used the following criteria to define a successful detection of H β lag: 1) non-zero lag from the CCF peak and 2) a maximum correlation coefficient larger than 0.5. Data for the light curves of the targets are given in Tables 2-4. All the measurements of the SEAMBH2014 sample are provided in Table 5. \nThe r \' -band light curves are generally consistent with the 5100 Å continuum light curve, but the former usually have small scatter, as shown in Figure 1. We calculated CCFs for the H β light curves with both r \' -band photometry and with 5100 Å spectral continuum for all objects. The quality of the \nH β -r \' CCFs is usually better than the H β -F 5100 CCFs. We show the H β -r \' CCFs for all objects in Figure 1, except for J080131. We use the H β -r \' lags in the following analysis. For J080131, the r \' -band light curve between 200 and 220 days does not match the 5100 Å continuum light curve, even though H β does follow 5100 Å continuum tightly. Notes to individual sources are given in Appendix D.', '3.2. Black Hole Masses and Accretion Rates': "There are two ways of calculating BH mass, base either on the RMS spectrum (e.g., Peterson et al. 2004; Bentz et al. 2009; Denney et al. 2010; Grier et al. 2012) or on the mean spectrum (e.g., Kaspi et al. 2005; Papers I-IV). Different studies also adopt different measures of the line width, typically either the line dispersion σ line (second moment of the line profile) or the FWHM. In this study, we choose to parameterize the line width using FWHM, as measured in the mean spectra. The narrow H β component may influence the measurement of FWHM. We adopt the same procedure as in Paper I to remove the narrow H β . We fix narrow H β /[O III] λ 5007 to 0.1, and measure FWHM from the mean spectra with narrow H β subtracted. Then we set H β /[O III] λ 5007 to 0 and 0.2 and \nFigure 2 plots distributions of L 5100, EW(H β ), ˙ M and M · of all the mapped AGNs (41 from Bentz et al. 2013 and the 18 \n<!-- image --> \nFigure 1 continued. \nrepeat the process to obtain lower and upper limits to FWHM. The relatively wide slit employed in our campaign (5 '' ) significantly broadens the emission lines by V inst ≈ 1200km s -1 , where V inst is the instrumental broadening that can be estimated from the broadening of selected comparison stars. As in Paper IV, we obtain the intrinsic width of the mean spectra from FWHM = ( FWHM 2 obs -V 2 inst ) 1 / 2 . The FWHM simply obtained here is accurate enough for BH mass estimation. Our procedure for BH mass estimation is based on FWHM measured from the mean spectrum (see explanation in Papers I and II). As shown recently in Woo et al. (2015), the scatter in the scaling parameter ( f BLR) derived in this method is very similar to the scatter in the method based on the RMS spectrum. We use Equations (2) and (3) to calculate accretion rates and BH masses for the five sources listed in Table 5. For convenience and completeness, Table 5 also lists H β lags, BH masses and accretion rates for the sources from SEAMBH2012 and SEAMBH2013. Our campaign has successfully detected H β lags for 18 SEAMBHs since October 2012. \nAs described in Paper II, there are some theoretical uncertainties in identifying a critical value of ˙ M to define a SEAMBH (Laor & Netzer 1989; Beloborodov 1998; Sadowski et al. 2011). Following Paper II, we classified SEAMBHs as those objects with η ˙ M ≥ 0 . 1. This is based on the idea that beyond this value, the accretion disk becomes slim and the radiation efficiency is reduced mainly due to photon trapping (Sadowski et al. 2011). Since we currently cannot observe the entire spectral energy distribution, we have no direct way to measure L bol / L Edd, and this criterion is used as an approximate tool to identify SEAMBH candidates. To be on the conservative side, we chose the lowest possible efficiency, η = 0 . 038 (retrograde disk with a = -1; see Bardeen et al. 1972). Thus, SEAMBHs are objects with ˙ M = 2 . 63. For simplicity, in this paper we use ˙ M min = 3 as the required minimum (Papers II and IV). We refer to AGNs with ˙ M ≥ 3 as SEAMBHs and those with ˙ M < 3 as sub-Eddington ones. Paper IV clearly shows that the properties of the R H β -L 5100 relation for ˙ M ≥ 3 and ˙ M < 3 are significantly different. \nTable 4 Light curves of J102339 \n| J102339 | J102339 | J102339 | J102339 | J102339 |\n|------------|--------------------|-----------|-------------------|-------------------|\n| Photometry | Photometry | Spectra | Spectra | Spectra |\n| JD | mag | JD | F 5100 | F H β |\n| 54.400 | 16 . 759 ± 0 . 009 | 102.431 | 5 . 455 ± 0 . 056 | 2 . 340 ± 0 . 049 |\n| 56.431 | 16 . 776 ± 0 . 011 | 105.309 | 5 . 430 ± 0 . 022 | 2 . 429 ± 0 . 050 |\n| 62.333 | 16 . 762 ± 0 . 008 | 111.450 | 5 . 443 ± 0 . 034 | 2 . 325 ± 0 . 046 |\n| 72.412 | 16 . 820 ± 0 . 007 | 115.406 | 5 . 374 ± 0 . 041 | 2 . 438 ± 0 . 064 |\n| 76.438 | 16 . 800 ± 0 . 021 | 119.396 | 5 . 397 ± 0 . 027 | 2 . 548 ± 0 . 044 | \nNote . - This table is available in its entirety in a machine-readable form in the online journal. A portion is shown here for guidance regarding its form and content. \nSEAMBHs from our campaign; see Table 7 in Paper IV and Table 5 here). As shown clearly in the diagrams, SEAMBH targets are generally more luminous by a factor of 2-3 compared to previous RM AGNs (Figure 2 a ). The BH masses of SEAMBHs are generally less smaller by a factor of 10 compared to previous samples, whereas, as a consequence of our selection, the accretion rates of SEAMBHs are higher by 2-3 orders of magnitude (Figures 2 c and 2 d ). However, EW(H β ) of SEAMBHs are not significantly smaller (Figure 2 d ). On average, the high -˙ M sources have lower mean EW(H β ), consistent with the inverse correlation between EW(H β ) and L bol / L Edd (e.g., Netzer et al. 2004).", '4. PROPERTIES OF H β LAGS IN SEAMBHS': 'The R H β -L 5100 correlation was originally presented by Peterson (1993; his Figure 10, only nine objects). It was confirmed by Kaspi et al. (2000) using a sample of 17 lowredshift quasars. Bentz et al. (2013) refined the R H β -L 5100 relation through subtraction of host contamination and found that its intrinsic scatter is only 0.13 dex. Paper IV (Table 7) provides a complete list of previously mapped AGNs, based on Bentz et al. (2013); we directly use these values 12 . As in Paper IV, for objects with multiple measurements of H β lags, we obtain the BH mass from each campaign and then calculate the average BH mass. Using the averaged BH mass, we apply it to get accretion rates of the BHs during each monitoring epoch, which are further averaged to obtain the mean accretion rates of those objects (Kaspi et al. 2005; Bentz et al. 2013). We call this the \'average scheme." On the other hand, we may consider each individual measurement of a single object as different objects (e.g., Bentz et al. 2013). We called this the \'direct scheme." Although the two approaches are in principle different, we obtain very similar results (see a comparison in Paper IV). \nAll correlations of two parameters shown in this paper are calculated with the FITEXY method, using the version adopted by Tremaine et al. (2002), which allows for intrinsic scatter by increasing the uncertainties in small steps until χ 2 reaches unity (this is typical for many of our correlations). We also emplot the BCES method (Akristas & Bershady 1996) but prefer not to use its results because it is known to give unreliable results in samples containing outliers (there are a few objects with quite large uncertainties of ˙ M ). \n4.1. The R H β -L 5100 relation \n12 NGC 7469 was mapped twice by Collier et al. (1998) and Peterson et al. (2014). While their H β lags are consistent, the FWHM of H β is very different. We only retain the later observation in the analysis. \nTable 5 H β Reverberations of the SEAMBHs \n| Objects | τ H β (days) | FWHM (km s - 1 ) | σ line (km s - 1 ) | log ( M · / M /circledot ) | log ˙ M | log L 5100 (erg s - 1 ) | log L H β (erg s - 1 ) | EW(H β ) (Å) |\n|--------------|--------------------------|--------------------|----------------------|------------------------------|--------------------------|---------------------------|--------------------------|------------------|\n| SEAMBH2012 | SEAMBH2012 | SEAMBH2012 | SEAMBH2012 | SEAMBH2012 | SEAMBH2012 | SEAMBH2012 | SEAMBH2012 | SEAMBH2012 |\n| Mrk 335 | 8 . 7 + 1 . 6 - 1 . 9 | 2096 ± 170 | 1470 ± 50 | 6 . 87 + 0 . 10 - 0 . 14 | 1 . 28 + 0 . 37 - 0 . 30 | 43 . 69 ± 0 . 06 | 42 . 03 ± 0 . 06 | 110 . 5 ± 22 . 3 |\n| Mrk 1044 | 10 . 5 + 3 . 3 - 2 . 7 | 1178 ± 22 | 766 ± 8 | 6 . 45 + 0 . 12 - 0 . 13 | 1 . 22 + 0 . 40 - 0 . 41 | 43 . 10 ± 0 . 10 | 41 . 39 ± 0 . 09 | 101 . 4 ± 31 . 9 |\n| Mrk 382 | 7 . 5 + 2 . 9 - 2 . 0 | 1462 ± 296 | 840 ± 37 | 6 . 50 + 0 . 19 - 0 . 29 | 1 . 18 + 0 . 69 - 0 . 53 | 43 . 12 ± 0 . 08 | 41 . 01 ± 0 . 05 | 39 . 6 ± 9 . 0 |\n| Mrk 142 | 7 . 9 + 1 . 2 - 1 . 1 | 1588 ± 58 | 948 ± 12 | 6 . 59 + 0 . 07 - 0 . 07 | 1 . 65 + 0 . 23 - 0 . 23 | 43 . 56 ± 0 . 06 | 41 . 60 ± 0 . 04 | 55 . 2 ± 9 . 5 |\n| IRAS F12397 | 9 . 7 + 5 . 5 - . | 1802 ± 560 | 1150 ± 122 | + 0 . 27 | + 0 . 98 | | | |\n| | 1 8 23 . 7 + 7 . 5 | 1942 ± 67 | | 6 . 79 - 0 . 45 | 2 . 26 - 0 . 62 | 44 . 23 ± 0 . 05 | 42 . 26 ± 0 . 04 | 54 . 2 ± 8 . 4 |\n| Mrk 486 | - 2 . 7 | | 1296 ± 23 | 7 . 24 + 0 . 12 - 0 . 06 | 0 . 55 + 0 . 20 - 0 . 32 | 43 . 69 ± 0 . 05 | 42 . 12 ± 0 . 04 | 135 . 9 ± 20 . 3 |\n| Mrk 493 | 11 . 6 + 1 . 2 - 2 . 6 | 778 ± 12 | 513 ± 5 | 6 . 14 + 0 . 04 - 0 . 11 | 1 . 88 + 0 . 33 - 0 . 21 | 43 . 11 ± 0 . 08 | 41 . 35 ± 0 . 05 | 87 . 4 ± 18 . 1 |\n| IRAS 04416 | 13 . 3 + 13 . 9 - 1 . 4 | 1522 ± 44 | 1056 ± 29 | 6 . 78 + 0 . 31 - 0 . 06 | 2 . 63 + 0 . 16 - 0 . 67 | 44 . 47 ± 0 . 03 | 42 . 51 ± 0 . 02 | 55 . 8 ± 4 . 7 |\n| SEAMBH2013 | SEAMBH2013 | SEAMBH2013 | SEAMBH2013 | SEAMBH2013 | SEAMBH2013 | SEAMBH2013 | SEAMBH2013 | SEAMBH2013 |\n| SDSS J075101 | 33 . 4 + 15 . 6 - 5 . 6 | 1495 ± 67 | 1055 ± 32 | 7 . 16 + 0 . 17 - 0 . 09 | 1 . 34 + 0 . 25 - 0 . 41 | 44 . 12 ± 0 . 05 | 42 . 25 ± 0 . 03 | 68 . 1 ± 8 . 6 |\n| SDSS J080101 | 8 . 3 + 9 . 7 - 2 . 7 | 1930 ± 18 | 1119 ± 3 | 6 . 78 + 0 . 34 - 0 . 17 | 2 . 33 + 0 . 39 - 0 . 72 | 44 . 27 ± 0 . 03 | 42 . 58 ± 0 . 02 | 105 . 5 ± 8 . 3 |\n| SDSS J080131 | 11 . 5 + 8 . 4 - 3 . 6 | 1188 ± 3 | 850 ± 12 | 6 . 50 + 0 . 24 - 0 . 16 | 2 . 46 + 0 . 38 - 0 . 54 | 43 . 98 ± 0 . 04 | 42 . 08 ± 0 . 03 | 64 . 0 ± 7 . 0 |\n| SDSS J081441 | 18 . 4 + 12 . 7 - 8 . 4 | 1615 ± 22 | 1122 ± 11 | 6 . 97 + 0 . 23 - 0 . 27 | 1 . 56 + 0 . 63 - 0 . 57 | 44 . 01 ± 0 . 07 | 42 . 42 ± 0 . 03 | 132 . 0 ± 23 . 7 |\n| SDSS J081456 | 24 . 3 + 7 . 7 - 16 . 4 | 2409 ± 61 | 1438 ± 32 | 7 . 44 + 0 . 12 - 0 . 49 | 0 . 59 + 1 . 03 - 0 . 30 | 43 . 99 ± 0 . 04 | 42 . 15 ± 0 . 03 | 74 . 4 ± 7 . 6 |\n| SDSS J093922 | 11 . 9 + 2 . 1 - 6 . 3 | 1209 ± 16 | 835 ± 11 | 6 . 53 + 0 . 07 - 0 . 33 | 2 . 54 + 0 . 71 - 0 . 20 | 44 . 07 ± 0 . 04 | 42 . 09 ± 0 . 04 | 53 . 0 ± 6 . 7 |\n| SDSS J075949 | 55 . 0 + 17 . 0 - 13 . 1 | 1807 ± 11 | 1100 ± 3 | 7 . 54 + 0 . 12 - 0 . 12 | 0 . 70 + 0 . 29 - 0 . 29 | 44 . 20 ± 0 . 03 | 42 . 48 ± 0 . 02 | 97 . 5 ± 9 . 1 |\n| SDSS J080131 | 11 . 2 + 14 . 8 - 9 . 8 | 1290 ± 13 | 800 ± 5 | 6 . 56 + 0 . 37 - 0 . 90 | 2 . 29 + 1 . 87 - 0 . 80 | 43 . 95 ± 0 . 04 | 41 . 96 ± 0 . 05 | 52 . 3 ± 7 . 7 |\n| SDSS J084533 | 15 . 2 + 3 . 2 - 6 . 3 | 1243 ± 13 | 818 ± 10 | 6 . 66 + 0 . 08 - 0 . 23 | 2 . 98 + 0 . 52 - 0 . 22 | 44 . 54 ± 0 . 04 | 42 . 58 ± 0 . 05 | 55 . 9 ± 7 . 5 |\n| SDSS J085946 | 34 . 8 + 19 . 2 - 26 . 3 | 1718 ± 16 | 1031 ± 14 | 7 . 30 + 0 . 19 - 0 . 61 | 1 . 51 + 1 . 27 - 0 . 43 | 44 . 41 ± 0 . 03 | 42 . 51 ± 0 . 02 | 63 . 1 ± 5 . 2 |\n| SDSS J102339 | 24 . 9 + 19 . 8 - 3 . 9 | 1733 ± 29 | 1139 ± 19 | 7 . 16 + 0 . 25 - 0 . 08 | 1 . 29 + 0 . 20 - 0 . 56 | 44 . 09 ± 0 . 03 | 42 . 14 ± 0 . 03 | 57 . 0 ± 5 . 9 | \nNote . -All SEAMBH2012 measurements are taken from Paper III, but 5100 Å fluxes are from I and II, SEAMBH2013 from Paper IV, and SEAM BH2014 is the present paper. MCG +06 -26 -012 was selected as a super-Eddington candidate in SEAMBH2012 but later was identified to be a sub-Eddington accretor ( ˙ M = 0 . 46); we discard it here. \nAs shown in Paper IV, the H β lags of the SEAMBH2013 sample were found to significantly deviate from the normal R H β -L 5100 relation, by a factor of a few. We plot the R H β -L 5100 relation of all samples in Figure 3. For sub-Eddington AGNs ( ˙ M ≤ 3) in the direct scheme, log ( R H β / ltd ) = (1 . 54 ± 0 . 03) + (0 . 53 ± 0 . 03)log /lscript 44 , with an intrinsic scatter of 0 . 15 (see Paper IV). Using FITEXY, we have \nlog R H β / ltd ) =', '(': ') (1 . 30 ± 0 . 05) + (0 . 53 ± 0 . 06)log /lscript 44 ( ˙ M ≥ 3) , (1 . 44 ± 0 . 03) + (0 . 49 ± 0 . 03)log /lscript 44 (for all ˙ M ) , (4) \nwith intrinsic scatters of σ in = (0 . 24 , 0 . 21). Clearly, the intrinsic scatter of SEAMBHs is much larger than the sample of sub-Eddington AGNs. In the averaged scheme, we have log ( R H β / ltd ) = (1 . 55 ± 0 . 04) + (0 . 53 ± 0 . 04)log /lscript 44 for subEddington AGNs, with an intrinsic scatter of 0 . 16 (see Paper IV), and \nlog ( R H β / ltd ) = (1 . 32 ± 0 . 05) + (0 . 52 ± 0 . 06)log /lscript 44 ( ˙ M ≥ 3) , (1 . 44 ± 0 . 03) + (0 . 49 ± 0 . 03)log /lscript 44 (for all ˙ M ) , (5) \nwith intrinsic scatters of σ in = (0 . 22 , 0 . 21). The slope of the correlation for the SEAMBH sample is comparable to that of sub-Eddington AGNs, but the normalization is significantly different. It is clear that the SEAMBH sources increase the scatter considerably, especially over the limited luminosity range occupied by the new sources. \nAs in Paper IV, we also tested the correlation between H β lag and H β luminosity, namely, the R H β -L H β relation. The scatter of the R H β -L H β correlation is not smaller than that of the R H β -L 5100 correlation, and we do not consider it further.', '4.2. ˙ M -dependent BLR Size': 'To test the dependence of the BLR size on accretion rate, we define a new parameter, ∆ R H β = log ( R H β / R H β , R -L ) , that specifies the deviation of individual objects from the R H β -L 5100 relation of the subsample of ˙ M < 3 . 0 sources (i.e., R H β , R -L as given by Equations 4b and 5b for ˙ M < 3 AGNs in Paper IV). The scatter of ∆ R H β is calculated by σ R H β = \n[ ∑ i ( ∆ R H β , i -〈 ∆ R H β 〉 ) 2 / N ] 1 / 2 , where N is the number of objects and 〈 ∆ R H β 〉 is the averaged value. Figure 3 provides ∆ R H β plots for comparison. \nFigure 4 shows ∆ R H β versus ˙ M , as well as ∆ R H β distributions for the ˙ M ≥ 3 and ˙ M < 3 subsamples in the di- \nFigure 2. Distributions of 5100 Å luminosity ( L 5100), BH mass ( M · ), dimensionless accretion rate ( ˙ M ), and equivalent width (EW) of all the mapped AGNs. These distributions show that the present sample of mapped AGNs is inhomogeneous. Only three luminous sources ( L 5100 /greaterorsimilar 10 45 erg s -1 ) have been mapped. The distribution of EW(H β ) in panel d shows that the SEAMBH sample tends to have low EW(H β ). \n<!-- image --> \ns a and b) and averaged (panels c and d) schemes. A Kolmogorov-Smirnov (KS) test shows that the probability that the two subsamples are drawn from the same parent distributions is p KS = 0 . 00029 for the direct scheme and p KS = 0 . 0094 for the averaged scheme. This provides a strong indication that the main cause of deviation from the normal R H β -L 5100 relation is the extreme accretion rate. Thus, a single R H β -L 5100 relation for all AGNs is a poor approximation for a more complex situation in which both the luminosity and the accretion rate determine R H β . From the regression for ˙ M ≥ 3 AGNs, we obtain the dependence of the deviations of R H β from the R H β -L 5100 relation in Figure 4: \n∆ R H β = \n (0 . 39 ± 0 . 09) -(0 . 47 ± 0 . 06)log ˙ M (direct scheme) , (0 . 34 ± 0 . 09) -(0 . 42 ± 0 . 07)log ˙ M (averaged scheme) , (6) \nwith σ in = (0 . 01 , 0 . 05), respectively. We have tested the above correlations also for ˙ M < 3. The FITEXY regressions give slopes near 0, with very large uncertainties: ∆ R H β ∝ ˙ M -0 . 055 ± 0 . 032 and ∆ R H β ∝ ˙ M -0 . 095 ± 0 . 050 for Figure 4a and 4c, respectively, implying that ∆ R H β does not correlate with ˙ M for the ˙ M < 3 group. All this confirms that ˙ M is an additional parameter that controls the R H β -L 5100 relation in AGNs with high accretion rates.', '5. A NEW SCALING RELATION FOR THE BLR': 'We provide evidence H β lags depend on luminosity and accretion rate. There are a total of 28 SEAMBHs (including those discovered in other studies). We now have an opportunity to define a new scaling relation for the BLR, one that properly captures the behavior of sub-Eddington and superEddington AGNs. Considering the dependence of ∆ R H β ∝ ˙ M -0 . 42 (Equation 6), a unified form of the new scaling law \ncan take the form 13 \nR H β = α 1 /lscript β 1 44 min [ 1 , ( ˙ M ˙ M c ) -γ 1 ] , (7) \nwhere ˙ M c is to be determined by data. Equation (7) reduces to the normal R H β -L 5100 relation for sub-Eddington AGNs and to R H β = α 1 /lscript β 1 44 ( ˙ M / ˙ M c ) -γ 1 for AGNs with ˙ M ≥ ˙ M c . There are four parameters to describe the new scaling relation, but only two ( ˙ M c and γ 1) are new due to the inclusion of accretion rates; the other two are mainly determined by subEddington AGNs. The critical value of ˙ M c , which is different from the criterion of SEAMBHs, depends on the sample of SEAMBHs. \nIn order to determine the four parameters simultaneously, we define \n( \nχ 2 = 1 N N ∑ i =1 ( R H β -R i H β ) 2 ∆ i R H β ) 2 , (8) \n) where ∆ i R H β is the error bar of R i H β . Minimizing χ 2 among all the mapped AGNs and employing a bootstrap method, we have \nα 1 = 29 . 6 + 2 . 7 -2 . 8 ; β 1 = 0 . 56 + 0 . 03 -0 . 03 ; γ 1 = 0 . 52 + 0 . 33 -0 . 16 ; ˙ M c =11 . 19 + 2 . 29 -6 . 22 . (9) \nThis new empirical relation has a scatter of 0 . 19, smaller than the scatter (0.26) of the normal R H β -L 5100 relation for all the mapped AGNs. The new scaling relation is plotted in Figure 5. \nEquation (7) shows the dependence of the BLR size on accretion rates, but it cannot be directly applied to single-epoch spectra for BH mass without knowning ˙ M . Iteration of Equation (7) does not converge. The reason is due to the fact that larger ˙ M leads to smaller R H β and higher ˙ M , implying that the iteration from Equation (1) does not converge. Du et al. (2016b) devised a new method to determine ˙ M from singleepoch spectra. Beginning with the seminal work of Boroson \n13 We have tried R H β = α 1 /lscript β 1 44 [ 1 + ( ˙ M / ˙ M c ) γ 1 ] δ 1 , which is continuous for the transition from sub- to super-Eddington sources. The fitting also yields a very rapid transition at ˙ M c ∼ 10, with γ 1 = 0 . 025 and δ 1 = 21 . 02 (the present sample is still dominated by sub-Eddington AGNs, with a ratio of 35/63). We prefer the form given by Equation (7). \nFigure 3. The R H β -L 5100 plot for all mapped AGNs. Left panel shows multiple-RM results as individual points, whereas the right panel shows the averaged results of AGNs with multiple-RM measurements. The dotted line is the regression of R H β -L 5100 relation for ˙ M < 3 AGNs (Equation 4); the dashed line is the regression for the ˙ M ≥ 3 objects. The scatter (standard deviations) of ∆ R H β is given in the upper left corner of Panels b and d . \n<!-- image --> \n&Green(1992), it has been well-known that R Fe ≡ F FeII / F H β , the flux ratio of broad optical Fe II to H β , correlates strongly with Eddington ratio (Sulentic et al. 2000; Shen & Ho 2014). At the same time, the shape of broad H β , as parameterized by D H β = FWHM /σ H β , where σ H β is the line dispersion, also correlates with Eddington ratio (Collin et al. 2006). Combining the two produces produces a strong bivariate correlation, which we call the fundamental plane of the BLR, of the form \nlog ˙ M = α 2 + β 2 D H β + γ 2 R Fe , (10) \nwhere \nα 2 = 2 . 47 ± 0 . 34; β 2 = -1 . 59 ± 0 . 14; γ 2 = 1 . 34 ± 0 . 20 . (11)', '6.1. Normalized BLR Sizes': 'In order to explore the relation between BLR size and accretion rate, we define a dimensionless radius for the BLR, r H β = R H β / R g, where R g = 1 . 5 × 10 12 m 7 cm is the gravitational radius. As in Paper IV, we insert Equation (3) into r H β to replace /lscript 44, to obtain r H β = 1 . 9 × 10 4 ˙ M 0 . 35 m -0 . 29 7 . This relation implies that r H β increases with accretion rates as r H β ∝ ˙ M 0 . 35 for sub-Eddington AGNs, whereas in SEAMBHs r H β ∝ ˙ M 0 . 29 ± 0 . 08 (as shown in Figure 6 a ) and r H β and tends toward a maximum saturated value of r max H β = f -1 BLR ( c / V min ) 2 = 9 × 10 4 f -1 BLR V -2 min , 3 , where V min , 3 = V min / 10 3 km s -1 is the minimumvelocity width of H β (see Equation 15 in Paper IV). We \nnote that the minimum observed FWHM values of H α (which is comparable to H β ) is ∼ 10 3 km s -1 among low-mass AGNs (Greene & Ho 2007; Ho & Kim 2016). Indeed, this limit is consistent with the saturation trend of r H β (Figure 6 a ). \nWe note that the relatively large scatter in Figure 6 a is mostly due to the uncertainties in BH mass. In order to better understand the relation between the BLR and the central engine, we define, as in Paper IV, the parameter Y = m 0 . 29 7 r H β , which reduces to \nY = 1 . 9 × 10 4 ˙ M 0 . 35 . (12) \nWe would like to point out that Equation (12) describes the coupled system of the BLR and the accretion disks. It is therefore expected that Y is a synthetic parameter describing the photoionization process including ionizing sources. \nIt is easy to observationally test Equation (12) using RM results. Figure 6 b plots Y versus ˙ M . It is very clear that the observed data for objects with ˙ M < 3 agree well with Equation (12). Furthermore, there is a clear saturation of Y for objects with ˙ M ≥ 3 objects. All these results strengthen the conclusions drawn in Paper IV. As in that work, we define an empirical relation \nY = Y sat min 1 , ( ˙ M ˙ M b ) b , (13) \nFigure 4. The deviation of H β lags from the normal R H β -L 5100 relation for sub-Eddington AGNs. Panel a shows the sample of all mapped AGNs (the repeatedly monitored AGNs are regarded as individual ones). Panel b gives the distribution of low- and high-accretion objects. Panels c and d are the same plots but for the average scheme. Sub-Eddington AGNs show a random distributions, but SEAMBHs correlate with accretion rates. We note that M · and ˙ M are calculated in exactly the same way for all objects, as indicated in Table 7 in Paper IV (i.e. using Equations 2 and 3). \n<!-- image --> \nwhere \nY sat = (3 . 5 + 0 . 6 -0 . 5 ) × 10 4 , ˙ M b = 15 . 6 + 22 . 0 -9 . 1 , b = 0 . 27 + 0 . 04 -0 . 04 . (14) \nIn fact, we can get Y by inserting Equation (3) into (7), and find that it is in agreement with Equation (13). From the saturated -Y , we have the maximum value of \nr H β , sat = (3 . 5 + 0 . 6 -0 . 5 ) × 10 4 m -0 . 29 7 or R H β , sat = (19 . 9 + 3 . 4 -2 . 8 ) m 0 . 71 7 ltd . (15) \nThis result provides a strong constraint on theoretical models of super-Eddington accretion onto BHs.', '6.2. The Shortened Lags': 'The shortened H β lags is the strongest distinguishing characteristic so far identified between super- and subEddington AGNs. Two factors may lead to shortened lags for SEAMBHs. First, Wang et al. (2014c) showed that, in the Shakura-Sunyaev regime, retrograde accretion onto a BH can lead to shorter H β lags. The reason is due to the suppression of ionizing photons in retrograde accretion compared with prograde accretion. The second factor stems from selfshadowing effects of the inner part of slim disks (e.g., Li et al. 2010), which efficiently lower the ionizing flux received by the BLR (Wang et al. 2014c). When ˙ M increases, the ratio of the disk height to disk radius increases due to radiation pressure; the radiation field becomes anisotropic (much stronger than the factor of cos i ) due to the optically thick funnel of the inner part of the slim disk. In principle, the radiation from a slim disk saturates ( ∝ ln ˙ M ), and the total ionizing luminosity slightly increases with accretion rate, but the selfshadowing effects efficiently suppress the ionizing flux to the BLR clouds. For face-on disks of type 1 AGNs, observers receive the intrinsic luminosity. If the ionization parameter is constant, the ionization front will significantly shrink, and hence the H β lag is shortened in SEAMBHs compared with sub-Eddington AGNs of the same luminosity. \nThe shortened H β lag observed in SEAMBHs cannot be caused by retrograde accretion. However, the strong dependence on accretion rate of the deviation from the standard lagluminosity relation implies that the properties of the ionizing \nsources are somehow different from those in sub-Eddington AGNs. According to the standard photoionization theory, the observed R H β ∝ L 1 / 2 5100 relation can be explained if L 5100 ∝ L ion and Qc = Une ¯ /epsilon1 is constant, where U = L ion / 4 π R 2 H β cne ¯ /epsilon1 , L ion is the ionizing luminosity, ne is gas density of BLR clouds, and ¯ /epsilon1 is the average energy of the ionizing photons (Bentz et al. 2013). The relation L 5100 ∝ L ion holds for sub-Eddington AGNs, and the constancy of Qc is determined by the clouds themselves. Qc is not expected to vary greatly as a function of Eddington ratio. Therefore, \nR H β = L 1 / 2 ion Qc = S R 0 H β , (16) \nwhere the factor S = ( L ion / L ion , 0 ) 1 / 2 describes the anisotropy of the ionizing radiation field, L ion is the shadowed ionizing luminosity received by the BLR clouds, and R 0 H β is the BLR radius corresponding to L ion , 0, the unshadowed luminosity. Based on the classical model of slim disks, Wang et al. (2014c) showed that, for a given accretion rate, S strongly depends on the orientation of the clouds relative to the disk, and that it range from 1 to a few tens. Therefore, the reduction of the H β lag can, in principle, reach up to a factor of a few, even 10, as observed. \nFurthermore, the saturatedY implies that the ionizing luminosity received by the BLR clouds gets saturated. The theory of super-Eddington accretion onto BHs is still controversial. Although extensive comparison with models is beyond the scope of this paper, we briefly discuss the implications of the current observations to the theory. Two analytical models, which reach diametrically extreme opposite conclusions, have been proposed. Abramowicz et al. (1988) suggested a model characterized by fast radial motion with sub-Keplerian rotation and strong photon-trapping. Both the shortened lags and saturated -Y may be caused by self-shadowing effects and saturated radiation from a slim disk. Both features are expected from the Abramowicz et al. model (Wang et al. 2014b). On the other hand, photon-bubble instabilities may govern the disk structure and lead to very high radiative efficiency (Gam- \nmie 1998). If super-Eddington accretion can radiate as much as L / L Edd /greaterorsimilar 470 m 6 / 5 7 (Equation 14 in Begelman 2002), the disk remains geometrically thin. In such an extreme situation, self-shadowing effects are minimal, H β lags should not be reduced, and Y -saturation disappears. \n<!-- image --> \nR H β \n= α 1 /lscript β 1 44 \nmin \n[ ( ) ] Figure 5. The best fit of the new scaling relation for all mapped AGNs. We find that α 1 = (29 . 6 + 2 . 7 -2 . 8 ) lt-d, β 1 = 0 . 56 + 0 . 03 -0 . 03 , γ 1 = 0 . 52 + 0 . 33 -0 . 16 and ˙ M c = 11 . 19 + 2 . 29 -6 . 22 . The scatter of the BLR size is greatly reduced to σ = 0 . 19. \n1 , \n˙ M / ˙ M c \n-γ 1 \nRecent numerical simulations that incorporate outflows (e.g., Jiang et al. 2014) and relativistic jets (Sadowski et al. 2015) also suggest that super-Eddington accretion flows can maintain a high radiative efficiency. However, most AGNs with high accretion rates are radio-quiet (Ho 2002; Greene & Ho 2006), in apparent contradiction with the numerical simulation predictions. Furthermore, evidence for Y -saturation also does not support the models with high radiative efficiency. Recent modifications of the classical slim disk model that include photo-trapping appear promising (e.g., Cao & Gu 2015; Sadowski et al. 2014), but the situation is far from settled. Whatever the outcome, the results from our observations provide crucial empirical constraints on the models.', '6.3. Inclination Effects on ˙ M': 'If the BLR is flattened, its inclination angle to the observer will influence M · , and hence ˙ M (see Equation 3). To zeroorder approximation, the observed width of the broad emission lines follow \n∆ V obs ≈ [ ( H BLR R ) 2 + sin 2 i ] 1 / 2 V K , (17) \nwhere V K is the Keplerian velocity and H BLR is the height of the flatten BLR (e.g., Collin et al. 2006). For a geometrically thin BLR, H BLR / R /lessmuch 1, ∆ V obs ≈ V Ksin i , and hence ˙ M ∝ (sin i ) -4 , which is extremely sensitive to the inclination can be severely overestimated for low inclinations. On the other hand, many arguments (e.g., Goad & Korista 2014) support H BLR / R /lessorsimilar 1, and the inclination angle significantly influences M · only for sin i /greaterorsimilar H BLR / R . Currently, the values of H BLR / R are difficult to estimate, but detailed modelling of RM data suggests H BLR / R ∼ 1 (Li et al. 2013; Pancoast et al. 2014). If true, this implies that the BLR is not very flattened, and hence the inclination angle only has a minimal influence on M · and ˙ M .', '6.4. Comparison with Previous Campaigns': 'The objects in our SEAMBH sample are very similar to NLS1s. As previous RM AGN samples include NLS1s, why have previous studies not noticed that NLS1s deviate from the R H β -L 5100 relation (e.g., Figure 2 in Bentz 2011)? We believe that the reason is two-fold. First, the number of NLS1s included in previous RM campaigns was quite limited (Denney et al. 2009, 2010; Bentz et al. 2008, 2009; see summary in Bentz 2011). The level of optical variability in NLS1s is generally very low (Klimek et al. 2004), and many previous attempts at RM have proved to be unsuccessful (e.g., Giannuzzo & Stirpe 1996; Giannuzzo et al. 1999). Second, not all NLS1s are necessarily highly accreting. Our SEAMBH sample was selected to have high accretion rates (see ˙ M listed in Table 7 of Paper IV), generally higher than that of typical NLS1s previously studied successfully through RM. As discussed in Wang et al. (2014b) and in Paper IV, high accretion rates lead to anisotropic ionizing radiation, which may explain the shortened BLR lags.', '6.5. SEAMBHs as Standard Candles': "Once its discovery, quasars as the brightest celestial objects in the Universe had been suggested for cosmology (Sandage 1965; Hoyle & Burbidge 1966; Longair & Scheuer 1967; Schmidt 1968; Bahcall & Hills 1973; Burbidge & O'Dell 1973; Baldwin et al. 1978). Unfortunately, the diversity of observed quasars made these early attempts elusive. After five decades since its discovery, quasars are much well understood: accretion onto supermassive black holes is powering the giant radiation, in particular, the BH mass can be reliably measured. Quasars as the most powerful emitters renewed interests for cosmology in several independent ways: 1) the normal R H β -L 5100 relation (Horn et al. 2003; Watson et al. 2011; Czerny et al. 2013); 2) the linear relation between BH mass and luminosity in super-Eddington quasars (Wang et al. 2013; Paper-II); 3) Eddington AGNs selected by eigenvector 1 (Marziani & Sulentic 2014); 4) X-ray variabilities (La Franca et al. 2014) and 5) α OX -LX relation (Risaliti & Lusso 2015). These parallel methods will be justified for cosmology by their feasibility of experiment periods and measurement accuracy. \nThe strength of SEAMBHs makes its application more convenient for cosmology. Selection of SEAMBHs only depends on single epoch spectra through the fundamental plane (Equation 10). BH mass can be estimated by the new scaling relation (Equation 7). We will apply the scheme outlined by Wang et al. (2014a) to the sample of selected SEAMBHs for cosmology in a statistical way (in preparation). On the other hand, the shortened H β lags greatly reduce monitoring \nFigure 6. The dimensionless BLR radius and Y -parameter versus accretion rate ˙ M . There is a trend of saturation of r H β as shown in panel a , but it is caused by the scatter of BH mass. Panel b shows much a tighter relation and unambiguous saturation of the Y -parameter. We should point out that NGC 7469 in this plot has been revised compared with Figure 5 in Paper IV, using the latest observation from Peterson et al. (2014). \n<!-- image --> \n<!-- image --> \n0 . 2 9 \nperiods if SEAMBHs are applied as standard candles, in particular, the reduction of lags govern by super-Eddington accretion can cancel the cosmological dilltion factor of (1 + z ). Otherwise, the monitoring periods of sub-Eddington AGNs should be extended by the same factor of (1 + z ) for measurements of H β lags. Such a campaign of using the normal R H β -L 5100 relation for cosmology will last for a couple of years, even 10 years for bright high -z quasars. Similarly to extension of the R H β -L 5100 relation to Mg II- and C IV-lines (Vestergaard & Peterson 2006), we can extend Equation (7) to Mg II and C IV lines for the scaling relations with luminosity as R MgII( L 3000 , ˙ M ) and R CIV( L 3000 , ˙ M ), respectively, where R MgII and R CIV are sizes of the Mg II and C IV regions, and L 3000 is the 3000 Å luminosity. Such extended relations conveniently allow us to investigate cosmology by making use of large samples of high -z quasars without time-consuming RM campaigns. It is urgent for us to make use of kinds of standard candles to test the growing evidence for dynamical dark energy (e.g., Zhao et al. 2012; Ade et al. 2015).", '7. CONCLUSIONS': "We present the results of the third year of reverberation mapping of super-Eddington accreting massive black holes (SEAMBHs). H β lags of five new SEAMBHs have been detected. Similar to the SEAMBH2012 and SEAMBH2013 samples, we find that the SEAMBH2014 objects generally have shorter H β lags than the normal R H β -L 5100 relation, by a factor of a few. In total, we have detected H β lags for 18 SEAMBHs from this project, which have accretion rates from ˙ M ∼ 10 to /lessorsimilar 10 3 . The entire SEAMBH sample allows us to establish a new scaling relation for the BLR size, which depends not only on luminosity but also on accretion rate. The new relation, applicable over a wide range of accretion rates from ˙ M ≈ 10 -3 to 10 3 , is given by R H β = α 1 /lscript β 1 44 min 1 , ( ˙ M / ˙ M c -γ 1 ] , where /lscript 44 = \n[ \n) \n( ] L 5100 / 10 44 erg s -1 is 5100 Å continuum luminosity, and coefficients of α 1 = (29 . 6 + 2 . 7 -2 . 8 ) lt-d, β 1 = 0 . 56 + 0 . 03 -0 . 03 , γ 1 = 0 . 52 + 0 . 33 -0 . 16 and ˙ M c = 11 . 19 + 2 . 29 -6 . 22 . \nWe thank an anonymous referee for critical comments that helped to improve the paper. We acknowledge the support of the staff of the Lijiang 2.4m telescope. Funding for the telescope has been provided by CAS and the People's Government of Yunnan Province. This research is supported by the Strategic Priority Research Program - The Emergence of Cosmological Structures of the Chinese Academy of Sciences, Grant No. XDB09000000, by NSFC grants NSFC11173023, -11133006, -11373024, -11503026, -11233003 and -11473002, and a NSFC-CAS joint key grant U1431228, and by the CAS Key Research Program through KJZDEW-M06, and by a China-Israel project through NSFC11361140347.", 'A. VALIDITY OF EQUATION (3)': 'The validity of Equation (3) can be justified for application to SEAMBHs. Solutions of slim disks are transonic and usually given only by numerical calculations (Abramowicz et al. 1988). When the accretion rate of the disk is high enough, the complicated structure of the disk reduces to a self-similar, analytical form (Wang & Zhou 1999). Using the self-similar solutions (Wang & Zhou 1999; Wang et al. 1999), we obtained the radius of disk region emitting optical (5100 Å) photons, \nR 5100 R Sch ≈ 4 . 3 × 10 3 m -1 / 2 7 , (A1) \nand the photon-trapping radius \nR trap R Sch ≈ 144 ( ˙ M 10 2 ) . (A2) \nWe used the blackbody relation kT eff = hc /λ , where k is the Boltzmann constant, T eff is the effective temperature of the disk surface, h is the Planck constant, and R Sch = 3 . 0 × 10 12 m 7 cm is the Schwartzschild radius. Equation (3) holds provided R 5100 /greaterorsimilar R trap, namely \n˙ M /lessorsimilar 3 × 10 3 m -1 / 2 7 . (A3) \nFigure 7. Photometric light curves of comparison stars in the slit. \n<!-- image --> \nIn this regime, optical radiation is not influenced by photontrapping effects. We would also like to point out that the BH spin only affects emission from the innermost regions of the accretion disk rather than the regions emitting 5100 Å photons. In the present campaign, no SEAMBH so far has been found to exceed this critical value. Beyond this critical value of accretion rate, optical photons are trapped by the accretion flow. We call this the hyper-accretion regime. \nHere the cited 10 -2 below Equation (3) is not a strict value of the ADAF threshold since it depends on several factors, such as viscosity and outer boundary conditions. There are a few mapped AGNs with ˙ M /lessorsimilar 10 -2 (NGC 4151, NGC 5273, 3C 390.3 and NGC 5548; see Table 7 in Paper IV), but we do not discuss them in this paper because they do not influence our conclusions. \nRecently, reprocessing of X-rays (e.g., Frank et al. 2002; Cackett et al. 2007) has been found to play an important role in explaining the variability properties of NGC 5548 (e.g., Fausnaugh et al. 2015). The fraction of X-ray emission to the bolometric luminosity strongly anti-correlates with the Eddington ratio (see Figure 1 in Wang et al. 2004). This result is usually interpreted to mean that the hot corona becomes weaker with increasing accretion rate, as a result of more efficiently cooling of the corona by UV and optical photons from the cold disk. This suggests that AGNs with high accretion rates will have less reprocessed emission, such that Equation (3) would be more robust in SEAMBHs.', 'B. LIGHT CURVES OF COMPARISON STARS': 'In order to avoid selecting variable stars as comparison stars, we examined their variability. To test the invariance of the comparison stars used in our spectroscopic observation, we performed differential photometry by comparing them with other stars in the same field. We typically use six stars for the differential photometry. The light curves of the stars are \nshown in Figure 7. On average, the standard deviations in the light curves of the comparison stars are 1%. This guarantees that they can be used as standards for spectral calibration. Figure 7 shows the light curves of the comparison stars for each SEAMBH targets.', 'C. AVERAGED AND RMS SPECTRA': "The averaged and RMS spectra of the SEAMBH2014 sample are provided in this Appendix. Following the standard way, we calculated the averaged spectrum as \n¯ F λ = 1 N N ∑ i =1 F i λ , (C1) \nand the RMS spectrum as \nS λ = [ 1 N N ∑ i =1 ( F i λ -¯ F λ ) 2 ] 1 / 2 , (C2) \nwhere F i λ is the i -th observed spectrum and N is the total number of observed spectra. They are shown in Figure 8. We note that both the averaged and RMS spectra are affected by the broadening effects of the 5 '' -slit on the observed profiles. Using the Richards-Lucy iteration, we can correct the observed profiles (averaged and RMS) for velocity-resolved mapping, which will be carried out in a separate paper (Du et al. 2015c).", 'D. NOTES ON INDIVIDUAL OBJECTS': "We briefly remark on individual objects for which H β lags have been successfully measured. We failed in getting lag measurements for the other five objects because either their flux variations are very small or the data sampling rate was inadequate. \nJ075949 : The detected H β lag arises from two major peaks in the light curves. \nFigure 8. The averaged and RMS spectra of the SEAMBH2014 sources. S λ and ¯ F λ are in units of 10 -16 ergs -1 cm -2 -1 . \n<!-- image --> \nJ080131 : The monitoring observations during 2013 -2014 did not yield a well-determined H β lag because of the lack of H β response to the second continuum flare (see Paper IV). During 2013 -2014, the first reverberation of H β , which can be clearly seen during the first 70 days of the light curves, yields a very significant lag, as shown in the CCF with a restframe centroid lag of 11 . 5 + 8 . 4 -3 . 6 days (with a very high coefficient of r max = 0 . 81). We monitored this object again in this observing season (Figure 1). We successfully measure τ H β = 11 . 2 + 14 . 8 -9 . 8 days, consistent with last season's result. \n-. J084533 : Its continuum slightly decreased before being monitored for ∼ 70 days, and steadily increased until ∼ 200 days and then decreased again. Although the CCF has a very flat peak close to ∼ 0 . 9, Monte Carlo simulations show that H β lag is around 20 days, which arises from the two peaks in the H β and r ' -band light curves. \nJ085946 : The CCF peaks near 0.6, which results from the two major dips in the H β and r ' -band light curves. There are two peaks with roughly the same correlation coefficients around ∼ 20 days and ∼ 70 days in the observed frame, respectively. Considering the relatively poor data quality of this object, it is difficult to distinguish which is the true response. The centroid lag represents the average of these two peaks (responses), and its uncertainties cover the distribution (Figure 1) obtained in FR/RSS method. So, we use it in the analysis of main text. \nJ102339 : The detected lag is from the dip feature around ∼ 150 days in light curves.", 'E. DESCRIPTION OF CCCD': 'For multiple-peaked CCFs with similar correlation coefficients, it is ambiguous as to which peak should be used to calculate the final lag. In such cases, we use the CCCD to determine the lag. However, there are two approaches to calculate the centroid time lag in the CCF, as illustrated in Figure 9. \n- · Approach 1 calculates the centroid using all peaks above some criterion, such as 0 . 8 r max.\n- · Approach 2 only uses the highest peak. \nIn Approach 1, the CCCD tends to be smoother than the CCPD (see Figure 1), whereas in in Approach 2 the CCCD and CCPD always have a similar distribution. We adopt Approach 1 in our analysis. If the CCF has two or even three peaks, the two approaches give different centroid lags. 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2015JHEP...11..157H
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Einstein-Born-Infeld-massive gravity: adS-black hole solutions and their thermodynamical properties
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2015-01-01
| 17 | 0.44 | 159 |
['black hole physics', '-', '-', '-', '-']
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[]
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In this paper, we study massive gravity in the presence of Born-Infeld nonlinear electrodynamics. First, we obtain metric function related to this gravity and investigate the geometry of the solutions and find that there is an essential singularity at the origin ( r = 0). It will be shown that due to contribution of the massive part, the number, type and place of horizons may be changed. Next, we calculate the conserved and thermodynamic quantities and check the validation of the first law of thermodynamics. We also investigate thermal stability of these black holes in context of canonical ensemble. It will be shown that number, type and place of phase transition points are functions of different parameters which lead to dependency of stability conditions to these parameters. Also, it will be shown how the behavior of temperature is modified due to extension of massive gravity and strong nonlinearity parameter. Next, critical behavior of the system in extended phase space by considering cosmological constant as pressure is investigated. A study regarding neutral Einstein-massive gravity in context of extended phase space is done. Geometrical approach is employed to study the thermodynamical behavior of the system in context of heat capacity and extended phase space. It will be shown that GTs, heat capacity and extended phase space have consistent results. Finally, critical behavior of the system is investigated through use of another method. It will be pointed out that the results of this method is in agreement with other methods and follow the concepts of ordinary thermodynamics.
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https://arxiv.org/pdf/1508.01311.pdf
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{'Einstein-Born-Infeld-Massive Gravity: adS-Black Hole Solutions and their Thermodynamical properties': 'S. H. Hendi 1 , 2 ∗ , B. Eslam Panah 1 † , and S. Panahiyan 1 ‡ \n1 Physics Department and Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran 2 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha, Iran \nIn this paper, we study massive gravity in the presence of Born-Infeld nonlinear electrodynamics. First, we obtain metric function related to this gravity and investigate the geometry of the solutions and find that there is an essential singularity at the origin ( r = 0). It will be shown that due to contribution of the massive part, the number, types and places of horizons may be changed. Next, we calculate the conserved and thermodynamic quantities and check the validation of the first law of thermodynamics. We also investigate thermal stability of these black holes in context of canonical ensemble. It will be shown that number, type and place of phase transition points are functions of different parameters which lead to dependency of stability conditions to these parameters. Also, it will be shown how the behavior of temperature is modified due to extension of massive gravity and strong nonlinearity parameter. Next, critical behavior of the system in extended phase space by considering cosmological constant as pressure is investigated. A study regarding neutral Einsteinmassive gravity in context of extended phase space is done. Geometrical approach is employed to study the thermodynamical behavior of the system in context of heat capacity and extended phase space. It will be shown that GTs, heat capacity and extended phase space have consistent results. Finally, critical behavior of the system is investigated through use of another method. It will be pointed out that the results of this method is in agreement with other methods and follow the concepts of ordinary thermodynamics.', 'I. INTRODUCTION': "Regarding experimental agreements of Einstein gravity (EN) in various area of astrophysics and cosmology, motivates one to consider it as an acceptable theory. In addition, adding a constant term Λ in the EN-Hilbert action may lead to agreement between the results of EN-Λ gravity with dark energy prediction. \nOn the other hand, general relativity is consistent with interaction of massless spin 2 fields, in which related gravitons are massless particles with two degrees of freedom. Since the quantum theory of massless gravitons is non-renormalizable [1], one may be motivated for modifying general relativity to massive gravity. In order to build up a massive theory with a massive spin 2 particle propagation, one can add a mass term to the EN-Hilbert action. This will result into graviton having a mass of m which in case of m → 0, the effect of massive gravity will be vanished. A class of massive gravity theory in flat and curved background which leads to absence [2] and existence [3] of ghost, have been investigated. Also, the quantum aspects of the massive gravity and a nonlinear class of massive gravity in ghost-free field [4] have been explored in Refs. [5, 6]. Generalization to nonlinearly charged massive black holes was done in Ref. [7]. More details regarding the motivations of massive gravity is given in Ref. [8]. \nIn this paper, we are interested in studying the nontrivial adS massive theory that was investigated in [9, 10]. The motivation for this consideration is due to fact that graviton shows similar behavior as lattice in holographic conductor [11]. In other words, a Drude like behavior is observed for the case of massless graviton in this theory which makes the role of graviton similar to lattice. Another interesting subject for study in this theory is metal-insulator transition [12]. Recently, charged massive black holes with consideration of this theory have been investigated in [13]. The P -V criticality of these solutions and their geometrical thermodynamic aspects have been studied [14, 15]. Also, the generalization to Gauss-Bonnet-Maxwell-massive gravity and its stability, geometrical thermodynamics and P -V criticality have been investigated [16]. \nOne of the main problems of Maxwell's electromagnetic field theory for a point-like charge is that there is a singularity at the charge position and hence, it has infinite self-energy. To overcome this problem in classical electrodynamics, Born and Infeld in Ref. [17] introduced a nonlinear electromagnetic field, with main motivation, to solve infinite self-energy problem by imposing a maximum strength of the electromagnetic field. Then, Hoffmann in Ref. [18] investigated EN gravity in the presence of Born-Infeld (BI) electrodynamics. In recent two decades, exact solutions of gravitating black objects in the presence of BI theory have been vastly investigated [19, 20]. Another interesting \nproperty of BI is that, BI type effective action arises in an open superstring theory and D-branes are free of physical singularities [21]. For a review of aspects of BI theory in the context of string theory see Ref. [22]. Recently, there has been growing interest in Eddington-inspired BI gravity in context of black holes and cosmology [23]. Also, it was proposed that one can consider BI theory as a gravitational theory [24]. Dualization of the BI theory and some of the special properties of this theory have been investigated in Ref. [25]. \nThere are several approaches for studying and obtaining critical behavior and phase transition points of black holes: First method is based on studying heat capacity. It was pointed out that roots and divergencies of the heat capacity are representing phase transition points. In other words, in place of roots and divergencies of the heat capacity system may go under phase transition. Another important property of the heat capacity is investigation of the thermal stability. Systems with positive heat capacity are denoted to be in thermally stable states. Therefore, the stability conditions are indicated by changes in sign of heat capacity [26]. This is known as canonical ensemble. \nIn the second method, by using the renewed interpretation of cosmological constant as thermodynamical variable, one can modify the thermodynamical structure of the phase space [27]. One of the most important property of this method is the similarity of critical behavior of the black holes and ordinary thermodynamical Van der Waals liquid/gas systems [28]. Recently, it was pointed out that the extended phase space should be interpreted as an RG-flow in the space of field theories, where isotherm curves codify how the number of degrees of freedom N (or the central charge c ) runs with the energy scale [29]. On the other hand, it was shown that variation of cosmological constant could be corresponded to variation of number of the colors in Yang-Mills theory residing on the boundary spacetime [30]. \nThe third method is using geometrical concept for studying critical behavior. In other words, by employing a thermodynamical potential and its corresponding extensive parameters, one can construct phase space. The divergencies of Ricci scalar in constructed metric are denoted as phase transition points. There are several metrics for this method that one can name: Weinhold [31], Quevedo [32] and HPEM [33] which has mass as thermodynamical potential and Ruppeiner [34] in which entropy is considered as thermodynamical potential. These metrics are used in context of heat capacity. Another set of metrics was introduced in Ref. [35] which can be used in context of extended phase space. \nFinally, a fourth method was introduced in Ref. [35] which is based on denominator of the heat capacity. In this method by replacing cosmological constant with pressure in denominator of the heat capacity and solving it with respect to pressure, a new relation is obtained for pressure. The existence of maximum for obtained relation, represents the critical pressure and volume in which phase transition takes place. The behavior of system in case of this method is consistent with ordinary thermodynamical concepts [15, 35]. \nThe outline of the paper will be as follow. In Sec. II, we introduce action and basic equations related to ENBI-massive gravity. Sec. III is devoted to obtain the black hole solutions of this gravity and investigation of the geometrical structure of them. In the next section, we calculate conserved and thermodynamic quantities related to obtained solutions and check the validation of the first law of thermodynamics. In section V, we study thermal stability of the EN-BI-massive black hole solutions in canonical ensemble. Next, we consider cosmological constant as pressure and study the critical behavior of the system. Then we employ the geometrical methods for investigating thermodynamical behavior of the system and extend this study by another method. In the last section we present our conclusions.", 'II. BASIC EQUATIONS': 'The d -dimensional action of EN-massive gravity with negative cosmological constant and a nonlinear electrodynamics is \nI = -1 16 π ∫ d d x √ -g [ R2Λ + L ( F ) + m 2 4 ∑ i c i U i ( g, f ) ] , (1) \nwhere R is the scalar curvature, Λ = -( d -1)( d -2) 2 l 2 is the negative cosmological constant and f is a fixed symmetric tensor. In Eq. (1), c i are constants and U i are symmetric polynomials of the eigenvalues of the d × d matrix K µ ν = √ g µα f αν which can be written as follows \nU 1 = [ K ] , U 2 = [ K ] 2 -[ K 2 ] , U 3 = [ K ] 3 -3 [ K ] [ K 2 ] +2 [ K 3 ] , U 4 = [ K ] 4 -6 K 2 ] [ K ] 2 +8 K 3 ] [ K ] + 3 K 2 ] 2 -6 K 4 ] . \n[ \n[ \n[ \n[ \n] ] ] ] Here, we want to study a particular model of nonlinear electrodynamics called BI theory which has attracted lots \nof attentions due to its relation to effective string actions. The function L ( F ) for BI theory is given as \nL ( F ) = 4 β 2 ( 1 -√ 1 + F 2 β 2 ) , (2) \nwhere β is the BI parameter, the Maxwell invariant is F = F µν F µν in which F µν = ∂ µ A ν -∂ ν A µ is the electromagnetic field tensor and A µ is the gauge potential. \nVariation of the action (1) with respect to the metric tensor g µν and the Faraday tensor F µν , leads to \nG µν +Λ g µν -1 2 g µν L ( F ) -2 F µλ F λ ν √ 1 + F 2 β 2 + m 2 χ µν = 0 , (3) \n∂ µ √ -gF µν √ 1 + F 2 β 2 = 0 , (4) \nwhere G µν is the EN tensor and χ µν is the massive term with the following form \nχ µν = -c 1 2 ( U 1 g µν -K µν ) -c 2 2 ( U 2 g µν -2 U 1 K µν +2 K 2 µν ) -c 3 2 ( U 3 g µν -3 U 2 K µν + 6 U 1 K 2 µν -6 K 3 µν ) -c 4 2 ( U 4 g µν -4 U 3 K µν +12 U 2 K 2 µν -24 U 1 K 3 µν +24 K 4 µν ) . (5)', 'III. BLACK HOLE SOLUTIONS IN EN-BI-MASSIVE GRAVITY': "In this section, we obtain static nonlinearly charged black holes in context of massive gravity with adS asymptotes. For this purpose we consider the metric of d -dimensional spacetime in the following form \nds 2 = -f ( r ) dt 2 + f -1 ( r ) dr 2 + r 2 h ij dx i dx j , i, j = 1 , 2 , 3 , ..., n , (6) \nwhere h ij dx i dx j is a ( d -2) dimension line element for an Euclidian space with constant curvature ( d -2) ( d -3) k and volume V d -2 . We should note that the constant k , which indicates that the boundary of t = constant and r = constant , can be a negative (hyperbolic), zero (flat) or positive (elliptic) constant curvature hypersurface. \nWe consider the ansatz metric [13] \nf µν = diag (0 , 0 , c 2 h ij ) , (7) \nwhere c is a positive constant. Using the metric ansatz (7), U i 's are in the following forms [13] \nU 1 = \nd 2 c r , \nU 2 = \nd 2 d 3 c 2 r 2 , \nU 3 = \nd 2 d 3 d 4 c 3 r 3 , \nU 4 = d 2 d 3 d 4 d 5 c 4 r 4 , \nin which d i = d -i . Using the gauge potential ansatz A µ = h ( r ) δ 0 µ in electromagnetic equation (4) and considering the metric (6), we obtain \nh ( r ) = -√ d 2 d 3 q r d 3 H , (8) \nin which H is the following hypergeometric function \nH = 2 F 1 ([ 1 2 , d 3 2 d 2 ] , [ 3 d 7 / 3 2 d 2 ] , -Γ ) , (9) \nwhere Γ = d 2 d 3 q 2 β 2 r 2 d 2 and q is an integration constant which is related to the electric charge. Also, the electromagnetic field tensor in d -dimensions is given by \nF tr = √ d 2 d 3 q r d 2 √ 1 + Γ . (10) \nNow, we are interested in obtaining the static black hole solutions. One may use components of Eq. (3) and obtain metric function f ( r ). We use the tt and x 1 x 1 components of the Eq. (3), which can be written as \ne tt = { d 2 m 2 c [ c 1 r 3 + d 3 c 2 cr 2 + d 3 d 4 c 3 c 2 r + d 3 d 4 d 5 c 4 c 3 ] -2Λ r 4 -d 2 d 3 r 2 f -d 2 r 3 f ' +4 β 2 r 4 + d 2 d 3 r 2 k } √ 1 -( h ' β ) 2 -4 β 2 r 4 = 0 , (11) \n√ \ne x 1 x 1 = d 3 m 2 c [ c 1 r 3 + d 3 d 4 c 2 cr 2 + d 3 d 4 d 5 c 3 c 2 r + d 3 d 4 d 5 d 6 c 4 c 3 ] -2Λ r 4 -2 d 3 r 3 f ' -d 3 d 4 r 2 f -r 4 f '' +4 β 2 r 4 -4 βr 4 β 2 -h ' 2 + d 3 d 4 r 2 k = 0 . (12) \nWe can obtain the metric function f ( r ), by using the Eqs. (11) and (12) with the following form \nf ( r ) = k -m 0 r d 3 + ( 4 β 2 -2Λ d 1 d 2 ) r 2 -4 β 2 r 2 d 1 d 2 √ 1 + Γ + 4 d 2 q 2 H d 1 r 2 d 3 + m 2 { cc 1 d 2 r + c 2 c 2 + d 3 c 3 c 3 r + d 3 d 4 c 4 c 4 r 2 } , (13) \nwhere m 0 is an integration constant which is related to the total mass of the black hole. It should be noted that, obtained metric function (13), satisfy all components of the Eq. (3), simultaneously. \nNow, we are in a position to review the geometrical structure of this solution, briefly. We first look for the essential singularity(ies). The Ricci scalar and the Kretschmann scalar are \nlim r -→ 0 R -→ ∞ , (14) \nlim r -→ 0 R αβγδ R αβγδ -→ ∞ , (15) \nand so confirm that there is a curvature singularity at r = 0. The Ricci and Kretschmann scalars are 2 d d 2 Λ and 8 d d 1 d 2 2 Λ 2 at r -→ ∞ . Therefore, the asymptotic behavior of these solutions are (a)dS for (Λ < 0 ). \n-→ ∞ On the other hand, in the absence of massive parameter ( m = 0), the solution (13) reduces to an d -dimensional asymptotically adS topological black hole with a negative, zero or positive constant curvature hypersurface in the following form \nf ( r ) = k -m 0 r d 3 -4 β 2 r 2 d 1 d 2 √ 1 + Γ + ( 4 β 2 d 1 d 2 + 1 l 2 ) r 2 + 4 d 2 q 2 H d 1 r 2 d 3 . (16) \nIn order to study the effects of the EN-BI-massive gravity on metric function, we have plotted various diagrams (Figs. 1 - 3 ). \nBy considering specific values for the parameters, metric function has different behaviors. Depending on the choices of the parameters, EN-BI-massive black holes can behave like Reissner-Nordstrom black holes. In other words, these black holes may have two horizons, one extreme horizon and without horizon (naked singularity) (see Fig. 1 for more details). On the other hand, by adjusting some of the parameters of EN-BI-massive black holes, we encounter with interesting behaviors. The solutions may have three or higher horizons (Figs. 2 and 3). The existence of three or higher horizons for black holes is due to the presence of massive gravity [15, 16]. In addition to the significant effects of massive term, we should note that the nonlinearity parameter can affect on the number of horizons. In addition, β can change the type of singularity. In other words, depending on the parameters, one can find a β c in which singularity is spacelike for β < β c , and it would be timelike for β > β c (see [20] for more details). \nNow, we give a brief discussion regarding Carter-Penrose diagrams. In order to study the conformal structure of the solutions, one may use the conformal compactification method through plotting the Carter-Penrose (conformal) diagrams. As we mentioned before, depending on the value of nonlinearity parameter, β , one may encounter with timelike or spacelike singularity. Penrose diagrams regarding to timelike singularity was discussed in [16] (see conformal diagrams in [16]). Here we focus on special case in which singularity is spacelike ( β < β c ). In other words, the singularity of this nonlinearly charged black holes behaves like uncharged Schwarzschild solutions (see Fig. 3). This means that, although massive and nonlinearity parts of the metric function can change the type of singularity and horizon structure of black holes, they does not affect asymptotical behavior of the solutions. Drawing the CarterPenrose diagrams, we find the causal structure of the solutions are asymptotically well behaved. \nFIG. 1: f ( r ) versus r for Λ = -0 . 8, q = 1, β = 0 . 6, m = 1 . 4, c = -0 . 8, c 1 = 2, c 3 = -4, c 4 = 0, k = 1, and d = 4. Left diagram for m 0 = 5, c 2 = 1 . 00 (dashed line), c 2 = 1 . 22 (continues line) and c 2 = 1 . 35 (dotted line). Right diagram for c 2 = 1 . 40, m 0 = 5 . 80 (dashed line), m 0 = 5 . 64 (continues line) and m 0 = 5 . 5 (dotted line). \n<!-- image --> \nFIG. 2: f ( r ) versus r for Λ = -1, q = 0 . 5, β = 7, m = 0 . 5, c = 0 . 4, c 1 = -40, c 2 = 60, c 3 = 1, c 4 = 0, k = 1 and d = 6. diagrams for m 0 = 1 . 75 (dashed line), m 0 = 1 . 67 (continues line) and m 0 = 1 . 61 (dotted line). \n<!-- image -->", 'IV. THERMODYNAMICS': 'In this section, we calculate the conserved and thermodynamic quantities of the static EN-BI-massive black hole solutions in d -dimensions and then check the first law of thermodynamics. \nBy using the definition of Hawking temperature which is related to the definition of surface gravity on the outer horizon r + , one can find \nT = m 2 c 4 πr 3 + [ c 1 r 3 + + d 3 c 2 cr 2 + + d 3 d 4 c 3 c 2 r + + d 3 d 4 d 5 c 4 c 3 ] + ( 2 β 2 -Λ ) r + 2 πd 2 + d 3 k 4 πr + -β 2 r + πd 2 √ 1 + Γ + , (17) \nFIG. 3: Metric functions and Carter-Penrose diagrams for the asymptotically adS black holes with spacelike singularity. Three horizons (continuous line of metric function and related Carter-Penrose diagram in right-up panel), two horizons which inner one is extreme (dotted line of metric function and related Carter-Penrose diagram in left-down panel) and two horizons which outer one is extreme (dashed line of metric function and related Carter-Penrose diagram in right-down panel). \n<!-- image --> \nwhere Γ + = d 2 d 3 q 2 β 2 r 2 d 2 + . The electric charge, Q , can be found by calculating the flux of the electric field at infinity, yielding \nQ = V d 2 √ d 2 d 3 4 π q. (18) \nIn order to obtain the entropy of the black holes, one can employ the area law of the black holes. It is a matter of calculation to show that entropy has the following form [36] \nS = V d 2 4 r d 2 + , (19) \nIt was shown that by using the Hamiltonian approach or counterterm method, one can find the mass M of the black hole for massive gravity as \nM = d 2 V d 2 16 π m 0 , (20) \nin which by evaluating metric function on the horizon ( f ( r = r + ) = 0), one can obtain \nM = d 2 V d 2 16 π ( kr d 3 + -2 r d 1 + d 1 d 2 Λ -4 β 2 r d 1 + d 1 d 2 [ √ 1 + Γ + -1 ] + 4 d 2 q 2 d 1 r d 3 + H + + cm 2 r d 5 + d 2 [ d 2 d 3 d 4 c 4 c 3 + d 2 d 3 c 3 c 2 r + + d 2 c 2 cr 2 + + c 1 r 2 + ] ) , (21) \nwhere H + = 2 F 1 1 2 , d 3 2 d 2 ] , 3 d 7 / 3 2 d 2 ] , -Γ + . \n] ] It is notable that, U is the electric potential, which is defined in the following form \n[ \n) \nU = A µ χ µ | r →∞ -A µ χ µ ∣ ∣ r → r + = √ d 2 d 3 q r d 3 + H + . (22) \n([ \nHaving conserved and thermodynamic quantities at hand, we are in a position to check the first law of thermodynamics for our solutions. It is easy to show that by using thermodynamic quantities such as charge (18), entropy (19) and mass (20), with the first law of black hole thermodynamics \ndM = TdS + UdQ, (23) \nwe define the intensive parameters conjugate to S and Q . These quantities are the temperature and the electric potential \nT = ( ∂M ∂S ) Q and U = ( ∂M ∂Q ) S , (24) \nwhich are the same as those calculated for temperature (17) and electric potential (22).', 'V. HEAT CAPACITY AND STABILITY IN CANONICAL ENSEMBLE': "Here, we study the stability conditions and the effects of different factors on these conditions. The stability conditions in canonical ensemble are based on the signature of the heat capacity. The negativity of heat capacity represents unstable solutions which may lead to following results: unstable solutions may go under phase transition and acquire stable states. This phase transition could happen whether when heat capacity meets a root(s) or has a divergency. Therefore, the roots of regular numerator and denominator of the heat capacity are phase transition points. In the other scenario, the heat capacity is always negative. This is known as non-physical case. But there is a stronger condition which is originated from the temperature. The positivity of the temperature represents physical solutions whereas its negativity is denoted as non-physical one. Therefore, in order to getting better picture and enriching the results of our study, we investigate both temperature and heat capacity, simultaneously. \nThe heat capacity is given by \nC Q = T ( ∂ 2 M ∂S 2 ) Q = T ( ∂T ∂S ) Q . (25) \nConsidering Eqs. (17) and (19), it is a matter of calculation to show that \n( ∂T ∂S ) Q = -d 3 k d 2 r d 1 + + 2 ( 2 β 2 -Λ ) πd 2 2 r d 3 + -d 3 m 2 c πd 2 r d 2 + [ 3 d 4 d 5 c 4 c 2 +2 d 4 c 3 cr + + c 2 cr 2 + ] -4 β 2 πd 2 2 r d 3 + (1 + Γ + ) 3 2 + 4 d 3 q 2 ( d 1 +Γ + ) πd 2 r 3 d -7 + √ 1 + Γ + . (26) \n√ In order to study the effects of different parameters on stability conditions and temperature, we have plotted various diagrams (Figs. 4 - 8). \nInterestingly, in the absence of the massive parameter (Fig. 4 right panel), temperature starts from -∞ and it is only an increasing function of horizon radius with a root. Therefore, we have two regions of physical and non-physical solutions. Adding massive gravity could modify the behavior of the temperature into an U shape diagram starting from + ∞ without any root. The extremum is an increasing function of massive parameter (Fig. 4 right panel) and dimensions (Fig. 8 right panel), whereas, it is a decreasing function of k (Fig. 6 right panel) and electric charge (Fig. 7 right panel). The only exception for this behavior is for strong nonlinearity parameter. Interestingly, for large values of nonlinearity parameter, a massive-less like behavior is observed (Fig. 5 right panel). In other words, the temperature starts from -∞ but the effect of the massive could be seen through two extrema. The root of temperature and smaller extremum are increasing functions of β and larger extremum is a decreasing function of it. Another interesting property of temperature is the effect of dimensions. Studying Fig. 8 (right panel) shows that the temperature for every two sets of dimensions will coincide with each other. In other words, there are places in which \nFIG. 4: For different scales: C Q (left panel) and T (right panel) versus r + for q = 1, Λ = -1 c = c 1 = c 2 = c 3 = 2, c 4 = 0, β = 0 . 5, d = 5 and k = 1; m = 0 (continues line), m = 0 . 25 (dotted line), m = 0 . 35 (dashed line) and m = 0 . 40 (dashed-dotted line). \n<!-- image --> \nFIG. 5: For different scales: C Q (left and middle panels) and T (right panel) versus r + for q = 1, Λ = -1, c = c 1 = c 2 = c 3 = 2, c 4 = 0, m = 0 . 30, d = 5 and k = 1; β = 2 (continues line), β = 3 (dotted line), β = 4 (dashed line) and β = 5 (dashed-dotted line). \n<!-- image --> \ndespite differences in dimensions, two black holes with two different dimensions and same values for other parameters will have same temperature in a special r + . \nIn absence of massive gravity, black holes could acquire temperature from zero to + ∞ , whereas adding massive, will cause the black holes never acquire some temperature. This effect is vanished in case of large nonlinearity parameter. In other words, the strength of nonlinearity parameter has opposing effects to massive's ones. Also, the U shape diagram indicates that for every temperature that black holes can acquire two horizons exist except for the extremum. Therefore, considering Hawking radiation, one is not able to recognize the size of these black holes by measuring their Hawking radiation. It is worthwhile to mention that extrema and root(s) of temperature are phase transition points of heat capacity. \nRegarding stability, it is evident that in absence of the massive gravity, there exists a region of the instability which is located where the temperature is negative. Therefore, this is a non-physical solution (Fig. 4 left panel). Interestingly, by adding massive gravity, the non-physical region is vanished and heat capacity acquires divergence point without any root. Before divergence point, the heat capacity is negative. Therefore, in this region black holes are unstable. In divergence point, black holes go under phase transition of smaller unstable to larger stable black holes. The divergence point is an increasing function of massive parameter (Fig. 4 left panel) and dimensions (Fig. 8 left panel), whereas, it is a decreasing function of k (Fig. 6 left panel) and electric charge (Fig. 7 left panel). \nFIG. 6: For different scales: C Q (left and middle panels) and T (right panel) versus r + for q = 1, Λ = -1, c = c 1 = c 2 = c 3 = 2, c 4 = 0, m = 0 . 4, d = 5 and β = 0 . 5; k = 1 (continues line), k = 0 (dotted line) and k = -1 (dashed line). \n<!-- image --> \nFIG. 7: For different scales: C Q (left panel) and T (right panel) versus r + for Λ = -1, c = c 1 = c 2 = c 3 = 2, c 4 = 0, β = 0 . 5, m = 0 . 4, d = 5 and k = 1; q = 0 (continues line), q = 0 . 5 (dotted line), q = 1 (dashed line) and q = 2 (dashed-dotted line). \n<!-- image --> \nInterestingly, in strong nonlinearity parameter, the mentioned behavior is modified. In this case black holes enjoy one root and two divergence points. Before root and between two divergencies, heat capacity is negative and between root and smaller divergence point and after larger divergence point, heat capacity is positive. According to thermodynamical concept, systems go under phase transition to acquire stable states. Therefore, following phase transitions take place: non-physical unstable to physical stable (in root), large unstable to smaller stable (in smaller divergence point) and smaller unstable to larger stable black holes (in larger divergency). Root and smaller divergence point are increasing functions of β (Fig. 5 left panel), whereas, larger divergency is a decreasing function of it (Fig. 5 middle panel). It is worthwhile to mention that larger divergency is not highly sensitive to variation of nonlinearity parameter. \nComparing obtained results for heat capacity (regarding phase transitions) and the behavior of the temperature, one can see that larger to smaller phase transition takes place at maximum (compare Fig. 6 left diagram with right) and smaller to larger one happens at minimum (compare Fig. 6 middle diagram with right) of temperature. Therefore, one is able to recognize the type and number of phase transition by only studying temperature's diagrams. \nFIG. 8: For different scales: C Q (left and middle panels) and T (right panel) versus r + for q = 1, Λ = -1, c = c 1 = c 2 = c 3 = 2, c 4 = 0, β = 0 . 5, m = 0 . 4 and k = 1; d = 5 (continues line), d = 6 (dotted line) and d = 7 (dashed line). \n<!-- image -->", 'VI. P -V CRITICALITY OF CHARGED BLACK HOLES IN EN-BI-MASSIVE GRAVITY': 'Now, we are in a position to study the critical behavior of the system through phase diagrams. Using the renewed interpretation of the cosmological constant as thermodynamical pressure, one can use following relation to rewrite thermodynamical relations of the solutions in spherical horizon [28] \nP = -Λ 8 π , (27) \nwhich results into following conjugating thermodynamical variable corresponding to pressure [28] \nV = ( ∂H ∂P ) S,Q . (28) \nDue to existence of the pressure in obtained relation for total mass of the black holes, one can interpret the total mass as thermodynamical quantity known as Enthalpy. This interpretation will lead to the following relation for Gibbs free energy [28] \nG = H -TS = M -TS. (29) \nNow by using Eqs. (20) and (27) with the relations of volume and Gibbs free energy (Eqs. (28) and (29)), one finds \nV = ω d 2 d 1 r d 1 + , (30) \nand \nG = r d 1 + d 1 d 2 P + m 2 c 2 r d 5 + 16 π ( 3 d 3 d 4 c 4 c 2 +2 d 3 c 3 cr + + c 2 r 2 + ) + d 2 2 q 2 H + 2 πd 1 r d 3 + + β 2 r d 1 + 4 πd 1 d 2 √ 1 + Γ + + r d 3 + 16 π . (31) \nObtained relation for volume indicates that volume of the black holes is only a function of the topology of the solutions and independent of electrodynamics and gravitational extensions, directly. \nIn order to obtain critical values, one can use P -V diagrams. In other words, by studying inflection point properties one can obtain critical values in which phase transitions may take place. Therefore, one can use \n( ∂P ∂r + ) T = ( ∂ 2 P ∂r 2 + ) T = 0 . (32) \nFIG. 9: P -r + (left), T -r + (middle) and G -T (right) diagrams for β = 0 . 5, q = 1, m = 0 . 1, c = c 1 = c 2 = c 3 = 0 . 2, c 4 = 0 and d = 5. \n<!-- image --> \nP -r + diagram, from up to bottom T = 1 . 1 T c , T = T c and T = 0 . 9 T c , respectively. \nT -r + diagram, from up to bottom P = 1 . 1 P c , P = P c and P = 0 . 9 P c , respectively. \n- G -T diagram for P = 0 . 5 P c (continuous line), P = P c (dotted line) and P = 1 . 5 P c (dashed line). \nConsidering obtained values for temperature (17) and pressure (27), one can obtain pressure as \nP = d 2 T 4 r + -m 2 c 16 πr 4 + [ c 1 r 3 + + d 3 c 2 cr 2 + + d 3 d 4 c 3 c 2 r + + d 3 d 4 d 5 c 4 c 3 ] -d 2 d 3 16 πr 2 + + β 2 4 π ( √ 1 + Γ + -1 ) . (33) \nNow, by considering Eq. (32) with obtained relation for pressure (33), one can obtain two relations for finding critical quantities. Due to economical reasons, we will not present them. Regarding the contribution of electromagnetic part, it is not possible to obtain critical horizon analytically, and therefore, we use numerical method. Considering the variation of β and massive parameter, one can draw following tables \nTable (1): q = 1, β = 0 . 5, c 1 = c 2 = c 3 = 0 . 2, c 4 = 0 and d = 5. \n| m | r c | T c | P c | P c r c T c |\n|---------------------------|-----------------------|-------------------------|----------------|----------------------|\n| 0 . 000000 1 | . 8264628 0 | . 1334354 0 | . 02200146 0 | . 3011558 |\n| 0 . 100000 1 | . 8263848 0 | . 1334835 0 | 02200495 0 | . 3010822 |\n| | | | . | |\n| 1 . 000000 1 5 . 000000 1 | . 7953522 0 . | . 1382092 0 | . 02233685 0 | . 2901581 . 2024811 |\n| 10 . 000000 0 | 6278897 0 . 7052643 0 | . 2562956 0 . 7329062 0 | . 03187871 0 . | 13205092 0 . 1270705 | \nTable (2): q = 1, m = 0 . 1, c 1 = c 2 = c 3 = 2, c 4 = 0 and d = 5. \n| β 1 . 000000 1 | r c 7819632 0 | T c . 1693410 0 | c . | P 0296043 0 | . | P c r c T c 3115245 |\n|------------------|-----------------|-------------------|-------------|---------------|---------------------|-----------------------|\n| 2 . 100000 1 | 8174387 0 | . | 1677996 0 . | 0290170 0 | . | 3142835 |\n| 3 . 500000 1 | 8235114 0 | . 1675323 0 | . | | 0289161 0 . 3147386 | |\n| 4 . 000000 1 | 8256045 0 | . | 1674399 0 | . 0288813 0 | . | 3148944 |\n| 5 . 000000 1 | 8265678 0 | . 1673974 0 | | . 0288653 0 | . 3149659 | | \nIn addition, we plot following diagrams (Figs. 9 - 12) to investigate that obtained values are the ones in which phase transition takes place or not. \nThe formation of swallow tail in G -T diagrams for pressure smaller than critical pressure (Figs. 9 and 11 right panels), subcritical isobars in T -r + diagrams for critical pressure (Figs. 9 and 11 middle panel) and isothermal diagrams in case of critical temperature in P -r + diagrams (Figs. 9 and 11 left panels), show that obtained values are critical ones in which phase transition takes place. \nFIG. 10: P -r + (left), T -r + (middle) and G -T (right) diagrams for β = 0 . 5, q = 1, c = c 1 = c 2 = c 3 = 0 . 2, c 4 = 0, d = 5, m = 0 (continuous line), m = 1 (dotted line) and m = 5 (dashed line). P -r + diagram for T = T c , T -r + diagram for P = P c and G -T diagram for P = 0 . 5 P c . \n<!-- image --> \nFIG. 11: P -r + (left), T -r + (middle) and G -T (right) diagrams for β = 2, q = 1, m = 0 . 1, c = c 1 = c 2 = c 3 = 2, c 4 = 0 and d = 5. \n<!-- image --> \nP -r + diagram, from up to bottom T = 1 . 1 T c , T = T c and T = 0 . 9 T c , respectively. T -r + diagram, from up to bottom P = 1 . 1 P c , P = P c and P = 0 . 9 P c , respectively. \n- G -T diagram for P = 0 . 5 P c (continuous line), P = P c (dotted line) and P = 1 . 5 P c (dashed line). \nFIG. 12: P -r + (left), T -r + (middle) and G -T (right) diagrams for m = 0 . 1, q = 1, c = c 1 = c 2 = c 3 = 0 . 2, c 4 = 0, d = 5, β = 1 (continuous line), β = 2 (dotted line) and β = 3 (dashed line). P -r + diagram for T = T c , T -r + diagram for P = P c and G -T diagram for P = 0 . 5 P c . \n<!-- image --> \nIt is evident that critical pressure (Fig. 10 left panel) and temperature (Fig. 10 middle panel) are increasing functions of the massive parameter, whereas the critical horizon (Fig. 10 left and middle panels) and universal ratio of P c r c T c are decreasing functions of this parameter. \nIt is worthwhile to mention that length of subcritical isobars (which is known as phase transition region) is a decreasing function of massive parameter (Fig. 10 middle panel). In opposite, the size of swallow tail and the energy of different phases are increasing functions of m (Fig. 10 right panel). \nInterestingly, the effects of variation of nonlinearity parameter is opposite of massive parameter. In other words, critical pressure (Fig. 12 left panel), temperature (Fig. 12 middle panel) and the size of swallow tail (Fig. 12 left panel) are decreasing functions of β , whereas, the critical horizon radius (Fig. 12 left and middle panels), length of subcritical isobars (Fig. 12 middle panel) and universal ration of P c r c T c are increasing functions of nonlinearity parameter. \nIt should be pointed that the length of subcritical isobars affects single regions of different states which in our cases are smaller and larger black holes. In other words, increasing the length of subcritical isobars (phase transition region) decreases the single state regions.', 'A. Neutral Massive black holes': 'In this section, by cancelling the electric charge ( q = 0), we will study the critical behavior of the system. Previously, it was shown that Schwarzschild black holes does not have any phase transition in context of extended phase space. Now, we are investigating the effects of massive gravity in case of EN-massive gravity. Using obtained relation for calculating critical horizon radius in previous part and setting q = 0, one can find following relation for calculating critical horizon radius \nm 2 6 d 4 d 5 c 4 c 4 +3 d 4 c 3 c 3 r + + c 2 c 2 r 2 + ) + r 2 + = 0 . (34) \n( \n) It is a matter of calculation to show that this relation has following roots which are critical horizon radii \nr c = -mc 2 ( 3 d 4 mc 3 c ± √ -3 d 4 [8 d 5 (1 + c 2 c 2 m 2 ) c 4 -3 d 4 m 2 c 2 3 c 2 ] ) 2 (1 + c 2 c 2 m 2 ) . (35) \nObtained relation shows that in absence of massive gravity, critical horizon radius will be zero which is not of our interest. This result consistent with Schwarzschild case. Now, for the simplicity, we consider the case of c 4 = 0. This leads into following critical horizon radius \nr cc = -3 d 4 mc 3 c 1 + c 2 c 2 m 2 . (36) \nIt is evident that for the cases of d = 4 and d > 4 with vanishing c 3 , the critical horizon radius will be zero. Therefore, there is no phase transition for these black holes. Interestingly for case of d > 4, the condition for having a positive critical horizon radius will be c 3 < 0 and 1 + c 2 c 2 m 2 > 0. By employing obtained value for critical horizon radius, one can find critical temperature and pressure in the following forms \nT cc = ( 3 d 4 c 1 c 3 -d 3 c 2 2 ) m 4 c 4 -d 3 ( 2 c 2 m 2 c 2 +1 ) 12 πd 4 m 2 c 3 c 3 , (37) \nP cc = -( c 2 m 2 c 2 +1 ) 3 d 2 d 3 432 πd 2 4 m 4 c 2 3 c 6 . (38) \nConsidering obtained values, one can show that following equality is hold \nP cc r cc T cc = ( c 2 m 2 c 2 +1 ) 2 d 2 d 3 12 (3 d 4 c 1 c 3 -d 3 c 2 2 ) m 4 c 4 -12 d 3 (2 c 2 m 2 c 2 +1) , (39) \nwhich shows that in this case, P cc r cc T cc is a function of massive parameter and coefficients. Using obtained critical values (Eqs. (36), (37) and (38)) with Eqs. (17), (27), (29), (33) and setting q = 0, we plot following diagrams for 5 and 6 dimensions (Figs. 13 and 14). \nIn Ref. [37], it was shown that in context of neutral Gauss-Bonnet black holes, no phase transition is observed in 6-dimensions. Here, the extension of the massive gravity enables the black holes to enjoy the existence of second order phase transition in 6-dimensions. Also, we observed that contrary to Gauss-Bonnet case, P cc r cc T cc is a function of massive gravity. \nFIG. 13: P -r + (left) and G -T (right) diagrams for m = 5, c = 0 . 2, c 1 = c 2 = 2, c 3 = -2, c 4 = 0 and d = 5. P -r + diagram, from up to bottom T = 1 . 1 T c , T = T c and T = 0 . 9 T c , respectively. G -T diagram for P = 0 . 5 P c (continuous line), P = P c (dotted line) and P = 1 . 5 P c (dashed line). \n<!-- image --> \nFIG. 14: P -r + (left) and G -T (right) diagrams for m = 5, c = 0 . 2, c 1 = c 2 = 2, c 3 = -2, c 4 = 0 and d = 6. P -r + diagram, from up to bottom T = 1 . 1 T c , T = T c and T = 0 . 9 T c , respectively. G -T diagram for P = 0 . 5 P c (continuous line), P = P c (dotted line) and P = 1 . 5 P c (dashed line). \n<!-- image -->', 'VII. GEOMETRICAL PHASE TRANSITION IN CONTEXT OF HEAT CAPACITY AND EXTENDED PHASE SPACE': 'In this section, we employ the geometrical concept for studying thermodynamical behavior of the obtained solutions. In order to do so, we employ HPEM method. In this method, the thermodynamical phase space is constructed by considering mass of the black holes as thermodynamical potential. By doing so, the components of the phase space will be extensive parameters such as electric charge, entropy and etc. The general form of HPEM metric is [33] \nds 2 New = SM S ( Π n i =2 ∂ 2 M ∂χ 2 i ) 3 ( -M SS dS 2 + n ∑ i =2 ( ∂ 2 M ∂χ 2 i ) dχ 2 i ) , (40) \nwhere M S = ∂M/∂S , M SS = ∂ 2 M/∂S 2 and χ i ( χ i /negationslash = S ) are extensive parameters which are components of phase space. Now, we will investigate whether the phase transition points that were obtained in section ( IV ) coincide with all divergencies of the Ricci scalar of HPEM metric. For economical reasons, we only plot diagrams correspond to variation of massive and nonlinearity parameters. To do so, we use Eqs. (18), (19) and (20) with HPEM metric (Eq. 40). This leads into following diagrams (Fig. 15). \nIt is evident that employed metric has consistent results with what were found in case of heat capacity (compare Figs. 4 and 5 with 15). In other words, the divergencies of the Ricci scalar are matched with phase transition points of the heat capacity. An interesting characteristic behavior of the diagrams is the different divergencies for different types of phase transition. In case of larger to smaller black holes phase transition, the divergency of the Ricci scalar is toward + ∞ (compare Fig. 5 left panel with Fig. 15 middle panel), whereas in case of smaller to larger phase transition, the divergency is toward -∞ (compare Fig. 5 middle panel with Fig. 15 right panel). This specific \nFIG. 15: For different scales: R versus r + diagrams for q = 1, c = c 1 = c 2 = c 3 = 2, c 4 = 0, d = 5 and k = 1. left: β = 0 . 5, m = 0 (continues line), m = 0 . 25 (dotted line), m = 0 . 35 (dashed line) and m = 0 . 40 (dashed-dotted line). middle and right: m = 0 . 3, β = 2 (continues line), β = 3 (dotted line) and β = 4 (dashed line). \n<!-- image --> \nFIG. 16: For different scales: R (continuous line), C Q (dashed line) diagrams for q = 1, c = c 1 = c 2 = c 3 = 2, c 4 = 0, β = 0 . 5, d = 5, k = 1 and m = 0 . 1. \n<!-- image --> \nP = 0 . 9 P c left and middle panels, P = P c left and right panels and P = 1 . 1 P c left panel. \nbehavior enables us to recognize the type of phase transition independent of heat capacity. \nNext, we employ another geometrical metric for studying the critical behavior of the system in context of extended phase space. In this metric, Due to consideration of the cosmological constant as thermodynamical pressure, we have three extensive parameters; electric charge, entropy and pressure. In order to construct phase space we employ following metric [35] \nds 2 = S M S M 3 QQ ( -M SS dS 2 + M QQ dQ 2 + dP 2 ) . (41) \nConsidering Eqs. (20), (26), (27) and (41), we plot following diagram (Fig. 16) with respect to Fig. 9. Due to existence of a root for heat capacity, in all plotted diagrams, a divergency is observed (Fig. 16 left panel). It is evident that for pressures smaller than critical pressure, system goes under two phase transitions with different horizon radii (Fig. 16 middle panel). This is consistent with what was observed in studying T -r + diagrams of Fig. 9. On the other hand, for critical pressure system goes under a phase transition. The place of this divergency is exactly located at the critical horizon which is obtainable through T -r + diagrams of Fig. 9 (Fig. 16 right panel). Finally, for pressures larger than critical pressure no phase transition is observed and the behavior of Ricci scalar will be what is plotted in Fig. 16 (left panel). These results are consistent with ordinary thermodynamical concepts and indicates that these three pictures (phase diagrams, heat capacity and geometrical thermodynamics) are in agreement. \nFIG. 17: P versus r + diagrams for q = 1, c = c 1 = c 2 = c 3 = 2, c 4 = 0, d = 5 and k = 1. left panel: β = 0 . 5 and m = 0 (bold-continues line), P = 0 . 02200146 (continues line), m = 5 (bold-dashed line) and P = 0 . 03187871 (dotted line). right panel: m = 0 . 1 and β = 1 (bold-continues line), P = 0 . 0296043 (continues line), β = 5 (bold-dashed line) and P = 0 . 0288653 (dashed line). \n<!-- image -->', 'VIII. HEAT CAPACITY AND CRITICAL VALUES IN THE EXTENDED PHASE SPACE': 'The final section of this paper is devoted to calculation of the critical pressure in extended phase space by using denominator of the heat capacity. It was shown that one can calculate critical pressures that were obtained in section ( V ) by using denominator of the heat capacity [35]. To do so, one should replace the cosmological constant in denominator of the heat capacity (26) with its corresponding pressure (27). Then, solve the denominator of the heat capacity with respect to pressure. This will lead into following relation \nP = d 2 d 3 c 4 m 2 c 2 16 πr 4 + ( 3 d 4 d 5 c 4 c 2 +2 d 4 c 3 cr + + c 2 r 2 + ) -d 2 d 2 3 q 2 8 πr 2 d 2 + √ 1 + Γ + -(√ 1 + Γ + -1 ) β 2 4 π 1 + Γ + + d 2 d 3 16 πr 2 + (42) \nObtained relation for pressure is different from what was obtained through use of temperature (33). In this relation, the maximum(s) of pressure and its corresponding horizon radius are critical pressure and horizon radius in which phase transition takes place. Now, by using indicated values in table 1 and Eq. (42), we plot following diagram (Fig. 17). \n√ \nIt is evident that obtained maximums are critical pressures in which phase transitions take place. The thermodynamical concept that was mentioned in last section (pressure being smaller than critical pressure leads to existence of two phase transitions and for pressures larger than critical pressure no divergency is observed) is also hold in case of this approach. In other words, this approach is an additional method for studying critical behavior of the system and the results of this approach is consistent with GTs, heat capacity and extended phase space.', 'IX. CONCLUSIONS': 'In this paper, we have considered EN-massive gravity in presence of BI nonlinear electromagnetic field. It was shown that considering this configuration leads to modification of the number and place of horizons that black holes can acquire. In other words, cases of multiple horizons were observed with different phenomenologies. Next, conserved and thermodynamical quantities were obtained and it was shown that first law of thermodynamics hold for these black holes. \nNext, we studied the thermodynamical behavior of the system. It was shown that temperature in the presence and absence of massive gravity presents different behaviors. Adding massive put limitations on values that temperature can acquire, while, there was no limitation for temperature in the absence of it. Interestingly, this behavior was modified in the presence of strong nonlinearity parameter. In strong nonlinearity parameter, the behavior of temperature \nreturned to a massive-less like behavior but the effects of the massive were observed in existences of extrema. It was also seen, that in case of different dimensions, for each pair of dimensions, one can find a point in which temperature for both dimensions are equal. \nRegarding the stability, it was seen that in the presence of massive gravity, black holes enjoy one phase transition of the smaller unstable to larger stable. The phase transition was related to the divergency of heat capacity. Then again, in strong nonlinearity parameter, this behavior was modified. In this case, black holes had three phase transitions of smaller non-physical unstable to larger physical stable (in place of root), larger unstable to smaller stable (in place of smaller divergency) and smaller unstable to larger stable (in place of larger divergency). \nClearly, one can conclude that nonlinear electromagnetic field has an opposing effect comparing to massive gravity. Strong nonlinearity parameter modifies the effects of the massive gravity and return the system to the massive-less like behavior, although the effects of massive still observed through extrema. \nIt was pointed out that at maximums of the temperature, larger unstable to smaller stable and at minimums, smaller unstable to larger stable phase transitions take place. Therefore, studying temperature provides an independent picture for studying phase transitions and stability of the solutions. \nNext, we extended phase space by considering cosmological constant as thermodynamical variable known as pressure. It was shown that volume of the black holes is independent of generalization of the electromagnetic field and extension of the massive gravity. Obtained values where critical points in which phase transitions took place. It was shown that the effects of variation of nonlinearity parameter was opposite of the massive parameter. In other words, these two factors put restrictions on each others effects. \nInterestingly, in Ref. [15] variation of massive gravity highly modified the critical temperature and pressure. In case of obtained solutions in this paper, the modification was not as considerable as what was observed in case of Gauss-Bonnet-Maxwell-massive black holes. This shows that generalization of electromagnetic field puts stronger restrictions on the effects of the massive gravity. In other words, in order to have stronger control over contributions of the massive gravity one should increase the nonlinearity of the electromagnetic sector. \nIn addition, a study in context of neutral solutions was conducted. It was shown that due to contribution of the massive gravity, the chargeless solutions of this gravity also enjoy the existence of phase transition. In other words, black holes in EN-massive gravity go under phase transitions in extended phase space. Also, it was shown that ratio of P c r c T c was a function of massive gravity. It was also pointed out that in 6-dimensions, contrary to case of Gauss-Bonnet black holes, these black holes enjoy second order phase transition. In addition, it was shown that in case of d = 4, no phase transition for massive black holes is observed. \nNext, geometrical approach was used for studying critical behavior of the system in context of heat capacity and extended phase space. It was shown that employed metrics for both cases have consistent results and follow the concepts of ordinary thermodynamics. The characteristic behavior of divergencies in Ricci scalar of the geometrical thermodynamical metrics, enabled us to recognize the type of phase transition (smaller to larger or larger to smaller). \nFinally, another method which was based on denominator of the heat capacity was used to calculate critical pressure and horizon radius. It was shown that this method has consistent results with extended phase space and follow the concepts of ordinary thermodynamics. In other words, this method provides an independent approach for investigating critical behavior of the system. \nDue to generalization of Born-Infeld for electromagnetic sector of the solutions, it will be worthwhile to study the effects of this generalization on conductivity of these black holes and their corresponding superconductors phase transition. Specially, it will be interesting to see how this generalization will affect the interpretation of graviton as lattice and the Drude like behavior. Also, it will be worthwhile to study the metal-insulator transition in context of these solutions.', 'Acknowledgments': "We thank Shiraz University Research Council. This work has been supported financially by the Research Institute for Astronomy and Astrophysics of Maragha, Iran. \n- [1] S. Deser, R. Jackiw and G. 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2001PhLB..521...87N
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Anti-de-Sitter black hole thermodynamics in higher derivative gravity and new confining-deconfining phases in dual CFT
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2001-01-01
| 4 | 0.44 | 158 |
['thermodynamics', '-', '-', 'phase', '-', '-']
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[]
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The thermodynamics of d5 AdS BHs with positive, negative or zero curvature spatial section in higher derivative (HD) gravity is described. HD contribution to free energy may change its sign which leads to more complicated regime for Hawking-Page phase transitions. Some variant of d5 HD gravity is dual to N=2Sp(N) SCFT up to the next-to-leading order in large N. Then, according to Witten interpretation the stable AdS BH phase corresponds to deconfinement while global AdS phase corresponds to confinement. Unlike to Einstein gravity in HD theory the critical N appears. It may influence the phase transition structure. In particular, what was confining phase above the critical value becomes the deconfining phase below it and vice-versa.
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https://arxiv.org/pdf/hep-th/0109122.pdf
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{'No Header': 'NDA-FP-?? September 2001', 'Anti-de Sitter Black Hole Thermodynamics in Higher Derivative Gravity and New Confining-Deconfining Phases in dual CFT.': "Shin'ichi NOJIRI 1 and Sergei D. ODINTSOV ♠ 2 \nDepartment of Applied Physics National Defence Academy, Hashirimizu Yokosuka 239-8686, JAPAN \n♠ Instituto de Fisica de la Universidad de Guanajuato, Lomas del Bosque 103, Apdo. Postal E-143, 37150 Leon,Gto., MEXICO", 'ABSTRACT': 'The thermodynamics of d5 AdS BHs with positive, negative or zero curvature spatial section in higher derivative (HD) gravity is described. HD contribution to free energy may change its sign which leads to more complicated regime for Hawking-Page phase transitions. Some variant of d5 HD gravity is dual to N = 2 Sp ( N ) SCFT up to the next-to-leading order in large N . Then, according to Witten interpretation the stable AdS BH phase corresponds to deconfinement while global AdS phase corresponds to confinement. Unlike to Einstein gravity in HD theory the critical N appears. It may influence the phase transition structure. In particulary, what was confining phase above the critical value becomes the deconfining phase below it and vice-versa.', '1 Introduction': 'It has been observed quite long ago by Hawking and Page [1] that Anti-de Sitter (AdS) Black Holes (BHs) thermodynamics admits phase transitions. These Hawking-Page phase transitions occur as the following: low temperature BHs are not stable and global AdS spacetime is then preferrable state. On the same time, high temperature BHs are stable and they do not decay to the global AdS spacetime. \nThe invention of AdS/CFT correspondence [2] increased the interest to phase transitions for AdS BHs. Indeed, Witten [3] demonstrated that Hawking-Page phase transition corresponds to a deconfinement-confinement transition in the largeN limit of an N = 4 SU ( N ) super Yang-Mills theory living on the boundary of 5d AdS BH. This is easily seen when calculating the expectation value of the temporal Wilson loop operator which is an order parameter for the spontaneous symmetry breaking for a subgroup of the gauge group center. When this expectation value is not zero (as is the case for stable AdS BHs phase) then deconfinement is realized. On the contrary, when it is zero (global AdS space) the confinement is realized. The same interpretation of phase transitions [4, 5, 6] survives for 5d AdS BHs with zero or negative curvature for spatial section. \nIn the present work we discuss (spherical, flat or hyperbolic) AdS BH thermodynamics and especially the role played by HD terms to Hawking-Page phase transitions. The free energy for such BHs is found and its dependence from HD terms coefficients is clarified. As a result, there appear several regimes for phase transitions. Using dual Sp ( N ) SCFT interpretation of specific HD gravity model one comes to complicated structure of Witten confinement-deconfinement phase transitions. In particular, there occurence of critical N is remarkable.', '2 Thermodynamics of bulk AdS black hole': "In this section, we review thermodynamics of AdS BH in bulk R 2 -gravity, based on [7]. We consider the case that HD terms contain the Riemann tensor square term, i.e. R µνξσ R µνξσ . As it will be shown the presence of this term will lead to interesting consequences for AdS BH thermodynamics. \nThe general action of d +1 dimensional R 2 -gravity is given by \nS = ∫ d d +1 x √ -ˆ G { a ˆ R 2 + b ˆ R µν ˆ R µν + c ˆ R µνξσ ˆ R µνξσ + 1 κ 2 ˆ R -Λ + L matter } . (1) \nWhen c = 0, Schwarzschild-anti de Sitter space is an exact solution of gravitational theory: \nds 2 = ˆ G µν dx µ dx ν = -e 2 ρ 0 dt 2 +e -2 ρ 0 dr 2 + r 2 d -1 ∑ i,j g ij dx i dx j , e 2 ρ 0 = 1 r d -2 ( -µ + kr d -2 d -2 + r d l 2 ) . (2) \nHere l is the radius of the asymptotic AdS space, given by solving the equation \n0 = d 2 ( d +1)( d -3) a l 4 + d 2 ( d -3) b l 4 -d ( d -1) κ 2 l 2 -Λ . (3) \nFor non-vanishing c , such an S-AdS BH solution may be constructed perturbatively. In this section, we only consider the case a = b = 0 for simplicity: \nS = ∫ d d +1 x √ -ˆ G { c ˆ R µνξσ ˆ R µνξσ + 1 κ 2 ˆ R -Λ } . (4) \nWhen we assume the metric (2) with µ = 0, the scalar, Ricci and Riemann curvatures are given by \nˆ R = -d ( d +1) l 2 , ˆ R µν = -d l 2 G µν , ˆ R µνξσ = -1 l 2 ( ˆ G µξ ˆ G νσ -ˆ G µσ ˆ G νξ ) , (5) \nwhich tell that the curvatures are covariantly constant. The equation of the motion derived from the action (4) (no matter) is: \n0 = -ˆ G ζξ 2 { c ˆ R µνρσ ˆ R µνρσ + ˆ R κ 2 -Λ } +2 c ˆ R ζµνρ ˆ R µνρ ξ + ˆ R ζξ κ 2 +4 cD ρ D κ ˆ R ρ κ ζ ξ . (6) \nThen substituting Eqs.(5) into (6), one finds the relation between c , Λ and l \n0 = 2 c l 4 d ( d -3) -d ( d -1) κ 2 l 2 -Λ , (7) \nwhich defines the radius l of the asymptotic AdS space even if µ /negationslash = 0. For d + 1 = 5 with µ /negationslash = 0, using Eq.(6), we get the perturbative solution from (2), which looks like: \ne 2 ρ = 1 r 2 { -µ + k 2 r 2 + r 4 l 2 + 2 µ 2 /epsilon1 r 4 } , /epsilon1 = cκ 2 . (8) \nSuppose that g ij (2) corresponds to the Einstein manifold, defined by r ij = kg ij , where r ij is Ricci tensor defined by g ij and k is the constant. For example, if k > 0 the boundary can be three dimensional sphere, if k < 0, hyperboloid, or if k = 0, flat space. Properly normalizing the coordinates, one can choose k = 2, 0, or 2. \nAfter Wick-rotating the time variable by t → iτ , the free energy F can be obtained from the action S (4) : F = -TS , where the classical solution is substituted. Multiplying ˆ G ζξ to (6) in case that D ρ D κ ˆ R ρ κ ζ ξ = O ( /epsilon1 ) as in the solution (8), one finds for d = 4 1 κ 2 ˆ R = -c 3 ˆ R µνρσ ˆ R µνρσ + 5 3 Λ + O ( /epsilon1 2 ). \n- \nThen the action (4) can be rewritten as \nS = ∫ d 5 x √ -ˆ G { 2 3 c ˆ R µνξσ ˆ R µνξσ + 2 3 Λ } . (9) \nSince ˆ R µνξσ ˆ R µνξσ = 40 l 2 + 72 µ 2 r 8 + O ( /epsilon1 ), by using (7) with d = 4, we obtain \nS = -∫ d 5 x √ -ˆ G ( 8 κ 2 l 2 -32 c l 4 -48 cµ 2 r 8 ) = -V 3 T ∫ ∞ r H drr 3 ( 8 κ 2 l 2 -32 c l 4 -48 cµ 2 r 8 ) . (10) \nHere V 3 is the volume of unit 3d sphere for k = 2 and we assume τ has a period 1 T . The expression for S contains the divergence coming from large r . In order to subtract the divergence, we regularize S (10) by cutting off the integral at a large radius r max and subtracting the solution with µ = 0 in a same way as in [8]: \nS reg = -V 3 T { ∫ r max r H drr 3 ( 8 κ 2 l 2 -32 c l 4 -48 cµ 2 r 8 ) -e ρ ( r = r max ) -ρ ( r = r max ; µ =0) ∫ r max 0 drr 3 ( 8 κ 2 l 2 -32 c l 4 )} . (11) \nThe factor e ρ ( r = r max ) -ρ ( r = r max ; µ =0) is chosen so that the proper length of the circle which corresponds to the period 1 T in the Euclidean time at r = r max coincides with each other in the two solutions. Taking r max →∞ , one finds \nF = V 3 {( l 2 µ 8 -r 4 H 4 ) ( 8 κ 2 l 2 -32 c l 4 ) -12 cµ 2 r 4 H } . (12) \nThe horizon radius r H is given by solving the equation e 2 ρ 0 ( r H ) = 0 in (8). We can solve r H perturbatively up to first order on c by putting r H = r 0 + cδr , where r 0 is the horizon radius when c = 0: \nr H = r 0 -cµ 2 κ 2 r 3 0 ( 2 µ -k 2 r 2 0 ) , r 2 0 = -kl 2 4 + 1 2 √ k 2 4 l 4 +4 µl 2 . (13) \nWe can also rewrite the black hole mass µ (using r H ) up to first order on /epsilon1 ( /epsilon1 = cκ 2 ): µ = k 2 r 2 H + r 4 H l 2 + 2 /epsilon1 r 4 H ( k 2 r 2 H + r 4 H l 2 ) 2 . Then F looks like \nF = V 3 κ 2 l 2 l 2 k 2 r 2 H -r 4 H + /epsilon1 l 2 k 2 2 + 6 r 4 H l 2 -12 l 2 r 4 H ( k 2 r 2 H + r 4 H l 2 ) 2 . (14) \nThe Hawking temperature T H is given by \nT H = (e 2 ρ ) ' | r = r H 4 π = 1 4 π 4 r H l 2 + k r H -8 /epsilon1 r 7 H ( k 2 r 2 H + r 4 H l 2 ) 2 , (15) \n where ' denotes the derivative with respect to r . Then the entropy S = -dF dT H = -dF dr H dr H dT H and the energy E = F + T H S have the following form: \nS = 4 πV 3 r 3 H κ 2 1 -1 /epsilon1 l 2 ( -8 -4 kl 2 r 2 H + 3 k 2 l 4 2 r 4 H )( 1 -kl 2 4 r 2 H ) -1 , (16) \n \n E = 3 V 3 κ 2 { 1 2 kr 2 H + r 4 H l 2 + /epsilon1 ( 34 r 4 H 3 l 4 -17 kr 2 H 6 l 2 -19 6 k 2 + k 3 l 2 24 r 2 H )( 1 -kl 2 4 r 2 H ) -1 . (17) \nIt is remarkable that the entropy S is not proportional to the area of the horizon when k /negationslash = 0 and the energy E is not to µ , either. We should note that the entropy S was proportional to the area and the energy E to µ even in R 2 -gravity if there is no the squared Riemann tensor term ( c = 0 in (1)) [7], where we have the following expressions: \nF = -V 3 8 r 2 H ( r 2 H l 2 -k 2 )( 8 κ 2 -320 a l 2 -64 b l 2 ) , (18) S = V 3 πr 3 H 2 ( 8 κ 2 -320 a l 2 -64 b l 2 ) , E = 3 V 3 µ 8 ( 8 κ 2 -320 a l 2 -64 b l 2 ) . \nHere a and b are given in (1). Thus, we demonstrated the role of non-zero c contribution in the thermodynamics of AdS BH.", '3 Phase transitions in higher derivative AdS gravity': 'It has been suggested by Hawking and Page [1], there is a phase transition between AdS BH spacetime and global AdS vacuum. BH is stable at high temperature but it becomes unstable at low temperature. From the point of view of the AdS/CFT correspondence [2], this phase transition could correspond to the confinement-deconfinement transition in dual gauge theory [3]. In this section, we investigate the phase structure in HD gravity by using the thermodynamical quantities obtained in the previous section. \nNote that there is a minimum T min = k πl for the Hawking temperature T H in (15) when k > 0 and /epsilon1 = 0 or c = 0 when the horizon radius r H is given by r 2 H = r 2 1 ≡ kl 2 4 . The existence of the minimum tells that BH cannot exist at low temperature T < T min . When we Wick-rotate the AdS metric, we can freely impose the periodic boundary condition for the Euclidean time variable. Then the temperature of the AdS can be arbitrary. Only global AdS can exist when T < T min . The free energy F of the AdS vanishes by the definition here. On the other hand, the free energy of BH is given in (12), which vanishes at r 2 H = r 2 2 ≡ l 2 k 2 when /epsilon1 = 0 or c = 0. The corresponding critical Hawking temperature is given by T H = T c = 3 √ 2 k 4 πl . Then when c = 0, there is a phase transition between AdS BH and AdS when T H = T c . When T H > T c , the BH free energy F with /epsilon1 = 0 is negative, and BH will be \n√ \npreferable. Since the phase transition is of the first order, BH can exist when T c > T H ≥ T min but it becomes unstable and it decays into global AdS. This also proves that the horizon radius has a minimum when r H = r 2 = l √ k 2 . \nWhat happens when /epsilon1 = 0 but small? Then \nT min = √ k 4 π [ 4 l -36 /epsilon1 l 3 + O ( /epsilon1 2 ) ] , r 2 1 = kl 2 4 -15 k/epsilon1 2 + O ( /epsilon1 2 ) , (19) \n/negationslash \nT c = √ 2 k 4 π [ 3 l + 12 /epsilon1 l 3 + O ( /epsilon1 2 ) ] , r 2 2 = l 2 k 2 +28 k/epsilon1 + O ( /epsilon1 2 ) . (20) \nIf /epsilon1 is positive (negative), the correction makes T min smaller (larger) and makes T c larger (smaller). \nWhen T H ∼ T min or T c , the essential behaviors are not changed by the corrections. The region corresponds to r H ∼ l . We should, however, note that the behaviors when r H is small might be changed. In Eq.(15), the Hawking temperature T H may become negative if /epsilon1 is positive. Although T should be positive, this means that the temperature of BH can be small for small r H . Assuming r 2 H = O ( /epsilon1 ) and /epsilon1 is small, one finds T H (15) vanishes when r 2 H = 2 k/epsilon1 + O ( /epsilon1 2 ). Then, the free energy (12) becomes \nF = -3 V 3 k 2 /epsilon1 2 κ 2 , (21) \nwhich is negative when /epsilon1 is positive. Since r 2 H = 2 k/epsilon1 + O ( /epsilon1 2 ), /epsilon1 should be positive if k is positive, as we assume here. Then the AdS black hole spacetime might be preferred than AdS spacetime for small r H again. When r 2 H = O ( /epsilon1 ), the correction becomes large and the perturbation with respect to /epsilon1 is not valid. The above results suggest that the essential behavior might be changed for small r H . \nSo far in this section, we have assumed k should be positive. When c = 0, the free energy (14) is always negative if k is not positive ( k ≤ 0) then the AdS black hole is always stable. Furthermore, when k < 0, the horizon radius (13) does not vanish even if µ = 0: \nr 2 H = r min -≡ | k | l 2 2 = -kl 2 2 . (22) \nEq.(22) does not change even if c /negationslash = 0. When r 2 H = r 2 min -, the free energy (14) has the following form: \nF = F L ≡ -V 3 l 4 k 2 2 κ 2 l 2 ( 1 -4 /epsilon1 l 2 ) . (23) \nThen if \n1 -4 /epsilon1 l 2 < 0 , (24) \nthe free energy can be positive. When r 2 H = r 2 min -, the Hawking temperature (15) has a minimum: T H = √ -2 k 4 πl . For large r H , the Hawking temperature (15) and the free energy (14) are: \nT H = r H πl 2 ( 1 -2 /epsilon1 l 2 ) , F → F H ≡ -V 3 κ 2 l 2 ( 1 -18 /epsilon1 l 2 ) . (25) \nEq.(25) tells that if /epsilon1 l 2 < 1 2 , the large (large r H ) black hole is at high temperature, as is usual for AdS BH (of course, it is not correct for the black hole in the flat background). Eqs.(23), (25) indicate that there are several critical points: \n- 1. If /epsilon1 l 2 < 1 18 , both of F L and F H are negative, then the black hole is always stable, as in /epsilon1 = 0 case. For Sp ( N ) SCFT (see next section) it gives N more than 288 and deconfining phase.\n- 2. If 1 4 > /epsilon1 l 2 > 1 18 , we find F L < 0 but F H > 0. Then at low temperature BH is stable but it becomes unstable at high temperature. For dual SCFT one gets deconfinement-confinement transition (64 < N < 288).\n- 3. If 1 2 > /epsilon1 l 2 > 1 4 , both of F L and F R are positive. Therefore BH becomes always unstable and decays into the AdS vacuum. For dual Sp ( N ) SCFT it corresponds to confining phase (32 < N < 64).\n- 4. When /epsilon1 l 2 > 1 2 , the structure becomes very complicated. It could be the perturbation with respect to /epsilon1 is not valid and we cannot come to any definite conclusion. \nAnyway the above observed phase transition does not occur when c = 0 ( /epsilon1 = 0) but it appears when /epsilon1 l 2 > 1 18 . Note, Eq.(25) does not depend on k , then if /epsilon1 l 2 > 1 18 , the black hole becomes always unstable. We should also note that the case 1 4 > /epsilon1 l 2 > 1 18 is the inverse of the Hawking-Page phase transition, where the black hole is stable at high temperature but unstable at low temperature. The above condition Eq.(24) in terms of AdS/CFT setup gives the bound for rank N of gauge group for dual CFT. For example, taking group Sp ( N ) of corresponding N = 2 dual superconformal theory (see \ndescription of the model at the end of next section) one arrives to critical value N = 288. \nHence, the situation is the following. For large N as we observed the free energy is always negative and AdS BH is stable. Then, only one phase (deconfinement) occurs for dual CFT living on such conformal boundary. On the same time, below the critical value on N there appears possibility of the inverse of the Hawking-Page phase transition thanks to non-zero c contribution. For dual theory it means that confinement-deconfinement transition occurs at some temperature. New (confining) phase may appear but only below the critical N ! This is purely higher curvature term effect. Note the conformal boundary of above 5d AdS is hyperbolic space.', '4 Phase transitions for AdS black holes with Ricci-flat horizons': 'In the last section, in order to regularize the action when the black hole solution is substituted we subtract the action of the AdS vacuum. In case of k = 0, however, there is an argument that it is easier to subtract the action of AdS soliton [9] instead of the vacuum AdS [6]. (Physical results on phase transitions are not changing). In this case, besides the temperature, the area of the horizon becomes an independent parameter on which the thermodynamical quantities depend. \nWe now start from the construction of AdS soliton in R 2 -gravity. When k = 0, the AdS BH (8) looks like \nds 2 BH = -e 2 ρ BH ( r ) dt 2 BH +e -2 ρ BH ( r ) dr 2 + r 2 dφ 2 BH + ∑ i =1 , 2 ( dx i ) 2 , e 2 ρ BH ( r ) = 1 r 2 { -µ BH + r 4 l 2 + 2 µ 2 BH /epsilon1 r 4 } , /epsilon1 ≡ cκ 2 . (26) \nIn (26), we choose a torus for the k = 0 Einstein manifold for simplicity. The coordinates of the torus are φ BH and { x 1 , x 2 } . One assumes φ BH has a period of η BH : φ BH ∼ φ BH + η BH . The black hole horizon r BH and the Hawking temperature T BH are given by \nr BH = l 1 2 µ 1 4 BH -1 2 /epsilon1l -3 2 µ 1 4 BH , T BH = 1 4 π { 4 r BH l 2 -8 /epsilon1r BH l 4 } . (27) \nThe Hawking temperature gives the periodicity of the time coordinate t BH when we analytically continue the time coordinate: it BH it BH + 1 T BH . \nThe AdS soliton solution can be obtained by exchanging the signature of t BH and φ BH as t BH → iφ s and φ BH → it s . Then the metric of the AdS soliton is given by \n∼ \nds 2 s = -r 2 dt s +e -2 ρ s ( r ) dr 2 +e 2 ρ s ( r ) dφ 2 s + r 2 ∑ i =1 , 2 ( dx i ) 2 , \ne 2 ρ s ( r ) = 1 r 2 { -µ s + r 4 l 2 + 2 µ 2 s /epsilon1 r 4 } . (28) \nIn the solution the radial coordinate is restricted to be: \nr ≥ r s , r s = l 1 2 µ 1 4 s -1 2 /epsilon1l -3 2 µ 1 4 s . (29) \nThe regularity at r = r s determines the periodicity of φ s : \nφ s ∼ φ s + 1 T s , T s = 1 4 π { 4 r s l 2 -8 /epsilon1r s l 4 } . (30) \nOne also assumes that the time coordinate t s has a periodicity η s when analytically continued: it s ∼ it s + η s . \n∼ The free energy of the AdS black hole may be calculated from the action as above. As in (11), we regularize the action S (10), where the AdS black hole solution (26) is substituted, by cutting off the integral at a large radius r max and subtracting the action where AdS soliton (28) is substituted. Then \nS reg = -[ η BH V 2 T BH ∫ r max r BH drr 3 -η s V 2 T s ∫ r max r s drr 3 ] ( 8 κ 2 l 2 -32 c l 4 -48 cµ 2 BH r 8 ) . (31) \nHere V 2 expresses the volume corresponding to the coordinates { x 1 , x 2 } . We now impose the following matching conditions at r = r max , which guarantee that the length of t BH ( φ BH ) direction on the cutted boundary is identical with that of t s ( φ s ) direction: \ne ρ BH ( r max ) T BH = r max η s , r max η BH = e ρ s ( r max ) T s . (32) \nThe conditions (32) determine the parameters µ s ( r s or T s ) and η s in terms of the parameters µ BH ( r BH or T BH ) and η BH . In the limit of r max →∞ , one gets \nη s → 1 lT BH , T s → 1 lη BH . (33) \nIn the limit, we obtain the following expression of S reg : \nS reg = -η BH V 2 T BH [ ( 1 κ 2 -4 c l 2 ) { µ BH -µ s -2 ( r 4 BH -r 4 s )} -12 c ( µ 2 BH r 4 BH -µ 2 s r 4 s )] . (34) \nUsing Eqs.(27), (29), and (30), one gets \nr i = l 2 ( πT i ) ( 1 + 2 /epsilon1 l 2 ) , µ i = l 6 ( πT i ) 4 ( 1 + 10 /epsilon1 l 2 ) , ( i = BH , s) . (35) \nBy using (35) and (33), we can rewrite the expression in (34) and obtain the free energy F in the following form: \nF = -η BH V 2 l 6 κ 2 ( 1 + 14 /epsilon1 l 2 ) { ( πT BH ) 4 -( πT s ) 4 } = -η BH V 2 l 6 κ 2 ( 1 + 14 /epsilon1 l 2 ) ( πT BH ) 4 -( π lη BH ) 4 . (36) \n When c = 0, the above expression reproduces the result in [6] 3 . Eq.(36) tells that there is a phase transion, as in c = 0 case [6], at \nT BH = 1 lη BH . (37) \nIf 1 + 14 /epsilon1 l 2 > 0, the situation is not so changed from c = 0 case and we find that when T BH > 1 η BH , the black hole is stable but when T BH < 1 η BH , the black \nβ b F = I = vol( F ) 16 πGl β b β s [ k n -1 s -k n -1 b ] \ncan be reproduced from (36) by replacing V 2 = vol( F ), κ 2 = 16 πG , η BH = β s l , l 6 ( πT BH ) 4 = µ BH = k n -1 b , l 6 ( πT s ) 4 = µ s = l 6 ( π lη BH ) 4 = k n -1 s . \nhole becomes unstable and the AdS soliton is preferred. Eq.(36), however, suggests that there appears a critical point at \n1 = -14 /epsilon1 l 2 . (38) \nIf 1 + 14 /epsilon1 l 2 < 0, the situation is changed and we find that when T BH > 1 η BH , the black hole is unstable and the AdS soliton is preferred but when T BH < 1 η BH , the black hole becomes stable. As we treat the correction from /epsilon1 perturbatively, it is not clear if the above critical point really occurs. In order to prove this fact, we consider the situation that c = 0 but a and/or b do not vanish in the action (1). When c = 0 but a , b /negationslash = 0, the Schwarzschild-AdS solution and the AdS soliton solution are exact solutions, whose metric are given by replacing e 2 ρ BH ( r ) in (26) and e 2 ρ s ( r ) in (28) by \ne 2 ρ BH ( r ) → 1 r 2 { -µ BH + r 4 l 2 } , e 2 ρ s ( r ) = 1 r 2 { -µ s + r 4 l 2 } . (39) \nThen by the procedure similar to the case of c /negationslash = 0, one finds the following expression for the free energy (analog of (36)): \nF = -η BH V 2 l 6 κ 2 ( 1 -40 aκ 2 l 2 -8 bκ 2 l 2 ) ( πT BH ) 4 -( π lη BH ) 4 . (40) \n We should note that the above expression (40) is valid for the arbitrary values of a and b although the expression (36) is valid for small /epsilon1 only. Eq.(40) tells again that there is a critical point (line), even for spherical AdS BH, when \n1 -40 aκ 2 l 2 -8 bκ 2 l 2 = 0 , (41) \nwhich is an analog of (38). The above results demonstrate that there should appear a critical point in R 2 -gravities. \nLet us consider the explicit example in the framework of AdS/CFT correspondence. The N = 2 theory with the gauge group Sp ( N ) arises as the low-energy theory on the world volume on N D3-branes sitting inside 8 D7branes at an O7-plane [10]. The string theory dual to this theory has been conjectured to be type IIB string theory on AdS 5 × X 5 where X 5 = S 5 /Z 2 [11], whose low energy effective action is given by \nS = ∫ AdS 5 d 5 x √ G { N 2 4 π 2 ( R -2Λ) + 6 N 24 · 16 π 2 R µνρσ R µνρσ } . (42) \nThen R 2 -term appears as 1 /N correction. Identifying /epsilon1 l 2 = 16 N , we find from (38) that there appears a critical point at \nN = 160 , (43) \nwhere N seems to be large enough. \nHence, the phase structure of such SCFT looks as following. For large N (above of critical value) there are two phases of AdS thermodynamics. Stable AdS BH phase corresponds to deconfinement of dual SCFT, at some critical temperature there occurs phase transition. The low temperature phase should correspond to the confining phase in the gauge theory from the viewpoint of the AdS/CFT correspondence. When one considers the same theory but with N less than critical value, then the situation is reversed. What before was deconfinement becomes confinement and vice-versa. It is remarkable the phase transition occurs formally at the same critical temperature as above. Hence, even for the low temperature depending on N , there may occur and confinement and deconfinement. \nHence, the role of next-to-leading correction in large N - expansion (higher derivative term) is to clarify the structure of confining- deconfining phases (and their reverse depending on N ) of dual SCFT. Forgetting these corrections would lead to wrong conclusion about the phase transition at dual SCFT. This consideration suggests that taking account of further corrections (say R 3 , R 4 , etc) would make the phase structure even more complicated. It could be that phase transitions with N as order parameter may be observed in such framework. \nIndeed as toy example let us consider the following HD action including say, R 4 -term. For simplicity, we assume the Lagrangian density is given by the arbitrary scalar function f ( ˆ G µν , ˆ R µν ) of the metric ˆ G µν and the Ricci tenosr ˆ R µν : \nS = ∫ d d +1 x √ -ˆ Gf ( ˆ G µν , ˆ R µν ) . (44) \nThe equation of motion derived from (44) has the following form: \n0 = f 2 ˆ G µν + ∂f ∂ ˆ G µν + 1 2 ( D ρ D µ ∂f ∂ ˆ R ρν + D ρ D ν ∂f ∂ ˆ R ρµ ) -1 2 D ρ D ρ ∂f ∂ ˆ R µν -1 2 g µν D ρ D σ ∂f ∂ ˆ R ρσ . (45) \nIf we assume the metric is given by the Schwarzschild-de Sitter (2), one finds that ∂f ∂ ˆ G µν and ∂f ∂ ˆ R ρσ are proportional to ˆ G µν with constant coefficients since the Ricci tensor ˆ R µν is proportional to ˆ G µν : ˆ R µν = -d l 2 ˆ G µν . Since ∂f ∂ ˆ G µν and ∂f ∂ ˆ R ρσ are covariantly constant, Eq.(45) becomes simple: 0 = f 2 ˆ G µν + ∂f ∂ ˆ G µν . By multiplying it to G µν , one gets \n0 = d +1 2 f + G µν ∂f ∂ ˆ G µν . (46) \nEq.(46) determines the length parameter l 2 in (2). Then the metric in the form of (2) is a solution of Eq.(45). Substituting the metric (2) into the function f ( g µν , R µν ), f ( g µν , R µν ) becomes a constant f ( g µν , R µν ) = f 0 and does not depend on the mass parameter µ . Using the method similar to that in Section 2, we find the following expression of the free energy for d = 4 and k = 0 case: \nF = -V 3 8 r 2 H ( r 2 H l 2 -k 2 ) f 0 . (47) \n/negationslash \nwhile for d = 4 and k = 0 case: \nF = -V 2 η BH 8 r 2 H f 0 { ( πT BH ) 4 -( πT s ) 4 } . (48) \nEqs.(47) and (48) tell that there is a critical point (even for spherical AdS BH) when f 0 = 0. For k = 0 case, as an example, when f 0 > 0 ( f 0 < 0), the black hole is stable (unstable) if T BH > 1 η BH but global AdS becomes unstable (stable). \nTo demonstrate the role of R 4 -term let us choose for d = 4: \nf ( g µν , R µν ) = a 1 R 2 + a 2 R 4 + 1 κ 2 R -Λ . (49) \nThen Eq.(46) has the following form: \n0 = 5 2 ( a 1 R 2 + a 2 R 4 + 1 κ 2 R -Λ ) -( 2 a 1 R 2 +4 a 2 R 4 + 1 κ 2 R ) = 200 a 1 l 4 -240000 a 2 l 8 -80 κ 2 l 2 -5 2 Λ . (50) \nand f 0 looks like \nf 0 = 400 a 1 l 4 + 160000 a 2 l 8 -40 κ 2 l 2 -Λ = 320 a 1 l 4 + 256000 a 2 l 8 -8 κ 2 l 2 . (51) \nHere we have deleted Λ by using (50). Then there is a critical point when \n0 = 40 a 1 l 4 + 32000 a 2 l 8 -1 κ 2 l 2 . (52) \nThe above result indicates to further modification of phase transition structure with account of higher powers of curvatures. In fact, in AdS/CFT set-up this suggests that above confinement-deconfinement phase transitions are deeply non-perturbative effect and (some?) non-perturbative technique should be used even when it is studied using SG dual description. \nAs a final remark let us note that above study may be generalized to charged AdS BH thermodynamics where more complicated phase diagrams (see, for example,[12]) appear due to presence of extra parameter (charge). Note added: Next day this work was in hep-th, there appeared in hep-th ref.[13] where similar question has been studied for AdS BHs in GaussBonnett theory. It has been demonstrated there also that phase transitions depend on Gauss-Bonnett parameter (in our case, this is combination of HD terms coefficients).', 'Acknowledgment': 'We are grateful to M. Cvetiˇc and I. Kogan for helpful discussions. The work by SDO has been supported in part by CONACyT (CP, Ref.990356). The authors are indebted to I. Neupane since he pointed out the mistakes in the previous version.', 'References': '- [1] S.W. Hawking and D.N. Page, Commun.Math.Phys. 87 (1983) 577.\n- [2] J.M. Maldacena, Adv.Theor.Math.Phys. 2 231 (1998), hep-th/9711200; E. Witten, Adv.Theor.Math.Phys. 2 253 (1998), hep-th/9802150; S. Gubser, I. Klebanov and A. Polyakov, Phys.Lett. B428 105 (1998), hep-th/9802109; O. Aharony, S. Gubser, J. Maldacena, H. Ooguri and Y. Oz, Phys.Repts. 323 183 (2000), hep-th/9905111.\n- [3] E. Witten, Adv.Theor.Math.Phys. 2 (1998) 505, hep-th/9803131. \n- [4] D. Birmingham, Class.Quant.Grav. 16 (1999) 1197, hep-th/9808032.\n- [5] R. Myers, hep-th/9903203; L. Cappiello and W. Muck, hep-th/0107238.\n- [6] S. Surya, K. Schleich and D.M. Witt, Phys.Rev.Lett. 86 (2001) 5231, hep-th/0101134.\n- [7] S. Nojiri, S.D. Odintsov and S. Ogushi, hep-th/0105117; hepth/0108172.\n- [8] S. Nojiri and S.D. Odintsov, Phys.Rev. D 62 (2000) 064018, hepth/9911152.\n- [9] G.T. Horowitz and R.C. Myers, Phys.Rev. D59 (1999) 026005, hepth/9808079.\n- [10] A. Sen, Nucl.Phys. B475 562 (1996), hep-th/9605150; T. Banks, M.R. Douglas and N. Seiberg, Phys.Lett. B387 278 (1996), hep-th/9605199; O. Aharony, C. Sonnenstein, S. Yankielowicz and S. Theisen, Nucl.Phys. B493 177 (1997), hep-th/9611222; M.R. Douglas, D.A. Lowe, J.H. Schwarz, Phys.Lett. B394 297 (1997), hep-th/9612062.\n- [11] A. Fayyazuddin and M. Spalinski, Nucl.Phys. B535 219 (1998), hepth/9805096; O. Aharony, A. Fayyazuddin and J.M. Maldacena, JHEP 9807 013 (1998), hep-th/9806159.\n- [12] M. Cvetic and S. Gubser, JHEP 9904 024 (1999), hep-th/9902195.\n- [13] R.-G. Cai, hep-th/0109133.'}
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2004CQGra..21.4511C
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How classical are TeV-scale black holes?
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2004-01-01
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We show that the Hawking temperature and the entropy of black holes are subject to corrections from two sources: the generalized uncertainty principle and thermal fluctuations. Both effects increase the temperature and decrease the entropy, resulting in faster decay and 'less classical' black holes. We discuss the implications of these results for TeV-scale black holes that are expected to be produced at future colliders.
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https://arxiv.org/pdf/hep-th/0404050.pdf
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{"Marco Cavagli'a": 'Dept. of Physics and Astronomy, The University of Mississippi, PO Box 1848, University, Mississippi 38677-1848, USA ∗', 'Saurya Das': "Dept. of Physics, University of Lethbridge, 4401 University Drive, Lethbridge, Alberta T1K 3M4, CANADA † \nWe show that the Hawking temperature and the entropy of black holes are subject to corrections from two sources: the generalized uncertainty principle and thermal fluctuations. Both effects increase the temperature and decrease the entropy, resulting in faster decay and 'less classical' black holes. We discuss the implications of these results for TeV-scale black holes that are expected to be produced at future colliders.", 'I. INTRODUCTION': "The possibility of the existence of large extra dimensions has recently opened up new and exciting avenues of research in quantum gravity [1, 2]. In particular, a host of interesting work is being done on different aspects of low-energy scale quantum gravity phenomenology . One of the most significant sub-fields is the study of black hole (BH) [3] and brane [4] production at particle colliders, such as the (Very) Large Hadron Collider [V(LHC)] [5] and the muon collider [6], as well as in ultrahigh energy cosmic ray (UHECR) airshowers [7]. (For recent reviews, see Refs. [8].) Several scenarios predict the fundamental Planck scale to be of order of the TeV. The simplest model postulates a number n of toroidally compactified extra dimensions with length ranging from a few microns ( n = 2) to a Fermi ( n = 7) [1]. Extra dimensions of infinite size and non-trivial 'warp-factor' may also lead to similar predictions [2]. In either case, particle collisions with center-of-mass (c.m.) energy above the fundamental Planck scale, and impact parameter smaller than the horizon radius corresponding to that energy, should produce BHs and branes [9]. (For criticisms, however, see Refs. [10].) Since the c.m. energy of next-generation particle colliders and UHECR primaries is as high as tens or hundreds of TeV, BH and brane production is likely to be observed. For this kind of event, the initial mass of the gravitational object is expected to be of the order of a few Planck masses. \nNewly formed BHs first lose hair associated with multipole and angular momenta, then approach classically stable Schwarzschild solutions, and finally evaporate via Hawking radiation [11]. Decay time and entropy completely determine the observables of the process. BH formation and decay can be described semiclassically, provided that the entropy is sufficiently large. The timescale for the complete decay of a BH to up to its supposed final Planck-sized remnant is expected to be of order of the TeV -1 . \nBH thermodynamic quantities depend on the Hawking temperature T H via the usual thermodynamic relations (Stefan-Boltzmann law). The Hawking temperature undergoes corrections from many sources, and these corrections are particularly relevant for BHs with mass of the order of the Planck mass. Therefore, the study of TeV-scale BHs in UHECR and particle colliders requires a careful investigation of how temperature corrections affect BH thermodynamics. In this article, we concentrate on the corrections due to the generalized uncertainty principle (GUP) and thermal fluctuations of thermodynamic systems. These corrections are not tied down to any specific model of quantum gravity; the GUP can be derived using arguments from string theory [12] as well as other approaches to quantum gravity [13, 14]. Similarly, corrections from thermal effects do not depend on the underlying quantum gravity theory since thermal fluctuations are present in any system. This generality provides in fact a strong motivation in studying GUP and thermal fluctuation effects. \nWe show below that the BH decay rate is increased by GUP and thermal fluctuation corrections, resulting in shorter decay times. Thus the BHs may not behave like well-defined resonances. We also show that a diminished entropy leads to smaller particle emission during the evaporation phase and 'less classical' BHs. The paper is organised as follows. In the next section, we review the connection between the uncertainty principle and Hawking radiation \n[15, 16]. In section III, we derive corrections to the Hawking decay rate, entropy, and multiplicity due to the GUP. We show that for BHs with mass M = 5 - 10 M Pl , the decay time and the entropy (or multiplicity) dramatically decrease if the GUP parameter is nonvanishing. In section IV we review the corrections to thermodynamic quantities due to thermal fluctuations. We apply these results to BHs in section V, where we show that BH decay time and entropy also decrease compared to their semiclassical value. We conclude with a summary of our results and a brief discussion of open questions in section VI.", 'II. UNCERTAINTY PRINCIPLE AND HAWKING RADIATION': "In this section we review and generalize to d dimensions the derivation of the Hawking radiation of Adler et al. [15]. A d -dimensional spherically symmetric BH of mass M (to which the collider BHs will settle into before radiating) is described by the metric \nds 2 = -( 1 -16 πG d M ( d -2)Ω d -2 c 2 r d -3 ) c 2 dt 2 + ( 1 -16 πG d M ( d -2)Ω d -2 c 2 r d -3 ) -1 dr 2 + r 2 d Ω 2 d -2 , (1) \nwhere Ω d -2 is the metric of the unit S d -2 and G d is the d -dimensional Newton's constant. Since the Hawking radiation is a quantum process, the emitted quanta must satisfy the Heisenberg uncertainty principle \n∆ x i ∆ p j > ∼ ¯ hδ ij , (2) \nwhere x i and p j , i, j = 1 . . . d -1, are the spatial coordinates and momenta, respectively. By modelling a BH as a ( d -1)-dimensional cube of size equal to twice its Schwarzschild radius r s , the uncertainty in the position of a Hawking particle at the emission is \n∆ x ≈ 2 r s = 2 λ d [ G d M c 2 ] 1 / ( d -3) , (3) \nwhere λ d = [16 π/ (( d -2)Ω d -2 )] 1 / ( d -3) . Using Eq. (2), the uncertainty in the energy of the emitted particle is \n∆ E ≈ c ∆ p ≈ M Pl c 2 2 λ d m -1 / ( d -3) , (4) \nwhere m = M/M Pl is the mass in Planck units and M Pl = [¯ h d -3 /c d -5 G d ] 1 / ( d -2) is the d -dimensional Planck mass. ∆ E can be identified with the characteristic temperature of the BH emission, i.e. the Hawking temperature. Setting the constant of proportionality to ( d -3) / 2 π we get \nT H = d -3 4 πλ d M Pl c 2 m -1 / ( d -3) . (5) \nThe energy radiated per unit time is governed by the Stefan-Boltzmann law. The surface gravity is constant over the horizon. Thus the Hawking temperature of the higher-dimensional BH and the temperature of the induced BH on the brane are identical. The BH temperature T H can be used in the calculation of the emission rate. Neglecting thermal emission in the bulk, and assuming that the brane has D spacetime dimensions (we will substitute D = 4 at the end, since BHs are supposed to radiate mainly on the brane [11]), the emission rate for a massless scalar particle on the brane is \ndM dt = -cσ D A D T D , (6) \nwhere A D = Ω D2 r D2 c is the horizon area of the induced BH with radius r c = [( d -1) / 2] 1 / ( d -3) [( d -1) / ( d -3)] 1 / 2 r s , and \nσ D = Ω D3 Γ( D ) ζ ( D ) ( D2)(2 π ¯ hc ) D1 ≡ ¯ σ D (¯ hc ) D1 (7) \nis the Stefan-Boltzmann constant in D -spacetime dimensions. If the BH evaporates into different particle species on the brane, the Stefan-Boltzmann constant has to be multiplied by the factor \n∑ i c i ( D )Γ s i ( D ) f i ( D ) , (8) \nand \nC 0 = -4 πλ d m ( d -2) / ( d -3) , (13) \nrespectively. The statistical total number of quanta emitted during the evaporation is proportional to the initial entropy of the BH. The exact relation is \nThe flavor multiplicity is \nN = S 0 ζ ( D1) ( D1) ζ ( D ) ∑ i c i ( D )Γ s i ( D ) f i ( D1) ∑ j c j ( D )Γ s j ( D ) f j ( D ) . (14) \nN i = N c i ( D )Γ s i ( D ) f i ( D1) j c j ( D )Γ s j ( D ) f j ( D1) . (15) \nEquation (15) gives the statistical number of particles per species produced during the evaporation process. \n∑", 'III. CORRECTIONS TO BH THERMODYNAMICS FROM THE GENERALIZED UNCERTAINTY PRINCIPLE': "We now determine the corrections to the above results due to the GUP. The general form of the GUP is \n∆ x i ≥ ¯ h ∆ p i + α 2 /lscript 2 Pl ∆ p i ¯ h , (16) \nwhere /lscript Pl = (¯ hG d /c 3 ) 1 / ( d -2) is the Planck length and α is a dimensionless constant of order one. There are many derivations of the GUP, some heuristic and some more rigorous. Equation (16) can be derived in the context of string theory [12], non-commutative quantum mechanics [13], and from minimum length [17] considerations [14]. The exact value of α depends on the specific model. The second term in r.h.s. of Eq. (16) becomes effective when momentum \nwhere the sum is over all particle flavors, c i are the degrees of freedom of the species i , Γ s i are the greybody factors for spin s i and f i = 1 ( f i = 1 -2 1 -D ) for bosons (fermions). (We neglect the energy dependence of the greybody factors. See, e.g., Refs. [11].) Expressing Eq. (6) in terms of m , we obtain \ndm dt = -µ t Pl m -2 / ( d -3) , (9) \nwhere t Pl = (¯ hG d /c d +1 ) 1 / ( d -2) is the Planck time, and \nµ = ( r c r s ) D2 ( d -3 4 π ) D ¯ σ D Ω D2 λ 2 d . (10) \nIntegration over t yields the decay time \n≈ \nτ 0 = µ -1 ( d -3 d -1 ) m ( d -1) / ( d -3) i t Pl , (11) \nwhere m i ≡ M i /M Pl , and M i is the initial BH mass. The decay time τ 0 is finite. Equation (9) implies that the end stage of Hawking evaporation is catastrophic, with infinite radiation rate and infinite temperature. However, a heuristic argument suggests that the final temperature and radiation rate are finite. At the last stage of evaporation, ∆ E in Eq. (4) must be of the order of the BH mass, and ∆ E = ∆ Mc 2 ≈ M end c 2 . This implies a minimum BH mass M end M Pl , a maximum Hawking temperature T max = O ( M Pl ), and a smaller decay time. \nO The thermodynamic properties of the BH can be computed via the usual thermodynamic relations. The entropy and the BH specific heat are \nS 0 = 4 πλ d d -2 m ( d -2) / ( d -3) = d -3 d -2 Mc 2 T H , (12) \nand length scales are of the order of Planck mass and of the Planck length, respectively. This limit is usually called 'quantum regime'. Inverting Eq. (16), we obtain \n∆ x i 2 α 2 /lscript 2 Pl [ 1 -√ 1 -4 α 2 /lscript 2 Pl ∆ x 2 i ] ≤ ∆ p i ¯ h ≤ ∆ x i 2 α 2 /lscript 2 Pl [ 1 + √ 1 -4 α 2 /lscript 2 Pl ∆ x 2 i ] . (17) \nThe left-inequality gives the correct /lscript Pl / ∆ x i → 0 limit and will be considered henceforth. The GUP implies the existence of a minimum length L min ≈ ∆ x = 2 α/lscript Pl . The string regime and the classical regimes are recovered by setting ∆ x i 2 α/lscript Pl and ∆ x i /greatermuch /lscript Pl in Eq. (16), respectively. \n/greatermuch BHs with horizon radius smaller than L min do not exist. Therefore, the minimum length implies the existence of a minimum BH mass \nM min = d -2 8Γ d -1 2 ) ( α √ π ) d -3 M Pl . (18) \n≈ \n) The minimum BH mass is a rapidly increasing function of the unknown parameter α for d ≥ 6; a value of α larger than unity may lead to a minimum BH mass M min M Pl . \nThe corrections to the BH thermodynamic quantities can be calculated by repeating the argument of the previous section. Setting ∆ x = 2 r s the GUP-corrected Hawking temperature is \n( /greatermuch \nT ' H = ( d -3) λ d 2 πα 2 m 1 / ( d -3) [ 1 -√ 1 -α 2 λ 2 d m 2 / ( d -3) ] M Pl c 2 . (19) \nEquation (19) may be Taylor expanded around α = 0: \nT ' H = ( d -3) 4 πλ d m -1 / ( d -3) [ 1 + α 2 4 λ 2 d m 2 / ( d -3) + · · · ] M Pl c 2 . (20) \nThe GUP-corrected Hawking temperature is higher than the semiclassical Hawking temperature T H of Eq. (5). The first-order correction is \n∆ T GUP ≡ T ' H -T H = d -3 16 π α 2 λ 3 d m 3 / ( d -3) M Pl c 2 . (21) \nFrom the first law of BH thermodynamics the first-order correction to the BH entropy is \n∆ S GUP = -πα 2 m ( d -4) / ( d -3) ( d -4) λ d d > 4 , = -πα 2 2 ln( m ) d = 4 . (22) \nThis follows from the exact expression: \nS GUP = 2 πλ d ( α λ d ) d -2 I (1 , d -4 , λ d m 1 / ( d -3) /α ) , (23) \nwhere \nI ( p, q, x ) = ∫ x 1 dzz q ( z + √ z 2 -1 ) p . (24) \nFrom Eq. (22) and Eq. (23) it follows that the GUP-corrected entropy is smaller than the semiclassical BekensteinHawking. The GUP-corrected Stefan-Boltzmann law is \ndm dt = -2 D µ t Pl m -2 / ( d -3) [ 1 + √ 1 -α 2 λ 2 d m 2 / ( d -3) ] -D . (25) \nTaylor expanding Eq. (25) we have \ndm dt = -µ t Pl m 2 / ( d -3) [ 1 + α 2 D 4 λ 2 d m 2 / ( d -3) + · · · ] . (26) \nThe relative GUP first-order correction to the Stefan-Boltzmann law is positive: \n∆ ( dm dt ) / ( dm dt ) 0 = α 2 D 4 λ 2 d m -2 / ( d -3) , (27) \nwhere ( dm/dt ) 0 is defined in Eq. (9). The Hawking evaporation ends at m min = M min /M Pl = ( α/λ d ) d -3 , where the emission rate becomes imaginary. The emission rate is finite at the end: \n( dm dt ) m min = -2 D µ t Pl ( λ d α ) 2 . (28) \nThis means that the end-point of Hawking radiation is not catastrophic. Since the final emission rate is finite, it might be argued that once the final stage has been reached, the BH evaporates completely by emitting a hard Planck-mass quantum in a finite time O ( t Pl ). However, the BH specific heat \nC ≡ T ∂S ∂T = -2 πλ d m ( d -2) / ( d -3) √ 1 -α 2 λ 2 d m 2 / ( d -3) · ( 1 + √ 1 -α 2 λ 2 d m 2 / ( d -3) ) (29) \nvanishes at the endpoint. Therefore, the BH cannot exchange heat with the surrounding space. The endpoint of Hawking evaporation in the GUP scenario is characterized by a Planck-size remnant with maximum temperature \n∣ \nT max = 2 T 0 ∣ ∣ M = M min . (30) \n∣ The GUP prevents BHs from evaporating completely, just like the standard uncertainty principle prevents the hydrogen atom from collapsing. The existence of BH remnants as a consequence of the GUP was pointed in Refs. [15] in the context of primordial BHs in cosmology [18]. BH remnants have also been predicted in string and quantum gravity models [19] and could play an important role in cosmology. (See, e.g., Refs. [20].) \nThe GUP implies a faster BH decay. The first-order decay time is \nτ 1 = µ -1 ( d -3 d -1 ) { [ m ( d -1) / ( d -3) i -D ( d -1) α 2 4( d -3) λ 2 d m i ] -[ 1 -D ( d -1) 4( d -3) ]( α λ d ) d -1 } t Pl . (31) \nIf the initial mass far exceeds the Planck mass, i.e. m i /greatermuch 1, the last term inside the curly brackets can be ignored. Using Eq. (11), we find \n∆ τ 1 τ 0 ≡ τ 1 -τ 0 τ 0 = -D ( d -1) α 2 4( d -3) λ 2 d m -2 / ( d -3) i . (32) \nThe GUP-corrected decay time is smaller than the semiclassical decay time. The GUP-corrected multiplicity is obtained from Eq. (14) with S 0 = S GUP . Table I shows the GUP-corrected parameters for two typical BHs produced at particle colliders or in UHECRs, with initial mass equal to 5 M Pl and 10 M Pl . The first row gives the standard Hawking parameters. The GUP effects on the thermodynamic parameters increase as the minimum BH mass becomes larger. It is interesting to note that decay time, entropy, and multiplicity are drastically reduced when α approaches unity. In the limiting case of a BH with initial mass 5 M Pl and GUP parameter α = 1, the minimum BH mass coincides essentially with the initial BH mass, and the BH does not evaporate. In contrast to the standard theory, the GUP-corrected entropy shows that a BH with a mass five times the fundamental Planck scale is not a classical object; quantum effects become manifest at an earlier stage of the BH evaporation phase than was predicted by the semiclassical Hawking analysis [8]. Therefore, GUP corrections have important consequences on the BH phenomenology in particle colliders and in UHECR airshowers.", 'IV. ENTROPY CORRECTIONS DUE TO THERMODYNAMIC FLUCTUATIONS': "Thermodynamic systems (including BHs) undergo small thermal fluctuations from equilibrium which affect thermodynamic quantities. In this section we calculate the corrections to entropy and Hawking temperature. We have seen that the GUP corrections affect the Hawking temperature of the BH while the BH energy remains constant. Therefore, we consider fluctuations around the equilibrium temperature instead of the equilibrium energy. This leads \nm = 5m = 10 \n| | | | Minimum mass Initial Temperature Final Temperature | Entropy | Decay time | Multiplicity |\n|-----------|------|-------------|------------------------------------------------------|-------------------------|--------------|----------------|\n| α = 0 | - | .553 | ∞ | .334 | 7.92 | 3 |\n| α = 0 . 5 | .037 | .591 (+7%) | 2.23 | .233 (-30%) | 7.18 (-9%) | 2 (-33%) |\n| α = 1 . 0 | 4.73 | .981 (+77%) | 1.11 | .002 (-99%) .269 (-97%) | | 0 (-100%) | \nTABLE I: GUP-corrected thermodynamic quantities for two ten-dimensional BHs with mass m = 5 and 10 (in fundamental units). The values in brackets give the percentage deviation from standard Hawking quantities. \n| | | | Minimum mass Initial Temperature Final Temperature | | Decay time | Entropy Multiplicity |\n|-----------|------|-------------|------------------------------------------------------|-------------|-------------------------|------------------------|\n| α = 0 | - | .500 | ∞ | .814 | 17.5 | 6 |\n| α = 0 . 5 | .037 | .529 (+6%) | 2.23 | .610 (-25%) | 16.2 (-7%) | 5 (-17%) |\n| α = 1 . 0 | 4.73 | .696 (+39%) | 1.11 | | .100 (-88%) 6.66 (-62%) | 2 (-66%) | \nto a decrease in the black hole entropy. Note that the Bekenstein-Hawking entropy is identified with the canonical entropy of the system 1 . Let us consider a canonical ensemble with partition function [22, 23]: \nZ ( β ) = ∫ ∞ 0 ρ ( E ) e -βE dE , (33) \nwhere β = 1 /T is the inverse of the temperature. The density of states can be obtained from Eq. (33) by the inverse Laplace transform (at fixed E ) \nρ ( E ) = 1 2 πi ∫ c + i ∞ c -i ∞ Z ( β ) e βE dβ = 1 2 πi ∫ c + i ∞ c -i ∞ e S ( β ) dβ , (34) \nwhere \nS ( β ) = ln Z ( β ) + βE . (35) \nTo evaluate the complex integral in Eq. (34) by the method of steepest descent, we expand S ( β ) around the saddle point β 0 (= 1 /T 0 ), where T 0 is the equilibrium temperature. Also using the fact that S ' 0 = ( ∂S ( β ) /∂β ) β = β 0 , we get [22, 24]: \nS = S 0 + 1 2 ( β -β 0 ) 2 S '' 0 + · · · , (36) \nwhere S 0 = S ( β 0 ) and S '' 0 = ( ∂ 2 S ( β ) /∂β 2 ) β = β 0 . Substituting Eq. (36) in Eq. (34), the density of states is \nρ ( E ) = e S 0 2 πi ∫ c + i ∞ c -i ∞ e ( β -β 0 ) 2 S '' 0 / 2 dβ = e S 0 2 πS '' 0 . (37) \nThe corrected entropy is given by the logarithm of the density of states ρ ( E ): \n√ \nS = ln ρ ( E ) = S 0 -1 2 ln S '' 0 + (higher order terms) . (38) \nFrom E = -( ∂ ln Z ( β ) /∂β ) β 0 and the definition of specific heat, C = ( ∂E/∂T ) β 0 , S '' ( β ) can be written as \nS '' ( β ) = 1 Z ( ∂ 2 Z ( β ) ∂β 2 ) -1 Z 2 ( ∂Z ∂β ) 2 = 〈 E 2 〉 - 〈 E 〉 2 = C T 2 . (39) \nEquation (39) shows that a non-vanishing S '' 0 is a consequence of thermal fluctuations. Substituting Eq. (39) in Eq. (38), we obtain \nS = ln ρ = S 0 -1 2 ln ( C T 2 ) + · · · . (40) \nThe above formula applies to any thermodynamic system in equilibrium. In particular, when applied to BHs, T is the Hawking temperature. For example, for a non-rotating three-dimensional BTZ BH, both T H and C are proportional to the BH entropy S 0 . In this case, Eq. (40) reads \nS = ln ρ = S 0 -3 2 ln S 0 + · · · . (41) \nSimilarly, for an AdS-Schwarzschild BH in d -dimensions, it can be shown that (see Ref. [22]) \nS = S 0 -d 2( d -2) ln S 0 + · · · . (42) \nAlthough Eq. (40) is not applicable directly to the Schwarzschild BHs because of its negative specific heat, the entropy corrections can be shown to be logarithmic by either assuming a small cosmological constant, or by putting the BH into a finite box. The result is: \nS Thermo = S 0 -k ln S 0 , (43) \nwhere k is a positive constant of order unity. We will apply these results to brane world BHs in the next section.", 'V. CORRECTIONS TO BH DECAY RATE DUE TO THERMODYNAMIC FLUCTUATIONS': "The corrected Hawking temperature is obtained from the first law of BH thermodynamics. The result is: \nT '' H = ( d -3) 4 πλ d m -1 / ( d -3) [ 1 + k ( d -2) 4 πλ d m -( d -2) / ( d -3) + . . . ] M Pl c 2 . (44) \nEquation (44) gives the first-order correction to the Hawking temperature \n∆ T Thermo ≡ T '' H -T H = k ( d -2)( d -3) 16 π 2 λ 2 d m -( d -1) / ( d -3) M Pl c 2 . (45) \nThe correction to the BH entropy follows from Eq. (43): \n∆ S Thermo = -k ln S 0 . (46) \nThe multiplicity is also reduced, and is now given by Eq. (14) with S 0 → S Thermo . Similarly to the GUP corrections, thermodynamic fluctuations reduce the number of degrees of freedom of the BH. Taking the ratio of Eq. (21) and Eq. (45), we find \n∆ T GUP ∆ T thermo = [ πα 2 λ d k ( d -2) ] m ( d -4) / ( d -3) . (47) \nThe GUP and the thermal fluctuation corrections are of the same order for d = 4. In d > 4, the situation is more complicated. If m /greatermuch 1, the GUP corrections far exceed the corrections due to thermodynamic fluctuations. However, \nwhen m ≈ 1, i.e. near the end stage of evaporation, the rates are comparable. The first-order corrected specific heat is: \nC = C 0 [ 1 -k ( d -1)( d -2) 4 πλ d m -( d -2) / ( d -3) ] , (48) \nwhere C 0 is given by Eq.(13). The first-order specific heat vanishes for the non-zero value of m \nm 0 = [ k ( d -1)( d -2) 4 πλ d ] ( d -3) / ( d -2) . (49) \nThis suggests that the BH becomes thermodynamically stable when the BH mass reaches m 0 . However, this conclusion should be interpreted with care; the thermodynamic fluctuations of a BH with mass m ∼ m 0 are large and the firstorder approximation (43) breaks down. The Stefan-Boltzmann law is obtained from Eq. (44): \ndm dt = -µ t Pl m 2 / ( d -3) [ 1 + k D ( d -2) 4 πλ d m -( d -2) / ( d -3) + . . . ] . (50) \nThe first-order correction to the Stefan-Boltzmann law due to thermal fluctuations is positive: \n∆ ( dm dt ) / ( dm dt ) 0 = k D ( d -2) 4 πλ d m -( d -2) / ( d -3) . (51) \nIntegrating Eq. (50) we obtain the expression for the time decay \nτ 2 = µ -1 ( d -3 d -1 ) m ( d -1) / ( d -3) i [ 1 -k D ( d -1)( d -2) 4 πλ d m -( d -2) / ( d -3) i + . . . ] t Pl . (52) \nSimilarly to the GUP case, it can be easily verified that the BH takes less time to decay when fluctuation corrections are taken into account: \n∆ τ 2 τ 0 ≡ τ 2 -τ 0 τ 0 = -k D ( d -1)( d -2) 4 πλ d m -( d -2) / ( d -3) i . (53) \nThe decay time is dramatically reduced by thermal fluctuations. Equation (52) implies a relation between the thermal fluctuation threshold m 0 and the BH initial mass. By imposing τ 2 > 0 we have \nm i > D ( d -3) / ( d -2) m 0 ≡ m ' 0 . (54) \nBH with initial mass smaller than m ' 0 form in a regime where thermal fluctuations dominate. A careful study of their thermodynamic properties should include higher-order terms in the expansion (36). Since the thermodynamic fluctuations prevent the analytical evaluation of the integral in Eq. (37), numerical techniques may have to be used to get accurate estimates of the thermodynamic quantities. In any case, our analysis shows that semiclassical Hawking theory is inadequate for the the description of these black holes. Note that for k = 0 . 5 (1), m ' 0 = 10 . 25 (18 . 8). Thermal fluctuations cannot be neglected in particle collider or UHECR BH events.", 'VI. DISCUSSION': 'We have examined the effects of the GUP and small thermal fluctuations on temperature, decay rate, and entropy of microscopic BHs. Although these effects are small under most circumstances, they can be significant in BH production at the TeV scale, where the BH mass is expected to be of the order of the fundamental Planck mass. The GUP and the thermal fluctuation corrections increase the BH temperature, and decrease decay time, entropy, and multiplicity of the evaporation phase: Quantum BHs are hotter, shorter-lived, and tend to evaporate less than classical BHs. The results described here are applicable to the ADD as well as the RS brane world scenarios [1, 2]. \nUnder the most favorable circumstances, the semiclassical cross section for BH formation at the TeV scale reaches hundreds of pb for proton-proton collision at the LHC, and millions of pb for neutrino-nucleon collision in the atmosphere. According to the semiclassical scenario, the Hawking evaporation mechanism will allow detection of microscopic BHs with mass of the order of a few Planck masses in next-generation particle colliders and UHECR detectors. However, our results seem to suggest that the semiclassical description could be inaccurate for this kind of events. \nFirstly, a shorter lifetime implies that the quantum BH may not behave like a well-defined resonance. Secondly, the classical picture breaks down if the degrees of freedom of the BH, i.e. its entropy, is small. The semiclassical entropy has been widely used in the literature to measure the validity of the semiclassical approximation. When the entropy is sufficiently large, the BH can be considered as a classical object [8]. If this is the case, i) the BH evaporation phase is described by a thermal spectrum with Hawking temperature; ii) the BH cross section for elementary particles is well approximated by the geometrical cross section; and iii) the total cross section for composite particles (e.g. nucleons) is obtained by integrating the geometrical cross section over the structure functions. A BH with mass equal to few Planck masses is usually assumed to have entropy above the threshold of validity of the classical description. However, the reduction in entropy by GUP and thermal fluctuation effects increases this threshold. Therefore, it may not be appropriate to treat these BHs as classical objects. (See also Ref. [10]). Thirdly, GUP physics implies the existence of a minimum BH mass given by Eq. (18). The existence of a minimum mass increases the lower cutoff for BH formation, thus reducing the rate of BH events. Even if a detectable signal is produced during the BH decay phase, it could prove very difficult to distinguish it from the background. Finally, GUP and thermodynamic fluctuations further decrease the already-weak lower bounds on the fundamental Planck scale that follow from the nonobservation of BH events up to date [25]. \nLet us conclude with a list of open problems and possible future research topics. It would be interesting to compute the effects of small residual charge Q and angular momentum J on the above results. In presence of nonzero charge and angular momentum, the corrections to the thermodynamic quantities are expected to depend on J and Q . However, the precise form of the corrections is yet to be determined. It would also be interesting to examine the relation of the GUP and thermodynamic corrections to other corrections that have been predicted for TeV-scale BHs (see e.g. Refs. [8], [10]). A detailed single-event analysis of particle collider and UHECR events when GUP and thermodynamic corrections are present, is also missing. We plan to report on these and other issues elsewhere.', 'Acknowledgements:': "We thank R. Maartens and M. Maggiore for interesting discussions. S.D. would like to thank R. K. Bhaduri, A. Dasgupta, J. Gegenberg, V. Husain, P. Majumdar and M. Walton for interesting discussions and comments. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada and funds of the University of Lethbridge. \n- [1] N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B 429 , 263 (1998) [arXiv:hep-ph/9803315]; I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B 436 , 257 (1998) [arXiv:hep-ph/9804398]; I. 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2005ApJ...625..699M
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The Murmur of the Sleeping Black Hole: Detection of Nuclear Ultraviolet Variability in LINER Galaxies
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2005-01-01
| 7 | 0.47 | 158 |
['galaxies active', 'galaxies nuclei', 'galaxies seyfert', 'galaxies starburst', 'galaxies quasars', 'astronomy uv', 'astrophysics']
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LINER nuclei, which are present in many nearby galactic bulges, may be the manifestation of low-rate or low-radiative-efficiency accretion onto supermassive central black holes. However, it has been unclear whether the compact UV nuclear sources present in many LINERs are clusters of massive stars, rather than being directly related to the accretion process. We have used the Hubble Space Telescope to monitor the UV variability of a sample of 17 galaxies with LINER nuclei and compact nuclear UV sources. Fifteen of the 17 galaxies were observed more than once, with two to five epochs per galaxy, spanning up to a year. We detect significant variability in most of the sample, with peak-to-peak amplitudes from a few percent to 50%. In most cases, correlated variations are seen in two independent bands (F250W and F330W). Comparison to previous UV measurements indicates, for many objects, long-term variations by factors of a few over decade timescales. Variability is detected in LINERs with and without detected compact radio cores, in LINERs that have broad Hα wings detected in their optical spectra (``LINER 1s''), and in those that do not (``LINER 2s''). This variability demonstrates the existence of a nonstellar component in the UV continuum of all types and sets a lower limit to the luminosity of this component. Interestingly, all the LINERs that have detected radio cores have variable UV nuclei, as one would expect from bona fide active galactic nuclei. We note a trend in the UV color (F250W/F330W) with spectral type-LINER 1s tend to be bluer than LINER 2s. This trend may indicate a link between the shape of the nonstellar continuum and the presence or the visibility of a broad-line region. In one target, the poststarburst galaxy NGC 4736, we detect variability in a previously noted UV source that is offset by 2.5" (~60 pc in projection) from the nucleus. This may be the nearest example of a binary active nucleus and of the process leading to black hole merging. <P />Based on observations with the Hubble Space Telescope, which is operated by AURA, Inc., under NASA contract NAS 5-26555.
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https://arxiv.org/pdf/astro-ph/0502347.pdf
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{'THE MURMUR OF THE SLEEPING BLACK HOLE: DETECTION OF NUCLEAR ULTRAVIOLET VARIABILITY IN LINER GALAXIES 1': 'Dan Maoz, 2 Neil M. Nagar, 3,4 Heino Falcke, 5,6 and Andrew S. Wilson, 7 Received 2004 December 22; accepted 2005 February 16', 'ABSTRACT': "LINER nuclei, which are present in many nearby galactic bulges, may be the manifestation of lowrate or low-radiative-efficiency accretion onto supermassive central black holes. However, it has been unclear whether the compact ultraviolet (UV) nuclear sources present in many LINERs are clusters of massive stars, rather than being directly related to the accretion process. We have used the Hubble Space Telescope to monitor the UV variability of a sample of 17 galaxies with LINER nuclei and compact nuclear UV sources. Fifteen of the 17 galaxies were observed more than once, with two to five epochs per galaxy, spanning up to a year. We detect significant variability in most of the sample, with peak-to-peak amplitudes from a few percent to 50%. In most cases, correlated variations are seen in two independent bands (F250W and F330W). Comparison to previous UV measurements indicates, for many objects, long-term variations by factors of a few over decade timescales. Variability is detected in LINERs with and without detected compact radio cores, in LINERs that have broad H α wings detected in their optical spectra ('LINER 1's'), and in those that do not ('LINER 2s'). This variability demonstrates the existence of a non-stellar component in the UV continuum of all types, and sets a lower limit to the luminosity of this component. Interestingly, all the LINERs that have detected radio cores have variable UV nuclei, as one would expect from bona fide AGNs. We note a trend in the UV color (F250W/F330W) with spectral type - LINER 1s tend to be bluer than LINER 2s. This trend may indicate a link between the shape of the nonstellar continuum and the presence or the visibility of a broad-line region. In one target, the post-starburst galaxy NGC 4736, we detect variability in a previously noted UV source that is offset by 2 . '' 5 ( ∼ 60 pc in projection) from the nucleus. This may be the nearest example of a binary active nucleus, and of the process leading to black hole merging. \nSubject headings: galaxies: active - galaxies: nuclei - galaxies: Seyfert - galaxies: starburst quasars: general - Ultraviolet: galaxies -", '1. INTRODUCTION': "Low-ionization nuclear emission-line regions (LINERs) are detected in the nuclei of a large fraction of bright nearby galaxies (Ho, Filippenko, & Sargent 1997a; Kauffmann et al. 2003). Since their definition as a class by Heckman (1980), they have elicited debate as to their nature and relation, if any, to active galactic nuclei (AGNs). Although the luminosities of most LINERs are unimpressive compared to 'classical' AGNs, a variety of observables point to similarities and continuities between AGNs and at least some LINERs. To list some of these, at least 10% of LINERs show weak, broad, Seyfert-1like H α wings in their spectra (Ho et al. 1997b), and Keck spectropolarimetry of several objects has revealed 'hidden BLRs' (Barth, Filippenko, & Moran 1999a,b), similar to those seen in some Seyfert 2s (Antonucci & Miller 1985; Tran 1995). Hubble Space Telescope (HST) \n- 1 Based on observations with the Hubble Space Telescope which is operated by AURA, Inc., under NASA contract NAS 5-26555.\n- 2 School of Physics and Astronomy, Tel-Aviv University, TelAviv 69978, Israel; [email protected]\n- 3 Kapteyn Institute, Postbus 800, 9700AV, Groningen, The Netherlands\n- 4 Departamento de F ' isica, Astronomy Group, Universidad de Concepci'on, Casilla 160-C, Concepci'on, Chile\n- 5 ASTRON, P.O. Box 2, 7990 AA, Dwingeloo, The Netherlands 6 Department of Astrophysics, Radboud University, Postbus 9010, 6500 GL Nijmegen, The Netherlands\n- 7 Astronomy Department, University of Maryland, College Park, MD 20742 \nimaging shows that some 25% of LINERs have compact, often unresolved (i.e., /lessorsimilar few pc), bright UV sources in their nuclei (Maoz et al. 1995; Barth et al. 1998). Optical imaging with WFPC2 on HST of 14 LINERs (Pogge et al. 2000) suggests that all nearby LINERs (including the 75% that are 'UV-dark', i.e., those that do not reveal a nuclear UV source at HST sensitivity, ∼ 10 -17 erg cm -2 s -1 ˚ A -1 ) likely have such a nuclear UV source, but that it is often obscured by circumnuclear dust. \nIn the radio, at Very Large Array (VLA) resolution (0 . '' 1 ≈ tens of pc), about half of LINERs display unresolved radio cores at 2 cm and 3.6 cm (Nagar et al. 2000, 2002). At 6 cm, with Very Long Baseline Interferometer (VLBI) resolution ( ∼ 1 pc), these cores remain unresolved, strongly arguing for the presence of an AGN (Falcke & Biermann 1999; Falcke et al. 2000). The radio core fluxes have been found to be variable by factors of up to a few in about half of the ∼ 10 LINERs observed multiple times over 3 years (Nagar et al. 2002). A radio survey for 1.3 cm water megamaser emission, an indicator of dense circumnuclear molecular gas, detected LINER nuclei at the same rate as type-2 Seyfert nuclei (Braatz et al. 1997). Such megamaser emission is seen only in AGNs. Some LINERs have indications of a Seyfert-like ionization cone oriented along their radio axis (Pogge et al. 2000). \nAt X-ray energies, Rosat HRI images showed compact ( < 5 '' ) soft X-ray emission in 70% of LINERs \nand Seyferts (Roberts & Warwick 2000) which, when observed with ASCA, were found to have a nonthermal 2-10 kev spectrum (e.g., Terashima, Ho, & Ptak 2000). Arcsecond-resolution Chandra observations by Terashima & Wilson (2003) of 11 LINERs, each of which was preselected to have a radio core, revealed an X-ray nucleus in all but one case, and the nuclei were generally (but not always) unresolved. \nThe super-massive black holes in the nuclei of most normal galaxies (e.g., Tremaine et al. 2002; Ferrarese & Merritt 2000), many of which are also LINERs, could be the remnants of ancient quasars/Seyferts, now accreting at a low rate and/or radiating inefficiently (e.g., Reynolds et al. 1996), and producing these multiwavelength signatures. If LINERs represent the low-luminosity end of the AGN phenomenon, then they are the nearest and most common examples, and their study is germane to understanding AGN demographics and evolution, and the X-ray background. \nHowever, an unambiguous optical/UV link between the LINER and AGN classes has remained elusive. Maoz et al. (1998) analyzed the HST UV spectra of seven 'UV-bright' LINERs and showed that, in at least some of them, most or all of the compact UV continuum emission is produced by a cluster of massive stars, whose energy output may be sufficient to account for the optical recombination lines. Even in the few objects showing broad, quasar-like emission lines, one cannot say conclusively whether the UV continuum source is stellar or nonstellar, because the broad emission lines coincide in wavelength with the main expected stellar absorptions. X-ray and radio data have provided convincing evidence for the presence of AGNs in some of these objects. However, they cannot identify the source of the optical emission lines, which are excited by UV radiation (beyond the Lyman limit), and the entire LINER definition rests on the ratios of these emission lines. There is thus the possibility that the LINER phenomenon and central black holes are not physically connected, but simply coexist in many galaxies because both are common. \nVariability is one of the defining properties of AGNs. Variability can reveal an AGN origin of an emission component, even when broad lines are not detected and much of the continuum emission is produced by a nuclear star cluster. For example, in both of the LINERs NGC 4569 and NGC 404, Maoz et al. (1998) showed that the UV spectrum has the broad absorptions in C IV λ 1549 and N V λ 1240 due to winds from O-type stars. However, the relative shallowness of these features in NGC 404 means the O-star light is diluted by another component. This component could be lower-mass stars (B and A dwarfs) in the same cluster, or it could be a featureless AGN component. Repeated observations could potentially detect UV variations, thereby exposing an AGN component in the UV, even when much of the continuum emission is produced by a nuclear star cluster. \nIndeed, there have been several reports of UV variability in LINERs. Barth et al. (1996) compared their HST/FOS spectrum of NGC 4579 to the HST/FOC F220W measurement of Maoz et al. (1995), and found a factor of 3 decrease in the flux of the central source over the 19-month period between the observations. Cappellari et al. (1999) compared FOC observations of NGC 4552 taken in 1991, 1993, and 1996, and found \na factor of 4.5 brightening between the first two epochs, followed by factor of 2 dimming between the last two epochs. While these detections of UV variability were suggestive, they were not conclusive from the technical aspect, because the observational setup was different at each epoch. In the case of NGC 4579, one is comparing a broad-band measurement to a spectroscopic one, where aperture misplacement is always a danger. In the case of NGC 4552, the three measurements were both pre- and post-HST-spherical-abberration correction, and in different FOC formats, having different dynamic and non-linearity ranges, and different fields of view. Other indications of UV variability in LINERs have been indirect, e.g., the appearance of broad (sometimes doublepeaked) Balmer emission-line wings in some galaxies with LINER spectra (e.g., Storchi-Bergmann et al. 1995). \n- If the reported UV variations in LINERs are real, it would mean that:\n- 1. Some of the UV emission of these LINERs is of an AGN nature;\n- 2. LINER variations are common (since UV variations were detected in the few galaxies that were observed more than once); and\n- 3. Large-amplitude (factor ∼ 3) variations on fewyear timescales are the norm. This would contrast with Seyferts 1s, where typical UV variations are /lessorsimilar 2, and quasars, where variations of only tenths of a magnitude are most common (e.g., Giveon et al. 1999). \nConfirming (or refuting) the above results on a carefullyselected sample could therefore provide the missing link between LINER emission and AGNs, and supply important new input to the phenomenology of AGN variability and its dependence on luminosity. \nQuantifying the stellar and AGN contributions to the UV is also important for correctly comparing the continuum emission of low-luminosity AGNs with models. While in quasars and Seyferts the UV continuum in generally attributed to a standard thin accretion disk (e.g., Shakura & Sunyaev 1973), in low-luminosity AGNs the nature of the UV continuum is still a matter of debate. In models such as advection/convection dominated accretion flows (ADAFs/CDAFs; e.g., Quataert et al. 1999), the emission is from the hot accretion flow itself. Alternatively, Yuan et al. (2002) and Falcke et al. (2004) have postulated that not only the radio emission, but also the optical/UV/X-rays in low-luminosity objects could be nonthermal emission, likely from the jet itself, with the jet emission becoming dominant as the disk becomes radiatively inefficient. In the latter picture, one postulates a transition from a thermally dominated spectrum to a non-thermal (possibly jet-dominated) spectrum as one goes from black holes accreting near the Eddington limit (quasars and Seyferts) to black holes with subEddington accretion. Stronger UV variability in lowluminosity AGNs could be a signature of this transition. \nIn the current paper, we present results from a monitoring program using HST to search for UV variability in a sample of LINERs over timescales of weeks to 10 years. Compared to previous, serendipitous, detections of variability, our program was designed to study this question systematically, using a stable observational setup and a representative sample of UV-bright LINERs.", '2.1. Sample': "Our sample includes all objects seen to have compact central UV sources in existing HST data as known by us in 2001, and classified optically as LINERs by Ho et al. (1997a) based on their optical emission line ratios. 8 The 17 LINERs in the sample include a variety of LINER subtypes: LINERs having broad H α wings (which we will designate 'LINER 1s'), and those having only narrow emission lines ('LINER 2s'); LINERs whose UV spectra show signatures of massive stars, and those that do not; and some LINERs that, in terms of optical classification, are borderline with Seyferts or with H II nuclei. We will refer to all these objects collectively as either LINER 1s or LINER 2s, depending on the presence or absence of broad H α wings. 9 VLA imaging at 2 cm and 3.6 cm has revealed a radio core in 11 of the objects, and variability has been detected (Nagar et al. 2002) in five of those that have been monitored. Tables 1 and 2 list the objects in the sample and summarize some of their previously known properties.", '2.2. Observations': "Imaging of the sample was carried out with the HST Advanced Camera for Surveys (ACS) with its High Resolution Camera (HRC) mode. The field of view of this CCD-based instrument is about 29 '' × 25 '' , with a scale of 0 . '' 0284 × 0 . '' 0248 pixel -1 . Each target was imaged in the F250W band ( λ central ≈ 2500 ˚ A, FWHM ≈ 550 ˚ A) with exposure times ranging from 5 to 25 min, depending on target brightness, and in the F330W band ( λ central ≈ 3300 ˚ A, FWHM ≈ 400 ˚ A) with an exposure time of 5 min. (The brightest target, NGC 4569, was exposed for just 1 min in each band.) The exposure time was split between two equal exposures that were used in the data reduction process to reject cosmic-ray events. Objects were repeatedly scheduled using HST's Snapshot mode, i.e., these short exposures were chosen by the HST schedulers, as dictated by convenience, in order to fill gaps left in the schedule after normal-mode observations had been scheduled. This means that both the number of epochs at which a given target was actually observed and the spacing between epochs were largely random. Between July 1, 2002 and July 2, 2003, 15 of the 17 LINERs were observed from two to five times each. Two objects, NGC 404 and NGC 1052, were observed only once. HST failed to acquire guide stars in two exposures, of NGC 3642 on December 17, 2002, and \n- 8 Our sample inadvertently excluded NGC 4303, a UV-bright nucleus (Colina et al. 1997, 2002), classified by Ho et al. (1997a) as an H II nucleus, but which higher spatial resolution spectroscopy by Colina & Arribas (1999) identified as a LINER \n9 Ho et al. (1997a) designated LINERs with broad H α wings as LINER 1.9 objects. Since this is the only kind of type-1 LINER in the sample of Ho et al. (i.e., there are no known examples of LINER 1.2, 1.5, etc.), we will simply refer to LINER 1.9s as LINER 1s. Three of the objects in our sample are classified by Ho et al. as Seyferts, since their narrow emission line ratios [OIII] λ 5007/H β are above the defining border between LINERs and Seyferts by ∼ 30 -40% (for M81 and NGC 3486) and by a factor ∼ 3 (for NGC 4258). The border is somewhat arbitrary (see, e.g., the distribution of emission-line nuclei from the Sloan Digital Sky Survey on the diagnostic diagrams shown by Kauffmann et al. 2003), so we consider the former two objects also as borderline LINER/Seyfert cases, and the latter as a low-luminosity Seyfert. \n<!-- image --> \nFig. 1.Sections of the ACS/HRC F250W images, 5 '' on a side, of each of the galaxies in the sample. The nuclei are at the center of each image, except in the case of M87 and NGC 4736, in which the nuclei are slightly offset to the lower left so as to include in the image the jet and the off-nuclear UV source, respectively. Note that an unresolved object is the only significant UV source in the central regions of most of these bright, nearby, galaxies. \n<!-- image --> \nof NGC 3486 on February 13, 2003. The resulting failed data will be ignored here. We supplemented our data with archival data obtained with the same observational setup for two of the objects, using the F330W filter: one epoch for NGC 3486 from June 3, 2003, and one for NGC 4258 from December 7, 2002. Table 1 lists the exposure times and epochs of each target. \nFigure 1 shows ACS image sections around each of the LINER nuclei. Compared with previous UV images of these objects with the FOC, WFPC1, and WFPC2 instruments, the improvement in resolution, sensitivity, and linearity reveals, in some cases, fine structures and details that were not clearly seen before. However, the morphologies characterized by isolated, compact or unresolved, nuclear UV sources are consistent with the previous images by Maoz et al. (1995) and Barth et al. (1998). This simple morphology also facilitates the photometric measurements described below.", '3.1. Data reduction': 'Data were reduced automatically by the Space Telescope Science Institute (STScI) pipeline. This includes bias and overscan subtraction, cosmic-ray rejection and combination of the two split exposures, dark subtraction, flat-fielding, and geometric distortion correction. To assure uniform reduction using the latest available flats and distortion-correction algorithms, the entire dataset was re-retrieved from the HST archive and processed on-thefly on May 4, 2004.', '3.2. ACS UV photometric stability': "Of great concern in a program such as this is the photometric stability of the camera. This concern is heightened by the fact that, given the small field of view, the short exposures, and the old stellar population of the galaxy bulges in which LINERS are preferentially found, the compact nucleus is generally the only bright feature in the image, and relative photometry is not possible. Indeed, WFPC2, the ACS's predecessor CCD imager on HST, suffers from severe and variable molecular contamination on its front window, which causes large variations with time in UV sensitivity. ACS was expected not to be afflicted by such a problem, since it does not have a cold window on which contaminants can condense. However, as our program was executed on the first observing cycle after the installation of ACS, the actual in-flight UV sensitivity stability was not known, and must be addressed when assessing the reality of any detected UV variations. \nFortunately, STScI staff carried out a program to monitor the UV photometric stability of ACS/HRC, including the two filters we used, contemporaneously with our program. As reported by Boffi, Bohlin, & de Marchi (2004), the open star cluster NGC 6681 was observed 19 times from May 2002 to July 2003. Observations were roughly bi-weekly until mid-November 2002, then paused, and resumed in mid-February 2003, roughly once a month. Not only was the observational setup identical to ours, but the total exposure time per filter (140 s split into two exposures) was comparable to the one we used (300 s for most targets), and the brightnesses of the stars (20,00030,000 total counts per star within an 8.5-pixel radius) in this test field were similar to those of the compact nuclei in our program. Boffi et al. show light curves for eight of the stars in the field, and report that the UV sensitivities in both F250W and F330W are stable 'to 1%'. \nTo obtain a more quantitative estimate of the photometric stability, we used the plots of Boffi et al. (2004) to calculate the actual rms scatter of each star's light curve in each filter. We also measured the brightness of each of \nthese stars in several of the reduced images of NGC 6681 that we retrieved from the HST archive. We then subtracted, in quadrature from the rms scatter of each light curve, the readout noise due to the pixels within an 8.5pixel radius in two exposures, and the Poisson noise due to the total counts, to obtain the remaining scatter due to other sources of noise. We designate this remaining noise as the 'photometric scatter'. We find that, among the 16 stellar light curves (8 stars in two bands each), the photometric scatter ranges from zero (i.e., the rms of a light curve is at the level expected from Poisson noise and readout noise alone) up to 1.1%. Four of the light curves have photometric scatter near zero, eight have scatter from 0.4% to 0.7%, and four have scatter of about 1%. We do not find a correspondence between the photometric scatter of a light curve and any obvious parameter, such as the identity of a star, its position on the chip, its brightness, or its color. In fact, some of the stars with the highest scatter in one band have the lowest scatter in the other band, even though at each epoch the two bands were obtained consecutively, with shifts of only a few pixels between exposures. Since a sizeable fraction (25%) of the stellar light curves show photometric scatter of about 1%, and since the first purpose of the present study is to test the null hypothesis that LINERs do not vary, we will conservatively assume that the ACS/HRC has a photometric rms scatter of 1% in both the F250W and the F330W bands. We will adopt this figure as the photometric calibration uncertainty of our measurements, to be combined with the other, statistical, sources of error.", '3.3. Photometry': "We used IRAF 10 to perform aperture photometry of the nuclear source in each image. Counts were summed within a 10-pixel-radius (0 . 27 '' ) aperture centered on the source. The background level was determined from the median counts in an annulus at radii of 14 to 18 pixels. Experimenting with the more constant among the nuclear sources (e.g., NGC 4569), we found that an aperture radius of > 8 pixels is required in order to obtain photometric stability of better than 1% between epochs. This is consistent with the use of 8.5-pixelradius apertures by Boffi et al. (2004) in the photometric stability tests described above. Errors were calculated by combining in quadrature the Poisson errors of the counts, the readout-noise errors from the pixels within the aperture in the two split exposures (assuming a readout noise of 4.71 e pixel -1 ), and the adopted photometric scatter of 1%. Counts and their errors were multiplied by 1.25 (for F250W) and by 1.18 (for F330W) to correct for the finite aperture radii, based on the point-source encircled energy curves in the ACS Data Handbook (Pavlovsky et al. 2004). Finally, count rates were converted to flux densities using the conversion given by the PHOTFLAM keyword in the image headers, 1 e s -1 = 4 . 781 × 10 -18 erg cm -2 s -1 ˚ A -1 (F250W), and 1 e s -1 = 2 . 237 × 10 -18 erg cm -2 s -1 ˚ A -1 (F330W). This conversion assumes a spectral shape that is flat in f λ , which is a reasonable approximation for these objects \n10 IRAF (Image Reduction and Analysis Facility) is distributed by the National Optical Astronomy Observatories, which are operated by AURA, Inc., under cooperative agreement with the National Science Foundation. \n- their UV colors (see below) imply a spectral slope in the range -0 . 4 < α < 0 . 4, for an assumed spectral shape f λ ∝ λ α . \nThree galaxies, NGC 4736, NGC 5055, and NGC 6500, merit separate mention. NGC 4736, apart from its nuclear UV source (which is clearly centered on a diffuse, centrally peaked stellar light distribution) displays a second UV point source, 2 . '' 5 north of the nuclear UV source. The flux from the off-nuclear source, which we designate NGC 4736b, was measured exactly as for the other nuclear sources. The off-nuclear source will be further discussed below. NGC 5055, has a central source which is resolved, with an observed full-width-half-maximum (FWHM) of about 5.5 pixels (0 . 15 '' ), as opposed to the 2-3 pixel FWHM typical of point sources. In its second epoch, on March 12, 2003, it is significantly more extended in both bands, with a FWHM of 7 pixels, perhaps due to spacecraft jitter. To prevent both the normal large width and the anomalous epoch from adversely affecting the photometry, we used a 13-pixel aperture in this case, which increases the flux by 20% but eliminates a spurious 3% decline at this epoch. NGC 6500 does not have a clearly defined nuclear source. Instead, it has a diffuse central light distribution, on which are superposed a number of faint sources, some compact and some extended. Although this structure was already known from previous imaging with WFPC2 by Barth et al. (1998), we included this galaxy in the sample since it has various known AGN properties (a radio core - Nagar et al. 2000; a possibly nonstellar UV spectrum - Barth et al. 1997), keeping in mind the possibility that one of the faint sources in the WFPC2 image could have been the active nucleus, perhaps temporarily in a low state. To encompass within the aperture the diffuse nuclear light from this galaxy, we used an aperture radius of 20, rather than 10, pixels.", '3.4. Light curves': "The fluxes at every epoch are included in Table 1. Figure 2 shows the light curves in F250W (filled circles) and F330W (empty circles) for each object. Every object is designated as L1, for type-1 objects (including transition LINER/Seyfert objects), or L2, for type-2 objects (including transition LINER/Seyfert objects and transition LINER/H II objects), and is marked with an 'R' if it has a detected compact flat-spectrum radio core. The Seyfert nucleus NGC 4258 and the LINER NGC 4552, unusual objects that have broad components in both the permitted and the forbidden transitions (see below, and in Notes on Individual Objects), are labeled S1/2 and L1/2, respectively. The horizontal dotted lines are plotted at the time-averaged mean flux value in each band. The solid lines show one of the flux levels measured previously (1993-2000) for these objects at bandpasses similar to the F250W band, usually at 2200 ˚ A. While straightforward comparison to the currently measured levels is difficult (see § 1), very large variations, of a factor of a few, between these 'historical' measurements and the current ones are probably real. Such a comparison is discussed in each case in § 4, 'Notes on Individual Objects'. \nIn Table 2 we list some statistics derived from the light curve of each object. These include: the number of epochs; the time-averaged mean flux in each band; the χ 2 per degree of freedom of the data in each \nFig. 2.UV light curves in F250W (filled circles) and F330W (empty circles) for each of the 17 nuclei in the sample, plus the offnuclear source NGC 4736b. The period shown corresponds to May 7, 2002 through July 30, 2003. Objects are labeled in parentheses as L1 for type-1 objects (including transition LINER/Seyfert objects), L2 for type-2 objects (including transition LINER/Seyfert objects and transition LINER/H II objects), and R for objects with a detected radio core. The Seyfert nucleus NGC 4258 and the LINER NGC 4552, have broad components in both the permitted and the forbidden lines, and are labeled S1/2 and L1/2, respectively. Dotted lines show the time-averaged mean flux in each band, and solid lines show one of the 'historical' (1991-1997) flux levels measured previously for these objects at bandpasses similar to the F250W band, usually at 2200 ˚ A . See § 4, Notes on Individual Objects, for details. A historical level is not shown for NGC 3998, as it is ∼ 5 times higher than the 2003 level. Many of the objects display significant short-term ( /lessorsimilar 1 yr) variations, correlated between both UV bands, and large-amplitude long-term variations. \n<!-- image --> \nband, relative to a model with a constant (non-variable) flux at the mean level - this number appears in boldface for the cases that are variable at > 95% confidence; the peak-to-peak variation, with the typical error subtracted in quadrature (if negative, the peak-topeak variation is set to zero); the time-averaged mean UV color f λ (F250W)/ f λ (F330W), after correction for Galactic reddening, assuming the B -band extinction values of Schlegel et al. (1998) and the Galactic extinction curve of Cardelli et al. (1995) with the parameter R V = 3 . 1; the color change between the two epochs with extreme fluxes - [ f max (F250W)/ f max (F330W)]/ \n[ f min (F250W)/ f min (F330W)]; and its uncertainty.", '3.5. Distances': "We have compiled from the literature recent distance estimates to all the galaxies in the sample. In 11/17 cases, the distances are based on 'modern' methods Cepheids, surface-brightness fluctuations, tip of the red giant branch, Tully Fisher, and maser proper motion. Several of the galaxies have distances from several different methods, which always agree to better than 10%, in which cases we have used the averages. Our adopted distances are listed in Table 2, along with the literature sources on which they are based. We have used these distances, and Galactic extinction corrections as described above, to compute monochromatic luminosities at 2500 ˚ A, which are also given in Table 2.", '3.6. UV Variability': "Inspection of Figure 2 and Table 2 reveals a number of new results. First, in the F250W band, among the 16 objects with multiple epochs (the 15 galaxies with multiple epochs, including the double nucleus in NGC 4736), significant variability at greater than the 95% confidence level, based on χ 2 , is detected in all but four cases: NGC 3368, NGC 3486, NGC 4569, and NGC 5055 (plus M87, that varied at 94% confidence. The apparently significant variations in NGC 6500 are uncertain - see below). In the F330W band, eight objects reveal significant changes, and all of these vary in F250W as well. Whenever significant variations are detected in both bands (in eight objects), the variations are correlated. Significant variations range in peak-to-peak amplitude (expressed as a fraction of the mean flux) from 3% to 46%, with a median of 7%, in F250W, and from 5% to 34%, with a median of 11%, in F330W. (Note that these statistics are affected by the different number of epochs and time intervals for each object.) \nAs summarized in § 1, UV variability in some of these LINERs has been reported before. However, this is the first time such variability is seen on relatively short timescales, and it is detected using an unchanging, and photometrically very stable, observational setup ( § 3.2). The variable flux provides a firm lower limit on the AGN contribution to the UV flux at each band. We see that variability, and hence an AGN contribution, exists in some members of both LINER classes, 1 and 2. This situation is distinct from the one in Seyfert galaxies. In Seyferts, the AGN continuum that is visible in type-1 objects is obscured in type 2s, in which the observed UV continuum is sometimes scattered AGN light and sometimes produced by young stars in the circumnuclear region (e.g., Gonz'alez-Delgado et al. 1998), and, in either case, is not expected to be variable on ≈ 1 year or shorter timescales. Our finding that variability is seen in at least some LINER 2s suggests that the 'unified scheme', believed to apply to Seyferts, may not always apply to LINERs. \nIn terms of longer timescale variability, comparison of the UV flux levels we measure to the 'historical' ones shown with a solid line in Figure 2 reveals likely largeamplitude variations even in some objects that were not seen to vary during the present campaign, either because they were sampled too closely or too infrequently, or because they were temporarily inactive. These include \nFig. 3.Histograms of Galactic-reddening-corrected UV color, f λ (F250W)/ f λ (F330W), for type-1 objects (solid line) and type 2 objects (dashed line). Type-1 objects appear to be generally bluer than type-2's. NGC 4569, the bluest type-2 object (see text), is labeled. \n<!-- image --> \nNGC 404, NGC 1052, NGC 3368, and M87 (see § 4, for details). Indeed, the nucleus of M87 was recently monitored by Perlman et al. (2003) using the ACS/HRC with the F220W filter, and was shown to vary in flux, consistent with previous reports of optical variability by Tsvetanov et al. (1998). This leaves only three out of the 17 LINER nuclei that appear to be constant on both short (months) and long (up to 10 years) timescales. These constant objects are NGC 3486, NGC 4569, and NGC 5055. We can conclude, therefore, that UV variability is a common property of the majority of LINERs. \nThe nuclear UV source in NGC 4736 appears variable in F250W, but only at the 95% confidence level, and is constant in F330W. However, a surprising result is that the off-nuclear source, NGC 4736b, is clearly variable in both bands. This raises the possibility that the offnuclear source is the active nucleus of a second galaxy in the final stages of a merger with NGC 4736, or perhaps it is related to jet activity originating in the nucleus. This is discussed in more detail below.", '3.7. UV color of type-1 and type-2 LINERs': "Inspection of Table 2 reveals another interesting result. In terms of the Galactic-reddening-corrected UV color f λ (F250W)/ f λ (F330W), there is an apparent trend that LINER 1s are, on average, blue, and LINER 2s are red. This is illustrated in Fig. 3.6, showing histograms of the UV color for LINER 1s and LINER 2s. The populations (at least as probed by our small sample) seem to overlap at f λ (F250W)/ f λ (F330W) ≈ 1. We have excluded four objects from this plot: \nNGC 4258 , which we have listed as an object of uncertain type, has the reddest nucleus in our sample. Pogge et al. (2000) have presented evidence that this object is a borderline case between UV-bright and UV-dark objects, and is likely partially obscured and reddened by foreground dust. Furthermore, its narrow [OIII]/H β ratio of 10 puts it firmly in the Seyfert regime (Ho et al. 1997a), distinct from the other nuclei in the sample, which are LINERs or borderline LINERs. Its broad H α wings, which could give it a type-1 classification, are also peculiar in the sense that both the permitted and the for- \ndden lines in its spectrum have broad bases, especially in polarized light (Barth et al. 1999b). \nNGC 4552 was classified by Ho et al. (1997a) as 'T2:', meaning a transition object between a LINER and an H II nucleus, with no evidence of a broad H α component, and an uncertain classification. The uncertainty is driven by the weakness of the emission lines, which in the ground-based optical spectrum are superposed on bright stellar emission from the center of the galaxy. Since the narrow emission lines are so faint, an even-weaker broad component would have been impossible to detect, and the classification of NGC 4552 as a type-1 or type2 LINER was limited by the signal-to-noise ratio of the spectral data (L.C. Ho, private communication). Indeed, Cappellari et al. (1999) analyzed an HST/FOS spectrum of NGC 4552, in which, at HST resolution, the nucleus can be better isolated from the surrounding starlight. They found that the emission lines have a significant broad component, with velocities of 3000 km s -1 , typical of type-1 objects. However, this broad component was present in both the permitted and the forbidden lines, in contrast to other type-1 AGNs, but reminiscent of the situation in NGC 4258. It is therefore unclear whether NGC 4552 should be considered a type-1 or a type-2, and we exclude it from the UV-color analysis. [Incidentally, Cappellari et al. (1999) classified NGC 4552, using the narrow-line ratios measured with HST, as a borderline case between a LINER and a Seyfert, rather than as a transition case between a LINER and an H II nucleus.] We note that a problem in detecting a faint broad component does not exist at the same level for the other LINER 2s in our sample, which are considerably brighter than NGC 4552. In the UV as well, NGC 4552 is, by far, the faintest source in our sample, and one of the least luminous. \nNGC 4736b , the off-nuclear source, was excluded from the UV-color analysis because we know nothing about its nature, let alone if it is a type-1 or type-2 object. \nNGC 6500 was excluded from the plot because it has no obvious nuclear source whose color can be measured. This leaves 14 objects in the plot - six LINER 1s and eight LINER 2s. \nOne LINER 2, NGC 4569, is very blue, contrary to the '1=blue/2=red' trend. However, it is a peculiar object in other respects as well. Maoz et al. (1998) have shown that its UV spectrum is completely dominated by the light from O-type stars, and it has failed to show AGN characteristics in any spectral band. In the present study, it is one of the three objects that are not variable even on decade timescales. Barth & Shields (2000) have calculated photoionization models specifically for this object, and have shown that, under particular conditions, a young stellar cluster can produce the observed optical emission-line spectrum. (Specifically, a sufficient number of Wolf-Rayet stars must be present, implying both an instantaneous starburst and a current age of 3-5 Myr.) Perhaps NGC 4569 is a peculiar case of a starburst-driven LINER, and therefore differs from other LINER 2s also in its UV color. \nIf our apologies for the excluded and the outlier nuclei are justified, and there is a true UV color distinction between LINER 1s and LINER 2s, this may be the first independent observable that can predict the existence of broad H α in a LINER, even if only for LINERs that are \nat the extreme red and blue ends of the color distribution. It is tempting to speculate that the redness of LINER 2s is produced by intervening dust, and that this dust somehow obscures the broad-line region, in some analogy to the unification schemes applicable to Seyfert 1s and 2s. However, such a scenario, at least in its simplest version, will not work. There is, at most, a factor ∼ 2 difference in the UV color ratio of the LINER 1s and LINER 2s (see Fig. 3.6). Assuming a dust screen with a standard Galactic extinction curve, a ratio of 2 between the continuum attenuations at 2500 ˚ A and 3300 ˚ A implies an attenuation of flux in the H α spectral region by only a factor of 2.4. If anything, one would expect the broad emission lines, which come from an extended region, to be even less attenuated than the continuum. In other words, in LINER 2s the 'hidden' broad H α flux would be lowered by a factor ∼ 2, relative to LINER 1's. Such a small reduction would probably not hinder the detection of broad components in type-2 objects, many of which are quite bright. To further investigate a possible role of dust, we have searched for correlations between the UV color we measure and the Balmer decrements of the objects in the sample, or the estimated host galaxy extinctions, as tabulated by Ho et al. (1997a). No trend was found. There is also no obvious relation between UV color and radio power, as tabulated by Nagar et al. (2002). Thus, the UV color may indicate some other link between the shape of the nonstellar continuum and the presence or the visibility of a broad-line region. \nOnthe other hand, even in high-luminosity AGN, there is seldom good agreement between dust extinction measures at different wavelengths, probably because of a combination of optical depth and geometry effects. Furthermore, considering the challenge of detecting weak broad H α wings, a factor 2 reduction may play some role after all. For example, by isolating galactic nuclei from the surrounding stellar light using the small spectroscopic apertures possible with HST, several LINERs, although previously classified as LINER 1s from the ground, have revealed also broad double-peaked H α profiles (Ho et al. 2000; Shields et al. 2000). Thus, further work is required both to confirm the suggested trend of UV color with type, and to understand the effect.", '3.8. UV color variation': "In terms of the temporal variation in UV color of the sources, one can see from Table 2 that the color remains constant, to within the errors, when the flux varies. The sole exception is NGC 4203, which is significantly bluer when it is brighter, as is commonly observed in many high-luminosity AGNs (e.g., Giveon et al. 1999). Note that this is also the object with the largest amplitude of variations. Thus, it may be that similar color changes occur in some of the other objects, but the color changes are too small to be seen when the flux variation amplitude is small. Indeed, Tsvetanov et al. (1998) have reported, for M87 at optical wavelengths, a continuum that bluens as it brightens. However, two of the galaxies with relatively large variation amplitudes, NGC 3998, NGC 4579, keep their color constant to within a few percent when they vary, so the brighter-bluer phenomenon is not universal. \nNGC 404 - HST UV spectroscopy of this LINER 2 nucleus by Maoz et al. (1998) showed clear absorption signatures of OB stars. However, the relative shallowness of the absorptions meant that the light from massive stars was diluted by another component, comparable in flux, which could be a featureless AGN continuum, or the light from less massive stars in an aging or continuous starburst. Nagar et al. (2000) did not detect a radio core in this galaxy, to a limit of 1 mJy. A Chandra X-ray image presented by Eracleous et al. (2002) shows a compact nuclear source with 0.5-8 keV luminosity of 1 × 10 37 erg s -1 , surrounded by some faint blobs. \nIn our new ACS images, the nucleus consists of a compact core on top of a diffuse halo of ∼ 0 . 5 '' diameter, and several surrounding faint sources. Since only one epoch was obtained for this object, we cannot say anything about short-term variability. However, the 2500 ˚ A flux is ≈ 60% of the level measured by the 1994 spectroscopy analyzed by Maoz et al. (1998; 115 × 10 -17 erg cm -2 s -1 ˚ A -1 ). This difference is likely real, given that: a) it is conceivable that slit losses could lead to an underestimate of the flux in a spectroscopic observation, but it is difficult to imagine what would lead to an overestimate. Indeed, from analysis of the FOS target acquisition records, Maoz et al. (1998) noted that NGC 404 had been at the edges of its peak-up scans, possibly leading to some light loss; b) The HST/FOS measurements by Maoz et al. (1998) for other objects (e.g., NGC 4569), taken with the same setup, do agree well with the new measurements, arguing against systematic calibration problems. Furthermore, the 1994 FOS spectroscopy indicated a UV flux of only 65% of that obtained with the HST/FOC imaging measurement in 1993 by Maoz et al. (1995; 180 × 10 -17 erg cm -2 s -1 ˚ A -1 at 2300 ˚ A). These measurements imply that the nucleus has faded by a factor ∼ 3 at 2500 ˚ A between 1993 and 2002. Again, for some objects there is excellent agreement between the FOC, FOS, and ACS measurements, lending credence to this conclusion. At most, 60% of the UV light in the spectrum obtained in 1994 was contributed by an AGN, with the rest coming from young stars (Maoz et al. 1998), and therefore it appears that the AGN component at 2500 ˚ A has faded by a large factor between 1994 and 2002. \nNGC 1052 -This galaxy hosts the archetypical LINER 1 nucleus, which has many AGN characteristics: weak broad H α wings, which become dominant in polarized light, revealing a 'hidden broad-line region' (Barth et al. 1999a); an 'ionization cone' (Pogge et al. 2000), reminiscent of those seen in some Seyfert galaxies, aligned with the direction of radio lobes and X-ray knots (Kadler et al. 2004); a variable radio core (Vermeulen et al. 2003); and H 2 O megamaser emission (Claussen et al. 1998). Unfortunately, in our HST snapshot program this object was imaged in the UV only once. Nevertheless, in 2002 the UV flux was at half its level in the 2200 ˚ A region of a 1997 HST/FOS spectrum (Gabel et al. 2000), as measured by Pogge et al. (2000; 15 × 10 -17 erg cm -2 s -1 ˚ A -1 ). As argued above for NGC 404, the sense of the difference (a lower flux in the imaging observation), and the reliability of the FOS calibration for other objects argue that this factor of 2 \ndecline is real. If so, there is a significant AGN contribution to the UV light of this nucleus. \nM81 - This nucleus is formally a Seyfert 1 (Ho et al. 1997a), since its [OIII]/H β ratio is 30% above the (rather arbitrary) border between LINERs and Seyferts. This is close enough that we can safely consider it a borderline LINER/Seyfert case. It has numerous AGN features, including a variable and double-peaked broad Balmer line component (Bower et al. 1996; Ho et al. 1996), a broad-line AGN-like UV spectrum (Ho et al. 1996; Maoz et al. 1998), and at VLBI resolution, a stationary radio core and a one-sided variable jet (Bietenholz et al. 2000). In our current observations, M81 was imaged at five epochs. At four of the epochs, the nucleus displays little or no variations in either F250W or F330W, with amplitudes of variation limited to < 2 -3%. The flux level at 2500 ˚ A , 200 × 10 -17 erg cm -2 s -1 ˚ A -1 , is similar to the one measured by Maoz et al. (1998) at 1500 ˚ A in the 1993 HST/FOS spectrum of Ho et al. (1996) - 150 × 10 -17 erg cm -2 s -1 ˚ A -1 . [From an analysis of the FOS target acquisition records, Maoz et al. (1998) deduced that M81 was located at the edges of its peak-up scans, possibly leading to some light loss.] The ACS-measured flux at 2500 ˚ A in these four epochs is also the same as the 2200 ˚ A flux estimated by Maoz et al. (1998) by extrapolating the 1996 WFPC2 measurement at ∼ 1600 ˚ A by Devereux et al. (1997). However, the nucleus brightened by ∼ 9% in both F250W and F330W filters at one epoch, February 2, 2003. The correlated variation in the fluxes in the two bands, and the lack of anything suspect on that date (e.g., the nucleus appears unresolved, just as at other epochs), favor that this variation is real. Therefore, of order 10% or more of the 2500 ˚ A and 3300 ˚ A UV continua in this object are nonstellar in nature. M81 is the bluest object in our sample in terms of f λ (F250W)/ f λ (F330W) color. \nNGC 3368 - There is no significant variation in either UV band between the two epochs at which this LINER 2 was observed. However, the 2500 ˚ A flux is a factor of 4.5 higher than the 2200 ˚ A flux measured in 1993 with HST/FOC by Maoz et al. (1996; 5 × 10 -17 erg cm -2 s -1 ˚ A -1 ). The large amplitude of this long-term variation makes it credible, despite the difficulties of photometry with the FOC. Note that for the nucleus of NGC 3486, which had a 2200 ˚ A flux in 1993 (10 × 10 -17 erg cm -2 s -1 ˚ A -1 ; Maoz et al. 1996) comparable to that of NGC 3368, there is good agreement, to 8%, between the old FOC measurement and the current ACS measurement. Thus, NGC 3368 appears to be another LINER 2 in which the UV is dominated by AGN emission, despite the fact that no other AGN features (e.g., a radio core; Nagar et al. 2002) have been detected to date. \nNGC 3486 - This nucleus is one of three objects in our sample (the others are NGC 4569 and NGC 5055) that show no significant short-term or long-term variations. It is a relatively high-ionization object, borderline between LINERs and Seyferts, and which Ho et al. (1997) have actually classified as a Seyfert 2 (its [OIII]/H β ratio is 40% above the formal LINER/Seyfert border). Its nonvariability may be related to its class, as the UV continuum of Seyfert 2's is dominated by either scattered \nnuclear light or starlight (Gonz'alez-Delgado et al. 1998). The 2500 ˚ A flux level, 10 . 8 × 10 -17 erg cm -2 s -1 ˚ A -1 , is in excellent agreement with the 2200 ˚ A level in 1993, 10 × 10 -16 erg cm -2 s -1 ˚ A -1 , measured with the HST/FOC by Maoz et al. (1996). Since this is another object without other AGN features (e.g., at a resolution of 1 '' , no radio core at 6 cm and 20 cm was detected to a 3 σ limit of 0.12 mJy beam -1 by Ho & Ulvestad 2001), it is tempting to label it as a non-AGN LINER. Note, however, that we imaged it on only two, closely spaced (by 1 month), epochs. For comparison, M81 and M87, which are clearly AGNs with variable UV flux, were also near their 'historical' UV level in the present observations and were constant in our two closely spaced epochs (for M87) or in four out of five epochs (for M81). Thus, detection of short-term variability in NGC 3486 might have been possible with better temporal sampling. \nNGC 3642 - The 8% peak-to-peak amplitude variations in F250W of this radio-undetected (Nagar et al. 2000) LINER 1 are significant at > 98% confidence, based on χ 2 . Fluctuations seen in the F330W band are not significant, for our assumed photometric uncertainty, although the sense of these fluctuations is correlated with those in F250W, suggesting they may also be real. The 30% increase in 2500 ˚ A flux compared to the 2200 ˚ A WFPC2 measurement in 1994 by Barth et al. (1998; 19 × 10 -17 erg cm -2 s -1 ˚ A -1 ) is not obviously significant, given the different bandpasses and the UV sensitivity fluctuations of WFPC2. The small, but significant, F250W variations show that at least a fraction of the UV flux is nonstellar. Interestingly, Komossa et al. (1999) found no evidence in this galaxy for X-ray variations on short (5 months) or long (years) timescales. NGC 3998 - This variable radio-cored (Filho et al. 2002) LINER 1 displayed a monotonic 20% decline in UV flux in both bands, F250W and F330W, over the 11 months we observed it. On long timescales, its mean 2500 ˚ A flux level in 2003 was about 5 times lower than reported by Fabbiano et al. (1994; 10 -14 erg cm -2 s -1 ˚ A -1 ) at 1740 ˚ A in 1992, based on FOC measurements. The large amplitude of this longterm variation makes it credible, despite the different bandpasses and the problems with FOC linearity and dynamic range. Thus, nonstellar light has dominated the UV output of the nucleus over the past decade, and likely still contributes a significant or dominant fraction. NGC4203 - This nucleus is a LINER 1 having a doublepeaked H α profile (Shields et al. 2000) and a variable radio core (Nagar et al. 2002). The radio core remains unresolved at the milli-arcsecond scale, and Anderson et al. (2004) have shown that its spectrum is most consistent with that of a jet pointed within < 45 · to our line of sight. At Chandra resolution its compact X-ray nucleus is embedded in soft diffuse emission of 50 '' diameter (Terashima & Wilson 2003). Its short-term UV variability was the largest in our sample, with a factor of 1.5 between maximum and minimum in F250W, and 1.4 in F330W, and with the variations in the two bands clearly correlated. Almost all of the variation occurred between the first two epochs, separated by 8 months. The 2500 ˚ A flux level in 2003 was 3-4 times higher than in the HST/WFPC2 2200 ˚ A measurement by Barth \net al. (1998; 21 × 10 -17 erg cm -2 s -1 ˚ A -1 ), obtained in 1994. This long-term variation is likely real, given its large amplitude, despite the different bandpasses and the UV photometric instability of WFPC2. As already noted, this is the only object in which we detect significant color changes, presumably because the variation amplitude is large enough to reveal the color changes. The sense of the color change, as in luminous AGN (e.g., Giveon et al. 1999), is that the nucleus is bluer when it is brighter. \nNGC 4258 - The galaxy with the famous masing water disk (Watson & Wallin 1994; Miyoshi et al. 1995), whose Keplerian rotation curve gives the second most accurately measured central black hole mass (after the Milky Way), has been variably classified as a LINER 1 or a Seyfert 1.9. In the spectrum of Ho et al. 1997a, its emission-line ratio of [OIII]/H β = 10, which is 3 times greater than the the formal border between LINERs and Seyferts and hence well in the Seyfert domain. Wilkes et al. (1995) and Barth et al. (1999b) showed that the spectrum in polarized light has emission lines that are broader than the lines in the total flux spectrum. However, this is seen not only in the Balmer lines but in most of the forbidden lines as well, with the width of the lines in the polarized spectrum depending on the critical density of the transition. The phenomenon is thus different from that of the hidden broad-line regions revealed in polarized light in some Seyfert 2 galaxies. Possible explanations for the effect are the presence of a structure that obscures and polarizes the inner parts of the narrowline region (Barth et al. 1999b), or a broadening of the lines due to the impact of the jet on the emission line gas (see, e.g., Wilson et al. 2000). \nOur measurements indicate significant fluctuations in nuclear flux, with a peak-to-peak amplitude of 16% in F250W and 8% in F330W. Contrary to the other objects with significant variations detected in both bands, the variations in this galaxy are not perfectly correlated between the bands, particularly on the third epoch. Pogge et al. (2000) estimated the 2200 ˚ A flux from a F218W WFPC2 image from 1997, at 7 × 10 -17 erg cm -2 s -1 ˚ A -1 . Given the photometric UV instability of WFPC2 (for which Pogge et al. made no correction), and the significant red leak in the WFPC2+F220W configuration, plus the fact that this is the reddest nucleus in our sample in terms of f λ (F250W)/ f λ (F330W) color, the WFPC2 flux level is consistent with the mean ACS F250W level, 5 . 2 × 10 -17 erg cm -2 s -1 ˚ A -1 . Thus, there is no evidence for long-term variation between 1997 and 2003. As already noted, this object is also anomalous in the sense that it has some type-1 characteristics but its UV color is red. Pogge et al. (2000) argued that it likely undergoes foreground reddening, perhaps by the dust in the molecular gas disk that produces this galaxy's observed water masers. \nM87 - Although a LINER 2 (Ho et al. 1997a), with no detected broad component to its Balmer lines, M87 has numerous AGN features, most notably its prominent radio-through-X-ray jet. We obtained only two epochs on this object, separated by 40 days, and showing only marginally significant (94% confidence, based on χ 2 ) variation in F250W and no significant variation in F330W. However, as noted above, Perlman et al. \n(2003) monitored M87 with ACS/HRC and the F220W filter at five epochs between November 2002 and May 2003. Their light curve, which includes the two epochs obtained by us, shows clear nuclear variability, with about 20% peak-to-peak amplitude. The 2500 ˚ A flux is about half that measured by Maoz et al. (1996; 1 × 10 -15 erg cm -2 s -1 ˚ A -1 ) using an archival HST/FOC image from 1991. There is thus no doubt that AGN emission contributes significantly to the nuclear UV flux from this object. The radio-to-X-ray emission in the jet is certainly synchrotron. Hence it is likely that the nuclear UV emission is also synchrotron emission from the jet, or at least has a very strong jet contribution. \nNGC 4552 - As already noted above, the classification of this nucleus is uncertain. Ho et al. (1997a) labeled it a type-2 object, on the border between LINERs and H II regions, with no detected broad H α component. Cappellari et al. (1999), analyzing an HST/FOS spectrum, found the narrow-line ratios were borderline between LINERs and Seyferts, and detected a broad component in both the Balmer lines and in the forbidden lines. It is therefore unclear whether this galaxy is more akin to type 1s or type 2s. Its blue f λ (F250W)/ f λ (F330W) color is certainly reminiscent of type 1s. Between the two epochs at which we observed it, the nucleus brightened by 20% in both UV bands. This confirms the previous reports of long-term variability in this object by Cappellari et al. (1999; see § 1). The mean of the two 2500 ˚ A data points, 2 × 10 -17 erg cm -2 s -1 ˚ A -1 , is the same as was measured with HST/FOS in 1996 by Cappellari et al. (1999), and close to their HST/FOC measurements obtained in 1993, 1 . 5 × 10 -17 erg cm -2 s -1 ˚ A -1 (in F220W) and 1 . 8 × 10 -17 erg cm -2 s -1 ˚ A -1 (in F275W). \nNGC 4569 - A 'transition' nucleus between an H II nucleus and a LINER 2 (Ho et al. 1997a), this object is one of three that showed no variations in either UV band. Its constancy is not surprising, since Maoz et al. (1998) showed, based on its UV spectrum, that at least 80% of the UV flux is stellar. (Of course, this left room for a 20% AGN component, but we have found no evidence for such a component in the present experiment). The mean 2500 ˚ A flux level we measure with ACS, (0 . 992 ± 0 . 005) × 10 -14 erg cm -2 s -1 ˚ A -1 , is in excellent agreement with previous measurements at ∼ 2200 ˚ A FOS: 1 . 05 × 10 -14 erg cm -2 s -1 ˚ A -1 (Maoz et al. 1998) FOC: 1 . 0 × 10 -14 erg cm -2 s -1 ˚ A -1 (Maoz et al. 1995) and WFPC2: 1 . 1 × 10 -14 erg cm -2 s -1 ˚ A -1 (Barth et al. 1998). This agreement lends credence to the detection of long-term variations in similar comparisons for the other objects in our sample. \nNGC 4579 - This nucleus, classified by Ho et al. (1997a) as a transition case between a LINER 1 and a Seyfert 1.9, has a radio core that remains unresolved at the milli-arcsecond scale, with a spectrum that is most consistent with that of a jet pointed within < 40 · to our line of sight (Anderson et al. 2004). Our new images show in sharp detail the disk/spiral arm around the point-like central source, already visible in the UV image by Maoz et al. (1995; 1996). Its X-ray morphology, as viewed with Chandra, is a very bright nucleus surrounded by soft diffuse emission from the circumnuclear \nstar-forming ring (Eracleous et al. 2002; Terashima & Wilson 2003). Barth et al. (1996) noted a factor 3 fading in UV flux between the 1993 FOC images of Maoz et al. (1995) and their own FOS observations in 1994, obtained 19 months later. This apparent large variation was further studied by Maoz et al. (1998), who again concluded it is likely real. Our current data leaves no doubt regarding the large variability of this object. Between the two epochs at which it was observed, separated by less than a month, it brightened by 7% in both UV bands. The mean flux level at 2500 ˚ A in 2003, 61 × 10 -17 erg cm -2 s -1 ˚ A -1 , is between, but significantly different from, the two previous measurements at ∼ 2200 ˚ A: 33 × 10 -17 erg cm -2 s -1 ˚ A -1 (Maoz et al. 1998) and 110 × 10 -17 erg cm -2 s -1 ˚ A -1 (Maoz et al. 1995). \nNGC 4594 - The LINER 2 nucleus of the Sombrero galaxy appears unresolved and isolated at 2500 ˚ A, as previously seen at 2400 ˚ A by Crane et al. (1994). The nucleus shows large short-term variations, with 20% peakto-peak amplitude in F250W and 11% in F330W, which are clearly correlated in the two bands. The 2500 ˚ A flux is at 2/3 of its level measured with the FOS in 1995 (Nicholson et al. 1998; Maoz et al. 1998), a change that is likely real. \nNGC 4736 - This ringed Sab galaxy has a LINER 2 nucleus, an exceptionally bright central surface brightness in the optical and infrared, and is thought to be an aging starburst (e.g., Waller et al. 2001, and references therein). VLA measurements with resolution 0 . '' 15 reveal an unresolved nuclear source with flux 1.7 mJy at 2 cm (Nagar et al. 2004). There is a 6 '' offset between the position of the radio core, and the position of the optical nucleus reported by Cotton et al. (1999), based on a measurement from the digitized Palomar Sky Survey (POSS). However, the optical position of the nucleus given by Cotton et al. (1999) is erroneous. We find agreement to better than 1 arcsecond between the 15 GHz radio core position, the radio core position reported by the VLA-FIRST survey (Becker et al. 1995), the optical position we measure from digitized POSS plates, and the near-IR nuclear position from the 2MASS survey (Jarrett et al. 2003). From all of these, the J2000 coordinates of the radio-through-IR nucleus are RA: 12 h 50 m 53 s . 06 DEC: 41 · 07 ' 12 . '' 7. \nMaoz et al. (1995) noted a second UV source, of brightness comparable to the nuclear one, 2 . '' 5 to the north of the nucleus (at position angle -2 . 4 · ). The distance to this galaxy is ∼ 4 . 9 Mpc, taking the mean between 5.2 Mpc found by Tonry et al. (2001) from surface brightness fluctuations, and 4.66 Mpc found by Karachentsev et al. (2003) based on the tip of the red-giant branch. At this distance, the separation between the two UV sources is 60 pc. Maoz et al. (1995) speculated that the off-nuclear source, which we designate NGC 4736b, could be the active nucleus of a galaxy that had merged with NGC 4736. Perhaps this merger triggered the past starburst and the peculiar morphological and kinematic features observed in this galaxy. \nMaoz et al. (1996) measured in 1993 the FOC 2200 ˚ A fluxes of the nuclear UV source, which we designate NGC 4736, and of NGC 4736b, as 19 × 10 -17 erg cm -2 s -1 ˚ A -1 and 26 × 10 -17 erg cm -2 s -1 ˚ A -1 , \nrespectively. NGC 4736 varied slightly during the current program, and only in the last measurement in F250W. Nevertheless, NGC 4736 was 2.5 times brighter in 2003 than in 1993. Given the large amplitude of the change in NGC 4736, and the reliability of the FOC photometry for several apparently non-variable objects, we believe the factor 2.5 brightening of the nucleus is real. As for NGC 4736b, its mean flux level is similar to that measured in 1993 by Maoz et al. (1996). However, in the present measurements this off-nuclear source shows significant, correlated F250W and F330W fluctuations with peak-to-peak amplitudes of 5%. We note that, due to the large brightnening of the nuclear UV source, NGC 4736b was 30% brighter than NGC 4736 in 1993, while in 2003 the nucleus was 70% brighter than NGC 4736b. We conclude that we have detected long-term UV variations in the nucleus of NGC 4736, and short-term variations in the off-nuclear source NGC 4736b, indicating significant nonstellar contributions to the UV fluxes of both sources. We also note that the two sources have quite different UV colors, with the nuclear source being red and NGC 4736b being blue. \nWe have searched the VLA archive for deep radio observations of this galaxy, to see if object b has, at any time, shown up as a radio source. We have found seven different VLA epochs of observation of NGC 4736 between December 1984 and February 2000, with useful data at 2 to 20 cm. In addition, NGC 4736 was monitored at eight epochs at 3.5 cm between June and October 2003 by Kording et al. (2005). In none of these radio maps do we see a signal at the position of the northern, off-nuclear, source. The deeper archival images are from December 1984 and January 1985, from which we can put a 3 σ upper limit of 150 µ Jy on the radio flux at 20 cm and 6 cm, respectively, and from December 1988, which puts a 3 σ upper limit of 300 µ Jy at 20 cm. The Kording et al. (2005) data from 2003 place a 3 σ upper limit of 150 µ Jy at each epoch, or an upper limit 60 µ Jy from the eight epochs combined. Interestingly, in the higher-resolution observations among these datasets, the nuclear source is resolved into two sources of comparable flux, separated by 0 . '' 99, at position angle -49 · . This second source, which is reported and described in detail by Kording et al. (2005), may be the result of jet activity in the nucleus, or some other radio source that is very close to the nucleus. \nHigh spatial resolution Chandra X-ray images of NGC 4736 were obtained by Eracleous et al. (2002). They detected an unresolved nuclear source with a 0.58 keV luminosity of 5 . 9 × 10 38 erg s -1 , and 39 other sources, presumably X-ray binaries and supernova remnants, distributed around the nucleus. However, they did not detect an X-ray source at the position of NGC 4736b, to a luminosity limit of 1 × 10 36 erg s -1 . Interestingly, variability data shown by Eracleous et al. (2002) suggest that nucleus may be variable in the X-rays on hour timescales, with a measured excess variance of 0 . 06 ± 0 . 04. \nAlternatively to the binary AGN scenario, the variable off-nuclear UV source NGC 4736b could be related to jet activity emerging from the nucleus. For example, the offnuclear source could be at a location where gas is heated by beamed radiation, or it could be synchrotron radiation from freshly accelerated particles at the end of a jet, with a spectrum that is hard enough to avoid detecting this \nknot in radio. Such a scenario would be reminiscent of NGC 1052, where Kadler et al. (2004) find spatial offsets between optical, X-ray, and radio knots associated with the jet. \nTo further investigate the nature of the two UV sources requires high spatial resolution optical-UV observations that will elucidate the spectral properties of each of the sources. At present, it is not clear whether the LINER 2 spectrum attributed to the central parts of this galaxy is emitted by the nuclear source NGC 4736a, the off-nuclear source NGC 4736b, or both. \nNGC 5055 - This transition H II/LINER 2 nucleus is not variable in our data. The nuclear UV source is clearly extended, with a FWHM of 0 . 14 '' . The 2500 ˚ A flux level, 76 . 4 × 10 -17 erg cm -2 s -1 ˚ A -1 , is virtually identical to that measured with the FOS in 1996 by Maoz et al. (1998), though 25% less than the 1993 FOC measurement by Maoz et al. (1995). The latter difference is also consistent with non-variability, considering the uncertainties in FOC photometry, which are further complicated by the extended nature of this source. Like the other non-variable LINER 2 in our sample, NGC 4569, the FOS UV spectrum of this object indicates a > 50% hot-star contribution to the UV flux (Maoz et al. 1998). Our results suggest that NGC 5055 is a member of the minority of LINERs whose UV flux is all or mostly from stars. \nNGC 6500 - This LINER 2 has several AGN traits, including a radio-core (Nagar et al. 2000), and a jet-like linear structure seen with the VLBA (Falcke et al. 2000) As seen in Fig. 1, and known from previous imaging with WFPC2 by Barth et al. (1998), NGC 6500 does not have a clear nuclear UV source. The UV morphology of the central region consists of a diffuse central light distribution, on which are superposed a number of faint sources, some compact and some extended, within a diameter of ∼ 0 . '' 5. It may actually be a 'UV-dark' LINER (like 75% of all LINERs; Maoz et al. 1995; Barth et al. 1998), that happens to possess some scattered circumnuclear star formation. Maoz et al. (1998) noted that its observed UV luminosity at 1300 ˚ A, extrapolated to the far UV, is insufficient to power its H α luminosity. Terashima & Wilson (2003) found that the nuclear X-ray emission observed with Chandra is anomalously faint given the H α luminosity and the X-ray vs. H α correlation observed in other AGNs, and proposed that the nucleus is heavily obscured in X-rays. Our large-aperture measurements for this galaxy (see § 2) indicate no variability in F330W, but fluctuations in F250W that are formally significant at 99% confidence, and driven mainly by a 5% drop at the last epoch. There is nothing suspect with the data at this last epoch. There is a compact source 9 '' east of the nucleus that appears in the first two epochs and the last epoch (in the third epoch it is obstructed by the HRC 'occulting finger'), and which can serve as a local calibrator. Its flux is constant to 0.7%, and its FWHM is similar at all three epochs. However, we are not certain of the reality of the 5% decline at the last epoch for a number of reasons: the large aperture and the diffuse nature of the object increase the susceptibility of the measurement to small fluctuations due to centering differences. We do not have an independent photometric stability check for such a diffuse source, as we do for the compact \nnuclear sources, through monitoring of the star cluster NGC 6681 by Boffi et al. (2004) - see § 3.2. Indeed, the amplitude of the decline on the last epoch depends on the choice of region used to determine the background level, and for some choices the decline is only 2%. The fairly large variation at one epoch, observed solely in one filter is contrary to what we have seen in all the other objects, where large variations are mirrored in the two bands. We have blinked and compared the images of the four epochs to try to identify a particular knot in the nuclear region that declined in brightness on the last epoch, but have not been able to reach a definitive conclusion. The 2500 ˚ A flux, 31 × 10 -17 erg cm -2 s -1 ˚ A -1 , is similar to that measured by Maoz et al. (1998) at 2200 ˚ A from the Barth et al. (1997) FOS spectrum taken in 1994, and to that measured with WFPC2 in 1995 by Barth et al. (1998). In both cases, the UV flux was 27 × 10 -17 erg cm -2 s -1 ˚ A -1 . The difference between our, and these earlier, measurements is not significant, especially considering the extended nature of the source.", '5. DISCUSSION AND CONCLUSIONS': "Our UV monitoring program has revealed, by means of variability, that an AGN component contributes to the UV emission of most UV-bright LINERs. Variability is detected irrespective of spectral type (1 or 2) and whether or not a nuclear radio source has been detected. LINERs are present in the majority of massive galaxies, and the true fraction of LINER galaxies that have a nuclear UV source, after accounting for extinction, is likely close to unity (Barth et al. 1998; Pogge et al. 2000). This conclusion implies that, not only do most galaxies have central black holes, but that the black holes are also accreting and emitting in the UV. The extreme-UV extension of the observed UV, beyond the Lyman limit, is the main ionizing agent in these objects, and it determines the optical line ratios that define LINERs and distinguish them from other nuclei (e.g., H II, Seyfert). Our results provide some of the strongest evidence to date that, in the majority of cases, a LINER spectrum in fact signals the presence of nonstellar activity (i.e., an AGN). Our data confirm previous reports of largeamplitude UV variability in several LINERs (see § 1), but now with a stable photometric setup, applied systematically to a moderate-sized sample. \nWe have identified only three galaxies with UV nuclei that show neither short-term ( /lessorsimilar 1 yr) nor long-term ( /greaterorsimilar 1 yr) variations. In all three cases, there is previous evidence that stars dominate the UV emission - based on UV spectra in NGC 4569 and NGC 5055 (Maoz et al. 1998), and on the Seyfert 2 classification (Ho et al. 1997a) in the case of NGC 3486, combined with the fact that the observed UV emission in Seyfert 2s often comes from stars (Gonz'alez-Delgado et al. 1998). However, even in these three cases, it is possible that UV variability would be detected in an experiment with denser or longer-term sampling. \nInterestingly, all three galaxies without detected UV variations have no detected radio cores, to 1 mJy sensitivity for NGC 4569 and NGC 5055 (Nagar et al. 2000) and to 0.12 mJy sensitivity for NGC 3486 (Ho & Ulvestad 2001). Conversely, all the LINERs that do have detected radio cores have variable UV nuclei. Of course, there are \nthree galaxies (NGC 404, NGC 3368, and NGC 3642) with no detected radio core that do display UV variations, but radio cores may be revealed by more sensitive observations of these objects. If deeper radio observations revealed cores in the three UV-variable LINERs, but not in the three UV-stable LINERs, a perfect correspondence would exist between the presence of radio cores and UV variability. \nCould stars produce the observed UV variations? The 2500 ˚ A luminosities of the variable nuclei in our sample, as seen in Table 2, are distributed more or less evenly in the range L λ (2500 ˚ A) ∼ 10 35 . 6 -10 37 . 7 ergs s -1 ˚ A -1 . The most luminous 'normal' stars are blue supergiants, among which those with effective temperatures of 1 -3 × 10 4 K (spectral classes B0-A0) have the highest near-UV luminosities. Bresolin et al. (2004) recently monitored 70 blue supergiants in the galaxy NGC 300 over a 5-month period, and found 15 of them to be variable, with V -band amplitudes of 8 -23%. The mean V band absolute magnitude of these variable supergiants is M V = -7 . 3, and the most luminous one (of spectral type B9) has M V = -8 . 7. Using spectral models by Kurucz for B9 supergiants to obtain the flux ratio between the V band and 2500 ˚ A, the UV luminosities for the typical and for the most luminous supergiants are L λ (2500 ˚ A) = 10 34 . 6 ergs s -1 ˚ A -1 and L λ (2500 ˚ A) = 10 35 . 2 ergs s -1 ˚ A -1 , respectively. Thus, even the least luminous galactic nuclei in our sample have luminosities about an order of magnitude larger than typical blue supergiants, but short-term variability amplitudes of 15-20%, comparable to those of the most variable supergiants. Furthermore, the large amplitude variations we see in many of the LINERs are not expected in supergiants. Therefore, individual supergiants in the galactic nuclei are not plausible candidates for producing the observed level of UV variability in the LINERs. \nStars more luminous than blue supergiants, by an order of magnitude, do exist - Wolf-Rayet (WR) stars (e.g., Conti 2000) and luminous blue variables (LBVs; e.g., Humphreys & Davidson 1994) with bolometric luminosities up to L bol ∼ 10 40 ergs s -1 . These are massive stars, nearing the end of their evolution and radiating near the Eddington limit. LBV variation amplitudes on timescales of months are /lessorsimilar 10% (Humphreys & Davidson 1994), whereas WN8 stars, which are the most variable among WRs, sometimes vary in the optical by a few percent on week-to-month timescales (Marchenko et al. 1998). Based on luminosity and variation amplitude alone, individual LBVs and WR stars could perhaps produce the UV variations in part of the lower-luminosity half of our sample. However, the variations seen in the most luminous objects in our sample certainly cannot be explained in this way, and, based on continuity, one can then argue that stars are not the source of variations in any of the LINERs. Nevertheless, we cannot rule out the possibility that individual evolved stars dominate the light output in a few of the low-luminosity objects. For example, spectroscopy of NGC 4736b, with a luminosity of L λ (2500 ˚ A) = 10 36 ergs s -1 ˚ A -1 , could reveal if it is a LBV or WR star, rather than a second merging nucleus. \nReturning to the AGN interpretation, the variable flux that we have measured in each UV band, in the absence \nof any extinction corrections, provides a firm lower limit to the AGN flux in that band. This observed lower limit can be used to test accretion models for each of these lowluminosity AGNs. It can also be argued that the total UV flux provides an upper limit on the UV emission, but the sensitivity of the UV to uncertain extinction corrections makes such an upper limit less robust. The VLBA images of Falcke et al. (2000) and Nagar et al. (2002) have shown that at least some of the radio emission in LINERs is contributed by jets, rather than by an actual accretion flow. Anderson et al. (2004) have obtained multifrequency VLBA spectra for the unresolved milli-arcsecond core in six low-luminosity AGNs (including three LINERs, two of which, NGC 4203 and NGC 4579, are in our sample). They showed that the radio spectra are inconsistent with expectations from accretion flows (cf. Nagar et al. 2001), but that the spectra, luminosity, and size limits are consistent with emission from jets that are pointed toward us to within /lessorsimilar 50 · . The observed radio flux must therefore constitute only an upper limit on radio emission from the accretion flow itself. \nStrictly speaking, X-ray data, too, provide only upper limits to the flux from the accretion flow, since even with the excellent spatial resolution of Chandra , non-nuclear X-ray sources (low-mass X-ray binaries, supernova remnants, diffuse emission) can be included in the beam, and may contribute to the X-ray flux. As an extreme example, in M32 the nuclear X-ray source produces only 1% of the total X-ray luminosity within a radius of 30 pc of the nucleus (Ho et al. 2003), yet this is the area covered by a ∼ 1 '' -diameter beam at 10 Mpc, the typical distance to a galaxy in our sample. Nevertheless, the excellent astrometric agreement ( /lessorsimilar 0 . '' 5) beween X-ray and radio positions in low-luminosity AGNs (e.g., Terashima & Wilson 2003), and the absence of close by ( < few arcseconds) X-ray sources indicates that, in most such AGNs, the X-rays originate from the active nucleus. \nAt optical and IR wavelengths, the nuclear emission cannot be detected, at present, in the face of the bright stellar backgrounds, and in the extreme-UV only indirect, model-dependent estimates of the SED can be obtained by attempting to reproduce the UV-through-IR emission line fluxes and ratios. The current flux limits (lower limits to the emission from the accretion flow in the UV, from our present results, and upper limits in the radio and X-rays, from previous observations) thus provide a potentially powerful test of accretion models. The observed SED can similarly be compared to the predictions of jet models. In this case, the UV emission should be connected to emission at other wavelengths and to the assumed black-hole mass by a relation analogous to the radio/X-ray correlation seen in low-accretion-rate black holes (Merloni et al. 2003; Falcke et al. 2004) We intend to carry out such comparisons to models in a future paper. \nOur data for NGC 4736 have revealed the first example of a variable off-nuclear UV source, giving new grounds to previous speculation (Maoz et al. 1995, 1996) that this is a 'wandering black hole' from the nucleus of another galaxy that has recently merged with this one. This hypothesis (and the alternative, that it is an individual WR or LBV star, see above) can be tested with straightforward high-spatial-resolution observations in radio, optical, UV, and X-ray bands. The observational evidence \nfor the existence of systems of massive black hole pairs has been reviewed recently by Komossa (2003). The best current candidate for a double AGN is NGC 6240, a relatively nearby (redshift z = 0 . 024) ultraluminous infrared galaxy. Both nuclei of NGC 6240 are probably LINERs (Raffanelli et al. 1997), both are compact radio sources at 1.4 and 5 GHz (Colbert et al. 1994; Gallimore & Beswick 2004), and both emit hard X-ray continua and Fe K α lines (Komossa et al. 2003). The 1 . 5 '' separation of the nuclei in NGC 6240, for an assumed distance of 100 Mpc, corresponds to a projected physical scale of ∼ 700 pc, compared to only 60 pc between the possible double nuclei of NGC 4736. Another, somewhat more ambiguous, case is NGC 3256, a merging galaxy system at a distance of about 40 Mpc. In this case, two nuclear sources, with a projected separation of 5 . '' 2, i.e., about 1 kpc, are detected in near-infrared and X-rays (Lira et al. 2002) and in radio (Neff et al. 2003), with a radio-toX-ray flux ratio that is characteristic of low-luminosity AGN. In the central CD galaxy of the cluster Abell 400, the twin-jetted double radio source 3C 75 (Owen et al. 1985), is a spectacular example of a binary AGN, though with a rather large separation of 7 kpc. Identification and study of new examples of binary AGN, especially as nearby as NGC 4736, can shed light on the issue of the rate of coalescence of supermassive black holes (e.g., Quinlan & Hernquist 1997). \nWe have found a possible UV-color-based indicator of whether a LINER is a type-1 or type-2 object. If confirmed, this would be the first LINER property that is found to be linked with the presence or absence of a broad Balmer emission line component. As we have argued above, reddening by dust is not obviously the mechanism behind the suggested trend - the factor ∼ 2 difference in the UV color ratio of type 1s and type 2s, if produced by a dust screen, would correspond to a mere factor of 2.4 increase in the extinction of a hypothetical broad H α line in LINER 2s. Instead, we argue that the instrinsic color of the UV continuum is related to the existence or the visibility of a BLR. For example, the physical conditions under which a BLR can form could depend on the present accretion mode, which might be reflected in the UV color. Nevertheless, as we have pointed out, some combination of dust, geometry, optical depth and selection effects may be, after all, behind the observed trend of UV color with LINER spectral type. Furthermore, the trend itself needs to be confirmed with a larger sample. This is not a simple task, since we have imaged all known LINERs that have a compact nucleus in the space-UV. A larger sample of such objects could be assembled by means of UV imaging (e.g., with GALEX) and subsequent optical spectroscopic classification to identify the LINERs. UV imaging need not necessarily be from space - the UV nuclei of our current sample are prominent in the F330W band, so such objects could potentially be indentified by ground-based observations near the atmospheric UV cutoff. Larger samples of LINERs could also be produced by studying a fainter sample of galaxies than that of Ho et al. (1997a), based, for example, on the Sloan Digital Sky Survey, or by surveying the Southern hemisphere. \nAfter we submitted this paper, Totani et al. (2005) reported discovering optical nuclear variability in a 'blind' variability search among ∼ 1000 massive galaxies at \nredshifts z ∼ 0 . 3 -0 . 4. They found six nuclei with estimated variability amplitudes of order unity over a one-month timescale, with marginal evidence for dayto-day variations. Spectroscopy of one of the six variable nuclei revealed a LINER spectrum at z = 0 . 33, with an H α flux implying a specific optical luminosity L λ = 2 × 10 37 erg s -1 ˚ A -1 , quite similar to the objects studied in this paper. It appears plausible that Totani et al. (2005) have discovered, at z ∼ 0 . 3, the large-amplitude-variability tail of the variations we have found in nearby LINERs. Assuming that of order onehalf of early-type galaxies are LINERs (Ho et al. 1997a), and that one-fourth of LINERs have unobscured optical/UV continua (Maoz et al. 1995; Barth et al. 1998), there would be in the data of Totani et al. of order 100 galaxies of the type we have studied here. Since only one galaxy among the 15 that we monitored with HST on month-long time scales showed a variability amplitude of order unity (NGC 4203, which varied by ∼ 40%), it is \nto be expected that Totani et al. would detect about six such cases. \nWe thank Tricia Royle for her expert assistance in the implementation of the observing program, and Luis Ho, Eva Schinnerer, Amiel Sternberg, Joe Shields, and an anonymous referee, for useful advice and input. This work was funded in part by grant GO-9454 from the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the JPL, Caltech, under contract with NASA. This publication also makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and IPAC/Caltech, funded by NASA and the NSF.", 'REFERENCES': "- Anderson, J. M., Ulvestad, J. 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B. 1988, 'Nearby Galaxies Catalog', Cambridge and New York, Cambridge University Press, 1988, 221 p. Vermeulen, R. C., Ros, E., Kellermann, K. I., Cohen, M. H., \nWilkes, B. J., Schmidt, G. D., Smith, P. S., Mathur, S., & McLeod, K. K. 1995, ApJ, 455, L13 Wilson, A. S., Yang, Y., & Cecil, G. 2001, ApJ, 560, 689 Wrobel, J. M. 1984, ApJ, 284, 531 \n- Yuan, F., Markoff, S., Falcke, H., & Biermann, P. L. 2002, A&A, 391, 139 \nTABLE 1 Observations and Photometry \n| Object | Exp. F250W | Exp. F330W | UT-Date | M.J.D. | f λ F250W | σ | f λ F330W | σ |\n|-----------|--------------|--------------|-----------------------|-------------|---------------|------------|---------------|-----------|\n| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) |\n| NGC 404 | 300 | 300 | 2002-10-28 | 175.1 | 73.57 | 0.86 | 85.27 | 0.90 |\n| NGC 1052 | 300 | 300 | 2002-10-18 | 165.4 | 7.65 | 0.27 | 8.97 | 0.16 |\n| | | | | | | | | 1.31 |\n| M81 | 300 | 300 | 2002-08-05 | 91.5 | 198.35 | 2.09 | 126.51 | |\n| | | | 2002-11-27 | 205.9 | 199.07 | 2.10 | 125.77 | 1.31 |\n| | | | 2003-02-02 2003-04-07 | 272.9 336.5 | 214.31 200.98 | 2.25 | 140.01 | 1.45 |\n| | | | 2003-06-12 | 402.3 | 202.54 | 2.12 2.13 | 130.01 128.93 | 1.35 1.34 |\n| NGC 3368 | 600 | 300 | 2002-10-25 | 172.1 | 22.53 | 0.30 | 30.00 | 0.36 |\n| | | | 2003-05-10 | 369.0 | 21.99 | 0.29 | 29.63 | 0.35 |\n| NGC 3486 | 600 | 300 | 2003-04-06 | 335.7 | 10.70 | 0.19 | 18.11 | 0.24 |\n| | | | 2003-05-10 | 369.4 | 10.82 | 0.19 | 18.11 | 0.24 |\n| | · · · | 1200 | 2003-06-03 | 393.5 | · · · 25.28 | · · · | 18.24 | 0.20 |\n| NGC 3642 | 300 | 300 | 2002-10-02 | 149.3 | 24.50 | 0.41 0.41 | 22.19 22.24 | 0.28 0.28 |\n| | | | 2003-01-20 | 259.3 | 24.62 | 0.42 | 22.34 | 0.28 |\n| | | | 2003-04-12 | 341.3 | | | | |\n| | | | 2003-05-26 | 385.5 | 23.55 | 0.40 | 21.74 | 0.28 |\n| NGC 3998 | 60 | 60 | 2002-07-01 | 56.0 | 220.46 | 2.91 | 168.38 | 1.96 |\n| | | | 2002-11-13 | 191.3 | 210.84 | 2.82 | 162.35 | 1.90 |\n| | | | 2003-03-05 | 303.3 | 193.63 | 2.66 | 145.96 | 1.74 |\n| | | | 2003-04-05 | 334.4 | 185.68 | 2.58 | 146.38 | 1.75 |\n| | | | 2003-05-29 | 388.4 | 182.69 | 2.56 | 140.14 | 1.69 |\n| NGC 4203 | 300 | 300 | 2002-07-03 | 58.7 | 77.51 | 0.90 | 46.45 | 0.52 |\n| | | | 2003-03-05 | 303.0 | 54.98 | 0.68 | 34.58 | 0.40 |\n| | | | 2003-04-07 | 336.8 | 54.20 | 0.68 | 35.27 | 0.41 |\n| | | | 2003-04-20 | 349.0 | 50.68 | 0.64 | 33.69 | 0.39 |\n| NGC 4258 | 600 | 300 | 2003-06-13 2002-07-06 | 403.8 61.1 | 53.79 5.02 | 0.67 0.15 | 35.95 | 0.41 |\n| | | | 2002-10-21 | 168.3 | 4.78 | 0.14 | 11.58 10.79 | 0.19 0.18 |\n| | · · · | 1140 | 2002-12-07 | 215.2 | · · · | · · · | 10.82 | 0.12 |\n| | | 300 | 2003-04-17 | 346.5 | 5.56 | 0.15 | 10.82 | |\n| | | | 2003-06-28 | 418.1 | 5.26 | 0.15 | 11.58 | 0.18 0.19 |\n| M87 | 300 | 300 | 2003-03-31 | 329.8 | 56.24 | 0.70 | 48.64 | 0.54 |\n| | | | 2003-05-10 | 369.0 | 54.44 | 0.68 | 48.19 | 0.53 |\n| NGC 4552 | 1500 | 750 | 2003-03-23 | 321.7 | 1.79 | 0.06 | 1.34 | 0.05 |\n| NGC 4569 | 60 | 60 | 2003-06-03 2002-07-03 | 393.2 58.1 | 2.15 999.53 | 0.06 10.55 | 1.63 841.30 | 0.05 8.65 |\n| | | | 2003-02-02 | 272.6 | 983.03 | 10.38 | 840.11 | 8.63 |\n| | | | 2003-03-31 | 329.8 | 988.35 | | | 8.56 |\n| | | | 2003-04-29 | | 999.23 | 10.44 | 832.59 | 8.58 |\n| NGC 4579 | 300 | 300 | 2003-03-17 | 358.0 315.2 | 59.05 | 10.54 0.72 | 834.64 40.34 | 0.46 |\n| | | | 2003-04-12 | 341.9 | 63.17 | 0.77 | 43.42 | 0.49 |\n| NGC 4594 | 300 | 300 | 2003-03-24 2003-05-05 | 322.5 | 6.93 8.43 | 0.27 0.28 | 14.37 | 0.21 0.23 |\n| | | | 2003-06-09 | 364.9 399.4 | 7.05 | 0.27 | 16.13 15.37 | 0.22 |\n| NGC 4736 | 300 | 300 | 2003-03-20 | 318.6 | 47.63 | 0.61 | 72.87 | 0.78 |\n| | | | 2003-04-03 | 332.8 | 47.09 | 0.61 | 74.03 | 0.79 |\n| | | | 2003-04-17 | 346.6 | 47.75 | 0.62 | 72.72 | 0.78 |\n| | | | 2003-06-21 | 411.2 | 49.48 | 0.63 | 73.19 | 0.78 |\n| NGC 4736b | 300 | 300 | 2003-03-20 | 318.6 | 28.21 | 0.44 | 20.74 | 0.27 |\n| | | | 2003-04-03 | 332.8 | 29.94 | 0.45 | 21.32 | 0.27 |\n| | | | 2003-04-17 | | 28.27 | 0.44 | 20.42 | 0.27 |\n| | | | 2003-06-21 | 346.6 | 29.64 | 0.45 | 21.50 | 0.28 |\n| NGC 5055 | 300 | 300 | 2002-07-19 | 411.2 74.0 | 77.57 | 0.92 | 91.32 | 0.97 |\n| | | | 2003-03-12 | 310.3 | 77.27 | 0.92 | 89.56 | 0.95 |\n| | | | 2003-03-31 | 329.9 | 76.38 | 0.91 | 90.48 | 0.96 |\n| | | | 2003-04-22 | 351.1 | 75.66 75.00 | 0.90 0.90 | 91.24 91.30 | 0.97 0.97 |\n| | | | 2003-07-02 | 422.5 | | | | |\n| NGC 6500 | 300 | 300 | 2002-07-06 | 61.0 | 31.49 | 0.62 | 27.11 | 0.37 |\n| | | | 2003-04-13 | 342.1 | 31.85 | 0.62 | 27.27 | 0.38 |\n| | | | 2003-04-20 2003-06-10 | 349.9 400.3 | 32.51 29.70 | 0.63 0.61 | 26.63 27.58 | 0.37 0.38 | \nNote . - (2)-(3)- Exposure time, in seconds. An empty entry indicates the same exposure time as above it; (5)- Modified Julian Date -2452400; (6)-(9)- Nuclear flux densities and 1 σ errors in units of 10 -17 erg cm -2 s -1 ˚ A -1 . NGC 4736b is the off-nuclear UV source in NGC 4736. See text for details of photometry and calibration. \nTABLE 2 Measured Properties \n| Object (1) | type (2) | radio (3) | D Mpc (4) | Ref (5) | ¯ f λ F250W (6) | ¯ f λ F330W (7) | log L λ F250W (8) | UV color (9) | n (10) | χ 2 dof F250W (11) | ∆ / ¯ f F250W (12) | χ 2 dof F330W (13) | ∆ / ¯ f F330W (14) | ∆ color (15) | σ (16) |\n|--------------|------------|-------------|-------------|-----------|-------------------|-------------------|---------------------|----------------|----------|----------------------|----------------------|----------------------|----------------------|----------------|----------|\n| NGC 404 | 2 | N | 3.05 | a,b | 73.57 | 85.27 | 36.11 | 0.97 | 1 | · · · | · · · | · · · | · · · | · · · | · · · |\n| NGC 1052 | 1 | Y | 18.03 | a | 7.65 | 8.97 | 36.57 | 0.9 | 1 | · · · | · · · | · · · | · · · | · · · | · · · |\n| M81 | 1 | Y | 3.63 | a,e | 203.05 | 130.25 | 36.77 | 1.83 | 5 | 9.28 | .08 | 18.36 | .11 | .97 | .02 |\n| NGC 3368 | 2 | N | 10.67 | a,f | 22.26 | 29.81 | 36.58 | 0.78 | 2 | 1.73 | .02 | .55 | .00 | .97 | .03 |\n| NGC 3486 | 2 | N | 7.4 | g | 10.76 | 18.15 | 35.93 | 0.62 | 2 | .20 | .00 | .15 | .00 | 1.00 | .03 |\n| NGC 3642 | 1 | N | 27.5 | g | 24.49 | 22.13 | 37.4 | 1.13 | 4 | 3.22 | .07 | .91 | .02 | 1.04 | .03 |\n| NGC 3998 | 1 | Y | 13.14 | a | 198.66 | 152.64 | 37.68 | 1.34 | 5 | 41.01 | .19 | 51.13 | .18 | 1.00 | .03 |\n| NGC 4203 | 1 | Y | 15.14 | a | 58.23 | 37.19 | 37.26 | 1.6 | 5 | 263.09 | .46 | 160.50 | .34 | 1.11 | .02 |\n| NGC 4258 | ? | Y | 7.3 | a,c,i | 5.16 | 11.12 | 35.59 | 0.48 | 5 | 5.08 | .15 | 5.20 | .07 | 1.08 | .05 |\n| M87 | 1 | Y | 15.42 | a,h | 55.34 | 48.42 | 37.29 | 1.19 | 2 | 3.52 | .03 | .35 | .00 | 1.02 | .02 |\n| NGC 4552 | ? | Y | 15.35 | a | 1.97 | 1.49 | 35.89 | 1.44 | 2 | 18.62 | .18 | 16.17 | .19 | .99 | .06 |\n| NGC 4569 | 2 | N | 11.86 | h | 992.54 | 837.16 | 38.38 | 1.3 | 4 | .60 | .01 | .24 | .00 | 1.01 | .02 |\n| NGC 4579 | 1 | Y | 20.99 | h | 61.11 | 41.88 | 37.65 | 1.58 | 2 | 14.62 | .07 | 20.00 | .07 | .99 | .02 |\n| NGC 4594 | 2 | Y | 9.08 | a | 7.47 | 15.29 | 35.97 | 0.52 | 3 | 9.32 | .19 | 16.25 | .11 | 1.08 | .06 |\n| NGC 4736 | 2 | Y | 4.89 | a,d | 47.99 | 73.2 | 36.21 | 0.68 | 4 | 2.69 | .05 | .56 | .01 | 1.03 | .02 |\n| NGC 4736b | ? | N | 4.89 | a,d | 29.01 | 20.99 | 35.99 | 1.43 | 4 | 4.03 | .06 | 3.33 | .05 | 1.01 | .03 |\n| NGC 5055 | 2 | N | 7.4 | g | 76.38 | 90.78 | 36.77 | 0.87 | 5 | 1.44 | .03 | .63 | .01 | 1.01 | .02 |\n| NGC 6500 | 2 | Y | 39.7 | g | 31.39 | 27.15 | 38.07 | 1.38 | 4 | 3.92 | .09 | 1.10 | .03 | 1.06 | .03 | \nReferences . - a - Tonry et al. (2001); b - Karachentsev et al. (2002); c - Newman et al. (2001); d - Karachentsev et al. (2003); e - Freedman et al. (2001); f - Tanvir et al. (1999); g - Tully (1988); h - Gavazzi et al. (1999); i - Herrnstein et al. (1999). \nNote . - (2)- Type-1 or type-2 object, depending on presence or absence, respectively, of broad H α . NGC 4258 and NGC 4552 do not fall easily into either category (see text), and the spectral type of the off-nuclear source NGC 4736b is unknown. These three are marked with a '?'; (3) - compact radio core detected (Y) or undetected (N); (4) - distance; (5) - distance reference (see below). When several measurements exist for a galaxy, their average was adopted; (6)-(7) - flux density, averaged over all epochs, in units of 10 -17 erg cm -2 s -1 ˚ A -1 ; (8) - log of monochromatic luminosity at 2500 ˚ A, corrected for Galactic extinction, in units of ergs s -1 ˚ A -1 ; (9) - UV color, f λ (F250W)/ f λ (F330W), after correction for Galactic reddening (see text); (10) - number of epochs at which observations were made; (11), (13) χ 2 per degree of freedom compared to a constant at the mean level. Values implying variability at > 95% confidence are in boldface; (12), (14) - peak-to-peak variation amplitude, with noise subtracted in quadrature, as a fraction of mean flux; (15)-(16) - UV color change between the two epochs with extreme fluxes - [ f max (F250W)/ f max (F330W)]/ [ f min (F250W)/ f min (F330W)] - and its uncertainty."}
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2009MNRAS.400..677Z
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Low-metallicity natal environments and black hole masses in ultraluminous X-ray sources
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2009-01-01
| 16 | 0.46 | 158 |
['accretion', 'accretion disks', 'black hole physics', 'astronomy x rays', '-', '-']
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[]
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We review the available estimates of the masses of the compact object in ultraluminous X-ray sources (ULXs) and critically reconsider the stellar mass versus intermediate-mass black hole (BH) interpretations. BHs of several hundreds to thousands of M<SUB>solar</SUB> are not required for the majority of ULXs, although they might be present in the handful of known hyperluminous (~10<SUP>41</SUP> erg s<SUP>-1</SUP>) objects and/or some sources showing timing features in their power density spectra. At the same time, however, stellar mass BHs may be quite a reasonable explanation for ULXs below ~10<SUP>40</SUP> erg s<SUP>-1</SUP>, but they need super-Eddington accretion and some suitable dependence of the beaming factor on the accretion rate in order to account for ULXs above this (isotropic) luminosity. We investigate in detail a `third way' in which a proportion of ULXs contain ~30-90M<SUB>solar</SUB> BHs formed in a low metallicity environment and accreting in a slightly critical regime and find that it can consistently account for the properties of bright ULXs. Surveys of ULX locations looking for a statistically meaningful relationship between ULX position, average luminosity and local metallicity will provide a definitive test of our proposal.
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[]
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https://arxiv.org/pdf/0909.1017.pdf
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{'L. Zampieri 1 and T. P. Roberts 2': "- 1 INAF-Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, I-35122 Padova, Italy\n- 2 Department of Physics, Durham University, South Road, Durham DH1 3LE, UK \nAccepted ... Received ...; in original form ...", 'ABSTRACT': "Wereview the available estimates of the masses of the compact object in Ultraluminous X-ray Sources (ULXs) and critically reconsider the stellar-mass versus intermediatemass black hole interpretations. Black holes of several hundreds to thousands of M glyph[circledot] are not required for the majority of ULXs, although they might be present in the handful of known hyper-luminous ( ∼ 10 41 erg s -1 ) objects and/or some sources showing timing features in their power density spectra. At the same time, however, stellar mass BHs may be quite a reasonable explanation for ULXs below ∼ 10 40 erg s -1 , but they need super-Eddington accretion and some suitable dependence of the beaming factor on the accretion rate in order to account for ULXs above this (isotropic) luminosity. We investigate in detail a 'third way' in which a proportion of ULXs contain ≈ 30 -90 M glyph[circledot] black holes formed in a low metallicity environment and accreting in a slightly critical regime and find that it can consistently account for the properties of bright ULXs. Surveys of ULX locations looking for a statistically meaningful relationship between ULX position, average luminosity and local metallicity will provide a definitive test of our proposal. \nKey words: accretion, accretion discs - black hole physics - X-rays: binaries", '1 INTRODUCTION': "these puzzling Super-Eddington sources constitute a very interesting class, that remains yet to be fully understood. \nWhen, at the beginning of the '80s, point-like, off-nuclear Xray sources were first detected in the field of nearby galaxies (see, e.g., Fabbiano 1989, 2006), it was immediately recognised that the luminosity of a subset of these objects was unusually large. If physically associated with their host galaxies, these sources had an isotropic luminosity well in excess of the Eddington limit for spherical accretion onto a 10 M glyph[circledot] compact object. These apparently Super-Eddington sources, later called UltraLuminous X-ray sources (ULXs), were first noticed in Einstein data (Long & van Speybroeck 1983; Helfand 1984; Fabbiano 2006). \nNowadays well in excess of 150 candidate ULXs have been detected and catalogued by a variety of X-ray observatories (see e.g. Roberts & Warwick 2000; Colbert & Ptak 2002; Swartz et al. 2004; Liu & Bregman 2005). A small fraction of ULXs are now known to be X-ray luminous interacting supernovae such as those described by Immler (2007). A much larger fraction have subsequently been identified with background AGNs ( ∼ 25%; Swartz et al. 2004; see also Foschini et al. 2002b; Masetti et al. 2003; Wong et al. 2008). This background contamination is stronger in ellipticals, where it accounts for ∼ 44% of all the ULXs, than in spirals ( ∼ 14%; Swartz et al. 2004). However, the majority of \nThe recent detection of a ∼ 62 day modulation in the light curve of M 82 X-1 has been interpreted as the orbital period of the system (Kaaret et al. 2006a,b; Feng & Kaaret 2007). A periodic modulation (12.5 hrs) has also recently been detected in another ULX (NGC 3379; Fabbiano et al. 2006). These results support the notion that ULXs are Xray binary systems, where mass transferred from a donor star falls onto a compact object via an X-ray luminous accretion disc. The high X-ray luminosities of ULXs suggest this compact object is most likely a black hole (BH). \nThe X-ray spectral properties of ULXs show similarities with those of X-ray binaries (XRBs) in our Galaxy (e.g. Foschini et al. 2002a). In many cases the spectrum can be well reproduced by a multicolour disc (MCD) blackbody, representing emission from an accretion disc, plus a power-law continuum (PL), with the latter nominally representing emission from a Compton-scattering corona. This is the same canonical model employed to describe the spectra of Galactic black hole X-ray binaries (cf. McClintock & Remillard 2006). Interestingly, the derived temperature of the MCD component in ULXs is often much lower than that observed in XRBs (e.g. Miller et al. 2003, 2004b; Feng & Kaaret 2005). However, for some of the brightest ULXs, \na possible curvature above 2-3 keV has been reported and equally acceptable fits of their spectra may be obtained with (physically) different models, that suggest the presence of hitherto unusual features such as an optically thick corona, a fast ionised outflow or a slim disc (e.g. Stobbart et al. 2006; Gon¸calves & Soria 2006; Mizuno et al. 2007). \nFast X-ray variability can also reveal much about accretion-powered sources. Although most ULXs show little variability on timescales of seconds to hours (Swartz et al. 2004; Roberts et al. 2004), the recent detections of quasi periodic oscillations in the power density spectra of M 82 X1 and NGC 5408 X-1 has shed new light on the timescales in the inner accretion disc of these systems (Strohmayer & Mushotzky 2003; Fiorito & Titarchuk 2004; Mucciarelli et al. 2006; Strohmayer et al. 2007; Casella et al. 2008). \nStellar optical counterparts have been discovered to be associated with a number of ULXs (Roberts et al. 2001; Goad et al. 2002; Liu et al. 2002, 2004; Kaaret et al. 2004a; Zampieri et al. 2004; Kaaret 2005; Mucciarelli et al. 2005; Soria et al. 2005; Pakull et al. 2006), although only some of them have been associated with stellar objects of known spectral type (e.g. Liu et al. 2002, 2004; Kaaret et al. 2004a; Mucciarelli et al. 2005). In almost all cases, the counterparts appear to be hosted in young stellar environments (e.g. Ramsey et al. 2006; Pakull et al. 2006; Liu et al. 2007) and have properties consistent with those of young, massive stars. However, some ULXs appear to be associated to older stellar populations and at least one possible later-type stellar counterpart is now known (Feng & Kaaret 2008; Roberts et al. 2008). Some ULXs are also associated with very extended optical emission nebulae, that may provide important information on the energetics and lifetime of these systems (Pakull & Mirioni 2002; Roberts et al. 2003). These nebulae are also beginning to be detected as extended radio sources (Miller et al. 2005; Lang et al. 2007). \nThe stellar environment of ULXs can also provide interesting constraints on the properties of ULX binary systems (ULXBs). Several ULXs are located in groups or clusters of OB stars. Isochrone fitting of the cluster colour-magnitude diagram has been attempted and provides cluster ages of tens of millions of years, although there is some disagreement among different authors (Ramsey et al. 2006; Pakull et al. 2006; Liu et al. 2007; Gris'e et al. 2008). These analyses translate into upper limits for the donor masses in ULXBs, assuming that they are coeval to their parent OB association. Typical values of these mass limits are in the range ∼ 10 -20 M glyph[circledot] . Comparison of stellar evolutionary tracks of ULXs with the photometric properties of their optical counterparts on the colour-magnitude diagram may also be used to constrain the masses of their donor stars (e.g. Soria et al. 2005; Copperwheat et al. 2005, 2007). If accurate photometry is available, this approach may also provide interesting clues to the BH mass, once binary evolution and X-ray irradiation effects are taken into account (Patruno & Zampieri 2008). \nULXs play a fundamental role in the framework of Xray source populations in nearby galaxies and can be detected and studied at larger distances than more 'normal' binary sources. An important tool to study the global properties of these populations is their X-ray Luminosity Function (XLF). Grimm et al. (2003) and Gilfanov et al. (2004) found that the XLF of high mass X-ray binaries in the Milky \nWay, Magellanic Clouds and nearby starburst galaxies has a smooth, single power-law behaviour in a broad luminosity range (10 36 -10 40 . 5 erg s -1 ; see also Kaaret & AlonsoHerrero 2008). This suggests that ULXs with luminosity up to a few 10 40 erg s -1 may simply represent the high luminosity tail of the high mass X-ray binary population. \nThese pieces of observational evidence, along with the long-term flux variability and the correlated luminosity/spectral variability (e.g. Kubota et al. 2001; La Parola et al. 2001; Zampieri et al. 2004), strongly suggest that a large fraction of ULXs are accreting BH X-ray binaries with massive donors. The very high luminosity demands that the accretion rate be very high, even in case of efficient, disc accretion. A massive donor is then needed to fuel persistent ULXs, irrespectively of the BH mass (e.g. Patruno et al. 2005; Rappaport et al. 2005; Patruno & Zampieri 2008), and the identification of blue, massive stars as the counterparts of some ULXs confirms this interpretation. However, transient ULXs associated with older stellar populations, may be fueled through the rapid accretion of material accumulated in the accretion disc over a relatively long period of time, and not necessarily be associated with massive companions, as it is the case for, e.g, the Galactic BH candidate GRS 1915+105 (King 2002). \nThe critical issue is then understanding what is responsible for the exceptionally high (isotropic) luminosity of these sources. Two main scenarios have been proposed. Firstly ULXs could be relatively normal stellar-mass BHs ( ∼ < 20 M glyph[circledot] ) that are either anisotropically emitting X-ray binaries in a peculiar evolutionary stage (King et al. 2001; King & Pounds 2003), or are truly emitting above the Eddington limit via a massive, modified accretion disc structure (e.g. photon bubble dominated discs, Begelman 2002; twophase super-Eddington, radiatively efficient discs, Socrates & Davis 2006; slim discs, Ebisawa et al. 2003), or perhaps via some combination of the two (Poutanen et al. 2007; King 2008). Secondly, the compact object could simply be bigger, and the accretion would be in the usual sub-Eddington regime. In this case the compact object would be an intermediate mass black hole (IMBH) with a mass in excess of 100 M glyph[circledot] (e.g. Colbert & Mushotzky 1999). Population synthesis calculations show that, in both scenarios, the mass transfer rates needed to supply the majority of the ULXs can be attained over a significant fraction of the life time of the systems and that the production efficiency of the two models are comparable (albeit with very large uncertainties on both), provided that stellar mass BHs can exceed the Eddington limit by a factor ∼ > 10 (Podsiadlowski et al. 2003; Rappaport et al. 2005; Madhusudhan et al. 2006, 2008). \nIn this Paper we review the available estimates of the masses of the compact object in ULXs and present a critical re-evaluation of the current evidence regarding the stellarmass versus intermediate-mass black hole interpretation. Here and in a companion investigation (Mapelli, Colpi & Zampieri 2009) we highlight an alternative formation scenario, already suggested before but never explored in depth, in which a proportion of ULXs contain ≈ 30 -90 M glyph[circledot] BHs formed in a low metallicity environment and accreting in a slightly critical regime. The plan of the paper reflects this approach. We start from a quite comprehensive review of the available estimates of the masses of the compact objects in ULXs ( § 2) and critically reconsider the 'traditional' in- \nTable 1. Masses of the BH hosted in some ULXs estimated using X-ray spectroscopic methods. \n| | L a X,max (10 40 erg s - 1 ) | Edd. limit b ( M glyph[circledot] ) | MCD fit ( M glyph[circledot] ) | Schwarzschild disc d ( M glyph[circledot] ) | KERRBB fit e ( M glyph[circledot] ) | Slim disc fit f ( M glyph[circledot] ) |\n|---------------|--------------------------------|---------------------------------------|----------------------------------|-----------------------------------------------|---------------------------------------------------|------------------------------------------|\n| M 81 X-1 | 0.66 | 130 | 400 - 40 +160 | 200 +40 - 40 100 +80 - 35 1200 +1000 | 49 +25 - 16 63 +127 - 60 73 +22 - 19 63 +113 - 31 | |\n| M 81 X-9 ∗ | 1.1 | 220 | 400 - 40 +160 | 200 +40 - 40 100 +80 - 35 1200 +1000 | 49 +25 - 16 63 +127 - 60 73 +22 - 19 63 +113 - 31 | |\n| M 101 X-2 | 0.41 | 82 | 400 - 40 +160 | 200 +40 - 40 100 +80 - 35 1200 +1000 | 49 +25 - 16 63 +127 - 60 73 +22 - 19 63 +113 - 31 | |\n| NGC 253 X-1 | 0.29 | 58 | 400 - 40 +160 | 200 +40 - 40 100 +80 - 35 1200 +1000 | 49 +25 - 16 63 +127 - 60 73 +22 - 19 63 +113 - 31 | |\n| NGC 253 X-3 | 0.1 | 20 | 400 - 40 +160 | 200 +40 - 40 100 +80 - 35 1200 +1000 | 49 +25 - 16 63 +127 - 60 73 +22 - 19 63 +113 - 31 | |\n| NGC 1313 X-1 | 2.5 | 500 | +40 200 - 60 2500 +1950 - 1100 | - 500 | 49 +25 - 16 63 +127 - 60 73 +22 - 19 63 +113 - 31 | |\n| NGC 1313 X-2 | 1.5 | 300 | +40 200 - 60 2500 +1950 - 1100 | - 500 | 49 +25 - 16 63 +127 - 60 73 +22 - 19 63 +113 - 31 | 16 ± 1 |\n| NGC 4559 X-7 | 2.1 | 420 | +40 200 - 60 2500 +1950 - 1100 | - 500 | 49 +25 - 16 63 +127 - 60 73 +22 - 19 63 +113 - 31 | 74 ± 5 |\n| NGC 4559 X-10 | 1.2 | 240 | +40 200 - 60 2500 +1950 - 1100 | - 500 | 49 +25 - 16 63 +127 - 60 73 +22 - 19 63 +113 - 31 | 31 +12 - 9 |\n| NGC 5204 X-1 | 0.5 | 100 | +40 200 - 60 2500 +1950 - 1100 | - 500 | 49 +25 - 16 63 +127 - 60 73 +22 - 19 63 +113 - 31 | 23 ± 3 | \npretations of the nature of these sources ( § 3). We then discuss the low-metallicity scenario ( § 4) and some observational tests to investigate it ( § 5). A conclusion section ( § 6) follows.", '2 MASS ESTIMATES IN ULXS: METHODS AND RESULTS': "Thanks to the identification of the optical counterparts of a handful of ULXs, the measurement of the mass function of ULXBs is now well on the way to becoming possible, and will provide direct constraints on the masses of individual sources. Unfortunately, given the observational difficulties associated with such measurements - most notably their faintness, with typical magnitudes in the range m v ∼ 22 -26 (Roberts et al. 2008), and the contamination of the counterpart spectrum by nearby stars - only one very recent measurement of optical periodicity has been made (Liu, Bregman & McClintock 2009), and a mass function is yet to be constrained. However, this is where the observational effort is focussed at present and where the definitive answer to the question of whether intermediate or stellar mass BHs power ULXs will come from. Claims of a radial velocity shift of ∼ 300 km s -1 in the He II λ 4686 line have been reported for the optical counterpart of NGC 1313 X-2 by Pakull et al. (2006), but this measurement may be uncertain (see e.g. Mucciarelli et al. 2005). \nUntil these measurements are performed, we have to rely on indirect methods to estimate the BH mass. Assuming that the emission of ULXs originates from accretion and that it is stationary and isotropic, a lower limit for the BH \nmass M BH is obtained for Eddington-limited accretion ( L ≈ L Edd ): \nM BH M glyph[circledot] glyph[similarequal] 80 ( L 10 40 erg s -1 ) glyph[similarequal] 200 ( L X 10 40 erg s -1 ) , (1) \nwhere L X typically refers to the [0.2-10] or [0.3-10] keV band and we consider a 'bolometric correction' of ∼ 2 to account for the flux emitted outside this band. In fact, for the typical spectral parameters of a bright ULX (colum density ∼ 3 × 10 21 cm -2 and power-law photon index ∼ 1 . 7) the bolometric flux is ≈ 2 times larger than the flux emitted in the 0.2-10 keV band (e.g. Patruno & Zampieri 2008). This equation is also sensitive to the material accreted; the Eddington limit will rise by a factor ∼ 2 for the accretion of helium (e.g. Grimm et al. 2003). Masses of some ULXs calculated from equation (1) are reported in Table 1. Estimates based on this argument may become more reliable if there is some other evidence that the emission is (almost) isotropic, as for example when ULXs appear to be responsible for the photo-ionization of their surrounding optical nebulae (e.g. Kaaret et al. 2004a; Abolmasov et al. 2007). \nSpectroscopic estimates of the BH masses have also been attempted assuming that the soft component observed in ULX X-ray spectra can be ascribed to emission from an accretion disc (e.g. Miller et al. 2003, 2004a). Assuming that this spectral component can be modelled with the so-called multicolour disc blackbody model (MCD; Mitsuda et al. 1984), it is possible to express the BH mass M BH as (e.g. Lorenzin & Zampieri 2009): \nM BH M glyph[circledot] = f 2 67 . 5 b ( D 1 Mpc ) ( K BB cos i ) 1 / 2 , (2) \nwhere D is the distance of the source, f is a colour correc- \nTable 2. Masses of the BH hosted in some ULXs estimated using timing methods. \n| | L a X,max (10 40 erg s - 1 ) | Edd. limit b ( M glyph[circledot] ) | QPO c ( M glyph[circledot] ) | Break/Lack of variability d ( M glyph[circledot] ) |\n|--------------|--------------------------------|---------------------------------------|--------------------------------|------------------------------------------------------|\n| Ho II X-1 | 1.7 | 340 | | ∼ < 100 |\n| M 82 X-1 | 17 | 3400 | 95-1300 | 25-520 |\n| NGC 5408 X-1 | 0.85 | 170 | 115-1300 | ∼ 100 |\n| NGC 4559 X-7 | 2.1 | 420 | | 38-1300 | \n- a From Cropper et al. (2004); Kaaret et al. (2004a); Mucciarelli et al. (2006); Strohmayer et al. (2007).\n- b M BH computed from eq. (1). \n- d Cropper et al. (2004); Dewangan, Titarchuk & Griffiths (2006); Goad et al. (2006); Soria et al. (2004). \ntion factor that accounts for transfer effects (e.g. Shimura & Takahara 1995; Zampieri et al. 2001; Davis et al. 2005; Hui et al. 2005), b is the inner radius in units of the gravitational radius, and K BB is the MCD normalization inferred from the spectral fit. In their work Miller et al. (2003) and Miller et al. (2004a) adopted b = 9 . 5 and f = 1 . 7 and derived estimates of M BH well in excess of 100 M glyph[circledot] for M 81 X-9, NGC 1313 X-1 and X-2. A similarly large value of M BH is obtained also for NGC 4559 X-7 using the MCD spectral parameters of Cropper et al. (2004) (see Table 1). However, recently Lorenzin & Zampieri (2009) computed appropriate correction factors for b in the case of a relativistic, standard accretion disc and showed that, unless the BH is maximally rotating, the BH masses inferred for the same sources can be significantly lower than the values estimated by Miller et al. (2003) and Miller et al. (2004a) (see again Table 1). Smaller BH masses have also been obtained through direct spectral fits of relativistic accretion disc models for a sample of disc-dominated ULXs by Hui & Krolik (2008). \nThe interpretation of the soft component in terms of emission from a standard accretion disc suffers from a high degree of equivocality. Spectral fits with a disc component and a comptonizing corona to the best available X-ray spectral data indicate that ULXs display distinct spectral curvature above 2 keV (Stobbart et al. 2006; Gladstone et al. 2009). This is because the corona is optically thick and cool, and hence hides the inner part of the accretion disc, in what is likely an extreme form of the so-called very high state of Galactic BH candidates (Done & Kubota 2006; Stobbart et al. 2006). In this case, the soft component is produced by the visible outer regions of the accretion disc. Only upper bounds to the inner disc radius can be obtained and, similarly, the spectroscopic estimates for M BH reported in Table 1 should be considered as upper limits. However, as the thick corona is probably only launched at extreme accretion rates, this implies that this 'ultraluminous' state is in the Super-Eddington regime, and hence the BH masses are relatively small ( < 100 M glyph[circledot] ; Roberts 2007). Further to this, Gladstone et al. (2009) consider the energy required to launch the thick coronae, and from that calculate the intrinsic (corona-less) disc temperatures, mainly recovering temperatures in the correct regime for the discs around stellar-mass BHs (0.7 - 1 keV). A subset of ULXs retain apparently cool discs even after this correction; however, Gladstone et al. (2009) argue this is because these are the highest accretion rate stellar-mass BHs, in which a strong \nwind is launched from the central regions, creating a cool photosphere. In a similar vein, for other models characterising super-Eddington accretion, such as the slim disc (e.g. Watarai et al. 2000; Ebisawa et al. 2003) or photon bubble models (Finke & Bottcher 2007), the entire 0.2-10 keV spectrum is produced in the accretion disc, although its physical state is completely different with respect to that of a standard disc. However, indirect estimates of M BH can again be obtained from X-ray spectral fits and give values typically in the range ≈ 15 -75 M glyph[circledot] (Vierdayanti et al. 2006, 2008; see again Table 1). \nThe mass estimates inferred from X-ray spectral fits of ULXs depend critically on the interpretation of their spectra which, as mentioned above, is not unique. The situation will improve in the future as our understanding of the spectral evolution of ULXs will increase and it will be possible to select spectral models on the basis of their consistency with the observed correlation patterns (see e.g. Feng & Kaaret 2009; Kajava & Poutanen 2009 for preliminary work on ULX spectral variability based on XMM-Newton data). Nonetheless, the available data points towards BH masses definitely smaller than those estimated from the early MCD spectral fits. \nIn the last few years, X-ray timing has provided a new opportunity to estimate BH masses in ULXs thanks in particular to the detection of broad band noise and quasi periodic oscillations (QPOs) in the power density spectrum (PDS) of some ULXs, as M 82 X-1 ( ν QPO = 54 -166 mHz; Strohmayer & Mushotzky 2003; Mucciarelli et al. 2006) and NGC 5408 X-1 ( ν QPO glyph[similarequal] 20 mHz; Strohmayer et al. 2007). Extrapolating timing and spectral-timing correlations known to exist for similar timing features in BH binaries and assuming that the frequency of the QPO scales inversely to M BH , various estimates have been obtained, which are consistent with a rather broad interval of values: 10 -1000 M glyph[circledot] for M 82 X-1 (Fiorito & Titarchuk 2004; Mucciarelli et al. 2006; Feng & Kaaret 2007), 600 -5000 for NGC 5408 X-1 (Strohmayer et al. 2007). Recently, a new timing approach has been presented to assess BH masses in ULXs, which is based the so called 'variability plane', populated by both Galactic black-hole candidates and active galactic nuclei. Assuming that the accretion flow in ULXs behaves in a similar way (which remains an open question) and taking into account the uncertainty on the efficiency of the accretion disc, Casella et al. (2008) find that M BH is in the interval ∼ 95 -1300 M glyph[circledot] for M 82 X-1 and ∼ 115 -1300 M glyph[circledot] for \nNGC 5408 X-1 (see Table 2). In combination with QPOs, the scaling of the break frequencies of the broad band noise, by comparison with the corresponding timing features of Galactic BH candidates, has also been proposed for estimating M BH . This method has been applied to NGC 5408 X-1 and gives a similar range of masses to QPOs (see Table 2). A tentative identification of a break at ∼ 28 mHz in the PDS of NGC 4559 X-7 has also been reported (Cropper et al. 2004), although Barnard et al. (2007) called it into question. The inferred BH mass is reported in Table 2. \nThe problem with using timing properties of ULXs to directly infer masses is that it remains to date unclear how exactly these quantities are related. All the estimates are based upon tentative identifications of the timing features and the use of scaling laws that are known to hold only for a limited number of objects, and are therefore highly uncertain. \nA further possibility is to use the non-detection of variability power in the PDS to limit the size of the BH, assuming that all power is at higher frequencies, as is seen in various XRB states. Adopting this approach Goad et al. (2006) found M BH ∼ < 100 M glyph[circledot] for Holmberg II X-1 (see again Table 2). In fact, Heil et al. (2009) have demonstrated that suppressed temporal variability (compared to the PDS of classic XRBs and AGNs) appears a common feature of ULXs. There are several possible explanations of this - the variability is limited to higher frequencies, the data in the XMM-Newton band pass are disc-dominated (and so any variability is heavily diluted) or the ULXs are in a new, super-Eddington accretion state in which the X-ray emission is very stable (note this is predicted in the hydrodynamic simulations of highly super-Eddington accretion by Ohsuga 2007). Again, the common thread running through all these models is that the BH is relatively small.", '3 INTERMEDIATE OR STELLAR-MASS BHS?': 'We have reviewed the mass estimates drawn from observations of individual ULXs; we now ask how these results fit into our more general understanding of the possible nature of their underlying engines. ULX models differ mainly in the assumptions on the physical state of the disc and its mode of accretion. If the accretion disc is in a standard regime, then emission is isotropic and the most straightforward interpretation of the exceptionally high luminosity of ULXs is that they contain BHs with large masses. Both the very high luminosity and low characteristic temperature (and high normalization, see previous Section) of the soft spectral component have been taken as evidence for this interpretation. \nBut how big is the BH mass? Early estimates based on equations (1) and (2) gave masses largely in excess of 100 M glyph[circledot] (up to several thousands; see Table 1). The obvious question is then how a BH this massive may form. It has been proposed that remnants from the collapse of Population III stars formed in cosmological epochs (Madau & Rees 2001) may trigger ULX activity, if they can capture a donor star to accrete from. A second formation route for IMBHs may be in globular clusters, through repeated mergers of stellar mass BHs (Miller & Hamilton 2002), or in young, dense stellar super clusters, from the dynamical collapse of \nsupermassive stars in their centres (e.g. Portegies Zwart et al. 2004). ULX activity would be sustained by binary companions captured in the cluster (Blecha et al. 2006). The latter may be a possible explanation for some ULXs (e.g. M82 X-1) but, in galaxies with starburst activity, the majority do not appear inside such supermassive clusters (e.g. Zezas et al. 2002; Kaaret et al. 2004b). Another difficulty with the IMBH interpretation is the apparent break in the power-law slope of the XLF of the high mass X-ray binary population in external galaxies at a luminosity ∼ 2 × 10 40 erg s -1 . If there is a population of large IMBHs contributing to the XLF, the break suggests that they turn off at this luminosity; yet this is at a rather low fraction of the Eddington limit for putative large IMBHs ( ∼ 10% for a 1000 M glyph[circledot] BH). This would be rather unusual behaviour, as no other accreting class switches off at only a fraction of their Eddington limit (Roberts 2007). Finally, the co-location of ULXs with regions of star formation, such as those in the Antennae and Cartwheel galaxies, implies that they must be (relatively) short-lived, which requires successive generations of ULXs to be formed over the duration of the star formation event. Thus, if all ULXs in these regions were IMBHs, an unfeasibly large fraction of star forming mass would end up in IMBHs (King 2004; Mapelli et al. 2008). In principle, these arguments rule out all but a small minority of ULXs from being IMBHs bigger than ∼ 100 M glyph[circledot] . \nSo can we explain ULXs as stellar-mass BHs? If the accretion flow in ULXs is in a different regime, the situation may be different and either the isotropy and/or the Eddington limit may be circumvented. This occurs if the accretion rate is at or above Eddington, so that radial advection of thermal and radiative energy (slim disc; Abramowicz et al. 1988; Ebisawa et al. 2003) and/or radiationdriven instabilities (photon bubble model; Begelman 2002, 2006) set in. A slim disc can sustain larger accretion rates than a standard disc and modest super-Eddington luminosities ( ∼ < 10 L Edd ). As the emission is isotropic, luminosity scaling arguments similar to those discussed above give M BH ∼ > 20( L X / 10 40 erg s -1 ) M glyph[circledot] . Therefore, the presence of slim discs would imply that only the brightest known ULXs (with L X ∼ > 3 × 10 40 erg s -1 ) need BHs significantly more massive than the stellar-mass BHs in our Galaxy (see also Table 1). Accretion discs dominated by photon bubble transport may also reach super-Eddington luminosities, while remaining geometrically thin. For a 20 M glyph[circledot] BH, the maximum luminosity is ∼ 30 L Edd (Begelman 2006). This may in principle account for the emission of all but the very brightest ( ∼ > 5 × 10 40 erg s -1 ) ULXs, but photon bubble-dominated discs are subject to the same thermal and viscous instabilities that characterize the inner region of radiation pressure dominated discs and hence may be significantly unsteady (with more than a factor of 10 variation in the emitted flux) on short time scales (e.g. Zampieri et al. 2001). However, bright ( ∼ > 10 40 erg s -1 ) ULXs typically show more limited fluctuations in the observed X-ray luminosity. An alternative scenario for generating steady, super-Eddington luminosities from stellar-mass BHs, is the radiatively efficient, two-phase super-Eddington accretion disc model by Socrates & Davis (2006). In this model the gravitational potential energy is not trapped in the disc, but effectively removed from it through magnetic buoyancy and dissipated in a corona. Fields anchored in the disc and/or Compton drag in the \nlow density corona prevent the launching of a wind, keeping the radiative efficiency high and assuring super-Eddington luminosities. However, we note that there are many assumptions and theoretical uncertainties that need to be clarified in this model, and a quantitative estimate of the maximum attainable luminosity is not yet available. \nIf accretion becomes largely super-Eddington, other processes may be present that complicate the picture. A thick disc may form and the emission becomes beamed (King et al. 2001; King 2002). At the same time, outflows and powerful ejection of matter along the axis perpendicular to the accretion disc may be produced (as in SS433; e.g. Poutanen et al. 2007). In these assumptions, emission is no longer isotropic. If L iso is the apparent isotropic luminosity, the BH mass inferred from equation (1) must then be corrected for the so called beaming factor b f , that represents the fractional opening angle of the beam: M BH glyph[similarequal] 20( b f / 0 . 1)( L 0 . 2 -10 ,iso / 10 40 erg s -1 ) M glyph[circledot] . Hence, simple beaming can not account for bright ( ∼ > 2 × 10 40 erg s -1 ) ULXs, unless one is willing to consider a beaming factor < 0 . 05 and therefore a very narrow opening angle ( < 18 · ), that appears to be more consistent with a jet rather than a geometrical funnel in a thick disc. However, in addition to having beamed emission, thick discs may also radiate at super-Eddington luminosities, reaching at most (1 + ln ( ˙ M/ ˙ M Edd )) L Edd (e.g. Poutanen et al. 2007; King 2008). A hypothetical ULX in this state would bear some similarity to a Galactic BH candidate in the very high state, albeit with the ULX being in a much more extreme version of this state, with powerful winds carrying away the excess matter and energy, potentially thickening the corona and even producing a cool photosphere (consistent with the X-ray spectral modelling of Gladstone et al. 2009). A combination of a beaming factor b f glyph[similarequal] 0 . 3 -0 . 5 and super-Eddington emission ( ∼ 10 L Edd ) may in principle explain ULXs with luminosities up to ≈ 10 40 erg s -1 , assuming accretion onto a stellar-mass BH. Furthermore, if a dependence of the beaming factor on the accretion rate is assumed ( b f ∝ ( ˙ M/ ˙ M Edd ) -2 ; King 2009), it might be possible to account also for the high luminosity tail of the ULX population. \nIn order to assess the viability of the different scenarios for the origin of ULXs, it is necessary to understand the possible evolutionary history of the various types of candidate binary systems and compare them with the available observations. Calculations of the evolutionary tracks of ULX binaries and model population studies of systems containing stellar-mass BHs and IMBHs have been carried out by several authors (e.g. Podsiadlowski et al. 2003; Patruno et al. 2005; Rappaport et al. 2005; Madhusudhan et al. 2006). Although calculations depend sensitively on uncertain parameters of the common envelope phase, it turns out that stellar-mass BHs accreting at super-Eddington rates may be able to account for most of the observed ULXs (except for the brightest), if they violate the Eddington limit by a factor ∼ 10 -30 (Podsiadlowski et al. 2003; Rappaport et al. 2005). Similarly, IMBH systems might produce bright ( L X ∼ > 10 40 erg s -1 ), persistent ULXs and have an acceptable production efficiency if the donor star is ∼ > 10 M glyph[circledot] and the initial orbital separation is small ( ∼ < 6 -40 times the initial main sequence radius of the donor; Patruno et al. 2005; Madhusudhan et al. 2006). \nAlong the same lines, theoretical calculations of the color-magnitude (CM) diagrams for systems containing stellar-mass BHs and IMBHs are used to constrain ULX models and their evolutionary history by comparison with observations of their optical counterparts (Madhusudhan et al. 2008; Patruno & Zampieri 2008). As already mentioned, in order to supply the mass transfer rates needed to fuel ULXs, rather massive donor stars are required. The evolutionary tracks of such systems are strongly affected by the binary interaction and the emission from the accretion disk, including X-ray irradiation. Numerical computations show that the regions of the CM diagram with the highest probability of finding ULX optical counterparts have B -V between ∼ -0 . 1 and -0 . 3 and correspond to the early phases of the evolution of massive donors, while they are on the main sequence or the subgiant branch (Madhusudhan et al. 2008). This result is consistent with the properties of the observed counterparts. Similarly, the most favourable orbital periods are between 1 and 10 days, corresponding again to the main sequence or subgiant phases. This is true for both stellar-mass BH and IMBH systems and depends on the fact that the donors spend most of their life time in these phases. In these conditions normal nuclear-driven mass transfer is effective and provides sufficiently high transfer rates to sustain the ULX emission, although in rare circumstances the evolution may be driven by the thermal timescale mass transfer during the giant phase.', '4 A DIFFERENT INTERPRETATION': 'A critical revaluation of the available observational evidence presented in Sections 2 and 3 indicates that BHs of several hundreds to thousands M glyph[circledot] are not required for the majority of ULXs. However, it is not possible to rule out that they are present in the handful of known hyper-luminous ( ∼ 10 41 erg s -1 ) objects and/or in the sources showing large amplitude broad band noise or QPOs in their PDS (such as M 82 X-1 and NGC 5408 X-1; see Table 2). At the same time, models with stellar mass BHs may work for a large fraction of the ULX population, if the accretion flow has some degreee of beaming and is super-Eddington. Bright ( ∼ > 10 40 erg s -1 ) ULXs may be accounted for if some form of modified beaming that accounts for a suitable dependence of b f on the parameters of the accretion flow is allowed (King 2009). Although none of these scenarios can be ruled out, the fact that the observational limits discussed in Section 2 are converging towards masses ∼ < 100 M glyph[circledot] , but bigger than stellar mass BHs, led us to consider an alternative interpretation. In our scenario bright ULXs may contain BHs with masses above 30-40 M glyph[circledot] and up to ∼ 80 -90 M glyph[circledot] , formed from ordinary stellar evolution of massive (30 -120 M glyph[circledot] ) stars in a low metallicity natal environment. While this idea has already been suggested before (e.g. Pakull & Mirioni 2002; Cropper et al. 2004; Zampieri et al. 2004), it has not yet been explored quantitatively in detail. \nStars with mass ∼ > 8 M glyph[circledot] produce compact remnants from the gravitational collapse of the iron core. For stars up to ∼ 25 -30 M glyph[circledot] the collapse is halted when the core reaches nuclear densities: the star explodes and a neutron star forms. For larger main sequence masses, the early acccretion of the inner mantle onto the core before shock passage and \nFigure 1. Final mass as a function of initial mass for stars of different metallicities as computed by Maeder (1990, 1992; black ), Maeder & Myenet (2001; blue ), Portinari et al. (1998, red ), and Eldridge & Tout (2004, cyan ). \n<!-- image --> \nthe fallback of material afterwards cause the newly formed proto-neutron star to collapse to a BH after the supernova explosion (Zampieri 2002 and references therein). At solar metallicity, these fallback BHs reach at most ∼ 10 M glyph[circledot] as, for very massive stars with mass ∼ > 40 M glyph[circledot] , the stellar envelope is in large part effectively removed through line-driven winds, while the remaining part is expelled by the supernova explosion. \nFor sub-solar metallicities, however, this mechanism becomes progressively less efficient and stars with masses above ∼ 30 -40 M glyph[circledot] may retain rather massive envelopes at the time of explosion. The supernova shock wave then loses more and more energy in trying to unbind the envelope until it stalls and most of the star collapses to form a BH with a mass comparable to that of the pre-supernova star (Fryer 1999). These may be the BHs hosted in some ULXs. Their mass would not be significantly larger than ∼ 80 -90 M glyph[circledot] as above ∼ 100 -120 M glyph[circledot] a star undergoes pulsational pairinstability in its core and most of the envelope mass is expelled. We note that the possibility of forming BHs in this mass range through a different channel (binary mergers of massive components) was also proposed few years ago (e.g. Belczynski, Sadowski & Rasio 2004). \nComputation of the evolution of massive stars up to advanced evolutionary stages for different metallicities and/or including mass loss have been performed by several authors (e.g. Hellings & Vanbeveren 1981; Maeder 1990, 1992; Portinari et al. 1998; Heger et al. 2003; Chieffi and Limongi 2004; Eldridge and Tout 2004; Hirschi 2007). These works adopted known empirical parameterizations of the mass loss rate for stars over the whole Hertzsprung-Russell diagram (e.g. de Jager et al. 1988; Nieuwenhuijzen & de Jager 1990). A cer- \ntain degree of uncertainty is introduced if the star enters some peculiar evolutionary stages, such as the Wolf-Rayet (WR) stage (e.g. Langer 1989; Wellstein & Langer 1999). In fact, mass loss rates in WR and O stars may be significantly reduced if the wind is clumpy as a consequence of, e.g., supersonic turbulence (Moffat & Carmelle 1994; Fullerton et al. 2006). In particular, during the WR phase a decrease by a factor of ∼ > 3 with respect to homogeneous wind models is attainable. This turns out to be compatible with some observational estimates and would clearly lead to more massive pre-supernova stars and hence more massive BHs. Additional uncertainty is caused by the dependence of mass loss on metallicity. A scaling law ∝ Z 0 . 5 is often adopted for hot stars (see e.g. Kudritzki et al. 1989; Nugis & Lamers 2000). For example, at the end of main sequence, the mass of a star with an initial mass of 100 M glyph[circledot] is ∼ 25 M glyph[circledot] at solar metallicity and ∼ 75 M glyph[circledot] for Z ≈ 0 . 1 Z glyph[circledot] . However, the mass lost during the He burning phase may be much more significant than that occurring during main sequence. According to the adopted mass loss history, the final mass of a star (after all nuclear burning stages are exhausted) may differ up to a factor of ∼ 2 (or even more for clumpy winds) for a given metallicity. Considering again a star with an initial mass of 100 M glyph[circledot] , Figure 1 shows that its final mass may be in the interval ∼ 3 -6 M glyph[circledot] for Z ≈ Z glyph[circledot] and ∼ 30 -70 M glyph[circledot] for Z ≈ 0 . 1 Z glyph[circledot] . \nAs already mentioned above, the final evolutionary stages of a star and, in particular, the outcome of the final collapse depend critically on how massive is the envelope that it retains at the time of explosion. Therefore, owing to their larger final masses, the fate of stars with sub-solar metallicity is likely to be be quite different from that of \nhigher metallicity stars. Although different authors obtain different results for the mass of the compact remnant, it is not unreasonable to think that, if an envelope more massive than ∼ 30 -40 M glyph[circledot] is retained at the time of explosion, a low metallicity ( Z ≈ 0 . 1 Z glyph[circledot] ) star may collapse directly to form a BH of comparable mass. Looking again at Figure 1, it is possible to see that, already at metallicities ∼ < 0 . 2 Z glyph[circledot] , stars with initial mass ∼ > 60 M glyph[circledot] appear to possess final envelope masses that overcome this threshold for direct BH formation. Significant stellar rotation (hundreds of km s -1 ) may change this picture somewhat, as it enhances the mixing of heavy elements throughout the star increasing the metal content of the envelope and consequently mass loss (e.g. Maeder & Meynet 2001; Meynet & Maeder 2005). Also, ejection of part of the envelope during the final collapse can not be ruled out. However, if the core is not rapidly rotating, there is no good reason why most of the star should not collapse into a BH. Therefore, the formation of a ∼ > 30 -40 M glyph[circledot] BH throughout the evolution of a low metallicity, slowly rotating star of 40 -120 M glyph[circledot] appears a viable possibility. The parameter space in the metallicity-main sequence mass plane where this formation channel may actually work corresponds to the black-shaded area between ∼ 40 and ∼ 100 M glyph[circledot] in Figure 1 of Heger et al. (2003) (see also Figure 5 in Eldridge and Tout 2004) 1 . \nAt variance with intermediate mass BHs, the formation of these very massive stellar remnant BHs does not require an exotic, new mechanism but is referable to ordinary stellar evolution. Also, the unbroken power-law slope of the X-ray binary population up to ∼ 2 × 10 40 erg s -1 is consistent with the fact that variations in metallicity may produce a continuum distribution of BH masses from the stellar-mass BHs of 10 -20 M glyph[circledot] up to the suggested BHs of ≈ 40 -90 M glyph[circledot] . Given the size of these BHs, no difficulty with the fraction of star-forming mass in large starbursts ending up in BHs would arise, and so the objection of King (2004) for ULXs as ∼ 1000 M glyph[circledot] IMBHs is circumvented. At the same time, only modest beaming ( b f ∼ 0 . 5) or slight violations of the Eddington limit (a factor of a few) would be needed to account for the luminosity of bright ( ∼ > 10 40 erg s -1 ) ULXs, at variance with the extreme accretion scenarios required by stellar mass BH models. Also the essentially isotropic irradiation of X-ray photoionised nebulae would find an explanation.', '5 OBSERVATIONAL TESTS': "Model population studies for our scenario are needed to determine the production efficiency of binary systems containing very massive BHs and to understand if they are in agreement, in a statistical sense, with the available X-ray and optical data of ULXs. However, there may be already some indications strengthening our present suggestion. In a parallel, preliminary investigation, we show that massive BHs formed in low metallicity environments might well explain most of the ULXs observed in the Cartwheel galaxy (Mapelli, Colpi & Zampieri 2009). Also, the optical luminosities of massive BH systems would be, on average, larger \nthan that of stellar-mass BHs, as the former allow for more massive donors ( ∼ > 25 M glyph[circledot] ) and have more extended accretion discs that dominate the optical emission. This would make them more consistent than stellar-mass BHs with the observed distribution of the luminosity of the ULX optical counterparts (e.g. Madhusudhan et al. 2008; Patruno & Zampieri 2008). \nA crucial aspect of the interpretation of ULXs in terms of BHs from the direct collapse of lowZ , massive stars is the metallicity of the environment in which ULXBs form. The available estimates appear to favour a low metallicity scenario, although there are some discrepancies. Optical observations appear to provide evidence of sub-solar metallicity in the environment of some ULXs. The emission nebula surrounding Ho II X-1 has a spectrum resembling that of a high-excitation H II region typical for low metallicity ( Z ∼ 0 . 1 Z glyph[circledot] ) star-forming regions (Mirioni 2002; Pakull & Mirioni 2002). The stellar environment of NGC 4559 X-7 shows a blue-to-red supergiant ratio (3 ± 1) and colours of the red supergiant population consistent with a low metal abundance environment with Z = 0 . 1 -0 . 4 Z glyph[circledot] , similar to that in the SMC and other nearby dwarf galaxies (Soria et al. 2005). Also the stellar field around NGC 1313 X-2 is characterised by a low metallicity ([ Fe/H ] = -1 . 9 ± 0 . 3), as inferred from the intrinsic colour of the red giant branch (Gris'e et al. 2008); studies of the metal abundance of H II regions in NGC 1313 also give low values of Z ( ∼ 0 . 008; Walsh & Roy 1997; Hadfield & Crowther 2007). \nWinter et al. (2007) analysed high signal-to-noise XMM spectra of a sample of 14 ULXs, trying to determine the Oxygen abundance from the detection of K-shell photoionization edges. They apparently find values that match the solar abundance. However, the comparison of the X-ray estimates with a compilation of [ O/H ] ratios determined through spectrophotometric studies of H II regions (Pilyugin 2004) and with the luminosity-metallicity relation derived from the Sloan Digital Sky Survey (Tremonti 2004) shows significant systematic differences, especially at low galaxy luminosities and sub-solar metallicities. We note that Ho II X-1 provides a clear example of this dichotomy for a ULX; its XMMNewton RGS spectrum suggests a significantly higher metallicity ( ∼ 0 . 6 times solar; Goad et al. 2006) than the optical data. At the same time, X-ray spectral fits in at least the case of NGC 4559 X-7 seem to provide evidence for a more subsolar metallicity ( Z ∼ 0 . 3 Z glyph[circledot] ; Cropper et al. 2004). \nIt will be possible also to test our proposal against the stellar-mass BH interpretation as it will lead to a different spatial distribution of bright ULXs. In fact, ULXs from stellar-mass BHs should essentially appear anywhere in regions of star formation or in young stellar environments, regardless of metallicity. Indeed there may even be a bias towards these objects appearing in low metallicity regions, as effects such as the low mass loss rate in stellar winds will tend to keep binaries close, allowing more high mass transfer systems. However, if we make the reasonable assumption that the modes of accretion and Eddington rates will be similar in both standard ∼ 10 -20 M glyph[circledot] black holes, and the ∼ 30 -90 M glyph[circledot] black holes examined in this paper, then the latter black holes may simply be on average brighter . Hence, in our proposed scenario ULXs should show some evidence of correlation (in terms of position and average luminosity) with low metallicity environments. So, one of the definitive \ntests of our proposal would be to survey ULX locations, and determine whether a relationship between ULX luminosity and local metallicity was evident in a large enough sample to provide statistically meaningful results. \nWe note that evidence supporting our argument is already available. Firstly, Swartz et al. (2008) have recently surveyed galaxies within the Local Volume and determined that the specific ULX frequency decreases with host galaxy mass above ∼ 10 8 . 5 M glyph[circledot] . This means that smaller, lower metallicity systems have more ULXs per unit mass than larger galaxies, consistent with the idea that BHs can at least form and/or feed more efficiently in low metallicity environments. Secondly, we note that there is interesting evidence from our own relative backyard that the brighter ULXs are more abundant in the late-type spiral galaxies we might expect to be low-metallicity systems. The ROSAT High Resolution Imager ULX survey of Liu & Bregman (2005) lists 15 ULXs within 5 Mpc 2 , four of which have observed X-ray luminosities in excess of 5 × 10 39 erg s -1 . All four of the very luminous ULXs reside in galaxies of Hubble type Sd or later. This compares to only 3 of the 11 lower-luminosity ULXs being hosted by similarly late systems - the remainder are in galaxies of type between Sab - Scd. Although the sample of ULXs is small, this clearly supports the case that brighter ULXs preferentially occur in the smaller, lower-metallicity systems where we might expect to find the very massive stellar remnant BHs. \nIt is worth noting that, recently, Prestwich et al. (2007) and Silverman & Filippenko (2008) succeeded in performing dynamical mass measurements using Gemini and Keck spectra of the Wolf-Rayet optical counterpart of IC 10 X-1, a variable X-ray source in the the Local Group metal poor starbust galaxy IC 10. They find a BH mass in the range 23 -33 M glyph[circledot] , which represents the most massive BH known to exist in a binary system and definitely corroborates our interpretation.", '6 CONCLUSIONS': 'In the last few years, X-ray and optical observations have significantly boosted our understanding of ULXs. We are now confident that the majority of these sources are X-ray binaries in external galaxies and we suspect that many may have massive binary companions. Yet, the most fundamental questions on ULXs still remain to be definitively answered: do they contain stellar or intermediate mass BHs? How do they form? \nAcritical revaluation of the available evidence presented indicates that BHs of several hundreds to thousands M glyph[circledot] are not required for the majority of ULXs, although they might be present in the handful of known hyper-luminous ( ∼ 10 41 erg s -1 ) objects and/or some sources showing timing features in their power density spectra. At the same time, however, stellar mass BHs may be quite a reasonable explanation for ULXs below ∼ 10 40 erg s -1 , but they need super-Eddington accretion and some suitable dependence of the beaming factor on the accretion rate in order to account for ULXs above this (isotropic) luminosity. \nWe investigated in detail an alternative scenario in which bright ULXs contain BHs with masses above ∼ 30 -40 M glyph[circledot] and up to ∼ 80 -90 M glyph[circledot] , produced by stars with initial, main sequence mass above ∼ 40 -50 M glyph[circledot] . At sub-solar metallicity, the explosion energy of these stars is not sufficient to unbind the envelope and most of the star collapses to form a BH with a mass comparable to that of the pre-supernova star. These may be the BHs hosted in bright ULXs. Above ∼ 100 -120 M glyph[circledot] pulsational instability becomes effective and most of the envelope is expelled from the star. \nThe formation of these very massive stellar remnant BHs does not require an exotic, new mechanism but is referable to ordinary stellar evolution. For luminosities ∼ 10 40 erg s -1 , this would imply only modest violations of the Eddington limit, attainable through very modest beaming ( b f ∼ 0 . 5) and/or slightly super-critical accretion. \nMeasurements of the metallicity of the environment of some ULXs appear to favour a low metallicity scenario, although there are some discrepancies. Surveys of ULX locations looking for a statistically meaningful relationship between between ULX number, position, average luminosity and local metallicity will provide a definitive test of our proposal.', '7 ACKNOWLEDGEMENTS': 'LZ acknowldges financial support from INAF through grant PRIN-2007-26. 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2003Sci...300.1119M
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Formation of a Black Hole in the Dark
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2003-01-01
| 14 | 0.48 | 158 |
['-', 'astrophysics']
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[]
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We show that the black hole in the x-ray binary Cygnus X-1 was formed in situ and did not receive an energetic trigger from a nearby supernova. The progenitor of the black hole had an initial mass greater than 40 solar masses, and during the collapse to form the ~10-solar mass black hole of Cygnus X-1, the upper limit for the mass that could have been suddenly ejected is ~1 solar mass, much less than the mass ejected in a supernova. The observations suggest that high-mass stellar black holes may form promptly, when massive stars disappear silently.
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https://arxiv.org/pdf/astro-ph/0305205.pdf
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{'Formation of a Black Hole in the Dark /a0': "I. F'elix Mirabel ✁✄✂ ☎ , Irapuan Rodrigues ✁ \n✆ Service d'Astrophysique - CEA-Saclay. 91191 Gif-sur-Yvette, France, ✝ Instituto de Astronom'ıa y F'ısica del Espacio/Conicet. Bs As, Argentina E-mail: [email protected]. \nWe show that the black hole in the x-ray binary Cygnus X-1 was formed in situ and did not receive an energetic trigger from a nearby supernova. The progenitor of the black hole had an initial mass greater than 40 solar masses and during the collapse to form the ✞✠✟☛✡ solar mass black hole of Cygnus X1, the upper limit for the mass that could have been suddenly ejected is ✞ 1 solar mass, much less than the mass ejected in a supernova. The observations suggest that high-mass stellar black holes may form promptly, when massive stars disappear silently. \nIt is believed that stellar black holes can be formed in two different ways: Either the massive star collapses directly into a black hole without a supernova (SN) explosion, or an explosion occurs in a protoneutron star, but the energy is too low to completely unbind the stellar envelope, and a large fraction of it falls back onto the short-lived neutron star, leading to the delayed formation of a black hole ( 1 ). Recently, it has been found that the black hole x-ray binary GRO J1655-40 was launched far from its birth place by an energetic SN explosion ( 2 ) at a runaway speed ( 3 ) of 120 km s GLYPH<0> GLYPH<1> . But so far there has been no observational evidence for a black hole formed without an energetic SN explosion. \nCygnus X-1 ( 4 ) is a well studied Galactic black hole candidate. The x-ray source was identified with a high mass binary system, where the radial velocity measurements indicate a compact object massive enough to be a black hole rather than a neutron star. Cygnus X-1 can be classified as a microquasar ( 5 ) with a persistent radio counterpart that has been resolved as a compact relativistic jet ( 6 ), which allows one to observe with high precision the motion of the x-ray binary in the sky. \nCygnus X-1 is moving as the association of massive stars Cygnus OB3 (Fig. 1). Despite the different observational techniques used to determine the proper motions of the high mass x-ray binary and the association of stars, the magnitudes are similar and both are directed to the Galactic plane. This supports the hypothesis that Cyg OB3 is the parent association of Cygnus X-1. Based on the proper motions and radial velocities presented in Table 1, at a distance of 2.0 GLYPH<2> 0.1 kpc the velocity of Cygnus X-1 relative to Cyg OB3 is GLYPH<3> GLYPH<2>GLYPH<5>GLYPH<4> kms GLYPH<0> ✆ , which is typical of the random velocities of stars in expanding associations ( 7 ). The proper motion of Cygnus X-1 relative to Cyg OB3 implies that the x-ray binary would have reached its projected distance of ✞ GLYPH<7>GLYPH<6> ✡ pc from the center of Cyg OB3 in GLYPH<8>GLYPH<10>GLYPH<9> GLYPH<2>GLYPH<11>GLYPH<4>GLYPH<13>GLYPH<12>GLYPH<15>GLYPH<14> ✟☛✡ GLYPH<17>GLYPH<16> years. \nA lower limit for the initial mass of the progenitor of the black hole can be estimated by assuming that all massive stars of the parent stellar association, including the progenitor of \nCygnus X-1, were formed over a short time span ( 8 ). The main-sequence star of higher mass found in Cyg OB3 is of spectral type O7 V and has a mass of 40 times the mass of the sun (M /a0 ) ( 8 ). Because more massive stars evolve faster, the lower limit for the initial mass of the progenitor is (40 GLYPH<2> 5) M /a0 . The upper limit for the initial mass would be equivalent to that of the highest mass stars found in Galactic associations, up to ✞ 100 ✁ /a0 . The time since the formation of Cyg OB3 and the progenitor of Cygnus X-1 as inferred from current models of stellar evolution ( 9 ) is GLYPH<8> ✄✂ GLYPH<2> ✟ ✆☎✝✂ GLYPH<12> GLYPH<14> ✟ ✡ GLYPH<16> years, which is -within the range of error- consistent with the GLYPH<8> GLYPH<9> GLYPH<2>GLYPH<11>GLYPH<4>GLYPH<13>GLYPH<12> GLYPH<14> ✟☛✡ GLYPH<16> years Cygnus X-1 would have taken to move from the center of Cyg OB3 to its present position. \nUsing the equations for symmetric mass ejection in black hole formation ( 10 ), we estimate the maximum mass that could have been suddenly ejected to accelerate the binary without disruption to a velocity of GLYPH<8> GLYPH<3> GLYPH<2>GLYPH<7>GLYPH<4> GLYPH<12> kms GLYPH<0> ✆ . From the properties of Cygnus X-1 (Table 1) it is found that not more than (1 GLYPH<2> 0.3) M /a0 was ejected in the core collapse of the massive progenitor. Indeed, there is no observational evidence for a SN remnant in the radio continuum, x-ray, or atomic hydrogen surveys of the region where Cygnus X-1 was most likely formed. \nBefore collapse the progenitor must have lost ✞ (30 GLYPH<2> 5) M /a0 , because the initial mass of the progenitor was ✞ (40 GLYPH<2> ✂ GLYPH<12> ✁ /a0 , and the estimated black hole mass is (10 GLYPH<2> 5) M /a0 ( 11 ). Some fraction of the missing mass may have been transfered to the binary companion, but because the later has a mass of ✞ 18 M /a0 , ✞ ✟ GLYPH<4> ✁ /a0 were lost by stellar winds. In such a case the progenitor of the black hole in Cygnus X-1 may have been a Wolf-Rayet star. \nThe formation of the black hole of Cygnus X-1 was not through a Type II SN, where hydrogen envelopes are blown away and the ejected masses are in the range of 10 to 50 M /a0 ( 12 ), much greater than the upper limit for the mass that could have been suddenly ejected in Cygnus X-1. Alternatively, the core collapse could have occurred in a progenitor that lost its hydrogenrich envelope (SN Ib), and even most of its helium envelope (SN Ic). Recent observations \nsuggest that the energy and luminosity of an explosion in a SN of type Ib or Ic increase with an increasing amount of ejected mass ( 12 ), so the core-collapse onto the black hole in Cygnus X-1 was either underluminous with respect to typical supernovae, or occurred without an explosion. Thus stellar black holes, such as in Cygnus X-1, may form without a SN. If there is no SN associated with the formation of the black hole, then observers would not see an increase in energy or luminosity from the region. The black hole would form in the dark. \nThe maximumlinear momentum and kinetic energy that could have been imparted to Cygnus X-1 by a SN trigger would be (250 GLYPH<2> 80) M /a0 kms GLYPH<0> ✆ and GLYPH<8> GLYPH<4> GLYPH<2> ✡ ☎ ✂ GLYPH<12> GLYPH<14> ✟☛✡ ✁ GLYPH<16> erg, respectively. The maximum linear momentum for Cygnus X-1 is 2.5 times smaller than the linear momentum imparted by the SN ( 2 ) to the runaway black hole system GRO J1655-40. The upper limit for the runaway kinetic energy of Cygnus X-1 is at least 20 times smaller than that estimated ( 3 ) for GRO J1655-40, and ✞ GLYPH<4> GLYPH<14> ✟☛✡ GLYPH<0> ✂ that of a SN of ✟☛✡ ✂ ✆ ergs. \nThe kinematics of Cygnus X-1 and GRO J1655-40 suggest that the black holes in these two x-ray binaries were formed through different evolutionary paths. The black hole in GRO J1655-40 has a mass of (5.4 GLYPH<2> 0.3) M /a0 ( 13 ) and was formed through an energetic SN explosion and fall-back on a neutron star. The black hole in Cygnus X-1 which has a mass of (10.1 GLYPH<2> 5) M /a0 ( 11 ) was formed through a low energy explosion or even by prompt implosion without a SN. These observations are consistent with the theoretical model ( 1 ) in which the energy of the explosion in the core collapse of massive stars decreases as a function of the increasing mass of the progenitor and black hole. Although the observations discussed here are of black holes in x-ray binaries, dim (or dark) formation of black holes should also occur in massive progenitors that are not in binary systems, where the massive hydrogen envelope has been retained by the progenitor, and if there is a SN it will appear as a low-luminosity type II event. \nThe formation of Galactic black holes can be used as a local template to gain insight into the physics of the gamma-ray bursts of long duration, which are believed to come from relativistic \njets produced during the formation of black holes in distant galaxies. The nature of the so called 'dark gamma-ray bursts' i.e., those without x-ray and/or optical counterparts, has been intriguing. The optical and x-ray counterparts of gamma-ray bursts are afterglows produced by the shocks of the jets with circumstellar material composed by the stellar wind from the progenitor and/or the ejecta from the SN explosion. It has been proposed that gamma ray bursts may be dark because of dust obscuration, very high redshifts, or rapidly decaying transients. The analysis of observations reported here suggests that some gamma-ray bursts could be intrinsically dark. Indeed, it is known that the metallicity decreases with increasing redshift, and massive stars in the distant universe may produce weak stellar winds, and collapse promptly into high mass black holes, where there would be no massive stellar winds or SN ejecta to be shocked by the jets. In such a case, some dark gamma-ray bursts may be jets from massive stellar black holes formed in the dark, like the black hole in Cygnus X-1.", 'References and Notes': "- 1. C. L. Fryer, V. Kalogera, Astrophys. J. 554 , 548 (2001).\n- 2. G. Israelian, R. Rebolo, G. Basri, J. Casares, E. L. Martin, Nature 401 , 142 (1999).\n- 3. I. F. Mirabel, et al. , Astron. Astrophys. 395 , 595 (2002).\n- 4. S. Bowyer, E. T. Byram, T. A. Chubb, H. Friedman, Science, Volume 147, Issue 3656, pp. 394-398 147 , 394 (1965).\n- 5. I. F. Mirabel, L. F. Rodr'ıguez, Nature 392 , 673 (1998).\n- 6. A. M. Stirling, et al. , Mon. Not. R. Astron. Soc. 327 , 1273 (2001).\n- 7. A. Blaauw, NATO ASIC Proc. 342: The Physics of Star Formation and Early Stellar Evolution, , C. J. Lada, N. D. Kylafis, eds. (Kluwer Academic Publishers, Dordrecht, 1991), p. 125.\n- 8. P. Massey, K. E. Johnson, K. DeGioia-Eastwood, Astrophys. J. 454 , 151 (1995).\n- 9. G. Schaller, D. Schaerer, G. Meynet, A. Maeder, Astron. Astrophys. Suppl. Ser. 96 , 269 (1992).\n- 10. G. Nelemans, T. M. Tauris, E. P. J. van den Heuvel, Astron. Astrophys. 352 , L87 (1999).\n- 11. A. Herrero, R. P. Kudritzki, R. Gabler, J. M. Vilchez, A. Gabler, Astron. Astrophys. 297 , 556 (1995).\n- 12. M. Hamuy, Astrophys. J. 582 , 905 (2003).\n- 13. M. E. Beer, P. Podsiadlowski, Mon. Not. R. Astron. Soc. 331 , 351 (2002). \n- 14. J.-F. Lestrade, et al. , Astron. Astrophys. 344 , 1014 (1999).\n- 15. A. K. Dambis, A. M. Mel'nik, A. S. Rastorguev, Astronomy Letters 27 , 58 (2001).\n- 16. J. LaSala, P. A. Charles, R. A. D. Smith, M. Balucinska-Church, M. J. Church, Mon. Not. R. Astron. Soc. 301 , 285 (1998).\n- 17. B. Margon, S. Bowyer, R. P. S. Stone, Astrophys. J. 185 , L113 (1973).\n- 18. A. M. Mel'nik, A. K. Dambis, A. S. Rastorguev, Astronomy Letters 27 , 521 (2001).\n- 19. M. Gierli'nski, et al. , Mon. Not. R. Astron. Soc. 309 , 496 (1999).\n- 20. N. R. Walborn, Astrophys. J. 179 , L123 (1973).\n- 21. C. D. Garmany, R. E. Stencel, Astron. Astrophys. Suppl. Ser. 94 , 211 (1992).\n- 22. C. Blaha, R. M. Humphreys, Astron. J. 98 , 1598 (1989).\n- 23. We thank J. Paul, J. Ballet, E. Le Floc'h, and S. Chaty for helpful comments. I.R. is a Fellow of the Conselho Nacional de Desenvolvimento Cient'ıfico e Tecnol'ogico (CNPq) of Brazil. \nTable 1: Data on CygnusX-1 and CygOB3. ✤ and ✥ are the Galactic longitude and latitude. The proper motion ✦★✧ and ✦✪✩ of Cygnus X-1 were determined by high-precision astrometry based on Very Long Baseline Interferometric (VLBI) observations of the compact radio counterpart in eight epochs between 1988 and 2001 ( 6, 14 ). The mean proper motions ✦✫✧ and ✦✪✩ of CygOB3 were determined with the Hipparcos proper motions of 18 stars. V ✬✮✭✰✯✲✱✴✳ for Cygnus X-1 is the heliocentric radial velocity of the center of mass of the binary, and V ✬✮✭✰✯✲✱✵✳ of Cyg OB3 is the mean heliocentric radial velocity based on the radial velocities of 30 stars ( 15 ). D in kpc units is the distance estimated by different methods. M ✶✸✷ , M ✹ and Spect. Type are the mass of the black hole, the mass of the donor star, and its spectral type, respectively. Bibliographic references are indicated in parentheses. \n| | | Cygnus X-1 | Cygnus X-1 | Cyg OB3 | Cyg OB3 |\n|-------------|---------------|-----------------|---------------|---------------|-----------|\n| | [ ] | 71.32 | ( 14 ) | 72.80 | ( 15 ) |\n| | [ ] | +3.09 | ( 14 ) | +2.00 | ( 15 ) |\n| ✄✆☎ | [mas yr ✝✟✞ ] | -4.2 0.2 | ( 6,14 ) | -3.9 0.3 | ( 15 ) |\n| ✄☛✡ | [mas yr ✝✟✞ ] | -7.6 0.2 | ( 6,14 ) | -6.7 0.3 | ( 15 ) |\n| V ☞✍✌✏✎ | [km s ✝✟✞ ] | -5.4 0.1 | ( 16 ) | -8.5 2.1 | ( 15 ) |\n| | | 2.5 0.4 | ( 17 ) | 2.3 0.4 | ( 18 ) |\n| D | [kpc] | 1.4 ✓☛✔✖✕ ✔✖✕ | ( 14 ) | 1.8 ✓☛✔✖✕ ✔✖✕ | ( 15 ) |\n| | | 2.0 | ( 19 ) | 2.0 0.1 | ( 8 ) |\n| M M | [M ] [M ] | 10.1 5 17.8 4.5 | ( 11 ) ( 11 ) | - - | |\n| Spect. type | Spect. type | O9.7Iab | ( 20 ) | - | | \nFigure 1: Optical image of the sky around the black hole X-ray binary Cygnus X-1 and the association of massive stars Cyg OB3. The red arrow shows the motion in the sky of the radio counterpart of Cygnus X-1 for the past 0.5 million years. The yellow arrow shows the average Hipparcos motion ( 15 ) of the massive stars of Cyg OB3 ( 21, 22, circled in yellow ) for the past 0.5 million years. Despite the different observational techniques used to determine the proper motions, Cygnus X-1 moves in the sky as Cyg OB3. At a distance of 2 kpc the space velocity of Cygnus X-1 relative to that of Cyg OB3 is GLYPH<3> GLYPH<2>GLYPH<11>GLYPH<4> km s GLYPH<0> ✆ . \n<!-- image -->"}
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2013EPJC...73.2645Z
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On the critical phenomena and thermodynamics of charged topological dilaton AdS black holes
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2013-01-01
| 24 | 0.45 | 158 |
['-', 'critical phenomena', '-', 'critical phenomena', 'critical phenomena', '-', '-']
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In this paper, we study the phase structure and equilibrium state space geometry of charged topological dilaton black holes in (n+1)-dimensional anti-de Sitter spacetime. By considering the pairs of parameters (P∼V) and (Q∼U) as variables, we analyze the phase structure and critical phenomena of black holes and discuss the relation between the two kinds of critical phenomena. We find that the phase structures and critical phenomena drastically depend on the cosmological constant l (or the static electric charge Q of the black holes), dimensionality n and dilaton field Φ.
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[]
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https://arxiv.org/pdf/1305.3725.pdf
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{'Ren Zhao': 'Institute of theoretical physics, Shanxi Datong University, 037009 Datong, China [email protected]', 'Hui-Hua Zhao': 'Institute of theoretical physics, Shanxi Datong University, 037009 Datong, China Department of Physics, Shanxi Datong University, 037009 Datong, China [email protected]', 'Meng-Sen Ma': 'Institute of theoretical physics, Shanxi Datong University, 037009 Datong, China Department of Physics, Shanxi Datong University, 037009 Datong, China [email protected]', 'Li-Chun Zhang': 'Institute of theoretical physics, Shanxi Datong University, 037009 Datong, China Department of Physics, Shanxi Datong University, 037009 Datong, China \nAbstract: In this paper, we study the phase structure and equilibrium state space geometry of charged topological dilaton black holes in ( n +1)-dimensional anti-de Sitter spacetime. By considering the pairs of parameters ( P ∼ V ) and ( Q ∼ U ) as variables, we analyze the phase structure and critical phenomena of black holes and discuss the relation between the two kinds of critical phenomena. We find that the phase structures and critical phenomena drastically depend on the cosmological constant l (or the static electric charge Q of the black holes), dimensionality n and dilaton field Φ.', 'Contents': '| 1. | Introduction | 1 |\n|------|-----------------------------------------------------|-----|\n| 2. | Charged Dilaton Black Holes in Anti-de Sitter Space | 3 |\n| 3. | Critical behaviour | 5 |\n| | 3.1 Q is an invariant parameter | 5 |\n| | 3.2 l is an invariant parameter | 9 |', '1. Introduction': 'Black hole physics is a subject at the intersection of general relativity, quantum mechanics and statistical physics and field theory. This makes the subject receive a lot of attention. Black holes have been used as the laboratory of many kinds of theories, specially the thermodynamics of black holes plays an important roles[1, 2, 3, 4, 5, 6]. The thermodynamic properties of black holes have been studied for several years, although the exact statistical explanation of black hole thermodynamics is still lacked. \nIt shows that black holes also have the standard thermodynamic quantities, such as temperature, entropy, even possess abundant phase structures like hawking-Page phase transition[7] and the critical phenomena similar to ones in the ordinary thermodynamic system. What is more interesting is the research on charged, non-rotating RN-AdS black hole, which shows that there exists phase transition similar to the van der Waals-Maxwell vapor-liquid phase transition[8, 9, 10]. \nMotivated by the AdS/CFT correspondence[11], where the transitions have been related with the holographic superconductivity[12, 13], the subject that the phase transitions of black holes in asymptotically anti de-Sitter (AdS) spacetime, has received considerable attention[14, 15, 16, 17, 18]. The underlying microscopic statistical interaction of the black holes is also expected to be understood via the study of the gauge theory living on the boundary in the gauge/gravity duality. \nThe studies[19, 20, 21, 22, 23, 24, 25, 26] on the phase transition and critical phenomena of black holes in AdS spacetime indicate the black holes are similar to the van der Waals vapor-liquid system. The ( Q ∼ U )(where Q is the static electric charge, U is the electrostatic potential on the horizon) phase diagram of black holes in AdS spacetime is almost the same as the ( P ∼ V ) phase diagram in van der Waals vapor-liquid system. \nAmong the gravity theories with higher derivative curvature terms, the Gauss-Bonnet (GB) gravity has some special features and gives rise to some interesting effects on the thermodynamics of black holes in AdS space[27, 28, 29, 30, 31, 32]. The phase structure of a GB-AdS black hole was briefly studied in [27, 33]. And in the grand canonical ensemble, the local and global thermal phase structure of a charged asymptotically AdS black hole with both GB and quartic field strength corrections were thoroughly researched [34]. In [35] the phase transition and critical phenamena of d-dimensional charged GB-AdS black hole is analyzed. The consequence show that the phase structure and critical temperature, critical electric charge, critical electrostatic potential of the black hole all depend on the cosmological constant Λ and the dimension of spacetime. The ( Q ∼ U ) critical conditions of d-dimensional charged GB-AdS black hole agree with the ( P V )ones in van der Waals vapor-liquid system. \nRecently, many interests focus on the studies of critical behaviors of AdS black holes [36, 37, 38, 39]by considering cosmological constant as thermal pressure \n∼ \nP = -1 8 π Λ = 3 8 π 1 l 2 , (1.1) \nand corresponding conjugate thermal volume as \nV = ( ∂M ∂P ) S,Q i ,J k . (1.2) \nThe complete analog model of vapor-liquid system for black holes is established in [36]. In [40] the relation (1.1) in higher dimensional spherically symmetric AdS spacetime first proposed which lays the foundation for the research of black holes thermodynamics. \nTheoretically, if regarding the black holes in AdS spacetime as thermodynamic systems the corresponding critical behaviors and phase transition should exist. However, until now the statistical explanation of black hole thermodynamics is still lack. Therefore it is a meaningful work to discuss the relations of thermodynamic properties for all kinds of black holes in AdS spacetime. This may help to recognize further black hole entropy, temperature, heat capacity and may help to improve the geometric theory of black holes thermodynamics . \nA scalar field called the dilaton appears in the low energy limit of string theory. The presence of the dilaton field has important consequences on the causal structure and the thermodynamic properties of black holes. Thus much interest has been focused on the study of the dilaton black holes in recent years[41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55]. \nIn this paper we study the phase transition and critical behaviors of ( n +1) dimensional charged topological dilaton AdS black hole. Firstly we consider the cosmological constant as thermodynamic pressure and the conjugate thermodynamic volume. We find that the phase structure and critical phenomena are dependent on the static electrostatic charge Q , dimension n and dilaton field Φ. The critical exponents are the same as the ones in van der Waals vapor-liquid system. Secondly we consider the pair of conjugate parameters ( Q ∼ U ) as the thermodynamic variables and study the phase transition and critical behaviors of ( n +1) dimensional charged topological dilaton AdS black hole again. The results show that \nthe phase structure and critical phenomena are dependent on the cosmological constant Λ, dimension n and dilaton field Φ. The critical exponents are also the same as the ones in van der Waals vapor-liquid system. Thus the two approaches are equivalent because of the same phase diagrams and critical behavior. \nThe paper is arranged as follows: in the next section we simply introduce the ( n + 1)dimensional charge Dilaton AdS black hole. In section 3 we will consider the parameters ( P ∼ V ) and ( Q ∼ U ) respectively and discuss the phase structure and critical phenomena of black holes. We will make some concluding remarks in section 4. (we use the units G n +1 = /planckover2pi1 = k B = c = 1)', '2. Charged Dilaton Black Holes in Anti-de Sitter Space': "The Einstein-Maxwell-Dilaton action in ( n +1)-dimensional ( n ≥ 3)spacetime is [49,50] \nS = 1 16 π ∫ d n +1 x √ -g ( R -4 n -1 ( ∇ Φ) 2 -V (Φ) -e -4 α Φ / ( n -1) F µν F µν ) , (2.1) \nwhere the dilaton potential is expressed in terms of the dilaton field and its coupling to the cosmological constant: \nV 2 Φ = n -1 8 ∂V ∂ Φ -α 2 e -4 α Φ / ( n -1) F λη F λη , (2.2) \n∇ µ ( e -4 α Φ / ( n -1) F µν ) = 0 , (2.3) \nwhere R is the Ricci scalar curvature, Φ is the dilaton field and V (Φ) is a potential for Φ, α is a constant determining the strength of coupling of the scalar and electromagnetic field, F µν = ∂ µ A ν -∂ ν A µ is the electromagnetic field tensor and A µ is the electromagnetic potential. The topological black hole solutions take the form [38,49,50] \nds 2 = -f ( r ) dt 2 + dr 2 f ( r ) + r 2 R 2 ( r ) d Ω 2 k,n -1 , (2.4) \nwhere \nf ( r ) = -k ( n -2)( α 2 +1) 2 b -2 γ r 2 γ ( α 2 -1)( α 2 + n -2) -m r ( n -1)(1 -γ ) -1 + 2 q 2 ( α 2 +1) 2 b -2( n -2) γ ( n -1)( α 2 + n -2) r 2( n -2)( γ -1) -n ( α 2 +1) 2 b 2 γ l 2 ( α 2 -n ) r 2(1 -γ ) , (2.5) \nR ( r ) = e 2 α Φ / ( n -1) , Φ( r ) = ( n -1) α 2(1 + α 2 ) ln ( b r ) , (2.6) \nwith γ = α 2 / ( α 2 +1). The cosmological constant is related to spacetime dimension n by \nΛ = -n ( n -1) 2 l 2 , (2.7) \nwhere b is an arbitrary constant and l denotes the AdS length scale . In the above expression, m appears as an integration constant and is related to the ADM (Arnowitt-Deser-Misnsr) mass of the black hole. According to the definition of mass due to Abbott and Deser [56, 57], the mass of the solution (2.5) is [58] \nM = b ( n -1) γ ( n -1) ω n -1 16 π ( α 2 +1) m. (2.8) \nthe electric charge is \nQ = qω n -1 4 π , (2.9) \nwhere ω n -1 represents the volume of constant curvature hypersurface described by d Ω 2 k,n -1 . The Hawking temperature of the topological black hole on the outer horizon r + can be calculated using the relation \nT = κ 2 π = f ' ( r + ) 4 π , (2.10) \nwhere κ is the surface gravity. It can easily show that \nT = -( α 2 +1) 2 π ( n -1) ( k ( n -2)( n -1) b -2 γ 2( α 2 -1) r 2 γ -1 + +Λ b 2 γ r 1 -2 γ + + q 2 b -2( n -2) γ r (2 n -3)( γ -1) -γ + ) = -k ( n -2)( α 2 +1) b -2 γ 2 π ( α 2 + n -2) r 2 γ -1 + + ( n -α 2 ) m 4 π ( α 2 +1) r ( n -1)( γ -1) + -q 2 ( α 2 +1) b -2( n -2) γ π ( α 2 + n -2) r (2 n -3( γ -1) -γ + . (2.11) \nTopological black hole entropy \nS = b ( n -1) γ ω n -1 r ( n -1)(1 -γ ) + 4 . (2.12) \nThe electric potential \nU = qb (3 -n ) γ r λ + λ , (2.13) \nwhere λ = ( n -3)(1 -γ ) + 1. From f ( r + ) = 0 and (2.8), we obtain \nM = q 2 b γ (3 -n ) ( α 2 +1) ω n -1 8 π ( α 2 + n -2) r -λ + -n ( n -1)( α 2 +1) b γ ( n +1) ω n -1 16 πl 2 ( α 2 -n ) r n -γ ( n +1) + -k ( n -2)( n -1)( α 2 +1) b γ ( n -3) ω n -1 16 π ( α 2 -1)( α 2 + n -2) r λ + . (2.14) \nOne may then regard the parameters S , Q and P as a complete set of extensive parameters for the mass M ( S, Q, P ) and define the intensive parameters conjugate to S , Q and P . These quantities are the temperature, the electric potential and volume. \nT = ( ∂M ∂S ) Q,P , U = ( ∂M ∂Q ) S,P , V = ( ∂M ∂P ) Q,S . (2.15) \nwhere \nIt is a matter of straightforward calculation to show that the quantities calculated by Eq. (2.16) for the temperature, and the electric potential coincide with Eqs. (2.11) and (2.13). Thus, the thermodynamics quantities satisfy the first law of thermodynamics \nP = n ( n -1) 16 πl 2 , V = -( α 2 +1) b γ ( n +1) ω n -1 ( α 2 -n ) r n -γ ( n +1) + . (2.16) \ndM = TdS + UdQ + V dp. (2.17) \nThe thermodynamic quantities above, energy M , entropy S , temperature T , volume V , pressure P , electrostatic potential U and electric charge Q satisfy Smarr formula: \nM = ( n -1)(1 -γ ) λ TS + UQ -n -λ λ V P. (2.18) \nIn what follows we concentrate on analyzing the phase transition of the ( n +1)dimensional charged topological dilaton AdS black hole system in the extended phase space while we treat the black hole charge Q as a fixed external parameter, or the cosmological constant is an invariant parameter, not a thermodynamic variable. We shall find that an even more remarkable coincidence with the Van der Waals fluid is realized in this case.", '3.1 Q is an invariant parameter': "For a fixed charge q , Eq. (2.11) translates into the equation of state for a charged topological dilaton black hole, P = P ( V, T ) \nP = T ( n -1) 4( α 2 +1) b -2 γ r 2 γ -1 + + k ( n -1)( n -2) 16 π ( α 2 -1) b -4 γ r 2(2 γ -1) + + Q 2 b 2(1 -n ) γ 2 π ω 2 n -1 r 2( n -1)( γ -1) + r + = ( V ( n -α 2 ) ( α 2 +1) ω n -1 b γ ( n +1) ) 1 / ( n -γ ( n +1)) . (3.1) \nWhere P and V are given by (2.16), V is the thermodynamic volume, given in terms of the event horizon radius r + , T is the black hole temperature, and Q its charge. The Van der Waals equation \nHere, v = V/N is the specific volume of the fluid, P its pressure, T its temperature, and k is the Boltzmann constant. \n( P + a v 2 ) ( v -˜ b ) = kT, (3.2) \nComparing with the Van der Waals equation, (3.2), we conclude that we should identify the specific volume v of the fluid with the horizon radius of the black hole as \nv = 4( α 2 +1) b 2 γ ( n -1) r 1 -2 γ + . (3.3) \nIn ( n +1) dimensions, the equation of (3.1) reads \nP = T v + k ( n -2)( α 2 +1) 2 π ( n -1)( α 2 -1) v 2 + Q 2 b 2(1 -n ) γ 2 π ω 2 n -1 ( v ( n -1) 4( α 2 +1) b 2 γ ) 2( n -1)( γ -1) 1 -2 γ = T v -A v 2 + B v 2( n -1)(1 -γ ) / (1 -2 γ ) , (3.4) \nwhere \nA = k ( n -2)( α 2 +1) 2 π ( n -1)(1 -α 2 ) , B = Q 2 b 2(1 -n ) γ 2 π ω 2 n -1 ( 4( α 2 +1) b 2 γ ( n -1) ) 2( n -1)(1 -γ ) / (1 -2 γ ) . (3.5) \nCritical points occur at points of inflection in the P -V diagram, where \n∂P ∂v = 0 , ∂ 2 P ∂v 2 = 0 , (3.6) \nSubstituting (3.4) into (3.6) we can derive the critical volume, temperature and pressure: \n2) ( \n) \nv x -2 c = ( x -1) xB 2 A , T c = 2 A v c -xB v x -1 c = 2 A ( x -2) ( x -1) ( 2 A ( x -1) xB ) 1 / ( x -2) , P c = A v 2 c -B v x c ( x -1) = A ( x -x 2 A ( x -1) xB 2 / ( x -2) , (3.7) \nx = 2( n -1)(1 -γ ) (1 -2 γ ) . (3.8) \nwhere \nVan der Waals equation critical temperature, volume and pressure are: \nv x -2 c = ( x -1) xB 2 A , T c = 2 A v c -xB v x -1 c = 2 A ( x -2) ( x -1) ( 2 A ( x -1) xB ) 1 / ( x -2) , T c = 8 a 27 ˜ b , v c = 3 ˜ b, P c = a 27 ˜ b 2 . (3.9) \nFrom (3.7), ( n +1)-dimensional charged topological dilaton black hole correspond to \n˜ b = x 4( x -1) ( ( x -1) xB 2 A ) 1 / ( x -2) , a = 27 Ax ( x -2) 16( x -1) 2 , ρ c = P c v c T c = x -1 2 x . (3.10) \nTherefore, the critical temperature, volume and pressure of ( n + 1)-dimensional charged topological dilaton black hole are \nv c = 4( x -1) x ˜ b, T c = 8 a 27 ˜ b , P c = a 27 ˜ b 2 . (3.11) \n<!-- image --> \n<!-- image --> \nFigure 1: p -v diagram of ( n + 1)-dimensional charge Dilaton AdS black hole in ( n = 3 , α = 0), ( n = 3 , α = 0 . 6) and ( n = 10 , α = 0 . 6) respectively. The temperature of isotherms decreases from top to bottom. The three upper dashed lines correspond to the 'ideal gas' one-phase behaviour for T > T c , the critical isotherm T = T c is denoted by the thick solid line, lower (red) solid lines correspond to two-phase state occurring for T < T c . We have set Q = 1 , b = 1 , k = 1. The behaviour for n > 3 and α > 0 is qualitatively. \n<!-- image --> \nFigure 2: The T c -α diagram shows the influence of dilaton field α on the critical temperature with different sapcetime dimensions. \n<!-- image --> \nIn Fig.1, (a) and (b) show the influence of dilaton field α on the isothermal curves with the same spacetime dimension. (b) and (c) represent the influence of spacetime dimension n on the isothermal curves with the same dilaton field. \nTo calculate the critical exponent α we consider the entropy S , (2.12), as a function of T and V . Using (3.1) we have \nS = S ( T, V ) = b -γ ( n -1) / ( n -γ ( n +1)) 4 ω (1 -2 γ ) / ( n -γ ( n +1)) n -1 ( V ( n -α 2 ) ( α 2 +1) ) ( n -1)(1 -γ ) / ( n -γ ( n +1)) . (3.12) \nSince this is independent of T , we have C V = T ( ∂S ∂T ) V = 0 and hence ˜ α = 0. Defining specific \nvariables \np = P P c , v = v v c , τ = T T c . (3.13) \nThus Eq.(3.4) turns into \np = τ v 2 x x -1 -1 v 2 x x -2 + 1 v x 2 ( x -1)( x -2) , (3.14) \nOne can write Eq.(3.14) in the form of van der Waals \n( x -1) v ( p + 1 v 2 x x -2 ) -2 v x -1 ( x -2) = 2 xτ. (3.15) \nIn Eq.(3.14) there exist no constants which depend on the properties of matter, but there is the quantity x which is dependent on n and γ . Therefore, when n and γ are the same, the equation can be simplified to \np = τ ρ c v + f ( v ) , (3.16) \nwhere ρ c stand for the critical ratio, which is given by (3.10). \nf ( v ) = -1 v 2 x x -2 + 1 v x 2 ( x -1)( x -2) (3.17) \nwhere ρ c stands for the critical ratio. Expanding this equation near the critical point \nτ = t +1 , v = ( ω +1) 1 /q , (3.18) \nwhere ˜ q > 0, and using the fact that from the definition of the critical point we have \n1 ρ c + f (1) = 1 , ρ c f ' (1) = 1 , ρ c f '' (1) = -2 , (3.19) \nAnd so obtain \np = 1 + t ρ c -tω ˜ qρ c -Cω 3 + O ( tω 2 , ω 4 ) , (3.20) \nwhere C = 1 ˜ q 3 ( 1 ρ c -f (3) (1) 6 ) .Differentiating the series for a fixed t < 0 we get \ndP = -P c ( t ˜ qρ c +3 Cω 2 ) dω. (3.21) \nEmploying Maxwell's equal area law, see, e.g., [35, 36, 38], while denoting ω g and ω l the 'volume' of small and large black holes, we get the following two equations: \np = 1 + t ρ c -tω l ˜ qρ c -Cω 3 l = 1 + t ρ c -tω g ˜ qρ c -Cω 3 g , 0 = ω g ∫ ω l ωdP = ω g ∫ ω l ω ( t ˜ qρ c +3 Cω 2 ) dω. (3.22) \nThe unique non-trivial solution is \nω g = -ω l = √ -2 t 3 C ˜ qρ c ∝ ( -t ) 1 . 2 , (3.23) \nwhich implies that the degree of the coexistence curve \nβ = 1 / 2 . (3.24) \nTo calculate the exponent ˜ γ , we use again (3.22), to get \nκ T = -1 V ( ∂V ∂P ) T ∝ 1 P c 1 ( t/ (˜ qρ c +3 Cω 2 ) , (3.25) \nThe set ω = 0, we get \nκ T ∝ 1 P c ˜ qρ c t . (3.26) \nThus the isothermal compressibility exponent \n˜ γ = 1 . (3.27) \nFinally, the 'shape of the critical isotherm' t = 0 is given by (3.23), i.e., \np -1 = -Cω 3 , (3.28) \nTherefore the critical exponent \nδ = 3 . (3.29) \nAccording to Eq.(3.11), when the electric charge of black holes is invariant, the equation of state of the charged topological dilaton AdS black hole in any dimension can be expressed as the form of Van der Waals equation. The critical exponents are the same as the ones for Van der Waals fluid. \nIn particular, the law of corresponding states, (3.14), takes the form (3.16). Taking ˜ q = n -γ ( n +1) 1 -2 γ (in which cases ω = V V c -1). We obtain the expansion (3.21) with \nC = x (1 -2 γ ) 3 3[ n -γ ( n +1)] 3 , (3.30) \nand so the discussion above applies.", '3.2 l is an invariant parameter': "In the case of l invariant, substituting Eq.(2.13) into (2.11), one can derive \nT = ( α 2 +1) 4 π k ( n -2) b -2 γ (1 -α 2 ) ( qb (3 -n ) γ λU ) (2 γ -1) /λ + n ( α 2 +1) b 2 γ 4 πl 2 ( qb (3 -n ) γ λU ) (1 -2 γ ) /λ \n-( α 2 +1) 2 π ( n -1) q 2 b -2( n -2) γ ( qb (3 -n ) γ λU ) ((2 n -3)( γ -1) -γ ) /λ = ˜ A ( qb (3 -n ) γ λU ) (2 γ -1) /λ + ˜ B ( qb (3 -n ) γ λU ) (1 -2 γ ) /λ -˜ Cq 2 ( qb (3 -n ) γ λU ) ((2 n -3)( γ -1) -γ ) /λ , (3.31) \nwhere \n˜ A = ( α 2 +1) 4 π k ( n -2) b -2 γ (1 -α 2 ) , ˜ B = n ( α 2 +1) b 2 γ 4 πl 2 , ˜ C = ( α 2 +1) 2 π ( n -1) b -2( n -2) γ . (3.32) \nThe critical points should satisfy the conditions \n( ∂q ∂U ) T = ( ∂ 2 q ∂U 2 ) T = 0 . (3.33) \nSubstituting Eq.(3.31) into Eq.(3.33), one can obtain the critical electric charge, critical electrostatic potential and critical temperature: \nq 2 c = -˜ A ' + ˜ B ' D ˜ C ' D [ nγ -3 γ -n +2] / (1 -2 γ ) = k ( n -2) 2( x -1) b 2( n -3) γ D λ/ (1 -2 γ ) = E \nλU c = E 1 / 2 b (3 -n ) γ D -λ/ (2 -4 γ ) , \nT c = ˜ AD -1 / 2 + ˜ BD 1 / 2 + ˜ C ˜ A ' + ˜ B ' D ˜ C ' D -1 / 2 = k ( n -2) b -2 γ ( x -2) 2 π (1 -2 γ )( x -1) D -1 / 2 , (3.34) \nwhere \n˜ A ' = ˜ A (1 -2 γ ) , ˜ B ' = ˜ B (2 γ -1) , ˜ C ' = ˜ C [(2 n -3)( γ -1) -γ ] , D = k ( n -2) b -4 γ l 2 2 n ( n -1) ( x -2) . \nIn Fig.3, (a) and (b) show the influence of dilaton field α on the isothermal curves with the same spacetime dimension. (b) and (c) represent the influence of spacetime dimension n on the isothermal curves with the same dilaton field. \nExpanding ground a critical point τ = β β c u = U U c ϑ = q q c . Eq.(3.31) can be expressed as \n1 τβ c = ˜ AD -1 / 2 ( ϑ u ) (2 γ -1) /λ + ˜ BD 1 / 2 ( ϑ u ) (1 -2 γ ) /λ + ˜ C ˜ A ' + ˜ B ' D ˜ C ' ϑ 2 ( ϑ u ) ((2 n -3)( γ -1) -γ ) /λ , (3.35) \nNear the critical points, defining t = τ -1, ω = u -1 and substituting them into the above equation, we can obtain \nϑ = ∑ m,n =0 a mn t m ω n , (3.36) \nwhere a mn = 1 m ! n ! ∂ ( m + n ) ϑ ∂t m ∂ω n ∣ ∣ t = 0 ω = 0 . According to Eq.(3.31) and (3.33), a 00 = 1, a 01 = 0, a 02 = \n0. This result holds for any values of k , l and γ . In the neighborhood of the critical points, \n∣ \n<!-- image --> \n<!-- image --> \nFigure 3: q -U diagram of ( n + 1)-dimensional charge Dilaton AdS black hole in ( n = 3 , α = 0), ( n = 3 , α = 0 . 6) and ( n = 10 , α = 0 . 6) respectively. The temperature of isotherms decreases from top to bottom. The three lower dashed lines correspond to the 'ideal gas' one-phase behaviour for T > T c , the critical isotherm T = T c is denoted by the thick solid line, upper (red) solid lines correspond to two-phase state occurring for T < T c . We have set b = 1 , k = 1 and the cosmological constant is given by Eqs.(3.5), (3.7). The behaviour for n > 3 and α > 0 is qualitatively. \n<!-- image --> \nwe have (3.36). The values of ω on either side of the coexistence curve can be found from the conditions that along the isotherm, Employing Maxwell's equal area law, see, e.g., [33,35], while denoting ω g and ω l the 'volume' of small and large black holes, we get the following two equations: \nϑ ( ω g ) = ϑ ( ω l ) , (3.37) \n0 = ω g ∫ ω l ( ω +1) dϑ. (3.38) \nFrom the first condition (3.37), we derive \na 11 t (˜ ω g + ˜ ω l ) + a 21 t 2 (˜ ω g + ˜ ω l ) + a 12 t (˜ ω 2 g -˜ ω 2 l ) + a 03 (˜ ω 3 g + ˜ ω 3 l ) + o ( tω 2 , ω 4 ) = 0 , (3.39) \nwhere we have denoted ˜ ω l = -ω l and ˜ ω g = ω g for different phases. The second condition (3.38) reduces to \na 11 t (˜ ω g + ˜ ω l ) + a 21 t 2 (˜ ω g + ˜ ω l ) + 1 2 ( a 11 +2 a 12 ) t (˜ ω 2 g -˜ ω 2 l ) + a 03 (˜ ω 3 g + ˜ ω 3 l ) + o ( tω 2 , ω 4 ) = 0 . (3.40) \nThe unique non-trivial solution is \n˜ ω 2 g = ˜ ω 2 l = -1 a 03 ( a 11 t + a 21 t 2 ) . (3.41) \nyielding \n˜ ω g = ˜ ω l ∼ √ -a 11 a 03 t ∼ t 1 / 2 ⇒ β = 1 2 . (3.42) \nTo calculate the exponent˜ γ , we use again (3.36), to get \nLet t = 0, (3.40) will be of the form \nκ T = -1 V ∂V ∂P ∣ ∣ ∣ ∣ T ∝ 1 P c 1 a 11 t ⇒ ˜ γ = 1 . (3.43) \nϑ = a 03 ω 3 + o ( ω 4 ) , (3.44) \nThus we get the degree of the critical isotherm \nδ = 3 . (3.45) \nThe heat capacity at fixed U is \nC U = T ( ∂S ∂T ) U = T ( ∂S ∂q ) U ( ∂T ∂q ) -1 U , (3.46) \nFrom Eq.(2.12) and Eq.(3.31) \n( ∂S ∂q ) U = b ( n -1) γ ω n -1 4 ( n -1)(1 -γ ) λ ( qb (3 -n ) γ λU ) ( n -1)(1 -γ ) /λ 1 q , ( ∂T ∂q ) U = -˜ A (1 -2 γ ) λq ( qb (3 -n ) γ λU ) (2 γ -1) /λ + ˜ B (1 -2 γ ) λq ( qb (3 -n ) γ λU ) (1 -2 γ ) /λ + ˜ Cq (1 -2 γ ) λ ( qb (3 -n ) γ λU ) ((2 n -3)( γ -1) -γ ) /λ = k ( n -2)( x -2) b -2 γ 2 πλq c x ( x -1) D -1 / 2 , (3.47) \nWhen n /negationslash = 2, and x /negationslash = 2, Eq.(3.47) is nonzero and C U is non-singular at the critical points. Thus the heat capacity exponent \n˜ α = ˜ α ' = 0 . (3.48) \nWhen the cosmological constant is invariant we conclude that the thermodynamic exponents associated with the charged topological dilaton AdS black holes in any dimension n ≥ 3 coincide with those of the Van der Waals fluid.", '4. Discussion and Conclusions': "In Sec.3 we discussed the phase structure and critical phenomena of charged topological dilaton AdS black holes in the case of k = 1 and the cases of electric charge Q and the cosmological constant l are invariant respectively. We obtain two pairs of critical temperature and critical pressure (critical electric charge) and critical volume(critical electric potential), \nwhich are represented by Eq.(3.7) and (3.34). Below we will analyze the relations between the two pairs of critical quantities. \nIn the case of invariant electric charge Q , from critical pressure Eq.(3.7) we know that \n1 l c = 4 ( πA ( x -2) n ( n -1) x ) 1 / 2 ( 2 A ( x -1) xB ) 1 / ( x -2) . (4.1) \nSubstituting Eq.(4.1) into the critical temperature Eq.(3.35) derived from the case of invariant cosmological constant l , one can get \nT c = ˜ AD -1 / 2 + ˜ BD 1 / 2 + ˜ C ˜ A ' + ˜ B ' D ˜ C ' D -1 / 2 = k ( n -2) b -2 γ ( x -2) 2 π (1 -2 γ )( x -1) D -1 / 2 \n= k ( n -2)( x -2) 2 π (1 -2 γ ) l ( x -1) ( 2 n ( n -1) k ( n -2)( x -2) ) 1 / 2 = 2 A ( x -2) ( x -1) ( 2 A ( x -1) xB ) 1 / ( x -2) . (4.2) \nFrom Eq.(4.2) and (3.7) we can find that the two pair of critical temperatures are the same. Because the critical temperature, critical pressure and critical volume in Eq.(3.7) are all dependent on B , and B is the function of electric charge of black hole, the critical quantities should be the function of electric charge of black hole. The critical temperature and critical electric charge in Eq.(3.34) are the function of l . The relations between the both quantities are given by Eq.(4.1), thus critical pressure and critical volume can also be expressed as the function of cosmological constant. The critical electric potential from Eq.(3.34) \nU c = 1 λ E 1 / 2 b (3 -n ) γ D -λ/ (2 -4 γ ) = 1 λ ( k ( n -2) 2( x -1) ) 1 / 2 . (4.3) \nFrom Eq.(4.3), the critical electric potential is dependent on the spacetime dimension n and dilaton field γ and is independent of the cosmological constant l . \nFrom above we find that when the relation (4.1) is satisfied for the charged topological dilaton AdS black hole the phase transition like van der Waals vapor-liquid one will turn up. The critical temperature, critical pressure, critical volume and critical electric potential are given by Eq.(3.7) and (3.34). The critical exponents are the same as the ones in van der Waals vapor-liquid phase transition. \nBecause of the relation (4.1), according to the critical temperature, critical pressure and critical volume derived in Sec.3 we can obtain the critical electric charge and critical electric potential. \nFrom Eq.(4.1) \nB = 2 A ( x -1) x ( 16 l 2 c πA ( x -2) n ( n -1) x ) ( x -2) / 2 . (4.4) \nSubstituting Eq.(4.3) into the above equation, one can get \nq 2 c = k ( n -2) 2( x -1) b 2( n -3) γ D λ/ (1 -2 γ ) . (4.5) \nwhich agrees with Eq.(3.35). From Eq.(3.3) and (3.7), the horizon of the black hole correspondent to the critical points is \nr + c = ( ( x -1) xB 2 A ) λ/ [( x -2)(1 -2 γ )] ( ( n -1)(1 -γ ) b -2 γ 4 ) λ/ (1 -2 γ ) . (4.6) \nSubstituting Eq.(4.5) into (2.13), one can obtain the consistent critical electric potential with Eq.(4.3). Thus for the ( n + 1)-dimensional charged topological dilaton AdS black hole, in Sec.3 and Sec.4 we can both derive the critical temperature, critical volume, critical pressure, critical electric potential and critical electric charge. For the two pairs of parameters ( P -V ) or ( Q -U ), the phase structure and critical phenomena of black hole are the same as the ones in van der Waals vapor-liquid system. \nDue to A ∝ k , the critical temperature, critical pressure, critical electric potential and critical electric charge derived above will tend to zero, however the critical volume will tend to infinity. Therefore when k = 0, for the ( n + 1)-dimensional charged topological dilaton AdS black hole there no exist similar phase transition to the one in the van der Waals system. \nWhen k = -1, the results are complicated. The critical temperature T c < 0, critical electric charge and electric potential are imaginary numbers. v c ∝ ( -1) 1 / ( x -2) , P c ∝ -( -1) 2 / ( x -2) , which is dependent on the value of x . Whether this process can happen or not? If happens, what physical mechanism it should correspond? These questions should be studied further. \nIn this paper we studied the phase structure and critical phenomena of the ( n + 1)dimensional charged topological dilaton AdS black holes. We find that the phase structure in the canonical ensemble significantly depends on the parameter k , dimensionality n , dilaton field γ and cosmological constant l or the electric charge q of black hole. We consider Hawking temperature, electric charge, the cosmological constant, electric potential and volume as the state parameters and analyzed the phase structure and the critical phenomena. 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2012PhRvD..85h4015L
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Gravitational recoil from accretion-aligned black-hole binaries
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2012-01-01
| 34 | 0.48 | 158 |
['-', '-', '-', '-', 'methods numerical', '-', 'perturbation theory', '-', 'waves', '-', '-', '-', '-', '-']
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[]
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We explore the newly discovered “hangup-kick” effect, which greatly amplifies the recoil for configurations with partial spin-/orbital angular momentum alignment, by studying a set of 48 new simulations of equal-mass, spinning black-hole binaries. We propose a phenomenological model for the recoil that takes this new effect into account and then use this model, in conjunction with statistical distributions for the spin magnitude and orientations, based on accretion simulations, to find the probabilities for observing recoils of several thousand kms<SUP>-1</SUP>. In addition, we provide initial parameters, eccentricities, radiated linear and angular momentum, precession rates and remnant mass, spin, and recoils for all 48 configurations. Our results indicate that surveys exploring peculiar (redshifted or blueshifted) differential line-of-sight velocities should observe at least one case above 2000kms<SUP>-1</SUP> out of 4000 merged galaxies. On the other hand, the probability that a remnant black hole recoils in any direction at a velocity exceeding the ∼2000kms<SUP>-1</SUP> escape velocity of large elliptical galaxies is 0.03%. Probabilities of recoils exceeding the escape velocity quickly rise to 5% for galaxies with escape velocities of 1000kms<SUP>-1</SUP> and nearly 20% for galaxies with escape velocities of 500kms<SUP>-1</SUP>. In addition the direction of these large recoils is strongly peaked toward the angular momentum axis, with very low probabilities of recoils exceeding 350kms<SUP>-1</SUP> for angles larger than 45° with respect to the orbital angular momentum axis.
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https://arxiv.org/pdf/1201.1923.pdf
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{'Gravitational Recoil From Accretion-Aligned Black-Hole Binaries': "Carlos O. Lousto, Yosef Zlochower, 1 Massimo Dotti, 2 and Marta Volonteri 3 \n1 Center for Computational Relativity and Gravitation, \nand School of Mathematical Sciences, Rochester Institute of Technology, \n85 Lomb Memorial Drive, Rochester, New York 14623 \n2 Universit'a di Milano Bicocca, Dipartimento di Fisica G. Occhialini, \nPiazza della Scienza 3, I-20126, Milano, Italy \n3 Astronomy Department, University of Michigan, Ann Arbor 48109, \nUSA and Institut d'Astrophysique de Paris, 98 bis Bd Arago, Paris, 75014, France \nWe explore the newly discovered 'hangup-kick' effect, which greatly amplifies the recoil for configurations with partial spin-/ orbital-angular momentum alignment, by studying a set of 48 new simulations of equal-mass, spinning black-hole binaries. We propose a phenomenological model for the recoil that takes this new effect into account and then use this model, in conjunction with statistical distributions for the spin magnitude and orientations, based on accretion simulations, to find the probabilities for observing recoils of several thousand km s -1 . In addition, we provide initial parameters, eccentricities, radiated linear and angular momentum, precession rates and remnant mass, spin, and recoils for all 48 configurations. Our results indicate that surveys exploring peculiar (redshifted or blueshifted) differential line-of-sight velocities should observe at least one case above 2000 km s -1 out of four thousand merged galaxies. On the other hand, the probability that a remnant BH recoils in any direction at a velocity exceeding the ∼ 2000 km s -1 escape velocity of large elliptical galaxies is 0 . 03%. Probabilities of recoils exceeding the escape velocity quickly rise to 5% for galaxies with escape velocities of 1000 km s -1 and nearly 20% for galaxies with escape velocities of 500 km s -1 . In addition the direction of these large recoils is strongly peaked toward the angular momentum axis, with very low probabilities of recoils exceeding 350 km s -1 for angles larger than 45 · with respect to the orbital angular momentum axis. \nPACS numbers: 04.25.dg, 04.30.Db, 04.25.Nx, 04.70.Bw", 'I. INTRODUCTION': "Speculations about the relevance of gravitational recoils in astrophysical black-hole binary (BHB) mergers can be traced back at least thirty years [1, 2]. The crucial scale of the problem is when those recoils reach velocities comparable to the escape velocities of the relevant structures, i.e. globular clusters, which have escape velocities of 10s of km s -1 , and dwarf, spiral, and giant elliptical galaxies, which have escape velocities from 100s to ∼ 1000 km s -1 for normal galaxies. For large galaxies undergoing major mergers the effective escape velocity can be up to a factor of a few higher at the time of coalescence, as the central potential well deepens rapidly at that time. Once the merger is complete and the stellar systems begins to relax, the potential becomes shallower [3]. \nEarly attempts to compute recoil velocities from BHB mergers used perturbative [4, 5] and post-Newtonian approximations (see [6] for a review up to 2005) and found recoils up to a few hundred km s -1 , but uncertainties in the computations were of the same order of magnitude as those velocities (see [7] for a more current review). The first computation that used full numerical simulations within the Lazarus approach produced similar results [8]. \nThe accurate computation of recoil velocities had to wait for the 2005 breakthroughs [9-11] in Numerical relativity (NR), since it proved to be a genuinely strongfield, highly nonlinear General Relativistic phenomenon. The first systematic study of recoil velocities considered \nunequal mass, nonspinning BHBs [12]. That study found that the maximum recoil velocity for non-spinning BHBs is 175 km s -1 , which occurs for a mass ratio near 1:3. \nUnexpectedly, spinning BHBs, with individual spins anti-aligned with each other and both parallel to the angular momentum direction were found to produce recoils of over a factor two larger than the unequal-mass maximum [13, 14], and a revolution occurred when it was discovered [15] that a configuration of the spins lying in the orbital plane led to recoils of almost [16, 17] 4000 km s -1 . This last figure caught the attention of observational astronomers who began to look for these highly-recoiling BHs by searching the spectral data of galaxies for differential redshifts of several thousand km s -1 . The idea there was that gas close to the BH would remain bound to it, while gas further out in the accretion disk would be left behind. The two gas components would then have different relative redshifts. Initial searches produced the first supermassive recoiling BH candidates [18-21] and now more thorough surveys have increased the numbers of potential candidates to several dozen [22, 23]. These observations may provide the first confirmation of a general relativistic strong field, highly-dynamical, full-numerical prediction. \nA recent study [17] pointed out that configurations with partially aligned spins, which we call the 'hangupkick' configuration, can lead to even larger recoil velocities, of nearly 5000 km s -1 . More importantly, these configurations are favored with respect to the 'spin in the orbital plane' configuration by the effects of accre- \ntion on the BHs during very early orbital (Newtonian) stages [24, 25]. We address this question in more detail in this paper. \nThis paper is organized as follows. In Sec. II we review the numerical methodology to perform the simulations. In Sec. III, we describe the initial configurations of a family of BHBs chosen to model 'hangup-kicks'. In Sec. IV, we provide the main results of these evolutions in tables of radiated energies, angular and linear momenta, as well as final remnant mass and spin. We then model recoils using empirical fitting formulas. In Sec. V we describe smoothed particle hydrodynamics (SPH) simulations that model accretion onto BHBs to obtain the spin magnitude and direction distribution of the individual spins in merging BHBs for our full numerical simulations. Using these spin direction and magnitude distributions, we give predictions for the recoil distribution and the probabilities of observing large recoils. We discuss the consequences and future extensions of these techniques in Sec. VIII.", 'II. NUMERICAL RELATIVITY TECHNIQUES': "We use the TwoPunctures thorn [26] to generate initial puncture data [27] for the black-hole binary (BHB) simulations described below. These data are characterized by mass parameters m p , which are not the horizon masses, of each BH, as well as the momentum and spin of each BH. We evolve these BHB data-sets using the LazEv [28] implementation of the moving puncture approach [10, 11] with the conformal function W = √ χ = exp( -2 φ ) suggested by Ref. [29]. For the runs presented here, we use centered, eighth-order finite differencing in space [30] and a fourth-order Runge Kutta time integrator. (Note that we do not upwind the advection terms.) \nOur code uses the Cactus / EinsteinToolkit [31, 32] infrastructure. We use the Carpet [33] mesh refinement driver to provide a 'moving boxes' style of mesh refinement. In this approach refined grids of fixed size are arranged about the coordinate centers of both holes. The Carpet code then moves these fine grids about the computational domain by following the trajectories of the two BHs. \nWe obtain accurate, convergent waveforms and horizon parameters by evolving this system in conjunction with a modified 1+log lapse and a modified Gamma-driver shift condition [10, 34, 35], and an initial lapse α ( t = 0) = 2 / (1 + ψ 4 BL ), where ψ BL is the Brill-Lindquist conformal factor and is given by \nψ BL = 1 + n ∑ i =1 m p i / (2 | glyph[vector]r -glyph[vector]r i | ) , \nwhere glyph[vector]r i is the coordinate location of puncture i . The lapse and shift are evolved with \n( ∂ t -β i ∂ i ) α = -2 αK, (1a) \n∂ t β a = (3 / 4) ˜ Γ a -ηβ a , (1b) \nwhere we use η = 2 for all simulations presented below. \nWe use AHFinderDirect [36] to locate apparent horizons. We measure the magnitude of the horizon spin using the Isolated Horizon algorithm detailed in Ref. [37]. Note that once we have the horizon spin, we can calculate the horizon mass via the Christodoulou formula \nm H = √ m 2 irr + S 2 H / (4 m 2 irr ) , (2) \nwhere m irr = √ A/ (16 π ) and A is the surface area of the horizon, and S H is the spin angular momentum of the BH (in units of M 2 ). In the tables below, we use the variation in the measured horizon irreducible mass and spin during the simulation as a measure of the error in these quantities. We measure radiated energy, linear momentum, and angular momentum, in terms of the radiative Weyl Scalar ψ 4 , using the formulas provided in Refs. [38, 39]. However, rather than using the full ψ 4 , we decompose it into glyph[lscript] and m modes and solve for the radiated linear momentum, dropping terms with glyph[lscript] ≥ 5. The formulas in Refs. [38, 39] are valid at r = ∞ . We extract the radiated energy-momentum at finite radius and extrapolate to r = ∞ using both linear and quadratic extrapolations. We use the difference of these two extrapolations as a measure of the error.", 'III. SIMULATIONS': "We evolved a set of 48 equal-mass, spinning, quasicircular, 'hangup-kick' configurations, with 30 simulations having individual BH spins of magnitude α = 1 / √ 2 and 18 simulations having BH spin magnitudes of α = 0 . 9, where glyph[vector] α is the dimensionless spin of the BH ( glyph[vector] α = glyph[vector] S H /M 2 H , where glyph[vector] S H is the spin angular momentum and M H is the mass of the BH). In the 'hangup-kick' configuration, the z components of the individual spins are equal, while the projections of the individual spins onto the orbital plane are equal in magnitude but opposite in direction. The α = 1 / √ 2 configurations were split into five sets of 6, where the runs in each individual set had the same initial angle θ between the spin direction and orbital angular momentum direction (here we chose θ = 22 . 5 · , 45 · , 60 · , 120 · , 135 · ). In each set with as given θ , we chose the initial orientation φ i between the in-plane spin and linear momentum to be 0 · , 30 · , 90 · , 130 · , 210 · , and 315 · . For the α = 0 . 9 runs, we used the same initial 6 φ i configurations for θ = 60 · , θ = 30 · , and θ = 15 · . We combine these results with the simulations of [40] (which have θ = 90 · ) in order to perform our analysis below. \nInitial data parameters for the 48 simulations are given in Table I. We denote these configurations by AsTHxxxPHyyy, where s indicates the approximate individual spin magnitude (7 for α i = 1 / √ 2 and 9 for α i = 0 . 9), xxx indicates the angle the spin makes with respect to the z axis, and yyy indicates the angle the spin makes with respect to the y axis. Here xxx and yyy \nFIG. 1: xy plane projections of the trajectories for various 'hangup-kick' configurations. (Top Left) Trajectory for the A7TH22.5PH0 configuration, (Bottom Left) trajectory for the A7TH135PH0 configuration, (Top Right) trajectory for the A9TH15PH0 configuration, (Bottom Right) trajectory for the A9TH60PH0 configuration. The plot shows the trajectories for configurations with the largest and smallest inclination angle for the α = 1 / √ 2 and α = 0 . 9 configurations. Note that the eccentricity is larger for large θ and that the eccentricity decreases more slowly. \n<!-- image --> \nThe orbital motion of these BHBs has an interesting property, the spin precession frequency and the orbital frequency agree right near merger (in these coordinates). Initially the spin precession frequencies are much lower than the orbital frequency, but ramp up dramatically near merger. Figure 3 shows this behavior for the different θ and α configurations (for clarity in the plot, we only show the φ i = 0 configurations). Interestingly, the strongest effect seems to be due to the inclination angle θ (which is measured with respect to the orbital angular momentum axis, i.e. the z axis) rather than on the projected z spin or total spin. For a given α and θ there are variations with φ i , but these are smaller than the variations with θ . This rapid increase of the precession frequency near merger (as measured with the techniques of [43]), increasing all the way up to the orbital frequency, is in contrast with the much milder increase in the spin magnitude due to weak tidal effects[44]. It also lends support to modeling of black hole merger assuming geodetic precession [45]. \nIn Table II we compare the radiated mass and angular momentum as calculated directly from ψ 4 to the corresponding quantities derived from the remnant mass and spin. The difference between these quantities is a better measurement of the error than those derived from variations in the final horizon mass and spin or in ex- \nre in degrees. An eccentricity reduction procedure like those given in [41, 42] could be used to generate configurations with very low eccentricity, but the amount of time required to reduce the eccentricity for 48 configurations would have been too long. We therefore chose to start from 3.5 Post-Newtonian (PN) quasicircular orbital parameters from further separations, such that each binary completed 5-6 orbits prior to merger, and then relied on the radiation of angular momentum during this 5-6 orbit inspiral to reduce the eccentricity. The initial separations varied between 10.16M and 8.2M, depending on the magnitude of the hangup effect, with smaller initial separations for configurations that exhibit larger hangups. Example trajectories for several of these configurations are given in Figs. 1 and 2. \ntrapolation of the waveform to infinity. That is, there are systematic errors due to truncation error and finite extraction radius, and the difference between these two measurements gives a lower bound to the error. \nThe dimensionless spin α merger , the orientation ϕ of the spin during the final orbit and plunge, as well as the remnant BH properties, including recoil velocity, are given in Table III. Note here that the orientation ϕ is the angle that the spin of BH1 (the BH originally located on the positive x axis) makes with the spin of BH1 in the corresponding AsTHxxxPH0 configuration in a rotated frame where infall directions all coincide (see [46]). Also note that the largest measured recoil for these runs is (4079 . 5 ± 10 . 1) km s -1 for the A9TH60PH30 and A9TH60PH210 configurations, which exceeds both the largest measured quasicircular 'superkick recoil' of 3300 km s -1 [47] and even the theoretical 'superkick maximum' recoil of 3680 ± 130 km s -1 [40]. The values given for the spins near merger should only be taken as an approximation. The spin-up apparent in the A9TH60 simulations (i.e. the difference between α merger and the initial spin of α = 0 . 9) was due to the lower resolution used in these simulations (simulations of A9TH45 with the same resolution as A9TH60 showed an even stronger spin-up at late times that converged away with higher resolution). Interestingly, these highly-spinning configu- \nTABLE I: Initial data parameters for the 48 'hangup-kick' configurations. In all cases the puncture masses were chosen such that the total ADM mass of the binary was 1 . 0 ± 10 -6 M . Here the punctures are located at ± ( x, 0 , 0) with momenta ± (0 , p, 0) and spins glyph[vector] S = ( ± S x , ± S y , S z ). The approximate initial eccentricities, eccentricities measured over the last orbit, and the number of orbits, are also given. \n| CONF | m p /M | x/M | p/M | S x /M 2 | S y /M 2 | S z /M 2 | ( e init , N orbits , e merge ) |\n|-----------------------|-------------------|----------|-------------------|--------------------|------------|------------|-----------------------------------|\n| A7TH22.5PH0 | 0.361001 | 4.141042 | 0.105976 | 0.000000 | 0.069426 | 0.167609 | ( 0.02, 5.5, 0.004) |\n| A7TH22.5PH30 | 0.361022 | 4.141042 | 0.105976 | -0.034713 | 0.060125 | 0.167609 | |\n| A7TH22.5PH90 | 0.361085 | 4.141042 | 0.105976 | -0.069426 | 0.000000 | 0.167609 | |\n| A7TH22.5PH130 | 0.361050 | 4.141042 | 0.105976 | -0.053183 | -0.044626 | 0.167609 | |\n| A7TH22.5PH210 | 0.361022 | 4.141042 | 0.105976 | 0.034713 | -0.060125 | 0.167609 | |\n| A7TH22.5PH315 | 0.361043 | 4.141042 | 0.105976 | 0.049092 | 0.049092 | 0.167609 | |\n| A7TH45PH0 | 0.360775 | 4.175510 | 0.106744 | 0.000000 | 0.128222 | 0.128222 | (0.027, 5, 0.0054) |\n| A7TH45PH30 | 0.360849 | 4.175510 | 0.106744 | -0.064111 | 0.111043 | 0.128222 | |\n| A7TH45PH90 | 0.361068 | 4.175510 | 0.106744 | -0.128222 | 0.000000 | 0.128222 | |\n| A7TH45PH130 | 0.360947 | 4.175510 | 0.106744 | -0.098224 | -0.082419 | 0.128222 | |\n| A7TH45PH210 | 0.360849 | 4.175510 | 0.106744 | 0.064111 | -0.111043 | 0.128222 | |\n| A7TH45PH315 | 0.360922 | 4.175510 | | 0.090667 | 0.090667 | 0.128222 | |\n| A7TH60PH0 | 0.360607 | 4.207527 | 0.106744 0.107470 | 0.000000 | 0.156971 | 0.090627 | (0.022, 4.5, 0.0052) |\n| A7TH60PH30 | 0.360718 | 4.207527 | 0.107470 | -0.078485 | 0.135941 | 0.090627 | |\n| A7TH60PH90 | 0.361052 | 4.207527 | 0.107470 | -0.156971 | 0.000000 | 0.090627 | |\n| A7TH60PH130 | 0.360868 | 4.207527 | 0.107470 | -0.120246 | -0.100899 | 0.090627 | |\n| A7TH60PH210 | 0.360718 | 4.207527 | 0.107470 | 0.078485 | -0.135941 | 0.090627 | |\n| A7TH60PH315 | 0.360830 | 4.207527 | 0.107470 | 0.110995 | 0.110995 | 0.090627 | |\n| A7TH120PH0 | 0.362448 | 5.295630 | 0.095864 | 0.000000 | 0.156161 | -0.090160 | (0.026, 5, 0.003) |\n| A7TH120PH30 | 0.362537 | 5.295630 | 0.095864 | -0.078081 | 0.135240 | -0.090160 | |\n| A7TH120PH90 | 0.362803 | 5.295630 | 0.095864 | -0.156161 | 0.000000 | -0.090160 | |\n| A7TH120PH130 | 0.362656 | 5.295630 | 0.095864 | -0.119627 | -0.100379 | -0.090160 | |\n| A7TH120PH210 | 0.362537 | 5.295630 | 0.095864 | 0.078081 | -0.135240 | -0.090160 | |\n| A7TH120PH315 | 0.362625 | 5.295630 | 0.095864 | 0.110423 | 0.110423 | -0.090160 | |\n| A7TH135PH0 | 0.362878 | 5.534525 | 0.093655 | 0.000000 | 0.127399 | -0.127399 | (0.02, 5, 0.005) |\n| A7TH135PH30 | 0.362934 | 5.534525 | 0.093655 | -0.063699 | 0.110331 | -0.127399 | |\n| A7TH135PH90 | 0.363104 | 5.534525 | 0.093655 | -0.127399 | 0.000000 | -0.127399 | |\n| A7TH135PH130 | 0.363011 | 5.534525 | | | -0.081890 | -0.127399 | |\n| A7TH135PH210 | 0.362934 | 5.534525 | 0.093655 0.093655 | -0.097593 0.063699 | -0.110331 | -0.127399 | |\n| A7TH135PH315 | 0.362991 | 5.534525 | 0.093655 | 0.090085 | 0.090085 | -0.127399 | |\n| A9TH15PH0 | 0.177282 | 4.094887 | 0.104887 | 0.000000 | 0.059803 | 0.223187 | (0.027, 6, 0.003) |\n| A9TH15PH30 | 0.177339 | 4.094887 | 0.104887 | -0.029901 | 0.051791 | 0.223187 | |\n| A9TH15PH90 | 0.177509 | 4.094887 | 0.104887 | -0.059803 | 0.000000 | 0.223187 | |\n| A9TH15PH130 | 0.177415 | 4.094887 | 0.104887 | -0.045811 | -0.038440 | 0.223187 | |\n| A9TH15PH210 | 0.177339 | 4.094887 | 0.104887 | 0.029901 | -0.051791 | 0.223187 | |\n| A9TH15PH315 | 0.177395 | 4.094887 | 0.104887 | 0.042287 | 0.042287 | 0.223187 | |\n| A9TH30PH30 | 0.176864 | 4.116022 | 0.105345 | -0.057748 | 0.100022 | 0.200045 | |\n| | | | | -0.115496 | | | |\n| A9TH30PH90 | 0.177505 | 4.116022 | 0.105345 | | 0.000000 | 0.200045 | |\n| A9TH30PH130 | 0.177152 | 4.116022 | 0.105345 | -0.088475 | -0.074239 | 0.200045 | |\n| A9TH30PH210 | 0.176864 | 4.116022 | 0.105345 | 0.057748 | -0.100022 | 0.200045 | |\n| A9TH30PH315 | 0.177078 | 4.116022 | 0.105345 | 0.081668 | 0.081668 | 0.200045 | |\n| A9TH60PH0 | 0.174838 | 4.190252 | 0.107000 | 0.000000 | 0.199841 | 0.115378 | (0.027, 5, 0.0055) |\n| | | 4.190252 | 0.107000 | -0.099920 | 0.173067 | 0.115378 | |\n| A9TH60PH30 A9TH60PH90 | 0.175510 0.177498 | 4.190252 | 0.107000 | -0.199841 | 0.000000 | 0.115378 | |\n| A9TH60PH130 | 0.176408 | 4.190252 | 0.107000 | -0.153087 | -0.128455 | 0.115378 | |\n| A9TH60PH210 | 0.175510 | 4.190252 | 0.107000 | 0.099920 | -0.173067 | 0.115378 | |\n| A9TH60PH315 | | | 0.107000 | | 0.141309 | | |\n| | 0.176177 | 4.190252 | | 0.141309 | | 0.115378 | | \ncan exhibit both spin-up and spin-down, depending on the location of the refinement boundaries, when not fully resolved. The other A9 runs used higher resolution and show much better spin conservation. \nWe evolve the A9TH15 and A9TH30 configurations with 10 levels of refinement and maximum resolution of \nh = M/ 153 . 6. The width of this level was 2 × 0 . 35 M , while the radius of the horizons grew to 0 . 24 M . Our initial explorations used grids that were smaller in radius, but we found that using larger grids improved the spin conservation considerably. The A9TH60 configuration were evolved with grids a factor of 1.2 coarser, and \nFIG. 2: The elevation of the trajectory as a function of time for the several of the 'hangup-kick' configurations. Note that the 'bobbing' amplitude does not necessarily correspond to a large recoil. The A7TH135PH0 configuration has a factor of 2 smaller recoil than he A7TH60PH0 configuration, but a slightly larger bobbing amplitude. \n<!-- image -->", 'IV. RESULTS AND MODELING OF RECOIL VELOCITIES': "With the discovery of very large recoil [15] velocities for certain configurations of merging spinning BHBs, the need for an empirical model for the recoil velocity as a function of the progenitor's parameters was apparent. Our approach to provide that phenomenological formula was based on the observation that the recoil of spinning BHs is largely generated around the time of merger of the two holes [48]; and that this nearly instantaneous burst of radiation of linear momentum can be modeled by a parametrized dependence of the leading (on spins and mass ratio) post-Newtonian (PN) expressions for the linear momentum radiated[49]. \nIn Ref. [50] we extended our original empirical formula for the recoil velocity imparted to the remnant of a BHB merger [15, 16] to include next-to-leading-order corrections (based on the PN work of [51]), still linear in the spins \nglyph[vector] V recoil ( q, glyph[vector]α ) = v m ˆ e 1 + v ⊥ (cos ξ ˆ e 1 +sin ξ ˆ e 2 ) + v ‖ ˆ n ‖ , (3) \nconsequently the spins near merger are not as accurate. Note that these A9TH60 runs were performed first, and based on the errors in these simulations, we refined the grid for the other A9 runs. \nwhere \nv m = A m η 2 (1 -q ) (1 + q ) [1 + B m η ] , v ⊥ = H η 2 (1 + q ) [ (1 + B H η ) ( α ‖ 2 -qα ‖ 1 ) + H S (1 -q ) (1 + q ) 2 ( α ‖ 2 + q 2 α ‖ 1 ) ] , v ‖ = K η 2 (1 + q ) [ (1 + B K η ) ∣ ∣ glyph[vector] α ⊥ 2 -qglyph[vector]α ⊥ 1 ∣ ∣ × cos( φ ∆ -φ 1 ) + K S (1 -q ) (1 + q ) 2 ∣ ∣ glyph[vector] α ⊥ 2 + q 2 glyph[vector] α ⊥ 1 ∣ ∣ × cos( φ S -φ 2 ) ] , (4) \nand η = q/ (1 + q ) 2 , with q = m 1 /m 2 the mass ratio of the smaller to larger mass hole, glyph[vector] α i = glyph[vector] S i /m 2 i , m i is shorthand for m Hi the mass of BH i , the index ⊥ and ‖ refer to perpendicular and parallel to the orbital angular momentum respectively, ˆ e 1 , ˆ e 2 are orthogonal unit vectors in the orbital plane, and ξ measures the angle between the unequal mass and spin contribution to the recoil velocity in the orbital plane. The angles φ ∆ and φ S are defined as the angle between the in-plane component glyph[vector] ∆ ⊥ = M ( glyph[vector] S ⊥ 2 /m 2 -glyph[vector] S ⊥ 1 /m 1 ) and glyph[vector] S ⊥ = glyph[vector] S ⊥ 1 + glyph[vector] S ⊥ 2 respectively and a fiducial direction at merger (see Ref. [46] for a description of the technique). \nTABLE II: A comparison of the radiated mass and angular momentum with the predictions based on the final remnant mass and spin and the initial ADM mass and angular momentum for the 48 'hangup-kick' configurations. \n| CONF | ( M ADM - M H ) /M | δE rad /M | ( J z ADM - S z H ) /M 2 | δJ z rad /M 2 |\n|---------------------------|-------------------------------------------------|-------------------------------------------------|-------------------------------------------------|-------------------------------------------------|\n| A7TH22.5PH0 | 0 . 065949 ± 0 . 000113 | 0 . 063479 ± 0 . 000169 | 0 . 452347 ± 0 . 001590 | 0 . 440182 ± 0 . 006183 |\n| A7TH22.5PH30 | 0 . 065440 ± 0 . 000125 | 0 . 063034 ± 0 . 000164 | 0 . 450602 ± 0 . 001706 | 0 . 438756 ± 0 . 006006 |\n| A7TH22.5PH90 | 0 . 065188 ± 0 . 000121 | 0 . 062822 ± 0 . 000171 | 0 . 450332 ± 0 . 001675 | 0 . 439398 ± 0 . 005161 |\n| A7TH22.5PH130 | 0 . 065842 ± 0 . 000105 | 0 . 063363 ± 0 . 000190 | 0 . 452581 ± 0 . 001493 | 0 . 440443 ± 0 . 005901 |\n| A7TH22.5PH210 | 0 . 065440 ± 0 . 000125 | 0 . 063035 ± 0 . 000165 | 0 . 450601 ± 0 . 001706 | 0 . 438766 ± 0 . 005996 |\n| A7TH22.5PH315 | 0 . 065910 ± 0 . 000105 | 0 . 063423 ± 0 . 000188 | 0 . 452761 ± 0 . 001487 | 0 . 440460 ± 0 . 006077 |\n| A7TH45PH0 | 0 . 058764 ± 0 . 000006 | 0 . 056538 ± 0 . 000181 | 0 . 417697 ± 0 . 000084 | 0 . 403669 ± 0 . 007395 |\n| A7TH45PH30 | 0 . 058594 ± 0 . 000008 | 0 . 056460 ± 0 . 000153 | 0 . 416460 ± 0 . 000088 | 0 . 403290 ± 0 . 006741 |\n| A7TH45PH90 | 0 . 056266 ± 0 . 000010 | 0 . 054394 ± 0 . 000120 | 0 . 408936 ± 0 . 000072 | 0 . 397875 ± 0 . 004794 |\n| A7TH45PH130 | 0 . 056959 ± 0 . 000010 | 0 . 054940 ± 0 . 000155 | 0 . 412016 ± 0 . 000078 | 0 . 399658 ± 0 . 005982 |\n| A7TH45PH210 | 0 . 058594 ± 0 . 000008 | 0 . 056460 ± 0 . 000153 | 0 . 416459 ± 0 . 000088 | 0 . 403290 ± 0 . 006739 |\n| A7TH45PH315 | 0 . 057167 ± 0 . 000008 | 0 . 055120 ± 0 . 000159 | 0 . 412756 ± 0 . 000078 | 0 . 400125 ± 0 . 006224 |\n| A7TH60PH0 | 0 . 050887 ± 0 . 000004 | 0 . 049046 ± 0 . 000145 | 0 . 379033 ± 0 . 000032 | 0 . 365962 ± 0 . 007685 |\n| A7TH60PH30 | 0 . 052125 ± 0 . 000003 | 0 . 050267 ± 0 . 000146 | 0 . 383455 ± 0 . 000029 | 0 . 371173 ± 0 . 006948 |\n| A7TH60PH90 | 0 . 051234 ± 0 . 000005 | 0 . 049628 ± 0 . 000094 | 0 . 378829 ± 0 . 000027 | 0 . 368250 ± 0 . 005342 |\n| A7TH60PH130 | 0 . 049522 ± 0 . 000004 | 0 . 047918 ± 0 . 000103 | 0 . 372896 ± 0 . 000025 | 0 . 361386 ± 0 . 006580 |\n| A7TH60PH210 | 0 . 052124 ± 0 . 000003 | 0 . 050266 ± 0 . 000146 | 0 . 383456 ± 0 . 000029 | 0 . 371168 ± 0 . 006945 |\n| A7TH60PH315 | 0 . 049445 ± 0 . 000004 | 0 . 047811 ± 0 . 000112 | 0 . 372866 ± 0 . 000030 | 0 . 360951 ± 0 . 006697 |\n| A7TH120PH0 | 0 . 033193 ± 0 . 000002 | 0 . 032297 ± 0 . 000048 | 0 . 303865 ± 0 . 000005 | 0 . 298884 ± 0 . 006258 |\n| A7TH120PH30 | 0 . 033107 ± 0 . 000002 | 0 . 032297 ± 0 . 000010 | 0 . 303095 ± 0 . 000005 | 0 . 295666 ± 0 . 008881 |\n| A7TH120PH90 | 0 . 031686 ± 0 . 000002 | 0 . 031037 ± 0 . 000015 | 0 . 297018 ± 0 . 000005 | 0 . 292110 ± 0 . 006177 |\n| A7TH120PH130 | 0 . 031974 ± 0 . 000002 | 0 . 031188 ± 0 . 000044 | 0 . 298771 ± 0 . 000005 0 . 303097 ± 0 . 000005 | 0 . 297030 ± 0 . 002584 0 . 295663 ± 0 . 008889 |\n| A7TH120PH210 A7TH120PH315 | 0 . 033107 ± 0 . 000002 0 . 032117 ± 0 . 000002 | 0 . 032298 ± 0 . 000010 0 . 031308 ± 0 . 000051 | 0 . 299419 ± 0 . 000005 | 0 . 297731 ± 0 . 002485 |\n| A7TH135PH0 | 0 . 029675 ± 0 . 000004 | 0 . 029004 ± 0 . 000024 | 0 . 287481 ± 0 . 000006 | 0 . 290758 ± 0 . 000188 |\n| A7TH135PH30 | 0 . 030091 ± 0 . | . 029449 ± 0 . | 0 . 289543 ± 0 . | 0 . 293205 ± 0 . 000038 |\n| | 000004 | 0 000043 | 000005 | |\n| A7TH135PH90 | 0 . 030000 ± 0 . 000004 | 0 . 029437 ± 0 . 000063 | 0 . 288696 ± 0 . 000006 | 0 . 294013 ± 0 . 001881 |\n| A7TH135PH210 | 0 . 030085 ± 0 . 000004 | 0 . 029448 ± 0 . 000048 | 0 . 289513 ± 0 . 000006 | 0 . 293050 ± 0 . 000149 |\n| A7TH135PH315 | 0 . 029385 ± 0 . 000004 | 0 . 028766 ± 0 . 000032 | 0 . 285792 ± 0 . 000006 | 0 . 290244 ± 0 . 001187 |\n| A9TH15PH0 | 0 . 086926 ± 0 . 000422 | 0 . 082160 ± 0 . 000612 | 0 . 540966 ± 0 . 004231 | 0 . 523516 ± 0 . 003980 0 . 524385 ± 0 . 004228 |\n| A9TH15PH30 | 0 . 087312 ± 0 . 000396 | 0 . 082564 ± 0 . 000612 | 0 . 541701 ± 0 . 003972 0 . 539095 ± 0 . 003287 | 0 . 523929 ± 0 . 003074 |\n| A9TH15PH90 A9TH15PH130 | 0 . 086773 ± 0 . 000341 0 . 086317 ± 0 . 000384 | 0 . 082188 ± 0 . 000602 0 . 081695 ± 0 . 000599 | 0 . 538447 ± 0 . 003722 | 0 . 522743 ± 0 . 002983 |\n| A9TH15PH210 | 0 . 087315 ± 0 . 000394 | 0 . 082564 ± 0 . 000612 | 0 . 541724 ± 0 . 003950 | 0 . 524384 ± 0 . |\n| A9TH15PH315 | 0 . 086325 ± 0 . 000394 | 0 . 081686 ± 0 . 000600 | 0 . 538588 ± 0 . 003835 | 004228 0 . 522588 ± 0 . 003187 |\n| A9TH30PH0 | 0 . 077615 ± 0 . 000339 | 0 . 073045 ± 0 . 000461 | 0 . 509243 ± 0 . 005934 | 0 . 484306 ± 0 . 004248 |\n| | 0 . 076957 ± 0 . 000316 | 0 . 072527 ± 0 . 000493 | 0 . 507180 ± 0 . 005340 | 0 . 483676 ± 0 . 004906 |\n| A9TH30PH30 | | | | |\n| A9TH30PH90 A9TH30PH130 | 0 . 078900 ± 0 . 000166 0 . 079891 ± 0 . 000209 | 0 . 074900 ± 0 . 000485 0 . 075617 ± 0 . 000486 | 0 . 510059 ± 0 . 002727 0 . 513344 ± 0 . 003666 | 0 . 496097 ± 0 . 000678 0 . 496170 ± 0 . 000939 |\n| A9TH30PH210 | 0 . 076956 ± 0 . 000316 | 0 . 072556 ± 0 . 000469 | 0 . 507180 ± 0 . 005341 | 0 . 483678 ± 0 . 004902 |\n| | 0 . 079810 ± 0 . 000224 | 0 . 075477 ± 0 . 000486 | 0 . 513380 ± 0 . 003973 | 0 . 495112 ± 0 . 001297 |\n| A9TH30PH315 | 0 . 056289 ± 0 . 000311 | 0 . 055634 ± 0 . 000322 | 0 . 387215 ± 0 . 007605 | 0 . 411510 ± 0 . 009728 |\n| A9TH60PH0 A9TH60PH30 | 0 . 059774 ± 0 . 000325 | | | |\n| A9TH60PH90 | 0 . 054809 ± 0 . 000087 | 0 . 058422 ± 0 . 000326 0 . 052769 ± 0 . 000184 | 0 . 402796 ± 0 . 005399 0 . 394238 ± 0 . 001762 | 0 . 422236 ± 0 . 007268 0 . 396247 ± 0 . 003734 |\n| | | 0 . 053567 ± 0 . 000241 | | |\n| A9TH60PH130 | 0 . 055249 ± 0 . 000168 | | 0 . 392036 ± 0 . 003341 | 0 . 399533 ± 0 . 008047 |\n| A9TH60PH210 | 0 . 059774 ± 0 . 000325 | 0 . 058422 ± 0 . 000326 | 0 . 402796 ± 0 . 005399 | 0 . 422236 ± 0 . 007268 |\n| A9TH60PH315 | 0 . 054902 ± 0 . 000201 | 0 . 053558 ± 0 . 000260 | 0 . 388072 ± 0 . 004485 | 0 . 400454 ± 0 . 008515 | \nNote that glyph[vector] ∆ = M ( glyph[vector] S 2 /m 2 -glyph[vector] S 1 /m 1 ) can be expressed as glyph[vector] ∆ = M 2 ( glyph[vector] α 2 -qglyph[vector]α 1 ) / (1+ q ). Phases φ 1 and φ 2 depend on the initial separation of the holes for quasicircular orbits (astrophysically realistic evolutions of comparable masses BHs lead to nearly zero eccentricity mergers). \nThe most recent published estimates for the above parameters can be found in [46, 52] and references therein. \nThe current best estimates are: A m = 1 . 2 × 10 4 km s -1 , B m = -0 . 93, H = (6 . 9 ± 0 . 5) × 10 3 km s -1 , K = (5 . 9 ± 0 . 1) × 10 4 km s -1 , and ξ ∼ 145 · , and K S = -4 . 254. Here we set B H and B K to zero, which is consistent with the findings in [50], where it was found that the uncertainties in the coefficients are of the same magnitude as the coefficients themselves. \nAlthough the post-Newtonian approximation fails to \nTABLE III: Merger and remnant BH properties of the 48 configurations. S H is the spin angular momentum of the remnant, M H is the Christodoulou mass, V z recoil is the recoil velocity, α merger is an approximate value of the dimensionless spin during the merger phase, and ϕ is the angle between the direction of the spin of BH1 (in the rotated frame) and the spin of BH1 in the corresponding PH0 configuration (see Section IV). \n| CONF | M H /M | S H /M 2 | V z recoil (km s - 1 ) | α merger | ϕ |\n|---------------|-------------------------------------------------|-----------------------------------------------|-------------------------------------|------------|----------|\n| A7TH22.5PH0 | 0 . 934051 ± 0 . 000113 | 0 . 760575 ± 0 . 001590 | - 925 . 3 ± 1 . 0 | 0.71 | 0 |\n| A7TH22.5PH30 | 0 . 934560 ± 0 . 000125 | 0 . 762320 ± 0 . 001706 | - 7 . 9 ± 2 . 3 | 0.71 | 31.53 |\n| A7TH22.5PH90 | 0 . 934812 ± 0 . 000121 | 0 . 762590 ± 0 . 001675 | 1531 . 5 ± 2 . 6 | 0.71 | 91.68 |\n| A7TH22.5PH130 | 0 . 934158 ± 0 . 000105 | 0 . 760341 ± 0 . 001493 | 1735 . 2 ± 1 . 2 | 0.71 | 131.14 |\n| A7TH22.5PH210 | 0 . 934560 ± 0 . 000125 | 0 . 762321 ± 0 . 001706 | 8 . 0 ± 2 . 3 | 0.71 | 211.52 |\n| A7TH22.5PH315 | 0 . 934089 ± 0 . 000105 | 0 . 760161 ± 0 . 001487 | - 1703 . 7 ± 1 . 1 | 0.71 | 315.97 |\n| A7TH45PH0 | 0 . 941235 ± 0 . 000006 | 0 . 730165 ± 0 . 000084 | 2527 . 7 ± 5 . 4 | 0.71 | 0 |\n| A7TH45PH30 | 0 . 941406 ± 0 . 000008 | 0 . 731403 ± 0 . 000088 | 1708 . 6 ± 1 . 2 | 0.71 | 29.33 |\n| A7TH45PH90 | 0 . 943734 ± 0 . 000010 | 0 . 738926 ± 0 . 000072 | - 1199 . 6 ± 4 . 8 | 0.71 | 91.22 |\n| A7TH45PH130 | 0 . 943041 ± 0 . | ± | | 0.71 | 131.57 |\n| A7TH45PH210 | 000010 | 0 . 735846 0 . 000078 ± | - 2486 . 4 ± 0 . 2 | | |\n| A7TH45PH315 | 0 . 941406 ± 0 . 000008 0 . 942833 ± 0 . 000008 | 0 . 731403 0 . 000088 0 . 735106 ± 0 . 000078 | - 1708 . 2 ± 1 . 3 2569 . 6 ± 0 . 7 | 0.71 0.71 | 209.35 |\n| A7TH60PH0 | 0 . 949113 ± 0 . 000004 | 0 . 706585 ± 0 . 000032 | - 2786 . 0 ± 4 . 6 | 0.71 | 316.39 0 |\n| A7TH60PH30 | 0 . 947875 ± 0 . 000003 | 0 . 702163 ± 0 . 000029 | - 2886 . 8 ± 2 . 9 | 0.71 | 27.78 |\n| A7TH60PH90 | 0 . 948766 ± 0 . 000005 | 0 . 706789 ± 0 . 000027 | - 968 . 8 ± 0 . 4 | 0.71 | 89.42 |\n| A7TH60PH130 | 0 . 950477 ± 0 . 000004 | 0 . 712722 ± 0 . 000025 | 1167 . 1 ± 4 . 6 | 0.71 | 129.89 |\n| A7TH60PH210 | 0 . 947876 ± 0 . 000003 | 0 . 702162 ± 0 . 000029 | 2886 . 7 ± 3 . 0 | 0.71 | 207.73 |\n| A7TH60PH315 | 0 . 950555 ± 0 . 000004 | 0 . 712752 ± 0 . 000030 | - 1495 . 7 ± 5 . 5 | 0.71 | 316.71 |\n| A7TH120PH0 | 0 . 966806 ± 0 . 000002 | 0 . 531135 ± 0 . 000005 | 1754 . 0 ± 6 . 7 | 0.71 | 0 |\n| A7TH120PH30 | 0 . 966893 ± 0 . 000002 | 0 . 531905 ± 0 . 000005 | 1370 . 2 ± 5 . 4 | 0.71 | 29.99 |\n| A7TH120PH90 | 0 . 968314 ± 0 . 000002 | 0 . 537982 ± 0 . 000005 | - 269 . 0 ± 1 . 3 | 0.71 | 86.11 |\n| 7TH120PH130 | 0 . 968026 ± 0 . 000002 | 0 . 536229 ± 0 . 000005 | - 1400 . 6 ± 3 . 5 | 0.71 | 129.11 |\n| A7TH120PH210 | 0 . 966893 ± 0 . 000002 | 0 . 531903 ± 0 . 000005 | - 1370 . 7 ± 5 . 4 | 0.71 | 209.98 |\n| A7TH120PH315 | 0 . 967883 ± 0 . 000002 | 0 . 535581 ± 0 . 000005 | 1495 . 8 ± 4 . 1 | 0.71 | 314.62 |\n| A7TH135PH0 | 0 . 970326 ± 0 . 000004 | 0 . 494390 ± 0 . 000006 | 1108 . 0 ± 1 . 1 | 0.71 | 0 |\n| A7TH135PH30 | 0 . 969909 ± 0 . 000004 | 0 . 492329 ± 0 . 000005 | 1328 . 0 ± 1 . 9 | 0.71 | 28.77 |\n| A7TH135PH90 | 0 . 970000 ± 0 . 000004 | 0 . 493176 ± 0 . 000006 | 775 . 6 ± 2 . 0 | 0.71 | 90.61 |\n| A7TH135PH130 | 0 . 970585 ± 0 . 000003 | 0 . 495978 ± 0 . 000006 | - 207 . 1 ± 0 . 6 | 0.71 | 133.14 |\n| A7TH135PH210 | 0 . 969915 ± 0 . 000004 | 0 . 492359 ± 0 . 000006 | - 1326 . 6 ± 1 . 9 | 0.71 | 208.32 |\n| A7TH135PH315 | 0 . 970615 ± 0 . 000004 | 0 . 496080 ± 0 . 000006 | 332 . 6 ± 0 . 4 | 0.71 | 318.33 |\n| A9TH15PH0 | 0 . 913073 ± 0 . 000422 | 0 . 764409 ± 0 . 004231 | 2028 . 2 ± 20 . 6 | 0.90 | 0 |\n| A9TH15PH30 | 0 . 912688 ± 0 . 000396 | 0 . 763674 ± 0 . 003972 | 1764 . 0 ± 23 . 2 | 0.90 | 30.06 |\n| A9TH15PH90 | 0 . 913227 ± 0 . 000341 | 0 . 766280 ± 0 . 003287 | 22 . 9 ± 8 . 4 | 0.90 | 91.51 |\n| A9TH15PH130 | 0 . 913683 ± 0 . 000384 | 0 . 766929 ± 0 . 003722 | - 1323 . 0 ± 9 . 6 | 0.90 | 130.98 |\n| A9TH15PH210 | 0 . 912685 ± 0 . 000394 | 0 . 763651 ± 0 . 003950 | - 1763 . 9 ± 23 . 2 | 0.90 | 210.02 |\n| A9TH15PH315 | 0 . 913676 ± 0 . 000394 | 0 . 766788 ± 0 . 003835 | 1455 . 4 ± 11 . 0 | 0.90 | 316.18 |\n| A9TH30PH0 | 0 . 922385 ± 0 . 000339 | 0 . 758052 ± 0 . 005934 | - 886 . 8 ± 13 . 6 | 0.895 | 0 |\n| A9TH30PH30 | 0 . 923043 ± 0 . 000316 | 0 . 760116 ± 0 . 005340 | - 2358 . 8 ± 6 . 0 | 0.895 | 28.07 |\n| A9TH30PH90 | 0 . 921100 ± 0 . 000166 | 0 . 757236 ± 0 . 002727 | - 3346 . 5 ± 34 . 1 | 0.895 | 80.64 |\n| A9TH30PH130 | 0 . 920108 ± 0 . 000209 | 0 . 753952 ± 0 . 003666 | - 2306 . 6 ± 39 . 5 | 0.895 | 122.75 |\n| A9TH30PH210 | 0 . 923043 ± 0 . 000316 | 0 . 760115 ± 0 . 005341 | 2360 . 4 ± 4 . 9 | 0.895 | 208.02 |\n| A9TH30PH315 | 0 . 920190 ± 0 . 000224 | 0 . 753916 ± 0 . 003973 | 2040 . 8 ± 38 . 7 | 0.895 | 308.95 |\n| A9TH60PH0 | 0 . 943710 ± 0 . 000311 | 0 . 740252 ± 0 . 007605 | 3792 . 7 ± 5 . 4 | 0.92 | 0 |\n| A9TH60PH30 | 0 . 940226 ± 0 . 000325 | 0 . 724671 ± 0 . 005399 | 4079 . 5 ± 10 . 1 | 0.92 | 36.83 |\n| A9TH60PH90 | 0 . 945191 ± 0 . 000087 | 0 . 733228 ± 0 . 001762 | - 2352 . 1 ± 1 . 6 | 0.92 | 146.22 |\n| A9TH60PH130 | 0 . 944751 ± 0 . 000168 | 0 . 735431 ± 0 . 003341 | - 3054 . 6 ± 0 . 7 | 0.92 | 160.22 |\n| A9TH60PH210 | 0 . 940226 ± 0 . 000325 | 0 . 724671 ± 0 . 005399 | - 4079 . 5 ± 10 . 1 | 0.92 | 216.90 |\n| A9TH60PH315 | 0 . 945098 ± 0 . 000201 | 0 . 739395 ± 0 . 004485 | 2772 . 5 ± 0 . 8 | 0.92 | 334.02 | \nprovide accurate amplitudes for each velocity component, the above parametrization and fitting to a set of full numerical simulations has shown its predictive power in a number of occasions; for instance by predicting the mass ratio dependence that was later confirmed by sets of lengthy numerical simulations[46]. The success of the \noriginal formula allowed the study of higher-order dependencies on the spin of the holes. In a previous study [40], we found that the 'superkick' recoil (where the two BHs have equal mass, equal intrinsic spin magnitudes α , and spins lying in the orbital plane in opposite directions) has the following dependence on the intrinsic spin α and \nFIG. 3: Spin precession frequency versus orbital frequencies for α = 1 / √ 2 and α = 0 . 9 configurations. In each case, only the φ i = 0 configuration is shown. The trend appears to be (but see θ = 30 for α = 0 . 9) that ω spin prec . increases as θ increases, with very little variation with the z component of the spin or even total spin α . Note the at late times (larger Ω orbit ) ω spin prec . approaches Ω orbit , indicating that the spin processes at nearly the same rate as the orbit during the final orbit and plunge. \n<!-- image --> \norientation φ (the angle between the in-plane spin vector and the infall direction near merger), \nV = V 1 cos( φ -φ 1 ) + V 3 cos(3 φ -3 φ 3 ) , V 1 = V 1 , 1 α + V 1 , 3 α 3 , V 3 = V 3 , 1 α + V 3 , 3 α 3 , (5) \nwhere V 1 , 3 = ( -15 . 46 ± 2 . 66) km s -1 , V 3 , 1 = (15 . 65 ± 3 . 01) km s -1 , and V 3 , 3 = (105 . 90 ± 4 . 50) km s -1 , while V 1 , 1 = (3681 . 77 ± 2 . 66) km s -1 . From that study, it was clear that in the 'superkick' configuration, the dominant contribution, even at large α , is linear in α and proportional to cos( φ ). Note that because of the small contributions of V 3 and V 1 , 3 , we neglect these terms in the statistical studies below (where we take a uniform distribution in φ -φ 1 ). \nIn Ref. [53] an alternative approach to fitting recoil and remnant mass and spin of a merged BHB was developed. It is based on a Taylor expansion in terms of the binary parameters and exploits all the symmetries of the problem (note that our approach also incorporates these symmetries because it is based on PN formulas for the instantaneous recoil). Using one of our previous set of six 'superkick' simulations in [16], the authors in [53] fitted the recoil velocities to terms in cos( φ ) and cos(3 φ ) to extract the cubic (in spin) dependence of the recoil from a single set of simulations with constant total spin. We also modeled that cubic dependence with more simulations including a range of spin magnitude in [40]. Ref. [53] also \nfitted the recoil velocity as a function of the angle θ that the spins make with the orbital angular momentum using the data from a series of runs reported in Ref. [54]. However, those results are inconclusive since they could not model the recoil as a function of φ or model the precession of the orbital plane using the data in Ref. [54]. \nIn order to analyze the results of the present simulations, we use the techniques developed in [46]. Briefly, we rotate each configuration such that the trajectories near merger overlap. We then calculate the spins in this rotated frame. The angle ϕ is then defined to be the angle between the AsTHxxPHyyy spin of BH1 (the BH originally located on the positive x axis) and the spin of BH1 in the corresponding AsTHxxPH0 configuration. Note that, for a given family of fixed spin and spin inclination angle θ , the angle ϕ and φ differ by a constant, which can be absorbed in the fitting constants φ 1 and φ 3 . We then fit the recoil to the form \nV rec = V 1 cos( ϕ -φ 1 ) + V 3 cos(3 ϕ -3 φ 3 ) (6) \nfor each set of configurations with the same spin and θ , and then fit the dominant V 1 coefficient as a function of ϕ . Results from these fits are given in Table IV and Figs. 4-6. Note that A9TH60 runs show the largest discrepancies in the fit, consistent with the larger errors in these simulations due to a coarser global resolution (see Sec. III above). \nBased on the 'superkick' formula (5), we expected that the recoil would have the form \nV 1 = ( V 1 , 1 + V A α cos θ + V B α 2 cos 2 θ + V C α 3 cos 3 θ ) × α sin θ, (7) \nwhere V 1 is the component of the recoil proportional to cos φ , V 1 , 1 arises from the 'superkick' formula, and the remaining terms are proportional to linear, quadratic, and higher orders in S z /m 2 = α cos θ (the spin component in the direction of the orbital angular momentum). Here, we do not consider terms higher-order in the in-plane component of glyph[vector] ∆ ∝ glyph[vector] α 2 -qglyph[vector]α 1 denoted by ∆ ⊥ (∆ ⊥ ∝ α sin θ here), where q is the mass ratio, because our previous studies showed that these terms were small at θ = 90 · . A fit to this ansatz (7) showed that the truncated series appears to converge very slowly with coefficients V 1 , 1 = (3677 . 76 ± 15 . 17) km s -1 , V A = (2481 . 21 ± 67 . 09) km s -1 , V B = (1792 . 45 ± 92 . 98) km s -1 , V C = (1506 . 52 ± 286 . 61) km s -1 that have relatively large uncertainties. In addition, we propose the modification \nV 1 = ( 1 + Eα cos θ 1 + Fα cos θ ) Dα sin θ (8) \nwhich can be thought of as a resummation of Eq. (7) with an additional term Eα cos θ , and fit to D , E , F (where we used the prediction of [40] to model the V 1 for θ = 90 · ) and find D = (3684 . 73 ± 5 . 67) km s -1 , E = 0 . 0705 ± 0 . 0127, and F = -0 . 6238 ± 0 . 0098. Note that E is approximately 1 / 10 of F , indicating that coefficients in \nTABLE IV: Fits of recoil velocities as a function of ϕ for each family of configurations with fixed α and θ to the form Eq. (5). \n| CONF | V 1 (km s - 1 ) | V 3 (km s - 1 ) | φ 1 | φ 3 |\n|---------------|-------------------|-------------------|------------------|------------|\n| A7TH22.5 1764 | ± 1 | 4 . 6 ± 0 . 1 | 58 . 36 ± 0 . 01 | 280 ± 1 |\n| A7TH45 | 2766 ± 1 | 41 . 6 ± 0 . 9 | 203 . 55 ± 0 . | 02 83 ± 2 |\n| A7TH60 | 2972 ± 3 | 54 ± 3 | 342 . 25 ± 0 . | 07 141 ± 4 |\n| A7TH120 | 1806 ± 1 | 27 . 4 ± 0 . 1 | 191 . 78 ± 0 . | 01 62 ± 1 |\n| A7TH135 | 1352 ± 1 | 16 . 6 ± 0 . 7 | 145 . 21 ± 0 . | 04 277 ± 4 |\n| A9TH15 | 2038 ± 2 | 27 ± 2 | 178 . 55 ± 0 . | 09 291 ± 7 |\n| A9TH30 | 3408 ± 2 | 42 ± 2 | 284 . 91 ± 0 . | 03 285 ± 3 |\n| A9TH60 | 4171 ± 14 | 87 ± 10 | 158 . 3 ± 0 . 7 | 17 ± 17 | \nFIG. 4: A fit of V recoil versus ϕ for the A7TH22.5PHyyy (Left) and A7TH60PHyyy (Right) configurations. \n<!-- image --> \nthis series get progressively smaller faster than in Eq. (7). Interestingly, a fit to just \nV 1 = ( 1 1 + Fα cos θ ) Dα sin θ \nfailed to produce sensible results (a badly conditioned matrix was encountered). The two formulas (7) and (8) give very similar results for a broad range of α (see Tables V, VI, and VII). We then use Eq. (8) to predict the recoil for higher spin α = 0 . 9 and test this formula for three angles θ = 90 · , θ = 60 · , and θ = 15 · , with very good agreement (see Fig. 6). In actuality, both Eq. (8) and Eq. (7) provide accurate predictions for our measured recoils at α = 0 . 9. We show the errors in the predictions for the α = 0 . 9 configurations in Table VI. \nWe also tried fits to \nV 1 = ( 1 + Eα cos θ + Gα 2 cos 2 θ 1 + Fα cos θ ) Dα sin θ, (9) \nbut found that the coefficients were not well determined. In this case, we found D = 3686 . 34 ± 8 . 87, E = 0 . 055 ± 0 . 056, F = -0 . 638 ± 0 . 051, G = -0 . 014 ± 0 . 050. The errors in both E and G in this case are larger than the values of the coefficients themselves. We therefore do not use Eq. (9) in the analysis below. Similarly large uncertainties are encountered if the quadratic F correction is in the denominator of Eq. (9) rather than the numerator.", 'V. BLACK HOLE SPIN EVOLUTION IN GAS RICH GALAXY MERGERS': "Full numerical simulations of BHBs typically start when the BHs are at distances of the order of 10 M from \nFIG. 5: A fit of V recoil versus ϕ for the A9TH15PHyyy (Left) and A9TH60PHyyy (Right) configurations. \n<!-- image --> \nFIG. 6: fit of the recoil ( V 1 ) to the form Eq. (8) for the α = 1 / √ 2 configurations, and predictions (based on this fitting) for the α = 0 . 91 recoils. Note how well the α = 0 . 91 curve matches the four measured values. For reference, curves corresponding to the original empirical formula prediction (which only had terms linear in ∆) for α = 1 / √ 2 and the new formula for α = 1 are also included. Note the skew in the velocity profile compared to the linear predictions. \n<!-- image --> \neach other. There are good reasons for this. Numerical runs are still extremely expensive, they need to run on hundreds of nodes for weeks at a time to obtain accurate computations of the gravitational radiation. Those few runs allow us to infer generic behaviors of the remnants, e.g. the modeling of the recoils by the phenomenological Eq. (3). While these initial separations are extremely close by astrophysical standards, most of the nonlinear general relativistic effects take place at these and closer separations. However, if one wants to study statistical distributions of recoils by astrophysical seeds one would like to input realistic spin and mass ratio distributions for the merging BHB. In a first study of such systems we assume an isotropic distribution of the spin direction of the BHs [55]. This could represent 'dry' binary mergers. We point out below the relevance of pre-merger accretion to partially align the BH spins with the orbital angular momentum. We then perform a preliminary study of an extended recoil formula (12) to see the differences between the predictions of the recoil formula based on only \nTABLE V: Predictions for α = 1 / √ 2 simulations based on Eq. (8) (denoted by pade) and Eq. (7) (denoted by FS), as well as the measured V 1 . Note that the θ = 90 · measured value comes from Ref. [40]. Velocities are in units of km s -1 .TABLE VI: Predictions for α = 0 . 9 simulations based on Eq. (8) (denoted by pade) and Eq. (7) (denoted by FS), as well as the measured V 1 . Note that the θ = 90 · measured value comes from Ref. [40]. When applying Eqs. (8) and (7) we use α merger . The relative error quoted here is the relative error in the prediction based on Eq. (8). Velocities are in units of km s -1 . \n| CONF | V 1 , pade | V 1 , FS | V 1 , meas |\n|--------|--------------|------------|--------------|\n| TH22.5 | 1760.46 | 1754.47 | 1764 |\n| TH45 | 2771.92 | 2777.4 | 2766 |\n| TH60 | 2967.09 | 2967.33 | 2972 |\n| TH90* | 2605.5 | 2600.57 | 2603.4 |\n| TH120 | 1802.6 | 1811.4 | 1806 |\n| TH135 | 1354.82 | 1348.48 | 1352 | \n| CONF | V 1 , pade | V 1 , FS | V 1 , meas | Rel. Error |\n|--------|--------------|------------|--------------|--------------|\n| TH15 | 1990 | 1905.23 | 2038 | -2.4% |\n| TH30 | 3367 | 3302.23 | 3408 | -1.2% |\n| TH60 | 4251 | 4258.62 | 4171 | 1.9% |\n| TH90* | 3353.1 | 3346.76 | 3350.41 | 0.1% | \nlinear terms in the spins and the new updated formula [17]. \nThe spins of massive BHs binding in binaries as a result of galaxy mergers can be deeply affected by gas accretion during the last stages of their orbital decay (for separations ∼ < 100 pc). This is due to the gas overdensities that the galaxy mergers are expected to convey into the nuclei of the galaxy remnants. Such dense gas structures have a disk like morphology ('circumnuclear disks'), reminiscent of the initial net angular momentum of the inflowing material, as observed in high resolution simulations [e.g. 56, 57] as well as in real merging systems [e.g. 58-60]. \nBogdanovic et al [24] proposed a physical process that could align the BH spins with the angular momentum of the nuclear disk in which the binary orbit is embedded, thus leading to slow recoils for the BH remnant. The evolution of the spin directions is due to the torques exerted by the gas accreting onto the BHs. Since this process happens on a timescale shorter than the orbital decay of the BHs in the remnant nucleus, the spins tend to align before the two BHs bind in a binary. As a consequence, the evolution of the spin of each BH in this earlier phase can be studied independently, neglecting the presence of the second BH. \nThe evolution of spin direction and magnitude is governed on small scales (milli-pc, much smaller than the circumnuclear disk within which BHs, and their accretion disks, are embedded). As shown by [61], if the orbital angular momentum of an accretion disk around the BH is misaligned with respect to the BH spin, the coupled \nTABLE VII: Angle θ that maximizes the recoil and the maximum recoil as a function of α for Eq. (8) and Eq. (7). Velocities are in units of km s -1 while angles are measured in degrees. \n| α | θ pade | V pade (km s - 1 ) | θ FS | V FS (km s - 1 ) |\n|-------|----------|----------------------|--------|--------------------|\n| 0.1 | 86 · | 369 | 86 · | 369 |\n| 0.5 | 70 · | 1961 | 70 · | 1955 |\n| 0.707 | 62 · | 2969 | 61 · | 2968 |\n| 0.91 | 54 · | 4225 | 54 · | 4232 |\n| 1 | 50 · | 4926 | 51 · | 4915 | \naction of viscosity and relativistic Lense-Thirring precession ('inertial frame dragging') causes important changes in the structure of an accretion disk, warping the disk. The equilibrium profile of the warped disk can be computed by solving the equation: \n1 R ∂ ∂R ( R glyph[vector] Lv R ) = 1 R ∂ ∂R ( ν 1 Σ R 3 d Ω dR ˆ l ) + + 1 R ∂ ∂R ( 1 2 ν 2 RL ∂ ˆ l ∂R ) + 2 G c 2 glyph[vector] S BH × glyph[vector] L R 3 (10) \n[see 62], where R is the distance from the BH, v R is the radial drift velocity, Σ is the surface density, Ω is the Keplerian angular velocity of the gas, and ν 1 ( ν 2 ) is the radial (vertical) viscosity. glyph[vector] L is the local angular momentum surface density of the disk, defined by its modulus L and the unit vector ˆ l that defines its direction. The disk profile that is a solution of Eq. (10) is composed of three regions, whose relative importance depends on the values of the specific disk parameters. In the outermost region the angular momentum of the gas is unperturbed by any relativistic effect, and therefore the direction of the disk's angular momentum is independent of the BH spin. In the inner region, the fluid is forced to rotate in the equatorial plane of the rotating BH, on either prograde or retrograde orbits. Therefore in this inner region the disk is either completely aligned or completely antialigned with respect to the BH spin. Finally, between the inner and outer regions, characterized by different directions of their angular momenta, there exists a transition region, centered at ∼ 100 -1000 gravitational radii, where the disk is warped connecting the inner and outer parts of the disk, misaligned one with respect to the other. \nThe spin of a BH embedded in a warped disk evolves under the influence of the disk itself. The BH spin evolution is described by the equation [62]: \nd glyph[vector] S BH dt = ˙ M Λ( R ISO ) ˆ l ( R ISO ) + 4 πG c 2 ∫ disk glyph[vector] L × glyph[vector] S BH R 2 dR. (11) \nThe first term in Eq. (11) accounts for the angular momentum deposited onto the BH by the accreted gas at the innermost stable orbit (ISO), where Λ( R ISO ) denotes the angular momentum per unit mass [Eq. 12.7.18 in 63] \nevaluated at R ISO and ˆ l ( R ISO ) the local disk angular momentum direction, which is parallel to glyph[vector] S BH as discussed above. The second term describes the interaction of the BH spin with the warped disk. It is responsible for the evolution of the BH spin direction, and tends to align the direction of the BH spin with the angular momentum of the outer regions of the accretion disk. \nThe efficiency of the alignment depends on the dynamics of the inflowing material that fuels the small scale accretion disks. It is particularly relevant to determine whether the accretion flow maintains a nearly constant direction of the angular momentum over the growth episode (i.e. the accretion is 'coherent'), or not. Only a substantial amount of gas (1 ∼ 10% of the BH mass) accreting from the same plane (for both the BHs) can significantly align the two spins. In order to constrain the degree of coherency of the gas accreting onto the BHs, and to predict the spin configurations in BH binaries, Dotti et al [25] performed numerical simulations of BH pairs in large scale nuclear disks with the N-Body/SPH code GADGET [64], upgraded to include the accretion physics. Here we give a short summary of the initial conditions for the different runs. For a more detailed discussion, we defer the reader to Refs. [25, 65]. \nThe two BHs are placed in the plane of a massive circumnuclear gaseous disk, embedded in a larger stellar spheroid. The disk is modeled with ≈ 2 × 10 6 gas particles, has a total mass M Disk = 10 8 M glyph[circledot] , and follows a Mestel surface density profile Σ( R ) ∝ R -1 , where R is the radial distance projected into the disk plane. Dotti et al. truncated the disk at an outer radius of 100 pc. The massive disk is rotationally supported in R and has a vertical thickness of 8 pc. Gas is evolved assuming a polytropic equation of state with index γ = 5 / 3 or γ = 7 / 5. In the former case, the disk is termed 'hot' as the temperature is proportional to a higher power of density than in the latter class of models ('cold' cases). The cold case has been shown to provide a good approximation to a gas of solar metallicity heated by a starburst [66, 67]. The hot case instead corresponds to an adiabatic monatomic gas, as if radiative cooling were completely suppressed during the merger, for example as a result of radiative heating after gas accretion onto the BHs [56]. The spheroidal component (bulge) is modeled with 10 5 collisionless particles, initially distributed as a Plummer sphere with a total mass M Bulge (= 6 . 98 × M Disk ). The mass of the bulge within 100 pc is five times the mass of the disk, as suggested by [58]. The BHs are equal in mass ( m BH = 4 × 10 6 M glyph[circledot] ), and their initial separation is 50 pc. A BH is placed at rest at the center of the circumnuclear disk, while the other is moving on an initially eccentric ( e 0 glyph[similarequal] 0 . 7) counter-rotating (retrograde BH) or corotating (prograde BH) orbit with respect to the circumnuclear disk. Given the large masses of the disk and the bulge, the dynamics of the moving BH (secondary) is unaffected by the presence of the primary until the BHs form a gravitationally bound system. Furthermore, the gravitational interaction between the orbiting BH and \nthe rotating gas forces the BH to corotate on almost circular orbits with the disk [68, 69], before the BHs bind in a binary. As a consequence, the initial orbital configurations of the BHs do not influence the final degree of alignment, that, as will be discussed in the following, depends only on the thermodynamical state of the disk. \nTo follow the evolution of the BH spins, it is necessary to track the dynamics of the gas accreting onto the two central objects. In the simulations discussed in [25] a gas particle can be accreted by a BH if the following two criteria are fulfilled: \n- · the sum of the kinetic and internal energy of the gas particle is lower than b -times the modulus of its gravitational energy (all the energies are computed with respect to each BH);\n- · the total mass accreted per unit time onto the BH every timestep is lower than the accretion rate corresponding to the Eddington luminosity computed assuming a radiative efficiency of 10%. \nThe parameter b is a constant that defines the degree to which a particle is bound to the BH in order to be accreted. Dotti et al [25] set b = 0 . 3. Note that due to the nature of the above criteria, the gas particles can accrete onto the BHs only if the time-varying Bondi-HoyleLyttleton radius is resolved in the simulations. Such a small radius can be resolved only by performing very high resolution simulations. The gravitational softening parameter of the BHs is 0.1 pc. The gravitational softening of the gas particles is set to the same value, in order to prevent numerical errors. This is also the spatial resolution of the hydrodynamical force in the highest density regions[84]. \nThe simulations discussed in Dotti et al [25] cannot follow the dynamics of the accreting gas on unresolved scales. Dotti and collaborators assume that, below the spatial resolution of the runs, gas settles on standard geometrically thin/optically thick α -disks [70]. The properties of those two unresolved disks (one surrounding each of the two BHs of the binary) embedded in the larger scale circumnuclear disk, are determined by the properties of the accreting material. Each gas particle accreted by the BH carries with it mass and angular momentum. These are data Dotti et al. used to model the unresolved accretion discs around the two BHs, becoming the outer boundary conditions for Eq. (10) and (11). Specifically, Dotti and collaborators define ˆ l edge as the unit vector defining the direction of the angular momentum of the accretion disk in its outermost region, i.e. where it is unaffected by any general relativistic effect. In a warped α -disk the two viscosities (radial, ν 1 and vertical, ν 2 ) can be described in terms of two different dimensionless viscosity parameters, α 1 and α 2 , through the relations ν 1 , 2 = α 1 , 2 Hc s , where H is the disk vertical scale height and c s is the sound speed of the gas in the accretion disk. Additionally, α 2 = f 2 / (2 α 1 ), with α 1 = 0 . 1 and f 2 = 0 . 6 [71]. Further details on the procedure used to evolve the BH spins can be found in [62]. \nThe resulting distributions of spin magnitudes and in- \nFIG. 7: The probability that the dimensionless spin of a BH in a merging binary has a given magnitude α for BHs in cold disks (squares) and in hot disks (circles). The fits to Beta functions are reasonably good. \n<!-- image --> \nclinations with respect to the angular momentum of the newly formed binary are shown in Figs. 7 and 8, respectively. Red circles refer to BHs embedded in hot disks, blue squares to BHs in cold disks. In both the cases, the well defined angular momentum of the large scale nuclear disk results in coherent accretion flows onto the two BHs, and, as a consequence, in high spins strongly aligned with the angular momentum of the BH binary. Note that, in absence of any alignment, the distributions in Fig. 8 should be ∝ sin( θ ) in the whole interval [0 , π ]. As discussed in [25], a 'hotter' disk, with a stiffer equation of state, is more pressure supported in the center, and, as a consequence, the degree of alignment is lower. Because of this additional support, the accretion rates onto the BHs in the hot runs are lower, corresponding to spin distributions less skewed towards high spin values. \nIn order to perform the statistical studies below, we fit the spin magnitude and inclination angle distributions above to Beta distributions, which have the form P ( x ) ∝ (1 -x ) ( b -1) x ( a -1) . Fits to the dimensionless spin magnitudes for BHs in hot and cold gaseous environments give a = 3 . 212 ± 0 . 258, b = 1 . 563 ± 0 . 093, and a = 5 . 935 ± 0 . 642, b = 1 . 856 ± 0 . 146, respectively. A comparison of these fits with the measured probabilities of a BH having a given spin magnitude is given in Fig. 7. Fits to the inclination angle for the hot and cold cases angular distributions give a = 2 . 018 ± 0 . 181, b = 5 . 244 ± 0 . 604, and a = 2 . 544 ± 0 . 198, b = 19 . 527 ± 2 . 075, respectively. Note that these distributions P ( θ ) are for θ in radians. The Beta distribution is not defined for θ > 1, but the data are consistent with near zero probabilities for angles larger than 1 radian. A comparison of these fits with the measured probabilities of a BH having a given spin direction is given in Fig. 8. \nFIG. 8: The probability that the spin of a BH in a merging binary is at an inclination angle θ with respect to the orbital angular momentum for cold (squares) and hot (circles) circumnuclear disks. The fits to Beta functions are reasonably good, but miss the small tail in the cold distribution. \n<!-- image -->", 'VI. EXTENDING THE HANGUP-KICK FORMULA': "While accretion will tend to align the spins of the two BHs in a BHB with the orbital angular momentum, it will not align the in-plane components of the spins. Additionally, the expected distribution of mass ratios [72-74] indicates that equal-mass mergers are rare. We therefore need a way to extend Eq. (7) to generic BHBs. \nUsing the same post-Newtonian analysis [51] as in [50], we can extend formula (7) to less symmetric configurations by replacing α sin θ by | glyph[vector] α ⊥ 2 -qα ⊥ 1 | / (1+ q ) and α cos θ by 2[ α z 2 + q 2 α z 1 ] / (1 + q ) 2 . Importantly, we are assuming that terms proportional to | glyph[vector] α ⊥ 2 -qglyph[vector]α ⊥ 1 ] n (for n > 1) are negligible. If this is not the case, then our expansion, which can be thought of as a Fourier sine series, would still converge, but our extension would contain errors that may not be small. For example, if a term like ( α ⊥ ) 2 α z were present, this would contribute to all even components of the Fourier sine series and when extending the series, we would have to take this into account. This would change the behavior of kick even in more symmetric configurations. This degeneracy in the interpretation of the sine series can be broken by examining configurations with constant α z (while varying α ⊥ ) and constant α ⊥ (while varying α z ). These, and other configurations, will be the subject of an upcoming paper. Our justification for not including these terms is that the higher-order α ⊥ terms are small in the 'superkick' configuration. Furthermore, the accuracy with which formula (7) predicts the results of our α = 0 . 91 simulations supports the conclusion that these terms remain small. This can be verified by confirming that formula (7) is accurate for all θ and α (a subject of our ongoing analysis that will be reported in a forthcoming paper). We emphasize that the proposed extension is an ansatz, that while reasonable as a starting point for the modeling, needs to be thoroughly \ntested and refined. \nOur new ansatz for the recoil velocity modifies Eq. (3) by changing the 'superkick' v ‖ term. The ansatz has the form (after dropping terms that previous studies indicated were small [50]): \nglyph[vector] V recoil ( q, glyph[vector]α ) = v m ˆ e 1 + v ⊥ (cos ξ ˆ e 1 +sin ξ ˆ e 2 ) + v ‖ ˆ n ‖ , v m = A m η 2 (1 -q ) (1 + q ) [1 + B m η ] , v ⊥ = H η 2 (1 + q ) [ ( α ‖ 2 -qα ‖ 1 ) ] , v ‖ = 16 η 2 / (1 + q ) [ V 1 , 1 + V A ˜ S z + V B ˜ S 2 z + V C ˜ S 3 z ] × ∣ ∣ glyph[vector] α ⊥ 2 -qglyph[vector]α ⊥ 1 ∣ ∣ cos( φ ∆ -φ 1 ) , (12) \nwhere glyph[vector] ˜ S = 2( glyph[vector] α 2 + q 2 glyph[vector] α 1 ) / (1 + q ) 2 , and the coefficients we use in the statistical studies below are H = 6 . 9 × 10 3 [48], A m = 1 . 2 × 10 4 , B m = -0 . 93 [12], and the remaining coefficients are obtained from Eq. (7) above.", 'VII. STATISTICAL STUDIES': "Using Eq. (12) and the above fitted spin magnitude ( α ), and direction distributions ( θ ), the mass ratio distribution suggested in [72-74], P ( q ) ∝ q -0 . 3 (1 -q ), and assuming that the two BHs can have arbitrary orientations for the in-plane component of the spin (i.e. uniform probability in the range 0 ≤ φ ≤ 2 π ), we obtain probabilities for the recoil velocity magnitude and direction. To perform our statistical studies, we choose 10 8 configurations (2 × 10 8 configurations in total) randomly chosen based on the above probability distributions and examine the predicted recoil magnitude and direction. Our results are summarized in Table VIII and Figs. 9 and 10. Although we include the pure unequal mass recoil [ v m in Eq. (12)], we note that the modeling of the angle ξ as a function of the binary's parameter is incomplete. However v m has normally a nonleading effect. We therefore chose a constant ξ = 145 · , as suggested in our previous study [48]. \nIn Fig. 11 we show the probabilities that the recoil has a given inclination angle (angle with respect to the orbital angular momentum) for hot and cold disks. Because of the θ → 180 · -θ symmetry, we map all recoil angles to the interval 0 · ≤ θ ≤ 90 · . Here P ( θ ) is the probability integrating over all possible recoil magnitudes, i.e. P ( θ ) = ∫ ∞ 0 P ( θ, v ) dv . The distribution is normalized such that ∫ 90 0 P ( θ ) dθ = 1. The angular distribution is broader for accretion in cold disks, since that tends to suppress the 'hangup-kick' and 'superkick', while the distribution is more sharply peaked near θ = 0 for hot disks. \nThis strong angular dependence of the recoil has particular relevance for the studies of the observational con- \nTABLE VIII: Recoil velocity probabilities (in percent) for BHs in hot and cold disks aligned binaries and the probabilities for the recoil along the line-of-sight having the given magnitude range (denoted by Obs.). For the hot case, there is a nontrivial probability of observing a recoil larger than 2000 km s -1 , but for cold disks, such recoils are suppressed. Velocities are in units of km s -1 . \n| Vel. (km s - | ) (Hot) | Obs. (Hot) | (Cold) | Obs. (Cold) |\n|----------------|--------------------|---------------------|------------|---------------------|\n| 0-100 | | 34.2593 % 60.1847 % | | 41.4482 % 71.2967 % |\n| 100-200 | | 21.1364 % 16.9736 % | | 28.3502 % 16.8471 % |\n| 200-300 | 11.6901 % 8.1110 % | | 12.503 % | 6.1508 % |\n| 300-400 | 7.8400 % | 4.8108 % | 7.0967 % | 2.8281 % |\n| 400-500 | 5.7590 % | 3.0913 % | 4.2490 % | 1.3973 % |\n| 500-1000 | 14.0283 % 5.6593 % | | 5.9309 % | 1.4258 % |\n| 1000-1500 | 4.0183 % | 0.9809 % | 0.4030 % | 0.0526 % |\n| 1500-2000 | 1.0309 % | 0.1638 % | 0.0185 % | 0.0015 % |\n| 2000-2500 | 0.2047 % | 0.0223 % | 0.0005 % | 2 × 10 - 5 % |\n| 2500-3000 | 0.0296 % | 0.0023 % | 1 × 10 - 5 | % 0.% |\n| 3000-3500 | 0.0032 % | 0.0002 % | 0. % | 0.% |\n| 3500-4000 | 0.0002 % | 4 . × 10 - 6 | % 0.% % | 0.% | \nFIG. 9: Probability distribution P ( v ) of the recoil magnitude for BHBs alignment configurations in hot (top curve, red circles) and cold disks BHBs (lower curve, blue squares). The velocity is in units of km s -1 . \n<!-- image --> \nsequences of merging and kicked BHs surrounded by preexisting gas disks [19, 75-80]. The merger of a BHB resulting in the remnant BH moving across the matter that surrounded the original BHB would greatly affect the dynamics of the gas and its thermodynamic state. This translates into distinctive electromagnetic signatures that could reveal the presence of recoiling BHs. The effect is very pronounced when the BH recoils in the orbital plane with a large magnitude. The strong preference for large recoils along the axis of the disk over those recoiling along the disk itself can strongly suppress the magnitude of such signatures. \nIn Fig. 12 we show the recoil velocity distribution for hot and cold disks integrated over 15 · intervals of the inclination angle θ (in Figs. 9 and 10 above, the integration is over 0 · ≤ θ ≤ 180 · ). While in Fig. 13, we show the integrated probability, Π( v ) of a recoil having velocity \nFIG. 10: Probability distribution P ( v ) of the recoil magnitude along the line of sight for BHBs in hot (top curve, red circles) and cold disks (lower curve, blue squares). The velocity is in units of km s -1 . \n<!-- image --> \nFIG. 11: The probability distribution of the inclination angle θ of the recoil (measured with respect to the axis of the angular momentum) for hot (narrower distribution, red circles) and cold environments (blue squares). Angles are measured in degrees. Note that P (180 · -θ ) = P ( θ ). These distributions were created by mapping θ → 180 · -θ for θ > 90 · . \n<!-- image --> \nv or larger [Π( v ) = ∫ ∞ v P ( ν ) dν ]. If we consider recoils within 15 · of the orbital axis, we see that velocities up to 900 km s -1 are likely (i.e. about 1% probability) for cold disks and up to 1600 km s -1 for hot disks (see also Fig. 13). If we look at angles between 15 · and 30 · we see the recoils are limited to less than 800 km s -1 (even for hot disks, a recoil of 600 km s -1 has < 0 . 1% probability). At larger θ angles, the maximum recoil drops below 270 km s -1 . \nSince the most striking effects are likely due to BHs recoiling with large magnitudes through the plane of the disk, it is interesting to examine the probabilities of such events occurring. Even with integrated probabilities of < ∼ 10 -4 , such events may be observed in large surveys of galaxies, e.g. the Sloan Digital Sky Survey DR7 contains ∼ 9 × 10 5 galaxies if the observable ef- \nTABLE IX: Recoil velocity direction ( θ ) probabilities for the hot and cold cases. Recoils with θ > 90 · have been remapped using the symmetry θ → 180 · -θ . \n| Range | (Hot) | (Cold) |\n|-------------|---------|----------|\n| 0 · - 30 · | 61.6% | 47.0% |\n| 30 · - 60 · | 22.6% | 31.4% |\n| 60 · - 90 · | 15.7% | 21.6% | \nFIG. 12: Recoil velocity distributions for hot (red solid lines) and cold (blue dashed lines) disks for recoils in angular intervals 15 degrees wide (Eq. (12) predicts equal probabilities for recoils at an angle θ and 180 · -θ ). Note how rapidly the maximum recoil decreases as a function of θ . Recoils as large as 1000 km s -1 must have θ < 15 · and in-plane recoils are less than 270 km s -1 (but see comment about unequal-mass recoils in the text). \n<!-- image --> \nlast long enough. We can see in Figs. 12 and 13 that while in hot disks we can observe recoils of nearly 3000 km s -1 , cold disks limit the maximum observable recoil to 2000 km s -1 . On the other end, if we require the recoiling hole to be within 30 · of the orbital plane (i.e. θ > 60 · ) we cannot observe recoils larger than 250 km s -1 , while at intermediate angles velocities seem limited to 400 km s -1 for both cold and hot disks. \nIn Fig. 14, we show the angular distribution for recoils in given velocity ranges. Again, because of the θ → 180 · -θ symmetry, we map all recoil angles to the interval 0 ≤ θ ≤ 90 · . For convenience, we plot the probabilities in degrees rather than radians. For these plots, we used 10 8 randomly chosen binaries consistent with the above distributions for spin magnitude, spin direction, and mass ratio. The maximum angle the recoil can make with the orbital angular momentum axis is very \nFIG. 13: The integrated probability Π( v ) of a recoil having velocity v or larger [Π( v ) = ∫ ∞ v P ( ν ) dν ] for hot (top) and cold (bottom) accretion disks for recoils in the ranges 0 · < θ < 15 · , 15 · < θ < 30 · , · · · , 75 · < θ < 90 · . In both cases, for recoils larger than 200 km s -1 , the recoil probabilities are smaller for larger values of θ . In the cold case, low-velocity ( < 200 km s -1 ) recoils with an angle 0 · < θ < 15 · are less probable than recoils with angle 15 · < θ < 30 · . \n<!-- image --> \nrestrictive for large velocities, as is shown in Table X.", 'VIII. DISCUSSION': "We studied in detail a family of BHB configurations with full numerical relativity that allowed us to single out the 'hangup-kick'. This effect is expected to be relevant in generic BBH mergers since it arises from a combination of generic properties of the orbital dynamics of spinning BHBs, namely the 'orbital-hangup effect' [81] and the 'superkick effect' [16]. We present evidence that this effect increases the maximum recoil velocity achievable from the merger of two orbiting BHs by up to 1200 km s -1 with respect to previous estimates, approaching nearly 5000 km s -1 . Even more importantly this maximum recoil is reached for spins at angles near 50 · with respect to the orbital momentum of the binary system. We have also shown evidence that accretion in the premerger stage of the binary tends to align spins with the angular momentum of the system, leading to \nTABLE X: Maximum recoil angle θ (angle with respect to the orbital angular momentum axis) for given recoil velocity ranges. Note here that θ max < δ means that θ must smaller than δ or larger than 180 · -δ . \n| Range | θ max (Hot) | θ max (Cold) |\n|----------------------|---------------|----------------|\n| 0 - 100 km s - 1 | 90 · | 90 · |\n| 100 - 200 km s - 1 | 90 · | 90 · |\n| 200 - 300 km s - 1 | < 80 · | < 70 · |\n| 300 - 400 km s - 1 | < 45 · | < 40 · |\n| 400 - 500 km s - 1 | < 33 · | < 30 · |\n| 500 - 600 km s - 1 | < 25 · | < 21 · |\n| 500 - 1000 km s - 1 | < 25 · | < 21 · |\n| 1000 - 1500 km s - 1 | < 11 · | < 8 · |\n| 1500 - 2000 km s - 1 | < 7 · | < 5 · |\n| 2000 - 2500 km s - 1 | < 5 · | < 4 · |\n| 2500 - 3000 km s - 1 | < 4 · | < 2 · |\n| 3000 - 3500 km s - 1 | < 3 · | *** | \nFIG. 14: Recoil angle probabilities P ( θ ) for hot (top) and cold (bottom) disks aligned binaries. The plots on the left show angular probabilities for velocities in the ranges 0 -100 km s -1 , 100 -200 km s -1 , · · · , 500 -600 km s -1 . The plots on the right show P ( θ ) for velocity ranges of 500 -1000 km s -1 , · · · , 3000 -3500 km s -1 . Probabilities for θ and 180 · -θ are equal. In the plots, closed circles correspond to the smallest range, followed by squares, diamonds, triangles (vertex up), triangles (vertex down), and open circles. The circles on the axis are an artifact of the visualization tool. \n<!-- image --> \ndistributions that favor the 'hangup-kick' configurations with respect to the purely in-plane 'superkick' ones. Due to depletion of nearby matter, the merger itself occurs in a 'dry' regime where accretion no longer affects the BH spins. \nIn an attempt to estimate the probability of observing such large recoils in real astronomical systems, we assumed accretion driven distributions for the spin magnitudes and directions, based on the two extreme scenarios of cold and hot disks, and assumed a mass ratio distribution based on independent estimates, to obtain non negligible probabilities of observing recoils of several thousand km s -1 . In particular, the results in Table VIII indicate that surveys exploring peculiar differential radial \nvelocities should observe at least one case of a 'line-ofsight' velocity above 2000 km s -1 out of four thousand merged galaxies (assuming 'hot' disks). The probability that a remnant BH receives a recoil exceeding the escape velocity (in any direction) of giant galaxies (2000 km s -1 ) is ten times larger. Probabilities of recoils exceeding the escape velocity quickly rise to 5% for galaxies with escape velocities of 1000 km s -1 and nearly 20% for galaxies with escape velocities of 500 km s -1 . These numbers indicate that recoil velocities and modeling the accretion of the supermassive BHs in centers of galaxies should be important ingredients in understanding the growth of supermassive BHs and large scale structure formation in the universe. \nOur initial study showed the relevance of recoil and accretion modeling in order to better understand how BHs evolve and grow in the universe. There are several aspects that deserve further study. The recoil formula needs to be further tested and developed in the intermediate mass ratio regime and for fully precessing BHBs. Accretion needs to be modeled at even smaller scales, i.e. at sub-milli-parsecs. In between the accretion regime governed by Newtonian physics and the fully nonlinear regime, a slow adiabatic inspiral occurs. 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2014PhRvD..89h4038C
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Asymptotically locally AdS and flat black holes in the presence of an electric field in the Horndeski scenario
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2014-01-01
| 21 | 0.44 | 158 |
['-', '-', '-', '-', '-', '-', '-']
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Asymptotically locally anti-de Sitter and asymptotically flat black hole solutions are found for a particular case of the Horndeski action. The action contains the Einstein-Hilbert term with a cosmological constant, a real scalar field with a nonminimal kinetic coupling given by the Einstein tensor, the minimal kinetic coupling, and the Maxwell term. There is no scalar potential. The solution has two integration constants related with the mass and the electric charge. The solution is given for all dimensions. A new class of asymptotically locally flat spherically symmetric black holes is found when the minimal kinetic coupling vanishes and the cosmological constant is present. In this case, we get a solution which represents an electric universe. The electric field at infinity is only supported by Λ. When the cosmological constant vanishes, the black hole is asymptotically flat.
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https://arxiv.org/pdf/1401.4479.pdf
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{'Asymptotically locally AdS and flat black holes in the presence of an electric field in the Horndeski scenario': "Adolfo Cisterna ∗ \nInstituto de F'ısica, Pontificia Universidad Cat'olica de Valpara'ıso, Av. Universidad 330, Curauma, Valpara'ıso, Chile.", "Cristi'an Erices †": "Departamento de F'ısica, Universidad de Concepci'on, \nCasilla, 160-C, Concepci'on, Chile. and \nCentro de Estudios Cient'ıficos (CECs), Casilla 1469, Valdivia, Chile. \n(Dated: January 17, 2014)", 'Abstract': 'Asymptotically locally AdS and asymptotically flat black hole solutions are found for a particular case of the Horndeski action. The action contains the Einstein-Hilbert term with a cosmological constant, a real scalar field with a non minimal kinetic coupling given by the Einstein tensor, the minimal kinetic coupling and the Maxwell term. There is no scalar potential. The solution has two integration constants related with the mass and the electric charge. The solution is given for all dimensions. A new class of asymptotically locally flat spherically symmetric black holes is found when the minimal kinetic coupling vanishes and the cosmological constant is present. In this case we get a solution which represents an electric Universe. The electric field at infinity is only supported by Λ. When the cosmological constant vanishes the black hole is asymptotically flat.', 'I. INTRODUCTION': "Scalar fields have a prominent role in high energy physics. At subatomic scales they are an essential part of the quantum description of the electroweak interaction. Indeed, a foundamental scalar field excitation is given by the well known Brout-Englert-Higgs particle, which allows a consistent mathematical description of the the short range of the weak force and lepton masses. \nAt the galactic and cosmological scales, scalar fields arise as the simplest candidate to the explanation of many phenomena. At these scales, the general theory of relativity successfully describes the gravitational interaction. However, despite the great success of the theory, it cannot give a satisfactory description of certain cosmological phenomena, such as the origin of the early Universe and its late time accelerated expansion, as well as the presence of dark matter and dark energy. The properties of these phenomena make the scalar field a suitable candidate able to solve such unknowns, giving rise to a wide variety of theories such as Brans-Dicke theory [1], inflation theories and several cosmological models [2][3][4]. \nMoreover scalar fields appear naturally in theories like Kaluza-Klein compactifications and in theories that intend to give a natural description of gravity at the quantum level, such as string theory, which includes the dilaton scalar field. \nWhile it is true that the study of scalar-tensor theories is not a new topic, currently a great interest resurfaced due to the study of galileon theories and their applications. This have revived the study of the most general scalar-tensor theory which has second order field equations and second order energy-momentum tensor, problem that was solved by Horndeski four decades ago [5]. Horndeski theory along with a big amount of interesting properties also includes galileon gravity [6] and massive gravity [7]. \nIf we focus our attention in a four dimensional curved spacetime, the most general Lagrangian which can be constructed with the above properties is given by \nL = λ 1 δ abdc efhi R ef ab R hi cd + λ 2 δ abc def ∇ a φ ∇ d φR ef cd + λ 3 δ ab cd R cd ab +Θ+ B/epsilon1 abdc R p qab R q pcd (1) \nwhere B is a constant, λ i are arbitrary functions of the scalar φ and Θ is and arbitrary function of the scalar field and its squared gradient, i.e. Θ = Θ( ∇ a φ ∇ a φ, φ ). \nAt this point, we can see that obtaining scalar field Lagrangians, whose kinetic term has non-minimal couplings with the curvature, is possible. In a cosmological context, theories \nwhere this non-minimal derivative coupling is given by the Einstein tensor, provides an expansion of the Universe without a scalar potential [8]. Accelerating behaviors were observed as well in the case of a coupling given by the Ricci tensor [9]. Many models appeared in this context [10][11][12]. \nLet us focus our attention on kinetic terms S which are quadratic in the derivatives of the field in arbitrary dimension n . Requiring second order energy-momentum tensor, as well as field equations for the field, single out S as a linear combination of the following terms \nS ( p ) = E ( p ) µν ∇ µ φ ∇ ν φ , (2) \nwhere E ( p ) µν is p -th order Lovelock tensors 1 [13] \nE ( p ) ν µ = δ να 1 ...α 2 p µβ 1 ...β 2 p R β 1 β 2 α 1 α 2 ...R β 2 p -1 β 2 p α 2 p -1 α 2 p , (3) \nBy setting p = 0, the standard kinetic term is therefore obtained. Since E (1) µν is proportional to the Einstein tensor, the first non-standard term in (2) already includes a nonminimal kinetic coupling of the scalar field and the curvature. \nIn this paper we shall focus on the study of black hole solutions and their properties that emerge from this theory. The action principle is given by \nI [ g µν , φ ] = ∫ √ -gd n x [ κ ( R -2Λ) -1 2 ( αg µν -ηG µν ) ∇ µ φ ∇ ν φ -1 4 F µν F µν ] . (4) \nThe strength of the non minimal kinetic coupling is controlled by η . Here κ := 1 16 πG . The possible values of the dimensionfull parameters α and η will be determined below requiring the positivity of the energy density of the matter field. \nThe first exact black hole solution to this system was found by Rinaldi in [14] for the case of vanishing cosmological constant Λ and without the Maxwell term. In that solution the scalar field becomes imaginary in the domain of outer communications, and the weak energy condition is violated outside of the horizon. \nA great interest has been generated by spacetimes which are asymptotically of constant curvature, particularly asymptotically AdS spacetimes. This interest is largely motivated by the AdS/CFT correspondence [15] which relates the observables in a gauged supergravity theory with those of a conformal field theory in one dimension less. In this way, black \nhole solutions with a negative cosmological constant are important because in principle they could provide the possibility of studying the phase diagram of a CFT theory. As we know, a black hole in an asymptotically flat spacetime is thermodynamically unstable. In order to solve this problem it is possible to put the black hole inside a cavity of finite size. However, there is an alternative method to stabilize such a black hole. It consists in adding a negative cosmological constant. The properties of the AdS spacetime stabilize the black hole simulating a reflecting cavity. \nTherefore, it seems natural to study the case where a negative cosmological constant is present. This was done in [16], where a real scalar field outside the horizon was found and where the positivity of the energy density is given by this reality condition. Recently in reference [17] it has been shown that allowing the scalar to depend on time permits to construct a black hole solution in which the scalar field is analytic at the future or at the past horizon. In a similar context exact solutions were found in [18]. \nThe aim of this work is to continue in this line and generalize the results in reference [16] by adding a Maxwell term given by a spherically symmetric gauge field A = A 0 ( r ) dt . \nA numerical solution in this case was found in [19], where phase transitions to charged black hole with complex anisotropic scalar hair were explored. We also extend the solution to the topological case in arbitrary dimension n ≥ 4 and show that it is also possible to obtain a non-trivial solution when α = 0. In this later case, when the black hole is spherically symmetric, we obtain an asymptotically locally flat black hole with Λ /negationslash = 0 and an asymptotically flat black hole (i.e the metric is Minkowski at spatial infinity 2 ) when Λ = 0. \nThe variation of the action (4) with respect to the metric tensor, the scalar field and the gauge field yields \nG µν +Λ g µν = α 2 κ T (1) µν + η 2 κ T (2) µν + 1 2 κ T em µν , (5) \n∇ µ [( αg µν -ηG µν ) ∇ ν φ ] = 0 , (6) \n∇ µ F µν = 0 , (7) \nrespectively. Here we have defined 3 \nT (1) µν = ∇ µ φ ∇ ν φ -1 2 g µν ∇ λ φ ∇ λ φ , T (2) µν = 1 2 ∇ µ φ ∇ ν φR -2 ∇ λ φ ∇ ( µ φR λ ν ) -∇ λ φ ∇ ρ φR µλνρ -( ∇ µ ∇ λ φ )( ∇ ν ∇ λ φ ) + ( ∇ µ ∇ ν φ ) /square φ + 1 2 G µν ( ∇ φ ) 2 -g µν [ -1 2 ( ∇ λ ∇ ρ φ )( ∇ λ ∇ ρ φ ) + 1 2 ( /square φ ) 2 -∇ λ φ ∇ ρ φR λρ ] , T em µν = F µ λ F νλ -1 4 g µν F 2 . \nWe will consider the family of spacetimes \nds 2 = -F ( r ) dt 2 + G ( r ) dr 2 + r 2 d Σ 2 K , (8) \nwhere d Σ K is the line element of a closed, ( n -2)-dimensional Euclidean space of constant curvature K = 0 , ± 1. The metric (8) corresponds to the most general static spacetime compatible with the possible local isometries of Σ K acting on a spacelike section. For K = 1, the space Σ K is locally a sphere, for K = 0 it is locally flat, while for K = -1 it locally reduces to the hyperbolic space. Hereafter we will consider a static and isotropic scalar field, i.e. φ = φ ( r ). \nThe outline of the paper is as follows: in section 2 the four-dimensional solution is given for arbitrary K , and the energy density is computed. In section 3, the spherically symmetric solution is described in detail and the constraints in the couplings parameters are described in order to obtain a real scalar field and positive energy density. We comment as well on some of the thermodynamical properties of the solution. In section 4, the solution in arbitrary dimension n is given. Finally in section 5 the solution in the special case when α = 0 is analyzed. In this paper we use the 'mostly plus signature' and Greek indices stand for indices in the coordinate basis.", 'II. FOUR DIMENSIONAL SOLUTION': "Using the ansatz (8) the equation of motion for the scalar field (6) admits a first integral, which implies the equation \nr F ' ( r ) F ( r ) = [ K + α η r 2 -C 0 η G ( r ) ψ ( r ) √ F ( r ) G ( r ) ] G ( r ) -1 , (9) \nwhere C 0 is an integration constant, ψ ( r ) := φ ' ( r ), and ( ' ) stands for derivation with respect to r . As it was done in reference [14], and then in [16] we (arbitrarily) set C 0 = 0, which allows to find a simple relation between the metric functions F ( r ) and G ( r ) \nG ( r ) = η F ( r ) ( rF ' ( r ) + F ( r ) r 2 α + ηK ) . (10) \nThe Maxwell equation admits a first integral as well, providing the following relation \nG ( r ) = r 4 q 2 F ( r ) ( A ' 0 ( r )) 2 , (11) \nwhere 1 q 2 is an integration constant. These two last equations allow us to find an expression for the first radial derivative of the electric potential \n( A ' 0 ( r )) 2 = q 2 η r 4 ( rF ' ( r ) + F ( r ) r 2 α + ηK ) . (12) \nIn this way, equations (10) and (12) together with the tt and rr components of (5), provide a consistent system which for K = ± 1 and η Λ = α , has the following solution \n/negationslash \nF ( r ) = r 2 l 2 + K α √ αηK ( α +Λ η + α 2 4 ηκK q 2 α -Λ η ) 2 arctan ( √ αηK ηK r ) -µ r + α 2 κ ( α -Λ η ) 2 q 2 r 2 + α 3 16 ηκ 2 K 2 ( α -Λ η ) 2 q 4 r 2 -α 2 48 κ 2 K ( α -Λ η ) 2 q 4 r 4 + 3 α +Λ η α -Λ η K , G ( r ) = 1 16 α 2 (4 κ ( α -η Λ) r 4 +8 ηκKr 2 -ηq 2 ) 2 r 4 κ 2 ( α -η Λ) 2 ( αr 2 + ηK ) 2 F ( r ) , ψ 2 ( r ) = -1 32 α 2 (4 κ ( α + η Λ) r 4 + ηq 2 )(4 κ ( α -η Λ) r 4 +8 ηκKr 2 -ηq 2 ) 2 r 6 ηκ 2 ( α -η Λ) 2 ( αr 2 + ηK ) 3 F ( r ) , A 0 ( r ) = 1 4 q √ α η 3 2 K 5 2 κ ( 4 βκK 2 ( α + η Λ) + α 2 q ( α -η Λ) ) arctan ( √ αηK ηK r ) + α ( 8 ηκK 2 + αq 4 ηκK 2 ( α -η Λ) ) q r -α 12 κK ( α -η Λ) q 3 r . \nHere we have defined the effective (A)dS radius l by l -2 := α 3 η . In the case of a locally flat transverse section ( K = 0) the system integrates in a different manner and the solution \ntakes the form \nF ( r ) = r 2 l 2 -µ r + α 2 κ ( α -η Λ) q 2 r 2 + αη 80 κ 2 ( α -η Λ) 2 q 4 r 6 , G ( r ) = 1 16 (4 κ ( α -η Λ) r 4 -ηq 2 ) 2 κ 2 ( α -Λ η ) r 8 F ( r ) , ψ ( r ) 2 = -1 32 (4 κ ( α + η Λ) r 4 + ηq 2 ) (4 κ ( α -η Λ) r 4 + ηq 2 ) 2 αηr 12 κ 2 ( α -Λ η ) 2 F ( r ) , A 0 ( r ) = -( 20 κ ( α -η Λ) r 4 -ηq 2 20 κ ( α -η Λ) r 5 ) q . \nIn the case when we set q → 0 we recover the result obtained in [16] for the cases K = ± 1 as well as for the case K = 0. The later case reduces to topological Schwarzschild solution with locally flat horizon [20]. \nIt can be seen that this solution is asymptotically locally dS or AdS for α/η < 0 or α/η > 0, respectively, since when r →∞ the components of the Riemann tensor go to \nR ab cd = r →∞ -α 3 η δ ab cd := -1 l 2 δ ab cd , \njustifying our previous definition of the effective (A)dS radius. The asymptotic expansion ( r →∞ ) of the metric functions and of the gauge field reads \ng tt = r →∞ r 2 l 2 + 3 α + η Λ α -η Λ K + K 2 α √ αηK ( ( α + η Λ) + α 2 q 2 4 ηκK 2 α -η Λ ) 2 πσ -2 µ r + O ( r -2 ) , g rr = r →∞ r 2 l 2 + 7 α + η Λ 3( α -η Λ) K + K 2 α √ αηK ( ( α + η Λ) + α 2 q 2 4 ηκK 2 α -η Λ ) 2 πσ -2 µ r + O ( r -2 ) , A 0 ( r ) = r →∞ a 0 -q r + O ( r -2 ) , \nwhere σ is the sign of ηK and a 0 is a constant. From here it is possible to see that our electric potential reproduces the Coulomb potential at infinity. There is a curvature singularity at r = 0 since for example the Ricci scalar diverges as \nR = r → 0 4 K r 2 + O (1) . (13) \nIf ρ ( r ) is the energy density, then the total energy E is given by \nE = V (Σ) ∫ drρ ( r ) , (14) \nwhere V (Σ) stands for the volume of Σ. Therefore \nρ ( r ) := r 2 √ G ( r ) F ( r ) -1 T tt . (15) \nNow, the tt component of the energy momentum tensor reads \nT tt = -( α +Λ η ) ηκ 2 F ( r ) [1 -H ( r ) F ( r )] , (16) \nwhere H ( r ) is the given by the expression \nH ( r ) = 64 η 2 r 2 ( α -Λ η ) 2 ( r 2 α + ηK ) α 2 κ 2 ( α +Λ η ) ( q 2 κ (2 r 2 α + ηK ) -4 K ( α +Λ η ) r 4 4( α -Λ η ) r 4 +8 r 2 ηK -ηκq 2 ) . \nIf we take the limit q → 0 we recover the T tt component of the uncharged case.", 'III. SPHERICALLY SYMMETRIC CASE': 'Now we study the particular case with a spherically symmetric transverse section K = 1. The solution for the metric components and for the square of the derivative of the scalar field reduces to \n( \n) r \nF ( r ) = r 2 l 2 + 1 α √ αη ( α +Λ η + α 2 4 ηκ q 2 α -Λ η ) 2 arctan ( √ αη η r ) -µ r + α 2 κ ( α -Λ η ) 2 q 2 r 2 + α 3 16 ηκ 2 ( α -Λ η ) 2 q 4 r 2 -α 2 48 κ 2 ( α -Λ η ) 2 q 4 r 4 + 3 α +Λ η α -Λ η , G ( r ) = 1 16 α 2 (4 κ ( α -η Λ) r 4 +8 ηκr 2 -ηq 2 ) 2 r 4 κ 2 ( α -η Λ) 2 ( αr 2 + η ) 2 F ( r ) , ψ 2 ( r ) = -1 32 α 2 (4 κ ( α + η Λ) r 4 + ηq 2 )(4 κ ( α -η Λ) r 4 +8 ηκr 2 -ηq 2 ) 2 r 6 ηκ 2 ( α -η Λ) 2 ( αr 2 + η ) 3 F ( r ) , A 0 ( r ) = 1 4 q √ α η 3 2 κ ( 4 βκ 2 ( α + η Λ) + α 2 q ( α -η Λ) ) arctan ( √ αη η r ) + α 8 ηκ + αq 4 ηκ ( α -η Λ) q -α 12 κ ( α -η Λ) q 3 r . \nIn order to analize the features proper of a black hole in our solution we need to analize the lapse function F ( r ). As we approach the origin, the lapse function goes to minus infinity. On the other hand, as we go to infinity along coordinate r , F ( r ) tends to plus infinity. Therefore, it is clear that this function being continuous has at least one cero. We can \nprove that this function has more than one cero. Since we know the existence of at least one cero r H , we can parametrize the function with r H as parameter. From the equation F ( r H ) = 0 we get µ ≡ µ ( r H ) which can be used to express the lapse function as F ( r, µ ( r H )). To prove the existence of the second event horizon, we can do the same as before but with the electric charge. We propose the existence of r h , then F ( r h ) = 0, and using this we get q 2 ≡ q 2 ( r h , r H ). It is possible to find two roots for F ( r h ) = 0 or in other words, two suitable values of q 2 for a possible r h . This values in some cases are both negatives, both positive or one positive and the second negative, but at least the existence of one positive root is enough to prove the existence of r h . As we said, due to the shape near the origin and at infinity of the lapse function, the existence of two zeros of the function implies the existence of a third zero for some range of paramaters. Therefore F ( r ) can have just one zero, two zeros 4 or three zeros. Each of these cases exist for a specific set of values of the coupling and cosmological constants. From hereafter and for simplicity, we will focus in the case when the lapse function has just one zero. \nReality condition of the lapse function requires αη > 0. Therefore l -2 := α 3 η is positive defined and the spacetime is asymptotically AdS. As it was noted in the uncharged case [16] without loss of generality it is possible to choose both parameters positive, since the solution with both α and η negative is equivalent to the former by changing µ →-µ . \nIn order to obtain a real scalar field in the domain of outer communications and satisfy the positivity of the energy, we need to impose some constraints in our parameters. In fact, the value of the cosmological constant is restricted to be \nΛ < -q 2 4 r 4 H κ -α η . (17) \nIt is important to note that we cannot switch off the scalar field. This implies that our solution is not continuously connected with the maximally symmetric background. Despite of this, setting µ = 0 and q = 0 we observe that the spacetime is regular, actually is the only regular spacetime that can be found within this family. Such a case describes an asymptotically AdS gravitational soliton. Close to r = 0 and after a proper reescaling on the time coordinate the spacetime metric takes the following form \nds 2 soliton = -( 1 -Λ 3 r 2 + O ( r 4 ) ) dt 2 + ( 1 -3 α +2Λ η 3 η r 2 + O ( r 4 ) ) dr 2 + r 2 d Ω 2 . (18) \nThe thermal version of this spacetime can be used as the background metric for obtain a regularized euclidean action which could be used to obtain the thermodynamical properties of the black holes in the Hawking-Page approach.', 'IV. N-DIMENSIONAL CASE': "In this section we analize the n -dimensional solution to the action principle defined by (4). For doing this, we take the variation of our Lagrangian with respect to all the functions involved F ( r ), G ( r ), φ ( r ) and A 0 ( r ). This procedure gives us the equations of motion of the system. \nTherefore, following the same strategy than in four dimensions, the equation of motion for the scalar field admits a first integral. Setting to zero the integration constant of this equation we obtain a relation between the metric coefficients, but now in arbitrary dimension \nG n ( r ) = η ( n -2) F n ( r ) ( F ' n ( r ) r + F n ( r )( n -3) 2 r 2 α + ηK ( n -2)( n -3) ) . (19) \nThe equation coming from the variation with respect to the electric field gives us the following relation \n( \nA ' 0 n ( r ) ) 2 = q 2 F n ( r ) G n ( r ) r (4 -2 n ) . \n) In the same spirit, and using the last result, it is possible to obtain a relation for ψ ( r ) 2 . Then \nψ n ( r ) 2 = -1 2 ( n -2) ( Ξ 1 n +Ξ 2 n Ξ 3 n ) , \nwhere we have defined \nΞ 1 n = ( n -3) 2 (4 κ Λ ηr 2 +4 κr 2 α + q 2 r ( -2 n +6) η ) F n ( r ) 2 +2( n -3)( q 2 r ( -2 n +7) η +4 κ Λ ηr 3 +4 αr 3 κ ) F ' n ( r ) F n ( r ) , Ξ 2 n = (4 κ Λ ηr 4 + q 2 r ( -2 n +8) η +4 αr 4 κ ) F ' n ( r ) 2 , Ξ 3 n = F n ( r )(2 r 2 αηKn 2 -5 ηKn +6 ηK ) 2 (( n -3) F n ( r ) + F ' n ( r ) r ) . \nUsing these expresions and the equation resulting from the variation with respect to the fuction F n ( r ), we can obtain a relation which allows to obtain the explict form of F n ( r ) for an arbitrary value of the dimension n , and in this way, the complete solution to our system. We checked the result from n = 4 to n = 10.", 'V. ASYMPTOTICALLY LOCALLY FLAT BLACK HOLES WITH CHARGE SUPPORTED BY THE EINSTEIN-KINETIC COUPLING': 'In this section we will study the particular case where the scalar field is coupled to the background only with the Einstein tensor. It is possible to do this by setting α = 0. Under the presence of an electric field, we obtain asymptotically locally flat black hole solutions in the case where the cosmological constant is present. Therefore, the action principle is given by \nI [ g µν , φ ] = ∫ √ -gd 4 x [ κ ( R -2Λ) + η 2 G µν ∇ µ φ ∇ ν φ -1 4 F µν F µν ] . (20) \nFollowing the same procedure (with α = 0 and K = 1) 5 we obtain \nds 2 = -F ( r ) dt 2 + 15[4 κr 2 (2 -Λ r 2 ) -q 2 ] 2 r 4 dr 2 F ( r ) + r 2 d Ω 2 , (21) \n/negationslash \nwhere \nF ( r ) = 48 κ 2 Λ 2 r 4 -320 κ 2 Λ r 2 +120 κ (8 κ +Λ q 2 ) -µ r +240 κ q 2 r 2 -5 q 4 r 4 , ψ ( r ) 2 = -15 2 (4 κ Λ r 4 + q 2 )(4 κr 2 (2 -Λ r 2 ) -q 2 ) 2 r 6 η 1 F ( r ) , A 0 ( r ) = √ 15 ( q 3 3 r 3 -8 κ q r -4 κ Λ rq ) . \nThis solution shows the following features: \n- · The solution is asymptotically locally flat, namely we have \nlim r →∞ R µν λρ → 0 . \n- · For a non degenerated horizon r = r H we have F ( r H ) = 0, then the scalar field vanishes at the horizon and is not analytic there.\n- · In order to obtain a real scalar field outside of the horizon we can impose two different conditions:\n- 1. Λ > 0 and η < 0 or \n2. Λ < -q 2 4 κr 4 H and η > 0. \n- · For any value of the integration constant µ we have the curvature singularities \nr 0 = 0 , r 1 , 2 = √ 2 κ Λ(2 κ ± √ 4 κ 2 -κ Λ q 2 ) 2 κ Λ . \nThen for Λ < 0 the only singularity is located at the origin of coordinates. If the cosmological constant is positive, in order to rule out the existence of singularities different than r = 0, we need to impose the following constraint in the value of Λ \nΛ > 4 κ q 2 . (22) \n- · We point out that in the limit r → ∞ our electric potential represents a constant electric field at that point supported by the cosmological constant, and in this way we obtain an asymptotically electric Universe.\n- · Finally the limit q → 0 we recover the results obtained in [16]. \nLet us put Λ = 0, then the solution takes the form \nds 2 = -F ( r ) dt 2 + 3(8 κr 2 -q 2 ) 2 r 4 dr 2 F ( r ) + r 2 d Ω 2 , \nwhere \nF ( r ) = 192 κ 2 -µ r +48 κ q 2 r 2 -q 4 r 4 , ψ ( r ) 2 = -15 2 (8 κr 2 -q 2 ) 2 r 6 η q 2 F ( r ) , A 0 ( r ) = √ 15 ( q 3 3 r 3 -8 κ q r ) . \nIn this case we have: \n- · The solution is asymptotically flat \nds 2 = -( 1 -µ r + O ( r -2 ) ) dt 2 + ( 1 + µ r + O ( r -2 ) ) dr 2 + r 2 d Ω 2 , \nwhich is razonable because when we have Λ = 0, the electric field at infinity vanishes. \n- · For a non degenerated horizon r = r H we have F ( r H ) = 0, then the scalar field vanish at the horizon, as in the previous cases, is not analytic there. \n- · In order to obtain a real scalar field outside of the horizon we impose \nη < 0 . \n- · For any value of the integration constant µ we have the curvature singularities \nr 0 = 0 , r 1 = √ 1 8 κ | q | . \n- · The electric field goes to zero at infinity.\n- · Taking the limit when q → 0 we obtain a trivial scalar field and then we recover the Schwarzschild solution.', 'VI. DISCUSSION': 'In this work a particular sector of the Horndeski theory was considered where the gravity part is given by the Einstein-Hilbert term, and where the matter source is represented by a scalar field which has a non minimal kinetic coupling constructed with the Einstein tensor. The main novelty of this work is the inclusion of the Maxwell field. We found exact solutions to this system for a spherically symmetric and topological horizons in all dimensions. The solution gives a new class of asymptotically locally AdS and asymptotically locally flat black hole solutions. \nThese solutions are obtained using two important observations. The first one, is the fact that the equation of motion for the scalar field admits a first integral, which after setting the integration constant to zero (arbitrarily) gives a simple relation between the two metric functions. The second one, is that the Maxwell equations are easily integrated for our ansatz and symmetry conditions, given a simple relation between the electric potential term and the metric functions. Mixing these two results we obtain a complete description of the system, obtaining in that way the exact solution for the topological case in n ≥ 4 dimensions. \nWe observe and point out that in the case of the asymptotically locally AdS solution, the cosmological constant at infinity is not given by the cosmological Λ term in the action but rather in terms of the coupling constants α and η that appear in the kinetic coefficients of the field. The electric field is well behaved and goes to the Coulombian one at infinity. \nThe solutions are not continuously connected with the maximally symmetric AdS or flat backgrounds since the scalar field cannot be turned off. Nevertheless, since our family of metrics contains a further integration constant, it is possible to show that within such a family there is a unique regular spacetime. Such spacetime is a gravitational soliton and it is useful in the four dimensional spherically symmetric case to define a regularized Euclidean action and to explore the thermodynamics of the black hole solution. A similar situation occurs with the AdS soliton, which can be considered as the background for the planar AdS black holes, as well as in gravity in 2+1 with scalar fields, where the gravitational solitons are the right backgrounds to give a microscopic description of the black hole entropies [22][23][24]. \nIn the particular case when the scalar field is only coupled to the metric through the Einstein tensor, namely, α = 0 we obtain an asymptotically locally flat black hole solution. When Λ /negationslash = 0 this solution presents some interesting properties. The solution exist in both cases, where the cosmological constant is positive and when is negative, given a real scalar field configuration depending on constraints imposed on the electric charge and on the coupling constant η . In any of these cases we obtain a constant electric field at infinity, representing in this way our solution a electric Universe. This constant electric field at infinity is just supported by the cosmological constant. \nIn the case where Λ = 0 we obtain a real scalar configuration just in case where the coupling constant is negative. The solution is asymptotically flat and the electric field vanishes at infinity when Λ = 0. If we switch off the electric field setting q = 0, we get a trivial scalar field and then we recover the Schwarzschild solution. \nIt is important to note that Horndeski theory offers the posibility of exploring its solutions in many different ways. In another context, using the same action principle, but without the Maxwell term an asymptotically Lifshitz solution was recently found in [25]. Moreover, even if it is not possible to obtain an analytic solution to the most general case of the Horndeski theory for the general static black hole solution, it would be interesting to study the cases where the non minimal coupling is given by more general tensors than the Einstein one, namely the Lovelock tensors.', 'VII. ACKNOWLEDGMENTS': "A. C. and C. E. would like to thank Andr'es Anabal'on for useful discussions and comments. We are grateful to Julio Oliva for the useful insight during the development of this work. The work of A. C. is supported by CONICYT and by the project FSM1204 of Internationalization of Ph. D. programs in physical science, biotechnology and electronics from the Universidad T'ecnica Federico Santa Mar'ıa. The work of C. E. is supported by CONICYT and Centro de Estudios Cient'ıficos (CECs) funded by the Chilean Government through the Centers of Excellence Base Financing Program of Conicyt. \n- [1] C. Brans and R. H. Dicke, Phys. Rev. 124 , 925 (1961).\n- [2] Daniele Bertacca, Sabino Matarrese and Massimo Pietroni [arxiv.org:0703259 [astro-ph]].\n- [3] Daniel J. H. Chung, Lisa L. Everett, Konstantin T. Matchev, [arxiv.org: 0704.3285 [hep-ph]].\n- [4] A. de la Macorra, [arxiv.org:0703702 [astro-ph]].\n- [5] G. W. Horndeski, Int. J. Theor. Phys. 10 , 363 (1974).\n- [6] C. Deffayet, G. Esposito-Farese and A. Vikman, Phys. Rev. D 79 (2009) 084003; C. Deffayet, S. Deser and G. Esposito-Farese, Phys. Rev. D 80 (2009) 064015.\n- [7] C. de Rham, G. Gabadadze and A. J. Tolley, Phys. Rev. Lett. 106 (2011) 231101, C. de Rham and L. Heisenberg, Phys. Rev. D 84 (2011) 043503.\n- [8] L. Amendola, Phys. Lett. B 301 (1993) 175.\n- [9] C. Deffayet, O. Pujolas, I. Sawicki and A. Vikman, JCAP. 1010 (2010) 026.\n- [10] J. -P. Bruneton, M. Rinaldi, A. Kanfon, A. Hees, S. Schlogel and A. Fuzfa, [arXiv:1203.4446 [gr-qc]].\n- [11] S. V. Sushkov, Phys. Rev. D 80 (2009) 103505.\n- [12] C. Germani and A. Kehagias, Phys. Rev. Lett. 106 (2011) 161302.\n- [13] D. Lovelock, J. Math. Phys. 12 , 498 (1971).\n- [14] M. Rinaldi, Phys. Rev. D 86 , 084048 (2012), [arXiv:1208.0103 [gr-qc]].\n- [15] J.M.Maldacena, [arXiv:9711200 [hep-th]].\n- [16] Andres Anabal'on, Adolfo Cisterna and Julio Oliva, [arXiv:1312.3597 [gr-qc]].\n- [17] E. Babichev and C. Charmousis, [arXiv:1312.3204 [gr-qc]]. \n- [18] Masato Minamitsuji, [arXiv:1312.3759 [gr-qc]].\n- [19] T. Kolyvaris, G. Koutsoumbas, E. Papantonopoulos and G. Siopsis, Class. Quant. Grav. (2012), [arXiv:1111.0263 [gr-qc]].\n- [20] J. P. S. Lemos, Phys. Lett. B 353 , 46 (1995), [arXiv:9404041 [gr-qc]].\n- [21] S. W. Hawking and D. N. Page, Commun. Math. Phys. 87 , 577 (1983).\n- [22] F. Correa, C. Martinez and R. Troncoso, JHEP 1202 , 136 (2012), [arXiv:1112.6198 [hep-th]].\n- [23] F. Correa, A. Faundez and C. Martinez, Phys. Rev. D 87 , 027502 (2013), [arXiv:1211.4878 [hep-th]].\n- [24] J. Zanelli, Class. Quant. Grav. 29 , 133001 (2012), [arXiv:1208.3353 [hep-th]].\n- [25] Moises Bravo-Gaete, Mokhtar Hassaine, [arXiv:1312.7736[hep-th]]."}
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2019CQGra..36p5002D
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Carrollian physics at the black hole horizon
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2019-01-01
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We show that the geometry of a black hole horizon can be described as a Carrollian geometry emerging from an ultra-relativistic limit where the near-horizon radial coordinate plays the role of a virtual velocity of light tending to zero. We prove that the laws governing the dynamics of a black hole horizon, the null Raychaudhuri and Damour equations, are Carrollian conservation laws obtained by taking the ultra-relativistic limit of the conservation of an energy-momentum tensor; we also discuss their physical interpretation. We show that the vector fields preserving the Carrollian geometry of the horizon, dubbed Carrollian Killing vectors, include BMS-like supertranslations and superrotations and that they have non-trivial associated conserved charges on the horizon. In particular, we build a generalization of the angular momentum to the case of non-stationary black holes. Finally, we discuss the relation of these conserved quantities to the infinite tower of charges of the covariant phase space formalism.
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https://arxiv.org/pdf/1903.09654.pdf
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{'Carrollian Physics at the Black Hole Horizon': 'Laura Donnay a,b , Charles Marteau c \n- a Center for the Fundamental Laws of Nature, Harvard University 17 Oxford Street, Cambridge, MA 02138, USA. \nb Black Hole Initiative, Harvard University 20 Garden Street, Cambridge, MA 02138, USA. \nc CPHT, Ecole Polytechnique, CNRS UMR 7644, Université Paris-Saclay 91128, Palaiseau, France.', 'Abstract': 'We show that the geometry of a black hole horizon can be described as a Carrollian geometry emerging from an ultra-relativistic limit where the near-horizon radial coordinate plays the role of a virtual velocity of light tending to zero. We prove that the laws governing the dynamics of a black hole horizon, the null Raychaudhuri and Damour equations, are Carrollian conservation laws obtained by taking the ultra-relativistic limit of the conservation of an energy-momentum tensor; we also discuss their physical interpretation. We show that the vector fields preserving the Carrollian geometry of the horizon, dubbed Carrollian Killing vectors, include BMS-like supertranslations and superrotations and that they have non-trivial associated conserved charges on the horizon. In particular, we build a generalization of the angular momentum to the case of nonstationary black holes. Finally, we discuss the relation of these conserved quantities to the infinite tower of charges of the covariant phase space formalism.', 'Contents': '| 1 | Introduction | 1 |\n|----------------|-----------------------------------------------------------------------------------|-----|\n| 2 | Near-horizon geometry and dynamics | 3 |\n| 2.1 | Intrinsic and extrinsic geometry of the horizon . . . . . . . . . . . . . . . . . | 3 |\n| 2.2 | Raychaudhuri and Damour equations . . . . . . . . . . . . . . . . . . . . . . | 5 |\n| 2.3 | Bulk symmetries and associated charges . . . . . . . . . . . . . . . . . . . . | 6 |\n| 3 | Near-horizon or ultra-relativistic limit | 9 |\n| 3.1 | Carrollian geometry: Through the Looking-Glass . . . . . . . . . . . . . . . | 9 |\n| 3.2 | Horizon dynamics as ultra-relativistic conservation laws . . . . . . . . . . . . | 11 |\n| 3.3 | Conserved charges on the horizon . . . . . . . . . . . . . . . . . . . . . . . . | 13 |\n| 4 Perspectives | 4 Perspectives | 17 |', '1 Introduction': "In the membrane paradigm formalism [1-3], the black hole event horizon is seen as a twodimensional membrane that lives and evolves in three-dimensional spacetime. This viewpoint was originally motivated by Damour's seminal observation that a generic black hole horizon is similar to a fluid bubble with finite values of electrical conductivity, shear and bulk viscosity [4-6]. It was moreover shown that the equations governing the evolution of the horizon take the familiar form of an Ohm's law, Joule heating law, and Navier-Stokes equation. The membrane paradigm developed by Thorne and Macdonald for the electromagnetic aspects, and by Price and Thorne for gravitational and mechanical aspects, combines Damour's results with the 3 + 1 formulation of general relativity, where one trades the true horizon for a 2+1-dimensional timelike surface located slightly outside it, called 'stretched horizon' or 'membrane'. The laws of evolution of the stretched horizon then become boundary conditions on the physics of the external universe, hence making the membrane picture a convenient tool for astrophysical purposes. In order to derive the evolution equations of the membrane, a crucial step in [3] was to renormalize all physical quantities (energy density, pressure, etc) on the membrane, as they turned out to be divergently large as one approaches the real horizon. We will show that a better approach to this issue is to interpret the near-horizon limit as an ultra-relativistic limit for the stretched horizon, where the radial coordinate plays the role of a virtual speed of light. This ultra-relativistic limit results in the emergence of Carrollian physics at the horizon. \nThe Carroll group was originally introduced in [7] as an ultra-relativistic limit of the Poincaré group where the speed of light is tending to zero (as opposed to the more familiar \nnon-relativistic limit leading to the Galilean group). Recently, there has been a renewed interest in Carrollian physics due to its relation to asymptotically flat gravity. The symmetry group of asymptotically flat spacetimes is the Bondi-Metzner-Sachs (BMS) group; it is an infinite-dimensional extension of the Poincaré group, and its connection with soft theorems has shead a new light on the infrared structure of gravitational theories [8]. Interestingly, the BMS group was also shown to be isomorphic to a conformal extension of the Carroll group in [9], while the dynamics of asymptotically flat spacetimes has been rephrased in terms of ultra-relativistic conservations laws on null infinity [10]. This leads to think that theories holographically dual to asymptotically flat gravity should be ultra-relativistic and enjoy a Carrollian symmetry [11]. Actually, it is now understood that any null hypersurface is endowed with a Carrollian geometry 1 [10,12-16] and that the associated constraint equations are ultra-relativistic conservation laws [17]. The aim of this paper is to give a complete analysis of this statement at the level of another physically interesting null hypersurface, the horizon of a black hole. The Carrollian symmetry emerging at the horizon was also used in [18] to explain the vanishing of Love numbers for the Schwarzschild black hole. \nThe recent focus on the symmetries of near-horizon geometries has been motivated by the fact that they exhibit, in some instances, a BMS-like algebra composed of supertranslations and superrotations [19-32]. Moreover, one can associate non-trivial charges to these large diffeomorphisms: they generate the so-called soft hair on black holes [22-25], which were pointed out to have implications for the information paradox. We will show that this rich symmetry structure is in fact naturally encoded in the Carrollian geometry of the horizon. To do so, we will interpret the near-horizon limit as an ultra-relativistic limit, where the radial coordinate ρ plays the role of a virtual speed of light for constant ρ hypersurfaces. This will allow to define proper, rather than ad hoc, finite quantities on the horizon. Moreover, we will prove that the laws governing the dynamics of the black hole horizon are Carrollian conservation laws. These are the ultra-relativistic equivalent of the conservation of an energymomentum tensor. Through the near-horizon analysis, we will derive the isometries of the induced Carrollian geometry on the horizon and show that they include supertranslations and superrotations. We will also construct associated conserved charges; in particular, the one associated with superrotations will provide a generalization of the angular momentum for very generic non-stationary black holes. Finally, the relation of these conserved quantities to the charges of the covariant phase space formalism will also be discussed. \nThe paper is organized as follows: in Sec. 2, we introduce a suitable coordinate system for the study of near-horizon geometries. We define the intrinsic and extrinsic objects of the horizon and write the constraint equations governing the dynamics, i.e. the null Raychaudhuri and Damour equations. We then review the set of vector fields preserving the near-horizon metric and the derivation of their associated surface charges defined in the covariant phase space formalism. In Sec. 3, we present the Carrollian geometry associated with the black hole horizon. By identifying the radial coordinate ρ as the square of a virtual speed of light for constant ρ hypersurfaces, we interpret the near-horizon limit ( ρ → 0 ) as an ultra-relativistic limit and compute the horizon Carrollian geometric fields. We then define \nthe energy-momentum tensor associated with a constant ρ hypersurface in terms of its extrinsic curvature. The analysis of its scaling w.r.t. the radial coordinate allows us to define the Carrollian momenta which are the ultra-relativistic equivalent of the energy-momentum tensor. We give a physical interpretation of those quantities in terms of energy density, pressure, heat current and dissipative tensor. Ultra-relativistic conservation laws are written in terms of the Carrollian momenta and are shown to match perfectly the null Raychaudhuri and Damour equations. Finally, we consider the Killing fields which preserve the Carrollian geometry induced on the horizon and construct associated conserved charges. The latter provides a generalization of the angular momentum for non-stationary black holes. We extend this analysis to conformal Killing vectors of the Carrollian geometry and show that the charges are now conserved provided a conformal state equation involving the energy density and the pressure is satisfied. We also write an interesting relation between these conserved charges and the one obtained in the covariant phase space formalism. We conclude in Sec. 4 with a discussion of open questions.", '2 Near-horizon geometry and dynamics': 'In this section, we describe the near-horizon geometry of a black hole and its dynamics. To do so, we introduce a coordinate system adapted to the study of the spacetime geometry near a null hypersurface. This will allow us to define the intrinsic and extrinsic geometry of the horizon. The projection of Einstein equations on the horizon gives rise to two constraint equations on the extrinsic geometry, the null Raychaudhuri equation and the Damour equation. These are the constraints that we ultimately want to interpret as ultra-relativistic conservation laws. Finally, we turn to the asymptotic symmetries preserving the form of the near-horizon geometry we have introduced, and present the associated charges computed through the covariant phase space formalism. They have the particularity of being generically non-integrable.', '2.1 Intrinsic and extrinsic geometry of the horizon': "We consider a D -dimensional spacetime whose coordinates are x a = ( x α , x A ) , with x α = ( v, ρ ) where v is the advanced time and ρ the radial coordinate. The surfaces of constant v and ρ are ( D -2) -dimensional spheres S v,ρ and parametrized by x A ( A = 3 , · · · , D ), the set of all these angular coordinates will be denoted x . Throughout the paper, when we refer to spatial objects, it will be with respect to the angular coordinates. The constant v surfaces are null, and constant ρ are timelike. Finally, we assume the existence of a horizon H sitting at ρ = 0 . \nIt is alway possible to find a coordinates system, usually called null Gaussian coordinates , such that the near-horizon geometry is given by [33] 2 \nds 2 = -2 κρdv 2 +2 dρdv +2 θ A ρdvdx A +(Ω AB + λ AB ρ ) dx A dx B + O ( ρ 2 ) , (2.1) \nFigure 1: The horizon is a null hypersurface situated at ρ = 0 and Σ ρ is a timelike constant ρ hypersurface near the horizon. We define also four vectors that are useful for our analysis, the null vector glyph[vector] L is the normal to the horizon while glyph[vector] N is transverse but also null. The spacelike vector glyph[vector]n is the normal to Σ ρ and the timelike vector glyph[vector] glyph[lscript] is the normal to a constant v section of Σ ρ . \n<!-- image --> \nwhere κ , Ω AB , λ AB , θ A in principle depend on the coordinates x and v . The spatial metric Ω AB will be used to raise and lower spatial indexes. We will sometime refer to the D -dimensional spacetime as the bulk . \nThere are now two types of geometrical objects we can define on H : the first ones are intrinsic and the others extrinsic. In a Hamiltonian perspective, they are canonical conjugate of each other. Moreover, the canonical momenta satisfy constraint equations that are imposed by the gravitational dynamics [35,36]. The induced geometry on H is degenerate and reads \nds 2 H = 0 · dv 2 +0 · dvdx A +Ω AB dx A dx B , (2.2) \nthe intrinsic geometry being then entirely specified by the spatial metric in this gauge. We now perform a decomposition of the bulk metric adapted to the study of null hypersurfaces: \ng ab = q ab + L a N b + N b L a , (2.3) \nwhere \nglyph[vector] L = L a ∂ a = ∂ v -ρθ A ∂ A + κρ∂ ρ and N = N a dx a = dv, (2.4) \nare respectively a null vector and a null form. They satisfy N ( glyph[vector] L ) = 1 and will allow us to define all the extrinsic curvature elements of H . The vector glyph[vector] L coincides with the normal to the horizon on H , and has the particularity of being also tangent to the horizon. On the other hand the vector glyph[vector] N ≡ g -1 ( N ) is transverse to the horizon and together with glyph[vector] L they define q ab , the projector perpendicular to glyph[vector] L and glyph[vector] N . In his work [5,6], T. Damour maps the black hole dynamics to the hydrodynamics of a fluid living on the horizon, and the vector glyph[vector] L defines the fluid's velocity through glyph[vector] L H = ∂ v + v A ∂ A . We have v A = 0 , as we have chosen \ncomoving coordinates, i.e. , in Damour's interpretation the fluid would be at rest but on a dynamical surface 3 . \nThe extrinsic geometry of the horizon is captured by a triple (Σ AB , ω A , ˜ κ ) where Σ AB is the deformation tensor (or second fundamental form), ω A is the twist field (Hajicek one-form) and ˜ κ the surface gravity, defined as follows: \nΣ AB = 1 2 q a A q b B L glyph[vector] L q ab , ω A = q a A ( N b D a L b ) and L b D b L a = ˜ κL a , (2.5) \nwhere L denotes the Lie derivative, and D a is the Levi-Civita associated with g ab . Using the bulk metric (2.1), these quantities become on H \nΣ AB = 1 2 ∂ v Ω AB , ω A = -1 2 θ A and ˜ κ = κ. (2.6) \nWe see that κ really plays the role of the surface gravity and that θ A is proportional to the twist. The deformation tensor gives rise to two new extrinsic objects: its trace and its traceless part, which are respectively the horizon expansion and the shear tensor: \nΘ = Ω AB Σ AB = ∂ v ln √ Ω , σ AB = 1 2 ∂ v Ω AB -Θ D -2 Ω AB , (2.7) \nwhere √ Ω is the volume form of the spatial metric. The scalar expansion Θ measures the rate of variation of the surface element of the spatial section of H . 4 It is possible to show, under the assumption that matter fields satisfy the null energy condition and that the null Raychaudhuri equation (see next section) is satisfied, that Θ is positive everywhere on H , which implies that the surface area of the horizon can only increase with time (see e.g. [37]).", '2.2 Raychaudhuri and Damour equations': 'Those quantities being defined, we can deduce from Einstein equations two conservation laws (or constraint equations) that belong to H : the null Raychaudhuri equation [38] and Damour equation [5,6], which are respectively \nL a L b R ab = 0 and q a A L b R ab = 0; (2.8) \nthey are thus given by projections of vacuum Einstein equations on the horizon. The first one is scalar and the second one is a vector equation w.r.t. the spatial section of H . Using the near-horizon geometry (2.1), the null Raychaudhuri equation becomes \n∂ v Θ -κ Θ+ Θ 2 D -2 + σ AB σ AB = 0 , (2.9) \nwhere σ AB = Ω AC Ω BD σ CD . This equation describes how the expansion evolves along the null geodesic congruence glyph[vector] L and is a key ingredient in the proofs of singularity theorems. Damour equation 5 becomes \n( ∂ v +Θ) θ A +2 ∇ A ( κ + D -3 D -2 Θ ) -2 ∇ B σ B A = 0 , (2.10) \nwhere ∇ A is the Levi-Civita connection associated with Ω AB . Damour has interpreted this last equation as a ( D -2) -dimensional Navier-Stokes equation for a viscous fluid; notice that the fluid velocity is not appearing here because we have chosen a comoving coordinate system as explained earlier. \nIt is useful to know what these equations become when considering the conformal gauge, i.e. when the spatial metric can be written as a conformal factor times a purely spatial metric: \nΩ AB = γ ( v, x ) ¯ Ω AB ( x ) . (2.11) \nOne can check that this is equivalent to asking the shear to be zero. If we make this choice, ¯ Ω AB disappears and the two conservation equations read \n∂ 2 v γ -1 2 γ -1 ( ∂ v γ ) 2 -κ∂ v γ = 0 , ∂ v θ A +2 ∂ A κ +( D -3) γ -1 ∂ A ∂ v γ -( D -3) γ -2 ∂ A γ∂ v γ + ( D -2) 2 γ -1 θ A ∂ v γ = 0 . (2.12) \nIn particular, one can verify that these equations reproduce the field equations studied in [20] in the D = 3 and D = 4 cases.', '2.3 Bulk symmetries and associated charges': 'We now turn our attention to the bulk symmetries of the near-horizon gauge. The vector fields χ = χ a ∂ a that preserve the shape of the metric (2.1) were shown in [20] to involve of a smooth arbitrary function f ( v, x ) , which depends on the advanced time and the sphere coordinates, and a vector field of the sphere Y A ( x ) ; they are given by \nχ v = f ( v, x ) , χ ρ = -∂ v fρ + 1 2 θ A ∂ A fρ 2 + O ( ρ 3 ) , χ A = Y A ( x ) + Ω AC ∂ C fρ + 1 2 λ AC ∂ C fρ 2 + O ( ρ 3 ) , (2.13) \nand in any dimension D . We will call them asymptotic Killing vectors even though the gauge introduced does not involve a notion of infinity. We notice an important feature, which is that these vector fields projected on the horizon become \nχ = f ( v, x ) ∂ v + Y A ( x ) ∂ A projected on H , (2.14) \nand as f and Y A are totally generic for the moment, this is exactly the infinitesimal version of a particular type of diffeomorphisms on the horizon that we will define in Sec. 3: the Carrollian diffeomorphisms. Following [19, 20], we will call f a supertranslation and Y A a superrotation. They act on the horizon fields in the following way: \nδ χ κ = Y A ∂ A κ + ∂ v ( κf ) + ∂ 2 v f, δ χ Ω AB = f∂ v Ω AB + L Y Ω AB , δ χ θ A = L Y θ A + f∂ v θ A -2 κ∂ A f -2 ∂ v ∂ A f + ∂ v Ω AB ∂ B f, δ χ λ AB = f∂ v λ AB -λ AB ∂ v f + L Y λ AB + θ A ∂ B f + θ B ∂ A f -2 ∇ A ∇ B f. (2.15) \nTo each of these vector fields preserving the near-horizon metric, one can associate a surface charge through the covariant phase space formalism [39]. 6 More precisely, the quantity which is constructed at first is not a charge, but rather the field-variation of a charge (namely a one-form in the configuration space). For an on shell metric g and variation h ≡ δg , it is given by: \nδ /Q χ [ g, h ] = ∮ S v,ρ k χ [ g, h ] , (2.16) \nwhere χ is an asymptotic Killing vector and k χ [ g, h ] is a one-form w.r.t. the field configuration space but a ( D -2) -form w.r.t. the spacetime. It is defined as follows: 7 \nk χ [ g, h ] = √ -g 8 πG ( d D -2 x ) ab ( χ a ∇ c h bc -χ a ∇ b h + χ c ∇ b h ac + 1 2 h ∇ b χ a -h cb ∇ c χ a ) , (2.17) \nwhere h = g ab h ab and ( d D -2 x ) ab = 1 2( D -2)! glyph[epsilon1] abc 1 ...c D -2 dx c 1 ∧ . . . ∧ dx c D -2 . The δ / is a notation that emphasizes the fact that the charges (2.16) are a priori non-integrable (namely not δ -exact). In the integrable case, Q χ represents the generator of the associated infinitesimal transformation χ . Computing δ /Q [ g, h ] for the metric written in the horizon gauge (2.1), the associated preserving vector fields (2.13) and integrated on a spatial section of H , one obtains [20]: \nδ /Q ( f,Y A ) [ g, δg ] = 1 16 πG ∮ S D -2 d D -2 x ( 2 fκδ √ Ω+2 ∂ v fδ √ Ω -2 f √ Ω δ Θ+ 1 2 f √ Ω ∂ v Ω AB δ Ω AB -Y A δ ( θ A √ Ω) ) . (2.18) \nWe can see that these charges are not integrable in full generality, due to the presence of the three following terms: 2 f √ Ω δ Θ , 2 fκδ √ Ω and 1 2 f √ Ω ∂ v Ω AB δ Ω AB . The authors of [20] circumvent this issue by restricting the phase space to the configurations where κ is a constant. They also use the fact that they work in four dimensions to choose a spatial metric related to the usual metric on the 2-sphere by a Weyl transformation. We would like instead for the moment to keep all possible dependencies of the fields. \nWhen surface charges are non-integrable, there is still a way to obtain a representation of the asymptotic Killing algebra through the definition of a modified bracket [41]. To do so, we split δ /Q χ into an integrable part Q int χ and a non-integrable part Ξ χ : \nδ /Q χ [ g, δg ] = δ ( Q int χ [ g ]) + Ξ χ [ g, δg ] , (2.19) \nwhere \nand \nQ int χ [ g ] = 1 16 πG ∮ S D -2 d D -2 x √ Ω ( 2 fκ +2 ∂ v f -2 D -2 f Θ -Y A θ A ) , (2.20) \nΞ χ [ g, δg ] = -1 8 πG ∮ S D -2 d D -2 x √ Ω f ( δκ + D -3 D -2 δ Θ -1 2 σ AB δ Ω AB ) . (2.21) \nFrom this splitting 8 we can see directly why, for three-dimensional bulk spacetimes, the condition δκ = 0 considered in [20] was sufficient to insure integrability of the charges (the shear vanishes by definition and the factor ( D -3) cancels the contribution of the expansion in (2.21)). We now define the following modified Dirac bracket \n{ Q int χ [ g ] , Q int η [ g ] } ∗ ≡ δ η Q int χ [ g ] + Ξ η [ g, L χ g ] . (2.22) \nIt was first introduced in [41] for the study of the BMS charges in four dimensions, which are also generically non-integrable. They also noticed that the splitting is not unique in the sense that for some N χ [ g ] we can always choose \n˜ Q int χ = Q int χ -N χ with ˜ Θ χ + δN χ . (2.23) \nHowever, we will see that the separation (2.20), (2.21) we have chosen happens to be relevant in the Carrollian anaysis that we perform in Sec. 3. This modified bracket defines a representation of the asymptotic Killing algebra: indeed, letting ( f 1 , Y A 1 ) and ( f 2 , Y A 2 ) to be two asymptotic Killing fields, one can show that \n{ Q int ( f 1 ,Y A 1 ) , Q int ( f 2 ,Y A 2 ) } ∗ = Q int ( f 12 ,Y A 12 ) , (2.24) \nwhere f 12 = f 1 ∂ v f 2 -f 2 ∂ v f 1 + Y A 1 ∂ A f 2 -Y A 2 ∂ A f 1 and Y A 12 = Y B 1 ∂ B Y A 2 -Y B 2 ∂ B Y A 1 . We notice that this algebra does not involve any central extension. A direct consequence of (2.24) is that the non-integrable part of the charges plays the role of a source for the non-conservation of Q int . Indeed choosing ( f 2 , Y A 2 ) to be (1 , 0) we obtain \nδ (1 , 0) Q int χ [ g ] + Q int ( ∂ v f, 0) [ g ] = -Ξ (1 , 0) [ g, L χ g ] , (2.25) \nmoreover δ (1 , 0) acts like a time derivative on the fields (2.15), so we finally obtain \nd dv Q int χ [ g ] = -Ξ (1 , 0) [ g, L χ g ] . (2.26)', '3 Near-horizon or ultra-relativistic limit': 'One of the particularity of null hypersurfaces is that they are equipped with a degenerate induced metric Ω in the sense that there exists a vector field glyph[vector]u that belongs to its kernel: \nΩ( ., glyph[vector]u ) = 0 . (3.1) \nIn the case of the horizon described above, Ω = Ω AB ( v, x ) dx A dx B and glyph[vector]u = f ( v, x ) ∂ v , for any function f on H . It was understood, for example in [13,14,18], that this defines a Carrollian geometry , the natural non-Riemannian geometry that ultra-relativistic theories couple to. This means that any null hypersurface can be thought of as an ultra-relativistic spacetime. In particular, for the near-horizon geometry presented above, we are going to show that the limit ρ → 0 , can be understood as an ultra-relativistic limit where √ ρ plays the role of a virtual velocity of light c . Notice that this parameter should not be confused with the physical velocity of light of the bulk spacetime that is set to 1 in (2.1). \nThis feature has strong consequences on the dynamics of the horizon, i.e. the null Raychaudhuri and Damour equations: indeed, we will show that they match ultra-relativistic conservation laws written in terms of the Carrollian geometry and the Carrollian momenta , sort of ultra-relativistic equivalent of the energy-momentum tensor. \nFinally, we will study the symmetries and charges associated with the horizon that we interpret as Carrollian Killing , defined as the vector fields on H that preserve the Carrollian geometry. In some instances, the symmetry algebra will be shown to have a BMS-like structure in the sense that it includes superrotations and supertranslations on the horizon [19,20,32].', '3.1 Carrollian geometry: Through the Looking-Glass': "Carrollian geometry emerges from an ultra-relativistic ( c → 0 ) limit of the relativistic metric and was shown to have a rich mathematical structure and interesting dynamics [7, 9, 10, 12, 13, 16, 42]. It was shown in [10, 16] that the c → 0 limit of relativistic generalcovariant theories is covariant under a subset of the diffeomorphisms dubbed Carrollian diffeomorphisms \nv ' = v ' ( v, x ) , x ' = x ' ( x ) , (3.2) \nwhose infinitesimal version is given by the vector fields \nξ = f ( v, x ) ∂ v + Y A ( x ) ∂ A , (3.3) \nfor any f and Y A . This suggests that space and time decouple and an adequate parametrization to study the ultra-relativistic limit is the so-called Randers-Papapetrou parametrization, where the metric is decomposed as 9 \na = ( -c 2 α 2 c 2 αb A c 2 αb B Ω AB -c 2 b A b B ) { dv,dx A } -→ c → 0 Ω AB dx A dx B . (3.4) \nAfter the limit is performed, one thus trade the metric a for α ( v, x ) the time lapse, b A ( v, x ) the temporal connection, and Ω AB ( v, x ) the spatial metric. These functions define the Carrollian geometry and one can check that they transform covariantly under Carrollian diffeomorphisms (see Sec. 2 of [10] for a complete presentation). Out of the Carrollian geometry, one can build the following first-derivative quantities: \nϕ A = α -1 ( ∂ v b A + ∂ A α ) , β = α -1 ∂ v ln √ Ω , ξ AB = α -1 ( 1 2 ∂ v Ω AB -Ω AB D -2 ∂ v ln √ Ω ) , ω AB = ∂ [ A b B ] + α -1 ( b [ A ∂ B ] α + b [ A ∂ v b B ] ); (3.5) \nthey are respectively, the Carrollian acceleration, expansion, shear and vorticity. They also transform covariantly under Carrollian diffeomorphisms, and will play an important role in the Carrollian conservation laws we will discuss in the next section. \nLet us come back to the black hole near-horizon metric (2.1). On each constant ρ hypersurface, called Σ ρ in Fig. 1, it induces a Lorentzian signature metric that becomes degenerate when taking the near-horizon limit: \na = ds 2 ρ = cst = ( -2 ρκ ρθ A ρθ B Ω AB + ρλ AB ) { dv,dx A } -→ ρ → 0 Ω AB dx A dx B . (3.6) \nIf we now compare this induced metric with the Randers-Papapetrou one, we are tempted to make the following identifications: 10 \nc 2 = ρ, α = √ 2 κ, and b A = θ A √ 2 κ . (3.7) \nWe thus identify the radial coordinate with the square of a virtual speed of light for the Lorentzian spacetime Σ ρ . As the horizon is located at ρ = 0 , it is an ultra-relativistic spacetime endowed with a Carrollian geometry given in terms of the surface gravity, the twist and the induced spatial metric Ω AB . After this identification, we can re-express the first-derivative Carrollian tensors (3.5) in terms of the extrinsic geometry of the horizon (2.6): \nϕ A = 1 2 κ ( ∂ A κ + ∂ v θ A -θ A 2 κ ∂ v κ ) , β = Θ √ 2 κ , ξ AB = 1 √ 2 κ σ AB , ω AB = 1 2 ( ∂ A θ B √ 2 κ + 2 θ A ∂ B κ + θ A ∂ v θ B (2 κ ) 3 / 2 ) -( A ↔ B ) . (3.8) \nWenotice that the Carrollian expansion and the Carrollian shear are proportional respectively to the expansion and the shear of the horizon defined extrinsically in Sec. 2.1.", '3.2 Horizon dynamics as ultra-relativistic conservation laws': "We now turn our attention to the gravitational dynamics of the horizon. Consider again the hypersurface Σ ρ near ρ = 0 . Its unit normal is given by \nn = dρ √ 2 κρ , (3.9) \nand allows us to define the extrinsic curvature and the momentum conjugate to the induced metric: \nT ab = 1 8 πG ( Ka ab -K ab ) , (3.10) \nwhere K a b = a c b D c n a is the extrinsic curvature of Σ ρ , K = K a a its trace and a ab = g ab -n a n b is the projector on the hypersurface perpendicular to n . 11 This hypersurface is sometimes referred to as the stretched horizon or membrane, while T ab is called the 'membrane energymomentum tensor' [2,3,28]. 12 Einstein equations ensure that it is conserved: \n¯ ∇ j T ji = 0 , (3.11) \nwhere the index i refers to { v, x } , and ¯ ∇ i is the Levi-Civita connection associated with the induced metric (3.6). The membrane is then interpreted as a fluid whose equations of motion are given by this conservation law. One notices that (3.11) describes the dynamics of a relativistic fluid that lies in the ( D -1) -dimensional spacetime given by the constant ρ hypersurface and equipped with the metric a . We are going to show that, to obtain the null Raychaudhuri (2.9) and Damour equations (2.10), one has to take the near-horizon limit of this conservation law which, at the level of the fluid, is interpreted as an ultra-relativistic limit through the identification ρ = c 2 . \nUsing (2.1), we compute the membrane energy-momentum tensor near the horizon, \n8 πGT vv = Θ 2 √ 2( ρκ ) 3 2 + O (1 / √ ρ ) , 8 πGT vA = -1 2 √ 2 ρκ 3 / 2 ( ∂ A κ + θ A ( κ +Θ) ) + O ( √ ρ ) , 8 πGT AB = -1 √ 2 ρκ ( Ω AB ( κ +Θ -∂ v κ 2 κ ) + 1 2 ∂ v Ω AB ) + O ( √ ρ ) . (3.12) \nWe now decompose T ij into the Carrollian momenta , which are defined such that they are independent of the speed of light and covariant under Carrollian diffeomorphisms [10], \n8 πGT vv = c -3 α -2 E + O ( c -1 ) , 8 πGT vA = c -1 α -1 ( π A -2 b B A AB ) + O ( c ) , 8 πGT AB = -2 c -1 A AB + O ( c ) , (3.13) \nwith E a scalar, π A a spatial vector and A AB a spatial symmetric 2-tensor. They are the ultrarelativistic equivalent of an energy-momentum tensor. They can be thought of respectively as the energy density, the heat current and the total stress tensor. The latter can be decomposed into its trace and traceless part \nA AB = -1 2 ( P Ω AB -Ξ AB ) , (3.14) \nwhich are interpreted respectively as the pressure and the dissipative tensor. \nComparing (3.12) with (3.13), we read the following Carrollian momenta: \nE = 1 √ 2 κ Θ , P = -1 √ 2 κ ( κ + D -3 D -2 Θ -∂ v κ 2 κ ) , Ξ AB = -1 √ 2 κ σ AB , π A = -1 2 ( ∂ A κ κ + θ B 2 κ ∂ v Ω BA + θ A 2 κ 2 ∂ v κ ) . (3.15) \nWe have obtained that the energy density is proportional to the expansion of the horizon. The pressure is related to the combination \nµ = κ + D -3 D -2 Θ , (3.16) \nwhich is referred to in [36] as the 'gravitational pressure' and receives corrections from the time evolution of the surface gravity. The dissipative tensor is proportional to the shear of the horizon (2.7). The heat current π A is harder to interpret but we notice that it receives a contribution from the gradient of κ , which can be thought of as a local temperature on the black hole horizon (see the discussion at the end of [31]). \nThese Carrollian momenta satisfy conservation equations that are given by the ultrarelativistic ( i.e. near-horizon) limit of the energy-momentum conservation (3.11). 13 Using the decompositions for the metric (3.4) and the energy-momentum tensor (3.13), we obtain: \n( α -1 ∂ v + β ) E - A AB α -1 ∂ v Ω AB = 0 , 2 ( ˆ ∇ A + ϕ A ) A A B -E ϕ B -( α -1 ∂ v + β ) π B = 0 . (3.17) \nThese equations 14 are covariant w.r.t. Carrollian diffeomorphisms, in the sense that the first one transforms like a scalar and the second one like a spatial vector and they are independant of c (or ρ , the radial coordinate). We have introduced a new object ˆ ∇ A , which is a Carrollcovariant derivative: \nˆ ∇ A v B = ˆ ∂ A v B + ˆ γ A BC v C , (3.18) \nwhere \nˆ ∂ A = ∂ A + b A α ∂ v and ˆ γ A BC = 1 2 Ω AD ( ˆ ∂ B Ω DC + ˆ ∂ C Ω DB + ˆ ∂ D Ω BC ) . (3.19) \nIf v A transforms like a spatial vector, i.e. v ' A = ∂x ' A ∂x B v B under a Carrollian diffeomorphism (3.2), then ˆ ∇ A v B will transform like a spatial 2 -tensor. One can check that this would not be the case for the usual Levi-Civita connection associated with Ω AB . The first equation of (3.17) can be interpreted as a conservation of energy on a curved background, but an exotic one: indeed, one would expect the gradient of the heat current to appear while here it is absent even when the heat current is non zero. This feature is a signature of the ultra-relativistic limit [16]. \nThe main result of this section is that, considering the Carrollian geometry (3.7) and the Carrollian momenta (3.15) and after a lenghty computation, one can show that the scalar equation is exactly the null Raychaudhuri equation (2.9) while the spatial one gives the Damour equation (2.10). This confirms that the dynamics of a black hole is mapped to ultra-relativistic conservation laws when the near-horizon radial coordinate is identified with a virtual speed of light.", '3.3 Conserved charges on the horizon': 'Using the results of the previous section we would like now to build conserved charges associated with the horizon. The idea is to use the techniques we know from relativistic physics to build charges on a constant ρ hypersurface and then send the radial coordinate to zero to obtain conserved charges on the horizon. The latter will be conserved on shell and associated to the symmetries of the induced Carrollian geometry on the horizon. At the end of this section, we discuss their relationship with the one obtained through the covariant phase space formalism in Sec. 2.3.', 'Charges associated to Carrollian Killing fields on the horizon': 'Consider again the energy-momentum tensor of the membrane (3.10): vacuum Einstein equations imply that it is conserved: \n¯ ∇ j T ji = 0 . (3.20) \nIt is thus possible to build a conserved current associated with any vector field of Σ ρ that satisfies the Killing equation for the induced metric a ij : \n¯ ∇ i ξ j + ¯ ∇ j ξ i = 0 , (3.21) \nwhere we recall that ¯ ∇ i is the Levi-Civita associated with a . This current is given by J i = ξ j T ji ; it is conserved \n¯ ∇ i J i = 0 , (3.22) \nand allows to build, for any small ρ , a conserved charge w.r.t. the v coordinate: \nQ ρ ξ = ∮ S v,ρ d D -2 x √ q glyph[lscript] i J i , (3.23) \nwhere \nq AB = Ω AB + ρλ AB + O ( ρ 2 ) and glyph[lscript] = √ 2 κρ dv + O ( ρ 3 2 ) , (3.24) \nare respectively the induced metric on a spatial section of the constant ρ hypersurface, i.e. S v,ρ , and the unit timelike normal to the spatial section in the constant ρ hypersurface, see Fig. 1. \nWe are now ready to perform the near-horizon limit of this construction. We consider first the Killing equation for the vector ξ that we decompose as ξ = f ( v, x ) ∂ v + Y A ( v, x ) ∂ A . The zeroρ limit of (3.21) becomes \n∂ v Y A = 0 , f∂ v κ + Y A ∂ A κ +2 κ∂ v f = 0 , f∂ v Ω AB + ∇ A Y B + ∇ B Y A = 0 . (3.25) \nThe first thing to notice is that the near-horizon limit of the Killing equation imposes the vector field ξ to be Carrollian! Moreover, these three equations have an interesting geometrical interpretation: indeed, consider the degenerate metric induced on the horizon Ω = Ω AB ( v, x ) dx A dx B and the vector field glyph[vector]v = α -1 ∂ v (where α is given by the identification (3.7)), they are equivalent to asking \nL ξ glyph[vector]v = 0 and L ξ Ω = 0 . (3.26) \nFollowing [13], the triple ( H , Ω , glyph[vector]v ) defines a non-Riemannian geometry called weak Carroll manifold . 15 The latter is the natural structure that appears when one wants to study ultrarelativistic symmetries. Things appear to be consistent: we have considered the symmetries of the relativistic metric a , i.e. its Killing vector fields, then we have taken the near-horizon limit, interpreted as an ultra-relativistic limit for ρ = c 2 , and we obtain the symmetries of the corresponding Carrollian geometry. These symmetries given by Eq. (3.26) will be called Carrollian Killing symmetries. \nWe can also perform the near-horizon limit of the charge (3.23) using the value of the membrane energy-momentum tensor derived in Sec. 3.2; we obtain \nQ ρ ξ -→ ρ → 0 C ξ = 1 16 πG ∮ S D -2 d D -2 x √ Ω ( -2 f Θ -Y A ( θ A + ∂ A κ κ )) . (3.27) \nThis charge is conserved provided that the null Raychaudhuri and the Damour equations are satisfied and the couple ( f, Y A ) satisfies the Carrollian Killing equations (3.25). Taking the trace of the last equation of (3.25) we obtain f Θ = -∇ A Y A , therefore the integration on the sphere of this term vanishes. The charge becomes \nC ξ = -1 16 πG ∮ S D -2 d D -2 x √ Ω Y A ( θ A + ∂ A κ κ ) . (3.28) \nThis is a sort of generalization of the angular momentum to the case of non-stationary black holes. We would like indeed to stress that in this formula, Ω AB , κ and θ A depend generically \non both v and x A , so the conservation of this charge is really non-trivial. Therefore, to any isometry of the induced Carrollian geometry on the horizon, we have associated a charge that is conserved on-shell. \nWhen we consider the case κ = cst and Ω AB = ¯ Ω AB ( x ) , the solutions to the Carrollian Killing equations are a supertranslation f = T ( x ) together with a real Killing of the metric ¯ Ω AB and if one considers the near-horizon geometry of a Kerr black hole and the spatial Killing Y = ∂ ϕ , this charge reproduces the constant angular momentum J [20].', 'The conformal case': 'The same analysis can be carried out for a conformal Killing on the constant ρ hypersurface Σ ρ , i.e. a vector ξ that satisfies \n¯ ∇ i ξ j + ¯ ∇ j ξ i = 2 λa ij , (3.29) \nwhere λ ( v, x ) is any function. We can build the same current by projecting ξ on the energymomentum tensor. However, if λ glyph[negationslash] = 0 , the associated charge will be conserved on-shell only if T ij satisfies the tracelessness condition \nT i i = 0 . (3.30) \nThe near-horizon limit of the conformal Killing equation is \n∂ v Y A = 0 , f∂ v κ + Y A ∂ A κ +2 κ∂ v f = 2 κλ, f∂ v Ω AB + ∇ A Y B + ∇ B Y A = 2 λ Ω AB . (3.31) \nAgain, it admits a nice interpretation as the conformal isometries of the weak Carroll manifold induced on the horizon. Indeed, (3.31) is equivalent to \nL ξ glyph[vector]v = -λglyph[vector]v and L ξ Ω = 2 λ Ω , (3.32) \nand, according to [13], this is the definition of the level-2 conformal isometries of ( H , g, glyph[vector]v ) ; we will call them conformal Carrollian Killing vectors. To any conformal Carrollian Killing ξ we can associate the following charge: \nC ξ = 1 16 πG ∮ S D -2 d D -2 x √ Ω ( -2 f Θ -Y A ( θ A + ∂ A κ κ )) , (3.33) \nwhich is the same as in the previous section, obtained through the near-horizon limit of Q ξ . The only difference is that, if λ glyph[negationslash] = 0 , this charge will not be generically conserved on-shell. It is generically conserved only if the near-horizon limit of the tracelessness condition (3.30) is satisfied, i.e. \nS ≡ Θ+ κ -∂ v κ 2 κ = 0 , (3.34) \nwhere the function S has been defined through \nT i i -→ ρ → 0 -1 8 πG √ 2 κ √ ρ S . (3.35) \nAsking S to be zero is a non-trivial additional constraint on the surface gravity and the expansion, that we will call the conformal state equation . Indeed, if we reintroduce the Carrollian momenta (3.15) we obtain that \nS = 0 ⇔ E = ( D -2) P . (3.36) \nWe recognize the usual state equation satisfied by the energy and the pressure of a conformal fluid (see [43] or [16]). \nWe consider now the case κ = cst and Ω AB = ¯ Ω AB ( x ) , the corresponding Carrollian Killings are given by \nξ = ( v D -2 ∇ A Y A + T ( x ) ) + Y A ( x ) ∂ A , (3.37) \nwhere T is a supertranslation and Y A is a conformal Killing of ¯ Ω AB . When the spatial metric is chosen to be the round metric on S D -2 we obtain the bms D algebra. The conformal state equation becomes κ = 0 . This constraint is obviously very restricting but actually, in this particular case, we will not have to impose it to obtain conserved charges. Indeed the charge C ξ becomes \nC ξ = -1 16 πG ∮ S D -2 d D -2 x √ ¯ Ω Y A θ A , (3.38) \nand the Damour equation becomes \n∂ v θ A = 0 . (3.39) \nSo, for any value of κ , this charge associated to a conformal Carrollian Killing of the type (3.37) is manifestly conserved on-shell, but insensitive to the supertranlsation T .', 'Relationship with the bulk analysis': 'Finally, in both the non-conformal and conformal case, we can relate C ξ to the integrable part of the charges obtained through the covariant phase space formalism in Sec. 2.3. Indeed, consider an asymptotic Killing ( f, Y A ) (2.13); as already stated in Sec. 2.3, its projection on the horizon is a generic Carrollian vector field. We can further ask the latter to be a (conformal-)Carrollian Killing, thus considering the subset of asymptotic Killings whose projection on the horizon provides an isometry of the induced Carrollian geometry. If we do so, one can show that \nC ( f,Y A ) = Q int ( f,Y A ) -1 8 πG ∮ S D -2 d D -2 x √ Ω f S , (3.40) \nwhere we notice the mysterious appearance of the function S that defines the conformal state equation (3.36). This equation holds up to boundary terms that are vanishing when integrated on the sphere and if the couple ( f, Y A ) satisfies the Carrollian Killing equations \n(3.25) or its conformal version (3.31). This equality is off-shell; if we further impose the equations of motion and perform a time derivative we obtain \nd dv Q int ( f,Y A ) = 1 8 πG ∮ S D -2 d D -2 x √ Ω [ f∂ v + ∂ v f -∇ A Y A ] S . (3.41) \nWe conclude that the non-conservation of Q int ( f,Y A ) , for (conformal-)Carrollian Killing vectors, will be sourced by the function S . Therefore we have established a connection between the conservation of the charges and the conformality of the Carrollian momenta associated with the horizon. A last remark is that these very compact results are valid for the splitting we have made in Sec. 2.3 between the integrable and non-integrable part of the charge, it would be interesting to determine how they get modified under the change of splitting (2.23).', '4 Perspectives': "This analysis sets an indubitable connection between Carrollian and near-horizon physics, the main result being that the dynamics of the black hole horizon is given by an ultrarelativistic conservation law. In the membrane paradigm, the 'fluid' describing the horizon is supposed to satisfy the Damour-Navier Stokes equation, which a priori is a non-relativistic equation but for a Galilean fluid ( i.e. when the speed of light is infinite). We want to point out that, instead, the fluid behaves more like a Carrollian one. This observation is emphasized by the fact that the energy conservation satisfied on the horizon seems very different from the one that a usual Galilean fluid would satisfy, as it does not involve the gradient of the heat current (see first equation of (3.17)), while it is perfectly interpreted in terms of an ultra-relativistic energy conservation. All these remarks lead to the conclusion that the ultra-relativistic approach seems to be more appropriate to the study of horizon dynamics. In [16], the authors study the ultra-relativistic limit of a relativistic fluid; it would be interesting to see how this translates in the horizon analysis. One could also study the thermodynamics of such a fluid, especially its entropy current, and see if we can relate it to the black hole entropy. \nAnother question is the role of the function S introduced to define the conformal state equation. It would be interesting to understand better its status at the level of the charges. Indeed, the exact same relationship was found in the context of asymptotically flat gravity between the Carrollian charges and the charges obtained through covariant phase space formalism [10]. In that case, the function S (called σ there) was representing the flux of gravitational radiation through null infinity and was therefore responsible of the non-conservation of the charges. At the level of the horizon, the function S could have the same kind of physical interpretation which would be worth clarifying. \nFinally, let us mention two other interesting directions. The first one would be to add other fields to source the bulk energy-momentum tensor and see how this analysis get modified, in particular their influence on the charges. The second one is the specific case of extremal black holes. We have not mentioned them in this paper since their study would require strong modifications in our analysis (for instance, Carrollian momenta for κ = 0 would diverge as \none can see from (3.15)). The study of Carrollian physics for extremal black holes will be the subject of future works.", 'Acknowledgments': 'We are grateful to L. Ciambelli, G. Giribet, R. Leigh, R. F. Penna, P. M. Petropoulos, and A.-M. Raclariu for useful discussions. LD acknowledges support from the Black Hole Initiative (BHI) at Harvard University, which is funded by a grant from the John Templeton Foundation. CM thanks the BHI for its hospitality while part of this work was done. This work was also partly funded by the ANR-16-CE31-0004 contract Black-dS-String.', 'References': "- [1] D. Macdonald and K. S. Thorne, Black-hole electrodynamics: an absolute-space/universal-time formulation , Monthly Notices of the Royal Astronomical Society 198 (1982), no. 2, 345-382\n- [2] K. S. Thorne, R. H. Price and D. A. 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